Frozen density embedding with hybrid functionalsS. Laricchia,1 E. Fabiano,2 and F. Della Sala1,2,a�
1Center for Biomolecular Nanotechnologies (CBN) of the Italian Institute of Technology (IIT), Via Barsanti,Arnesano, I-73100 Lecce, Italy2National Nanotechnology Laboratory (NNL), Istituto Nanoscienze-CNR, Distretto Tecnologico,Università del Salento, Via per Arnesano, I-73100 Lecce, Italy
�Received 23 July 2010; accepted 8 September 2010; published online 27 October 2010�
Modeling the electronic structure of large systems �e.g.,biomolecules� is a current important research topic in quan-tum chemistry and computational material science. However,quantum simulations are often limited by the exceedinglyhigh computational cost, which is mostly related to the con-struction and/or diagonalization of the Hamiltonian matrixwhich scales as O�N3�. Direct approaches to overcome theproblem are based on linear-scaling methods,1–4 which aimat reducing of the scaling factor by taking advantage of thesparsity of the density matrix and/or avoiding direct diago-nalization of the Hamiltonian. Alternatively, a strong reduc-tion of the computational cost, at expense of accuracy, isoften obtained by employing multiscale approaches, such asthe hybrid quantum mechanics/molecular mechanics5–10 andour-own N-layer integrated molecular orbital molecularmechanics,11,12 which use different levels of accuracy fordifferent parts of the system.
A different route is followed by methods based on thesubsystem approach, which is related to the nearsightednessprinciple:13 the system is partitioned in smaller subsystemssuch that the interaction between subsystems can be includedexactly or well approximated. In the past decades differentmethods have been developed in this direction, such as thedivide and conquer method of Yang,14,15 originally derivedwithin the framework of the density functional theory16,17
�DFT� and later extended to Hartree–Fock18 �HF� andpost-HF correlation methods;19,20 the fragment molecular or-
bital method of Kitaura21–23 originally derived for HF andlater extended to DFT �Refs. 24 and 25� and to wave-function based correlation methods;26–28 the linear scalingthree-dimensional fragment approach;29,30 and severalothers.31–33
Another approach which is currently attracting growinginterest is the frozen density embedding (FDE) method, firstproposed by Senatore and Subbaswamy to study rare-gascrystal34 and by Cortona35 to study solid-state systems, andsuccessively improved by Wesolowski and Warshel36,37 whodeveloped the Kohn–Sham equations with constrained elec-tron density �KSCED�. Numerous applications of the FDEmethod have appeared in recent years, ranging from thestudy of solvated molecules,38–40 to the calculation of opticalexcitations41–43 and interaction energies,44–46 geometryoptimization,47 and simulation of molecular dynamics.48,49
For a recent review, see Ref. 50.The FDE was originally developed in the framework of
DFT making use of conventional exchange-correlation �XC�functionals based on the local densityapproximation16,17,51–53 or the generalized gradient approxi-mation �GGA�.16,17,54–56 With this choice the nonadditive XCembedding potential can be computed exactly, leaving theexpression of the nonadditive kinetic energy term as the onlyapproximation.37 However, the use of local or semilocal XCfunctionals, although it is rather successful for some weaklyinteracting systems,57,58 is not satisfactory in general becauseof their well known shortcomings, in particular for the de-scription of electronical properties59,60 of different molecularsystems and their interaction energies.45,61 Thus more accu-rate XC functionals within the FDE are required.
Jacob et al.62 pointed out that FDE can be used to treata�Author to whom correspondence should be addressed. Electronic mail:
different subsystems with different XC functionals �includ-ing orbital-dependent ones� to improve the dipole and/or thepolarizability of the subsystems: it is in fact well known thatthe description of dipole moments and polarizabilities aregreatly improved using hybrid63,64 ororbital-dependent39,64–67 DFT methods. Recently extensionsof the FDE method to wave function correlated methods,68–72
multideterminant wave function methods,73 and to thedensity-matrix functional theory74 have been proposed.
In the present article we extend the FDE formalism tohybrid functionals75,76 by developing a FDE theory in theframework of the generalized Kohn–Sham �GKS� scheme,77
which provides a formal definition of hybrid functionals inDFT. We derived the general formalism and we implementedthe method using a semilocal approximation for the resultingXC embedding potential, to study the contribution of thenon-local exact-exchange admixture to the description ofelectronic properties of different systems.
