Hartree-Fock methods of
Quantum Chemistry
János Ángyán
Contents
1 Introduction
4
2 Historical Overview
5
3 Variation theorem
3.1 Variation theorem II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Stationary-value conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
6
6
4 Separation of electronic and nuclear motions
4.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Breakdown of the Born-Oppenheimer approximation . . . . . . . . . . . .
7
8
9
5 Electron Hamiltonian and wave function
5.1 Electron spin . . . . . . . . . . . . . . . . .
5.2 Spin operators and spin functions . . . .
5.3 Total spin operators . . . . . . . . . . . . .
5.4 Spin functions of a two-electron system
5.5 Space and spin orbitals . . . . . . . . . . .
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6 Density matrices
6.1 Reduced density matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
12
7 Construction of the wave function
7.1 Expansion theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Configuration Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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14
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8 Method of determinants
8.1 Permutation (symmetric) group . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Antisymmetrizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Why antisymmetrizer? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 Determinant wave function
9.1 Invariance properties of the determinant wave function . . . . . . . . . . .
9.2 Physical meaning of the determinant wave function . . . . . . . . . . . . .
17
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18
10 Matrix elements between determinant wave functions
10.1 Laplace expansion formulae . . . . . . . . . . . . . .
10.2 Overlap of two determinants . . . . . . . . . . . . .
10.3 Matrix elements of one-electron operators . . . .
10.4 Matrix elements of two-electron operators . . . .
10.5 Notations for two-electron integrals . . . . . . . .
10.6 Summary of matrix element rules . . . . . . . . . .
10.7 Slater rules . . . . . . . . . . . . . . . . . . . . . . . .
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10.8 Variation of a determinant wave function . . . . . . . . . . . . . . . . . . . .
24
11 Method of Second Quantization
11.1 Representations of determinant wave functions . . . . . . . . .
11.2 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Creation of electrons . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 Annihilation of electrons . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Commutation relations . . . . . . . . . . . . . . . . . . . . . . . . .
11.6 Adjoint relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.7 Non-orthogonal functions . . . . . . . . . . . . . . . . . . . . . . .
11.8 Properties of creation and annihilation operators: Summary .
11.9 Particle number operator . . . . . . . . . . . . . . . . . . . . . . .
11.10Representation of operators . . . . . . . . . . . . . . . . . . . . . .
11.11One-electron operators . . . . . . . . . . . . . . . . . . . . . . . . .
11.12Two-electron operators . . . . . . . . . . . . . . . . . . . . . . . .
11.13Reduced density matrices . . . . . . . . . . . . . . . . . . . . . . .
11.14Hamiltonian and energy expressions . . . . . . . . . . . . . . . .
11.15Calculation of matrix elements . . . . . . . . . . . . . . . . . . . .
11.16Matrix elements between single determinants . . . . . . . . . .
11.17Fermi vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.18Slater rules – an example . . . . . . . . . . . . . . . . . . . . . . .
11.19Energy of a single determinant . . . . . . . . . . . . . . . . . . . .
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25
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12 Hartree-Fock theory
12.1 Brillouin theorem . . . . . . . . . . . . . . . . . .
12.2 Unrestricted Hartree-Fock equations . . . . . .
12.3 Canonical UHF equations . . . . . . . . . . . . .
12.4 Unrestricted Hartree-Fock energy expression
12.5 Spin properties of the UHF wave function . . .
12.6 Symmetry dilemma . . . . . . . . . . . . . . . . .
12.7 Restricted Hartree-Fock equations . . . . . . .
12.8 RHF energy expression . . . . . . . . . . . . . . .
12.9 Orbital energies . . . . . . . . . . . . . . . . . . .
12.10Koopmans theorem . . . . . . . . . . . . . . . . .
12.11Excitation operators - Generalities . . . . . . .
12.12Generalized Koopmans theorem . . . . . . . . .
12.13How good is the Koopmans theorem? . . . . .
12.14Excited states . . . . . . . . . . . . . . . . . . . . .
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13 EOM for excited states
13.1 Hartree-Fock density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Natural orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Two-particle density matrix for a HF wave function . . . . . . . . . . . . . .
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49
14 Hartree-Fock-Hall-Roothan equations
14.1 P-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Population analysis . . . . . . . . . . . . . . . . . . . . . .
14.3 Potential-fitted charges . . . . . . . . . . . . . . . . . . .
14.4 Covalent bond orders . . . . . . . . . . . . . . . . . . . . .
14.5 Implementation of the HFR equations: SCF procedure
14.6 Direct SCF procedure . . . . . . . . . . . . . . . . . . . . .
14.7 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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14.8 Product theorem for Gaussian functions . .
14.9 Contracted Gaussian functions . . . . . . . .
14.10Second Quantized Hamiltonian in AO basis
14.11Hellmann-Feynman theorem . . . . . . . . .
14.12Hartree-Fock gradients . . . . . . . . . . . . .
14.13BSSE . . . . . . . . . . . . . . . . . . . . . . . .
14.14Response properties . . . . . . . . . . . . . .
14.15General wave function variation . . . . . . .
14.16Coupled Hartree-Fock equations . . . . . . .
15 Comparison of RHF and UHF solutions
15.1 RHF solution of minimal basis H2 .
15.2 UHF orbitals of minimal basis H2 .
15.3 UHF solutions for minimal basis H2
15.4 Stability of the RHF/UHF solutions
15.5 Behaviour at dissociation . . . . . .
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63
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16 Configuration interaction method
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17 Structure of the CI matrix
17.1 Doubly excited CI . . . . . . . . . . . . . . . . .
17.2 Approximate solutions of the DCI equations
17.3 Natural orbitals . . . . . . . . . . . . . . . . . . .
17.4 Minimal basis H2 molecule . . . . . . . . . . .
17.5 DCI of the minimal basis H2 . . . . . . . . . .
17.6 Approximate CI solution of minimal basis H2
17.7 Size-consistency problem . . . . . . . . . . . .
17.8 Full CI of a non-interacting H2 dimer . . . . .
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18 Rayleigh-Schroedinger perturbation theory
18.1 Schroedinger equation with perturbation . . . .
18.2 Iterative solution of the perturbed Schroedinger
18.3 H-atom in electric field . . . . . . . . . . . . . . . .
18.4 Rayleigh-Schroedinger perturbation expansion
18.5 Explicit formulae at low orders . . . . . . . . . . .
18.6 Energy with the first-order wave function . . . .
18.7 Deformation energy and perturbation energy .
18.8 Dalgarno’s (2n+1) theorem . . . . . . . . . . . . .
18.9 Pertubational correction of expectation values .
18.10Partitioning method . . . . . . . . . . . . . . . . . .
