Quantum Theory Project, Departments of Chemistry and Physics, University of Florida,Gainesville, Florida 32611-8435, USA
�Published 22 February 2007�
Today, coupled-cluster theory offers the most accurate results among the practical ab initioelectronic-structure theories applicable to moderate-sized molecules. Though it was originallyproposed for problems in physics, it has seen its greatest development in chemistry, enabling anextensive range of applications to molecular structure, excited states, properties, and all kinds ofspectroscopy. In this review, the essential aspects of the theory are explained and illustrated withinformative numerical results.
I. Perspective on the Molecular Electronic Problem 291II. Plan for Review 293
III. Some Essential Preliminaries 294A. Configuration interaction 295B. Perturbation theory 296C. Coupled-cluster theory 297
IV. Normal-Ordered Hamiltonian 299V. Coupled-Cluster Equations 302
A. Double excitations in CC theory 304B. Choice of single determinant reference function 305C. Single excitations in CC theory 307D. Triple and quadruple excitations in CC theory 308E. Noniterative approximations 313
VI. Survey of Ground-State Numerical Results 315A. Equilibrium properties 315B. Basis-set issue 319C. Bond breaking 321
VII. The Coupled-Cluster Functionaland the Treatment of Properties 327
VIII. Equation-of-Motion Coupled-Cluster Method forExcited, Ionized, and Electron Attached States 330A. Numerical results 333
IX. Multireference Coupled-Cluster Method 338A. Hilbert-space formulation of the MRCC approach 340B. Fock-space formulation of the MRCC approach 342C. Fock-space MRCC based on an intermediate
Hamiltonian 346Acknowledgments 347References 347
I. PERSPECTIVE ON THE MOLECULAR ELECTRONICPROBLEM
As the recent developments in coupled-cluster �CC�theory have been mostly accomplished in quantum
chemistry circles, we begin with a quote from Ken Wil-son �1989�:
“Ab initio quantum chemistry is an emerging com-putational area that is fifty years ahead of latticegauge theory…and a rich source of new ideas andnew approaches to the computation of many fer-mion systems.”Driving these developments are the types of problems
addressed by quantum chemists, as shown in Fig. 1. Pri-mary among these are potential-energy surfaces �PES�which describe the behavior of the electronic energywith respect to the locations of the nuclei, subject to theunderlying Born-Oppenheimer or clamped nuclei ap-proximation. As shown, these PES can be for ground orelectronic excited states. At the equilibrium geometry,�E�R�=0 defines the molecular structure. The secondderivatives ��E�R� determine whether the critical pointis a minimum or a saddle point.
In addition, from the ground- and excited-state wavefunctions one obtains all properties that arise from asolution to the vibrational Schrödinger equation thatgives the frequencies, and, with the derivatives of thedipole moment, the infrared intensities. The derivativesof the dipole polarizability define the Raman intensities.Electronic excited states are also accessible along withelectronic and photoelectron spectra.
For the more kinetic aspects of chemistry, the basicconcept is a reaction path that is defined as a multidi-mensional path along which all vibrational degrees offreedom are optimum except one, which defines a pathtoward products. The latter might have a saddle point asshown, which defines a transition state and its activationbarriers. From that information, it is possible to obtainrate constants and state-to-state cross sections.
In addition, properties that arise from the one-particledensity matrix, such as dipole moments, hyperfine cou-pling constants, and electric-field gradients, are readilyavailable. Also, one obtains second- and higher-orderproperties such as the dipole polarizability and NMRchemical shifts and coupling constants. From evenhigher-order electric-field derivatives, one obtains hy-perpolarizabilities, which determine nonlinear optical
*Permanent address: Institute of Chemistry, University ofSilesia, Szkolna 9, 40-006 Katowice, Poland.
REVIEWS OF MODERN PHYSICS, VOLUME 79, JANUARY–MARCH 2007
behavior. From derivatives relative to atomic displace-ments in molecules, one obtains anharmonic effects onvibrational-rotational spectra. The two-particle densitymatrix, besides having a role in the energy, is also essen-
tial for spin operators such as S2 and spin-orbit effects inparticular.
Consequently, the objective is an accurate solution ofthe Schrödinger equation for molecules composed ofcomparatively light elements. When relativistic effectsare essential, the solution of the Dirac equation mightbe preferred. Of course, Darwin, mass-velocity, andspin-orbit effects can be added to the nonrelativistic so-lution to provide a wealth of approximations lying be-tween the Schrödinger equation and the four-component Dirac equation.
For modest-sized molecules of up to �15 light atomsor �100 electrons and small-molecule relativistic calcu-lations, coupled-cluster theory �Coester, 1958; Coesterand Kümmel, 1960; Cížek, 1966, 1969; Paldus et al., 1972;Bartlett and Purvis, 1978, 1980; Bishop and Lührmann,1978� has become the preeminent tool to introduce theinstantaneous effects of electron correlation that are notincluded in a mean-field approximation �Urban et al.,1987; Bartlett, 1989, 1995; Paldus, 1991; Lee and Scuse-ria, 1995; Gauss, 1998�. There are many textbook andsimilar accounts, each with a different focus �Lindgrenand Morrison, 1986; Harris et al., 1992; Bartlett andStanton, 1994; Bishop, 1998; Crawford and Schaefer,2000; Helgaker et al., 2000; Bishop et al., 2002; Shavittand Bartlett, 2006�.
In short, from the viewpoint of a physicist, coupled-cluster theory offers a synthesis of cluster expansions,Brueckner’s summation of ladder diagrams �Brueckner,
1955�, the summation of ring diagrams �Gell-Mann andBrueckner, 1957�, and an infinite-order generalization ofmany-body perturbation theory �MBPT� �Kelly, 1969;Bartlett and Silver, 1974a, 1976�. Hence, it is a very pow-erful method for correlation in many-electron systems.Its principal rationale compared to other quantumchemical methods is its correct scaling with size, termedsize extensivity �Bartlett and Purvis, 1978, 1980�. Thismeans it is a purely linked diagram theory that guaran-tees correct scaling with the number of particles or unitsin a system, and facilitates accurate relative energiesalong a potential-energy surface or between differentelectronic states. Only with this property are applica-tions to polymers, solids, or the electron gas possible,and, even for small molecules, its effects are numericallyquite significant. Configuration interaction methods,long the focus of the correlation problem in quantumchemistry �Shavitt, 1998�, do not, in general, have thisproperty which is responsible for the emphasis on CCtheory and its MBPT approximations �Kelly, 1969;Bartlett and Silver, 1974a, 1974b; Pople et al., 1976� inchemistry.
The CC theory was introduced in 1960 �Coester andKümmel, 1960� for calculating nuclear binding energiesin nuclei that could be treated in the first approximationby a single configuration of neutrons or protons. Thedetailed equations for electrons were first presented in1966 �Cížek, 1966� and its initial applications to elec-tronic structure were reported �Cížek, 1966; Paldus et al.,1972�. Starting in 1978 general purpose programs andapplications of CC theory were developed �Bartlett andPurvis, 1978, 1980; Pople et al., 1978; Purvis and Bartlett,1982�. Early in the history of the CC, it was shown to beremarkably accurate for describing the correlation en-ergy of the electron gas compared to the random-phaseapproximation �RPA� �Freeman, 1977; Bishop and Lühr-mann, 1978�.
For more details, the history of CC theory is best toldfrom the viewpoint of some of its principal developers.In the proceedings from the workshop “Coupled ClusterTheory of Electron Correlation” papers of Kümmel�1991� and Cížek �1991� address this issue. More re-cently, papers on this topic �Bartlett, 2005; Paldus, 2005�are pertinent. For the physics viewpoint, in the previousproceedings see Arponen �1991� and Bishop �1991�.
For larger molecules and solids, far more approximatebut more easily applied methods such as density-functional theory �DFT� or from the wave-functionworld the simplest correlated model MBPT�2� �alsosometimes known as MP2 when using a Hartree-Fockreference function� are preferred. There are CC solu-tions for some simple polymers �Hirata, Grabowski, etal., 2001; Hirata, Podeszwa, et al., 2004�. For nuclei, afteran appropriate regularization of the strong interaction,CC theory can be applied almost as it is for molecules�Coester and Kümmel, 1960; Kümmel et al., 1978;Guardiola et al., 1996; Bishop et al., 1998; Heisenbergand Mihaila, 1999; Mihaila and Heisenberg, 2000; Kow-alski et al., 2004; Włoch et al., 2005�.
FIG. 1. The nature of quantum chemical problems.
292 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
Heavy-atom relativistic CC theory is also widely ap-plied in several variants. These range from simply usingrelativistic pseudopotentials to describe the passive�Darwin and mass-velocity� relativistic effects �Pereraand Bartlett, 1993; Sari et al., 2001�, to generalizedpseudopotentials that also address spin-orbit effects�Mosyagin et al., 2001�, to one- and two-componentDouglas-Kroll-Hess methods �Kaldor and Hess, 1994;Hess and Kaldor, 2000�, and to a full four-componentmethod �Eliav and Kaldor, 1996; Visscher et al., 1996;Eliav et al., 1998�.
An essential element to understand when quantumchemists and electronic-structure physicists attempt tocommunicate �to paraphrase George Bernard Shaw,“one discipline divided by a common language”� is thatchemists are typically interested in molecules whosewave functions satisfy square-integrable boundary con-ditions, instead of infinite systems whose wave functionssatisfy periodic boundary conditions. Since wave func-tions for individual molecules go to zero at infinity, thissuits the Gaussian basis-set expansions used for mol-ecules. For periodic boundary conditions, Gaussians arestill often used, but there is also the prospect of usingplane waves that are not tied to specific atoms, yet thesehave not yet found a role in the wave-function correla-tion treatment of individual molecules �see, however,Chawla and Voth, 1998; Sorouri et al., 2006�. Also purelynumerical solutions can be considered and are foratomic �Lindgren and Salomonsen, 2002� and some di-atomic applications of CC theory �Adamowicz et al.,1985�, but because of the multicenter nature of mol-ecules, and the need for explicit consideration of two-particle effects in CC and related methods �as opposedto DFT�, such grid-based solutions have not been shownto be as feasible.
On the other hand, finite Gaussian basis sets intro-duce an inherent error in any solution of theSchrödinger equation that has to be considered in itsapplications. High-level calculations will typically use aconverging series of Gaussian basis functions such as thecc-pVXZ sets �Dunning, 1989; Kendall et al., 1992;Woon and Dunning, 1993� or atomic natural orbital sets�Almlöf and Taylor, 1987�, where cc means “correlationconsistent,” the pV indicates “polarized valence,” mean-ing that the basis will have higher angular momentumorbitals than those required to describe an atom’sground state, and X will range from D for double zeta,meaning two linear combinations of Gaussian atomic or-bitals per electron, to T, Q, 5, 6, etc. For the N atom, forexample, there are 14 contracted Gaussian functions inthe DZ form, 30 for TZ, 55 for QZ, 91 for the 5Z, and140 for 6Z. Calculations followed by extrapolation canthen be argued to approach the basis-set limit. A betterbut more complicated solution for the basis set is theinclusion of explicit r12 interactions with coupled-clustermethods �R12-CC� �Noga et al., 1992�. However, mostmolecular applications make little attempt to achievethe true basis-set limit, but instead depend upon the ap-proximate cancellations known for the relative energiesin spectroscopy, or along a reaction path, or breaking a
bond into its fragments, causing much of the commonbasis-set correlation error to cancel from the computa-tion. In fact, without this effect there would scarely be acomputational chemistry.
II. PLAN FOR REVIEW
The intent of this review is to systematically developthe ideas that have enabled CC theory to become thepredictive method for electron correlation in molecules,as supported by its numerical results. In Sec. III, wepresent an overview of solutions to the correlation prob-lem, using elementary configuration space concepts todiscuss its description using configuration interaction�CI�. Configuration interaction is exact in the full CIlimit, but lacks size extensivity with any truncation ofthe configuration space, such as to single and double
excitation C1 and C2. This is the wave function �CISD
= �1+ C1+ C2��0, with �0 an independent particle �mean-field� reference function. See Fig. 2 for the definition ofsuch excitations.
We then relate CI to Rayleigh-Schrödinger perturba-tion theory �RSPT� as a way to extract the CI eigen-value. Rayleigh-Schrödinger perturbation theory has thesame failings as truncated CI, but once all configurationsthat can contribute in a given order are considered,RSPT becomes many-body perturbation theory�MBPT�, which as a fully linked method has to be size
FIG. 2. Graphical examples of the selected single ��ia�, double
��ijab�, and triple ��ijk
abc� excitations due to the T1 �C1�, T2 �C2�,and T3 �C3� operators, respectively. Electrons from all possibleoccupied orbitals can be excited to all possible unoccupied or-bitals, so T3�0=�
a�b�ci�j�k tijk
abc�ijkabc.
293Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
extensive. In practice, MBPT mostly offers a finite-orderapproximation to the correlation problem, with the in-herent failings of such expansions.
On the contrary, coupled-cluster theory offers a veryconvenient and powerful resummation of MBPT dia-grams, providing an infinite-order approximation in se-lected cluster operators. In particular, �MBPT=exp�T��0=�CC for T=T1+T2+ ¯ +Tn, where Tp is aconnected cluster operator that corresponds to p-foldexcitations �Fig. 2�. Correct scaling with the number ofparticles �extensivity� is ensured due to the exponentialform with the condition that T be connected. Limitingthe resummations to all terms that arise from single anddouble excitation cluster operators T1 and T2, e.g., de-fines �CCSD=exp�T1+T2��0, which adds all the productslike T2
2 /2 and T1T2 to the CI wave function. As shown inFig. 3, numerical results of CI with higher excitation op-erators compared to orders of MBPT and CC withhigher excitation clusters clearly demonstrate the nu-merical superiority of CC theory.
In Secs. IV and V, we introduce the unambiguous eas-ily applied diagrammatic approach that has been devel-oped to derive the detailed form of the CC equationsincluding T2, coupled-cluster doubles �CCD�, T1 and T2�CCSD�, and with triples �CCSDT� and connected qua-druple excitations �CCSDTQ�. The computational diffi-culty and the expense of the latter necessitate some sim-plified approximations, and these can be iterative likeCCSDT-1, or noniterative like CCSD�T�. Section VI de-
fines and documents how well all these approximationswork for real world applications in comparison with ex-periment.
Section VII introduces the � deexcitation operator,which allows the treatment of energy derivatives onpotential-energy surfaces �E�R�= �0��1+��exp�−T���H�R�exp�T��0� and for the density matrices �pq
= ��0��1+��exp�−T�cp†cq exp�T���0� and �pqrs= ��0��1
+��exp�−T�cp†cq
†cscr exp�T���0�, where cp†cq are occupa-
tion number operators described in Sec. V. � and exp�T�are dual. Together they define the CC functional E= �0��1+��e−THeT�0�. Numerical results for molecularstructures and vibrational frequencies are pertinenthere.
In the next section, we consider excited, ionized, and
electron attached states by using the ansatz Rk�CC=�k,
where Rk=R0�k�+R1
�k�+R2�k�+¯ is another excitation op-
erator which after a constant creates single, double, etc.excitations from the CC reference solution. Insertingthis ansatz into the Schrödinger equation for the kthexcited state and subtracting the ground state leads to
the EOM-CC method �H , Rk�0�=�kRk�0�, with �k=Ek
−E0, and Rk is a right-hand eigenvector. The left-hand
eigenvector �Lk is a deexcitation operator� LkH= Lk�k,
�0�LkRl�0�=kl; L0= �1+�� and R0=1, making theconnection with the ground-state functional. Excitedstates have the associated density matrices, �pq
kl
= �0�Lk exp�−T�cp†cq�T�Rl�0�. The spectrum of eigenvec-
tors also defines second- and higher-order responseproperties. Many numerical results are presented.
The final section considers the generalization of thereference function ��0� to multireference form by re-placing it with a linear combination of important deter-minants �1 ,�2 , . . . ,�m�, and redefines the cluster ex-pansion operator accordingly to define two differentmultireference �MRCC� methods. These are pertinentwhen there are quasidegeneracies as would occur inopen-shell atoms, or as bonds are broken in dissociation.One then obtains an effective Hamiltonian matrix andassociated eigenvector equations that can treat severalstates at once, or be reduced to a multireference, state-specific form. Numerical illustrations of Hilbert-spaceand Fock-space MRCC applications are discussed, in-cluding the application of an intermediate Hamiltonianto eliminate intruder states.
III. SOME ESSENTIAL PRELIMINARIES
The Hamiltonian in the Born-Oppenheimer approxi-mation is
H�R� = − 12�
i=1
n
�2�ri� − �
Z/�ri − R�
+ 12 �
i,j=1
n
1/�ri − rj� + 12�,�
ZZ�/�R − R�� �1�
FIG. 3. �Color online� Performance of theories for the corre-lation energy in small molecules. Graphed is the percentage ofthe full correlation energy achieved by the CI, CC, and MBPTtheories, as a function of the level of approximation. To facili-tate comparisons, the ordinate gives the size-scaling parameterof the approximation =n+N+it in the computational costfunction nnNNNit
it. Shown are MBPT �solid circles�, approxi-mations �2�–�6�; CI �solid squares�, approximations SD-SDTQ;and CC �stars�, approximations SD-SDTQ. The correlation en-ergy is defined with respect to the Hartree-Fock energy for thegiven basis set, and the full correlation energies are obtainedfrom the FCI calculations quoted in Table I.
294 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
where ri� locates the electrons and R� the nuclei. Z isthe atomic number of the nucleus . ENN is a constant atany geometry, so it needs no further considerationwithin the clamped nuclei approximation.
Choosing to use expressions in terms of spin orbitalsx1=1=r�1�1 with �1 the spin coordinate for electron1, the independent particle wave function isA„ 1�1� 2�2�¯ n�n�…, where A= �1/�n!��P�−1�pP, withP providing the permutation and �−1�p its parity.
Separating the electronic Hamiltonian that is para-metrically dependent on R into an unperturbed and per-turbed part,
H�r ;R� = H0 + V , �3�
H0 = �i
�h + u�i� = �i
heff�i� , �4�
V = 12�
ijg�ij� − �
iu�i� , �5�
where u�i� represents an average �mean� field potential
as in the Hartree-Fock case, where u�1�= J�1�−K�1�= d�2�j
n j*�2� j�2� / �r1−r2�− d�2�j
n j*�2�P12 j�2� / �r1−r2�,
and heff�1�= f�1� is the Fock operator. x and R representpositions of all electrons and nuclei, respectively. Thespace-spin volume element is d�2, and P12 indicates thepermutation of electrons 1 and 2. Unless stated other-wise, throughout this review we will assume the spin-orbital form of the one-particle equations. This is thesimplest for formal manipulations and means all equa-tions will be equally applicable to closed-shell and high-spin �i.e., maximum unpaired spin� open-shell atoms andmolecules. In the simplest case, � will reduce to either for spin Sz= +1/2 or � for spin −1/2. For generalizedspin orbitals �1=N�1+c�1�, which would sacrifice theSz quantum number.
With the solution to the one-particle equation
heff�1� p�1�=�p p�1�, the solution of the unperturbedSchrödinger equation is given by H0�0=E0�0, whereE0=�i�i. �0 is the single-determinant approximation tothe electronic wave function Eref= ��0�H��0�=E0+E�1�.
In addition, in any finite basis-set solution of the one-particle equation, the spatial part of the orbitals �=�c,where � is the underlying, typically Gaussian atomic-orbital basis set of dimension, M, and c is a vector of thecoefficients. These functions are determined for differ-ent elements and are located on all atoms in a molecule.This leads to the equation heffc=Sc�, where S is an over-lap matrix Sij= ��i ��j�. There are M solutions to thisequation, so for p�n, we have occupied orbitals �thosebelow a Fermi level�, and for p�n, M−n=N unoccu-pied orbitals, sometimes called virtual since they are aby-product of the finite basis calculation and have norole in defining heff.
A. Configuration interaction
The essence of the electron correlation problem in aone-particle basis set is to give the n-particle wave func-tion the flexibility to keep electrons apart by the admix-ture of the higher excitations. The traditional route inquantum chemistry that goes back to Slater �1929�, Parr�Parr and Crawford, 1948�, and Boys �1950� has beenconfiguration interaction �CI�. That is, build ann-particle wave function from the singly excited, doublyexcited, etc., determinants, �i
a� , �ijab� , . . ., where each
such excitation means a determinant where some of then occupied orbitals, i , j ,k , l , . . . ,n are replaced by theM−n=N virtual orbitals, a ,b ,c ,d , . . . ,N when the orbit-als have the same spin. We use the indices p ,q ,r ,s whenthe orbitals are unspecified. We choose intermediatenormalization ��0 ��CI�=1. The exact wave function in afinite basis set is the full CI �FCI�, which means includeall same-spin excitations up to n-tuple ones for n elec-trons,
�CI = �0 + �a,i
Cia�i
a + �i�k,a�b
Cijab�ij
ab + ¯
+ �i�j�k�¯
a�b�c�¯
Cijk¯nabc¯M�ij¯n
ab¯N �6�
=�1 + �p
n
Cp��0, �7�
where the coefficients Cij¯ab¯�, of which there are
��nN�m, where m indicates the excitation level, are nor-mally determined variationally. The full CI is an impos-sibility for any but quite small molecules in small basissets since the number of determinants is ��nN�n�Mn
for M basis functions. However, the full CI is an unam-biguous reference model for the correlation problem asit is the best possible solution in any finite basis set. It isvariational ECI�Eexact, invariant to all orbital rotations,and is size extensive. In a complete basis, the full CI,then termed the complete CI, gives the exact solution tothe Schrödinger equation. When one refers to a trun-cated CI, one means that one is limited to some subsetof the possible excitations, like all single and double ex-citations, or CISD. It is variational, invariant to orbitaltransformations among just the occupied or unoccupiedorbitals, but it is not size extensive.
Paying a little more attention to the eigenvalue prob-lem for a truncated CI like CISD, we have the equations
��ia��H − �E�C1 + HC2�0� = 0, �8�
��ijab�HC1 + �H − �E�C2�0� = 0, �9�
ECISD = ��0�H��0� + �E , �10�
295Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
The size-extensivity problem in CI arises from the−�ECn terms, which will remain in any less than full CIcalculation. These, of course, contribute unlinked dia-grams �discussed in the next section� into the CISD en-ergy, since �E is represented by closed diagrams, caus-ing its product with Cn to be unlinked. Such unlinkedterms can only be removed from the CI equations byformal consideration of some kinds of contributionsfrom higher excitations beyond those in the particulartruncated CI. The proper inclusion of such contributionsis accomplished in MBPT and coupled-cluster theory.
B. Perturbation theory
The connection between CI and perturbation theoryis readily apparent when perturbation theory �PT� isused to extract the CI eigenvalue. Using the CISD ma-trix problem for the ground-state energy as an example,
���0�H��0� ��0�H�h��h�H��0� �h�H�h� �� 1
C� = � 1
C�E ,
where �h� indicates the single ��ia� and double ��ij
ab� ex-citations in configuration space. Solving for C in the ma-trix equation, a little manipulation gives the ground-state eigenvalue
E = ��0�H��0� + ��0�H�h��h�E − H�h�−1�h�H��0� . �12�
From this expression, Rayleigh-Schrödinger PT emergesfrom a separation of H=H0+V, where H0=�ih
eff�i�, V=H−H0, and a recognition that E=E0+E�1�+E�2�+¯.Then, R= �h�E0−H0+V−�E�h�−1=R0+R0�V−�E�R.This leads to the energy E= ��0�H��RSPT�, and wave-function corrections �RSPT=�0+��1�+��2�+¯ �Löwdin,1968�. Within the configuration space �h�,
E0 = �i=1
n
�i, �13�
E�1� = ��0�V��0� , �14�
E�2� = ��0�H�h��h�E0 − H0�h�−1�h�H��0�
= ��0�VR0V��0� = ��0�V���1�� , �15�
E�3� = ��0�VR0�V − E�1��R0V��0� �16�
=��0�V���2�� = ���1��V − E�1����1�� , �17�
E�4� = ��0�V���3��
= ��0�VR0�V − E�1�����2�� − E�2����1����1�� �18�
=���2��E0 − H0���2�� − E�2����1����1�. . . ,
. . . . �19�
The resolvent operator R0= �h��h�E0−H0�h�−1�h�= �h�R0�h�. We also have �E0−H0���ij¯
ab¯�= ��i+�j+ ¯−�a
−�b− ¯ ���ijab�, which makes the resolvent matrix diago-
nal. With a Hartree-Fock �HF� reference, only doubleexcitations contribute to E�2� and E�3�.
Note that as long as the configuration space is re-stricted to a subset of possible excitations, like singleand double excitations, renomalization terms like−E�2����1� ���1�� remain. Since E�2� has to scale linearlywith the number of particles, and it may be shown that���1� ���1�� does likewise, such renormalization termshave the potential �except for exclusion principle violat-ing �EPV� terms, discussed later to scale as �n2 withthose in higher orders of PT scaling with higher powersof n. So any truncated CI eigenvalue will retain suchterms. However, in the full CI they would not. Theirelimination is achieved in a given order when all higherexcitations are included in the space of configurations�h� that can contribute to that order. That is, in fourthorder we have to consider �h�= �h1h2h3h4� where we alsohave triple and quadruple excitations. Once we includethe latter, with some algebra �Bartlett and Silver, 1975� itcan be shown that −E�2����1� ���1�� is solely determinedby double excitations and is removed from E�4� by therole of quadruple excitations in the lead term of E�4�.This cancellation between different categories of excita-tions was first shown by Brueckner �1955�. This is thesubstance of the linked-diagram theorem, as proved toall orders by Goldstone �1957� using time-dependent,diagrammatic techniques. It may also be proven in atime-independent way �Paldus and Cížek, 1975; Manne,1977; Shavitt and Bartlett, 2006�. Hence, if we do notrestrict our configuration space but allow all excitationsthat can contribute in a given order �that is always lessthan the full CI space�, we make the transition fromRSPT to MBPT and dispense with any renormalizationor unlinked terms. In MBPT, we can then write
��MBPT� = ��0� + �k=1
n
�R0�V − E�1��k��0�L, �20�
E�n+1� = ��0�V���n��L, �21�
E�2n+1� = ���n��V − E�1����n��L, �22�
E�2n� = ���n��E0 − H0���n��L, �23�
where L indicates the restriction to linked diagrams. Inthis form, MBPT assumes the formal simplicity ofBrillouin-Wigner perturbation theory �Löwdin, 1968�,but without the dependence on an unknown energy.
This simple example leads to several important conse-quences. �i� Rayleigh-Schrödinger perturbation theorywhen allowing all categories of excitations that can oc-cur in a given order of perturbation theory becomesmany-body perturbation theory �MBPT�, where themany-body terminology emphasizes that the theory hasto provide correct scaling with the number of particlesor size-extensive results for the energy, wave function,and density matrices �i.e., there are no unlinked dia-grams�. �ii� The converse is if we restrict the excitationsin RSPT to subsets like single and double excitations, as
296 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
in CISD, infinite-order RSPT=CISD, and any such trun-cated CI is not size extensive. Only the full CI is.
An important benefit of the linked-cluster factoriza-tion is that the contribution of �h4� into E�4� might ap-pear to require eight-index denominators since R0
would contain �h4��h4�E0−H0�h4�−1�h4�� ��ijklabcd���i+�j
+�k+�l−�a−�b−�c−�d�−1��ijklabcd�, but all such terms are
replaced by a product of two four-index denominators��i+�j−�a−�b���k+�l−�c−�d� by simply putting twoterms over a common denominator �Bartlett and Silver,1975�, a simple application of the Frantz-Mills factoriza-tion theorem �Frantz and Mills, 1960�. The effect of sucha factorization is important numerically, since an eight-index denominator would require an �n4N6 computa-tional procedure, while four-index denominators, evenas products, never require more than an �n2N4 proce-dure. This is the same as CISD, but now the effects ofessential quadruple excitations are included.