We restricted our attention to electronic properties ofweakly interacting and hydrogen bonded systems, which canbe well described by simple kinetic energyapproximations78,79 such as Thomas–Fermi16,17 and secondorder gradient expansion,16,17 and thus we do not considerthe large variety of kinetic energy approximations.79,80 Simi-larly we focused exclusively on ground-state electronic prop-erties, i.e., electron density distribution and dipole moments,and we did not consider total energies, as these are stronglydependent from the kinetic energy approximation used78 and,for the examined systems, would require the proper inclusionof dispersion interactions, which is beyond the current stateof the art for DFT.
The article in organized as follows: in Sec. II we developthe FDE theory in the framework of the GKS scheme; inSec. III we report the details of our implementation; in Sec.IV the results of FDE-GKS calculations are reported anddiscussed. Finally conclusions are drawn in Sec. V.
II. THEORY
We consider a system �total system� of N interactingelectrons, moving in a local external potential vext�r�, withdensity ��r�. We partition the total system in two subsystemswith densities �A�r� �embedded density� and �B�r� �back-ground density�, such that �A+�B=�, �B�0 and
where vemb is a local external embedding potential �to bedetermined� including the effects due to the presence of sub-system B. Minimization with respect to �A under the con-straint �3� gives the Euler–Lagrange equation
�FHK��A���A�r�
+ vextA �r� + vemb�r� = � . �7�
Because Eqs. �4� and �7� define the ground state densityof the same system, we can equate them to find the form ofthe embedding potential
vemb�r� =�FHK
nadd��A;�B���A�r�
+ vextB �r� , �8�
where
�FHKnadd��A;�B���A�r�
��FHK��A + �B�
��A�r�−
�FHK��A���A�r�
. �9�
Equation �7� cannot be used in practice to determine theground-state density �A because the functional derivative ofFHK is not known.
Here we follow the GKS �Ref. 77� scheme to obtain asingle set particle equations yielding the same density as Eq.�7�. We consider an auxiliary system represented by a single�N−NB�-electron Slater-determinant made of orthonormalembedded orbitals ��A. We define the following functionalof ��A:
S���A� � K���A� + �EEXX���A� , �10�
where the orbital dependent kinetic energy functional is
K���A� =� 2i
occ.
�iA�r��−
1
2�2��i
A�r�dr , �11�
EEXX is the exact-exchange �EXX� energy, and � is the EXXcontribution �0���1�. In this work we consider, for sim-plicity, closed-shell systems. Note that our definition of Srestricts our work to hybrid functionals, nevertheless differ-ent choices can also be made for S.77 We then define the
164111-2 Laricchia, Fabiano, and Della Sala J. Chem. Phys. 133, 164111 �2010�
direct interaction functional
FS��A� � min��A→�A
S����A� = S���minA ��A�� , �12�
where ��minA indicate the embedded orbitals which minimize
the functional FS and such that iocc.2 �i
A�r� 2=�A�r�. Theresidual interaction77 is defined as the difference between theHohenberg–Kohn and the S functional
RS��A� � FHK��A� − S���minA ��A��
= T��A� + Vee��A� − K���minA ��A��
− �EEXX���minA ��A�� , �13�
where T is the exact kinetic energy and Vee is the exactelectron-electron interaction. In conventional hybrid func-tionals Eq. �13� is approximated as
with Ex,GGA and Ec,GGA GGA versions of the residual ex-change and correlation energy, respectively.
The energy functional �6� can now be minimized, fol-lowing the constraint-search method,16,17 as
E0 = min�A→N−NB
�FS��A� + RS��A� +� �A�r��vextA �r�
+ vemb�r��dr�= min
��A�S���A� + RS�2
i
occ.
�iA�r� 2�
+� 2i
occ.
�iA�r� 2�vext
A �r� + vemb�r��dr� . �15�
The following single particle equations are thus obtained:77
�− 12�2 + �v̂x
NL + v̂J + �1 − ��v̂x + v̂c + v̂extA + v̂emb��i
A
= iA�i
A �16�
with
�v̂J�iA��r� =
�J��A���A�r�
�iA�r� =� �A�r��
r − r� dr��i
A�r� , �17�
�v̂xNL�i
A��r� = −� j
occ.� j
A�r�� jA�r��
r − r� �i
A�r��dr�, �18�
�v̂x�iA��r� =
�Ex,GGA��A���A�r�
�iA�r� , �19�
�v̂c�iA��r� =
�Ec,GGA��A���A�r�
�iA�r� . �20�
Equations �17�–�20� define the Coulomb, the nonlocal exact-exchange, the residual exchange, and correlation operators,respectively.
