Computational Chemistry Ab initio methods seek to solve ...jtaylor/teaching/Fall2010/... · Quantum...

Quantum Chemistry Theory

Computational Chemistry

Ab initio methods seek to solve the Schrodinger equation.

Molecular orbital theory expresses the solution as a linear combinationof atomic orbitals.Density functional theory (DFT) attempts to solve for the electrondensity function.

Semi-empirical methods use approximate Hamiltonians that are partlyparameterized using experimental data.

Molecular mechanics uses Newtonian mechanics and empiricallyparameterized force fields.

Jay Taylor (ASU) APM 530 - Lecture 3 Fall 2010 1 / 56


The Time-Dependent Schrodinger Equation

The time-dependent Schrodinger equation can be written as

i~∂Ψ(x , t)

∂t= HΨ(x , t)

where

~ = 1.055× 10−34J s is the reduced Planck’s constant;

the operator H is the system Hamiltonian;

Ψ(·, t) is the wavefunction of the system at time t.

The modulus of the wavefunction, |Ψ(·, t)|2, is often interpreted as theprobability density of the system at time t.



The Time-Independent Schrodinger Equation

If H is time-indepedent, then we can seek an eigenfunction expansion

Ψ(x , t) =∞∑

n=1

cnΨn(x)e−i En~ t

where Ψn satisfies the eigenvalue problem

HΨn = EnΨn

and En is interpreted as the energy of the stationary configuration Ψn.The eigenfunction Ψ0 corresponding to the smallest eigenvalue E0 is calledthe ground state of the system.



The Hamiltonian of a Molecule

The Hamiltonian H can be written as the sum of the kinetic and potentialenergy operators

H = T + V,

where for a (non-relativistic) molecule

T = −~2

2

∑i

1

mi

(∂2

∂x2i

+∂2

∂y2i

+∂2

∂z2i

)V =

1

4πε0

∑i<j

eiej

rij(electrostatic potential).

Here i , j range over nuclei and electrons, mi and ei are the mass andcharge of the i ’th particle, and ε0 is the vacuum permittivity.


Quantum Chemistry Hydrogen-like atom

Hydrogen-like Atom

In a few cases, Schrodinger’s equation can be solved analytically. For asingle nucleus with mass M and charge +Ze with a single electron (ofmass m), we have:(

− ~2

2(M + m)∇2

CM −~2

2µ∇2 − Ze2

4πε0r

)Ψ = EΨ,

where

∇2CM is the Laplacian for the center-of-mass coordinates

RCM =Mra + mre

M + m

∇2 is the Laplacian for the coordinates of the electron relative to thenucleus: r = re − ra

µ is the reduced mass µ = (M−1 + m−1)−1 ≈ m.



Reduction of Dimension

Because the potential energy depends only on the distance of the electronto the nucleus, the center-of-mass can be separated out of the previousequation, leaving

− ~2

2µ∇2Ψ− Ze2

4πε0rΨ = EΨ

where r = |r|. Transforming to polar coordinates (r , θ, φ) gives

− ~2

2m

1

r2 sin(θ)

[sin(θ)

∂

∂r

(r2 ∂

∂r

)+

∂

∂θ

(sin(θ)

∂

∂θ

)+

1

sin(θ)

∂2

∂φ2

]Ψ

− Ze2

4πε0rΨ = EΨ.



Separation of Variables

The last equation can be solved by separation-of-variables and leads tothe following set of eigenfunctions and eigenvalues

Ψnlm(r , θ, φ) = CnlRnl(r)Y ml (θ, φ)

Enlm = −(

Ze2

2ε0h

)2µ

2n2,

which are indexed by

the principal quantum number n = 1, 2, 3, · · · ;the azimuthal quantum number l = 0, 1, 2, · · · , n − 1;

the magnetic quantum number m = −l ,−l + 1, · · · , l − 1, l .



Radial Component

The radial part of the wavefunction is

Rnl(r) =

(2r

n

)l

L2l+1n+l

(2r

n

)e−r/n

where

the associated Laguerre polynomial Lβα(x) is defined by

Lβα(x) =dβ

dxβLα(x)

Lα(x) = ex dα

dxα[xαe−x

]r = (µe2Z/mε0h

2)r ;

Cnl is a normalization constant chosen so that∫|Ψnlm|2dr = 1.