One final lesson in this simple example is that for twoelectrons �i , j� we have no quadruple excitations, sorather than canceling the renomalization term, the exactresult would have to retain it. However, we can still al-low the formal quadruple excitations like �h4� to be in-cluded in the equations, even though we know that thesedeterminants violate the exclusion principle for twoelectrons making their contribution zero. However, afterthe cancellation of the renormalization term, we are leftwith nonvanishing contributions of those determinantsthat appear as part of the quadruple excitation linkeddiagrams that remain, even for two electrons. For twoelectrons their value is equal to that of the renormaliza-tion term, thereby accounting for it. These residualterms are somewhat misleadingly called EPV for exclu-sion principle violating �Kelly, 1962�, though, of course,there is no violation. It is just a different way of counting�Szalay and Bartlett, 1992�. This will also be the reasonwhy all equations in MBPT and CC theory will haveunrestricted summation indices, which is very differentfrom CI, where the exclusion principle is enforced forevery determinant in the wave function. This, too, offersa distinct computational advantage.
C. Coupled-cluster theory
The basic equations of coupled-cluster theory are de-ceptively simple. We start from the fact that the exact,linked-diagram wave function above �as will be shown inthe next section� can be written as
�MBPT = �CC = exp�T��0 = ��0, �24�
�CC = �1 + T + T2/2 + T3/3! + ¯ ��0, �25�
where � is often called the wave operator as it takes anunperturbed solution into the exact solution. The clusteroperator T is composed of a series of connected opera-tors that can be expanded in terms of its componentsthat introduce single �i
a, double �ijab, triple �ijk
abc, etc.excitations into the wave function as in Fig. 2,
T = T1 + T2 + T3 + ¯ + Tn, �26�
Tn = �n!�−2 �i,j,. . .
a,b,. . .
tij¯ab¯ca
†cb†¯ cjci, �27�
T1�0 = �i,a
tia�i
a, �28�
T2�0 = �i�j,a�b
tijab�ij
ab, �29�
T3�0 = �i�j�k,a�b�c
tijkabc�ijk
abc, �30�
T4�0 = �i�j�k�l,a�b�c�d
tijklabcd�ijkl
abcd, �31�
. . . .
These Tn contributions are referred to as connectedsince they cannot be reduced further. However, by virtueof the nonlinear terms in the exponential expansion, wehave, in addition, the disconnected �but linked� compo-nents of the exact wave function,
12T2
2�0 = �i�j,a�b
k�l,c�d
tijabtkl
cd�ijklabcd, �32�
12T1
2�0 = �i,a
j,b
tiatj
b�ijab, �33�
T1T2�0 = �ia
k�l,c�d
tiatkl
cd�ijkabc, �34�
. . . .
Note that even though these terms such as T22 /2 intro-
duce quadruple excitations into the wave function, theyare greatly simplified as their coefficients are composedof products of just double excitation coefficients, or�n2N2 coefficients instead of the �n4N4 associated withT4.
The equations for the CC amplitudes tij¯ab¯� are ob-
tained by insertion into the Schrödinger equation, fol-lowed by projection onto a sufficient number of excita-tions,
exp�− T�H exp�T��0 = H�0 = E�0, �35�
��ij¯ab¯�H��0� = QHP = 0, �36�
E = ��0�exp�− T�H exp�T���0� = ��0�H��0� . �37�
From the well-known Hausdorff expansion,
297Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
which has to terminate after fourfold commutators whenH has no more than two-electron operators in it. Hence,the CC equations �36� are connected, as any terms thatwould not have indices in common between H and Twould be eliminated. This makes the CC wave function�CC=exp�T���0� linked, and consequently the energy inEq. �37� is too. This requires that the energy be sizeextensive �Bartlett and Purvis, 1978�, which is an essen-tial requirement of the theory. Note that Eq. �36� is ofthe general form Tp= f�T1 ,T2 , . . . ,Tp−1�+g�Tp ,Tp+1 ,Tp+2�. The contribution of the first term is always�npNp+1, while Tp into Tp is always one N higher, withTp+1 being nN2 and Tp+2 being n2N3 higher. Hence, forCCSD where T3=T4=0, we have a computational scal-ing of �n2N4; CCSDT is �n3N5, andCCSDTQ is �n4N6.
Comparing the exact CC solution with the full CI, wesee that
C1 = T1, �39�
C2 = T2 + 12T1
2, �40�
C3 = T3 + T1T2 + T13/3!, �41�
C4 = T4 + T22/2 + T1T3 + T1
2T2/2 + T14/4!, �42�
] ,
which demonstrates the cluster decomposition of thewave function. As the Hamiltonian has at most two-particle interactions in it, logically one might expect thatthe simultaneous correlation of two electrons in differ-ent parts of a molecule, as represented by T2
2 /2, is moreimportant in the wave function than the true, connectedfour-particle cluster interactions associated with T4, andfor electronic structure this is indeed the case. Hence, acoupled-cluster wave function limited to connecteddouble excitations �CCD� �CCD=exp�T2��0, the sim-plest CC approximation, already includes the discon-nected parts of quadruples, hextuples, and higher even-ordered excitations. Such disconnected products areresponsible for the size-extensivity property of themethod. That is, it is appropriate for many electrons.Without this, CC theory, like truncated CI, could not beapplied to infinite systems such as crystalline solids orthe electron gas.
Besides the obvious application to infinite systems, amanifestation of size extensivity in chemistry is the en-ergy released �or absorbed� in a reaction, called the heatof the reaction. For example, for the reaction A+B→C+D, �Erxn=�E�C�+�E�D�−�E�A�−�E�B�. Whenthese energies are obtained from CC theory, it is appro-priate to add them as above since �assume closed shells
for simplicity� HA+B�CC�A+B�=HA�CC�A�HB�CC�B�and EAB=EA+EB, but for a nonextensive method suchas CI, separate calculations have to be made for A+Band C+D far apart, to get meaningful energy differ-ences, since HA+B�CI�A+B��HA�CI�A�HB�CI�B�.Hence, there cannot be any such thing as a table of en-ergies computed by truncated CI for a variety of mol-ecules, which can then be added to evaluate energies�heats� of chemical reactions.
Another manifestation is obtaining consistent, relativeenergies along a PES. That is, the theory should givemeaningful energy differences for activation barrierswhere bonds are being formed and broken, or even forthe detailed vibrational frequencies for a molecule in itsequilibrium geometry. Size extensivity is absolutely es-sential in today’s quantum chemistry, and that has led tothe emphasis on CC and its MBPT approximations.
Since the full CI has to be the exact result in a basisset, it provides an unambiguous measure of how well agiven approximation does for electron correlation. InTable I and Fig. 3, we illustrate results from CI subjectto higher excitation operators and finite-order MBPTand CC theory with higher connected operators. Theplot shows convergence to 100% of the correlation en-ergy with an excitation level or order of perturbationtheory. All methods first require an integral transforma-tion from atomic to molecular orbitals. For M=n+N ba-sis functions, this scales as �M5. In terms of computa-tional scaling, the CI and CC methods without furtherrestrictions scale as �nlNl+2Nit with the level of excita-tion l, where n means the number of occupied orbitalsand N is the number of unoccupied ones, and Nit indi-cates the number of iterations required to converge. Inother words, to do CISD requires �n2N2 coefficients in
C2 and at least one summation of N2 in the evaluation of
��ijab�HC2��0�. Adding quadruple excitations into CI re-
quires �n2N2 more time and computational resources.As even in a small calculation, n=10, N=100, the exten-sion to quadruple excitations is �106 times as difficult.The most important parts of the quadruple excitations inCI are those that account for the unlinked diagrams.Hence, the transition to MBPT has already eliminatedsuch terms to all orders, making even the low-order re-sults better in some cases. The MBPT�2� approximationis limited by the integral transformation unlike the othermethods, as it has a very quick �n2N2 evaluation. TheMBPT�3� approximation scales the same as CISD, but itis noniterative as are all the MBPT approximations. Thepower of CC theory is shown when considering the scal-ing of CCSD which is the same as CISD, but unlike thelatter CCSD already benefits from the elimination ofunlinked quadruple excitation diagrams, and the largestpart of the remainder of the linked ones is convenientlyintroduced by the disconnected term T2
2 /2. But the scal-ing of this term is only �n2N4, compared to �n4N6 forthe quadruples in CISDTQ, which would include con-nected, disconnected, and unlinked terms. This is obvi-ously an enormous savings. Full MBPT�4� scales as�n3N4, since the rate-determining step is for connected
298 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
triple excitations. If they are eliminated leaving SDQ-MBPT�4�, then we have the noniterative ��n2N4� ap-proximation to CCSD. The noniterative approximationMBPT�4� to CCSDT has one N higher in scaling insteadof two because there is no T3→T3 coupling in MBPT�4�.Those terms would arise in fifth order, along with thefirst contribution from connected quadruple excitationsT4. The T4→T4 part of connected quadruples first arisesin sixth order. By combining MBPT and CC theory,noniterative approximations such as CCSD�T� are de-rived. These are discussed in Sec. V. Because of thiscombination, CCSD�T� scales as �n2N4Nit+n3N4, whichis a significant savings on CCSDT, and there is no stor-age of the �n3N3tijk
abc amplitudes. An iterative methodsuch as CCSDT-3 �Sec. V.D� scales as �n3N4Nit, and alsodoes not require storage of triple excitation amplitudes.
IV. NORMAL-ORDERED HAMILTONIAN
The CC equations can be developed systematically us-ing the unifying and compact diagrammatic develop-ment that has evolved over several years �Cížek, 1969;Paldus et al., 1972; Kucharski and Bartlett, 1986�. Dia-grammatic methods begin with the second-quantizedform of the electronic Hamiltonian in Eq. �1�. In thefollowing, we have no time or frequency dependence so
creation operators p† , ap† ,Xp
† , cp† and annihilation opera-
tors p , ap ,Xp , cp work in occupation number space, suchthat
cb†�0� = �− 1�mb�111
ijk¯ 010
abc¯ �
with �0� the Fermi vacuum and
ck�0� = �− 1�mk�110ijk
¯ 000abc
¯ � .
The parity of the operation is determined by the numberof occupied orbtials mb �mk� to the left of the b �k� lo-cation. The important part of these operators, regardlessof chosen notation, is their indices, so the most compactnotation is simply to use p† and p in the operators. Thenthe Hamiltonian can be written in terms of field opera-
tors, �†�x1�=�p p�x1�p†,
H =� �†�x1�h��x1�d�1
+12 � d�1� d�2�
†�x1��†�x2�1
r12���x1���x2�
− ��x2���x1� , �43�
H = �pq
�p�h�q�p†q + 14 �
pqrs�pq��rs�p†q†sr . �44�
Here �pq �rs� represents an antisymmetrized integraldefined as �pq �rs�− �pq �sr�, where �pq �rs�= �p
*�1��q*�2�1/r12�r�1��s�2�d�1d�2. See also definitions
listed in Table II. From the definition of p† and p, con-sider all the possibilities p†q†=−q†p†, pq=−qp, and p†q=pq− qp†, which give the standard anticommutation re-
TABLE I. Correlation corrections �in mH� with various CC methods relative to FCIa values.
N2 Re 13.465 1.626 0.192 0.016C2 Re 29.597 3.273 0.622 0.103
mean abs. err. 9.17 1.069 0.092 0.010
aBauschlicher and Taylor, 1986, 1987a, 1987b; Bauschlicher et al., 1986; Kucharski and Bartlett 1993; Christiansen et al., 1996.bKucharski and Bartlett, 1998a.cMusiał et al., 2002b.
299Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
lation p†q+ qp†=pq. Once p and q refer to occupiedorbitals i , j ,k , . . . and unoccupied ones a ,b ,c , . . ., the nor-
mal order indicated by � is defined to have all hole i†
and particle a operators moved to the right of the other
operators, to facilitate i†��0�= i†�0�=0 and a�0�=0. So
a†i�= a†i, ia†�=−a†i. Now introducing the contractiondefinition
and considering all the hole-hole, particle-particle, etc.forms, we immediately see that this contraction vanishesunless
where p† and q represent hole operators i , j , k , l, while
similarly vanishes unless the operators correspond to
particles a , b , c , d. The form of the Hamiltonian abovecan be put into normal-ordered form by virtue of Wick’stheorem �Wick, 1950; Cížek, 1966; Paldus and Cížek,1975�. The particular form used here, which is integral tothe diagrammatic development that is the cornerstone ofthis review, is the time-independent mixed particle-holeoperator form of the theorem �Bogoliubov and Shirkov,1959�. Wick’s theorem says any product of second-quantized operators,
TABLE II. The rules to interpret the diagram algebraically.
Each upgoing line is labeled with a “particle” label a ,b ,c ,d , . . . and each downgoing line with a “hole” label i , j ,k , l , . . ..Open lines should be labeled in sequence as a , i ;b , j ;c ,k, etc.
Each one-particle vertex in the diagrammatic equation should be interpreted as the integral�left out �operator� right in�
Each two-particle vertex corresponds to the antisymmetrized integral�left out, right out �left in, right in�
Similarly, the cluster vertices occurring in the diagrammatic equations correspond to
etc., and are antisymmetric as well; hence tijab=−tji
ab=−tijba= tji
ba and similartly for tijkabc.
All the orbital labels are summed over “internal” lines, i.e., lines terminating below the last HN.The sign of the diagram is obtained from �−1� raised to the power of the sum of hole lines and loops: �−1�h+l. For the
purpose of getting the sign, open lines are closed into fictitious loops by paring i ,a ; j ,b; etc.
The weigh factor for the diagram is specified by � 12
�m, where m is the number of pairs of “equivalent” lines. A pair ofequivalent lines is defined as being two lines originating at the same vertex and ending at another, but identical vertex, andgoing in the same direction.
To maintain full antisymmetry of an amplitude, the algebraic expression for a diagram should be preceded by apermutation operator permuting the open lines in all distinct ways, �P�−1�PP.
A factor of 12 is also required for each pair of equivalent Tn vertices �a pair of T vertices is considered equivalent if they
have the same number of line pairs and are connected in equivalent ways to the interaction vertex�, i.e.,
300 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
Using this theorem �and now suppressing the ˆ on theoperators for simplicity� defines the normal-ordered op-erator
�46�
For the two-particle part,
Hence, the normal-ordered Hamiltonian becomes
H = �p
fppp†p� + �p�q
fpqp†q� + 14 � �pq��rs�p†q†sr�
+ �0�H�0� . �47�
Choosing to subtract the constant �0 �H �0�=E0+E�1�
=Eref, we have a slight redefinition
HN = H − �0�H�0� , �48�
fN = f − �0�f�0� , �49�
WN = W − �0�W�0� . �50�
In this way, all internal contractions are removed fromthe operators themselves, making all subsequent deriva-tions dependent upon contractions among products of
normal-order operators. For purposes of perturbationexpansions, in the most general case, the normal-ordered HN=H0+VN,
H0N= �
pfppp†p� + �
i�jfiji†j� + �
a�bfaba†b� , �51�
VN = �a,i
faia†i + i†a� + 14 � �pq��rs�p†q†sr� , �52�
VN = fov + WN, �53�
where fov represents the occupied-virtual part of fN andWN is the two-electron operator.
Hence, the Schrödinger equation now becomes HN�=�E�, where �E=E−Eref, the correlation correction�Löwdin, 1959�. This eliminates E�1� from expressionspreviously containing V−E�1� once we understand thatVN is normal ordered as above.
Note that the appearance of the Fock operator in theHamiltonian is a consequence of normal ordering anddoes not presuppose HF orbitals or a HF reference func-tion. The Hamiltonian is completely general. However,for the canonical HF case fpq=�ppq, giving the simpleform HN=�p�pp†p�+WN. For any other choice of orbit-als and reference determinant, such as Kohn-Sham�where the density is given by a single determinant��Kohn and Sham, 1965�, natural �Löwdin, 1955� �wherethe first natural determinant gives the best single deter-minant approximation to the density matrix�, andBrueckner �Brueckner, 1955� �where the Brueckner de-terminant has maximum overlap with the exact wavefunction�, we retain the fij and fab parts in H0 and fai inVN. That is, we insist upon orbital invariance for anyrotation of the occupied orbitals among themselves, orthe virtual orbitals among themselves. For non-Hartree-Fock orbitals fai�0, so those effects, which rotate thevirtual space into the occupied space, will change theresults �except for full CI� and are introduced in VN.Many-body perturbation theory will be invariant to anyrotations in the occupied or virtual space, as long as H0does not change. Hence, this choice of H0 has this prop-erty and is the only one that does. Generalized MBPTpresented elsewhere �Bartlett, 1995� assumes this H0,and as a special case of CC theory it naturally allowsMBPT to be done with any single determinant referenceand associated orbitals. As an infinite-order method, theCC solutions are formally independent of H0, except thechoice of H0 will suggest a natural iterative scheme forthe solution of the nonlinear equations. Coupled-clustertheory will also be invariant to rotations in the occupiedor virtual space, and at the CCSD level and beyond,noninvariant, but insensitive to orbital rotations that mixthe two spaces, beginning to approach full CI’s invari-ance. This feature �discussed later� gives CC theory avery high degree of flexibility for orbital choices notshared by MBPT or truncated CI.
301Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
We choose the diagrammatic notation shown in Fig. 4for the Hamiltonian HN. Note that two �four� lines con-nected with the one- �two-� electron operator indicatetwo �four� creation-annihilation operators occurring inthe definition of the Hamiltonian, Eq. �48�. Our dia-grams are based upon normal-ordered operators, spinorbitals, and use antisymmetrized two-electron integralsand cluster amplitudes, but are drawn in the Goldstoneform with one antisymmetrized diagram representingseveral conventional Goldstone diagrams, as in the Hu-genholtz convention. We also identify the various termsin the perturbation VN by their excitation level, meaningthat if it is 0, there is no change, but +1 increases theexcitation, and �2 decreases it by that amount. Thethree particle-hole pairs in T3 and higher cluster ampli-tudes have to also be understood to be treated equiva-lently, as there is no diagrammatic distinction betweenthe center pair and the other two. Using these diagrams,we can immediately write the linked diagrams of theMBPT wave function,
�MBPT = �0 + �k=1
�
�R0V�k��0�L = �0 + ��1� + ��2� + ¯
where we choose not to indicate the line directions.These skeleton diagrams are sufficient for our currentformal manipulations, but the line directions would haveto be introduced in a computational formula. The en-ergy diagrams E�n+1�= �0 �V ���n�� are closed, with ex-amples shown in Fig. 5. The wave functions are neces-sarily linked, meaning that there are no closed energydiagrams. But the remaining open linked diagrams haveboth connected and disconnected parts. Now cluster op-erators are introduced,
to sum all connected terms that result in a net doubleexcitation and
FIG. 4. Diagrammatic form of the fN, WN, and Tn operators.The labels at the bottom of the fN and WN operators refer tothe changes in the excitation level caused by that form of theoperator.
FIG. 5. Skeleton diagrams for the second- and third-orderMBPT energies.
302 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
to do the same for single excitations, and T3,T4, etc. forthe higher connected terms. The linked, but discon-nected terms like
arise from the higher terms in the exponential expansiononce it is understood that the factorization theorem�Frantz and Mills, 1960� provides
That is, we exploit the factorization theorem to removethe eight-index denominator that occurs via
in using all distinct time orders of the two-particle inter-actions in favor of a product of two four-index ones, andfor each T2 cluster operator. Generalization to any ordercan be established with mathematical induction �Shavittand Bartlett, 2006� so that it is apparent that the linked-diagram MBPT wave function may be written as
�MBPT = �0� + �n=1
�
�R0V�n�0�L = exp�T��0� , �54�
where R0 is the resolvent defined in Sec. III.B. This wasfirst stated by Hubbard �1957�. This also tells us that thewave operator �=exp�T� takes the approximation �0�into ���.
We find the choice of spin-orbital, antisymmetrizedtwo-electron integrals and amplitudes to be the mostconvenient for formal derivations, while at the sametime accounting for closed- and �single determinant�open-shell molecules. The equations for the various spe-cial cases �discussed below� can be derived from the gen-eral form by adding line directions and carrying out theappropriate spin integrations. Also, based upon antisym-
metrized spin-orbital forms, we present below and inFig. 6 easily applied tools that enable an unambiguousgeneration of all the diagrams with no redundancy,eliminating the uncertainty often associated with dia-grammatic derivations.
As discussed in Eqs. �21�–�23�, the CC equations are
�0�H�0� = E�0� , �55�
� ij¯ab¯�H�0� = 0. �56�
Individual, normal-ordered excitations will be indicated
by �ij¯ab¯�= �0�i†aj†b¯ �. The quantity
HN = HN + �H,T +12†�H,TT‡ +
13!
�†�H,TT‡T
+14!†�†�H,TT‡TT‡ , �57�
HN = exp�− T�HN exp�T� = �HN exp�T�C �58�
obtained from H by a similarity transformation, Eq. �58�,is a critical one in CC theory since it terminates afterfourfold commutators. That is, because the Hamiltonianhas only one- and two-particle operators, the maximumnumber of T operators that can lead to nonvanishingcontributions to the CC amplitude equations is four, re-gardless of their excitation level. This feature will causethe CC equations to always have a finite number ofterms, or be in closed form, despite the fact that the CCwave function remains an unterminated exponential inT. The commutators necessarily eliminate any terms inH and T that have no indices in common. Hence, theequations for the energy and amplitudes in CC arelinked, and the amplitudes, themselves, necessarily con-nected. The concept of an exponential wave function forfermions was considered by Coester and Kümmel�1960�, with the first workable equations presented byCížek �1966� for the simplest model, coupled-clusterdoubles �CCD�, then called coupled-pair many-electrontheory �CPMET�.
Using the occupation number representation and rec-ognizing that we can only have fully contracted opera-tors in a vacuum matrix element of normal-ordered op-erators to be nonvanishing, we can derive the energy asa simple illustration,
Note the sign rule, which means if the number of linescrossed in the full contraction is even, the term is posi-tive; if odd, it is negative. Though straightforward, theredundancy of contractions that lead to identical expres-sions makes such an exercise tedious. With the diagrams,however, we immediately have the answer.
A. Double excitations in CC theory
For the CCD wave function ���CCD�=exp�T2��0�, wehave
0 = � ijab�HN exp�T2��0�C ∀ a,b,i,j �64�
=� ijab�HN + HNT2 + HNT2
2/2�0�C �65�
=� ijab�W + H0T2 + WT2 + WT2
2/2�0�C. �66�
We used the fact that H0 cannot couple a double excita-tion with the reference function �0�, or a double with aquadruple that would derive from T2
2. If we persist inusing occupation number tools above to obtain the re-sults, we would need to evaluate the several vacuumcontractions,
14 �
pqrs�pq��rs��0�i†aj†b�p†q†sr��0� , �67�
�pq
fpqtklcd�0�i†aj†b�p†q�c†kd†l��0� , �68�
14 �
pqrs�pq��rs�tkl
cd�0�i†aj†b�p†q†sr�c†kd†l��0� , �69�
116 �
pqrs�pq��rs�tkl
cdtmnef �0�i†aj†b�p†q†sr�c†kd†l�e†mf†n��0� .
�70�
However, the nonlinear term begins to challenge our pa-tience �one might even go back to using determinantsand Slater’s rules, which can be done through CCD atleast �Cížek and Paldus, 1971; Hurley, 1976�. We prefermore powerful diagrammatic ways to derive the CCequations for higher than double excitations.
The diagrams introduced in Fig. 4 and shown in Fig. 6immediately tell us what the CCD diagrams are, wherewe use the excitation and deexcitation level indicatedunder the f and W operators. The diagrams take advan-tage of the fact that �i� the particular double excitationprojection need never be explicitly included, and �ii� thefactor of 1
4 in the antisymmetrized W operator will besubsummed into the diagrams’ numerical factors, ratherthan having to be associated with four different contrac-tions that amount to the same term, and �iii� the sign ofthe terms will be automatic. Furthermore, we follow theprescription shown in Fig. 6 where we identify the netexcitation level, the excitation level of the products of Tamplitudes, and the relevant part of the perturbationthat will reduce or enhance the excitation level to that
304 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
desired. Then we make all possible, simple combinationsthat will allow us to achieve the final, net excitationlevel. Since there are only a handful of combinations,and each leads uniquely to one and only one distinctantisymmetrized diagram �regardless of how it is drawn�,anyone can follow this prescription confidently to deriveunique diagrams. When the diagram is interpreted usingthe rules listed in Table II, we easily have the algebraicform in terms of spin-orbital, antisymmetrized integralsand antisymmetrized amplitudes, which are shown inFig. 6.
B. Choice of single determinant reference function
From this point, the actual equations programmed de-pend upon the choices made for the spin-orbital form.The possibilities are shown in Fig. 7. If we are describinga closed-shell molecule with doubly occupied spatial or-bitals as in Fig. 7�a�, i.e., spin orbital p=p=P and p+1=q=P�, we can go from antisymmetrized diagramsto the usual Goldstone form by drawing all the exchange
variants as shown in Fig. 8. This allows the immediateinterpretation of the CC equations in terms of the spa-tial orbitals P as all closed loops now account for a fac-tor of 2 in the equations, and there is a vertical symme-try 1
2 rule but no equivalent line rule. The diagrams andthe interpretation are shown in Fig. 8.
If we want to describe a closed-shell but spin-polarized system like Fig. 7�b�, then we require differentorbitals for different spins �DODS�, p=P ,p+1=P��,where P�P�. For the HF case, this is termed unre-stricted Hartree-Fock �UHF�. The diagrams and compu-tational equations are shown in Fig. 9. We present thediagrams for � and spin blocks, since the �� blockcan be easily obtained from Fig. 9�a�. Note that all work-ing equations take advantage of spin integration, sinceotherwise the spin-orbital calculations would be �26
slower than the closed-shell calculation. Once the spinintegration is made, the difference in time for DODSversus doubly occupied is �3. Such a single UHF deter-minant is not an eigenfunction of spin, unlike the doublyoccupied case. The more common use for a UHF refer-ence is for open shells �Fig. 7�c� when the number of electrons exceeds those for �. Though also not an eigen-function of spin, it will be much closer for high-spincases. In contrast, for the closed-shell spin-polarized ex-ample, the UHF solution can be a 50-50 mixture of thesinglet and triplet, like for H2 at large R separation.
The intermediate situation is shown in Fig. 7�d�. Here
FIG. 6. Generation of the WT2 and WT22 /2 contributions to
the T2 amplitude.
FIG. 7. The choices made for the spin-orbital form: �a� RHF�closed-shell molecule with doubly occupied spatial orbitals�,�b� UHF �closed-shell spin polarized situation�, �c� UHF open-shell triplet, and �d� ROHF �triplet maximum double occu-pancy�.
FIG. 8. Diagrammatic representation of the CCD method inthe closed-shell spatial orbital form together with the corre-sponding algebraic expression. Summation over repeated up-per and lower indices assumed. P�ia / jb� implies a sum of twocomponents differing by permutaion of ia and jb.
305Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
we have maximum double occupancy up to open-shellorbitals. This kind of wave function is usually referred toas a restricted open-shell Hartree-Fock �ROHF� func-tion. Here the single determinant approximation is a
spin eigenfunction S2�ROHF�=S�S+1��ROHF�. How-ever, using that as the reference determinant does notmean that the corresponding CC method will provide aspin eigenfunction. The complication occurs due to therole of nonlinear terms. If T is spin adapted, then itsdirect product with a second spin-adapted T leads to areducible representation that has to be properly reducedto give the spin eigenfunction. And in CC theory, unlikeCI where there are no nonlinear terms, this is not easy.Our approach �Lauderdale et al., 1992; Bartlett, 1995� tousing ROHF is to build the Fock matrix f in terms ofthose orbitals, and then exploit the fact that any rotationof the occupied-occupied and virtual-virtual block is al-lowed, to semicanonicalize the matrix to the form
From this point, the calculation is like any other UHFcalculation, except we have to have fai terms in the per-turbation, V. In this way, any kind of reference determi-nant and orbitals can be used in CC theory. However,because of the essential role of fai, which now will arisein second-order perturbation theory like W �see Fig. 5�,it is not meaningful to use such arbitrary orbitals withoutat least the CCSD approximation discussed below.
For completeness, consider a low-spin situation, likethat for an open-shell singlet. Here we require that twodeterminants with the same Sz=0 value be coupled to-gether to get a spin eigenfunction 1
�2 ��AB��± �A�B�.
