Molecular Quantum Mechanics

Lecture #5:

Electronic Structure CalculationsHartree-Fock & Electron Correlation

Contents• The Born-Oppenheimer approximation• The Hartree-Fock approximation

– Spin orbitals– Exchange correlation

• Hartree-Fock equation– Operators and interpretation of solutions

• Restricted Hartree-Fock– Roothaan equations– SCF procedure

• Selection of basis set• Electron correlation

– Configuration interaction– Many-body perturbation theory

The Born-Oppenheimer Approximation

N

i

M

A

N

i

M

A

M

AB AB

BAN

ij ijiA

AA

M

A Ai

N

i R

ZZ

rr

Z

MH

1 1 1 1

2

1

2

1

1

2

1

2

1

N

i

N

i

M

A

N

i

N

ij ijiA

Aielec rr

ZH

1 1 1 1

2 1

2

1• One can consider the electron in a molecule to be moving in the field of fixed nuclei.

}){};({})({}){};({ AielecAelecAielecelecH RrRRr

• The electronic wave function depends explicitly on the electronic coordinates but parametrically on the nuclear coordinates, as does the electronic energy.

The Born-Oppenheimer Approximation

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A

M

AB AB

BAelectot R

ZZ

1

M

1AAR })({

2

1 2totA

Anucl M

H

nuclnuclnuclH

• The total energy for fixed nuclei is the sum of the electronic energy and the constant nuclear repulsion.

• The total energy will then be the potential in which we solve the nuclear Schrödinger equation.

• The solutions of the nuclear Schrödinger equation will Kerstin & Martin talk about in lecture 7 &8

The Hartree-Fock Approximation

• The simplest antisymmetric wave function, which can be used to describe the ground state of an N-electron system, is a single Slater determinant,

• The variation principle states that the best wave function of this functional form is the one which gives the lowest possible energy.

• The variational flexibility in the wave function is in the choice of spin orbitals, i. By minimizing the energy with respect to the choice of spin orbital, one gets the Hartree-Flock equation, which determines the optimal spin orbitals.

N 210

000 elecH

Spin Orbitals

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i

N

i

M

A

N

i

N

ij ijiA

Aielec rr

ZH

1 1 1 1

2 1

2

1• The electronic Hamiltonian depends only on the spatial coordinates.

• In the nonrelativistic theory we specify the spin by introducing two spin functions () and (), corresponding respectively to spin up and spin down.

• The spatial distribution of a single electron is determined by the spatial orbitals , i(r).

• From each spatial orbital, we can form two different spin orbitals by multiplying the spatial orbital by or spin function,

},{ ,

)()(

or

)()(

)(

rx

r

r

x

• The spin functions and are othonormal

• If the spatial orbital are orthonormal , so are the spin orbitals.

Slater Determinants• The wave function for a two electron system is the antisymmetric wave

function,

))()()()((2),( 21212/1

21 xxxxxx ijji

)()(

)()(2),(

22

2/121 xx

xxxx

11

ji

ji

• For an N-electron system we have following determinant,

)()()(

)()()(

)()()(

)!(),...,,(2222/1

21

NkNjNi

kji

kji

N N

xxx

xxx

xxx

xxx

111

NN 2121 ),...,,( xxx

Exchange Correlation

)()()(

)()()(

)()(),(

22222

111

22121

rx

rx

xxxx

11

1

• For a system of two electrons with opposite spins we have the expressions to the right.

• The probability of finding the electrons 1 & 2 in a small volume around the coordinates r1 & r2 is given by,

21

2

12

2

21

2

22

2

1121

2

212121 )()()()(2

1),( rrrrrrrrrrrr ddddddddP

• For the case when 1=2, we have P(r1,r2)=| 1(r1) |2| 1(r2) |2.

• The conclusion is that there is a finite probability of finding two electrons with opposite spins at the same point in space or there is no exchange correlation between two electrons with opposite spin.

Exchange Correlation

)()()(

)()()(

22222

111

rx

rx 11

• If the electrons have parallel spin we

will get an extra cross term in the expression for the probability.

)()()()()()()()(

)()()()(2

1),(

2*1221

*211212

*2121

*1

2

12

2

21

2

22

2

1121

rrrrrrrr

rrrrrr

P

• This cross term is the exchange correlation between electrons of parallel spin.

• Note that P(r1, r1) = 0, and thus the probability of finding two electrons with parallel spins at the same point in space is zero.

Hartree-Fock Equation

• For the general case with an N-electron system the energy will be

baabababahaEN

a

N

a

N

ab

0

• In the derivation of the Hartree-Fock equation one vary the spin orbitals under the constraint that they remain orthonormal until the energy E0 is a minimum. By doing so one obtains the Hartree-Fock equation that defines the best spin orbitals.