Angular Component

The angular part of the wavefunction is a spherical harmonic:

Y ml (θ, φ) = P

|m|l (cos(θ))

1

2πe imφ,

where the associated Legendre polynomial Pml (x) is defined by

Pml (x) = (1− x2)|m|/2

d |m|

dx |m|Pl(x)

Pl(x) =1

2l l!

d l

dx l

[(x2 − 1)l

].

Notice that, in general, this function is complex except when l = 0.



s-orbitals

When l = 0, we must also have m = 0. In this case, the wavefunctionΨn,0,0 is radially-symmetric and is said to represent the ns orbital.

Notice that as n increases:

the energy increases;

the diffuseness of theorbital also increases;

nodes appear.



p-orbitals

For larger values of l , radial symmetry is lost. For example, if l = 1, thenm = −1, 0, 1 and there are three p orbitals that can be aligned along thecoordinate axes. If n = 2, then these have the following appearance:


Quantum Chemistry Separation of timescales

Born-Oppenheimer Approximation

Since analytical solutions are usually unavailable, we try to simplify theproblem. The Born-Oppenheimer approximation assumes that:

the nuclei are fixed on the timescale of electronic motion (since nucleiare much heavier than electrons);

we can treat the electronic wavefunction and energy as functions ofthe nuclear coordinates R;

the total wavefunction can be factored as

Ψtot(r,R) = Ψel(r,R)Ψnuc(R).

This leads to the electronic Schrodinger equation:

Hel(R)Ψel(r,R) = E el(R)Ψel(r,R).



The Electronic Hamiltonian

The electronic Hamiltonian Hel is equal to the sum of the electronickinetic energy and the coulomb potential energy:

T el = − ~2

2m

elec∑i

(∂2

∂x2i

+∂2

∂y2i

+∂2

∂z2i

)

Vel =1

4πε0

− elec∑i

nuc∑s

Zse2

ris+

elec∑i<j

e2

rij

where Zi is the atomic number of the i ’th nucleus.



Effective Potential Energy Surface

Typically, we are interested in the electronic ground state energy, E el0 (R),

which can be used to estimate the potential energy surface for the nuclearconfiguration:

Enuc(R) ≈ E el0 (R) +

1

4πε0

nuc∑s<t

ZsZte2

Rst

This can be used to:

identify the minimum energy molecular structure;

characterize reaction trajectories;

solve for the nuclear wavefunction: HnucΨnuc(R) = EtotΨnuc(R).

However, even with this approximation, the Schrodinger equation stillmust be solved numerically.


Quantum Chemistry Molecular orbital theory

Molecular Orbital Theory

Molecular orbital theory attempts to approximate the full wavefunctionΨ using a collection of one electron functions called spin orbitalsχ(x , y , z , ξ). These usually have the form

ψ(x , y , z)α(ξ) or ψ(x , y , z)β(ξ)

where ξ denotes the spin angular momentum of the electron along thez-axis and the spin wavefunctions for spin-up and spin-down particles are

α

(+

1

2

)= 1, α

(−1

2

)= 0

β

(+

1

2

)= 0, β

(−1

2

)= 1.



Fermi-Dirac Statistics

Because electrons have half-integer spin, the following two conditionsmust be satisfied:

The wavefunction must be antisymmetric: if πij(X) denotes theconfiguration obtained by permuting the positions and spins of thei ’th and j ’th electrons, then

Ψ(πij(X)) = −Ψ(X).

Pauli exclusion principle: no two electrons can occupy the samespin orbital function. Thus each spatial orbital function can containat most two electrons, one with spin +1/2 and the other with spin−1/2.



Slater Determinants

Given a collection of linearly independent spin orbitals, χ1, · · · , χn, anantisymmetric n-electron wavefunction can be constructed by setting

Ψdet =1√n!

∣∣∣∣∣∣∣∣∣χ1(X1) χ2(X1) · · · χn(X1)χ1(X2) χ2(X2) · · · χn(X2)

......

...χ1(Xn) χ2(Xn) · · · χn(Xn)

∣∣∣∣∣∣∣∣∣=

1√n!

∑π∈Sn

(−1)|π|n∏

i=1

χi (Xπ(i))

where Xi = (xi , yi , zi , ξi ) denotes the location and spin of the i ’th electron.Ψdet is called a Slater determinant.



