Coupled Perturbed Hartree-Fock (CPHF)

Massimo MalagoliSummer Lecture Series 08/10/2010

● Perturbation theory applied to the Hartree-Fock equations.

● It is used in many contexts, here we will focus on the calculation of response properties to an external perturbation (electromagnetic field).

● Density Matrix formalism. AO based method mainly developed by R. McWeeny in the '60.

References

● R. McWeeny, Rev. Mod. Phys. 32, 335 (1960)● R. McWeeny, Phys. Rev. 126, 1028 (1962)● G. Diercksen and R. McWeeny, J. Chem. Phys.

44, 3554 (1966).● R. McWeeny, “Methods of Molecular Quantum

Mechanics”, Second Edition (Academic Press, London 1992)

● R. Ditchfield, Mol. Phys. 27, 789 (1974)● K. Wolinski, J.F. Hinton and P. Pulay, J. Am.

Chem. Soc. 112, 8251 (1990)

H el=∑i

h i∑i j

1rij

V NN

f x1=h x1∑j

J j x1−K j x1

f KS x1=h x1∫ x2r12

V XC x1

Introduction: review of HF theory

● Electronic Hamiltonian

● Fock operator

● Kohn-Sham operator (CPHF theory can be directly applied to KS DFT)

f x1i x1=ii x1

● Hartree-Fock equations

● Atomic basis set: LCAO MO matrix equations

i=∑=1

K

C i

FC=SC

C=c1c2⋯cn

R=∑i=1

n

c i c iT

● Density Matrix

F=hG R

h pq=⟨ p∣h∣q⟩

G pq=∑rs

Rrs ⟨ pr∣qs⟩−⟨ pr∣sq ⟩

● Fock Matrix. One- and two-electron part

● Building G is the most computationally demanding step of an HF calculation

E=2Tr F ' R

F '=h 12G R

FRS−SRF=0

● HF energy

● Commutation relation between F and R and idem potency of R

RSR=R

S−1 /2

c i S1 /2c i

R S1 /2R S1 /2

F S−1 /2F S−1 /2

● Löwdin orthogonalization

pi=c i c iT

pi2= pi ∑

i

pi=1

R1≡R= ∑i=occ

c i c iT R2≡1−R1= ∑

j=vir

c j c jT

● Density matrix as a projection operator

● Occupied and virtual subspace projectors

F=∑i=1

m

i c i c iT

F−1=∑i=1

m

i−1c i c i

T

M=M 11M 12M 21M 22

M ab=RaM Rb

● A generic matrix associated with an operator on the full space can be separated in projection components

● Spectral resolution of the Fock matrix

F=F 0F 12F 2

F n=hnG Rn

R=R0R12R2

● Perturbation expansions

FR−RF=0

F 0R0−R0F 0=0

F 0R1−R1F 0F 1R0−R0F 1=0

RR=R

R0R0=R0

R0R1R1R0=R1

R1= ∑i=occ

c i0c i

1Tc i1c i

0T

R1=R12R21=XX T

F 0 X−X F 0−F 121=0

● First order density matrix

● (1,2) projection of the first order commutator

X=F 0−1X F 0F 121

X=∑n=0

∞

F 0−n1F 121F 0n

X= ∑j=occ

∑k=vir

1 j−k

c jT F 1ck c j ck

T

● Solution of the first order equation

● Iteration starting with X=0

● substituting the spectral resolution of F(0) and F(0)-1

E=2Tr F ' R

F '=h 12G R

E 1=2Tr F 0 ' R1F 1' R0

Tr AG B=Tr BG A

E 1=2Tr F 0R1h1R0=2Tr h1R0

● Expansion of the energy

● First order energy

E 2=2Tr 12h1R1h2R0

E 3=2Tr F 1R112R22

2h2R1h3R0

● Higher order energies

● 2n+1 theorem of perturbation theory

H= H 0 H 1= H 0

W=W 0−a Ea−12

ab Ea E b−16

abc E a EbE c

− 124

abcd E a E bE c E d

● Electric properties: polarizabilities

Nab=Tr hN

0bRa0hNabR00

hNab= 1

2c2

[r⋅r−R−ra r−Rb]

∣r−R∣3

hN0b=−i

c

[r−R×∇]b∣r−R∣3

ha=−ic

[r×∇]a

● Magnetic properties: nuclear magnetic shieldings

p=exp [−i2c

H×R p⋅r ] p0

F 1R0S 0F 0R1S0F 0R0S 1

−S 1R0F 0−S 0R1F 0−S 0R0F 1=0

● Perturbation dependent basis set: Gauge Including Atomic Orbitals (GIAO)

F 1=h1G R1 , g 0G R0 , g 1

h pq1=⟨ p1∣h∣q⟩⟨ p∣h1∣q⟩⟨ p∣h∣q1⟩