FIG. 9. Diagrammatic representation of the CCD method in the spin-expanded open-shell form together with correspondingalgebraic expressions. Summation over repeated upper and lower indices assumed. The antisymmetrizer is defined as P�pq /rs�=1+ �qp��sr�− �qp��rs�− �pq��sr�. The capital letters A, B, I, J, etc. refer to spin-orbitals, and lower case letters a, b, i, j, etc.denote � spin-orbitals.
306 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
For the Sz=0 component of the triplet, we can describeequally well the state starting from a single determinant�AB� ,Sz=1; but for the singlet, we are forced to useboth determinants, which would require the equivalenttreatment of two determinants. Single determinant ref-erence CC cannot describe such states appropriatelywithout modification �Balkova and Bartlett, 1992�. Interms of configurations instead of determinants, where aconfiguration means a combination of determinants thatare spin eigenfunctions, the open-shell singlet is a singleconfiguration, and a proper orthogonal spin-adaptedsingle reference CC theory is possible �Li and Paldus,2004�, but for more general cases it is formally and com-putationally much more difficult than might be ex-pected, due to the complications introduced by the non-linear terms discussed above. Consequently, the singledeterminant reference CC methods in ACES II �Stantonet al., 1993; Bartlett and Watts, 1998� will allow for UHF,ROHF, RHF, or their equivalents with Kohn-Sham,natural, Brueckner, or quasi-Hartree-Fock �QRHF� or-bital references. The latter means orbitals taken fromsome related system like using neutral orbitals to de-scribe an N2
+ state or F− orbitals to describe the F atom.Such almost arbitrary orbital choices are possible inCCSD and beyond because of the theory’s orbital insen-sitivity, as CCSD does not depend much upon a varia-tionally optimum reference function �see below�. Foropen-shell singlets a two-determinant CC method, whichis the simplest realization of multideterminant CC, hasbeen developed and is included in ACES II �Balkovaand Bartlett, 1992; Szalay et al., 1995�.
The CCD equations in their most general and—at thesame time—most compact form are shown in Fig. 10.Here we do not separate the spin part from the spacepart of the spin orbital, which means that we do not
perform the spin integration. Each line is summed overthe whole range of one-particle functions: the hole linesover all occupied spin orbitals and the particle lines overall virtual spin orbitals. Such a situation occurs when wetreat nucleons with the CC approach �Kowalski et al.,2004�. Naturally, for that case we need a new Hamil-tonian with properly defined one-nucleon and two-nucleon integrals transformed to the spin-orbital basis,but in other respects such calculations require us simplyto evaluate the diagrams in Fig. 10. The algebraic ex-pressions coresponding to the latter are given in Eq. �67��Fig. 10.1�, Eq. �68� �Fig. 10.2–3�, Eq. �69� �Fig. 10.4–6�,and Eq. �70� �Fig. 10.7–10�.
The generic form of the CCD equations as given inFig. 10 also has another important application, namely,in relativistic calculations. Solving, e.g., the full four-component Dirac equation �Pisani and Clementi, 1995;Visscher et al., 1996� requires the general, i.e., non-spin-integrated formulas for CC equations. Also for othersomewhat simpler calculations using the two-componentDouglas-Kroll-Hess Hamiltonian �Kaldor and Hess,1994�, the spin integration cannot be performed sincehere j is the appropriate quantum number instead of sz,which indicates that the generic form of the CC equationshould be used, although some symmetry simplificationis still possible �Visscher, 1996�.
Hence, the generic CC equations enable the treat-ment of two- and four-component relativistic problemsas well as nucleons in the same manner as electrons, aslong as no inappropriate further simplifications of thegeneric formulas are made. This generality recommendsusing the underlying spin-orbital framework for bothformal and computational aspects of CC theory.
C. Single excitations in CC theory
The role of single excitations in CCSD �Purvis andBartlett, 1982� theory, �CCSD=exp�T1+T2��0�, is quiteimportant. Whereas the contribution to the energy fromsingle excitations subject to a RHF or UHF referencefunction first appears in the fourth-order energy, for anon-HF reference, singles appear in second order justlike doubles, and can be quite large. Also, whereas CCDsubject to HF is correct through the third-order energyof perturbation theory, the one-particle density matrix,and thus all properties, is already wrong in second order.CCSD fixes this.
Another important point is that we know that singledeterminant references can be related by �0�=exp�T1���� �Thouless, 1961; Flocke and Bartlett, 2003�;so rather than trying to get some kind of optimum ref-erence function for CC theory, it is frequently preferableto simply solve the CCSD equations and allow theexp�T1� operator to account for orbital changes in thereference function, passively. As we go to even higherlevels of CC theory, like �CCSDT=exp�T1+T2+T3��0�,the additional coupling between T1, T2, and T3 furtherenhances this effect, causing the CC result to begin toapproach the complete orbital invariance of full CI.
FIG. 10. Diagrammatic representation of the CCD method inthe generic spin-orbital form together with the correspondingalgebraic expression. Summation over repeated upper andlower indices is assumed. The antisymmetric permutation op-erator P�pq /rs� is defined as P�pq /rs�=P�pq��rs�=1+ �qp��sr�− �qp��rs�− �pq��sr�.
307Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
However, the CCSD equations have 45 diagrams com-pared to only 10 for CCD, and they would be difficult toderive without diagrammatic �or computer-aided� proce-dures �Hirata, 2003�. Using the diagrams, we easily getexpressions and additional diagrams shown in Fig. 11.We will not interpret these diagrams, but that could beeasily done using the rules in Table II. All interpreta-tions and diagrams through CCSDTQ have been pre-sented �Shavitt and Bartlett, 2006�. CCSD was first for-mulated and implemented in 1982 �Purvis and Bartlett,1982�. With this foundation, higher-connected-clusteroperators could be introduced to rapidly approach thefull CI solution.
Another way to get the benefit of exp�T1� but limit thecomputation to just the few diagrams of CCD is to ex-ploit the flexibility to make orbital rotations such thatT1=0. The orbitals that accomplish this are termedBrueckner orbitals. The determinant composed of theseorbitals, �B, is also guaranteed to have maximum over-lap with the exact wave function,
�����B�� = max. �71�
If a reference determinant ��0� is chosen, then a deter-minant different from ��0� is obtained by rotating virtualorbitals into occupied orbitals, represented by the op-erator ti
aa†i�, which leads to
����ia� = 0 ∀ i,a . �72�
The condition of vanishing T1 amplitudes partitionsthe orbital space into a new occupied and virtual space.Since they are obtained by rotating virtual orbitals intothe occupied space, the T1 equation, where all diagramscontaining T1 amplitudes are removed, defines thevirtual-occupied block of an effective, Brueckner,Hamiltonian,
�Heffai = fai + �jb
fjbtijab + 1
2�jbc
�aj��bc�tijbc
− 12�
jkb�jk��ib�tjk
ab. �73�
Diagonalization of the Brueckner effective Hamiltonianprovides updates to ti
a until at convergence T1=Haieff=0.
Brueckner orbital-based CC theory and its extensionshave been used in a variety of CC applications �Chilesand Dykstra, 1981; Adamowicz and Bartlett, 1985;Handy et al., 1989; Stanton et al., 1992; Krylov et al.,2000� where the hope, particularly for properties otherthan the energy, is that higher-order products involvingT1 with T3 and higher clusters that are implicitly set tozero will help for some classes of problems �Watts andBartlett, 1994�. It also offers an important link to a cor-related effective one-particle theory that offers an exact,correlated analog to Hartree-Fock theory, including aKoopmans-type theorem that would give exact ioniza-tion potentials for its orbital energies �Lindgren andSalomonsen, 2002; Beste and Bartlett, 2004�.
D. Triple and quadruple excitations in CC theory
Triple excitations in CC theory are also important, asthey contribute to the fourth-order MBPT energy. Atthe CCD level, we already have the disconnected contri-butions of quadruple excitations in the wave function.Once we add singles, we have further disconnected con-tributions to the CI triples and quadruples, including allthrough fourth order in MBPT except for the connectedtriple excitations T3. This is quite different from CI, be-cause CI retains large contributions from unlinked dia-grams whose initial contributions are not canceled untilquadruple excitations are included in the CI. Hence,triple excitations are comparatively less important. Thisis shown in Fig. 3, where we illustrate the convergenceof single reference CI, MBPT, and CC theory toward thefull CI limit �see also Table I�. Note the abrupt changewhen quadruple excitations are introduced in CI. Due tothis cancellation of the fourth-order unlinked diagramsdiscussed in Sec. III.B, MBPT has the advantage that allthe unlinked diagrams are removed from the beginning,but suffers from being finite-order approximations.
FIG. 11. Diagrams representing the single excitation andsingle contributions to the double-excitation CCSD equations.Diagrams occurring in the T2 equation plus those present inthe CCD model �Fig. 10� form the full T2 equation in theCCSD method.
308 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
Coupled-cluster theory combines the linked-diagram ad-vantages of MBPT with infinite-order summations, so ithas to give the best convergence of these three ex-amples.
Drawing all the 99 diagrams for CCSDT and 180CCSDTQ tells us little �see Tables III and IV for these
numbers�. Instead we can summarize the equations forhigher-level CC theory in terms of the quasilinearizedform, Figs. 12 and 13, based upon the intermediates
from H that are actually computed in such programs.These are shown diagrammatically in Fig. 14 and alge-
TABLE III. General formulas for the number of the diagrams in the CC equation for Tn amplitude�valid for n�1�;a Na=1/24n3+1 5/8n2+11 1/12n+1; Ng=1/12n3+3 1/2n2+24 1/6n−9.
Model
Diagrams in the Tn equation Diagrams in the Tn equation
braically in Table V. The quasilinearized form is ob-tained using the factorization procedure, which can beexplained with the following example. Taking the inter-mediate
defined in Fig. 14.3a and applying it in the term
�third term in Fig. 12�a�, we obtain the following threediagrams:
which can be identified easily as those contributing tothe T1 equation in Fig. 11. Taking other intermediatesdefined in Fig. 14 and applying them to the terms in theequation shown in Figs. 12�a�–12�c�, we can reproduceall terms contributing to the T1, T2, and T3 equations inthe CCSDT model, respectively. For completness, wealso present the quasilinearized form of the CCSD equa-tions in Fig. 15 �cf. Figs. 10 and 11�. Use of the recur-sively computed intermediates �Kucharski and Bartlett,1991a� makes the concept of a high-order CC program
quite straightforward, and that, plus using various auto-matic �Hirata, 2003� or other computer-generated pro-grams �Olsen, 2000; Kallay and Surjan, 2001; Kallay andGauss, 2004�, has now made it possible to go to quitehigh levels of excitation through hextuples, i.e.,CCSDTQPH, for example. Their application to chemi-cally interesting problems, though, is limited by otherfactors.
As discussed briefly earlier, CCSD requires no morethan �n2N2tij
ab amplitudes, and the rate-determining stepis the evaluation of the ladder diagram, �n2N4, thefourth diagram in Fig. 9, which already arises in CCD.Once we go to CCSDT, we have �n3N3 amplitudes anda computational dependence of �n3N5. In general, wehave �nlNl amplitudes and an �nlNl+2 computationaldependence for the level of excitation l. For our modest-sized example from before n=10, N=100 functions,CCSDT would require storing some �109 amplitudes. Alower bound to its time for the evaluation of a singleiteration would be �10�1011 operations ��1 h for a 2gigaflop processor�. Doubling the size of molecule adds�212 to the time attesting to the extreme nonlinear de-pendence of high-level CC calculations if no further sim-plification is made. Yet, today, CCSD �Purvis and Bart-lett, 1982�, CCSDT-1 �Lee et al., 1984�, CCSDT-3 �Nogaet al., 1987�, CCSD�T �Urban et al., 1985�, and CCSD�T��Ragavachari et al., 1989; Bartlett et al., 1990� calcula-tions are done with �300 basis functions and have used�600. The first two triples models include the principaleffects of triple excitation iteratively, which requiresonly an �n3N4 step and is done without storing the tijk
abc
amplitudes. The last two are further perturbative ap-proximations of triples based upon CCSDT-1 that are
FIG. 12. Diagrammatic CCSDT equations with total factoriza-tion of nonlinear terms: �a�, �b�, and �c� represent the T1, T2,and T3 equations, respectively.
FIG. 13. Diagrams representing connected quadruple contri-butions to the double-, triple-, and quadruple-excitation equa-tions in the CCSDTQ method.
310 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
used to augment the CCSD solution �see Sec. V.E�, butreduce the computational dependence down to a nonit-erative �n3N4 step augmenting an iterative �n2N4
CCSD calculation. At each level some accuracy is sacri-ficed for a broader range of applications.
In addition to the CCSD diagrams shown in Fig. 10,the additional diagrams that define CCSDT-1 �Lee et al.,1984� are shown in Fig. 16, truncating the T3 equation to
its lead, second-order term and then adding the tripleexcitation contribution to the T1 and T2 equations. Inthis way, the T3 amplitudes, which would appear on theright-hand side of the full CCSDT equations, are notallowed to contribute to T3. However, the T3 obtaineddoes contribute to T1 and T2 and T2 then updates the T3
amplitudes, until reaching convergence. The tijkabc�2
„the
FIG. 14. Diagrammatic form of intermediates introduced in CCSDT and CCSDTQ models with total factorization of nonlinearterms. The * indicates that the intermediate is preceded by a 1/2 to avoid overcounting.
311Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
TABLE V. Algebraic expression for the intermediates used in the CCSDT and CCSDTQ equations.
Intermediate Expressiona
1. Iai fa
i + tnf vaf
in
2. Iba
fba + tn
f vbfan−
12
tnoaf vbf
no− tnaIb
n
3. Iji Ij�
i+ tjfIf
i
3a. Ij�i
fji+ tn
f vjfin+
12
tjnfgvfg
in
4. Ibcai
�bcai −
12
tnavbc
ni
4a. �bcai
vbcai −
12
tnavbc
ni
5. Ikaij
�kaij +
12
tkf vfa
ij
5a. �kaij
vkaij +
12
tkf vfa
ij
6. Icdab
Icd�ab+
12
tnoabvcd
no
6a. Icd�ab vcd
ab−P�a /b��cdantn
b
7. Iklij
vklij +P�k / l��kf
ij tlf+
12
tklfgvfg
ij
8. Ibjia
�bj�ia+vbf
intjnaf+
12
Ibfia tj
f
8a. �bjia
vbjia −
12
vbjintn
a +�bfia tj
f
8b. �bj�ia
vbjia −
12
vbjintn
a +12�bf
ia tjf
8c. �bj�ia
vbjia −vbj
intna +
12
Ibfia tj
f
9. Iciab
vciab+vcf
abtif−P�a /b�tn
a�cinb−Ic
ntniab+P�a /b�Icf
antinbf+
12
tnoabIci
no−12
tnoiafbvcf
no
9a. Ici�ab
vciab+
12
vcfabti
f−P�a /b�tna�ci�
nb
10. Ijkia Ijk�
ia+Ifitjk
fa
10a. Ijk�ia
vjkia −vjk
intna +P�j /k�tj
f�fk�ia+P�j /k�Ijf
intknaf +
12
Ifgiatjk
fg+12
vfgintjnk
fga
10b. Ijk�ia
vjkia −
12
vjkintn
a
11. Idij�abc 12
P�ab /c�Idfabtij
fc+P�a /bc�Idfantijn
bcf+12
P�i / j�Idinotnoj
abc−12
vdfnotnoij
afbc
12. Ijkl�iab
−12
P�jk / l�Ijkintnl
ab−P�a /b�P�kl / j�Ifjiatkl
fb+P�j /kl�Ijfintk ln
abf
+12
P�a /b�Ifgiatjkl
fgb+12
vfgintjnkl
fgab
13. Iklm�ija 12
vfgij tklm
fga
14. Icjk�iab 12
vcfintjkn
abf
aSummation over repeated indices assumed. P�i / j� or P�a /b� implies the sum of two components differing by permutation of i,j and a, b indices, respectively. P�ab¯ /c¯ � indicates that in addition to the identity permutation, the summation should includeall possible permutations exchanging labels between subsets �ab¯ � and �c¯ �. The same refers to P�ij¯ /k�.
312 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
�2 instead of �2� indicates a generalized second order
since the T2 amplitudes contain infinite-order terms…amplitudes are used on the fly to avoid any storage. An-other convenience, if not a necessity in this evaluation, isthat we are free to dismiss the off-diagonal part of thediagram
by exploiting the fact that CCSDT-1, like any other it-erative CC approximation that always evaluates com-plete diagrams as opposed to using just selected parts ofthem, will be invariant to rotations in the occupied orvirtual space. Hence, we are free to make the semica-nonical transformation of the Fock matrix discussed inSec. V.B to make the off-diagonal fij= fab=0. Withoutdoing so, we could be left with an extra �n3N4 iterativestep in the determination of the above diagram. Fornon-HF cases, in the equations below, we will assumethat this transformation has already been made. Asshown in Figs. 16 and 17, to make the transition fromCCSDT-1 to CCSDT-3 �Noga et al., 1987�, which meansthat all possible contributions of T2 and T1 to T3 areincluded instead of just their lead terms, it is apparentthat this requires nothing but the replacement of the
diagram units in CCSDT-1 by their corresponding H in-
termediates. Another way of saying this is
�ijkabctijk
abc�T − 3� = � ijkabc��HN exp�T1 + T2�C�0� �74�
instead of just using linear T2 as is done in CCSDT-1. Sothis too is an iterative �n3N4 method, yet it has theenhanced orbital insensitivity that accrues due to the in-clusion of higher-order terms in T1 and T2. This methodstill avoids the storage of the �n3N3 tijk
abc amplitudes andgains �N in speed over the full CCSDT, making it apractical, yet highly accurate level of approximation formany molecular problems.
E. Noniterative approximations
Short circuiting the iterative procedure of CCSDT-1and simply using converged T2 and T1 amplitudes fromCCSD, and using the expectation value energy formulainstead of that in Eqs. �59�–�63� �Urban et al., 1985�,gives
E = �0�exp�T†�H exp�T��0�/�0�exp�T†�exp�T��0� �75�
=�0��exp�T†�H exp�T�C�0� �76�
=�0�exp�T†��H exp�T�C�0� . �77�
This formula recognizes that the HN expectation valueof the full CC wave function also gives the energy. Byinserting the resolution of the identity, it is easy to seethat
E = �0�exp�T†�exp�T�exp�− T�HN exp�T��0�/�0�exp�T†�exp�T��0� �78�
which is the usual, sometimes called, transition formula.
This formula follows from the fact that QH�0�=0 as theCC equations would be satisfied for any excitation in Q.Whereas the transition formula is always in closed form,the expectation value form is not, either requiring divi-sion by the denominator in Eq. �75�, or, as in Eq. �76�,the division has already been incorporated by the re-striction to the connected form in the numerator �Cížek,1966� which similarly leads to an infinite series regard-less of the number of electrons in the problem. For spe-cific CC approximations like CCSD, the energy equiva-lence does not hold, and in fact it can be shown thatimportant contributions from connected quadruple exci-tations will arise from the expectation value of CCSD�Bartlett and Noga, 1988�. The third form, Eq. �80�, isintermediate and particularly useful for the followinganalysis, as it imposes connectedness on the �HNeT�cproducts.
Slightly generalizing the original derivation �Urban etal., 1985� to the non-HF case �Watts et al., 1993�, we usethe tijk
abc�2 amplitude evaluated from the first iteration ofCCSDT-1 to obtain the noniterative approximationCCSD�T by computing the fourth-order terms that canarise from connected T3,
ET�4 = �0�T3
�2†�H0T3�2�C + T3
�2†�WT2�C + T2†�WT3
�2�C
+ T1†�WT3
�2�C + T2†�fvoT3
�2�C�0� . �81�
Using the T3�2 equation, � ijk
abc�H0T3�2+WT2�0�=0, it is ap-
parent that the first and second terms cancel, and thethird term, the complex conjugate of the second, can bereplaced by the negative of the first and the extra de-nominator indicated, to give the diagrams
313Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
If we count orders relative to HF, then T1 would firstarise in second order. So the appropriate purely fourth-order approximation is given by diagram 1, which is theoriginal CCSD�T �Urban et al., 1985�.
Once we consider non-HF cases, then the second dia-gram corresponds to a fourth-order term, and adding itwe obtain CCSD�T� �Ragavachari et al., 1989�. Its evalu-ation is as follows:
�iati
a�3�T3� = � ia��WT3�C�0� , �82�
EST�5 = 1
4�i,a
tia*ti
a�3. �83�
So when used for HF cases this selects only one of manyfifth-order terms, yet the numerical results tend to im-prove since this diagram usually acts in an opposite di-rection to the first term, which is necessarily negative; so
the second term can help to avoid overshooting the en-ergy in difficult cases.
Finally, the completely general form of Eq. �81� alsoadds diagram three �and presupposes the semicanonicaltransformation to avoid the off-diagonal diagram previ-ously mentioned� �Watts et al., 1993�. This final, nonit-erative form has exactly the same invariance propertiesas iterative CC methods, as it is invariant to orbital ro-tations within the occupied or within the virtual space, ahighly desirable benefit of the approximation. Amongother properties, the invariance facilitates analytical de-rivatives of the energy as discussed in Sec. VII. Thepresence of the denominator in Eq. �75� can be ex-ploited to derive renormalized approximations for non-iterative corrections. This will be discussed in Sec. VI.C.
An even more general derivation of iterative andnoniterative triple and quadruple excitation contribu-tions can be made based upon the CC functional pre-sented in Sec. VII, and derived there.
We can achieve even a lower dependence for some ofthe T4 effects as follows. Assume we are improving uponan existing CCSDT solution, and for non-HF cases in-voke the semicanonical transformation. Focusing on thelead term in the T4 equations,
0 = � ijklabcd�H0T4�0� + � ijkl
abcd��WT22/2�C�0�
+ � ijklabcd��WT3�C�0� ∀ a,b,c,d,i,j,k,l , �84�
T4�3 = R4��WT2
2/2�C + �WT3�C , �85�
T2�4�T4� = R2WT4
�3, �86�
E�5�T4� = �0�WT2�4�T4��0� , �87�
where the last term arises for the usual transition energyformula �R2 means the double excitation part of the re-solvent and R4 the quadruples excitation part�. Furthermanipulations of the energy expression give
E�5�T4� = �0�WR2WR4��WT22/2�C + �WT3�CC��0� �88�
FIG. 15. Diagrammatic CCSD equations with total factoriza-tion of nonlinear terms.
FIG. 16. Additional diagrams to the CCSD model that definethe CCSDT-1 method.
FIG. 17. Additional diagrams to the CCSD model that definethe CCSDT-3 method.
314 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
We have identified WR0 as the first approximation toT2
† ,T2�1�†. Then using the factorization theorem in the
last line, 12 �WR2W�+WR2�W��= 1
2 �W�2−1W�+W��2
−1��W�=WW���2+�2�� /�2�2� eliminates R4��4� from the energyexpression. As the eight-index �4 denominator would re-quire in this case an �n4N5 computational step, its elimi-nation reduces the computation to �n2N5 for the T2
2
term and to �n3N4 for T3.Though this can be done rigorously in the fifth-order
transition energy expression, it cannot be done in theamplitude expression. However, as we know that the T4
�3
amplitudes inserted into the T2 equation will introducethis initial energy correction �Kucharski and Bartlett,1998c�, we can modify the T2 equation by adding to it,
T2Qf
�4 �T4� = 12R2T2
�1�†�W�T22/2 + T3�C� . �90�
This can be incorporated into its iterative solution in thenormal way, coupled to the T1 and T3 equations. Its en-ergy contribution would then arise in the usual energyformula, Eq. �80�. This factorization preserves the samecomputational simplicity of the factorized approxima-tion into the amplitude equation as it only adds an�n3N4 and an �n2N5 step to an already �n3N4 CCSDT-n or �n3N5 full CCSDT method. It is also fundamen-tally different from the usual CC disconnected simplifi-cation as it approximately factorizes a connected T4. In aseries of comparisons for H2O, HF, and BH molecules intheir equilibrium geometry, the use of the T2Qf
�4 ampli-tude approximation differs from using the regular qua-druple contribution by �0.003 mH �Kucharski andBartlett, 1998c�.
Finally, using the expectation value energy expressionas we did above for triples and isolating the contributionof quadruples, we have
EQf
�5 = 12 �0�T2
†T2�1�†�W�T2
2/2 + T3�C�0� , �91�
which defines a noniterative, fifth-order factorized qua-druple contribution in analogy to that for T3, calledCCSDT�Qf� �Kucharski and Bartlett, 1998c�. One canalso use a lower approximation than CCSDT with its�n3N5 step, by evaluating Qf from an underlyingCCSDT-n approximation, or go all the way toCCSD�TQf� where the usual T is combined with Qf. Thelatter is the simplest possible initial approximation forconnected T3 and T4. All such approximations, particu-larly the noniterative ones, make applications possiblethat could not be done otherwise. Several such nonitera-tive approximations have been considered that are cor-rect through fifth �Kucharski and Bartlett, 1998a� andsixth order �Kucharski and Bartlett, 1998b�, using threedifferent types of energy formulas: that from the expec-tation value, the normal expression, and the CC func-tional to be discussed in Sec. VII. Other such nonitera-tive approximations that include quadruple excitationshave been suggested �Gwaltney and Head-Gordon,
2001; Hirata, Nooijen, et al., 2001; Hirata, Fan, et al.,2004; Bomble et al., 2005�. The latter, termedCCSDT�Q�, is correct to sixth order and the most com-plete to date. It is discussed in Sec. VII.
The above approximations have established the para-digm of converging, size-extensive approximations forelectron correlation from
MBPT�2� � CCSD� CCSD�T� � CCSDT
� CCSDT�Qf� � CCSDT�Q�
� CCSDTQ� full CI.
Coupled with an adequate basis set for the phenomenaof interest, this paradigm provides predictive results towithin reasonable error bars.
One word about applications to nuclei. The abstractof Kowalski et al. �2004� says, “the quantum chemistryinspired coupled-cluster approximations provide an ex-cellent description of ground and excited states of nu-clei. The bulk of the correlation effects is obtained at theCCSD level. Triples, treated non-iteratively, provide vir-tually the exact description.”
For those readers who prefer to continue with newtheory developments, the next section focuses primarilyon CC numerical results at equilibrium and some of thelimitations in single reference CC �SR-CC� in breakingmolecular bonds.
VI. SURVEY OF GROUND-STATE NUMERICALRESULTS
A. Equilibrium properties
In the vicinity of the equilibrium geometry for mol-ecules, single determinant CC theory is exceptionally ac-curate. However, all ab initio results depend upon thequality of the basis set as well as the correlation correc-tions. So we have three levels of meaningful numericalcomparison for coupled-cluster theory with MBPT andCI: �a� comparison of CC with full CI; �b� comparison ofCC with experiment as a function of basis set; and �c�comparison of CC with experiment at the extrapolatedbasis-set limit, or, alternatively, using CC-R12 wheremost of the basis-set dependence is removed due to ex-plicit R12 inclusion.
Full CI comparisons are the least ambiguous, as oneadvantage of finite basis-set methods is that the exactresults in the basis is given by the full CI that includes allexcitations through n-fold for n electrons. The limita-tion, of course, is that the full CI itself cannot generallybe obtained except for few electrons in small basis sets.Subject to this caveat, in Fig. 3 we demonstrate how theevaluation of the correlation energy converges with in-creasing CI excitation, with orders of MBPT diagrams,and with the addition of higher connected excitations inthe infinite-order CC method. Whereas nonpolarizedbases cannot be expected to offer meaningful measuresof behavior, once polarization functions are included,the correlation effects for small molecules are indicative.
315Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
The next level of comparison is to experiment as afunction of basis set. The most extensive set of resultsfor CC theory has been presented by Helgaker and co-workers �Bak, Gauss, et al., 2000; Bak, Jörgensen, et al.,2000; Helgaker et al., 2000; Coriani et al., 2005�. We re-plot some of their results for bond lengths for a selectionof 31 molecules in Figs. 18�a� and 18�b�. The structure ofbound molecules is typically the easiest quantity to cor-rectly describe in quantum chemistry, with evenHartree-Fock often being adequate. In each case theHartree-Fock, MP2, CCSD, and CCSD�T� values with atriple-zeta �cc-pVTZ� basis and then a quadruple zetabasis are shown. In the triple-zeta basis, the HF distri-bution of errors centers at about −2 pm �−0.02 Å�,and varies from −6 to +4 pm, while MP2 centers atabout +2 pm and varies from −3 to +6. For the largerbasis, the HF distribution is virtually the same, as theHF limit has likely been achieved already for this prop-erty, while the simplest correlated method MP2 is im-proved somewhat centering at +1 pm. CCSD further im-proves upon the MP2 distribution, with it being centeredat exactly the experimental value in the QZ basis.CCSD�T� is not necessarily an improvement over CCSDfor this example because of very small errors encoun-tered and core correlation effects. The cc-pVXZ basesdo not have functions explicitly chosen to correlate coreelectrons. The core effect is shown in Fig. 19 as a func-tion of basis set where the cc-pCVQZ adds core corre-lation functions. Then the error in CCSD�T� is less than±2 pm.
Energies are more difficult to obtain accurately. Theworst energy for a basis-set-dependent quantum chem-istry calculation is the heat of atomization for a poly-atomic molecule, because separating a molecule into itsatomic fragments maximizes the basis-set errors in thecalculation. For 16 small molecules, the heat of atomiza-tion in a cc-pVQZ basis is shown in Fig. 20. CCSD�T�is a notable improvement over CCSD, which has anarrower distribution than MP2, but is not as well cen-tered as the latter. The range of errors in CCSD�T�is about −13 to +6 kcal/mol �1 kcal/mol=4.184 kJ/mol=0.043 349 eV�.
Bond dissociation energies are somewhat easier to de-scribe accurately, because breaking one bond in a poly-atomic molecule instead of all retains much of the basis-set error cancellation in the calculation, except fordiatomic molecules. For a set of 13 molecules, the reac-tion enthalpies compared to experiment are illustratedin Fig. 21 for a core-corrected, cc-pCVQZ basis. Theerrors in CCSD range from −8 to +6 kcal/mol, and cen-ter just a little below experiment. CCSD�T� narrows thedistribution somewhat �±6 kcal/mol� and is centeredjust slightly above experiment. There is substantial im-provement from HF�MP2�CCSD�CCSD�T�, as onewould anticipate.
For a property other than the energy, CCSD�T� dipolemoments in Debyes as a function of basis sets are com-pared to reference values for 11 polar molecules in Fig.22. All such properties are evaluated analytically fromthe response density matrices discussed in the next sec-
tion, which are the CC analogs �Bartlett, 1995� of thestandard density matrices associated with variationalwave functions. In good basis sets, the errors are lessthan ±0.2 D, compared to an even larger augmented QZ
FIG. 18. Normal distribution functions of the deviations fromexperiment of the calculated bond distances �pm� for a set of31 molecules in the �a� cc-pVTZ basis set and �b� cc-pVQZbasis set �Coriani et al., 2005�.
316 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
basis. The augmentation adds several more diffuse basisfunctions than in the standard basis, and these functionswill be important in accurately describing the dipole op-erator �ieiri.
To summarize, with a triply polarized basis like cc-pVTZ, the CCSD�T� standard deviations are for struc-ture ��0.0024 Å�, dissociation energies for single bonds��3.5 kcal/mol� �Helgaker et al., 2000�, and harmonicvibrational frequencies ��5–20 cm−1� �Bartlett, 1995�.Hence, this kind of accuracy for modest-sized moleculescan be expected. To quote from the CI community,Thom Dunning has stated that “…of the methods inwidespread use today, the CCSD�T� method is the onlyone that provides a consistently accurate description ofmolecular interactions for all interaction scales investi-gated, from more than 200 kcal/mol to 0.02 kcal/mol.”Better basis sets, plus basis extrapolation, and better lev-els of CC theory like CCSDT-3 or CCSDT will typically�but not always� give even more accurate, but more ex-pensive results. At the highest levels, we might also haveto concern ourselves with achieving balance between T4
and T3, recommending a level like CCSDT-3�Qf� �Ku-charski and Bartlett, 1999� or CCSDT�Q� �Bomble et al.,2005�.
As illustrated in Tables VI and VII, if we want toobtain the harmonic vibrational frequency of a smallmolecule to an accuracy of �1 cm−1, we can onlyachieve that for N2 �Kucharski et al., 1999; Musiał et al.,2001� and C2 �Kucharski et al., 2001� when using someconsideration of pentuple excitations �Pf� on top ofCCSDTQ, in a core-corrected, cc-pCV6Z basis. That ba-sis set has 460 contracted Gaussian functions for N2 andC2. Note the convergence with basis set and the neces-sity of extrapolation to fill in the tables. In Table VIII weshow results where further effects of basis sets as mea-sured with explicit R12-CCSD �the best current approxi-mation to removing the basis-set error in CC theory�and full CCSDTQP are used, plus adiabatic and relativ-istic corrections �Pawlowski et al., 2003; Ruden et al.,2004�. The core-correlation effects are listed separately.Extremely high-accuracy CC extrapolated thermochem-istry �HEAT� values have been presented by Tajti et al.�2004� to resolve discrepancies in the thermochemistry
FIG. 19. Normal distribution functions of the deviations fromexperiment of the calculated bond distances �pm� for a set of31 molecules for the CCSD�T� method in the cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pCVQZ basis sets �Coriani et al.,2005�.
FIG. 20. Normal distribution of errors �kcal/mol� for calcu-lated equilibrium atomization energies compared to experi-ment for a set of 16 molecules containing first-row atoms in thecc-pCVQZ basis set �Bak, Jörgensen, et al., 2000�.
317Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
data bases for some small molecules and radicals. Otherworkers �Pawlowski et al., 2003; Alexeev et al., 2005�have provided similar ultimate accuracy CC results toanswer questions that depend upon an accuracy of lessthan 1 kcal/mol. The fact that this can be done attests tothe power of CC theory.
For a more demanding example for vibrational fre-quencies, consider the ozone molecule. Ozone has twonearly degenerate MO’s �a2 and b1� of which only one�a2a2� is doubly occupied to define the Fermi vacuum.�The a2 means it has � spin.� This is an example of aproblem with some multireference character, since amultireference approach would try to treat both orbitalsequivalently. Instead, SR theory requires that the secondbe introduced via the double excitation operator in thecluster expansion, and since it will have a comparativelylarge weight in the final wave function compared to thatfor �a2a2�, that can lead to slower convergence. Moreimportantly, a proper wave function might be expectedto include both determinants into a MR space from thebeginning, which would then introduce excitations fromboth determinants equivalently, which the SR-CC theory
does not do. As Figs. 23 and 24 show, for the symmetricA1 vibrational frequencies the symmetric stretch and thebend CC theory still does very well, with the bettermethods showing some improvement over the lowerones. The first figure uses a DZP basis, while the seconduses a cc-pVTZ basis. The effect of the larger basis set ispronounced, though the general behavior of the SR-CCmethods is still quite good for symmetric vibrations.However, in all cases the antisymmetric vibration is nottoo well described.
This is an interesting feature of the O3 problem dis-cussed long ago �Stanton et al., 1989�. When the antisym-metric vibration is considered, two other determinants
abruptly enter into the calculation, �a2b1� and �a2b1�,which together provide another singlet configurationthat only contributes when the C2v symmetry of O3 isbroken, allowing the formal a2 and b1 orbitals now tomix. This abrupt change causes some lack of balance indetermining this antisymmetric vibrational frequencycompared to the symmetric ones, which makes it a moredemanding test for the theory. Both the effects of basis
FIG. 21. Normal distribution of errors �kcal/mol� for calcu-lated reaction enthalpies compared to experiment for a set of13 reactions containing first-row atoms in the cc-pCVQZ basisset �Bak, Jörgensen, et al., 2000�.
FIG. 22. �Color online� Normal distribution of dipole momenterrors �D� related to the CCSD�T�/aug-cc-pVQZ referencenumbers for a set of 11 polar closed-shell molecules for theCCSD�T� method in the cc-pVXZ �X=D ,T ,Q� and cc-pCVQZ basis sets �Bak, Gauss, et al., 2000�.
318 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
set and level of theory are important. The DIP-STEOMmethod �Nooijen and Bartlett, 1997c�, which we havenot yet defined but will in Sec. VIII, treats these deter-minants equivalently, and shows some improvementover other SR-CC methods. We will further considerthis example using MR methods in Sec. IX.
A great deal of chemistry pertains to modest changesfrom equilibrium geometries like the vast majority ofstructure determinations, spectroscopic measurements,and heats of reaction. Transition states where bonds be-gin to break and form occur somewhat farther fromequilibrium, and place some extra demands upon thetheory. However, of all readily applicable quantumchemical and DFT methods, only CCSD�T� can reducethe error in the activation barrier for simple reactions to�1 kcal/mol �Lynch and Truhlar, 2003�. Using a cc-
pVTZ basis, the mean error for their benchmark ex-amples is 0.91 kcal/mol, while with the augmented basisaug-cc-pVTZ the error is 0.24. All DFT methods cur-rently applicable have errors more negative than−4 kcal/mol. In a rate constant, an error of �1 kcal/molin the barrier changes the rate by a factor of �5.
B. Basis-set issue
The basis-set limitations in molecular applications arewell known. For two-electron interactions, the solutionof the correlation problem ultimately depends upon theMO product approximation � p�1� q�2�. Hence, thetwo-electron basis is much poorer than the one-electronbasis set. It has been known since the time of Hylleraas
TABLE VI. Computed and extrapolated harmonic frequencies with coupled-cluster methods for theN2 molecule.
�1929� that wave functions that are constructed with ex-plicit r12 dependence will be much more accurate. Such awave function satisfies the so-called �singlet� electroncusp condition defined by Kato �1957�,
lim� �r12
� =12��r12 = 0� . �92�
Kutzelnigg, Klopper, and Noga �Kutzelnigg, 1985;Noga et al., 1992; Noga and Kutzelnigg, 1994� have de-veloped such an explicit R12-dependent CC approachinto a practical computational method, CC-R12. Withinthe CC-R12 ansatz, the exponential operator consists ofthe conventional, standard exponential T operator and anew R operator that takes care of the correlation cusp,and more importantly the short-range correlation. Thefinal wave function ����� is expressed as
��� = eReT��� = e�R+T����
and R is defined as
R = 14ckl
ij Rijkl, �93�
Rijkl = 1
2R�kl aij
� = 12 �r�
kl aij� − rab
kl aijab� , �94�
rpqrs = �rs�r12�pq� − �rs�r12�qp� , �95�
where ,� , . . . denote virtual orbitals within a completebasis; ars
pq= p†q†sr�; cklij are a set of amplitudes defining
the r12 contribution to the double excitation operatorand r12 is an operator representing the interelectronicdistance.
An R12 theory is most effective when applied to thecoupled-cluster approach since it makes it possible tocombine the fast convergence of the CC approach to-ward the full correlation limit with more robust satura-tion toward the complete basis-set limit �Noga et al.,1992, 2001; Noga and Kutzelnigg, 1994; Noga and Va-liron, 2000, 2002�. From the computational point of view,the effort required to solve the additional set of equa-tions connected with the R12 amplitudes is not muchmore demanding than normal CC, once the new three-electron integrals that arise in the theory are approxi-mated by resolutions of the identity �ROI� approxima-tions. However, ROI imposes a restriction to large basissets. The performance of the CC-R12 approach has beensummarized by Noga and Valiron �2002�. The mean de-viation from estimated complete basis-set limitCCSD�T� correlation energies for a number of ten-electron systems is equal to 16.14 mH, while the samequantity for the CCSD�T�-R12 method is reduced to1.69 mH. A significant improvement due to the R12 an-satz is also observed for atomization energies and elec-trical properties �Franke et al., 1995�. Further improv-ments in the R12 theory arise from using it in anexponential form, exp�−�r12
2 � �Hino et al., 2002�, thatcuts off the inappropriate long-range effect of R12.
The advantage of the CC-R12 theory can be appreci-ated most fully for small molecular systems where verylarge basis sets can be used. Also, the extensions re-quired for analytical gradients and EOM-CC excitedstates have not yet been made.
TABLE VIII. Contributions to the best estimates of harmonicfrequencies �in cm−1�.
The other important issue refers to the use of thecoupled-cluster method in calculations for large mol-ecules. The construction of such computational schemesis guided by the idea of retaining the most essential sub-sets of CC amplitudes and neglecting the less importantones, which eventually leads to substantial reduction inthe number of amplitudes. Such procedures are builtupon either the singular value decomposition �SVD� ap-proach �Kinoshita et al., 2003; Hino et al., 2004�, theCholesky decomposition �Beebe and Linderberg, 1977;Koch et al., 2003�, or they try to exploit localization ar-guments �Schuetz and Werner, 2000; Flocke and Bart-lett, 2003�. The singular value decomposition applies toany matrix, while Cholesky requires a positive-definiteone. In the first case, we can replace the matrix by anexpansion in vectors weighted by their singular values,which measures their importance. Then we can make acontraction of the usual MO indexed amplitudes tij¯
ab¯ bya contracted set tX
Y as determined, in principle, by theirsingular values. �In practice, we have to obtain theweight factors from some simpler, related problem likeMBPT�2�. Now the effective dimension of the CC prob-lem is greatly reduced, again dramatically diminishingthe high scaling of the unmodified calculation. This iscalled compressed coupled cluster. Very impressive lo-calized orbital results for quite large molecules havebeen reported �Schuetz and Werner, 2000�, some evenwith explicit R12 effects �Werner and Manby, 2006�.
The coupled-cluster method CCSD has also been ap-plied to polymers for its first numerical example for aninfinite, translationally invariant system �Hirata,Grabowski, et al., 2001; Hirata, Podeszwa, et al., 2004�.Prior CCD work was reported by Förner et al. �1997�.Such correlated methods for periodic systems are diffi-cult to do, primarily because of having to converge thelattice sums, even though long-range interaction termscan be effectively grouped together and the integral cal-culation limited to the symmetrically unique integralsthat extend over several neighbor unit cells. The mainnew element in the cluster amplitudes is that in additionto the usual i , j¯a ,b¯ orbital indices, one has to alsoadd the wave vector k to the Bloch spin-orbitals. How-ever, not all wave-vector indices are varied indepen-dently, because the t-excitation amplitudes will vanishwhen such excitations do not conserve momentum. Inthis respect, the polymer problem is closer to that de-scribed in standard physics texts on many-body theorythan is the rest of this review. The CCSD equations thenbecome
�0�H�0� = KEunit cell, �96�
��iki
aka�H�0� = 0 ∀ a,i,ki, �97�
��ikijkj
akabkb�H�0� = 0 ∀ a,b,i,j,ki,kj,ka, �98�
where �ka−ki� ·a=2�m and �ka+kb−ki−kj� ·a=2�n,with m and n integers and a the fundamental vector thatdefines the unit cell. K is the number of wave-vector
sampling points. The detailed equations have beenshown by Hirata, Grabowski, et al. �2001�, and Hirata,Podeszwa, et al. �2004�. Results have been presented forpolyethylene, polyacetylene, polyyne �C�C��, as well asprototypes �LiH�� and �HF��. In particular, the decay ofthe integrals and amplitudes as a function of the numberof unit cells has been studied extensively.
If the CC paradigm of converging methods,MBPT�2� � CCSD� CCSD�T� �CCSDT�CCSDT�Q��CCSDTQ� full CI, and its EOM-CC variants for ex-cited, ionized, etc. states discussed in Sec. VIII could beeffectively applied to 1D, 2D, and 3D systems, then CCtheory would be able to offer significant insights intoseveral interesting phenomena in solid-state physics,such as band gaps, phonon spectra, optical and magneticproperties, and superconductivity.
C. Bond breaking
When insisting upon correct separation all the way tothe asymptotic dissociation limit for a molecule, therecan be difficulties. This is not a problem with CC meth-ods as such, but with the reference function used in thesingle determinant based CC. An RHF reference for aclosed-shell molecule cannot separate correctly to open-shell fragments. So for even H2→2H it is well knownthat the RHF energy will go to an average of that for aproton H+ and a hydride ion H−. For the fragmentationof an open-shell molecule, ROHF will have similarproblems to RHF. The unrestricted Hartree-Fock willusually correctly separate, but at the cost of breaking thesymmetry of the wave function to permit an electron tolocalize on each H atom. Though the wave function iswrong, the energy is correct for such a solution at thedissociation limit. The density is also correct at separa-tion. In the interest of developing CC methods that caneven transcend the deficiencies of an incorrectly separat-ing reference function, one often intentionally uses aRHF reference to test the CC approach. Obviously,CCSD, which is the full CI for H2, would give the rightanswer for two electrons with any possible single deter-minant reference, including RHF and UHF.
Figure 25 shows the behavior of the F2 potential-energy curve at various levels of CC theory �Musiał andBartlett, 2005� with a RHF and a UHF reference. F2 isan interesting case since, as shown in Fig. 26, the RHFcurve is artificially bound because of its incorrect sepa-ration, while contrary to fact, the UHF curve shows nobinding, as the two F atoms are lower in energy than theF2 molecule. At the CCSDT level for RHF and CCSDfor UHF, both curves are qualitatively correct as wouldbe expected for a single bond, showing that CC theoryhas the capacity to overcome a misbehaving referencedeterminant. Closer inspection shows difficiences in thatthe CCSDT result slips below the asymptotic limit in theRHF reference case, while CCSD is too high. Addingthe quadruple excitations CCSDTQ mostly correctsthese features for the RHF reference. Results for thewidely used CCSD�T� approximation are also shown.
321Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
Obviously, any perturbative correction to the infinite-order CCSD for a RHF reference will invariably failsince as the bond breaks, the highest occupied MO�homo�, �n, and the lowest unoccupied MO �lumo�, �n+1,energies can become degenerate, causing the term to goto minus infinity. For the UHF reference, CCSD�T� hasto behave better because the UHF function is separatingcorrectly. Similarly, the differences between CCSD andCCSDT are less pronounced at separation, since theUHF is energetically correct there, but as the RHF so-lution is the correct, lowest-energy single-determinantsolution at the equilibrium geometry, shortly beyondthat the UHF becomes lower in energy than the RHFand there is a bifurcation. This region is sometimescalled the spin recoupling region as the closed shell isbeginning to take on the characteristics of two doubletopen shells through the UHF localization, causing thespin-eigenfunction property of the RHF-CC method tobe lost. However, insofar as we are converging onto thefull CI solution for F2, the actual eigenfunction wouldhave to eventually become a spin eigenfunction even ifwe do not impose it a priori. However, we can monitor
this behavior by evaluating �S2�.
We can compute �S2� in two ways. �i� As a projectedvalue in analogy to the transition formula for the energy,
where �0�S2���= S�S+1�, with an average multiplicity
obtained from the approximate spin eigenvalue S. This
gives a quadratic equation where 2S+1=�1+4�0�S2����Purvis et al., 1988�. �ii� As an expectation value
���S2���= S�S+1�, which is in general an infinite seriesin CC theory, but it will be shown in the next sectionthat it can be written in closed form as �0��1+��exp�−T�S2 exp�T��0� �Stanton, 1994�. The formerprojected value is consistent with how the energy is de-termined in CC theory, and is a convenient index for
UHF-based CC results, as 2S+1 will typically be a num-ber like 3.0001 for a triplet state, attesting to the factthat there is little spin contamination. For a ROHF ref-
erence, however, it follows that �0�S2�0�=S�S+1�, whichguarantees that the projected value would be exactly2S+1 regardless of ���. So if we want to more defini-tively assess the residual spin contamination in the CCwave function, the second choice is preferred. Severalnumerical results for the spin multiplicity for open shellsare shown elsewhere �Purvis et al., 1988; Stanton, 1994;Bartlett, 1995�.
FIG. 25. �Color online� Potential-energy curves for the F2 mol-ecule obtained with various CC methods in a cc-pVDZ basisset.
FIG. 26. The SCF potential curves for the F2 molecule.
322 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
The more difficult example for bond breaking is of-fered by the triple bond in N2. The curve needs tochange from a normal closed-shell RHF-based solutionat Req to two 4S states for the N atoms, each with threeunpaired electrons, whose multiplicity could be as high
as a septet. Here, as shown in Fig. 27�a� �Chan et al.,2004�, the extreme change makes the RHF-based CCSDand CCSDT poorer in the large R region, with the latterfalling beneath the asymptotic limit. Once again, theUHF-based CC curves are qualitatively correct. Closerinspection of the recoupling region shows the difficultieswith the latter, however. This effect is nicely summarizedin the paper of Chan, Kallay, and Gauss, see Fig. 27�b��Chan et al., 2004�. Even though the UHF-CCSD�T�looks qualitatively right in Fig. 27�a�, as shown in Fig.27�b�, between about 2 and 4 bohr it can be in error byup to about 19 mH, compared to UHF-CCSD’s error of26. As shown in Fig. 27�b�, the maximum error is re-duced to about 10 with UHF-CCSDT, while T4 furtherreduces it to 4 mH. Pentuples �T5� reduce that to2.3 mH and hextuples �T6� to 0.8 mH.
At the levels of CC theory that are available for seri-ous molecular application, CCSD, CCSD�T�, CCSDT-3,and maybe CCSDT-3 �Qf� beyond some simple singlebond-breaking examples, we cannot expect that thatlevel of CC theory without further modification canovercome the deficiencies of an incorrectly separatingsingle-determinant reference. And even if the UHF doesseparate correctly, there will still be regions of the PESthat are less well described. Most would say that correctbond breaking requires a multireference treatmentwhere more than one determinant is used in the refer-ence function. As such, CI linear combinations can bebuilt to ensure that all elements required for correctseparation are present in the wave function. We discussmultireference CC in the last section of this review, butdespite substantial effort �Lindgren, 1979; Jeziorski andMonkhorst, 1981; Lindgren and Mukherjee, 1987; Jezi-orski and Paldus, 1989; Mukherjee and Pal, 1989; Ku-charski and Bartlett, 1991b; Bartlett, 2002� and encour-aging recent progress �Pittner, 2003; Li and Paldus, 2004�there is no generally applicable MR-CC for PES yetavailable. Instead, extensions of the more easily appliedsingle reference methods are being pursued further. Theessence of the problem, then, is how to get the criticaleffects of higher-order cluster contributions that facili-tate bond breaking into a practical CC computationalmethod with the accuracy and wide applicability thatcurrently exists for molecules near their equilibrium ge-ometries.
To illustrate the problem, consider the ansatz that thecorrect CC wave function for bond breaking shouldhave a part that accounts for the so-called nondynamiccorrelation, which roughly means the quasidegeneracyencountered in bond breaking; and a second part thataccounts for the dynamic correlation, which means tokeep electrons apart. The full CI obviously has both, sothis is an artificial separation but one found to be usefulwhen discussing the problem, as the two effects are de-scribed somewhat differently. The nondynamic part typi-cally requires a few or several highly weighted determi-nants to effect correct separation, while the dynamicpart is composed of small contributions from very manydeterminants. The CC method does the latter extremelywell, which is why such accurate, correlated results can
FIG. 27. �Color online� Potential-energy curves for the N2molecule. �a� CC results compared to full CI in a cc-pVDZbasis set �frozen core�. �b� Errors compared to CI for UHF-CCmethods as a function of R.
323Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
be obtained near a molecule’s equilibrium geometry.To illustrate this point we assume a CC ansatz
�Piecuch et al., 1993; Kinoshita et al., 2005� that will forcea solution to be composed of a nondynamic part,exp�Text��0�, and a dynamic part, exp�Tint�,
��� = exp�Tint�exp�Text��0� , �99�
�Tint,Text = 0, �100�
��� = exp�Text��0� , �101�
��� = exp�Tint���� , �102�
ECAS-CI = Eref = �0�H��� , �103�
ETCCSD = Eref + �ECCSDint , �104�
where Text will be taken from a correct nondynamic so-lution, and Tint will be subsequently determined by theusual CC equations. Li and Paldus �1994� have used suchan ansatz called externally corrected to add effects forquadruple and higher excitations into CCSD. Here weuse a different variant that is limited to just single anddouble excitations that leads us to tailored CCSD�TCCSD�. The simplest solution to bond breaking is todo the full CI in a small, active orbital space that has theessential elements for the bond breaking. For a singlebond like the HF molecule, that means the occupied �orbital and the unoccupied �*. Using two orbitals, a lin-ear combination of the two determinants gives the cor-rect, bond-breaking behavior shown in Fig. 28. Notehow much higher the energy is than in the correlatedresults. However, this correct behavior is all that is re-quired to enable the TCCSD result to show correct bondbreaking. Extracting the external amplitudes from thefull �sometimes called complete active space� CI in thetwo-orbital problem, by using the cluster decomposition,T1
ext=C1 and T2ext=C2− �T1
ext�2 /2. So by inserting the twoT1,2
ext amplitudes involving the � and �* orbitals, while allthe rest �T1,2
int � are obtained from the standard CCSDequations, in effect, the reference function in the ansatzbecomes the two-determinant one. Usually that wouldrequire the tools of MR-CC, which would also have theadvantage that the two functions would be treated com-pletely equivalently, but by virtue of fixing the externalamplitudes and determining the internal ones separately,some of the MR effect is introduced without operation-ally changing the vacuum from the single determinant.Tailored coupled-cluster TCCSD greatly improves theseparation, yet the method only uses a single determi-nant reference and has much the same ease of applica-tion as the usual single reference CC theory. Once thecoupling between Text and Tint is permitted, one movestoward the true MR-CC problem. One intermediatepoint is to add in the higher excitations into the singlereference problem that would correspond to single anddouble excitations out of a second reference determi-nant �Oliphant and Adamowicz, 1992�, which are tripleand quadruple excitations out of the first. To make this
feasible also requires some kind of active orbital restric-tion to limit these excitations. Finally, if CC is built upona generalized valence bond �GVB� reference, it wouldachieve the same bond-breaking behavior for a singlebond, but now both components in the reference func-tion would be treated equivalently �Balkova and Bart-lett, 1995�. Also, because of its fixed form, it is possibleto do GVB-based CC �Van Voorhis and Head-Gordon,2001� without all the complications of general MR-CC.Single bonds, however, are not really the problem.
Application to N2 requires breaking a triple bond, sothe active orbital space consists of the three valence oc-cupied orbitals �g, �u and the three corresponding un-occupied orbitals �u��*�, �g��*�. Figure 29 shows thecorrect behavior for this very small full CI. Here some of
the effects of the higher-order excitations C3 through C6are subsummed from cluster analysis into just the T1 andT2 external amplitudes, and once the remaining ampli-tudes are determined from the CCSD equations, there isagain correct separation. The comparison here is to avery large �14-electron, 10-orbital� MR-CI result. Thecomparative timings for the two calculations are �407 sfor the MR-CI to �1.25 s for the TCCSD. When in-creasing the basis from pVDZ to pVTZ, it is not practi-cal to do the MR-CI but the TCCSD requires only�36 s, which is essentially the same as for the ordinaryCCSD itself. So in a calculation that is much faster thanwould be the case for even CCSDT, we achieve qualita-tively correct bond breaking. The conclusion is that if wecan hide into the T1 and T2 amplitudes in CCSD theessential effects required for the bond breaking, whichare only known here from some consideration of thehigher-order CI coefficients, it would give a viable ap-proach. The down side is that if we allow the additionalcoupling between Text and Tint, we have to move to aMR-CC framework. Even the alternative of adding
FIG. 28. �Color online� Potential-energy curves for the HFmolecule in the cc-pVDZ basis set. The horizontal axis repre-sents the internuclear distance normalized by the equilibriumbond length �Re=1.733 a.u.� �Kinoshita et al., 2005�.
324 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
some additional higher excitations in an augmentedsingle reference approach is now quite limited, as thereare 52 D2h reference determinants in the six-orbital–six-electron CAS-CI for N2.
It is clear that improvements on the basic single refer-ence CC framework will have to arise from some kind ofconsideration of higher excitations, since QnH exp�T1+T2�P�0, n�2. This prohibits the equivalence shownin Eqs. �78�–�80�. Of course, this is the basic idea ofmethods like CCSD�T�, as discussed in Sec. V.E, butwhat other ways can this information be incorporated?
