Computer programs are written to solve the Schrödinger equation 2 Max Born (German, 1882-1970) Julius Robert Oppenheimer (Berkeley- Los alamos, 1904.

Hartree-Fock Theory

Patrick TamukongNorth Dakota State UniversityDepartment of Chemistry & BiochemistryFargo, ND 58108-6050 U.S.A.June 10, 2015

Computer programs are written to solve the Schrödinger equation

EΨΨH ˆ

2

|||| AiiAiA Rrrr

|||| jiijij rrrr

|||| BAABAB RRRR

nneeenne V

M

1A

M

AB AB

BA

V

N

1i

N

ij ij

V

N

1i

M

1A iA

A

T

M

1A

2A

A

T

N

1i

2i R

ZZr1

rZ

2M1

21

H

Max Born (German, 1882-

1970)

Julius Robert Oppenheimer (Berkeley- Los alamos, 1904 –1967)

Szabo, A. Ostlund, N. S. McGraw-Hill, New York, 1989, p. 40.

Electronic Structure Problem

Born-Oppenheimer Approximation

The approximation used in solving the electronic structure problem

nneeenne V

M

1A

M

AB AB

BA

V

N

1i

N

ij ij

V

N

1i

M

1A iA

A

T

M

1A

2A

A

T

N

1i

2i R

ZZr1

rZ

2M1

21

H

=0 constant

N

i

N

i

N

ij ij

M

A iA

AN

iielec rr

ZH

1 111

2 1

2

1

M

A

M

AB AB

BAelectot R

ZZEE

1elecelecelecelec EH

Aielecelec Rr ;

})({2

1

})({2

1

1

2

1

2

1

1

2

11

2

11 1 1 1 1

22

Apottot

M

AA

A

M

A

M

AB AB

BAAelec

M

AA

A

M

A

M

AB AB

BAM

A

N

i

N

i

M

A

N

i

N

ij ijiA

AiA

Anucl

REM

R

ZZRE

M

R

ZZ

rr

Z

MH

Szabo, A. Ostlund, N. S. McGraw-Hill, New York, 1989, p. 40. 3

Atomic UnitsChosen for convenience such that (me=1, e=1, = h/2 = 1, ao=1, and the potential energy in the hydrogen atom (e2/ao = 1).

re

4π1

Δ2m

2

0

2

r

1

2

1

Other frequently used energy units:

1a.u. = 27.212 eV = 627.51 Kcal/mol = 2.1947·105 cm-1

1Kcal/mol = 4.184KJ/molBoltzmann’s constant: k = 1.38066·10-23J/K

Avogadro’s number: NA= 6.02205·1023mol-1

Rydberg constant: R∞= 1.097373·107m-1

Compton wavelength of electron: λC= 2.426309·10-

12m

Stefan-Boltzmann constant: σ = 5.67032·108W/(m2K4)

ΦEΦrε4π

e2m 0

22

e

2

ΦEΦrε4π

e2m 0

22

e

2

2

a0

2

2e

2

Eε4π

em

λλ

02e

20 aem

ε4π

4

Our Main Concern Static Electron

Correlation

Dynamic Electron Correlation

5

1*u

1gz

2g

1uyz

1uxz 5sσσ4d5sσπ4dπ4d 2

The need to include more than one electron configuration in the description of the total wave function

2YY2

contributes some 80% to the total wave function at 2.80 Å but only 55% at 4.4 Å

The need to account for the coulomb hole

Electron Correlation in He

6Helgaker et al. Molecular Electronic-Structure Theory, John Wiley & Sons Ltd, Chichester, England, 2000, p. 257.

HF Approximation

Full Treatment

Full Treatment - HF

Correlation Effect

7Tamukong, P. K.; Theis, D.; Khait, Y. G.; Hoffmann, M. R. J. Phys. Chem. A 2012, 116, 4590.

MCSCF GVVPT2

Cr2 Ground State

Conceptual Picture

RHF

RMP2Higher

MCSCF

8

Hartree-Fock Theory lacks electron correlation and represents the total wave function as a single electron configuration

Use of Molecular Orbitals A MO is a wavefunction associated with a single

electron. The use of the term "orbital" was first used by Mulliken in 1925.

MO theory was developed, in the years after valence bond theory (1927) had been established, primarily through the efforts of Friedrich Hund, Robert Mulliken, John C. Slater, and John Lennard-Jones. The word orbital was introduced by Mulliken in 1932. According to Hückel, the first quantitative use of MO theory was the 1929 paper of Lennard-Jones. The first accurate calculation of a molecular orbital wavefunction was that made by Charles Coulson in 1938 on the hydrogen molecule. By 1950, MO were completely defined as eigenfunctions of the self-consistent field

Robert Sanderson Mulliken 1996-1986Nobel 1966

Friedrich Hund 1896-1997

Charles Alfred Coulson 1910-1974

Sir John Lennard-Jones 1894-1954

9

John Clarke Slater

1900-1976

LCAO = MO

It was introduced in 1929 by Lennard-Jones with the description of bonding in the diatomic molecules of the first main row of the periodic table, but had been used earlier by Pauling for H2

+.