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19 RSPT treatment of intramolecular correlation
19.1 Third order MBPT energy . . . . . . . . . . . . .
19.2 Comparison with the partitioning method . . .
19.3 Partial summation of MBPT series . . . . . . .
19.4 MBPT correlation energy of minimal basis H2
19.5 Size-consistency . . . . . . . . . . . . . . . . . . .
19.6 Size-consistent methods . . . . . . . . . . . . . .
19.7 Local MP2 . . . . . . . . . . . . . . . . . . . . . . .
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1
Introduction
Explicit use of the wave function
• strict definition of any electronic state with any multiplicity
• density matrices (1st and 2nd order) calculated from wave function
Hierarchy of “infinitely improvable” approximations
• one-electron basis set
• correlation energy
Bibliography
A. Szabo and N.S. Ostlund Modern Quantum Chemistry: Introduction to Advanced Electronic
Structure Theory McMillan, New York (1982)
R. McWeeny Methods of Molecular Quantum Mechanics, Academic Press, London (1989)
P.R. Surján Second Quantized Approach to Quantum Chemistry, Springer Verlag, Heidelberg,
(1989)
Mayer István Fejezetek a kvantumkémiából, BME, Budapest (1987)
B. Silvi Cours de DEA, Université P.& M. Curie, Paris (1992)
2
Historical Overview
Schrödinger 1926
Hartree 1928
Thomas Fermi 1927
Fock 1930
Dirac 1930
Moller-Plesset 1934
Weizsäcker 1935
Slater 1951
Cizek 1968
Kohn Sham 1965
Sham 1971
GGA
Becke 1993
HF,CC,MBPT,CI
3
Hybrid
KS DFT
True DFT
Variation theorem
The expectation value of the energy is equal to the eigenvalue of the Schrödinger equation
ĤΨ = EΨ
4
(1)
Multiply by Ψ∗ and integrate:
hΨ|Ĥ|Ψi = EhΨ|Ψi
E=
hΨ|Ĥ|Ψi
hΨ|Ψi
(2)
The lowest energy belongs to the exact ground state wave function, i.e. for any approximate Ψ̃
hΨ̃|Ĥ|Ψ̃i
≥ E0
hΨ̃|Ψ̃i
E=
(3)
Eigenfunctions Ψk of the Hamiltonian form a complete, orthonormal basis
X
ck Ψk
Ψ̃ =
(4)
k
hΨ̃|Ψ̃i =
XX
i
hΨ̃|Ĥ|Ψ̃i =
X
j
XX
i
c∗i cj hΨi |Ψj i =
|ci |2
(5)
i
c∗i cj Ej hΨi |Ψj i =
X
j
|ci |2 Ei
(6)
i
Energy
P
2
hΨ̃|Ĥ|Ψ̃i
i |ci | Ei
E=
= P
2
hΨ̃|Ψ̃i
i |ci |
(7)
Since Ei ≥ E0
E=
P
2
hΨ̃|Ĥ|Ψ̃i
i |ci | E0
≥ P
= E0
2
hΨ̃|Ψ̃i
i |ci |
(8)
The energy of an approximate wave function is upper bound of the exact energy.
3.1
Variation theorem II
The Schrödinger equation can be considered as a result of the variation theorem: we are looking
for the wave function that makes stationary the energy functional
"
#
hΨ̃|Ĥ|Ψ̃i
δE = δ
=0
(9)
hΨ̃|Ψ̃i
First variation
δE =
hδΨ|Ĥ|ΨihΨ|Ψi − hΨ|Ĥ|ΨihδΨ|Ψi
+ c.c. = 0
hΨ|Ψi2
(10)
This expression and its complex conjugate should be separately zero
hδΨ|Ĥ|ΨihΨ|Ψi
hΨ|Ĥ|Ψi
− hδΨ|Ψi
=0
hΨ|Ψi
hΨ|Ψi
(11)
we obtain another form of the variation theorem
hδΨ|Ĥ − E|Ψi = 0
(12)
and since δΨ is arbitrary
(Ĥ − E)Ψ = 0
5
(13)
3.2
Stationary-value conditions
The variation theorem provides a necessary but not sufficient condition to have a minimum of the
energy. Let us examine the general Taylor expansion of the energy functional.
Denote the wave function, containing the independent parameters p1 , p2 , . . ., by
Ψ(p1 , p2 , . . . |x1 , x2 , . . . xN )
(14)
hΨ|Ĥ|Ψi
hΨ|Ψi
(15)
and the energy functional
E = H(p1 , p2 , . . .) =
The differential dH is:
(
dH =
X
∂Ψ
hΨ|Ĥ| ∂p
ji
hΨ|Ψi
j
−
∂Ψ
hΨ|Ĥ|ΨihΨ| ∂p
ji
hΨ|Ψi2
)
dpj + c.c. + . . .
(16)
The term in curly brackets is just the j-th component of the general “gradient vector” of the energy
surface E = H(p1 , p2 , . . .). With the notations
D
E
∂Ψ
Ψ|(Ĥ − H)| ∂p
j
(H∇)j =
hΨ|Ψi
D
(∇H)j =
∂Ψ
∂pj |(Ĥ
E
− H)|Ψ
hΨ|Ψi
(17)
the stationary-value conditions can be formulated as
(H∇)j = (∇H)j = 0
Using the shorthand notation
(∇∇H)jk =
(∇H∇)jk =
∂Ψ
∂pj
for all j
(18)
= Ψj etc. the second derivatives are
hΨjk |(Ĥ − H)|Ψi − hΨj |Ψi(∇H)k − (∇H)j hΨk |Ψi
hΨ|Ψi
hΨj |(Ĥ − H)|Ψk i − hΨj |Ψi(H∇)k − (∇H)j hΨ|Ψk i
hΨ|Ψi
(0)
(19)
(20)
(0)
The variation around a point p0 = (p1 , p2 , . . .) with E = H(p0 ) = H0 can be written in terms
(0)
of the above-defined tensors. Let pj = pj + dj then
X
∗
∗
δE =
dj (∇H)j + (H∇)j dj
j
1X ∗ ∗
∗
+
dj dk (∇∇H)jk + 2dj (∇H∇)jk dk + (H∇∇)jk dj dk + . . .
2
(21)
jk
The generalized ”forces” will be denoted by a and a∗ and the blocks of the generalized Hessian
matrix by
Qjk = (∇H∇)jk
and
Mjk = (∇∇H)jk
The Taylor expansion of the energy around the point p0 is
† † 1 d
d
a
M
E = H0 +
+
∗
d
Q∗
d∗
a∗
2
Q
M∗
d
d∗
(22)
In obvious notations:
1
E = H0 + D † A + D † BD
2
This relationship can serve to minimize the wave function using the Newton-method
a
M
Q
d
+
=0
a∗
Q∗ M ∗
d∗
6
(23)
(24)
4
Separation of electronic and nuclear motions
Total Hamiltonian
X ~2
X 1
X X Zα
X Zα Zβ
~2 X
4i +
−
+
4α −
2Mα
2me i
r
riα
Rαβ
α
α
i<j ij
i
α<β
|
{z
}|
{z
} |
{z
}
TN
Te
V
H(R, r) = −
(25)
Let us introduce M an “average nuclear mass”, the dimensionless number µα = M/Mα characterizing the nuclear mass of atom α, and the electron/nuclear mass ratio parameter
κ=
m 1/4
(26)
M
Total Hamiltonian in atomic units
H(R, r) = −
1X
κ4 X
4i + V(r, R) −
µα 4α
2 i
2 α
= He (R; r) + κ4 TN (R)
Consider κ4 TN as perturbation. At zero order we obtain the electronic Schrödinger equation
He (r; R)Ψn (r; R) = Un (R)Ψn (r; R)
(27)
with the electronic Hamiltonian
He = Te + V
(28)
The solutions of the electronic Schrödinger equation form a complete basis of the eigenfunctions
Ψn (r; R) on which the total wave function can be expanded
X
Υ(r, R) =
Ξn (R)Ψn (r; R)
(29)
n
with the nuclear coordinate dependent expansion coefficients, Ξn (R).