Basis Set Expansions

In practice, the molecular orbitals ψi (x) are expressed as linearcombinations of one-electron functions known as basis functions,

ψi (x) =N∑

s=1

csiφs(x), 1 ≤ i ≤ N,

where the csi are the molecular orbital expansion coefficients.

Notice that

2N ≥ n, if we allow for double occupancy of orbitals;

if 2N > n, then there are unoccupied orbitals that do not appearin the Slater determinant.



Linear Combination of Atomic Orbitals

Usually, the basis elements φs are taken to be ‘atomic orbitals’ associatedwith individual nuclei present in the molecule (LCAO method). In thiscase, a set of orbitals is assigned to each nucleus such that these orbitals

are centered at the nucleus;

depend on the type of nucleus (e.g., H vs. C);

allow for symmetric and polarized electron distributions.

It is often the case that several such basis elements are used to representan ‘actual’ atomic orbital, for example, by allowing for different degrees ofdiffuseness of the orbital about the nucleus.



Slater-type atomic orbitals

One choice for the basis functions are the Slater-type orbitals (STOs):

φ1s =

(ζ31

π

)1/2

exp(−ζ1r)

φ2s =

(ζ52

96π

)1/2

r exp(−ζ2r/2)

φ2px =

(ζ52

32π

)1/2

x exp(−ζ2r/2)

...

where ζ1, ζ2, · · · are parameters that control the size of the orbitals. WhileSTOs are good approximations for atomic orbitals, integrals of theirproducts must be evaluated numerically.



Gaussian-type atomic orbitals

Gaussian orbital functions are polynomials multiplied by exp(−αr2):

gs =

(2α

π

)3/4

exp(−αr2)

gx =

(128α5

π3

)1/2

x exp(−αr2)

gy =

(128α5

π3

)1/2

y exp(−αr2)

...

Although integrals involving Gaussian-type functions can be evaluatedexplicitly, these functions are poor approximations for atomic orbitals.



Contracted Gaussians

A compromise can be reached by using linear combinations of Gaussianfunctions (called contracted Gaussians):

φs(x) =∑α

dsα gα(x)

where each gα(x) is a Gaussian function and the coefficients dsα are fixedin advance, i.e., do not depend on the Hamiltonian. One approach is tochoose the coefficients to minimize the least squares distance betweenthe contracted Gaussian and a STO:

εnl =

∫ (φSTO

nl − φCGnl

)2dx.


Quantum Chemistry Hartree-Fock theory

Hartree-Fock Theory

Having chosen a set of basis functions, Hartree-Fock theory seeks toestimate the ground state energy E0 by finding a determinantalwavefunction Φ that minimizes the quantity:

E ′ =

∫Φ∗HΦdX ≥ E0.

This leads to variational conditions on the expansion coefficients

∂E ′

∂cri= 0 1 ≤ r , i ≤ N.

Thus, we have reduced an infinite-dimensional linear problem to afinite-dimensional non-linear one (see below).



Roothan-Hall Equations

For a closed-shell system (i.e., two electrons per orbital), the variationalconditions lead to a system of non-linear equations

N∑s=1

(Frs − εiSrs)csi = 0 1 ≤ r , i ≤ N,

where εi is the one-electron energy of molecular orbital ψi and Srs is theoverlap of the atomic orbitals φr and φs :

εi =

∫ψ∗i (x)Hψi (x)dx

Srs =

∫φ∗r (x)φs(x)dx.



The Fock matrix

The Fock matrix F = (Frs) is defined as

Frs = hrs +AO∑p,q

Pqp

[(rp|sq)− 1

2(rp|qs)

]where the summation goes over atomic orbitals and hrs is a component ofthe one-electron energy in a field of bare nuclei:

hrs =

∫φ∗r (x)h(1)φs(x)dx

h(1) = − ~2

2m∇2 − 1

4πε0

Nuc∑a

Za

ra1.



Bond-order Matrix

The matrix P = (Ppq) is called the bond-order matrix or theone-electron density matrix:

Pqp = 2MO∑

i

c∗picqi ,

where the summation is over occupied molecular orbitals and the factor of2 reflects the closed-shell assumption.



Two-electron Integrals

Much of the computational burden in the H-F theory comes from the needto compute the two-electron repulsion integrals

(rp|sq) =

∫ ∫φ∗r (x1)φ∗p(x2)

(1

r12

)φs(x1)φq(x2)dx1dx2,

where r12 = |x2 − x1|.