Another partial solution that has been suggested�Kowalski and Piecuch, 2000a� is to renormalize theSR-CC equations. The issue of renormalization might beapproached from Eq. �75�, where there is potentially adenominator. As shown in the derivation of CCSD�T�above, we used the connected expectation form for theenergy without any denominator. However, we limitedthe numerator to just fourth-order terms. It might beargued that rather than cancelling the denominator com-pletely from the equation, which leads to just connectedterms in the numerator, we should retain the denomina-tor to the same order as the numerator �Meissner andBartlett, 2001�. Once that is done, we have the generalrenormalized correction �ER=�E / �1+S� form, where Sis an overlap and �E would be the triples correction,e.g., in CCSD�T�. Expansion shows that �ER=�E−S�E+S2�E−¯. Clearly, except for the so-called EPVterms, which will arise as parts of linked diagrams,all such terms have unlinked character, just asE�2����1� ���1�� did in our introductory fourth-order PTexample in Sec. III. This violates size extensivity, theguiding principle of CC theory, which raises argumentsabout how far such methods should be pursued. How-ever, in the limit of the full CC, the expression is exact.
When quasidegeneracy is not present, �S��1, but itgrows rapidly as bonds are broken which offsets the factthat �E tends to go to minus infinity. This keeps thecurve from turning over as in the N2 and F2 examplesshown previously.
Many different kinds of approximations to this basicstructure have been considered, termed renormalizedand completely renormalized. To show just one, theirCR-CCSD�T� correction �Kowalski and Piecuch, 2000b�is given by
�ECR-CCSD�T�
=�0��T1 + T2
†�WR3�W exp�T1 + T2�C�0�
�0��1 + �T1† + T2
†��1 + WR3�exp�T1 + T2��0�, �105�
where the denominator is composed of second- andthird-order corrections �counting T1 as first order� to ac-commodate the fourth-order terms in the numerator.
Although the details might be critical in finding theoptimum way to renormalize, they all tend to have simi-lar numerical behavior. For example, a different, butmore easily evaluated one �Meissner and Bartlett, 2001�has
�ER-CCSD�T�M�
=�0��T1
† + T12/2 + T2
†�WR3WT2�0�
�0��1 + �T1† + T1
2/2 + T2†��1 + WR3�exp�T1 + T2��0�
.
�106�
Results are shown for RHF reference N2 compared tothe full CI result in Fig. 30. Due to the denominator, thecurve cannot turn over like the unmodified CCSD�T�did in Fig. 27�a�. However, closer inspection of the twosets of curves shows that in the vicinity of the equilib-rium geometry, the unmodified CC methods are superiorto their renormalized versions. This is another manifes-tation of the size inextensivity of such models, whichalso guarantees that the vibrational frequencies ob-tained from some such model will not be nearly as goodas the usual CC results. In fact, it can be argued that thesize-extensive UHF-based CC results that can separatecorrectly are actually better for many applications.
If the denominator is limited to just provide EPVterms instead of more general denominators, then onecan regain size extensivity. Some attempts at doing thishave been considered �Kowalski and Piecuch, 2000b;Nooijen, 2005�. The main objection to this approach isthat when EPV parts of diagrams are retained, insteadof the whole diagram that contains the EPV part �Bart-lett and Musiał, 2006�, the orbital invariance of CCmethods is compromised. If localized orbitals are used,
FIG. 29. Potential-energy curves for the N2 molecule in thecc-pVDZ basis set. The horizontal axis represents the internu-clear distance normalized by the equilibrium bond length �Re=2.074 a.u.� �Kinoshita et al., 2005�.
325Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
good virtually extensive results are obtained at separa-tion, but a rotation of the orbitals destroys this behavior.Also, any particular choice of orbitals would have toapply at any R, and that further complicates the issue.So at this time, such renormalized CC approaches havesignificant formal limitations. There is also no denomi-nator in iterative CC �like CCSDT-n� as opposed tononiterative CC approximations. So renormalizationcannot be achieved by denominators, per se, but there is
still room for other ways to exploit the higher QnHPequations such as by modification of the left-hand eigen-vector of the CC functional �Moszynski et al., 2005;Piecuch et al., 2006�.
Another possibility for a better description of bondbreaking is offered by the so-called extended CC �ECC�method of Arponen and Bishop �Arponen, 1983; Ar-ponen et al., 1987a, 1987b�. The main element of theECC theory relies on using a double similarity transfor-mation defined through two sets of amplitudes � and T.The ECC energy functional in the singles and doublesapproximation �ECCSD� is expressed as
E�ECCSD� = �0�e�†e−THeTe−�†
�0� , �107�
where T is a standard cluster operator defined in Eqs.�26�–�29� and �† is a deexcitation operator of the samecluster structure as T†,
�† = �1† + �2
† �108�
and
�n† = �
ij¯
ab¯
�ab¯ij¯ i†j†
¯ ba . �109�
Taking advantage of the transformation �Arponen,1983� and remembering that �†�0�=0, we may rewriteEq. �107� as
E�ECCSD� = �0��e�†�HeT�cdc�0� , �110�
where dc �doubly connected� indicates that the � opera-tor is connected with H or at least two T operators. Thedouble connectedness property ensures that the equa-tions for � and T contain only connected contributions.
The usefulness of the ECCSD theory in the descrip-tion of the bond-breaking situation has been considered
FIG. 30. �Color online� Potential-energy curves for the N2molecule in the cc-pVDZ basis set obtained with the FCI andR-CCSD�TQf� methods.
FIG. 31. �Color online� Potential-energy curves for the N2molecule in a STO-3G basis set obtained with the FCI,ECCSD, ECCSD�TQ�, CCSD, and CR-CCSD�TQ� methods.
326 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
�Van Voorhis and Head-Gordon, 2000; Gwaltney et al.,2002; Fan et al., 2005�. It can be observed from Fig. 31,taken from Fan et al. �2005�, that the ECCSD curveshows correct behavior in the region of 2Re unlike thestandard variants such as CCSD or even renormalizedversions with inclusion of the noniterative triples andquadruples CR-CCSD�TQ�. Apparently the best curveis obtained by combining the so-called renormalizednoniterative corrections with the ECCSD iterative solu-tion �Fan et al., 2005�; see the ECCSD�TQ� curve in Fig.31. It needs to be stressed, however, that the rigorousformulation of the ECC approach even in its simplestsingles and doubles approximation is a complicated andcostly method. One has to carry out simultaneous itera-tions of the two sets of the amplitude equations: � andT. The terms occurring in both equations are cumber-some and require a high-rank computational procedure.Some of the diagrams originating from the term��ij
ab��2†2HT2
4�0� /48 contributing to the T2 amplitude af-ter proper factorization scale as M10, which is prohibitivefor wider applications of the method. An inclusion ofhigher clusters, such as �3 and T3, which are necessary toobtain a method correct through fourth order of pertur-bation theory, is still more complex and only feasible inapproximate versions.
VII. THE COUPLED-CLUSTER FUNCTIONALAND THE TREATMENT OF PROPERTIES
A molecule has 3N−6 vibrational degrees of freedomfor N atoms, or, in terms of Cartesian forces that aremost frequently obtained in calculations, 3N atomic de-rivatives. Absolutely essential to the wide range of CCapplications that are made is the existence of analyticalprocedures to obtain the 3N forces, �E�R�=F�R�, tosearch a PES to determine the points where the forcesvanish, which will give a molecule’s structure and saddlepoints for transition states. Furthermore, to know thatthe molecule’s geometry is indeed a minimum energyconfiguration requires the matrix of second derivatives,��E�R�, the Hessian. This defines the harmonic forceconstants from which solution of the vibrationalSchrödinger equation yields the vibrational frequencies.If at a point where F�R�=0 all frequencies are real, thenthe geometry on the PES is a minimum, while if onlyone is imaginary, then we have a saddle point that cor-responds to a transition state on a reaction coordinate.Without some procedure to evaluate the forces in aboutthe same amount of time as the wave function and en-ergy at a point, applications of theory to PES for poly-atomic molecules would be hopeless. The fact that todaythis can be done for CC theory �Adamowicz et al., 1984;Bartlett, 1986; Scuseria et al., 1987; Salter et al., 1989� iscritical to its role as the reference method for most mo-lecular applications.
The essential new idea required can be readily de-rived in a few steps. We know that the CC energy at
each point of a PES is E�R�P=PH�R�P. If we differen-tiate it with respect to R, we obtain the forces. The key,
then, is to arrange this expression for the forces into acomputationally convenient form that can be evaluatedin about the same amount of time as can the wave func-tion itself. This is done through the following steps:
E�R�P = PH�R�P , �111�
�E�R�P = P � H�R�P , �112�
�E�R�P = P exp�− T���H�R�exp�T�P
+ P�H,�T�R�P �113�
=P�H�R�P + PH�P + Q� � T�R�
− �T�R��P + Q�HP �114�
=P�H�R�P + PHQ � T�R�P , �115�
where after inserting the resolution of the identity 1=P+Q, the last line uses the fact �i� that �T�R� has tocorrespond to an excitation from the reference determi-nant �0� or the P space, P= �0��0�, to its orthogonal
complement Q; and �ii� that QHP=0 by virtue of the CCequations being solved; the bar over the operator indi-
cates its similarity transformed form H=e−THeT, Eq.�58�. We also need to consider the derivative of the am-
plitude equations QHP=0, which gives
Q�H�R�P + Q�H,�T�R�P = 0. �116�
Again inserting the resolution, we can write
Q�H�R�P + Q�H�P + Q� � T�R�
− �T�R��P + Q�HP = 0, �117�
Q�H�R�P + Q�HQ � T�R�
− �T�R�PHP = 0. �118�
The form of the last term can be recognized as that fromfirst-order perturbation theory, since we have
�E − QHQ� � T�R� = Q�H�R�P , �119�
�T�R� = �E − QHQ�−1Q�H�R�P , �120�
�T�R� = RQ�H�R�P , �121�
where we introduced the resolvent operators for H, R.Inserting this into the energy formula and defining �
=P�HRQ leads to
�E�R� = �0��1 + ���H�R��0� . �122�
The quantity � is independent of the 3N perturbations.Notice also that it is a deexcitation operator, as �
=P�Q=�i�j�¯�a�b�¯
�ab¯ij¯ i†aj†b¯ �. It is also essential
to recognize that �H�R�=exp�−T��H exp�T� has re-moved any T dependence from the gradient, leaving
327Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
only the gradient of the Hamiltonian. Hence, combiningthese features we have a generalized Hellman-Feynmanformula for the forces. If we specify to a single degree offreedom � /�X, the formula becomes
E�R� = �0��1 + ��H�R��0� . �123�
This can be viewed as a generalization of the inter-change theorem of double perturbation theory �Dal-garno and Stewart, 1958; Bartlett, 2004�.
We can further generalize Eq. �123� using the integralHellman-Feynman theorem, which simply means inte-grate the above expression to give the CC functional E= 0
R��E /�R�dR,
E�R� = �0��1 + ��H�0� . �124�
The functional is a very important quantity in CCtheory. As shown here, it arises as a natural consequenceof simply asking for energy derivatives along with theenergy �Adamowicz et al., 1984; Bartlett, 1986; Salter etal., 1989�. From another viewpoint, it arises from a gen-eralization of CC theory, where NCC �meaning normalCC in this designation� would be the usual CC equationsand adding � is viewed as an extension �Arponen et al.,1987a�. A third way to obtain this formula is to intro-duce constraints via a Lagrangian multiplier when deriv-ing the CC gradient equations �Koch et al., 1990; Szalayet al., 1993�. The functional immediately shows that
E /�=0 gives the CC equations, QHP=0, and varia-tion with respect to T gives the � equations,
P�QHQ + PHQ − EP�Q = 0, �125�
P��,HQ + PHQ + PHQ�Q = 0. �126�
Note that Eq. �125� has E in it, as it is CI-like and linearin �. This leads to disconnected contributions to the �2equations, as shown in Fig. 32. The second form of the �equations formally eliminates E by introducing the com-mutator and facilitates its diagrammatic derivation. Fig-ure 32 presents the skeleton �undirected diagrams� for�1 and �2 in the CCSD approximation. Through linearterms in T, �=T†.
The functional also allows the immediate definition ofCC one-particle and two-particle response density ma-trices as
�pq = �0��1 + ��e−Tp†qeT�0� , �127�
�pqrs = �0��1 + ��e−Tp†q†sreT�0� . �128�
Notice that these density matrices apply equally well tomethods like CCSD�T� that do not have associated wavefunctions. This is very different from that in ordinaryquantum mechanics where the density matrix is definedas an expectation value of the wave function. It is true,however, that �pq= �0�eT†p†qeT �0� / �0�eT†
eT�0� in the limit
of all Tn, and similarly for the two matrix; but the expec-tation value is an infinite series and the �-based expres-sion is always in closed form. In second-order perturba-tion theory, the two forms are equivalent. Detailedexpressions for the particular blocks of the one-particledensity matrix can be derived from the diagrams givenin Fig. 33. For example, the particle-particle block canbe expressed as �Fig. 33�a� �ab=�m�a
mtmb + 1
2�mne�eamntmn
eb
+�mne�eamntm
e tnb. Similar expressions correspond to the re-
maining blocks of �. Using the same diagram rules, wecan readily write down the response correlation correc-tions for the CC density matrices from just the normalordered parts p†q�, p†q†sr� of the density matrix opera-tors. See Shavitt and Bartlett �2006� for the details. Mostfirst-order properties, meaning those defined by first-order perturbation theory where a one-particle operator
h is added to the one-particle part of the Hamiltonian
h+ h �like moments, field gradients, etc.�, arise from thegeneralized expectation value,
h = �0��1 + ��e−TheT�0� , �129�
FIG. 32. Skeleton diagrams for the �1 and �2 equations forCCSD.
328 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
and can be readily evaluated to quite high accuracy byCCSD and its modifications due to triples �Bartlett,1995�.
However, though any CCSD and higher approxima-tion benefits from the fact that �0�=exp�T1���� relatestwo determinants and introduces substantial orbital in-sensitivity, the response density matrices do not explic-itly contain the effects of orbital relaxation that occurfor the reference determinant problem, which by virtueof changing the orbitals propagates through the correla-tion calculation. That is, it allows the occupied and vir-tual orbitals in the reference function to change to ac-commodate a perturbation due to h. For example, ifh=�p,q�p�z�q�p†q, which would be the z component of aperturbation due to a static electric field, the orbitalsthat define the Fermi vacuum would be changed by thisperturbation. Such changes are normally introduced intothe theory by orbital perturbation theory. That is, as-suming a HF reference determinant one seeks the solu-tion of the coupled-perturbed Hartree-Fock equationsto define �=�d, subject to � � �+ � � �=0. Eachorbital equation written in terms of the unperturbed andperturbed Fock operators is
��p�0� − f�0�� = �f − �p
� �0�. �131�
Expanding out the Fock operators and isolating the par-ticular coefficient, the CPHF equations can be written interms of the familiar A and B matrices from RPA�TDHF� theory,
�A + B�d = − h, �132�
Aai,bj = ��a − �i�ijab − �aj��bi� , �133�
Bai,bj = �ab��ij� . �134�
Note that here too, we run into the problem of comput-ing every d unless we once again use the interchangetheorem �Dalgarno and Stewart, 1958� by formally tak-ing the inverse of �A+B�−1, the resolvent operator inCPHF theory, to isolate d �Handy and Schaefer, 1984�.Then putting this together with Eq. �123�, we can isolateall effects of the true operator to define a MO relaxeddensity matrix Dpq such that we evaluate a property as
h = �p,q
hpq Dpq. �135�
This density matrix obviously accounts for the appropri-
ate orbital relaxation regardless of what the actual h
might be.The most general kind of first-order property is the
analytical gradient problem, which adds a third elementthat arises from the changes in the finite AO basis forthe problem. For such a property, the first-quantized op-erator h=�h /�X and the second-quantized form willadd, in addition to the changes in the MO coefficientsthat introduce orbital relaxation, changes in the AO ba-sis functions, as they follow the atoms in the molecule.That is, HN
is
H�R0� = �p,q
�/�X�p�f�q�R0p†q�
+ 14 �
pqrs�/�X��pq��rs��R0
p†q†sr� �136�
and then
��/�X� p�R� = ��
���/�X����R��R0c�p�R0�
+ ��
����R0���/�X�c�p�R��R0�137�
=��
���/�X����R��R0c�p�R0�
+ �q�p
� q�R0���/�X�dqp�R��R0,
�138�
FIG. 33. Diagrams representing the one-particle reduced den-sity matrix for CCSD: �a� particle-particle elements, �b� hole-hole elements, �c� hole-particle elements, and �d� particle-holeelements.
329Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
In any analytical gradient calculation where we chooseto use a basis set of atomic orbital functions attached tothe atoms, we have to compute all the one- and two-electron atomic orbital integrals at each R to account forthe AO derivative term. This, however, is simply an�M4 calculation, which is much faster than the CC cal-culation itself. The CPHF problem now has to beslightly modified to accommodate the new terms thatarise from the AO derivatives, and those also affect theHF �reference� part of the analytical gradient calcula-tion. Incorporated within the most general Dpq are thetwo interchange theorems, one to avoid T and theother to avoid d. Only with both does the analyticalgradient formula assume a form that permits a compu-tationally convenient evaluation of all 3N gradients for aCC wave function in about the same amount of time asthe CC calculation itself. The evaluation of �, as a linearequation, is somewhat faster than that for T, while theadditional evaluation of the derivative integrals addsoverhead and, if they are stored rather than recomputedfor a displacement, substantial disk space requirements.The rather involved analysis that leads to these final CCderivative equations is presented in detail elsewhere�Salter et al., 1989; Shavitt and Bartlett, 2006�. Analyticalsecond derivatives were formulated some years ago�Salter and Bartlett, 1989�, with an implementation byGauss and Stanton �2002�. Stanton and Gauss have alsoimplemented analytical derivatives for excited states�Stanton and Gauss, 1995�, as discussed in Sec. VIII.
Before we finish this section, we reconsider nonitera-tive triple and quadruple excitation corrections to CCbased upon the CC functional �Kucharski and Bartlett,1998a; Crawford and Stanton, 1998� instead of the CCexpectation value. First we assume an underlying CCSDsolution and then isolate all terms that would dependupon T3 or �3,
ET�4 = �0��3�H0T3
�2�C + �3�WT2�2�C + �2�WT3
�2�C
+ �1�WT3�2�C + �2�fvoT3
�2�C�0� .
Then using the fact that the T3�2 amplitudes are defined
by �3��H0T3�2�C+ �WT2�C�0�=0, we are left with
ET�4 = �0��2�WT3
�2�C + �1�WT3�2�C + �2�fvoT3
�2�C�0� .
The �-based diagrams are basically the same as thosefor CCSD�T� except that the top vertex represents �and because of the different symmetry we cannot fur-ther simplify the first term. So finally the diagrammaticterms are
This �CCSD�T� modification has been proposed �Craw-ford and Stanton, 1998; Kucharski and Bartlett, 1998a�and shown to have prospects of being a better approxi-mation than CCSD�T�.
Starting from CCSDT and doing precisely the samething for �4 and T4
�3, and recognizing that �4��H0T4�3�C
+ �WT22 /2�C+ �WT3�C�0�=0, we have
EQ = �0��3�WT4�3�C + �2�WT4
�3�C + �3�fvoT4�3�C�0�
or in diagrammatic form as
to define �CCSDT�Q�. Unlike �CCSD�T�, which onlycontains �generalized� fourth-order corrections,�CCSDT�Q� has both fifth- and sixth-order terms. Thismakes the rate-determining step �n4N5 compared to�n3N4 for the triple variant. If the �second diagram�fifth-order term itself is separated �CCSDT�Qf�, then asshown in Sec. VI it can be factorized into an �n2N5 termfor Qf. Odd-order contributions, however, tend to beless stable than those based on even orders of perturba-tion theory. See Hirata, Fan, et al. �2004a� for a nonfac-torized fifth-order approximation and Bomble et al.�2005� for a sixth-order approximation. The latter re-places � in the above diagrams by T†.
VIII. EQUATION-OF-MOTION COUPLED-CLUSTERMETHOD FOR EXCITED, IONIZED, AND ELECTRONATTACHED STATES
A convenient approach to excited states in CC theoryis offered by the equation-of-motion �EOM-CC�method, whose basic ideas were presented in Reviews ofModern Physics �Rowe, 1968�. The general idea wasused in various quantum chemical contexts �Dunningand McKoy, 1967; Simons and Smith, 1973; McCurdyet al., 1982�.
It derives from simultaneously considering twoSchrödinger equations, one for an excited state k andone for a ground or reference state,
H�k = Ek�k, �140�
330 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
We introduce an operator �k, such that it will create theexcited state from the reference state,
�k = �k�0, �142�
and after subtracting one equation from the other, andleft multiplying by �, we obtain
�H,�k�0 = �k�k�0, �143�
�k = Ek − E0. �144�
At this point there are two decisions to make: what tochoose for �k and what to choose for �0. Answering thesecond question first, we want �0 to be a CC wave func-tion �0=exp�T��0� �Emrich, 1981�. The simplest optionfor �k, if we want to create an electronic excited state, isto make it an excitation operator,
�k = Rk = r0�k� + �a,i
ria�k�ai� + �
a�b,i�jrij
ab�k�a†ib†j�
+ ¯ �145�
as we know that has to give the exact excited-state wavefunction. Also as an excitation operator, �� ,T=0.Hence, using the CC wave function for the referencestate, the EOM Eq. �143� becomes �Sekino and Bartlett,1984; Geertsen et al., 1989�
�H,Rk�0� = �kRk�0� , �146�
�HRk�C�0� = �kRk�0� , �147�
�HRk� = Rk�k, �148�
�HR�C = R� . �149�
We use the symbol Rk for the operator �not to be con-fused with the resolvents from earlier� to indicate that it
is a right-hand eigenvector because the matrix H= �h�e−THeT�h�, where �h� is the excitation manifold, isnon-Hermitian. Limiting Rk to single and double excita-tions, �h�= �h1h2�, defines EOM-CCSD �Comeau andBartlett, 1993; Stanton and Bartlett, 1993b; Geertsen etal., 1989� �see Figs. 34 and 35�.
For the HR product diagrams, the EOM-CCSD ma-trix will have a rank of �n2N2+nN. For the n=10, N=100 example from earlier, we would have to extracteigenvalues and vectors for a matrix with a rank of amillion. Adding triples to define EOM-CCSDT, it be-comes a billion. In a typical problem, about 20 or soroots are obtained using large-scale non-Hermitian gen-eralizations of Lanczos-type methods �Davidson, 1975;Hirao and Nakatsuji, 1982�.
FIG. 34. Diagrammatic form of the EE-EOM-CCSD equa-tions.
FIG. 35. Diagrammatic form of the three-body elements of Hused in the EE-EOM-CCSD equations.
FIG. 36. The diagrammatic representation of the one-, two-,and three-body elements of H.
331Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
The diagrammatic form of the H elements is com-posed of 1, 2, 3, 4, and higher particle number operators,
H=�k=02 Ik
1 +�k=04 Ik
2 +�k=03 Ik
3 +�k=03 Ik
4 +¯. The quantity Ikn
means an n-body element, while subscript k indicatesthat there are k annihilation lines �or k second-quantized annihilation operators�. Due to the two-bodynature of the electronic interaction, we have the relationthat k�3. Formally we can consider k=4, but in thiscase the intermediate I4
2 is reduced to the two-electron
integral. All are shown in Fig. 36. The H elements thatrepresent a complete set of diagrammatic contributionsare indicated by a wiggly line. Elsewhere we present thegeneral formulas for the number of the diagrams con-
tributing to the H elements �Musiał et al., 2002a� and thenumber of diagrammatic terms occurring in the com-plete form of the Ik
n element for n=1, 2, 3, and 4.In EOM-CC we can also consider excited states as
those that correspond to an ionization into orbital a,where a is considered to be in the continuum. In thesudden approximation an electron in a continuum plane-wave orbital has no interaction with those in the bound,square-integrable orbitals. This makes the operator thationizes the electron from the mth orbital become
Rk = �m
rm�k�m� + �b,m�i
rima �k�b†im� + ¯ , �150�
where we have one-hole and two-hole one-particle op-erators in the equivalent, IP-EOM-CCSD, approxima-tion. We can equally well do an electron attachment, i.e.,bring in an electron from the continuum,
�a
ra�k�a†� + �a�b,i
riab�k�a†b†i� + ¯ , �151�
to define an EA-EOM-CCSD approximation. Triple ex-citation operators introduce three-hole, two-particle andthree-particle, two-hole operators �Musiał and Bartlett,2003; Musiał et al., 2003�, respectively. We can considerdouble ionization, electron attachment �Nooijen andBartlett, 1997c�, etc., processes as we are simply exploit-ing the Fock space structure of the theory to obtaineigenvectors and eigenvalues for any sector of Fockspace.
Since the matrix H is not Hermitian, it also has left-
hand eigenvectors, �0�Lk, with the same eigenvalue. Thatis,
�0�LkH = �0�Lk�k, �152�
LH = �L . �153�
These are chosen to be normalized such that �0�LkRl�0�=kl. Unlike the right-hand eigenvector, the left-hand
one is not necessarily connected to H. We now shouldrealize that the ground-state CC functional is simply aspecial case of the EOM-CC problem for excited states,as L0= �1+�� and R0=1. So EOM-CC simply extendsthe concept into any stationary state.
To complete the specification of the excited state k,we require the corresponding density matrix, which inthe EOM-CC framework is
�pqk = �0�Lke−Tp†qeTRk�0� �154�
just as was used for the ground state. For oscillatorstrengths, we also require the transition density matrix
�pqkl = �0�Lke−Tp†qeTRl�0� , �155�
which provides the dipole strength,
��pqkl �2 = �0�Lke−Trpqp†qeTRl�0��0�Lle
−Trpqp†qeTRk�0� ,
�156�
that gives the intensity of the transition. This is pre-ferred since the individual transition density matricesare not self-adjoint in EOM-CC.
An alternative development of the excited-state prob-lem is offered by the time-dependent linear-response CCmethod �Monkhorst, 1977�, sometimes called CCLR�Sekino and Bartlett, 1984; Koch and Jörgensen, 1990�.From this viewpoint, instead of dealing with stationarystates, which are a characteristic of EOM-CC, one startswith the time-dependent Schrödinger equation andseeks the frequency-dependent polarizability. From itspoles, one obtains the same EOM-CC excitation ener-gies. The residue at the pole provides the dipolestrength. However, there are some differences betweenthe treatment of properties like dipole strengths in theCCLR compared to the EOM treatment outlined above.A discussion with applications of EOM-CC to the dy-namic polarizability addresses these, formally and nu-merically �Rozyczko et al., 1997; Sekino and Bartlett,1999�. A related approach to EOM-CC and CCLR is theSAC-CI method �Nakatsuji, 1978� whose differences arediscussed elsewhere �Bartlett, 2005�.
When any EOM-CC state is determined from aclosed-shell reference CC calculation, whether singlet,doublet, or triplet, it too will be a spin eigenstate. WhenEOM-CC is used relative to an open-shell reference,that is not necessarily the case, but a measure of
�0�Lk exp�−T�S2 exp�T�Rk�0� will typically show that thespin contamination is minor except for pathologicalcases. For ROHF-based EOM-CC calculations of exci-tation energies starting from the open-shell referencedeterminant, the approach of Szalay and Gauss �2000�,though computationally based upon spin-orbital equa-tions, returns an exact spin target state. This procedurethereby maintains the attractive generality of the under-lying spin-orbital framework of CC theory presented inthis review.