Substitute the expansion into the Schrödinger equation
X
X
{TN + He }Ξn (R)Ψn (r; R) = E
Ξn (R)Ψn (r; R)
n
(30)
n
and use that Ψn are eigenfunctions of He
X
X
{TN + Un (R)}Ξn (R)Ψn (r; R) = E
Ξn (R)Ψn (r; R)
n
(31)
n
The nuclear kinetic energy operator leads to electron-nuclear coupling terms related to the variation
of the electronic wave function with respect to the nuclear coordinates
1X
µα 4α Ψn (r; R)Ξn (R) =
2 α
1X −
µα Ψn 4α Ξn + 2(∇α Ψn ∇α Ξn ) + Ξn 4α Ψn
2 α
−
(32)
Multiply from left by Ψ∗k (r; R), integrate over the electronic variables and denote the off-diagonal
coupling matrix elements by
X
Bkn = hΨk |TN |Ψn i −
µα hΨk |∇α |Ψn i∇α
(33)
α
7
The exact solution of the nuclear problem
{TN + Uk (R) + hΨk |TN |Ψk i − E}Ξk (R) =
X
Bkn Ξn (R)
(34)
n6=k
requires the complete set of electronic eigenfunctions Ψn .
4.1
Born-Oppenheimer approximation
Provided the energy gap of electronic states is large, the matrix elements of T and ∇α are small. In
this case the nuclei move on a single potential energy surface, Un (R), and the total wave function
can be written in a simple product form
Υ(R, r) = Ξ(R)Ψ(r; R)
Born-Oppenheimer approximation
⇒ all electron-nuclear coupling terms are neglected
{TN + Ek (R)}Ξk (R) = EΞk (R)
Adiabatic approximation
⇒ diagonal coupling term is retained
{TN + Ek (R) + hΨk |TN |Ψk i}Ξk (R) = EΞk (R)
4.2
(35)
(36)
Breakdown of the Born-Oppenheimer approximation
The hΨi |∇α |Ψj i coupling terms can be neglected only if the electronic states are
well-separated.
Differentiate the electronic Schrödinger equation, multiply from left by Ψi , i 6= j and integrate
hΨi |
∂He
∂Ψj
∂Ui
∂Ψj
|Ψj i + hΨi |He |
i=
hΨi |Ψj i + Uj hΨi |
i
∂Rα
∂Rα
∂Rα
∂Rα
(37)
∂Ui
∂He
|Ψj i + Ui hΨi |∇α |Ψj i =
δij + Uj hΨi |∇α |Ψj i
∂Rα
∂Rα
(38)
after rearranging
hΨi |
leading to the condition
hΨi |∇α |Ψj i = hΨi |
∂He
|Ψj i/(Uj − Ui )
∂Rα
(39)
which indicates that a smaller energy gap leads to an enhanced coupling term (e.g. avoided crossings).
5
Electron Hamiltonian and wave function
Electronic Schrödinger equation
or
Te + VeN + Vee + VN N Ψ = EΨ
X
i
ĥ(i) +
X
0
ĝ(i, j) + VN N Ψ = EΨ
i,j
The many-electron wave function Ψ(x1 , x2 , . . . , xN )
8
(40)
(41)
1. describes a definite spin state
Ĥ Ŝ 2 − Ŝ 2 Ĥ = 0
2. is normalized
Z
therefore
Z
∗
Ψ Ψdx =
Ŝ 2 Ψ = S(S + 1)Ψ
|Ψ|2 dx = 1
(42)
(43)
3. satisfies the antisymmetry requirement
P̂ Ψ = εP Ψ
(44)
where εP = ±1 for a permutation of the arguments x1 , x2 , . . . , xN comprising an even/odd
number of interchanges.
In the following we discuss briefly some implications of these properties on the construction of the
many-electron wave function.
5.1
Electron spin
Although the molecular Hamiltonian is usually spin-independent, we know from experiment (Zeeman effect) that electrons have an intrinsic magnetic moment. Therefore we introduce
spin angular momentum operators ŝα acting on the spin variable, σ
ŝz η(σ) = λη(σ)
with two permitted solutions:
1
α(σ)
2
1
ŝz β(σ) = − β(σ)
2
ŝz α(σ) =
5.2
(45)
Spin operators and spin functions
Spin variable σ with possible values
σ = ±1
(46)
Two spin functions
α( 21 ) = 1
α(− 12 ) = 0
β( 12 ) = 0
β(− 12 ) = 1
(47)
orthonormalized
Z
α(σ)α(σ)dσ =
X
β(σ)β(σ)dσ =
X
σ
Z
1
1
α2 (σ) = α2 ( ) + α2 (− ) = 02 + 12 = 1
2
2
β 2 (σ) = 12 + 02 = 1
(48)
σ
Z
α(σ)β(σ)dσ =
X
α(σ)β(σ) = 0 × 1 + 1 × 0 = 0
σ
eigenfunctions of ŝz and ŝ2 operators (~ = 1)
ŝ2 α(σ) = 34 α(σ)
ŝz α(σ) = 12 α(σ)
ŝ2 β(σ) = 34 β(σ)
ŝz β(σ) = − 12 β(σ)
(49)
9
behave as follows with the ŝ± operators
ŝ+ α(σ) = 0
ŝ− α(σ) = β(σ)
(50)
ŝ+ β(σ) = α(σ) ŝ− β(σ) = 0
5.3
Total spin operators
Cartesian components of the total spin of an N-particle system
 (i)  

Ŝx
X
X  ŝx 
(i)
ŝ(i) =
Ŝ =
 ŝy  =  Ŝy 
(i)
i
i
Ŝz
ŝz
(51)
Satisfy the commutation rules
[Ŝα , Ŝβ ] = iŜγ
(αβγ = xyz, yzx, zxy)
(52)
Ŝ− = Ŝx − iŜy
(53)
Step-up and step-down operators
Ŝ+ = Ŝx + iŜy
in terms of which the Ŝ 2 = Ŝx2 + Ŝy2 + Ŝz2 operator is
Ŝ 2 = Ŝ− Ŝ+ + Ŝz + Ŝz2 = Ŝ+ Ŝ− − Ŝz + Ŝz2
(54)
The simultaneous eigenfunctions Θ(S, MS ) of Ŝ 2 and Ŝz satisfy
Ŝ 2 Θ(S, MS ) = S(S + 1)Θ(S, MS )
Ŝz Θ(S, MS ) = MS Θ(S, MS )
where § is a positive half-integer and MS = S, S − 1, . . . , −S.