Because the elements of the Fock matrix are themselves functions of themolecular orbital expansion coefficients, the Roothan-Hall equations arenon-linear and must be solved iteratively (e.g., Newton-type methods).



Self-Consistent Field Method

A simple iterative scheme for solving for the coefficient matrix c is:

1 Use c(n) to form the bond-order matrix P(n).

2 Use P(n) to form the Fock matrix F(n).

3 Find c(n+1) and ε(n+1) by solving the secular equation(F(n) − ε(n+1)S

)c(n+1) = 0.

4 Choose the N/2 molecular orbitals of lowest energy to be occupied.

5 Repeat steps (1)-(4) until ||c(n+1) − c(n)|| < δ.


Quantum Chemistry Electron correlation

Electron Correlation

There are two sources of negative correlation between the locations ofdifferent electrons.

The Coulomb hole results from electrostatic repulsion.

The exchange hole is a consequence of the Pauli exclusionprinciple: electrons with the same spin cannot occupy the sameorbital.

The most important limitation of the H-F method is that it fails toaccount for the Coulomb hole. This is particularly problematic whenmodeling large molecules, for which about half of the interaction energycan be due to electron correlation.



Hartree-Fock wavefunction of H2

Example: The H-F wave function for the hydrogen molecule H2 withdouble occupancy of a single molecular orbital φ(·) is

ψ(1, 2) =1√2

∣∣∣∣ χ1(1) χ2(1)χ1(2) χ2(2)

∣∣∣∣=

1√2φ(x1)φ(x2)

[α(ξ1)β(ξ2)− β(ξ1)α(ξ2)

],

where (xi , ξi ) is the location and spin of electron i , ξ2 = −ξ1, and α(·)and β(·) are the spin-up and spin-down functions introduced earlier.



The H-F wave function for H2 neglects electronic correlation.

The joint probability density of the locations of the two electrons is

|ψ(1, 2)|2 = Cφ2(x1)φ2(x2)

where C is a normalizing constant. Since this density factors into theproduct of two one-electron densities, it follows that under the H-F wavefunction, the locations of the two electrons are independentof one another.

This result is unphysical, but is also observed in H-F calculations formore complicated molecules.



Post-Hartree-Fock Methods

Several methods have been devised to better account for electroniccorrelation:

Configuration Interaction (CI) method

Coupled Cluster (CC) method.

Many Body Perturbation Theory (MBPT), including theMøller-Plesset perturbation theory.

However, these are even more computationally intensive than the H-Fmethod.



Occupancy

Given a collection of spin-orbitals, χ1, χ2, · · · and a Slater determinantΨ, we define the occupancy of spin-orbital χi to be

ni =

{1 if χi appears in Ψ0 otherwise ,

and we write Ψ = Ψn(n1, n2, · · · ). Here we require that

∞∑k=1

nk = n.



Creation and Annihilation Operators

The creation and annihilation operators are defined as:

k†Ψn(· · · nk · · · ) = θk(1− nk)Ψn+1(· · · 1− nk · · · ),kΨn(· · · nk · · · ) = θknkΨn−1(· · · 1− nk · · · ),

where θk = (−1)P

j<k nj . Thus k† creates an electron and places itin the unoccupied spin-orbital χk , while k annihilates the electron inthe occupied spin-orbital χk .



Configuration Interaction

In the CI method, the H-F wave function is replaced by a linearcombination of Slater determinants formed from different sets of nspin-orbitals:

ΨCI = c0Ψ0 +∑a,p

cap p†aΨ0 +

∑a<b,p<q

cabpq q†p†abΨ0 + · · · .

Here, Ψ0 is usually the H-F wave function and the coefficients cap , · · · are

chosen to minimize the energy:

ECI =

∫Ψ∗CIHΨCIdX ≤

∫Ψ∗0HΨ0dX ≡ EHF .



Partial Configuration Interaction

As a rule, the full CI calculation is not feasible, since(2N

n

)different

n-electron determinants can be formed from a set of N molecular orbitalsand we usually take N � n. Instead, the CI expansion is usually restrictedto lower order terms (called excitations):

CID includes all doublet excitations cabpq q†p†abΨ0;

CISD includes all singlet and doublet excitations;

frozen shell CI only allows outer shell orbitals to be substituted.