A. Conventional semilocal functionals
For conventional GGA functionals, we have �=0. Thus
FS��A� = min��i
A→�A
K���A� = Ts��A� , �21�
RS��A� = J��A� + Ex,GGA��A� + Ec,GGA��A� , �22�
where Ts is the usual noninteracting kinetic energy func-tional. In this case the GKS scheme reduces to the standardKohn–Sham procedure and the usual KSCED method istherefore recovered.37
Using Eqs. �21� and �22� in Eqs. �8� and �9� we can writethe embedding potential as
vemb�r� =�Ts
nadd��A;�B���A�r�
+ vJ�r;��B�� + vextB �r�
+�Ex,GGA
nadd ��A;�B���A�r�
+�Ec,GGA
nadd ��A;�B���A�r�
, �23�
where vJ�r ; ��B�� is the Coulomb potential generated by theelectron density �B and the nonadditive kinetic, exchange,and correlation potential are, respectively
�Tsnadd��A;�B���A�r�
=�Ts��A + �B�
��A�r�−
�Ts��A���A�r�
,
�Ex,GGAnadd ��A;�B�
��A�r�=
�Ex,GGA��A + �B���A�r�
−�Ex,GGA��A�
��A�r�,
�Ec,GGAnadd ��A;�B�
��A�r�=
�Ec,GGA��A + �B���A�r�
−�Ec,GGA��A�
��A�r�.
�24�
As the exact explicit dependence of Ts from the densityis not known, the nonadditive kinetic potential in Eq. �24�cannot be computed exactly. However using a GGAapproximation16,57,62,80 to the kinetic energy functional wehave
�Tsnadd��A;�B���A�r�
� vTGGA�r;��A + �B�� − vT
GGA�r;��A�� ,
�25�
where vT�r�=�Ts��� /���r� is the local kinetic potential. Notethat, within a GGA level of computation for the XC func-tional, Eq. �25� is the only approximation determining thequality of the embedding procedure: without this approxima-tion the FDE would exactly reproduce the KS density of thetotal system, as a sum of the densities of the embedded sub-systems.
B. Hybrid functionals
For hybrid functionals we have ��0 and
FS��A� = S���minA ��A��
= K���minA ��A�� + �EEXX���min
A ��A�� , �26�
164111-3 Embedding with hybrid functionals J. Chem. Phys. 133, 164111 �2010�
as the orbitals which minimize K or S are, in general, differ-ent. In fact Ts is the noninteracting kinetic energy functional�see Eq. �21��, while in GKS the electrons are �partially�interacting.77
Using Eqs. �26� and �27� in Eqs. �8� and �9� the GKSembedding potential can be written
vemb�r� = vembF �r� + vemb
R �r� , �29�
where the direct interaction part is
vembF �r� =
FS��A + �B���A�r�
−FS��A���A�r�
, �30�
=�K���min
tot ��A + �B����A�r�
−�K���min
A ��A����A�r�
+ ���EEXX���mintot ��A + �B��
��A�r�
−�EEXX���min
A ��A����A�r�
� �31�
and the residual part is
vembR �r� =
RS��A + �B���A�r�
−RS��A���A�r�
= vJ�r;��B�� + vextB �r� + �1 − ��
�Ex,GGAnadd ��A;�B�
��A�r�
+�Ec,GGA
nadd ��A;�B���A�r�
. �32�
In Eq. �31� ��mintot ��A+�B� indicates the orbitals of the total
system which yields the total density �A+�B and minimizethe functional S, i.e., the GKS orbitals of the total system.
While the XC-GGA contribution in Eq. �32� can beevaluated as described in the previous subsection, the termsin Eq. �31� are not explicit functionals of the density and thefunctional derivative cannot be carried out directly. Note alsothat the first two term terms in Eq. �31� differs from thenonadditive kinetic energy potential of Eq. �24�, as Eq. �28�holds.
The fourth term in Eq. �31� is proportional to the localEXX potential81 and it can be computed using the optimizedeffective potential �OEP� method81 or its approximatedforms.82–85 Thus in the FDE-GKS, while the nonlocal EXXoperator is considered in the embedded subsystem, a localEXX potential is used for the construction of the embeddingpotential; see also Ref. 74.
The third term in Eq. �31� resembles a local EXX poten-tial but requires the orbitals ��min
tot ��A+�B� of the total sys-tem, not available in FDE. One possibility is to use ap-proaches based on the inverse-Kohn–Sham,85,86 already usedfor FDE,87–89 to compute the orbitals which yield the density�A+�B: this is however not a trivial task. As both techniques
�inverse-method and OEP� are numerically complicated andcomputationally very demanding,87,88,90 approximations arerequired.