Step-down and step-up operators act as
p
Ŝ± Θ(S, MS ) = S(S + 1) − MS (MS ± 1)Θ(S, MS ± 1)
5.4
(55)
(56)
Spin functions of a two-electron system
Form four possible spin-functions
α(σ1 )α(σ2 );
α(σ1 )β(σ2 );
β(σ1 )α(σ2 );
β(σ1 )β(σ2 )
(57)
One antisymmetric function
1
Θ(0, 0) = √ (α(σ1 )β(σ2 ) − β(σ1 )α(σ2 ))
2
(58)
Θ(1, 1) = α(σ1 )α(σ2 )
1
Θ(1, 0) = √ (α(σ1 )β(σ2 ) + β(σ1 )α(σ2 ))
2
Θ(1, −1) = β(σ1 )β(σ2 )
(59)
1 1
(2)
Ŝz α(σ1 )α(σ2 ) = (s(1)
z + sz )α(σ1 )α(σ2 ) = ( + )α(σ1 )α(σ2 )
2 2
(60)
Three symmetric functions
For instance
In the case of two electrons, an antisymmetric total wave function can always be constructed as
antisymmetric/symmetric combination of the space- and spin functions.
10
5.5
Space and spin orbitals
spatial orbitals φi (r)
• wave functions of a single particle
ĥ(r)φi (r) = i φi (r)
• describe the spatial distribution of an electron %i (r) = |φi (r)|2
• form an orthonormal set
Z
drφ∗i (r)φj (r) = hφi |φj i = δij
spin orbitals ψi (x)
• simultaneous eigenfunctions of ĥ(r) and ŝz
• products of spatial and spin functions
ψi (x) = φi (r)ηi (σ)
• orthonormality
Z
6
dxψi∗ (x)ψj (x) =
Z
drφ∗i (r)φj (r)
Z
dσα(σ)∗ β(σ) = δij δαβ
Density matrices
We do not necessarily need N-electron all information contained in the 4N -variable wave function
to obtain the energy and properties of the system. We introduce a hierarchy of functions that
depend explicitly on one, two, three, etc. particle coordinates.
The probability of finding electron 1 in dx1 = dr 1 dσ1 , while other electrons are anywhere, is
Z
dx1 Ψ∗ (x1 , x2 , . . . , xN )Ψ(x1 , x2 , . . . , xN )dx2 . . . dxN
(61)
and the probability of finding any of the N electrons in dx1 is N times this, which can be written
as P (x1 )dx1 , where
Z
P (x1 ) = N Ψ∗ (x1 , x2 , . . . , xN )Ψ(x1 , x2 , . . . , xN )dx2 . . . dxN
(62)
The probability of finding an electron in r 1 , without making reference to the spin coordinate is
Z
ρ(r 1 ) =
P (x1 )dσ1
This is the electron density function measured by X-ray crystallography.
The probability of finding any two electrons simultaneously in x1 and x2 is
Z
Γ(x1 , x2 ) = N (N − 1) Ψ∗ (x1 , x2 , . . . , xN )Ψ(x1 , x2 , . . . , xN )dx3 . . . dxN
(63)
(64)
and in the spinless version
Z Z
γ(r 1 , r 2 ) =
Γ(x1 )dσ1 dσ2
11
(65)
6.1
Reduced density matrices
For the purposes of evaluating one- and tow-electron operator expectation values, we need to
generalize the density functions as follows
Z
P (x1 ; x01 ) = N Ψ∗ (x1 , x2 , . . . , xN )Ψ(x01 , x2 , . . . , xN )dx2 . . . dxN
(66)
and
Γ(x1 , x2 ; x01 , x02 ) =
Z
Ψ∗ (x01 , x02 , . . . , xN )Ψ(x01 , x02 , . . . , xN )dx3 . . . dxN
N (N − 1)
The expectation value of a one-electron operator Ĥ1 =
hΨ|Ĥ1 |Ψi =
N Z
X
P
i
(67)
ĥ(i) is
Z
···
dx1 dxi−1 dxi+1 . . . dxN
i=1
Z
dxi Ψ∗ (x1 , x2 , . . . , xN )ĥ(i)Ψ(x01 , x2 , . . . , xN )
×
(68)
which can be written with the 1RDM as
Z
ĥ(1)P (x1 ; x01 )dx1
hΨ|Ĥ1 |Ψi =
(69)
x01 =x1
where we put x01 = x1 after the action of the operator on P (x1 ; x01 ), before performing the integral.
Similarly, for a two-electron operator
ZZ
hΨ|Ĥ2 |Ψi =
ĝ(1, 2)Γ(x1 ; x01 )dx1 dx2
(70)
x01 =x1
x02 =x2
Higher than 2RDMs are not needed to describe the interaction in a molecular system.
7
Construction of the wave function
Consider a simplified problem, by neglecting the electron-electron interaction, leading to a sum of
one-electron Hamiltonians
N
X
Ĥ =
ĥ(i)
(71)
i
which has as eigenfunctions
ΨHP = ψi (x1 )ψj (x2 ) . . . ψn (xN )
(72)
i.e. the product of one-electron functions, satisfying the following eigenequations
ĥ(i)ψj (xi ) = εj ψj (xi )
(73)
where the electron i occupies the spin orbital ψj (xi ) with energy εj . The energy of this Hartree
product is
hΨ|Ĥ|Ψi = εi + εj + εk + . . . + εn = E HP
(74)
This wave function is obviously not a physically appropriate one (violates the Pauli principle)
and the complete neglect of electron repulsion is a very crude approximation. Nevertheless, we
can examine whether the idea of writing the N-electron wave function as a linear combination of
product formed by one-electron functions, can lead to the exact solution.
12
7.1
Expansion theorem
Assume we have a complete set of functions {χi (x1 )} of a single variable x1 . Then any arbitrary
function of that variable can be expanded exactly as
X
Φ(x1 ) =
ai χ(x1 )
(75)
i
We can expand also a two-variable function of x1 and x2 on the same domain by holding x2 fixed
X
ai (x2 )χ(x1 )
(76)
Φ(x1 , x2 ) =
i
The ensemble of expansion coefficients ai (x2 ) for each fixed x2 coordinates, can be considered now
as a new function of a single variable, that can be expanded as
X
ai (x2 ) =
bij χj (x2 )
(77)
j
and substituting this expansion into that of Φ(x1 , x2 ), we have
X
Φ(x1 , x2 ) =
bij χi (x1 )χj (x2 )
(78)
ij
This process can be extended to functions Φ(x1 , x2 , . . . , xN ) of any number of variables.
Let us collect the space and spin coordinates of an electron to x. The one-electron functions, the
spin orbitals, χ(x) enter into the general expansion
X
Φ(x1 , x2 , x3 , . . . , xN ) =
bijk...N χi (x1 )χj (x2 )χk (x3 ) . . . χN (xN )
(79)
ijk...N
However, the wave function should be antisymmetric with respect to the exchange of coordinates
of two electrons. For two electrons
Φ(x1 , x2 ) = −Φ(x2 , x1 )
(80)
which implies for the expansion coefficients that bij = −bji and bii = 0, or
Φ(x1 , x2 ) =
X
bij {χi (x1 )χj (x2 ) − χj (x1 )χi (x2 )} =
i>j
X√
2bij det [χi (x1 )χj (x2 )]
(81)
i>j
More generally, an arbitrary N -electron function can be expanded as linear combination of all
possible N -electron determinants, formed from a complete set of one-electron spin orbitals, {χi (x)}.
Note that any complete set of one-electron spin orbitals can be used, in principle. Other possibilities
...
