The Hartree-Fock Self-Consistent Field Method, Lecture notes of Quantum Chemistry

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The Hartree-Fock Self-Consistent-

Field Method

Chem 561

Lecture Outline

1. The Hartree-Fock Energy (Levine 11.1)

a) Coulomb and exchange integrals

2. The Hartree-Fock Equations

a) The LCAO Approximation

3. The Hartree-Fock-Roothaan Equations
4. The Self-Consistent-Field Procedure (Levine 13.16)
5. Examples

a) H 2

b) He

HF SCF

" Recall the electronic Hamiltonian for molecules

" Define

€

ˆ H el

= −

1

2

∇ i

2

i

−

Z α

r i i^ α

α

1

r i > j ij

j

€

ˆ f i

= −

1

2

∇ i

2 −

Z α

r α i^ α

g^ ˆ ij

r ij

H

el

f

i

i

+ g ˆ

ij

i > j

j

one-electron operator two-electron operator

HF SCF

" No analytical solution to this differential equation

where

" Use a trial-and-error, iterative approach

H

el

el

= E

el

el

H

el

f

i

i

+ g ˆ

ij

i > j

j

Using the Variational Method

" searching for a Slater determinant of spin orbitals (D)

that minimizes the variational integral

" Since D is normalized

€

W =

D

ˆ H el

Dd τ

D

Dd τ

€

W = D

ˆ H el

Dd τ

= D

ˆ H el

D

€

u i (^1 ) u^ j (^1 ) =^ δ ij

Using the Variational Method

" according to the previous definition

W = D

H

el

D = D

f i

D

i

∑ +^ D

g ˆ ij

D

i > j

i

€

ˆ H el =

ˆ f i

i

∑ +^

g ˆ ij

i > j

j

Using the Variational Method

" cross terms disappear because of orthonormality

D

f 1

D = φ

i

(^1 )

f 1

i

(^1 )

i

W = D

f i

D

i

∑ +^ D

g ˆ ij

D

i > j

i

i and j label electrons

i labels spin orbitals

D

g ˆ 12

D = φ i

(^1 )φ^

j

(^2 ) g ˆ

12

φ i

(^1 )φ^

j

(^2 ) −^ φ

i

(^1 )^ φ^

j

(^2 ) g ˆ

12

φ j

(^1 )^ φ

i

(^2 )

[ ]

i > j

i

i and j label spin orbitals from Slater determinant

Coulomb and Exchange Integrals

" Define

D

g ˆ 12

D = φ i

(^1 )^ φ^ j

(^2 ) g ˆ 12

φ i

(^1 )φ^ j

(^2 ) −^ φ i

(^1 )φ^ j

(^2 ) g ˆ 12

φ j

(^1 )^ φ i

(^2 ) [ ]

i > j

∑

i

∑

€

J ij

= φ i (^1 )φ^ j (^2 )

1

r 12

φ i (^1 )^ φ^ j (^2 )

K ij

= φ i (^1 )^ φ^ j (^2 )

1

r 12

φ j (^1 )^ φ i (^2 )

Coulomb integral

Exchange integral

€

g^ ˆ ij

=

1

r ij

Hartree-Fock Energy

" The Hartree-Fock energy is our best approximation to

the energy eigenvalue

D

ˆ H el

D = 2 φ

i

(^1 )

f 1

i

(^1 )

i = 1

n 2

+ 2 J

ij

− K

( ij )

j = 1

n 2

i = 1

n 2

D

ˆ H el

D = E = E

HF

E

HF

i

(^1 )

f 1

i

(^1 )

i = 1

n 2

∑ +^ (^2 Jij −^ Kij )

j = 1

n 2

i = 1

n 2

Douglas Hartree

" English mathematician and

physicist (1897-1958)

" used numerical analysis to

solve differential equations

for the calculation of atomic

wavefunctions

" the Hartree unit of energy is

named after him

Hartree-Fock Energy

E

HF

i

(^1 )

f 1

i

(^1 )

i = 1

n 2

∑ +^ (^2 Jij −^ Kij )

j = 1

n 2

i = 1

n 2

€

J ij

= φ i (^1 )φ^ j (^2 )

1

r 12

φ i (^1 )φ^ j (^2 )

€

K ij

= φ i

(^1 )φ^

j

(^2 )

1

r 12

φ j

(^1 )^ φ

i

(^2 )

Coulomb integral

Exchange integral

Coulombic (electrostatic)

interaction between two electrons

A nonclassical interaction

J ij

=J ji

K ij

=K ji

J ii

=K ii

Hartree-Fock Energy

" Define

the one-electron core Hamiltonian

E

HF

i

(^1 )

f 1

i

(^1 )

i = 1

n 2

∑ +^ (^2 Jij −^ Kij )

j = 1

n 2

i = 1

n 2

H

core

(^1 ) =^

f 1

and H ii

core

i

(^1 )

f 1

i

(^1 )

E

HF

= 2 H

ii

core

i = 1

n 2

+ 2 J

ij

− K

( ij )

j = 1

n 2

i = 1

n 2

+ V

NN

for a system of n electrons in n /2 occupied spin orbitals

The Hartree-Fock Equations

" The spin orbitals are an orthonormal set

" We are now ready to optimize with respect to each

variational parameter {c si

}

€

φ i

= c si

s = 1

b

∑^ χ s

€

φ i

(^1 )^ φ^

j

(^1 ) =^ δ

ij

= c si

t = 1

b

∑

s = 1

b

∑ ctj^ χ i (^1 )^ χ^ j (^1 )

∂ D

H D

∂ c

si

" We define

The Hartree-Fock Equations

€

ˆ F (^) ( (^1) ) =

ˆ H

core (^1 ) +^2

ˆ J j (^1 ) −^

ˆ K j (^1 ) [ ]

j = 1

n 2

∑

€

ˆ J j (^1 ) f^ (^1 ) ≡^ f^ (^1 )^ φ^ j (^2 ) ∫

(^2 )

r 12

dv 2

ˆ K j (^1 ) f^ (^1 ) ≡^ φ^ j (^1 )

φ j

(^2 ) f^ (^2 )

r 12

dv ∫ 2

Coulomb operator

exchange operator

Fock operator