In particular, Eqs. �26� and �30� suggest that approxima-tions for the kinetic and exchange terms together can be in-troduced. Note that the need of an additional approximationfor both the kinetic and EXX terms is not fortuitous, butreflects the conjointness conjecture80,91,92 which relates thekinetic and exchange functionals. Such kinetic-exchangefunctionals have not been yet investigated, but the presentFDE-GKS theory suggests future research in this direction.Using currently available approximations we can set in Eq.�26�
FS��� � TsGGA��� + �Ex,GGA��� �33�
so that Eq. �31� becomes
vembF �r� � vT
GGA�r;��A + �B�� − vTGGA�r;��A��
+ �vx,GGA�r;��A + �B�� − �vx,GGA�r;��A�� .
�34�
In this way a practical computational embedding scheme forhybrid functionals is obtained. The quality of the embeddingprocedure depends now on two approximations: the approxi-mate expression of the kinetic term and exchange term in Eq.�34�. Note in addition that current approximations to the ki-netic energy functional are related to the noninteracting ki-netic energy, and not to the partially interacting K, see Eq.�28�. The two approximations might compensate each other,possibly on the ground of the conjointness conjecture be-tween the exchange and the kinetic functionals.80,91,92
Note that the employed approximations only enter thedefinition of the embedding potential �34�, while the singleparticle Eq. �16� are still solved using the exact nonlocalexchange functional for both �active and frozen� subsystems.Therefore the electronic structure of the subsystems and theirresponse to the interaction potential are computed using thenonlocal exchange operator. This implies that electronicproperties, and in particular those related to the response ofthe subsystem to an external perturbation �e.g., dipole mo-ments�, can be expected to be described significantly betterwith the present embedding scheme, which takes advantageof hybrid functionals, than in other KSCED implementationswhich are limited to GGA functionals.
III. COMPUTATIONAL METHOD
We implemented the embedding method described inSec. II in a development version of the TURBOMOLE
program.93 In the present work only the case of two sub-systems was considered. Extension to multiple subsystems isstraightforward.
Full relaxation of the density of both subsystems wasobtained by using a freeze-and-thaw procedure,94 until dipolemoments converged to 10−3 a.u.. Note that this is a tightconvergence criterion, as the dipole moment is highly sensi-tive to small changes in electron density. The required matrixelements of the nonadditive kinetic and XC potentials werecomputed by numerical integration on the total system mo-lecular grid.
164111-4 Laricchia, Fabiano, and Della Sala J. Chem. Phys. 133, 164111 �2010�
The calculations were performed using the B3LYP,56,75
BHLYP,95 and PBE0 �Ref. 76� hybrid functionals, and thecorresponding GGA functionals BLYP �Refs. 55 and 56� andPBE �Ref. 54� for comparison.
To compute the embedding potential, the kinetic energyterm was approximated using the regular gradientexpansion16 truncated either at the zeroth order �GEA0�, cor-responding to the Thomas–Fermi functional,16 or at the sec-ond order �GEA2�. The difference between GEA2 and GEA0is 1/9 of the von Weizsäcker kinetic functional.16,37 In TableS1 �Ref. 96� we also report results for the LC94 kineticfunctional.97
In case of hybrid functionals, the exact-exchange term inEq. �31� was approximated using the Becke-exchangefunctional55 for B3LYP and BHLYP and the PBE exchangefunctional54 for PBE0. This is a natural choice, but differentGGA exchange potential can be used in Eqs. �34� and �32�.Note that with this choice the computation of the XC contri-bution to the embedding potential simplifies to
vembxc �r;��A,�B�� = vxc
GGA�r;��A + �B�� − vxcGGA�r;��A�� .
�35�
Thus in the present implementation a multilevel formalism isobtained: a hybrid functional is used for the computation ofall subsystems, while the corresponding nonhybrid func-tional is used for the computation of the embedding poten-tial. As pointed out in Ref. 62, the FDE can also be used toconsider different XC functionals in different subsystems.