7.2
Configuration Interaction
Let us denote the N -electron determinants as ΦI , which will be supposed to be real and orthonormal. The exact wave function, Ψ expanded on the fixed basis of these determinants
X
Ψ=
cI ΦI
(82)
I
can be obtained by minimizing the expectation value of the Hamiltonian
X
X
hΨ|Ĥ|Ψi =
cI cJ hΦI |Ĥ|ΦJ i =
cI cJ HIJ
IJ
IJ
13
(83)
We suppose that Ψ is normalized, i.e.
hΨ|Ψi =
X
cI cJ hΦI |ΦJ i =
IJ
X
c2I = 1
(84)
I
Since we would like to preserve the normalization of the wave function, the constrained minimization method with Langrange multipliers is used, i.e. we minimize the function
L(c1 , c2 , . . . , cK , E) = hΨ|Ĥ|Ψi − E hΨ|Ψi − 1
X
X
=
cI cJ HIJ − E
c2I − 1
IJ
(85)
I
The minimum of this function can be found from the condition
X
∂L
=2
cJ HIJ − 2EcI = 0
∂cI
(86)
J
which is just the standard eigenvalue problem of the Hamiltonian matrix
Hc = Ec
(87)
Consider all the possible eigenvalues Eα of the Hamiltonian matrix
Hcα = Eα cα
(88)
Let us introduce the C and the diagonal matrix of the eigenvalues, E, then we have
HC = CE
The α-th eigenvalue is the expectation value of the Hamiltonian with respect to Φα
X β
X
hΦβ |Ĥ|Φα i =
cI hΨI |Ĥ|ΨJ icα
(cβ )† Hcα = Eα (cβ )† cα = Eα δαβ
J =
IJ
(89)
(90)
IJ
The lowest eigenvalue is upper bound to the ground state energy. Higher eigenvalues are upper
bounds to the energies of the excited states.
8
Method of determinants
We have seen that the exact N -electron wave function, satisfying the Pauli principle, can be
expanded in terms of antisymmetrized products of one-electron functions.
Such wave functions correspond to well-known mathematical objects, the determinants, which we
shall study in more details.
8.1
Permutation (symmetric) group
Consider N ordered indices
{1, 2, 3, . . . , N }
Define a permutation as
P {1, 2, 3, . . . , N } = {p1 , p2 , p3 , . . . , pN }
which means that 1 is replaced by p1 , etc.
14
Notation
1
p1
P =
2
p2
... N
. . . pN
Product of two permutations is also permutation
1 2 ... N
1 2 ... N
PQ = p p ... p
q1 q2 . . . qN =
1
2
N
q1 q2 . . . qN
1 2 ... N
= r r ... r
q1 q2 . . . qN =
1
2
N
1 2 ... N
= r r ... r
=R
1
2
(91)
N
Inverse permutation:
if
P =
1
p1
2
p2
...
...
N
pN
then
P −1 =
p1
1
p2
2
...
...
pN
N
(92)
Unity:
I=
1
1
2 ... N
2 ... N
(93)
Associativity:
(P Q)R = R(RQ)
(94)
The ensemble of the possible permutations of N objects forms SN , a symmetric group of rank N
and of order N !.
8.2
Antisymmetrizer
Projection operator, selects the antisymmetric component of a many-electron function.
1 X
 = √
P P
N ! P ∈SN
(95)
with
P = (−1)P
Idempotent: ÂÂ = const × Â
ÂÂ =
1
√
N!
2 X
P ∈SN
P P
X
Q Q =
1 X
N!
X
(−1)p+q P Q
(96)
P ∈SN Q∈SN
Q∈SN
for a given P , if Q runs over SN , the product R = P Q runs over all the N ! elements of SN :
"
#
X
√
1 X
r
(−1) R = N !Â
(97)
N!
P ∈SN R∈SN
|
{z
}
√
N !Â
Hermitian: † = Â, i.e. hÂΨ|Ψi = hΨ|ÂΨi
hÂΨ(1, 2, . . . N )|Ψ(1, 2, . . . N )i =
X
P hP −1 P Ψ(1, 2, . . . N )| P −1 Ψ(1, 2, . . . N )i
P ∈SN
15
(98)
Insertion of P −1 changes only the order of variables of integration. If P runs over all permutations,
then so does P −1 = Q too:
X
P hΨ(1, 2, . . . N )| P −1 Ψ(1, 2, . . . N )i =
P ∈SN
X
Q hΨ(1, 2, . . . N )| QΨ(1, 2, . . . N )i =
(99)
Q∈SN
hΨ(1, 2, . . . N )| ÂΨ(1, 2, . . . N )i
8.3
Why antisymmetrizer?
Consider a Q ∈ SN then
1 X
1 X
QÂ = √
(−1)q (−1)q (−1)p QP = (−1)q √
(−1)r R = (−1)q Â
N ! P ∈SN
N ! R∈SN
(100)
Apply Q, a permutation of the N variables in Φ(1, 2, . . . , N ) to
Ψ(1, 2, . . . , N ) = ÂΦ(1, 2, . . . , N )
, where Φ(1, 2, . . . , N ) is an arbitrary function.
QΨ(1, 2, . . . , N ) = QÂΦ(1, 2, . . . , N )
= (−1)q ÂΦ(1, 2, . . . , N )
= (−1)q Ψ(1, 2, . . . , N )
(101)
After the action of Â, the function Φ(1, 2, . . . , N ) becomes antisymmetric.
9
Determinant wave function
The wave function obtained after the action of the antisymmetrizer on a Hartree-product can be
written as a determinant:
DK = Â[φk1 (1)φk2 (2) . . . φkN (N )]
1
= √ det [φk1 (1)φk2 (2) . . . φkN (N )]
N!
φk1 (1) φk2 (1) . . . φkN (1) 1 φk1 (2) φk2 (2) . . . φkN (2) DK = √ .
..
..
..
N ! ..
.
.
.
φk (N ) φk (N ) . . . φk (N )
1
2
N
9.1
(102)
(103)
Invariance properties of the determinant wave function
Mixing of the orbitals constituting the determinant leaves it invariant (up to a constant
factor).
Let us consider the wave function
1
Φ = Â [φ1 (2)φ2 (2) . . . φN (N )] = √ det {A}
N!
16
(104)
where Aij = φj (i). Apply a linear transformation on the set of orbitals{φk }
X
ψj =
Tkj φk
(105)
k
The determinant wave function built from the new set of orbitals is
1
Ψ = Â [ψ1 (2)ψ2 (2) . . . ψN (N )] = √ det {B}
N!
(106)
where Bij = ψj (i).
Using the transformation equation
ψj (i) = Bij =
X
Tkj φk (i) =
X
k
Aik Tkj = (AT)ij
(107)
k
Determinant of the product of two matrices is equal to the product of the determinants
1
1
1
Ψ = √ det B = √ det AT = √ det A det T = det T · Φ
N!
N!
N!
(108)
If the transformation is unitary U−1 = U† , then | det U| = 1 and the wave function is invariant
with respect to the unitary transformation of the spin orbitals.
Corollary: any determinant wave function can be written in terms of orthonormal spin orbitals.