Coupled Cluster Method

The coupled cluster method seeks to find an operator T such that theexact ground state wave function can be written as

Ψ = eT Ψ0

where Ψ0 is usually the H-F wave function and the cluster operator T isa sum of excitation operators

T = T1 + T2 + T3 + · · ·

with

T1 =∑a,r

tra r†a (single excitations)

T2 =1

4

∑a,b,r ,s

trsab s†r †ab (double excitations).



The amplitudes tra , · · · can be characterized in the following way. First, we

writeHeT Ψ0 = EeT Ψ0

which, upon multiplying both sides by e−T , gives

e−THeT Ψ0 = EΨ0.

Next, we use the commutator expansion to write

e−THeT = H+ [H, T ] +1

2![[H, T ], T ] +

1

3![[[H, T ], T ] , T ]

+1

4![[[[H, T ], T ] , T ] , T ]

where [A,B] ≡ AB − BA. The key observation is that the commutatorexpansion terminates because H only has two-particle interactions.



Calculating the CC Amplitudes

If we then substitute the commutator expansion into (*) and multiply bythe function

Ψm1···mla1···al

= m†l · · · m†1a1 · · · alΨ0

and finally integrate over the spin and spatial coordinates, then we obtainthe identity∫ (

Ψm1···mla1···al

)∗(H+ [H, T ] + · · ·+ 1

4![[[[H, T ], T ] , T ] , T ]

)Ψ0dX = 0.

The right-hand side vanishes because Ψ0 and Ψm1···mla1···al

are orthogonal.

This leads to a system of fourth-order polynomial equations in the CCamplitudes which in principle can be solved iteratively. As with CI, thecluster expansion is usually truncated to obtain CCD or CCSD.



Møller-Plesset (MP) Perturbation Theory

In the Møller-Plesset perturbation theory, we write the Hamiltonian as

H = H(0) +H(1) =∞∑i=0

εi i† i +H(1),

where εi is the one-electron energy of the i ’th molecular orbital and theH-F wave function Ψ0 is the ground state eigenfunction for H(0). We canthen expand the true ground state wave function and energy as series

Ψ0 =∞∑

n=0

Ψ(n)0 , E0 =

∞∑n=0

E(k)0

where the terms in these series can be evaluated recursively, usually up tosecond (MP2) or sometimes fourth (MP4) order.



More formally, if Ψ(0)k and E

(0)k form a complete set of eigenvalues and

eigenfunctions for H(0), then we can define the reduced resolvent

R0 =∞∑

n=1

(1

E(0)0 − E

(0)n

)Pn.

where Pn is the orthogonal projection onto Ψ(0)n . It can then be shown that

Ψ0 =∞∑

n=0

SnΨ(0)0

E0 = E(0)0 + 〈Ψ(0)

0 ,H(1)∞∑

n=0

SnΨ(0)0 〉,

where S ≡ R0(E(0)0 − E0 +H(1)). The difficulty here is that S also

depends on E0.


Quantum Chemistry Applications

Sponer et al. (2004). Accurate Interaction Energies of Hydrogen-BondedNucleic Acid Base Pairs. JACS 126: 10142-10151.

Background

Base pair interactions in DNA and RNA depend largely on H-bonding.

Experimental measurement of the H-bond energies is difficult.

This motivates ab initio calculations of these energies.



Interaction Energies

The interaction energy of dimer A · · ·B is defined as

∆EA···B = EA···B − (EA + EB) + EDef

where

EA···B is the energy of the optimized dimer;

EA,EB are the energies of the isolated bases, with the geometriesof the optimized dimer, calculated using the dimer basis set;

EDef is the deformation energy of the two isolated bases.

Thus, the interaction energy is equal to the energy of the H-bonds lessthe energy required to deform the geometries of the isolated bases.



Deformation Energy

The deformation energy of dimer A · · ·B is defined as

EDef = (EA′ − EA′mon) + (EB′ − EB′

mon)

where

EA′,EB′

are the energies of the isolated bases, with the geometriesof the optimized dimer, calculated using the monomer basis sets

EA′mon,E

B′mon are the energies of the isolated bases, with the geometries

optimized in isolation using the monomer basis sets.

Here the monomer basis sets are used in both calculations to avoid basisset superposition error (BSSE).