In addition, reference values for the dipole moments ofthe investigated total systems and subsystems were com-puted at the coupled cluster singles and doubles �CCSD�level by using the finite field method. All calculations wereperformed with the TURBOMOLE program.93
A. Geometries
In this work, we studied the ground-state electronicproperties of a set of six weakly interacting and/or hydrogen-bonded systems, including H2–NCH, HF–NCH, benzene-HCN in T-shape configuration,98 microhydrated complex ofthymine with six water molecules99 �T-6H2O�, guanine-cytosine �GC�, and adenine-thymine �AT� base pairs in theirWatson-Crick arrangement.98 These systems constitute a rep-resentative selection of typical systems of interest for embed-ding applications. The geometries of the investigated systemswere taken from literature,98,99 with the exception of the lin-ear H2–NCH and HF–NCH complexes that were optimizedat the B3LYP/TZVPP level56,75,100 using TURBOMOLE.93
B. Analysis of the electronic density
To analyze the quality of the embedding calculations wecompared the deformation densities resulting from the GKScalculation on the total system and the embedding procedure.The analysis of the deformation densities was frequently em-ployed to assess the approximations made within the FDEscheme.89,101–103
They are defined as
�GKS�r� � �totGKS�r� − �A�r� − �B�r� , �36�
�emb�r� � �Aemb�r� + �B
emb�r� − �A�r� − �B�r� , �37�
where �totGKS is the electron density of the total system, �A and
�B are the electron densities of the two �isolated� subsystems,and �A
emb and �Bemb are the total densities of the embedded
subsystems. The embedding-density error is defined as
�err�r� � �emb�r� − �GKS�r�
= �Aemb�r� + �B
emb�r� − �totGKS�r� �38�
and it is thus a measure of the error in the density due to theembedding procedure.
Connected to the density, we considered the bond di-poles
�� GKS � �� totGKS − �� A − �� B, �39�
�� emb � �� Aemb + �� B
emb − �� A − �� B �40�
and the corresponding dipole error
�� err = �� emb − �� GKS. �41�
In addition we considered the plane-averaged valuesalong a principal axis �z� of the system
�̄GKS�z� =� �GKS�x,y,z�dxdy , �42�
�̄emb�z� =� �emb�x,y,z�dxdy , �43�
�̄err�z� =� �err�x,y,z�dxdy . �44�
As a global test on the density we consider the relative error
� =1
V� �err�r�
�totGKS�r�
� 1000, �45�
where V is the volume �enclosed by the quadrature grid� ofthe total system. Note that � is an adimensional number. The� is very sensible to the density variation, in particular in thenear asymptotic region, i.e., in the bonding region betweenweakly bonded systems. To avoid numerical problems in thefar asymptotic region we limit the values of the density to10−5 a.u..
A more detailed analysis was also obtained by partition-ing the grid points into atomic basins j according to a grid-based atoms-in-the-molecule procedure,104 implemented as astandalone tool interfaced with TURBOMOLE. Theembedding-density relative error for the jth atom was thendefined as
� j =1
Vj�
j
��r��r�
�totGKS�r�
� 1000. �46�
The following relation between Eqs. �45� and �46� holds:
� =1
Vj=1
N
Vj� j �47�
with N the number of atoms.
164111-5 Embedding with hybrid functionals J. Chem. Phys. 133, 164111 �2010�
C. Basis set
All calculations were performed using the TZVPP �Ref.100� basis set. The calculations on embedded subsystemswere performed considering only basis functions centered onthe subsystem’s atoms. This choice is appropriate for thepresent investigation, where only weakly interacting systemsare considered �small overlap between the two subsystems�,and the relatively large basis set employed for each sub-system is fully capable to represent the subsystem electrondensity.62,102,105 Additional basis functions centered on back-ground atoms �or a complete supermolecule basis set� mightbe needed in systems with charge transfer79,105 and/or withstrong hydrogen bonds, such as in F–H–F− �Ref. 102�, toaid the �partial� description of intramolecular chargetransfer.106,107 In other cases a supermolecule basis set is notstrictly needed and can even result in an enhancement of theshortcomings of the employed approximated kinetic energyfunctionals.62,106
To test the role of basis set effects on our results weperformed calculations using the TZVPP, the aug-TZVPPand the aug-QZVPP basis sets: the augmented basis sets havebeen obtained by adding the diffuse basis functions of thecorrelation consistent basis set108 to the TZVPP �QZVPP�basis set.
In Fig. 1 we report the difference between the embed-ding and the GKS deformation density resulting from BH-LYP and BLYP calculations. The addition of diffuse func-tions yields a moderate improvement in the bond region andis negligible in the subsystems. Moving from a triple- to aquadruple-zeta basis set has almost no effects. More impor-tantly, the trends between different methods are not affectedby the increase of the basis set: results for BHLYP showsmaller error than BLYP for all basis set. We also report aquantitative density analysis in Table S1 �Ref. 96� for differ-ent basis sets and with the LC94 kinetic functional.
Therefore, the use of a supermolecular or a very large�and computationally very expensive, when hybrid func-tional are considered� basis sets is not required in the presentwork, which mainly deals with a method comparison.