)1()1()2()2()1()2()1()1( 112

*2

112

2

2 aabab

abaab

ba rdrdh

xx

The Fock, Coulomb & Exchange Operators• The first of the two-electron terms represents the

interaction between the electron in a and the average potential arising from the other N-1 electrons in the other electron states.

• We define the Coulomb operator to represent the average local potential at x1 arising from an electron b.

)1()2()2()1()1( 112

*2 babab rdK x

)1()2( 112

2

2 aab

b rd

x

112

2

2 )2()1( rdJ bb x

• The second two-electron term arise from the antisymmetric nature of the single determinant.

ab

bab

b KJhf )1()1()1()1(• We define the Fock operator to be the

sum of the one-electron operator h and the two two-electron operators, Ja & Ka.

Interpretation of Solutions

• When we have solved the Hartree-Fock equation we will get a set of the ”best” spin orbitals with corresponding energy.

• The N spin orbitals with the lowest energies are the spin orbitals occupied in the ground state. For these orbitals we use the indices a,b,… .

• The remaining orbitals with higher energies are called virtual or unoccupied spin orbitals, which we label with the indices r,s,… .

• The orbital energies are the diagonal elements of the Fock operator, f.

,,2,1 jf jjj

b

ibiibiiiiii KJhf

The Orbital Energies• The energy of the ground state is not the sum of the N lowest orbital

energies, because we will include the average interaction between the electrons twice.

N

a

N

babaaba

N

aaa

N

a

N

babaaba

N

aaa

N

aa

KJh

KJh

2

10

0

• The occupied spin orbital energy a represents the energy (with opposite sign) required to remove an electron from that spin orbital.

• The negative virtual spin orbital energy r is instead the electron affinity for adding an electron to the virtual spin orbital r.

• These two interpretations of the orbital energies are also called Koopmans’s Theorem, which gives us a way of calculating approximate ionisation potential and electron affinities, if we neglects the relaxation of the spin orbitals.

Restricted & Unrestricted Hartree-Fock• Before one starts to solve the Hartree-Fock equation one has to choose which

kind of spin orbitals one is going to use.

• The choice is between restricted spin orbitals which are constrained to have the same spatial function for (spin up) and (spin down) spin functions; and unrestricted spin orbitals , which have different spatial functions for and spins.

• Example for atomic lithium,

ion wavefunctedunrestrict

ion wavefunctrestricted

0

2110

cba

sss

Closed-Shell Hartree-Fock:Restricted Spin Orbitals

• The molecules are allowed to have only an even number N of electrons, with all electrons paired such that n=N/2 spatial orbitals are doubly occupied.

• One has to convert the spin orbital Hartree-Fock equation to a spatial eigenvalue equation where each of the occupied spatial molecule orbitals is doubly occupied, this is done by integrating out the spin functions in the spin orbitals.

2/

)1()1(2)1()1(

)1()1()1(

N

aaa

jjj

KJhf

f

)1()2()2()1()1(

)2()2()1(

112

*2

112

*2

aiaia

aaa

rdK

rdJ

r

r

Introduction of a Basis• Numerical solutions of this differential equation are common in atomic

calculations, but no practical procedures are available for obtaining numerical solutions for molecules of the spatial differential equation,

)1()1()1( jjjf

• Roothaan and Hall showed how, by introducing a set of known spatial basis functions, the differential equation could be converted to a set of algebraic equations and solved by standard matrix techniques.

• One choose a set of K known basis functions {(r)| =1,2,…,K} and expand the unknown molecular orbitals in the linear expansion,

KiCK

ii ,...,2,1 , 1

The Roothaan Equations• The goal is now to find the set of expansion coefficients Ci,

)1()1()1(

iii CCf

• By multiplying by *(1) on the left and integrating, we turn the differential equation into a matrix equation,

)1()1()1()1()1( *1

*1

rr dCfdC iii

• The overlap matrix, S,

)1()1(*1 rdS

• The Fock matrix, F,

)1()1()1(*1 fdF r

• The integrated Hartree-Fock equations are the Roothaan equations

SCFC

KiCSCF iii ,...,2,1 ,

The SCF Procedure• Because the Fock matrix

depends on the expansion coefficients, the Roothaan equations are nonlinear and they will need to be solved in an iterative fashion.