Let the overlap matrix
Sij = hψi |ψj i =
6 δij
and choose T a the following orthogonalization transformation (Löwdin orthogonalization)
T = S−1/2 V
(109)
where V is an arbitrary unitary matrix. In fact,
X
X
X
hφi |φj i = h
T`i ψ` |
Tkj ψk i =
T`i∗ hψ` |ψk iTkj
=
`
k
X
Ti`† S`k Tkj
`k
†
= T ST ij
`k
= V† S−1/2 SS−1/2 V
= V† V ij = δij
9.2
(110)
ij
Physical meaning of the determinant wave function
Consider a two-electron system with spin orbitals ψ1 (x1 ) and ψ2 (x2 )
◦ Hartree product
1
Ψ(1, 2) = √ ψ1 (x1 )ψ2 (x2 )
2
Two particle density matrix
Γ(x1 , x2 ) = 2Ψ∗ (x1 , x2 )Ψ(x1 , x2 ) = |ψ1 (x1 )|2 |ψ2 (x2 )|2
(111)
the simultaneous probability of finding electron-one in dx1 at x1 and in x2 at x2 is a simple
product of the densities associated with the two electrons and the indistinguishability of the
electrons is not respected.
17
◦ Determinant with electrons having opposite spins
|Ψ(1, 2)i = Â|ψ1 (x1 )ψ2 (x2 )i = Â|φ1 (r 1 )α(σ1 )φ2 (r 2 )β(σ2 )i
Spin-dependent two-particle reduced density matrix
Γ(x1 , x2 ) =
1
2
{φ1 (r 1 )α(σ1 )φ2 (r 2 )β(σ2 ) − φ1 (r 2 )α(σ2 )φ2 (r 1 )β(σ1 )}
2
The spinless 2RDM is obtained by integration of the spin variables
Z
Z
1
γ(r 1 , r 2 ) = dσ1 dσ2 Γ(x1 , x2 ) =
|φ1 (r 1 )|2 |φ2 (r 2 )|2 + |φ1 (r 2 )|2 |φ2 (r 1 )|2
2
(112)
(113)
The indistinguishability of the electrons is reflected by the average of two terms, but the two
electrons are uncorrelated. In particular, if the space part of the orbitals is the same,
γ(r 1 , r 1 ) 6= 0
(114)
i.e. there is a finite probability of finding two electrons of opposite spins at the same point
in space.
◦ Determinant with electrons having parallel spins
|Ψ(1, 2)i = Â|ψ1 (x1 )ψ2 (x2 )i = Â|φ1 (r 1 )β(σ1 )φ2 (r 2 )β(σ2 )i
After integration the spin dependent 2RDM, we obtain
Z Z
γ(r 1 , r 2 ) =
dσ1 dσ2 Γ(x1 , x2 )
1
=
|φ1 (r 1 )|2 |φ2 (r 2 )|2 + |φ1 (r 2 )|2 |φ2 (r 1 )|2
2
(115)
− [φ∗1 (r 1 )φ2 (r 1 )φ∗2 (r 2 )φ1 (r 2 ) + φ1 (r 1 )φ∗2 (r 1 )φ2 (r 2 )φ∗1 (r 2 )]
Electrons of the same spin are correlated (Fermi hole) as it can be seen by calculating the
probability of finding two electrons at the same point r 1
γ(r 1 , r 1 ) = 0
(116)
In a Slater-determinant the motion of electrons with parallel spin is correlated but the motion of
electrons with opposite spin is not.
10
Matrix elements between determinant wave functions
Two determinants constructed from two sets of spin orbitals {ui } and {vj }:
U ∗ = Â [u∗1 (1)u∗2 (2) . . . u∗N (N )]
and
V = Â [v1 (1)v2 (2) . . . vN (N )]
Consider the product U ∗ V
U ∗ V = Â [u∗1 (1)u∗2 (2) . . . u∗N (N )] · Â [v1 (1)v2 (2) . . . vN (N )]
√
It can be written using that † =  and Â2 = N ! as
U ∗ V = [u∗1 (1)u∗2 (2) . . . u∗N (N )] · †  [v1 (1)v2 (2) . . . vN (N )]
√
= N ![u∗1 (1)u∗2 (2) . . . u∗N (N )]Â [v1 (1)v2 (2) . . . vN (N )]
= [u∗1 (1)u∗2 (2) . . . u∗N (N )] det [v1 (1)v2 (2) . . . vN (N )]
18
(117)
The determinant det {V } is multiplied N times by factors of type u∗i (i).
Since the multiplication of a determinant by a number is equivalent to multiplying one of its rows
by the same number, we can arrange this expression by attributing each multiplicative factor to
one of the rows in the following way
U ∗ V = det [u∗1 (1)v1 (1) · . . . u∗i (i)vi (i) · . . . u∗N (N )vN (N )]
= det {u∗i (i)vk (i)}
(118)
The product of two Slater determinants is also a determinant, formed from products of the oneelectron functions.
Matrix element of an operator Ô
Z
Z
hU |Ô|V i = · · · dx1 . . . dxN Ô(1, 2, . . . , N )
× det [u∗1 (1)v1 (1) · . . . u∗i (i)vi (i) · . . . u∗N (N )vN (N )]
10.1
(119)
Laplace expansion formulae
An arbitrary determinant can be expanded according to the Laplace expansion formulae.
1. First order expansion formula
det{A} =
X
ak (i)D(i|k)
(120)
k
where ak (i) = Aik , the minor, formed by the i-th row and k-th column, and D(i|k) is the
corresponding cofactor, i.e. the signed determinant of the matrix obtained by deleting from
the original one the k-th row and i-th column:
A12 . . . A1,k−1
A1,k+1 . . .
A1N A11
A
A22 . . . A2,k−1
A2,k+1 . . .
A2N 21
.
.
.
.
.
.
..
..
..
..
..
..
..
.
k+i A
Ai−1,N (−1)
i−1,1 Ai−1,2 . . . Ai−1,k−1 Ai−1,k+1 . . .
Ai+1,N Ai+1,1 Ai+1,2 . . . Ai+1,k−1 Ai+1,k+1 . . .
.
..
..
..
..
..
..
.
.
.
.
.
.
.
.
A
AN 2 . . . AN,k−1 AN,k+1 . . . AN N (N )
N1
Applied to the product of two determinants:
U∗ · V =
X
u∗i (i)vk (i) D(k|i)
(121)
k
2. Second-order Laplace expansion formula
1 X Aik
det{A} =
Ajk
2
k`
Ai` D(ij|k`)
Aj` (122)
where the minor is a 2×2 determinant formed by the k and `-th rows and i and j-th columns,
and D(ij|k`) is the corresponding cofactor, i.e. the signed determinant of the matrix obtained
by deleting from the original one the k and `-th row and i and j-th column.