IV. RESULTS AND DISCUSSION
Table I lists the magnitude of bond dipoles of investi-gated total systems and the magnitude of dipole moments ofthe corresponding isolated subsystems, computed with differ-ent XC functionals. Reference CCSD results are also re-ported; bold and italic styles indicate the DFT value in bestand worst agreement with the reference, respectively.
Inspection of Table I shows that conventional GGA func-tionals underestimate the dipole moments for all the �rela-tively small� molecules considered. This is an opposite trendwith respect to what was found for larger conjugatemolecules.60,67
Larger values, in better agreement with the CCSD refer-ence, are found for hybrid functionals, with BHLYP yielding
-10 -8 -6 -4 -2 0 2 4 6
-0.01
0
0.01Δρ
err
(a.u
.)
BHLYP/TZVPPBHLYP/aug-TZVPP
BHLYP/aug-QZVPP
-10 -8 -6 -4 -2 0 2 4 6Position (Bohr)
-0.01
0
0.01
Δρer
r(a
.u.)
BLYP/TZVPPBLYP/aug-TZVPP
BLYP/aug-QZVPP
F H
N C H
F H
N C H
FIG. 1. HF–NCH complex: plane-averaged embedding density error�̄err�z� resulting from BHLYP �upper panel� and BLYP �lower panel� cal-culations with the TZVPP, aug-TZVPP, and aug-QZVPP basis set. TheGEA2 kinetic functional was used in embedding calculations.
TABLE I. Magnitude of dipole moments ��� and bond dipoles ��� computed with different XC DFT func-tionals and the CCSD method. Dipole moments refer to isolated subsystems. All values are in Debye. Bold anditalic styles indicate the DFT value agreeing best and worst with CCSD, respectively.
164111-6 Laricchia, Fabiano, and Della Sala J. Chem. Phys. 133, 164111 �2010�
the best performance for all systems. It is well known thatfunctionals with larger nonlocal HF contribution yield betterdipole moments.63
One apparent exception is observed for adenine. Thisdepends however on the fact that the table only reports thevalue of the magnitudes of dipole moments, and does notprovide any information about the direction of the dipolemoment vector. A better comparison with the reference isthus obtained by calculating the norm of the vectorial differ-ence between DFT and CCSD dipoles, i.e., �� CCSD−�� DFT .This is reported for the DNA bases in Fig. 2.
The largest deviations are in this case always observedfor the GGA functionals and a clear improvement is recog-nized when larger amounts of nonlocal exact-exchange areincluded in the functional. The BHLYP functional yields thebest agreement with the reference for all bases, with a maxi-mum deviation smaller than 0.05 D. In particular, a remark-able improvement of BHLYP with respect to BLYP is foundfor cytosine.
For the bond dipoles the comparison with CCSD valueswas only possible for the smallest systems, since for DNAbase pairs and the T-6H2O the CCSD calculations were un-affordable. The results in Table I show that for the linearHF–NCH and H2–NCH systems the GGA functionals givean overestimation of the bond dipole while smaller values,closer to the reference, are obtained with hybrid functionals.An opposite trend, although very weak, is instead observedfor the C6H6–HCN system. These results are rationalized forthe linear HF–NCH and H2–NCH systems in terms of anoverestimated longitudinal polarizability of HF, NCH, andH2 at the GGA level59,109 �see Table II�.
In fact, in first approximation we can consider the bonddipole as arising from the polarization of each subsystemunder the dipole moment of the other one, so that an over-estimation of the polarizability leads to overestimated bonddipoles. For the C6H6–HCN system instead a partial depo-larization effect occurs because the two systems are not col-linear but on T-shape configuration. The depolarization ishowever smaller for hybrid functionals �which also havehigher HCN dipoles� because of a smaller polarizability ofbenzene at this level of theory. Therefore, the bond dipole isslightly larger for hybrid functionals than for GGA.
The use of hybrid functionals appears therefore to be offundamental importance to accurately describe the electronicproperties of interacting subsystems. In particular, the inclu-sion of nonlocal exact-exchange strongly improves both thedescription of the electronic structure and the response of asubsystem to the perturbations induced by the other one. Theembedding scheme described in Sec. II, including hybridfunctionals, might thus be expected to provide accurate re-sults in this context. However, the accuracy of approximation�34� must be first verified.
In order to assess the accuracy of our GKS embeddingscheme, we start with the comparison �see Fig. 3� between�GKS�r� and �emb�r� �computed along intermolecularbond axis� for the well studied HF–NCH complex. Similarresults �not reported� were found for the other systems.