Extrapolated Energies

Extrapolation of energies to the complete basis set (CBS) limit wasdone using Helgaker’s formula

E corrX = E corr

CBS + BX−3

where

E corrCBS is the extrapolated energy;

X is the number of basis functions used to represent each valenceorbital (X = 2, 3, 4);

E corrX is the calculated energy, using the MP2 perturbation theory

with an aug-cc-pVXZ basis set;

The aDZ → aTZ values reported in Table 2 were obtained byextrapolating from X = 2 and X = 3.



Results

dimer ∆EA···B ∆ESCF ∆E corr EDef AMBER

GC (WC) -27.5 -20.0 -7.4 3.6 -28.0

AT (WC) -15.0 -7.0 -8.0 1.5 -12.8

GU (wobble) -15.8 -9.7 -6.1 3.0 -16.0

GA 1 -17.5 -8.2 -9.3 1.9 -14.7

AA 1 -13.1 -5.1 -8.0 1.4 -10.8

AMBER is a molecular mechanics package that uses a particular set offorce fields.



Conclusions

Comparison with higher-order extrapolations suggests that theaDZ → aTZ are within ∼ 1 kcal/mol of the true values.

Electron correlation (dispersion attraction + intramolecularcorrelation) contributes significantly to the interaction energies.

H-bond energies estimated using AMBER are within ∼ 3 kcal/mol ofthe the QM estimates, with greater discrepancies for weak base pairs.

Base pair stability is mainly determined by electrostatic interactionsthat can be approximated by atom-centered charges.



Schreier et al. (2007). Thymine Dimerization in DNA Is an UltrafastPhotoreaction. Science 315: 625-9.

Boggio-Pasqua et al. (2007). Ultrafast Deactivation Channel for ThymineDimerization. JACS 129: 10996-7.

Background

Thymine dimerization occurs through UV irradiation of DNAsequences containing adjacent thymine bases (TT dinucleotides).

Thymine dimers are usually repaired by photoreactivation or bynucleotide excision pathways.

Unrepaired dimers are mutagenic and are believed to be a majorcontributor to melanoma.



Photocycloaddition

Absorption of a photon by thymineexcites a π electron to a π∗ orbital.This can then:

Return to the ground state(S0) via internal conversionof the excitation energy toheat.

Attack the double bond on anadjacent thymine, leading todimerization.

Thermal activation of the cycloaddition reaction has low yield because ofimproper symmetry of the interacting orbitals.



Femtosecond Time-Resolved IR Spectroscopy

Schreier et al. (2007) studied thymine dimerization in UV-irradiated(dT )18 using IR spectroscopy.

This indicates that TD formation occurs within 3 ps of UV absorption.



Quantum Mechanical Analysis of Thymine Dimerization

Boggio-Pasqua et al. (2007) used molecular orbital theory to characterizethe likely reaction pathways for thermal and photochemical thyminedimerization.

The QM calculations used multi-configurational SCF (CASSCF) andperturbation theory (CASPT2) to calculate energies of intermediatestates.

These methods allow for electron excitation to higher-energy orbitals.

The potential energy surface near the UV-excited state has thegeometry of a conical intersection (CI).



Thermally-driven TD formation has low yield.

The thermal reaction pathway passes through two transition states to leadto a thymine dimer that is higher in energy than the TT dinucleotide.Thymine dimerization is unlikely to occur through this mechanism.



The photochemical reaction is barrierless.

In contrast, following photon absorption, the excited system will relax to aCI from which it can either rapidly relax to the thymine dimer or return tothe TT ground state.



The conical intersection.

Relaxation of the excited state occurs so rapidly that the neighboringdimers must have a suitable conformation at the time of photoexcitationfor thymine dimerization to occur.


Quantum Chemistry Summary

Quantum Chemistry: Scope and Limitations

Quantum mechanical calculations are important whenever:

Empirical data relevant to molecular energetics are lacking or havequestionable accuracy;

We wish to study processes involving bond formation and breaking(chemical reactions).

However, all of these methods are computationally intensive and so areusually restricted to aperiodic systems containing at most a tens of atoms.


Quantum Chemistry Summary

References

Hehre, W. J., Radom, L., Schleyer, P. v. R., and Pople, J. A. (1986)Ab Initio Molecular Orbital Theory. Wiley.

Piela, L. (2007) Ideas of Quantum Chemistry. Elsevier.


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