Applied to the product of two determinants:
X u∗ (i)vk (i) u∗ (i)v` (i) i
∗i
U∗ · V =
uj (j)vk (j) u∗j (j)v` (j) D(ij|k`)
k<`
19
(123)
10.2
Overlap of two determinants
Applying the general result for Ô = 1
Z
Z
hU |V i = · · · dx1 . . . dxN det [u∗1 (1)v1 (1) · . . . u∗i (i)vi∗ (i) · . . . u∗N (N )vN (N )]
Z
Z
Z
∗
∗
∗
= det
dx1 u1 (1)v1 (1) · . . . dxi ui (i)vi (i) · . . . dxN uN (N )vN (N )
(124)
which is an antisymmetrized product of N one-particle overlap integrals.
hU |V i = det [hu1 |v1 ihu2 |v2 i . . . huN |vN i] = det {hui |vj i}
(125)
determinant of the overlap matrix Sui vj formed from the two sets of spin orbitals
hU |V i = det {S}
10.3
(126)
Matrix elements of one-electron operators
Let H1 a the symmetric sum of h(i) one-electron operators:
X
Ĥ1 =
ĥ(i)
(127)
i
First-order Laplace expansion formula of the product of determinants and put into the general
result
X XZ
∗
hU |Ĥ1 |V i =
dxi ui (i)ĥ(i)vk (i) ×
i
k
Z
×
Z
···
dxi . . . dxi−1 dxi+1 . . . dxN D(k|i)
(128)
The first part is just a one-electron integral. The second part, analogous to the overlap integral
discussed above, is the ak (i) = Aik cofactor of the determinant of the overlap matrix hui |vk i.
D(k|i) is the determinant of the overlap matrix obtained by deleting the i-th row and k-th column.
The standard expression of an inverse matrix
S−1
ik
=
D(k|i)
det{S}
(129)
the matrix elements of a one-electron operator can be written in terms of the one-electron integrals,
the determinant of the overlap matrix and its inverse
X
hU |H1 |V i =
hui |h|vk i S−1 ik det{S}
(130)
i,k
10.4
Matrix elements of two-electron operators
Let us consider the two-electron symmetric operator
1 X0
Ĥ2 =
ĝ(i, j)
2 i,j
(131)
Taking the second-order Laplace expansion of the determinant product in the matrix element
expression and using the expression for the integral of the second-order cofactor, D(ij|k`)
D(ij|k`) = S−1 ki S−1 `j − S−1 kj S−1 `i det{S}
(132)
20
we have the following expression
X XZ Z
hU |Ĥ2 |V i =
dxi dxj [u∗i (i)vk (i)ĝ(i, j)u∗j (j)v` (j)−
i<j k<`
−
u∗i (i)v` (i)ĝ(i, j)u∗j (j)vk (j)]
×
10.5
−1
S
−1
ki
S
−1
`j
− S
−1
kj
S
`i
det{S}
(133)
Notations for two-electron integrals
Two-electron integral of spin orbitals ψ(x) = φ(x)η(σ) in the bra-ket notation
Z Z
dx1 dx2 ψi∗ (1)ψk (1)ĝ(1, 2)ψj∗ (2)ψ` (2) = hψi∗ (1)ψj∗ (2)|ψk (1)ψ` (2)i
(134)
= hij|k`i
This is the physicist’s notation, respecting the conventions for the bra and ket vectors
hij| = ψi∗ (1)ψj∗ (2)
|k`i = ψk (1)ψ` (2)
The antisymmetrized two-electron integral, appearing in the matrix elements of the the two-electron
operators, is usually denoted as
hij|k`i − hij|`ki = hij||k`i
(135)
The chemist’s notation emphasizes that a two-electron integral is the electrostatic interaction of
two charge distributions
Z Z
dx1 dx2 ψi∗ (1)ψk (1) ĝ(1, 2) ψj∗ (2)ψ` (2) = [ik|j`]
(136)
{z
}
|
| {z }
ρik (1)
ρj` (2)
In numerical calculations is more advantageous to express integrals directly in spatial orbitals. The
transformation is obvious using the orthogonality of the spin functions
[ik|j`] = (φi φk |φj φ` )δηi ηk δηj η`
(137)
In particular,
(
parallel spin
hij||iji =
antiparallel spin
10.6
[ii|jj] − [ij|ij] = (ii|jj) − (ij|ij)
[ii|j̄ j̄] − [ij̄|ij̄] = (ii|jj)
(138)
Summary of matrix element rules
◦ overlap
hU |V i = det {S}
(139)
◦ one-electron operator
hU |Ĥ1 |V i =
X
hui |ĥ|vk i S−1
ik
det{S}
(140)
i,k
◦ two-electron operator
XX
hU |Ĥ2 |V i =
hij||k`i S−1 ki S−1 `j − S−1 kj S−1 `i det{S}
i<j k<`
21
(141)
10.7
Slater rules
We need these results applied to the special case of determinants built from a unique set of orthonormal spin orbitals, {ψi }. Let D such a determinant, any other determinant can be generated
by replacing one, two, or more spin orbitals {a, b, c, d, . . .} occupied in D by other spin orbitals,
{r, s, t, u, . . .} not occupied in D
D(a → r) = Dar
rs
D(a → r, b → s) = Dab
rst
D(a → r, b → s, c → t) = Dabc
Overlap
hD|Di = det {S}
(142)
since Sab = δab
hD|Di = 1
(143)
If the two determinants differ in at least one orbital
hD|Dar i = 0
(144)
since due to the orthogonality of the orbitals Sar = 0 ∀ a, i.e. one row in the determinant
det {S} is zero.
One-electron matrix elements
hU |H1 |V i =
X
hui |h|vk i S−1
ik
det{S}
i,k
• U = V = D, since Sik = δik and det {S} = 1
hD|Ĥ1 |Di =
X
hψi |ĥ|ψi i
(145)
i
• V =D
and U = Dar only one cofactor is nonzero
hDar |Ĥ1 |Di = hψr |ĥ|ψa i
• V =D
(146)
rs
and U = Dab
all cofactors are zero
rs
hDab
|Ĥ1 |Di = 0
(147)
Two-electron matrix elements
hU |Ĥ2 |V i =
XX
−1
−1
−1
−1
hij||k`i S
S
− S
S
det{S}
ki
`j
kj
`i
i<j k<`
• U = V = D then D(ij|k`) = δik δ`j since i = ` and k = j cases cannot occur
hD|Ĥ2 |Di =
XX
=
XX
hik||j`i δik δj` − δjk δi`
i<j k<`
(148)
hij||k`iδik δj` =
i<j k<`
22
X
i<j
hij||iji
• U = Dar apply the general formula
hU |Ĥ2 |V i =
XX
hui uj ||vk v` iD(ij|k`)
(149)
i<j k<`
the S matrix is diagonal, excepted the Saa element.
• i = k = a case: D(aj|a`) = δj` and sum over ` = j > a
X
hr`||a`i
`>a
• j = ` = a case: D(ia|ka) = δik and sum over k = i < a
X
X
hkr||kai =
k<a
hr`||a`i
`<a
Summarizing these two cases
hDar |Ĥ2 |Di =
N
X
hrj||aji
(150)
j=1
j6=a
rs
• U = Dab
only one non-zero cofactor, D(ab|ab) = 1
rs
hDab
|Ĥ2 |Di = hrs||abi
10.8
(151)
Variation of a determinant wave function
The determinant
|Ψi = A[ψ1 (1)ψ2 (2) . . . ψN (N )]
(152)
can be varied through the variation of the one-electron orbitals:
ψa → ψa + δψa = ψa + pa ψa 0
(153)
where ψa0 is an arbitrary spin orbital and pa is a complex variation parameter.