In the figure negative �positive� � values indicate thereduction �increase� of the electron density following the for-mation of the complex. The plot is directly comparable withFig. 5 of Ref. 57. Our GKS embedding procedure yields a
System0
0.1
0.2
0.3
0.4|μCCSD-μ|(Debye)
BHLYPB3LYPBLYPPBE0PBE
A T C G
FIG. 2. Values of �� CCSD−�� DFT for the DNA bases as computed with dif-ferent DFT functionals.
TABLE II. Polarizability �a.u.� along the molecular axes of HCN, HF, and H2 computed with different DFTmethods.
FIG. 3. HF–NCH complex: deformation densities along the main axis, re-sulting from GKS calculation of the total system ��GKS� and from the FDEcalculation ��emb� using the BHLYP XC-functional. The GEA2 kineticfunctional was used in embedding calculations.
164111-7 Embedding with hybrid functionals J. Chem. Phys. 133, 164111 �2010�
result in very good agreement with the full GKS calculation,with only a small underestimation of the deformation densitynear the F atom.
To highlight the differences between embedding resultsobtained using functionals including different amounts ofnonlocal exact-exchange, and thus to analyze the role of suchcontributions in the embedding, we plot in Fig. 4 the �̄err�z��see Sec. III B for definition� resulting from the BHLYP,B3LYP, and BLYP embedding simulations of the HF–NCHsystem. Note that this plot differs from that of Fig. 3: hereplane-averaged quantities are considered. Observation of thefigure reveals that all simulations yield similar deformationdensities and the differences between them are much lowerthan the difference between embedding and GKS deforma-tion densities. This indicates that the main source of inaccu-racy in the embedding procedure has to be related to the�common� approximation of the kinetic term and that only aminor role is played by the additional approximation of non-
additive exchange used in Eq. �34�. In particular, an overes-timation of the density at nitrogen lone pair and an underes-timation in the H–F bond is observed. The use of hybridfunctionals, although introducing an additional approxima-tion in the embedding procedure, results, unexpectedly, in aslight improvement, with BHLYP performing better thanB3LYP and the latter outperforming BLYP. This observationmight indicate that an error cancellation probably occurs be-tween the kinetic and exchange non-additive terms.
In Table III we report the magnitude of bond-dipole er-rors and the relative errors in embedding densities for all thesystems considered. Results for both the GEA2 and GEA0kinetic energy functionals are shown.
Similar conclusions, as those drawn above for the HF–NCH complex, apply to all systems reported. The quality ofthe embedding is in general improved by the hybrid func-tionals, i.e., the differences between full GKS and embed-ding results are smaller for hybrid functionals. The meanabsolute error �MAE� on the bond dipoles �computed usingGEA2 kinetic energy functional� reduces from 0.125 to0.077 D in going from BLYP to BHLYP and from 0.127 to0.097 D going from PBE to PBE0. Larger errors are foundfor the GEA0 kinetic functional but the trends between XC-functional is fully preserved, highlighting the independenceof our discussion from the employed kinetic approximation.We also obtained the same conclusions with the LC94 ki-netic functional, see Table S1.96
Similarly, the MAE of relative errors � in the embeddingdensities reduces from 0.60 to 0.51 when BHLYP is usedinstead of BLYP, and from 0.61 to 0.52 when PBE0 is usedin place of PBE. The � MAE for B3LYP and BHLYP arequite close, and somehow smaller values are obtained withthe GEA0 kinetic functional. The results of embedding cal-culations using BHLYP are always the closest to GKS resultsfor the bond-dipoles and the best for most embedding-density errors.
The only exceptions are found for the H2–NCH com-
-10 -8 -6 -4 -2 0 2 4 6Position (Bohr)
-0.015
-0.01
-0.005
0
0.005
0.01
0.015Δρ
err
(a.u
.)BHLYPB3LYPBLYP
F H N C H
FIG. 4. HF–NCH complex: plane-averaged embedding density error�̄err�z� from BHLYP, B3LYP, and BLYP embedding simulations. TheGEA2 kinetic energy approximation was used for the embeddingcalculations.
TABLE III. Magnitude �Debye� of the bond-dipole errors � �� err = �� emb−�� GKS � and relative error in embedding densities � �see text for definition� fordifferent XC-functionals and kinetic functionals. The MAEs are also reported. The smallest errors are in bold.