GH

CCH

KJdhdF

core

N

aaa

core

aa

N

a

2/

*

*2/

1*

1

2

)1()1()1(2)1()1()1()1( rr

1. Specify the molecule and the basis set {}.

2. Calculate all required molecular integrals.

3. Obtain a guess at the expansion coefficients.

4. Calculate the matrix G from the coefficients and the two-electron integrals and create the Fock matrix, F = Hcore + G.

5. Solve the secular equation

to obtain C and .6. Determine whether the procedure has

converged. If the procedure has not converged return to step (4) with the new expansion coefficients.

7. If the procedure has converged, then use the resultant solution to calculate expectation values and other quantities of interest.

0det SF a

The Selection of Basis SetThe choice of a basis set for quantum chemical calculations is

mainly an art rather than science.• By computational reason we are restricted to a finite set of K basis functions.

It’s therefore important to chose a basis that will provide, as far as possible, a reasonably accurate expansion for the exact molecular orbits, particularly, for those molecular orbitals {a} which are occupied in and determine the ground state.

• As the basis set becomes more and more complete, the expansion leads to more and more accurate representation of the exact molecular orbitals.

• The number of two-electron integrals is of order K4, so even small basis sets for moderately sized molecules the number of two-electron integrals can rapidly approach millions.

• The number of integrals can be reduced by using the symmetry properties of the integrals.

0

Different Types of Orbitals• The two types of orbitals mentioned in Atkins are the Slater type orbitals

(STO) and the Gaussian type orbitals (GTO).

• The STOs are impractical for calculations of many two-electron integrals. STOs are instead often used in atomic calculation because they give good representation of orbitals at the atomic nuclei.

• The GTOs give poor representation at the atomic nuclei, but their advantage is that the product of two Gaussians at different centres is equivalent to single Gaussian function centred at a point between the two centres.

• The Gaussian functions are often grouped together to form contracted Gaussian functions.

Electron Correlation

• So far we have only considered the electron-electron interaction in an average way. The correct interaction between the electrons comes from the instantaneous Coulomb interaction.

• In the configuration interaction and the many-body perturbation theory one often uses the solutions from the Hartree-Fock equation to get more accurate values.

Configuration Interaction• The results from our restricted Hartree-Fock calculations are a finite set of

2K spin orbitals with corresponding orbital energy. The N orbitals with lowest energy will occupy , the determinant representing the ground state.

• From remaining 2K-N virtual orbitals we can construct excited determinants.

• A single excited determinant corresponds to one for which a single electron in the occupied spin orbital a has been promoted to a virtual spin orbital r.

0

Nba 210

Nbrra 21

• We write the exact many-electron wave function as the sum of the ground state and the excited states. This linear combination of determinants is called configuration interaction.

0

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rstabc

rsab

rsab

ra

ar

ra

cbatsr

basr

cccc 000

Configuration Interaction

• The number of determinants become rapidly very large, even for systems with small number of electrons.

• The number of determinants can be reduced by not include the determinants with the wrong symmetries.

• The expansion must almost always be truncated.

• A calculation is classified as full CI if all determinants of appropriate symmetry are used for a given finite basis set.

• One example where we not have to truncate the expansion and can perform a full CI calculation is H2.

CI Calculations• The basic idea in CI calculations is to diagonalize the N-electron

Hamiltonian in the basis of the determinants.

• We expand the wave function we are looking for as a linear combination of N-electron determinants and use this expansion in the electronic Schrödinger equation.

L

JJJs

L

JJJs CECH

11

• By multiplying with I* on the left and integrate over all coordinates we

turn the differential equation to a matrix equation.

ESCHC • The Slater determinants form an orthonormal set, the matrix equation

becomes ECHC

Potential Energy Curves for H2

• To make STO-NG function one fit a STO to a linear combination of N Gaussian functions. One get good approximation of the atomic orbitals from the STO, while still evaluating integrals only with Gaussian functions.

Potential Energy Curves for H2

• The 6-31G** basis consists of contracted Gaussian functions which been polarized.

• By adding polarization functions to a basis set, we directly accommodate the effect of the other nuclei.

Many-Body Perturbation Theory • In perturbation theory we separate the Hamiltonian into

a unperturbed part and a perturbation part. In the Møller-Plesset perturbation theory described in Atkins, we use the sum of one-electron Fock operators as the unperturbed Hamiltonian.

n

ii

n

ii

fHH

fH

1

'

10

• The unperturbed wave function will be the ground state Hartree-Fock wave function . The Hartree-Fock energy can now be evaluated as,0

)1(0

)0(00

'0

0'

0000000

H

HHH

aa

• The first correction to the ground state is given by second-order perturbation theory as,

0)0(

0''

0)2(

J J

JJ HH