Other properties of interest that are amenable toEOM-CC are second-order properties, which meansthat they arise from second-order perturbation theoryinstead of first-order properties discussed in the last sec-tion that come from the generalized expectation value.For second-order properties, we require the first-orderperturbed wave function and cannot make the simplifi-cations made for first derivatives. Consequentially, for
332 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
2h�, we have fromsecond-order perturbation theory that ��CC=�CC
�0� �
E� = ��CC�h���CC� + ��CC�h − E����
+ ��CC�h� − E���� , �157�
��� = Re−TheT�0� = Rh�0� , �158�
E� = �0��1 + ��h��0� + �0��1 + ���h − E�Rh��0�
+ �0��1 + ���h� − E��Rh�0�
= �0��1 + ��h��0� + �E�. �159�
Use was made that the left-hand eigenvector ��CC�= �0��1+��e−T. Note the contribution of the second-order h� perturbation is a generalized expectationvalue. For the other two terms, we use the resolvent R,introduced in Sec. VII,
R = Q�E − H�−1Q . �160�
Since we know the eigenfunctions of H consist of �0�Lkand Rk�0��, to refer the resolvent to the general state CCsolution instead of the �0� reference implicit in R, wehave to exclude the left- and right-hand reference func-
tions to define a modified resolvent R0,
R0 = R − �0��0��1 + �� = �k�0
Rk�0��0�Lk
�k. �161�
This enables representing the perturbed CC wave func-tion in a generalized sum-over-states �SOS� form�Sekino and Bartlett, 1999�,
�E� = �k
��0��1 + ���h − E�Rk�0��k−1�0�Lkh��0�
+ �0��1 + ���h� − E��Rk�0��k−1�0�Lkh�0� ,
�162�
to make the connection with ordinary perturbationtheory. In practice, there can be no meaningful evalua-tion by using the SOS form, which for the examples wehave discussed even at the EOM-CCSD level easilyhave �106 states. So, in practice, to avoid evaluation ofthe inverse resolvent, one solves the linear equation�Stanton and Bartlett, 1993a�
��h1�E − H�h1� �h1�E − H�h2�
�h2�E − H�h1� �h2�E − H�h2���C1
C2�
= ��h1��1 + ��h�0�
�h2��1 + ��h�0�� ,
which can be accomplished much more efficiently usingthe same kind of numerical methods employed for thediagonalization of the EOM-CC equations, to get solu-tions for C. The second-order property can then beconveniently obtained from the matrix product
�E� = �0��1 + ���h − E��h1h2�C�
+ �0��1 + ���h� − E���h1h2�C. �163�
When the energies E and E� remain in the expression,just as in ordinary RSPT there is the potential for sizeinextensivity, since their scaling for N units will beE�N�=NE, and without their cancellation, the finalsecond-order result will not be size extensive exceptin the limit of all excitations �Kobayashi et al., 1994;Rozyczko and Bartlett, 1997; Sekino and Bartlett, 1999�.Hence, there is a slight difference between this pertur-bation �or propagator� -type approximation to a second-order property and the second derivative with respect to� in the finite limit. The latter introduces a quadraticterm in the perturbed T’s,
E� = �0��1 + ���h − E�Rh��0� + �0��1 + ���h�
− E��Rh�0� + �0��1 + ���H − E��f��f�TT��0� ,
�164�
where �f� indicates all determinants beyond those in�h1h2�. As a derivative of a size-extensive quantity, itmust be size extensive. However, its evaluation is moredifficult. For two electrons there is no �f�, so there is noquadratic term. Yet for more than two electrons, thequadratic term effects the cancellation of the energiesE in the lead term among other things. To maintain thedistinction between the two different approximations,we refer to the perturbation form as the CI-like approxi-mation since we are using the EOM-CC eigenstates that
are CI solutions to the H matrix. As a CI approximationfor excited states, unlike a CC approximation that woulduse an exponential operator as in Fock space CC �seebelow�, it will not be completely size extensive. We cansimply remove the offending term and get excellent ex-tensivity �Sekino and Bartlett, 1999�, but that introducesan approximation. Results using the quadratic approxi-mation, the perturbation, CI-like approximation, and itslinearized, extensive form are shown elsewhere for dy-namic polarizabilites �Rozyczko and Bartlett, 1997;Sekino and Bartlett, 1999�. In general, for individualmolecules the CI-like and quadratic form give quiteclose results. For NMR coupling constants they are vir-tually indistinguishable �Perera et al., 1996�.
It is a failing of standard CC theory that there are twoways to define second- and higher-order properties thatare equivalent in the limit. We have recently shown thatthe more general unitary CC, exp����0�, where �=T−T†
�Bartlett et al., 1989�, has to make them equivalent�Taube and Bartlett, 2006�. However, the nontermina-tion of the UCC ansatz has to still be addressed.
A. Numerical results
A few numerical examples of how well the EOM-CCmethods perform for vertical excitation energies com-pared to full CI are shown in Tables IX–XII. For satu-rated atoms and molecules like Ne, HF, and H2O, all
333Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
excited states are Rydberg in character, meaning thatthey can be approximated by a one-electron atom en-ergy formula −Zeff / �n−�2 in terms of an effectiveatomic number with n the principal quantum numberand the quantum defect. It also means that the elec-tron is comparatively far from the molecule as measuredby �r2�, for example. When this is computed from theexcited state’s density matrix for a Rydberg state, it willbe much larger than that in the ground state. On theother hand, for a valence excited state they will be com-parable in size. To have lower-lying valence excitedstates typically requires some unsaturation in the mol-ecule or other quasidegenerate behavior. Hence, Ryd-berg excited states are quite sensitive to the diffusenessin the Gaussian basis to describe them, but somewhatless sensitive to the description of electron correlation,while the valence states tend to reverse this behavior. Toshow one example of the former, results for EOM-CCSD and EOM-CCSDT for Ne are shown in Table IXalong with Fock space �FS� and STEOM results to bediscussed later. At the CCSD level, the error in a cc-pVDZ basis is a little more than �0.2 eV. The EOM-CCSDT result reduces this to �0.02 eV. Comparisons toexperiment would not be good in this basis, as it lacks
suitable diffuse functions, but when compared to full CIthere is no ambiguity. If we make the same comparisonsfor the HF molecule’s lowest 14 excited states and H2O’slowest 9, we obtain average errors of 0.123 for EOM-CCSD and 0.035 for EOM-CCSDT, respectively.
For the more correlation-demanding excited states asin CH+, shown in Table X, we see very large errors atthe CCSD level for the third 1�+ and 1� states. Thereason for this is that these states have a large compo-nent of double excitation character, which can be judgedby the large weight of double excitations in the excited-state wave function, perhaps most easily measured bythe magnitude of the amplitudes from R1 and R2 �Stan-ton and Bartlett, 1993b�. For these three states thesingles contribution is, respectively, 0%, 0.22%, and1.04%. Once we add the triple excitations fully into thecalculation via EOM-CCSDT, the errors are greatly re-duced to a few hundreths of an eV. The lowest-lyingexcited 1� state is mostly a double excitation, as its %singles is 0.23, and as a valence state, its error at theCCSD level, �0.5 eV, is reduced to �0.07 with triples.Similar behavior is shown by the second and third 1�states.
Table XI shows how higher-order cluster amplitudescontinue to modify the excitation energy for the meth-ylene radical. In this case, the reference state is not theground state, but the excited 1A1. To get the groundstate from EOM-CC, we deexcite to 3B1. For all states,singlets or triplets, most of the error that remains at theCCSD level is removed by triple excitations. For thelowest 1A1, 1B2, and 3B2 states, there is a significant con-tribution of double excitations that are again handled bythe inclusion of triples.
For the C2 molecule �Table XII�, the inclusion oftriples significantly improves the CCSD results, reducingthe deviations from the FCI values even by several eV�cf. the 3�g state�. But the interesting observation is thatEOM-CCSDT results in some cases are off by �0.1 eVand only inclusion of quadruples cures the problem �see,e.g., the 1�g state�. For molecules like C2, which hasquasidegeneracy even at this equilibrium geometry, thecluster convergence of the EOM results is much slower
TABLE IX. Vertical excitation energies �in eV� of Ne with a cc-pVDZ basis set augmented byadditional diffuse functions with exponents s�0.04�, p�0.03�. The active orbitals �lowest-lying unoccu-pied, highest occupied� are �4,4� for the FS and STEOM calculations.
than the previous examples, and to achieve good agree-ment with the reference values higher EOM modelsneed to be considered.
Recognizing that we have different numbers of resultsat the various levels, and without further separation intoRydberg or valence, for all excited states we obtain av-erage deviations from FCI shown in Table XIII. Therewe also show results from other approximations likeCC2 and CC3 �Christiansen et al., 1995a, 1995b�, whichare defined to be consistent through second- and third-order perturbation theory based upon the CCLR analy-sis.
In Tables XIV and XV, we show the same kind of fullCI comparison for the EA-EOM-CC for CH+ and theIP-EOM-CC for BH and C2, where the accuracy is simi-lar to that for excitation energies.
As we only have full CI results for small molecules insmall basis sets, the more meaningful step toward ex-periment is offered by basis-set extrapolated results. Formolecules, of course, unless one obtains at least the op-timum geometry in each of the bound excited states toobtain an adiabatic excitation energy, which is what thespectroscopists should see, or better, even simulate thespectrum �Stanton and Gauss, 1999�, theory compari-sons are at the mercy of comparing with what the spec-troscopists say is the correct vertical excitation energy.Also, the theorist has to be able to trust the basis-setextrapolation even though there are variants that willgive different answers. With those caveats, such com-parisons can be made and are illustrated for vertical ex-citation energies, vertical ionization potentials, and ver-tical electron affinities in Tables XVI–XIX. To offer onenumber to characterize the quality of the results, themean absolute errors are 0.241 for CCSD and 0.089 forCCSDT for excitation energies, which are close to thosededuced from the full CI comparisons. C2, however,which is a very difficult molecule to treat due to itsquasidegeneracies, does not compare as well as theother, very well studied systems. This might suggest aninaccurate vertical excitation energy for comparison.When the same C2 molecule is studied in a small basis
TABLE XI. Vertical excitation energies �in eV� of CH2 �R=1.102 Å, �HCH=104.7°� with 6-31G*
basis set. The lowest and highest orbitals were frozen.
TABLE XII. Vertical excitation energies �in eV� of C2 �R=2.348 a.u.� with a cc-pVDZ basis set augmented by additionaldiffuse functions with exponents s�0.0469�, p�0.04041�. The 1sorbitals were frozen.
Statesymmetry
EOM
FCICCSD CCSDT CCSDTQ
1�u 1.475a 1.419a 1.386a 1.385b
1�g 4.347a 2.700a 2.317a 2.293b
1�u+ 5.799a 5.715a 5.615a 5.602b
1�g 6.202a 4.582a 4.487a 4.494b
a 3�u 0.280c 0.317c 0.305c
c 3�u+ 0.777c 1.143c 1.214c
b 3�g− 3.566c 1.889c 1.385c
d 3�g 4.290c 2.723c 2.589c
g 3�g 10.712c 6.847c 6.670c
aHirata, 2004.bChristiansen et al., 1996.cLarsen et al., 2001.
TABLE XIII. Average deviations �in eV� from FCIa excitationenergies for several methods. Results averaged over the seriesof small systems: Ne, HF, CH2, H2O, N2, and C2.
CC2 EOM-CCSD CC3 EOM-CCSDT
Singlet states0.50 0.25 0.07 0.05
Triplet states0.42 0.13 0.03 0.02
aKoch et al., 1995; Christiansen et al., 1996; Hald et al., 2001;Larsen et al., 2001.
335Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
set where the full CI comparison is possible �Hirata etal., 2000�, 1�u and 1�u
+ at the EOM-CCSDT level differby 0.03 and 0.11 eV, respectively, suggesting that at leastthe former vertical excitation energy is likely to be mis-assigned. At the EOM-CCSDTQ level, the differencefrom the full CI is 0.001 eV for the first state and0.013 eV for the second. To make basis-set limit EOM-CCSDTQ calculations is beyond the state of the art atpresent.
As shown in Table XVII, the valence ionization po-tentials show less dependence on triple excitations re-gardless of whether the basis sets used in the extrapola-tion are the normal ones or those augmented with extradiffuse functions. The results appear to be quite close tothe experimental ones. In Table XVIII, we have poly-atomic molecules where the chances of missing the ori-gin of the band in the experimental photoelectron spec-trum are greater, potentially causing more apparentdifferences in the results. However, for the case of eth-ylene, where several different experimental results�Bieri and Asbrink, 1980; Holland et al., 1997; Davidsonand Jarzecki, 1998� have reached entirely different con-clusions as to what the values are, we chose to turn theproblem around and use theory to predict accurate ver-tical ionization potentials �Yau et al., 2002; Musiał andBartlett, 2004a�. In Table XIX, we go to complete basis-set extrapolations for the difficult electron affinities ofC2 and the ozone molecule.
Matrices with rank �109 for small molecules forEOM-CCSDT and �1012 for EOM-CCSDTQ clearly re-quire simplifications before they can be widely applied.One can use EOM-CCSDT-3 as for the ground state, butexcept for a savings in that calculation there is little togain for the excited state, as that would still require di-
agonalization using all triple excitations in the problem.Noniterative approximations should still be possible andwere proposed �Watts and Bartlett, 1995; Watts et al.,1996� to try to get workable methods. So far, however,none seem to work nearly as well as does CCSD�T� forground states. The basic reason is that the description ofthe excited state has different demands from that for theground state, and the demands for both the ground andexcited states need to be satisfied to get accurate excita-tion energies. Using the same kind of analysis as shown
in Sec. VI, an EOM-CCSD�T� model can be extractedfrom EOM-CCSDT-3 for excited states �Watts and Bart-lett, 1996�. It tends to lower the excitation energy fromEOM-CCSD and appears to improve the description ofthe excitation energy of valence excited states, but closerinspection shows that it also inappropriately lowers theexcitation energy of Rydberg states. Hence, there is nogood solution along these lines yet.
A better alternative is simply to limit higher excita-tions to those for a set of active orbitals, that is, to in-clude triples iteratively, but only those that correspondto excitations involving the highest occupied orbitalsinto the lowest unoccupied ones. Then the rate-determining step in the EOM-CC calculation is almostthat for CCSD. We call that EOM-CCSDt �Piecuch etal., 1999� to indicate that the triple excitations are re-stricted to just those that can be made for a small num-ber of active orbitals. Extensions to EOM-CCSDtq canobviously be done. Applications have been reported forsuch approximations �Kowalski and Piecuch, 2001;Piecuch et al., 2002� with excellent results.
Another element that should be appreciated aboutEOM-CC methods is their use as a multireference targetstate method. That is, by adding or removing electrons,the method acquires the capability of treating multipledeterminants in the target state. Consider two quaside-generate MO’s a and b. The target wave function has
four Sz=0 determinants—aa, bb, ab, and ab—whichshould be treated equivalently. This is the essence of aMR-CC problem. Yet standard single-determinant refer-ence CC would use aa as the reference, and the otherdeterminants’ weight in the wave function would have tobe introduced from the Q space into the CC equations.This means their values would have to be attainedthrough the iterations of the CC equations, and if some
TABLE XIV. Vertical electron affinities �in eV� of CH+ �R=1.120 � with 6-31G* basis set. The lowest and highest or-bital were frozen.
TABLE XV. Vertical ionization potentials �in eV� of C2 �R=1.262 � with 6-31G basis set and BHwith 6-31G** basis set. The two lowest and two highest orbitals were frozen.
are as large as that for the aa determinant �after ac-counting for the intermediate normalization�, this wouldresult in a badly converging calculation. Furthermore,there would still be a bias toward the aa reference sincethe excitations, say singles and doubles, are being takenfrom it, but not the same single and double excitationsfrom the other three determinants. But because of thematrix diagonalization in EOM-CC, when these four de-terminants are in the manifold of EOM excitations, theircoefficients have the flexibility to immediately becomewhatever they require. To achieve this in EOM, it mightbe useful to define the reference function to be a deter-minant with two fewer electrons, and then use the EOM
double �DEA� operator RDEA to add two electrons tothe problem. In that way, these four determinants would
all be treated equivalently in H, being able to have theirweights determined by diagonalization to whatever isappropriate. Similarly, we could start with a closed-shellreference determinant that has two more electrons than
the target, and use the double �DIP� operator to elimi-nate two of them, which, once again, gives an equivalenttreatment of all four determinants in the target state.This is the essence of a multireference problem as al-ready discussed for ozone. This result for ozone wasshown earlier. Because of CCSD’s insensitivity to orbitalchoice, using orbitals for the N−2 determinant or theN+2 determinant reference is relatively reasonable forthe description of the target state. However, as theEOM-CC equations are CI-like for the target state, asopposed to the CC ground state, we cannot always ex-pect the same degree of orbital invariance as we havewith help of the exponential exp�T1� operator; see To-bita et al. �2003� for examples where orbital choice is stillimportant.
Finally, another useful variant on the EOM-CC themeis that one can use high-spin reference determinants forCC, which may be UHF, ROHF, or QRHF, and are typi-cally as good for such states as are the RHF ground-state references �Watts et al., 1992; Comeau and Bartlett,1993�, and then deexcite from them to get a descriptionof lower-spin states. This is the spin-flip approach ofKrylov �Levchenko and Krylov, 2004�. To illustrate,starting from a ROHF triplet whose reference determi-
nant is �AB�, a deexcitation operator for RSF
=rB
B�cB�† cB+rA
A�cA�† cA will give the linear combination
rBB� �AB��+rA
A��A�B�, which depending upon the com-parative signs of the amplitudes will provide the singleteigenstates. In fact, the EOM-CC approach allows manyroutes to many different states by excitation, adding andremoving electrons, and adding the spin-flipped compo-nents.
For an illustration of the EOM-CCSD approach for asecond-order property, consider the scalar spin-spin cou-pling component of NMR. Its origin arises from interac-tions between nuclei that have magnetic moments andelectrons. In the Ramsey nonrelativistic theory �Pereraand Bartlett, 2005�, there are four contributions: the dia-magnetic spin-orbit �DSO� term
TABLE XVI. Extrapolated vertical excitation energies �ineV�.
MoleculeNominal
state
CBSa
Expt.EOM-CCSD EOM-CCSDT
N21�g 9.546b 9.301b 9.31b
1�u− 10.167b 9.844b 9.92b
1�u 10.604b 10.251b 10.27b
CO 1� 8.697b 8.540b 8.51b
1�− 10.243b 10.044b 9.88b
1� 10.379b 10.178b 10.23b
C21�u 1.287 1.268c 1.041c
1�u+ 5.523 5.501c 5.361c
aFor N2 and CO: cc-pV�Z; for C2: aug-cc-pV�Z.bMusiał and Kucharski, 2004.cMusiał et al. 2004.
TABLE XVII. Extrapolated vertical ionization potentialsa �in eV�.
Here �0 and �B are the permeability of the vaccum andthe Bohr magneton; �N and �N� are the nuclear magne-togyric ratios of the nuclei N and N�, respectively; andIN and IN� are the respective nuclear magnetic moments,k labels the electrons, and rkN and Sk are the positionvector and the total spin operator of the electron k. Pre-dictive theory requires that all terms be evaluated. TheDSO term is obtained from the generalized expectationvalue of h,� in Eq. �159�. The others require the treat-ment of Eqs. �162� and �163�.
In Table XX, we show the NMR spin-spin couplingconstants using the CI-like approximation for severalsmall molecules. We observe—comparing absolute er-rors for the coupled-perturbed HF �CPHF� and EOM-CCSD columns—that electron correlation effects are es-sential for the adequate description of the couplingconstants. The CPHF average error of 28.1 Hz for thewide variety of coupling constants shown is significantlyreduced to the value of 3.5 Hz. Before this work �Pereraand Bartlett, 2000�, such coupling constants could not beadequately described by electronic structure theory.EOM-CC now enables essentially predictive results tobe obtained for situations that are not amendable to ex-perimental observation. These include filling in the gapsin the experimental observation of carbocations like the2-norbornyl cation that was instrumental in the long-term classical versus nonclassical carbocation contro-versy between Georg Olah and H. C. Brown �Perera andBartlett, 1996�; or in providing a universal curve thatrelates all secondary N-H-N coupling constants across Hbonds to their distance apart, whether they belong tocations or neutrals �Del Bene et al., 2000�. The lattermight have a role in identifying where protons are inbiochemical structures, since they cannot be seen inx-ray crystallography.
The vector term in NMR, the chemical shift, has alsobeen accurately described by CC theory �Gauss andStanton, 1995�. Unlike the scalar term it is necessary touse gauge including atomic orbitals �GIAO�, which addssome complexity at the basis-set level.
IX. MULTIREFERENCE COUPLED-CLUSTER METHOD
The multireference �MR� formulation of the coupled-cluster theory is an important extension of the standard�single reference �SR� approach. Basically, one employs
TABLE XVIII. Extrapolated vertical ionization potentialsa �ineV�.
the MR formalism for systems that are degenerate orquasidegenerate. In the single-determinant approach,the reference function is very often the Hartree-Fockdeterminant. The MR formulation assumes that insteadof the single determinant we have a number of themforming a subspace denoted as M0. Thus the whole con-figurational space M is divided into two subspaces: themodel space M0 defined through the projection operatorP and the orthogonal one M� defined through the ac-tion of the operator Q. We may express the P operatoras
P = �m
m0
��m���m� ,
where �m is a model determinant and m0 is the size ofthe model space. The projectors P and Q are related toeach other through
P + Q = 1.
We have to introduce also the notion of the model func-tion �k
0 defined through the action of the operator P onthe exact wave function �k,
�k0 = P�k.
Thus, �k0 is a component of the function �k residing
within the model space. This can be expressed as a linearcombination of the model determinants,
�k0 = �
m
m0
Cmk�m.
A principal quantity in the MR theories is the waveoperator �, which is used to construct the exact wavefunction �k from the model function �k
0,
�k = ��k0 = ��k. �165�
Usually one also introduces the assumption of interme-diate normalization, which can be expressed in operatorlanguage as
P� = P . �166�
The advantage of the multireference formalism relieson the fact that finding the exact energy of the systemdoes not require solving the full Schrödinger equation,
H�k = Ek�k, �167�
but instead solving the eigenvalue equation involving aneffective Hamiltonian Heff,
Heff�k0 = Ek�k
0 . �168�
The Heff operator is defined as
Heff = PH�P . �169�
Equation �168� is obtained by applying the operator P toboth sides of the Schrödinger equation, Eq. �167�, andby exploiting the relations in Eqs. �165� and �166�.
The last element of the multireference formalism isthe Bloch equation �Bloch and Horowitz, 1958�, ob-tained by the action of the operator � on theSchrödinger equation,
TABLE XX. Comparison of CPHF and EOM-CCSD indirect nuclear spin-spin coupling constantswith experiment �in Hz�.
and by subtracting from the last equation Eq. �167� toobtain the Bloch equation,
H�P = �PH�P �171�
or using the Heff operator,
H�P = �PHeffP . �172�
The Bloch equation, as given by Eq. �172�, is generalenough to be applied to various forms of the multirefer-ence formalism. The text of Lindgren and Morrison�1986� discusses such approaches in depth. Their focus isthe quasidegeneracy that occurs in open-shell atomsrather than the quasidegeneracy that occurs in bondbreaking. The latter introduces what most call nondy-namic correlation, also sometimes called left-right corre-lation. In our terminology �Bartlett and Stanton, 1994�,static correlation means correlation encountered in thetreatment of open-shell atoms and molecules that in-volves proper spin combinations of determinants andtheir interactions. Hence, static is not a synonym fornondynamic. However, both involve quasidegeneracy,although of a different kind. The multireference prob-lem is meant to superimpose the essential effects ofstatic or nondynamic correlation on top of the extremelyefficient characterization of the dynamic correlation in-troduced by CC theory.
In order to define a model space, we need to intro-duce a formal classification of the one-particle states aspresented in Fig. 37. The active space includes Nah activehole levels and Nap active particle levels. Since the num-ber of active hole levels is equal to the number of elec-trons assigned to the active space Nae, the completeNae-electron model space is constructed by consideringdeterminants corresponding to all possible distributionsof Nae electrons among Nah+Nap active levels. If some ofdeterminants are excluded from the model space, thenwe have an incomplete model space.
Within the coupled-cluster theory, there are two prin-cipal definitions of the wave operator �, which leads totwo different MRCC methods: the Hilbert-space MRCCapproach and the Fock-space MRCC approach.
A. Hilbert-space formulation of the MRCC approach
In the first approach, the wave operator is a sum ofoperators defined independently for each model func-tion �Jeziorski and Monkhorst, 1981; Lindgren andMukherjee, 1987; Mukherjee and Pal, 1989; Balkova etal., 1991a, 1991b; Kucharski and Bartlett, 1991b, 1992;Balkova, Kucharski, et al., 1991; Kucharski et al., 1992;Paldus et al., 1993; Mašik and Huba~, 1999; Pittner et al.,1999; Huba~ et al., 2000; Pittner, 2003; Li and Paldus,2004�. This can be expressed as
� = ��
�� = ��
eS�P�, �173�
where P� is a projector onto the Nae-valence-electronmodel determinant and the sum over � runs over allmodel determinants ��. So the model space is an ex-ample of an Nae-valence-electron Hilbert space and thisapproach is called a Hilbert-space MRCC �HS-MRCC�or state-universal MRCC �SU-MRCC� method.
The operator S� is a cluster operator,
S� = S1� + S2
� + ¯ �174�
and
Sk� =
1
�n!�2�ij¯ab¯� sij¯ab¯a†b†
¯ ji , �175�
where the prime sign indicates that terms that generateexcitations within the model space are excluded fromthe summation. The summation over i , j , . . . runs over alllevels occupied in the �� determinants while a ,b , . . .represents levels unoccupied in ��. Each of the �� de-terminants becomes, in turn, a Fermi vacuum that allowsdefining the appropriate particle �a ,b , . . . � and hole�i , j , . . . � levels. Because of the complete active space, thedefinition of the wave operator given in Eqs. �173�–�175�is very similar to the definition of the Tn cluster operatorin the single reference coupled-cluster theory. The maindifference is that all terms in the summation in Eq. �175�that lead to the excitations within a model space areexcluded. The equations for the amplitudes sij¯
ab¯ are ob-tained from the Bloch equation, Eq. �172�, using thedefinition of the wave operator, Eqs. �173�–�175�, andprojecting against determinants �ij¯
ab¯ outside of themodel space,
��ij¯ab¯����HeS����� = �
!
��ij¯ab¯����eS!��!���!�Heff���� .
�176�
The effective operator’s elements are immediately ob-tained from Eq. �175�, upon projection against the refer-ence determinant �!,
��!�HeS����� = ��!�Heff���� = H!�eff . �177�
As a matrix, the effective Hamiltonian is trying to de-scribe m0 states simultaneously. The set of amplitudeequations given by Eq. �176� is solved for each referencedeterminant ��. Note that due to the renormalizationterm, i.e., that occurring on the right-hand side of Eq.
FIG. 37. Classification of the one-particle states.
340 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
�176�, the equations for each set of amplitudes arecoupled and should be iterated simultaneously. The en-ergy eigenvalues are obtained by diagonalization of theeffective Hamiltonian matrix defined within the modelspace.
The problem of completness versus incompleteness ofthe model space is crucial in formulations of the MRCCtheory. The completeness of the model space guaranteesthat the amplitude equations as well as the effectiveHamiltonian elements include only connected terms,which ensures the size extensivity of the method. On theother hand, construction of the complete model spacerequires consideration of a large number of model de-terminants, particularly when large active spaces areused. In addition, large model spaces are plagued by theintruder state problem. The latter arises when determi-nants from the model space are close energetically tothose belonging to the orthogonal space, and this causessevere divergences in the iteration process. Thus thepreferable choice would be a development of an ap-proach that would allow the use of a general modelspace �GMS�, i.e., either complete or incomplete de-pending upon the problem.
Such a method was advocated by Mukherjee, whichrequired relaxing the intermediate normalization in CCtheory �Mukherjee and Pal, 1989�. An alternative wasproposed by Li and Paldus �2004�, who introduced ageneral model space �GMS� SU-MRCC scheme basedon connectivity �C� conditions. The C conditions recog-
nize that in CI language ��= �1+ C1+ C2+ ¯ �P�. How-ever, the intermediate normalization �=�P requires theexclusion of internal excitations, i.e., those within the
model space, from each C�. However, the exclusion of
several excitations in Cn� is not equivalent to their exclu-
sion from Sn�, because of the usual relationship that C2
�
= 12 �S1
��2+S2�, etc. If the model space is incomplete, some
of the lower excitations in the disconnected products cangenerate Q space determinants. The C conditions prop-erly handle such terms. Hence, they force the cancella-tion of disconnected terms within both the amplitudesand effective Hamiltonian, which ensures the size-extensive property �see Refs. 20, 21, 27, 28 from Li andPaldus �2004� for details.