10

Sir John Lennard-Jones 1894-1954

Linus Carl Pauling 1901-1994 Nobel 1962

iν

K

1ννi Cψ

ωβrψ

ωαrψxχ

i

ii

The Variational Principle Method of choice for approximate solutions to

physical problems The raison d'être for the LCAO Define an approximate solution (with same

boundary conditions as the eigenvectors of the Hamiltonian) to the Shrodinger equation as a LC of its unknown eigenvectors

ααα ΦεΦH α

αα CΦΦ~

0εΦHΦ ~ˆ~then

The task then is to determine the optimal set of expansion coefficients which is accomplished by Lagrange’s method of undetermined multipliers

11

Lagrangian Method In the Lagrangian method, a function (or functional) is minimized (or maximized) subject to given equality constraints Nrdrρ

α

β α

*βαβ CCΦΦ1ΦΦ ~~

Example: One wishes to minimize subject to

yxyx,f 23yx 22

Construct the Lagrangian

3yxλyxλy,x,Λ 222

Solve the equations

03yx2yλx2xλ2xy

0λ

λy,x,Λ

y

λy,x,Λ

x

λy,x,Λ

222

2x

3y

and y = ±1

ORand x = 0

12

The Secular Equation Use the above philosophy

αββαβαβαβ

αβα

N21

1CCΦΦEΦHΦCC

1ΦΦEΦHΦE,C,,C,CΛ

ˆ

~~~ˆ~

α

αα CΦΦ~

E,C,,C,CΛ N21 Consider small variations in

0C

Λ

C

Λ

21

β β

βαββαβ CSECH where βααβ ΦHΦH ˆ

βααβ ΦΦS HC = ESC

All quantum chemistry methods solve this secular equation using different approximations. It is a matrix problem and reduces to that of matrix diagonalization. Often it is transformed into an eigenvalue problem 13

Matrix Eigenvalue Problem

Transform the secular equation into an eigenvalue problem by rewriting C as

14

CSC

2

1

HenceECSSSCHSS

2

1†

2

1

2

1†

2

1

HC = ESC

ECCH Eigenvalue problem

ISSS

2

1†

2

1

where

The above transformation is known as the symmetric (Löwdin) orthonormalization

The thus obtained matrix eigenvalue problem is the final problem solved in quantum chemistry

Hartree Approximation The total Hamiltonian is approximated as a sum of

one-electron operators and the wave function as a product of eigenvectors of those operatorsc

N

1i

N

ij1j

ij

N

1ii J

2

1HE

ii

M

1A iA

A2iiiiiii (i)dτΦ

r

Z

2

1(i)Φ(i)dτΦ(i)hΦH

212j

12

2iij dτ(2)dτΦ

r

1(1)ΦJ

)(r)...Φ(r)Φ(rΦ)r,...,r,(rΨ Nn2j1iN21HP

iiHH

The variational principle then leads to

15

E=εi+εj+…+εn

The Problem)(r)...Φ(r)Φ(rΦr,...,r,rΨ Nn2j1iN21

HP )(

Φi – spin orbitals• The form of ΨHP suggests the independence of Φi • Probability density given by ΨHP is equal to the product of monoelectronic probability densities • This is true only if each electron is completely independent of the other electrons• ΨHP - independent electron model

A ♥ A♥

 PA=1/13 P♥=1/4 PA♥=1/52=PAP♥

PA is uncorrelated (independent) with P♥.

Uncorrelated probabilitiesCorrelated probabilitiesIn a n-electron system of electrons the motions of the electrons is correlated due to the Coulomb

repulsion (electron-one will avoid regions of space occupied by electron two).

E=εi+εj+…+εn

Electronic Hamiltonian can be rewritten:

ee

N

1iiE VhH

i2ii v

21

h

Where:

is the monoelectronic operator

N

1i

v

M

1A iA

AN

1iieN

i

rZ

vV

vi is the monoelectronic term of the external potential:

In HP, hi will act only on the wavefunction corresponding to the i-th electron. However, Vee depends on pairs of electrons so that we can not separate the variables in Schrödinger equation.

16

Slater Determinants The Hartree product ignores electron correlation

completely and ignores Pauli’s exclusion principle To fix this, the wave function is often written as a

slater determinant or their linear combination

NNN2N1

2N2221

1N1211

N21

xχxχxχ

xχxχxχ

xχxχxχ

N!

1x,,x,xΨ

ωβrψ

ωαrψxχ

i

ii

0

0)()()()(

1

1)()()()(

**

**

dd

and

dd

N is a normalization factor

17

In Hartree-Fock Theory, the wave function is a single slater determinant

Hartree-Fock Theory Write the Hamiltonian as a sum of Fock

operators

18

i

ifH HFi

M

1A iA

A2ii v

r

Z

2

1f

ECCF iν

K

1νν

iν

K

1ννi CφεCf

where the Hartree-Fock potential is defined in terms of coulomb and exchange operators

b

bbHFi iKiJv

2χ2χr1χ1χdxdx1χ1J1χ bb1

12aa21aba Coulomb integral

2χ2χr1χ1χdxdx1χ1K1χ ab1

12ba21aba Exchange integral

ωβrψ

ωαrψxχ

i

ii

Use the variational and langrangian

methods to arrive at

Douglas Rayner Hartree English

1897-1958

Vladimir Aleksandrovich Fock Russian 1898–1974