164111-8 Laricchia, Fabiano, and Della Sala J. Chem. Phys. 133, 164111 �2010�
plex and for the AT base pair. In these cases the large amountof nonlocal exchange �50%� included in BHLYP appears tobe excessive to guarantee a good description of the ground-state electronic properties. On the other hand, an improve-ment with respect to GGA is found when a more limitedamount of nonlocal exchange is considered, as in B3LYP andPBE0.
To conclude our analysis we examine in closer detail theGC system, which was also a test-system in previous FDEstudies.79,102,105 This is in fact a relatively large system andrepresents a tighter test for our GKS embedding methodbased on nonlocal hybrid functionals. The GC base pair ischaracterized by three strong hydrogen bonds, therefore theinteraction between the two subsystems results from a com-plex pattern. From data of Table III we know that the BH-LYP functional yields overall the best embedding density�and bond dipole� for this system, with a significant improve-ment over BLYP. This finding is confirmed by the plot re-ported in Fig. 5 where �̄GKS and �̄emb computed along thebonds direction are compared.
The two deformation densities agree very well except fora small underestimation of the H-bond polarization by theBHLYP-embedding calculation.
A deeper understanding is provided by inspection of Fig.6, which reports the relative embedding-density errors forindividual atoms. The main contribution to the embedding-density error is given by atoms at the interface between thetwo subsystems, which are the most perturbed by the basepair formation. The BHLYP functional yields in this casesmaller errors with respect to BLYP, in particular for the Hatoms, possibly because of its superior ability to describe theH-bond polarization. Smaller errors are found on other at-oms, where BHLYP and BLYP yield roughly the same re-sults, with slightly better BHLYP results. Exceptions arefound on few hydrogen atoms not involved in H-bonding. Inthese cases rather large errors are occasionally found at ei-ther the BHLYP or the BLYP level, related to oscillations inthe tail of the density. Note however that these atoms giveonly a minor contribution to the total error because of theirlow weight in the sum of Eq. �47�, thus these oscillations arealmost negligible.
V. CONCLUSIONS
In this work we introduced the GKS-FDE method. Equa-tion �31� defines how to construct the embedding potential inthe case of global hybrid functionals: for an exact construc-tion inverse-GKS methods and the OEP scheme is required.
The approximation introduced in Eq. �34� for the nonad-ditive nonlocal exchange embedding term, provides a practi-cal and efficient tool for accurate subsystem DFT calcula-tions. The possibility of using hybrid functionals allows infact to accurately describe ground-state electronic propertiesof a large variety of systems, overcoming the limitationstypical of GGA functionals.
Our results show that freeze-and-thaw embedding calcu-lations with hybrid functionals can well reproduce the corre-sponding GKS calculations on the total system. This assessesthe quality of the embedding procedure and provides a vali-dation of the underlying approximations. Moreover, wefound that our GKS scheme leads to smaller density-embedding error the one employing GGA functionals. Thissuggests that an error cancellation probably occurs in theformer case because both the partially interacting kinetic en-ergy and nonlocal exchange term are approximated at thesame level, resembling the conjointness conjecture.80,91,92
Thus the presented FDE-GKS theory suggests this path asfuture development of nonadditive kinetic-exchange func-tionals.
Our GKS embedding method opens the way to embed-ding calculations using hybrid functionals. This will allow toextend the applicability of the frozen density formalism andto increase the accuracy of embedding calculations for com-plex systems. In addition, the GKS embedding theory pre-sented in this work provides a firm ground for recent embed-ding applications, where orbital-dependent methods wereused in combination with a GGA embedding potential62 andconstitutes the basis for future developments beyond hybridfunctionals, based on different choices of the direct and re-sidual interaction terms in the GKS scheme.
ACKNOWLEDGMENTS
We thank TURBOMOLE GmbH for providing the TURBO-
MOLE program package and M. Margarito for technical sup-
-12 -8 -4 0 4 8 12Position (Bohr)
-0.1
-0.05
0
0.05Δρ
(z)
(a.u
.)GKSemb
FIG. 5. Plane-averaged deformation densities ��̄�z�� resulting from GKSand BHLYP-embedding calculations for the GC system. The GEA2 kineticenergy approximation was used for the embedding calculations.
FIG. 6. Relative error in embedding density � j for individual atoms of theGC system. Values for the BHLYP simulation are reported; values from theBLYP simulation are reported in brackets. The guanine molecule in theright, the cytosine on the left. Colors of atoms: carbon, cyan; nitrogen, blue;oxygen, red; hydrogen, white.
164111-9 Embedding with hybrid functionals J. Chem. Phys. 133, 164111 �2010�
port. This work was funded by the European Research Coun-cil �ERC� Starting Grant FP7 Project DEDOM �Grant No.207441�.
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