Li and Paldus �2004� tested the performance of the SUCCSD approach for several choices of model space bycomparing the SU CCSD results with reference values�FCI or CISDTQ�. In all cases studied �CH+, HF, F2,H2O, and others�, the SU-CCSD values remain close tothe reference. Figure 38 presents potential-energycurves for the ground state and three excited states ofCH+ previously considered, obtained with 5R �five ref-erences in the model space� -SU CCSD. Comparisonswith FCI PES �taken from Li and Paldus �2004� showexcellent agreement between FCI and SU CCSD results.
Returning to the problem of ozone’s vibrational fre-quencies, we can compare the MR-CC results with thosefrom SR-CC in Fig. 39 in the same DZP basis. The DIP-STEOM method has already been discussed as it falls
between a true multireference approach and single ref-erence theory, which it operationally is. Yet it is MR inthat it treats all four of the revelant determinants �threeconfigurations� equivalently. The MR�3�-CCSD resultsof Li and Paldus correspond to the true SU-CCSD re-
FIG. 38. �Color online� CH+ potential-energy curves obtainedwith the FCI and 5R-SU CCSD methods.
FIG. 39. �Color online� Vibrational frequencies of ozone �mul-tireference methods, DZP basis set�. See Fig. 23 for the experi-mental values.
341Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
sults, and they, too, are quite good and generally betterthan the SR-CC approximations. The poorest results aregiven by the MR�2�-CISD. The CASSCF includes non-dynamic correlation, but no dynamic correlation isadded. The final result is from a method termed MR-AQCC, which means multireference averaged quadraticCC �Szalay and Bartlett, 1993; Fusti-Molnar and Szalay,1996; Szalay, 1997�. This method follows the older mul-tireference linearized CC method �Laidig et al., 1987�and is close to the MR-ACPF �multireference averagedcoupled-pair� �Gdanitz and Ahlrichs, 1988� method.Both are built upon the MR-CI structure and programs,as they benefit from the long-term development of CItechnology, except in a few critical ways. The MR-AQCC method corrects for CI’s lack of extensivity andintroduces a functional, which makes it easy to doanalytical gradients. The method is available in theCOLUMBUS program system �Lischa et al., 2001�. It hasbeen used in an extensive number of applications forcomplicated systems with outstanding success �Szalay,1997�. The two-reference variant shown here does ratherwell for ozone frequencies. As a final word on ozonefrequencies, the tailored TCCSD discussed in Sec. VI.Cfor bond breaking has also been recently applied to thisproblem. Unlike for bond breaking, it was found �Hinoet al., 2006� that CAS-CI alone was not adequate fortailoring the SR-CC. Instead, the orbitals had to be ro-tated optimally too, to go from CAS-CI to CAS-SCF.Once this was done with a rather large CAS space con-sisting of 12 electrons into 9 orbitals, the final results in acc-pVTZ basis were �1 �1137 cm−1 versus 1135 cm−1
expt.�, �2 �1137 cm−1 versus 716 cm−1 expt.�, and �3�1098 cm−1 versus 1089 cm−1 expt.�. All are less than 1%in error. It is unfortunate that no complete basis-set limitresults are yet available for O3 to bring the basis-setissue under control and to focus on the correlation prob-lem unambiguously, but this attests to the limitations ofsuch basis-set extrapolation techniques as they are en-countered for even three atoms. This problem would bea worthy test for quantum Monte Carlo �QMC� calcula-tions.
B. Fock-space formulation of the MRCC approach
The second MR-CC approach assumes the wave op-erator to be of the same form for all reference determi-nants, i.e., it is defined universally with respect to thewhole model space. This version of the MRCC theory isoften called a valence universal MRCC �VU-MRCC�approach �Lindgren, 1979; Haque and Mukherjee, 1984;Sinha et al., 1986; Pal et al., 1987, 1988; Jeziorski andPaldus, 1989; Rittby and Bartlett, 1991; Landau et al.,1999, 2000�.
The universal definition of the wave operator bearsanother important consequence: In order to have a well-determined set of amplitude equations, one has to in-clude into the model space also determinants corre-sponding to variable numbers of active electrons, i.e., fora complete model space we have to include determi-nants with a number of valence electrons from 0 to
Nah+Nap distributed among all active levels. This type ofHilbert space includes, in addition to the Nah-electrondeterminants, also those that correspond to the ionized�up to Nah-tuple ionization� and electron attached �up toNap attached electrons� states. This is a Fock-spaceMRCC �FS-MRCC� approach.
In fact, it is more convenient to consider the modeldeterminants as corresponding to a certain number ofquasiparticles instead of a number of electrons �par-ticles�. That means that having Nah electrons in the sys-tem with all hole levels occupied, i.e., �0, we have zeroquasiparticles or a zero-valence situation. Similarly, re-moving one electron from any occupied valence level inthe �0 or adding one electron to any unoccupied valencelevel, we have one quasiparticle in the system or one-valence problem; moving one electron from the occu-pied valence level to the unoccupied valence level, wehave a two-valence problem �two quasiparticles: onehole and one particle�.
In the FS-MRCC approach, the wave operator isgiven as
� = eS�P , �178�
S = S1 + S2 + ¯ + Sn, �179�
and the Sn operator in its most general form is expressedas
Sn =1
�n!�2 �a�b�¯ı�j�¯
�sı�j�¯
a�b�¯a�†b� †
¯ j�ı�, �180�
where the summation over a� ,b� , . . . �ı� , j� , . . . � runs over in-active particles �inactive holes� and all valence levels, seeFig. 37, and the prime indicates that excitations withinmodel determinants are excluded from the summation.It is important to note that the summation ranges for
creation operators �a� ,b� , . . . � and for annihilation ones
�ı� , j� , . . . � overlap within the valence levels, and because
of this, contractions among S operators are possible. Toprevent that, a normal-ordered ansatz was introduced byLindgren �1978�—indicated by �—of the creation-
annihilation operators in the expansion of eS.We may separate the P operator into zero-valence,
one-valence, two-valence, etc., sectors depending on thenumber of valence quasiparticles present in the modeldeterminant,
P = P�0� + P�1� + P�2� + ¯ . �181�
The n-valence sector can be further separated into thek-valence particle, l-valence hole �k+ l=n� sectors,
P�0� = P�0,0�, �182�
P�1� = P�1,0� + P�0,1�, �183�
342 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
As an example, we provide an explicit form of the pro-jectors P�1,0�, P�0,1�, P�1,1�,
P�1,0� = �
������ , �185�
P�0,1� = ��
�������� , �186�
P�1,1� = ��
�������
� . �187�
The same sector structure distinguishes the Sn operator.
Let us consider, for example, S2,
S2 = 14 �
a�b� ı�j�
�sı�j�a�b�
a�†b� †j�ı�
= 14 �
abijsij
aba†b†ji + 12 �
abjsj
aba†b†j + 12 ��bij
sij�b�†b†ji
+ ��bj
�sj�b�†b†j + ¯
= S2�0,0� + S2
�1,0� + S2�0,1� + S2
�1,1� + ¯ ,
where the terms for the sectors �2,0�, �0,2�, �2,1�, �1,2�,and �2,2� are skipped. To establish the summation rangesof the a ,b , i , j , ,�, indices, Fig. 37 should be consulted.In the general case, we have the following sector struc-
ture of the S operator:
S�k,l� = �i=0
k
�j=0
l
S�i,j� �188�
and by definition the S�i,j� operator includes i annihila-tion valence particle lines and j annihilation valence holelines. It follows from that that there is a hierarchicalstructure of the S operator �Haque and Mukherjee,1984; Chaudhuri et al., 1989�, which can be written as
S�m,n�P�i,j� = 0 if m � i or n � j . �189�
For example,
S�1,0�P�0,0� = 0.
The maximum valence rank of the sector occurringfor the Nah valence holes and Nap particles is Nah+Nap,i.e., for the complete model space we would need toconsider the �Nap ,Nah� sector. However, in the majority
of applications the valence rank of the S�k,l� operator,i.e., k+ l, is much lower than the size of the model spaceNa, which means that we deal with an incomplete modelspace.
In Fig. 40, we give algebraic and diagrammatic defini-tions of the S1 and S2 operators in the sectors �0,0�, �1,0�,�0,1�, and �1,1�. Note that the S operators for the �0,0�sector are identical with the T operator introduced in
the single reference theory. The need for considering the
lower sectors arises as follows. The S operator for the�1,1� sector is defined as
S�1,1� = S�0,0� + S�1,0� + S�0,1� + S�1,1�.
The amplitude equations within the �1,1� sector are un-derdetermined, i.e., the number of equations is smallerthan the number of the unknown amplitudes,
���a �S���
� = ���a ��
b�
s�bb†� + �
b�!s�!
b!b†!†!�����
= sa + s�
a� . �190�
The above equation does not provide a single amplitude,unlike SR, but instead determines a sum of two ampli-tudes s
a +s�a� . However, the value of s
a can be deter-mined from the lower sector, i.e., according to the rela-tion
��a�S��� = ��a��b�
s�bb†���� = s
a ,
and due to the relation �189� the operators from the �1,1�sector do not enter the last equation. Thus from the so-lution of the FS-MRCC equations in the �1,0� and �0,1�sector, we obtain s
a and sı
�, respectively, and we can thentreat them as known quantities when solving the respec-tive equations within the �1,1� sectors. Hence, s�
a� is de-termined in Eq. �190� uniquely. This is a general featureof the FS-MRCC approach that in order to solve theequations for the �m ,n� sector, solutions for all lowersectors �i , j� �i=0,m and j=0,n� must be known. For ex-ample, for the �1,1� sector, solutions for �0,0�, �0,1�, and�1,0� sectors are required. This has sometimes beencalled the subsystem embedding condition �Chaudhuri etal., 1989�.
The general FS-MRCC equation for the �k , l� sectorformulated in the operator form can be written as
FIG. 40. A graphical representation and algebraic expressionsfor the S�0,0�, S�1,0�, S�0,1�, and S�1,1� operators at the CCSDlevel. In the definition of the S2
�1,1� operator, the case in whichboth creation lines are active is excluded �this is denoted by ��.
343Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
It is convenient to separate from the S operator thatcorresponding to the �0,0� sector,
S�k,l� = S��k,l� + S�0,0� = S��k,l� + T . �192�
Rewriting Eq. �191� on the basis of the equality in Eq.�192� and multiplying from the left with e−T, we obtain
e−THeTeS��k,l��P�k,l� = eS��k,l�
�P�k,l�HeffP�k,l�, �193�
where e−THeT=H is the quantity introduced in previoussections. Thus the final form of the FS-MRCC equationis obtained upon projection of the last equation on theexcited determinants in each sector represented by theprojector Q�k,l�,
Q�k,l�HeS��k,l��P�k,l� = Q�k,l�eS��k,l�
�P�k,l�HeffP�k,l�, �194�
where S�k,l� is defined according to Eq. �188�, i.e., it in-cludes all S components coming from the lower sectors.
The Heff operator is defined in an analogous manneras in the HS variant, and, for instance, for the �k , l� sec-tor we have
P�k,l�HeffP�k,l� = P�k,l�HeS��k,l��P�k,l�. �195�
In Fig. 41, we present the diagrammatic equation for theone-valence sectors, i.e., �1,0� and �0,1�. Providing up-ward �downward� arrows to the annihilation lines in Fig.41�a�, we obtain equations for the s
a �si�� amplitudes;
applying the same procedure to Fig. 41�b�, we obtainequations for the sj
ab and sij�b amplitudes. In Fig. 41�c�,
we present diagrammatic contributions to the effectiveHamiltonian elements also for the one-valence sector.Note that only the linear term is present in the one-valence sector, as nonlinear terms contain more thanone annihilation operator which makes them vanishwhen applied to the model determinants in the one-valence sector. The wiggly lines represent the elements
of the H operator.
The equation for the S2 amplitudes in the �1,1� sectorof the FS-CCSD model is given in Fig. 42. Here the S1and S2 amplitudes determined in the one-valence sectorsare used and they occur both in linear and quadraticterms, whereas the S2 amplitudes in the �1,1� sector ap-pear only via linear terms. This is a general feature ofthe FS equations that the S�i,j� amplitudes in the equa-tions for the �i , j� sector occur linearly. The solution ofthe FS-MRCC equations presented in Figs. 41 and 42 isobtained in an iterative manner. In each iteration, a setof the S amplitudes is constructed as well as the ele-ments of the Heff operator �see Figs. 41�c� and 43, whichare further diagonalized to obtain energy eigenvalues.The majority of applications of the FS-MRCC are lim-ited to the �1,1� sector of the Fock space, while the clus-ter expansion, Eq. �179�, is truncated at the S2 operator�MRCCSD model�. Just recently, an extension of the FS-MRCCSD model to full inclusion of the connected tripleexcitations has been reported �Musiał and Bartlett,2004b�. The FS-MRCCSDT approach requires determi-nation of T1 ,T2 ,T3 operators at the zero valence level,S1 ,S2 ,S3 operators at the one-valence level, and S2 andS3 operators for the �1,1� sector. One should note thatthe rank of the computational procedure �based on thesummation of inactive indices� decreases for higher sec-tors, i.e., we have an M8 ,M7 ,M6 procedure for the FS-
FIG. 41. The diagrammatic representation of the FS-CCSDequations for the �1,0� and �0,1� sectors in the skeleton form:�a� an equation for the S1, �b� an equation for the S2, and �c� anexpression for the one-particle part of the Heff. Active lines aredesignated by a circle.
FIG. 42. The diagrammatic representation of the S2 equationfor the �1,1� sector in skeleton form.
FIG. 43. The diagrammatic expression for the two-particlepart of Heff.
344 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
CCSDT model at zero-, one-, and two-valence sectors,respectively.
In Tables IX and XXI, we compare the performanceof the EOM and FS methods up to triples �relative tothe FCI values�. For the Ne atom, the FS and EOMvalues are very close; for the N2 molecule, the CCSDmodel provides slightly better results for the FS ap-proach. In both cases, the inclusion of triples improvesagreement with the FCI values remarkably. It should beemphasized, however, that the FS-MRCC results are rig-orously size extensive, while the EOM ones, because of
their dependence on a linear operator R instead of anexponential as in the Fock space case, mean that it re-tains certain elements of CI in its solutions. While EOM
is fine for �AB�*→A*+B or A+B*, it does not gosmoothly to A++B−; FS-CC does �Meissner and Bartlett,1995�.
An intermediate approach between FS-CC andEOM-CC is the similarity transformed EOM �STEOM��Nooijen and Bartlett, 1997a� method. In this approach,the �0,1� and �1,0� sector problems are solved to define a
second similarity transformation eS that is then applied
to H, i.e., e−SHeS=G. By virtue of the second transfor-mation, it can be shown that the single-excitation blockof the �1,1� �excited-state� sector is approximately de-coupled from both double and triple excitations. Thisgives a matrix of excited-state solutions whose rank is no
TABLE XXI. Vertical excitation energies �in eV� of N2 �R=2.068 a.u.� with cc-pVDZ basis set. The1s orbitals were frozen. The active orbitals �lowest-lying unoccupied, highest occupied� are �2,4� forthe FS and STEOM calculations.
larger than the very small ��nN� number of single exci-tations. In a sense this is an exact, size-extensive CIsingles theory for excited states that correspond to singleexcitations. The STEOM method also controls intruderstates without further modification, unlike the FS-CC orHS-CC methods. Applications to free-base porphyrin il-lustrate its applicability �Nooijen and Bartlett, 1997b;Gwaltney and Bartlett, 1998�. In Tables IX, XXI, andXXII, we compare the performance of the FS, EOM,and STEOM methods at the CCSD level relative to theFCI values. As we can see, the FS and STEOM methodsgive results that are of an overall quality comparablewith EOM, but at substantially less cost because of thesmall matrix diagonalization.
The Fock-space approach, in particular, is plagued bythe same complications as its Hilbert-space equivalent,i.e., by divergence problems in the iterative processwhen intruder states occur. That situation is more likelyto occur when large model spaces are used. Intruderstates constitute a serious limitation of the FS-MRCCapproach.
C. Fock-space MRCC based on an intermediate Hamiltonian
In order to avoid the intruder state problem, a newversion of the FS-MRCC method has been establishedthat is based on the intermediate Hamiltonian theory ofMalrieu et al., �1985� and Meissner �Meissner and Bart-lett, 1995; Meissner and Nooijen, 1995; Meissner, 1998;Landau et al., 2000; Meissner and Malinowski, 2000; Ma-linowski et al., 2002; Musiał and Meissner, 2005; Musiałet al., 2005�. The main idea of the intermediate Hamil-tonian approach relies on selecting a part of the or-thogonal space M� as an intermediate space MI, con-nected with the projector PI. The MI space is achievedby the action of the operator Z operating on the modelspace P. The remaining part of the orthogonal spaceMI� is connected with the projector QI related to PI by
Q = PI + QI. �196�
Within the FS-MRCCSD model in the expansion of the
eS��1,1�
operator,
eS��1,1�
�P = �1 + Z + Y�P , �197�
Z = �S1�1,0� + S1
�0,1� + S1�1,0�S1
�0,1� + S2�1,1���P�1,1�, �198�
TABLE XXIII. Vertical excitation energies �in eV� for N2 as obtained with the IH-FS and EOMmethods at the CCSD levela �R=2.068 a.u.�
Statesymmetry
aug-cc-pVQZ aug-cc-pVQZ+b
Expt.
IH-FS
EOMIH-FS�18,4� EOM�6,4� �18,4�
Singlet states1�g 10.526 9.194 9.489 9.285 9.489 9.311�u
We see that the operator Z operating on the modelspace P�1,1� generates determinants belonging to MI
space, while the Y operator connects the P�1,1� and MI�
�1,1�
subspaces. Note that amplitudes of the S�1,1� operatorthat are sought in the equation for the �1,1� sector do notoccur in the Y operator. We define the operator HI,
HI = P0e−Y�HeY�P0 = P0�1 − Y�H�1 + Y�P0, �200�
where P0=P+PI is a projection operator onto themodel+intermediate space. Taking into account that Ygenerates configurations belonging to the MI�
�1,1� space,we replace the last equation with
HI = P0H�1 + Y�P0. �201�
Diagonalization of the HI operator within the P0 spaceprovides a subset of eigenvalues that are identical withthose obtained by diagonalization of the Heff operatorwithin the model space. We also observe that the Y op-erator used to construct the intermediate Hamiltonian isexpressed exclusively through the operators known fromlower sectors in the current case: S�1,0� and S�0,1�. As aresult, the FS-MRCC approach formulated in terms ofthe intermediate Hamiltonian does not require an itera-tive procedure and is free of the difficulties mentioned atthe end of the previous subsection connected with in-truder states. The diagrammatic contributions to the HIelements are given in Fig. 44. Note that here, also, dis-connected terms contribute but the result is size exten-sive due to the cancellation of disconnected terms in thediagonalization process.
The intermediate Hamiltonian formulation of the FS-MRCC theory has great advantages over the traditionalFS approach. Being free from intruder states and diver-gence problems, it allows consideration of larger modelspaces for which iterative solutions are impossible. InTable XXIII, we present IH-FS-MRCCSD results forthe N2 molecule obtained for two relatively large basissets: the standard aug-cc-pVQZ basis set �Kendall et al.,1992� and that with additional diffuse s and p functions.Two model spaces were considered: one containing 24reference determinants and the other including 72 deter-minants. It may be seen from Table XXIII that the re-sults depend on the type of model space. For the largermodel space, the theoretical values are much closer tothe experimental ones. Particularly good agreement withexperiment is observed for the larger basis set: the dif-fuse basis functions allow for a proper treatment of theRydberg states, and that combined with the sufficientlylarge model space gives results much closer to experi-mental data than the EOM values obtained for the samebasis set. We believe that the intermediate Hamiltonianversion of the FS-MRCC theory opens new and interest-ing possibilities for multireference CC theory.
ACKNOWLEDGMENTS
R.J.B. would like to acknowledge his teachers Per-Olov Löwdin, whose influence, many years later, can beseen throughout this paper; and John C. Slater, who firstintroduced him to electronic structure theory. Our con-tribution to the development of coupled-cluster theoryhas critically depended upon the exceptional students,postdocs, and visiting professors that joined us in thiseffort, many of whom are today’s leaders in the field.Their names appear throughout the bibliography. M.M.owes a debt of gratitude to Professor Stanisław Kuchar-ski, Dean of the School of Sciences, University of Sile-sia, Katowice, Poland, who first introduced her to dia-grammatic CC methods. The authors particularlyappreciate the long term support from the Air ForceOffice of Scientific Research, who supported theBartlett’s group development of coupled-cluster theoryfrom the mid-1970s to the present. Special thanks go toDr. Michael Berman, Dr. Larry Burgraff, Dr. LarryDavis, and Dr. Ralph Kelley.
REFERENCES
Adamowicz, L., and R. J. Bartlett, 1985, Int. J. QuantumChem. 19, 217.
Adamowicz, L., R. J. Bartlett, and E. A. Mccullough, 1985,Phys. Rev. Lett. 54, 426.
Adamowicz, L., W. D. Laidig, and R. J. Bartlett, 1984, Int. J.Quantum Chem. 18, 245.
Alexeev, Y., T. L. Windus, C. G. Zhan, and D. A. Dixon, 2005,Int. J. Quantum Chem. 104, 379.
Almlöf, J., and P. R. Taylor, 1987, J. Chem. Phys. 86, 4070.Arponen, J. S., 1983, Ann. Phys. �N.Y.� 151, 311.Arponen, J. S., 1991, Theor. Chim. Acta 80, 149.Arponen, J. S., R. F. Bishop, and E. Pajanne, 1987a, Phys. Rev.
A 36, 2519.Arponen, J. S., R. F. Bishop, and E. Pajanne, 1987b, Phys. Rev.
A 36, 2539.Bak, K. L., J. Gauss, T. Helgaker, P. Jörgensen, and J. Olsen,
2000, Chem. Phys. Lett. 319, 563.Bak, K. L., P. Jörgensen, J. Olsen, T. Helgaker, and W. Klop-
per, 2000, J. Chem. Phys. 112, 9229.Balkova, A., and R. J. Bartlett, 1992, Chem. Phys. Lett. 193,
364.Balkova, A., and R. J. Bartlett, 1995, J. Chem. Phys. 102, 7116.Balkova, A., S. A. Kucharski, and R. J. Bartlett, 1991, Chem.
Phys. Lett. 182, 511.Balkova, A., S. A. Kucharski, L. Meissner, and R. J. Bartlett,
1991a, Theor. Chim. Acta 80, 335.Balkova, A., S. A. Kucharski, L. Meissner, and R. J. Bartlett,
1991b, J. Chem. Phys. 95, 4311.Barbe, A., C. Secroun, and P. Jouve, 1974, J. Mol. Spectrosc.
49, 171.Bartlett, R. J., 1986, in Geometrical Derivatives of Energy Sur-
faces and Molecular Properties, edited by P. Jörgensen and J.Simons �Reidel, Dordrecht, The Netherlands�, pp. 35–61.
Bartlett, R. J., 1989, J. Phys. Chem. 89, 93, feature article.Bartlett, R. J., 1995, in Modern Electronic Structure Theory,
edited by D. R. Yarkony �World Scientific, Singapore�, Vol. 2,pp. 1047–1131.
Bartlett, R. J., 2002, Int. J. Mol. Sci. 3, 579.
347Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
Bartlett, R. J., 2004, in Molecular Quantum Mechanics: Se-lected Papers of N.C. Handy, edited by D. C. Clary, S. M.Colwell, and H. F. Schaefer III �Taylor & Francis, London�,pp. 127–130.
Bartlett, R. J., 2005, in Theory and Applications of Computa-tional Chemistry: The First Forty Years, edited by C. E. Dyk-stra, G. Frenking, K. S. Kim, and G. E. Scuseria �Elsevier,Amsterdam�, pp. 1191–1221.
Bartlett, R. J., S. A. Kucharski, and J. Noga, 1989, Chem. Phys.Lett. 155, 133.
Bartlett, R. J., and M. Musiał, 2006, J. Chem. Phys. 125,204105.
Bartlett, R. J., and J. Noga, 1988, Chem. Phys. Lett. 150, 29.Bartlett, R. J., and G. D. Purvis III, 1978, Int. J. Quantum
Chem. 14, 561.Bartlett, R. J., and G. D. Purvis III, 1980, Phys. Scr. 21, 225.Bartlett, R. J., and D. M. Silver, 1974a, Phys. Rev. A 10, 1927.Bartlett, R. J., and D. M. Silver, 1974b, Chem. Phys. Lett. 29,
199.Bartlett, R. J., and D. M. Silver, 1975, Int. J. Quantum Chem.,
Symp. 9, 183.Bartlett, R. J., and D. M. Silver, 1976, J. Chem. Phys. 64, 4578.Bartlett, R. J., and J. F. Stanton, 1994, Rev. Comput. Chem. 5,
65.Bartlett, R. J., and J. D. Watts, 1998, ACES II, in Encyclopedia
of Computational Chemistry, edited by P. von Schleyer et al.�Wiley, New York�.
Bartlett, R. J., J. D. Watts, S. A. Kucharski, and J. Noga, 1990,Chem. Phys. Lett. 165, 513.
Bauschlicher, C. W., Jr., S. R. Langhoff, P. R. Taylor, N. C.Handy, and P. J. Knowles, 1986, J. Chem. Phys. 85, 1469.
Bauschlicher, C. W., Jr., and P. R. Taylor, 1986, J. Chem. Phys.85, 2779.
Bauschlicher, C. W., Jr., and P. R. Taylor, 1987a, J. Chem. Phys.86, 1420.
Bauschlicher, C. W., Jr., and P. R. Taylor, 1987b, J. Chem. Phys.86, 2844.
Beebe, N. H. F., and J. Linderberg, 1977, Int. J. QuantumChem. 12, 683.
Beste, A., and R. J. Bartlett, 2004, J. Chem. Phys. 120, 8395.Bieri, G., and L. Asbrink, 1980, J. Electron Spectrosc. Relat.
Phenom. 20, 149.Bishop, R. F., 1991, Theor. Chim. Acta 80, 95.Bishop, R. F., 1998, in Microscopic Quantum Many-Body
Theories and Their Applications, Lecture Notes in PhysicsVol. 510, edited by J. Navarro and A. Polls �Springer, Berlin�,pp. 1–70.
Bishop, R. F., T. Brandes, K. A. Germnoth, N. R. Walet, and Y.Xianm, 2002, Recent Progress in Many-Body Theories, Ad-vances in Quantum Many-Body Theories �World Scientific,Singapore�, Vol. 6.
Bishop, R. F., R. Guardiola, I. Moliner, J. Navarro, M. Partesi,A. Puente, and N. R. Walter, 1998, Nucl. Phys. A 643, 258.
Bishop, R. F., and K. H. Lührmann, 1978, Phys. Rev. B 17,3757.
Bishop, R. F., and K. H. Lührmann, 1981, Physica B & C 108,873.
Bloch, C., and J. Horowitz, 1958, Nucl. Phys. 8, 91.Bogoliubov, N. N., and D. V. Shirkov, 1959, Introduction to the
Theory of Quantized Fields �Wiley, New York� �Englishtransl.�.
Bomble, Y. J., J. F. Stanton, M. Kallay, and J. Gauss, 2005, J.Chem. Phys. 123, 054101.
Boys, S. F., 1950, Proc. R. Soc. London, Ser. A 200, 542.Brueckner, K. A., 1955, Phys. Rev. 97, 1353.Chan, G. K.-L., M. Kallay, and J. Gauss, 2004, J. Chem. Phys.
121, 6110.Chaudhuri, R., D. Mukhopadhyay, and D. Mukherjee, 1989,
Chem. Phys. Lett. 162, 393.Chawla, S., and A. G. Voth, 1998, J. Chem. Phys. 108, 4697.Chiles, R. A., and C. E. Dykstra, 1981, J. Chem. Phys. 74, 4544.Christiansen, O., H. Koch, and P. Jörgensen, 1995a, Chem.
Phys. Lett. 243, 409.Christiansen, O., H. Koch, and P. Jörgensen, 1995b, J. Chem.
