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⟩