Phys. 103, 7429.Christiansen, O., H. Koch, P. Jörgensen, and J. Olsen, 1996,
Chem. Phys. Lett. 256, 185.Cížek, J., 1966, J. Chem. Phys. 45, 4256.Cížek, J., 1969, Adv. Chem. Phys. 14, 35.Cížek, J., 1991, Theor. Chim. Acta 80, 91.Cížek, J., and J. Paldus, 1971, Int. J. Quantum Chem. 5, 359.Coester, F., 1958, Nucl. Phys. 1, 421.Coester, F., and H. Kümmel, 1960, Nucl. Phys. 17, 477.Comeau, D., and R. J. Bartlett, 1993, Chem. Phys. Lett. 207,
414.Coriani, S., D. Marchesan, J. Gauss, Ch. Hattig, T. Helgaker,
and P. Jörgensen, 2005, J. Chem. Phys. 123, 184107.Crawford, T. D., and H. F. Schaefer, III, 2000, in Reviews of
Computational Chemistry, edited by K. B. Lipkowitz and D.B. Boyd �Wiley, New York�, Vol. 14, pp. 33–136.
Crawford, T. D., and J. F. Stanton, 1998, Int. J. QuantumChem. 70, 601.
Dalgarno, A., and A. L. Stewart, 1958, Proc. R. Soc. London,Ser. A 247, 245.
Davidson, E. R., 1975, J. Comput. Phys. 17, 87.Davidson, E. R., and A. A. Jarzecki, 1998, Chem. Phys. Lett.
285, 155.Del Bene, J. E., S. A. Perera, R. J. Bartlett, and J. Elguero,
2000, J. Phys. Chem. A 104, 7165.Dunning, T. H., Jr., 1989, J. Chem. Phys. 90, 1007.Dunning, T. H., and V. McKoy, 1967, J. Chem. Phys. 47, 1735.Eliav, E., and U. Kaldor, 1996, Chem. Phys. Lett. 248, 405.Eliav, E., U. Kaldor, and B. A. Hess, 1998, J. Chem. Phys. 108,
3409.Emrich, K., 1981, Nucl. Phys. A 351, 379.Fan, P.-D., K. Kowalski, and P. Piecuch, 2005, Mol. Phys. 103,
2191.Flocke, N., and R. J. Bartlett, 2003, Chem. Phys. Lett. 367, 80.Förner, W., R. Knab, J. Cizek, and J. Ladik, 1997, J. Chem.
Phys. 106, 10248.Franke, R., H. Müller, and J. Noga, 1995, J. Chem. Phys. 114,
7746.Frantz, M., and R. L. Mills, 1960, Nucl. Phys. 15, 16.Freeman, D. L., 1977, Phys. Rev. B 15, 5512.Fusti-Molnar, L., and P. G. Szalay, 1996, J. Phys. Chem. 100,
6288.Gauss, J., 1998, in Encyclopedia of Computational Chemistry,
edited by P. von R. Schleyer �Wiley, New York�, Vol. I, pp.615–636.
Gauss, J., and J. F. Stanton, 1995, J. Chem. Phys. 102, 251.Gauss, J., and J. F. Stanton, 2002, J. Chem. Phys. 116, 4773.Gdanitz, R. J., and R. Ahlricks, 1988, Chem. Phys. Lett. 143,
413.Geertsen, J., M. Rittby, and R. J. Bartlett, 1989, Chem. Phys.
Lett. 164, 57.Gell-Mann, M., and K. A. Brueckner, 1957, Phys. Rev. 106,
348 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
364.Goldstone, J., 1957, Proc. R. Soc. London, Ser. A 239, 267.Guardiola, R., P. I. Moliner, J. Navarro, R. F. Bishop, A.
Puente, and N. R. Walter, 1996, Nucl. Phys. A 609, 218.Gwaltney, S. R., and R. J. Bartlett, 1998, J. Chem. Phys. 108,
6790.Gwaltney, S. R., E. F. C. Byrd, T. Van Voorhis, and M. Head-
Gordon, 2002, Chem. Phys. Lett. 353, 359.Gwaltney, S. R., and M. Head-Gordon, 2001, J. Chem. Phys.
115, 2014.Hald, K., Ch. Hattig, J. Olsen, and P. Jörgensen, 2001, J. Chem.
Phys. 115, 3545.Handy, N. C., J. A. Pople, M. Head-Gordon, K. Raghavachari,
and G. W. Trucks, 1989, Chem. Phys. Lett. 164, 185.Handy, N. C., and H. F. Schaefer III, 1984, J. Chem. Phys. 81,
5031.Haque, M. A., and D. Mukherjee, 1984, J. Chem. Phys. 80,
5058.Harris, T. E., H. J. Monkhorst, and D. L. Freeman, 1992, Al-
gebraic and Diagrammatic Methods in Many-Fermion Theory�Oxford University, New York�.
Heisenberg, J. H., and B. Mihaila, 1999, Phys. Rev. C 59, 1440.Helgaker, T., P. Jörgensen, and J. Olsen, 2000, Molecular
Electronic-Structure Theory �Wiley, New York�, pp. 817–883.Hess, D. K., and U. Kaldor, 2000, J. Chem. Phys. 112, 1809.Hino O., T. Kinoshita, and R. J. Bartlett, 2004, J. Chem. Phys.
121, 1206.Hino, O., T. Kinoshita, G. K.-L. Chan, and R. J. Bartlett, 2006,
J. Chem. Phys. 124, 114311.Hino, O., Y. Tanimura, and S. Ten-no, 2002, Chem. Phys. Lett.
353, 317.Hirao, K., and H. Nakatsuji, 1982, J. Comput. Phys. 45, 246.Hirata, S., 2003, J. Phys. Chem. A 107, 10154.Hirata, S., 2004, J. Chem. Phys. 121, 5.Hirata, S., P.-D. Fan, A. A. Auer, M. Nooijen, and P. Piecuch,
2004, J. Chem. Phys. 121, 12197.Hirata, S., I. Grabowski, M. Tobita, and R. J. Bartlett, 2001,
Chem. Phys. Lett. 345, 475.Hirata, S., M. Nooijen, and R. J. Bartlett, 2000, Chem. Phys.
Lett. 326, 255.Hirata, S., M. Nooijen, I. Grabowski, and R. J. Bartlett, 2001,
J. Chem. Phys. 114, 3919.Hirata, S., R. Podeszwa, M. Tobita, and R. J. Bartlett, 2004, J.
Chem. Phys. 120, 2581.Holland, D. M. P., D. A. Shaw, M. A. Hayes, L. G. Shpinkova,
E. E. Rennie, L. Karlsson, P. Baltzer, and B. Wannberg, 1997,Chem. Phys. 219, 91.
Huba~, I., J. Pittner, and P. Carsky, 2000, J. Chem. Phys. 112,8779.
Hubbard, J., 1957, Proc. R. Soc. London, Ser. A 240, 539.Huber, K. P., and G. Herzberg, 1979, Constants of Diatomic
Molecules �Van Nostrand Reinhold, New York�.Hurley, A. C., 1976, Electron Correlation in Small Molecules
�Academic, New York�.Hylleraas, E. A., 1929, Z. Phys. 54, 347.Jeziorski, B., and H. J. Monkhorst, 1981, Phys. Rev. A 24,
1668.Jeziorski, B., and J. Paldus, 1989, J. Chem. Phys. 90, 2714.Kaldor, U., and B. A. Hess, 1994, Chem. Phys. Lett. 230, 1.Kallay, M., and J. Gauss, 2004, J. Chem. Phys. 120, 6841.Kállay, M., and P. Surjan, 2001, J. Chem. Phys. 115, 2945.Kato T., 1957, Commun. Pure Appl. Math. 10, 15.
Kelly, H. P., 1962, Ph.D. thesis, University of California, Ber-keley.
Kelly, H. P., 1969, Adv. Chem. Phys. 14, 129.Kendall, R. A., T. H. Dunning, Jr., and R. J. Harrison, 1992, J.
Chem. Phys. 96, 6796.Kinoshita, T., O. Hino, and R. J. Bartlett, 2003, J. Chem. Phys.
119, 7756.Kinoshita, T., O. Hino, and R. J. Bartlett, 2005, J. Chem. Phys.
123, 074106.Kobayashi, R., H. Koch, and P. Jörgensen, 1994, Chem. Phys.
Lett. 219, 30.Koch, H., O. Christiansen, P. Jörgensen, and J. Olsen, 1995,
Chem. Phys. Lett. 244, 75.Koch, H., A. S. de Meras, and T. B. Pedersen, 2003, J. Chem.
Phys. 118, 9481.Koch, H., H. J. Aa. Jensen, P. Jörgensen, and T. Helgaker,
1990, J. Chem. Phys. 93, 3333.Koch, H., and P. Jörgensen, 1990, J. Chem. Phys. 93, 3333.Kohn, W., and L. J. Sham, 1965, Phys. Rev. 140, A1133.Kowalski, K., D. J. Dean, M. Hjorth-Jensen, T. Papenbrock,
and P. Piecuch, 2004, Phys. Rev. Lett. 92, 132501.Kowalski, K., and P. Piecuch, 2000a, J. Chem. Phys. 113, 18.Kowalski, K., and P. Piecuch, 2000b, J. Chem. Phys. 113, 5644.Kowalski K., and P. Piecuch, 2001, J. Chem. Phys. 115, 643.Krylov, A. I., C. D. Sherill, and M. Head-Gordon, 2000, J.
Chem. Phys. 113, 6509.Kucharski, S. A., A. Balkova, P. G. Szalay, and R. J. Bartlett,
1992, J. Chem. Phys. 97, 4289.Kucharski, S. A., and R. J. Bartlett, 1986, Adv. Quantum
Chem. 18, 281.Kucharski, S. A., and R. J. Bartlett, 1991a, Theor. Chim. Acta
80, 387.Kucharski, S. A., and R. J. Bartlett, 1991b, J. Chem. Phys. 95,
9271.Kucharski, S. A., and R. J. Bartlett, 1992, Int. J. Quantum
Chem., Quantum Chem. Symp. 26, 107.Kucharski, S. A., and R. J. Bartlett, 1993, Chem. Phys. Lett.
206, 574.Kucharski, S. A., and R. J. Bartlett, 1998a, J. Chem. Phys. 108,
5243.Kucharski, S. A., and R. J. Bartlett, 1998b, J. Chem. Phys. 108,
5255.Kucharski, S. A., and R. J. Bartlett, 1998c, J. Chem. Phys. 108,
9221.Kucharski, S. A., and R. J. Bartlett, 1999, J. Chem. Phys. 110,
8233.Kucharski, S. A., M. Kolaski, and R. J. Bartlett, 2001, J. Chem.
Phys. 114, 692.Kucharski, S. A., J. D. Watts, and R. J. Bartlett, 1999, Chem.
Phys. Lett. 302, 295.Kümmel, 1991, Theor. Chim. Acta 80, 81.Kümmel, H., K. H. Lührmann, and J. Zabolitzky, 1978, Phys.
Rep., Phys. Lett. 36, 1.Kutzelnigg, W., 1985, Theor. Chim. Acta 68, 445.Laidig, W. D., P. Saxe, and R. J. Bartlett, 1987, J. Chem. Phys.
86, 887.Landau, A., E. Eliav, Y. Ishikawa, and U. Kaldor, 2000, J.
Chem. Phys. 113, 9905.Landau, A., E. Eliav, and U. Kaldor, 1999, Chem. Phys. Lett.
313, 399.Larsen, H., K. Hald, J. Olsen, and P. Jörgensen, 2001, J. Chem.
Phys. 115, 3015.Lauderdale, W. J., J. F. Stanton, J. Gauss, J. D. Watts, and R. J.
349Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
Bartlett, 1992, J. Chem. Phys. 97, 6606.Lee, T. J., and G. E. Scuseria, 1995, in Quantum Mechanical
Electronic Structure Calculations with Chemical Accuracy, ed-ited by S. R. Langhoff �Kluwer, Dordrecht, The Netherlands�,pp. 47–108.
Lee, Y. S., S. A. Kucharski, and R. J. Bartlett, 1984, J. Chem.Phys. 81, 5906.
Levchenko, S. V., and A. I. Krylov, 2004, J. Chem. Phys. 120,175.
Li, X., and J. Paldus, 1994, J. Chem. Phys. 101, 8812.Li, X., and J. Paldus, 2004, J. Chem. Phys. 120, 5890.Lindgren, I., 1978, Int. J. Quantum Chem. S12, 33.Lindgren, I., 1979, Int. J. Quantum Chem. S12, 3827.Lindgren, I., and M. Morrison, 1986, Atomic Many-Body
Theory, 2nd ed. �Springer, Berlin�.Lindgren, I., and D. Mukherjee, 1987, Phys. Rep. 151, 93.Lindgren, I., and S. Salomonsen, 2002, Int. J. Quantum Chem.
90, 294.Lischa, H., R. Shepard, I. Shavitt, R. M. Pitzer, M. Dallos, Th.
Muller, P. G. Szalay, F. B. Brown, R. Ahlirchs, H. J. Bohm, A.Chang, D. C. Comeau, R. Gdanitz, H. Dachsel, C. Ehrhardt,M. Ernzerhof, P. Hochtl, S. Irle, G. Kedziora, T. Kovar, V.Parasuk, M. J. M. Pepper, P. Scharf, H. Schiffer, M. Schindler,M. Schuler, M. Seth, E. A. Stahlberg, J.-G. Zhao, S.Yabushita, and Z. Zhang, 2001, COLUMBUS, an ab initio elec-tronic structure program, release 5.8.
Löwdin, P.-O., 1955, Phys. Rev. 97, 1474.Löwdin, P.-O., 1959, Adv. Chem. Phys. 2, 207.Löwdin, P.-O., 1968, Int. J. Quantum Chem. 2, 867.Lynch, B. J., and D. G. Truhlar, 2003, J. Phys. Chem. A 107,
3898.Malinowski, P., L. Meissner, and A. Nowaczyk, 2002, J. Chem.
Phys. 116, 7362.Malrieu, J.-P., Ph. Durand, and J.-P. Daudey, 1985, J. Phys. B
18, 809.Manne, R., 1977, Int. J. Quantum Chem. Y-11, 175.Masik A., and I. Huba~, 1999, Adv. Quantum Chem. 31, 75.McCurdy, C. W., Jr., T. N. Resigno, D. L. Yeager, and V.
McKoy, 1982, in The Equation of Motion Method: An Ap-proach to the Dynamical Properties of Atoms and Molecules,edited by G. A. Webb �Academic, New York�, Vol. 12, pp.339–386.
Meissner, L., 1998, J. Chem. Phys. 108, 9227.Meissner, L., and R. J. Bartlett, 1995, J. Chem. Phys. 102, 7490.Meissner, L., and R. J. Bartlett, 2001, J. Chem. Phys. 115, 50.Meissner, L., and P. Malinowski, 2000, Phys. Rev. A 61, 062510.Meissner, L., and M. Nooijen, 1995, J. Chem. Phys. 102, 9604.Mihaila, B., and J. Heisenberg, 2000, Phys. Rev. C 61, 054309.Monkhorst, H. J., 1977, Int. J. Quantum Chem., Quantum
Chem. Symp. 11, 421.Mosyagin, N. S., A. V. Titov, E. Eliav, and U. Kaldor, 2001, J.
Chem. Phys. 115, 2007.Moszynski, R., P. S. Zuchowski, and B. Jeziorski, 2005, Collect.
Czech. Chem. Commun. 70, 1109.Mukherjee, D., and S. Pal, 1989, Adv. Quantum Chem. 20, 291.Musiał, M., and R. J. Bartlett, 2003, J. Chem. Phys. 119, 1901.Musiał, M., and R. J. Bartlett, 2004a, Chem. Phys. Lett. 384,
210.Musiał, M., and R. J. Bartlett, 2004b, J. Chem. Phys. 121, 1670.Musiał, M., and R. J. Bartlett, 2005, J. Chem. Phys. 122,
224102.Musiał, M., and S. A. Kucharski, 2004, Struct. Chem. 15, 421.Musiał, M., S. A. Kucharski, and R. J. Bartlett, 2001, J. Mol.
Struct.: THEOCHEM 547, 269.Musiał, M., S. A. Kucharski, and R. J. Bartlett, 2002a, Mol.
Phys. 100, 1867.Musiał, M., S. A. Kucharski, and R. J. Bartlett, 2002b, J. Chem.
Phys. 116, 4382.Musiał, M., S. A. Kucharski, and R. J. Bartlett, 2003, J. Chem.
Phys. 118, 1128.Musiał, M., S. A. Kucharski, and R. J. Bartlett, 2004, Adv.
Quantum Chem. 47, 209.Musiał, M., and L. Meissner, 2005, Collect. Czech. Chem.
Commun. 70, 811.Musiał, M., L. Meissner, S. A. Kucharski, and R. J. Bartlett,
2005, J. Chem. Phys. 122, 224110.Nakatsuji, H., 1978, Chem. Phys. Lett. 59, 362.Noga, J., R. J. Bartlett, and M. Urban, 1987, Chem. Phys. Lett.
134, 126.Noga, J., and W. Kutzelnigg, 1994, J. Chem. Phys. 101, 7738.Noga, J., W. Kutzelnigg, and W. Klopper, 1992, Chem. Phys.
Lett. 199, 497.Noga, J., and P. Valiron, 2000, Chem. Phys. Lett. 324, 166.Noga, J., and P. Valiron, 2002, in Computational Chemistry:
Reviews of Current Trends, edited by J. Leszczynski �WorldScientific, Singapore�, Vol. 7, pp. 131–185.
Noga, J., P. Valiron, and W. Klopper, 2001, J. Chem. Phys. 115,2022.
Nooijen, M., 2005, unpublished.Nooijen, M., and R. J. Bartlett, 1997a, J. Chem. Phys. 106,
6441.Nooijen, M., and R. J. Bartlett, 1997b, J. Chem. Phys. 106,
6449.Nooijen, M., and R. J. Bartlett, 1997c, J. Chem. Phys. 107,
6812.Oliphant, N., and L. Adamowicz, 1992, J. Chem. Phys. 96,
3739.Olsen, J., 2000, J. Chem. Phys. 113, 7140.Pal, S., M. Rittby, and R. J. Bartlett, 1987, Chem. Phys. Lett.
137, 273.Pal, S., M. Rittby, R. J. Bartlett, D. Sinha, and D. Mukherjee,
1988, Chem. Phys. 88, 4357.Paldus, J., 1991, in Methods in Computational Molecular Phys-
ics, NATO Advanced Study Institute �Plenum, New York�.Paldus, J., 2005, in Theory and Applications of Computational
Chemistry: The First Forty Years, edited by C. E. Dykstra, G.Frenking, K. S. Kim, and G. E. Scuseria �Elsevier, Amster-dam�, pp. 115–147.
Paldus, J., and J. Cížek, 1975, Adv. Quantum Chem. 9, 105.Paldus, J., J. Cízek, and I. Shavitt, 1972, Phys. Rev. A 5, 50.Paldus, J., P. Piecuch, L. Pylypow, and B. Jeziorski, 1993, Phys.
Rev. A 47, 2738.Parr, R. G., and B. L. Crawford, 1948, J. Chem. Phys. 16, 526Pawlowski, F., A. Halkier, P. Jörgensen, K. L. Bak, T. Hel-
gaker, and W. Klopper, 2003, J. Chem. Phys. 118, 2539.Perera, S. A., and R. J. Bartlett, 1993, Chem. Phys. Lett. 216,
606.Perera, S. A., and R. J. Bartlett, 1996, J. Am. Chem. Soc. 118,
7849.Perera, S. A., and R. J. Bartlett, 2000, J. Am. Chem. Soc. 122,
1231.Perera, S. A., and R. J. Bartlett, 2005, Adv. Quantum Chem.
48, 435.Perera, S. A., M. Nooijen, and R. J. Bartlett, 1996, J. Chem.
Phys. 104, 3290.Peterson, K. A., A. K. Wilson, D. E. Woon, and T. H. Dunning,
350 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
Jr., 1997, Theor. Chim. Acta 97, 251.Piecuch, P., K. Kowalski, I. S. Pimienta, and M. J. McGuire,
2002, Int. Rev. Phys. Chem. 21, 527.Piecuch, P., S. A. Kucharski, and R. J. Bartlett, 1999, J. Chem.
Phys. 110, 6103.Piecuch, P., N. Oliphant, and L. Adamowicz, 1993, J. Chem.
Phys. 99, 1875.Piecuch, P., M. Włoch, J. R. Gour, and A. Kinal, 2006, Chem.
Phys. Lett. 418, 467.Pisani, L., and E. Clementi, 1995, J. Chem. Phys. 103, 9321.Pittner, J., 2003, J. Chem. Phys. 118, 10876.Pittner, J., P. Nachtigall, P. Carsky, J. Mašik, and I. Huba~,
1999, J. Chem. Phys. 110, 10275.Pople, J. A., J. S. Binkley, and R. Seeger, 1976, Int. J. Quan-
tum Chem., Symp. 10, 1.Pople, J. A., R. Krishnan, H. B. Schlegel, and J. S. Binkley,
1978, Int. J. Quantum Chem., Quantum Chem. Symp. 14, 545.Purvis, G. D., III, and R. J. Bartlett, 1982, J. Chem. Phys. 76,
1910.Purvis, G. D., III, H. Sekino, and R. J. Bartlett, 1988, Collect.
Czech. Chem. Commun. 53, 2203.Ragavachari, K., G. W. Trucks, J. A. Pople, and M. Head-
Gordon, 1989, Chem. Phys. Lett. 157, 479.Rittby, M., and R. J. Bartlett, 1991, Theor. Chim. Acta 80, 649.Rowe, D. J., 1968, Rev. Mod. Phys. 40, 153.Rozyczko, P., and R. J. Bartlett, 1997, J. Chem. Phys. 107,
10823.Rozyczko, P., S. A. Perera, M. Nooijen, and R. J. Bartlett,
1997, J. Chem. Phys. 107, 6736.Ruden, T. A., T. Helgaker, P. Jörgensen, and J. Olsen, 2004, J.
Chem. Phys. 121, 5874.Salter, E. A., and R. J. Bartlett, 1989, J. Chem. Phys. 90, 1767.Salter, E. A., G. W. Trucks, and R. J. Bartlett, 1989, J. Chem.
Phys. 90, 1752.Sari, L., Y. Yamaguchi, and F. Schaefer III, 2001, J. Chem.
Phys. 115, 5932.Schuetz, M., and H. J. Werner, 2000, Chem. Phys. Lett. 318,
370.Scuseria, G. E., A. C. Sheiner, T. J. Lee, J. E. Rice, and H. F.
Schaefer III, 1987, J. Chem. Phys. 86, 2881.Sekino, H., and R. J. Bartlett, 1984, Int. J. Quantum Chem.,
Quantum Chem. Symp. 18, 255.Sekino, H., and R. J. Bartlett, 1999, Adv. Quantum Chem. 35,
149.Shavitt, I., 1998, Mol. Phys. 94, 3.Shavitt, I., and R. J. Bartlett, 2006, Many-Body Methods in
Quantum Chemistry: Many-Body Perturbation Theory andCoupled-Cluster Theory �Cambridge University Press, inpress�.
Simons, J., and W. D. Smith, 1973, J. Chem. Phys. 58, 4899.Sinha, D., A. Mukhopadhyay, M. D. Prasad, and D. Mukher-
jee, 1986, Chem. Phys. Lett. 125, 213.Slater, J. C., 1929, Phys. Rev. 34, 1293.Sorouri, A., W. M. C. Foulkes, and N. D. M. Hine, 2006, J.
Chem. Phys. 124, 064105.Stanton, J. F., 1994, J. Chem. Phys. 101, 371.Stanton, J. F., and R. J. Bartlett, 1993a, J. Chem. Phys. 99,
5178.Stanton, J. F., and R. J. Bartlett, 1993b, J. Chem. Phys. 98,
7029.Stanton, J. F., and J. Gauss, 1995, J. Chem. Phys. 103, 8931.Stanton, J. F., and J. Gauss, 1999, J. Chem. Phys. 110, 6079.Stanton, J. F., J. Gauss, and R. J. Bartlett, 1992, J. Chem. Phys.
97, 5554.Stanton, J. F., J. Gauss, J. D. Watts, M. Nooijen, N. Oliphant,
S. A. Perera, P. G. Szalay, W. J. Lauderdale, S. A. Kucharski,S. R. Gwaltney, S. Beck, A. Balková, D. E. Bernholdt, K. K.Baeck, P. RóSyczko, H. Sekino, C. Hober, J. Pittner, and R. J.Bartlett, 1993, ACES II program is a product of the QuantumTheory Project, University of Florida. Integral packages in-cluded are VMOL �J. Almlöf and P. Taylor�; VPROPS �P. R.Taylor�; and a modified version of the ABACUS integral de-rivative package �T. U. Helgaker, H. J. Aa. Jensen, J. Olsen,P. Jörgensen, and P. R. Taylor�.
Stanton, J. F., W. N. Lipscomb, D. H. Magers, and R. J. Bar-tlett, 1989, J. Chem. Phys. 90, 1077.
Szalay, P. G., 1997, in Modern Ideas in Coupled-Cluster Meth-ods, edited by R. J. Bartlett �World Scientific, Singapore�, pp.81–123.
Szalay, P. G., and R. J. Bartlett, 1992, Int. J. Quantum Chem.S26, 85.
Szalay, P. G., and R. J. Bartlett, 1993, Chem. Phys. Lett. 214,481.
Szalay, P. G., and J. Gauss, 2000, J. Chem. Phys. 112, 4027.Szalay, P. G., J. Gauss, and J. F. Stanton, 1993, Theor. Chim.
Acta 100, 5.Szalay, P. G., M. Nooijen, and R. J. Bartlett, 1995, J. Chem.
Phys. 103, 281.Tajti, A., P. G. Szalay, A. G. Csaszar, M. Kallay, J. Gauss, E. F.
Valeev, B. A. Flowers, J. Vazquez, and J. F. Stanton, 2004, J.Chem. Phys. 121, 11599.
Tanaka, T., and Y. Morino, 1970, J. Mol. Spectrosc. 33, 538.Taube, A., and R. J. Bartlett, 2006, Int. J. Quantum Chem. �in
press�.Thouless, D. J., 1961, The Quantum Mechanics of Many Body
Systems �Academic, New York�, p. 121.Tobita, M., S. A. Perera, M. Musiał, R. J. Bartlett, M. Nooijen,
and J. S. Lee, 2003, J. Chem. Phys. 119, 10713.Urban, M., I. Cernusak, V. Kello, and J. Noga, 1987, in Meth-
ods in Computational Chemistry, edited by S. Wilson �Ple-num, New York�, Vol. I, p. 117.
Urban, M., J. Noga, S. J. Cole, and R. J. Bartlett, 1985, J.Chem. Phys. 83, 4041.
Van Voorhis, T., and M. Head-Gordon, 2000, Chem. Phys.Lett. 330, 585.
Van Voorhis, T., and M. Head-Gordon, 2001, J. Chem. Phys.115, 7814.
Visscher, L., 1996, Chem. Phys. Lett. 253, 20.Visscher, L., J. Styszynski, and W. C. Nieuwpoort, 1996, J.
Chem. Phys. 105, 1987.Watts, J. D., and R. J. Bartlett, 1994, Int. J. Quantum Chem.
28, 195.Watts, J. D., and R. J. Bartlett, 1995, Chem. Phys. Lett. 233, 81.Watts, J. D., and R. J. Bartlett, 1996, Chem. Phys. Lett. 258,
581.Watts, J. D., J. Gauss, and R. J. Bartlett, 1992, Chem. Phys.
Lett. 200, 1.Watts, J. D., J. Gauss, and R. J. Bartlett, 1993, J. Chem. Phys.
98, 8718.Watts, J. D., S. R. Gwaltney, and R. J. Bartlett, 1996, J. Chem.
Phys. 105, 6979.Werner, H. J., and F. R. Manby, 2006, J. Chem. Phys. 124,
054114.Wick, G. C., 1950, Phys. Rev. 80, 268.Wilson, K., 1989, Ab Initio Quantum Chemistry: A Source of
Ideas for Lattice Gauge Theorists, Nuclear Physics
351Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry
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