0
|Ψ + δΨi = A [{ψ1 (1) + p1 ψ10 } {ψ2 (2) + p2 ψ20 } . . . {ψN (N ) + pN ψN
}]
(154)
This is a sum of 2N determinant. To first order in the pa parameters the variation is a sum of
singly excited determinants:
X
|δΨi =
pa |Ψ(ψa → ψa0 )i
(155)
a
Let the Q̂ the projection operator on the subspace of the orbitals that are occupied in the determinant and P̂ the projector to its orthogonal complement
Q̂ =
N
X
|ψa ihψa |
P̂ = 1 − Q̂ =
i
X
|ψr ihψr |
(156)
r∈
/{a}
The arbitrary spin orbital ψa0 can be decomposed as:
pa |ψa0 i =
N
X
qba |ψb i +
X
r
b
23
pra |ψr i
(157)
where the set of orbitals {ψr } are taken from the orthogonal complement of the occupied set {ψa }.
PN
b qba ψb does not change the determinant, up to a factor of normalization.
An arbitrary variation of a determinant can be written as a sum of singly excited determinants
|δΨi =
N X
X
a
11
pra |Ψra i
(158)
r
Method of Second Quantization
Up to now, we satisfied the antisymmetry requirement by using determinant wave function. The
same goal can be achieved by transferring this antisymmetry property on the algebraic properties
of electron creation and annihilation operators.
No essentially new physics, just a convenient way to handle wave functions, operators and matrix
elements.
11.1
Representations of determinant wave functions
The determinant wave function can be considered as an ordered product of occupied spin orbitals
D(x1 , x2 , . . . , xN ) ⇔ |ψ1 ψ2 . . . ψa . . . ψb . . . ψN |
(159)
The set of occupied spin orbitals is the electron configuration.
Particle number representation:
the number of electrons filling each of the orbitals occupation number is given explicitly (ni =
0 or 1)
D(x1 , x2 , . . . , xN ) ⇔ |n1 n2 . . . nk . . . n` . . . nM i
(160)
For instance:
11.2
|Di
=|11111 . . . 1 1 00 . . . 000 . . . 000i
|Dar i
=|11101 . . . 1 1 00 . . . 100 . . . 000i
N
a
N
r
Fock space
The Fock space is the vector space constituted of all the possible “kets”, corresponding to the total
number of electrons between 0 and M .
The vacuum state is an abstract state with 0 electron:
|vaci = |01 02 03 . . . 0M i
(161)
Properties of the vacuum:
• normalization
hvac|vaci = 1
(162)
• orthogonal to all other states
Second quantized operators manipulate the occupation numbers of the vectors in the Fock-space.
24
11.3
Creation of electrons
A one-electron state, ψk (1) can be represented by the creation of an electron on the spin orbital
ψk
|ki = a+
(163)
k |vaci
A two-electron state, Ψ(1, 2) is represented in the wave function language as a Slater determinant
1 ψi (1) ψk (1)
(164)
D(1, 2) = √ 2 ψi (2) ψk (2)
The same state is written in a second quantization language as
+
D(1, 2) ⇔ |iki = a+
i ak |vaci
(165)
As a consequence of the antisymmetry property of D(1, 2)
D(1, 2) = −D(2, 1)
(166)
we have
+
+ +
a+
i ak |vaci = −ak ai |vaci
(167)
The antisymmetry of the wave function is translated as an algebraic property of the creation
operators:
+
+ +
a+
(168)
i ak + ak ai = 0
The creation operators of electrons (fermions) anticommute:
+
[a+
k , ai ]+ = 0
(169)
If k = i
+
+ +
a+
i ai = −ai ai = 0
(170)
We are not allowed to create electrons twice on the same spin orbital:
Pauli principle.
11.4
Annihilation of electrons
The operator that removes an electron from a spin orbital is called the annihilation operator
ai |ii = |vaci
(171)
We are not allowed to remove an electron from the vacuum
ai |vaci = 0
(172)
In order to annihilate an electron on the orbital ψi , this orbital should be immediately to the right
of the annihilator, e.g.
+
+
ai |iki = ai a+
(173)
i ak |vaci = ak |vaci = |ki
If the position of the spin orbital ψi was not appropriate, we must do transpositions until it is
placed to the right of ai
+
+ +
+
ai |kii = ai a+
k ai |vaci = −ai ai ak |vaci = −ak |vaci = −|ki
(174)
The sign depends on the number of transpositions:
• even number of transpositions: + sign
+ +
+ + +
+ + +
+ +
a3 a+
2 a1 a3 |vaci = −a3 a2 a3 a1 |vaci = +a3 a3 a2 a1 |vaci = a2 a1 |vaci
(175)
• odd number of transpositions: − sign
+ +
+ + +
+ +
a3 a+
2 a3 a1 |vaci = −a3 a3 a2 a1 |vaci = −a2 a1 |vaci
25
(176)
11.5
Commutation relations
Let us consider an N -electron determinant in the particle number representation
|Di = |n1 n2 n3 . . . ni . . . nk . . . nM i
(177)
and reorder the orbitals so that ni and nk be in the first two positions
|Di = ±|ni nk n1 n2 n3 . . . nM i
(178)
Study the effect of ai a+
k on this determinant
ai a+
k |ni nk n1 n2 n3 . . . nM i
(179)
This expression is zero, if
◦ nk = 1 cannot create e− on an already occupied orbital (Pauli principle)
◦ ni = 0 cannot annihilate an e− on an empty orbital
Without loosing the generality of our conclusions, we can study the effect of ai a+
k on |ii
ai a+
k |ii = ai |kii = −ai |iki = −|ki
(180)
Similarly, for the operators in the reverse order
+
a+
k ai |ii = ak |vaci = |ki
(181)
This is an “excitation operator”, i.e. removes an electron from i and puts it on k.
By the virtue of the above results, for an arbitrary determinant
+
(ai a+
k + ak ai )|Di = 0
(182)
which is equivalent to the following relationship between the operators (i 6= k)
+
ai a+
k + ak ai = 0
(183)
Let us examine the case of i = k. Two possibilities
ni = 0
ni = 1
a+
a
|n
.
.
.
n
.
.
.
n
i
=
0
|n
.
.
.
ni . . . nM i
i
1
i
M
1
i
ai a+
|n
.
.
.
n
.
.
.
n
i
=
|n
.
.
.
n
.
.
.
n
i
0
1
i
M
1
i
M
i
After summing the two equations we obtain
+
ai a+
i + ai ai = 1̂
(184)
The two results can be summarized in the anticommutation relation
[ai , a+
k ]+ = δik
11.6
(185)
Adjoint relations
The role of the operator ai is to annihilate the effect of a+
i
ai a+
k |vaci = δik |vaci
(186)
Let us consider the functions ψi and ψk , members of an orthonormal set of functions {ψi }M
i=1 , i.e.
Sik = hi|ki = δik
26
(187)
The ψk function can be written as
|ki = a+
k |vaci
(188)
and the adjoint of the analogous definition for ψi = |ii
hi| = hvac| a+
i
†
(189)
The scalar product, i.e. the overlap integral is
Sik = hvac| a+
i
Since Sik = δik is diagonal
a+
i
†
†
a+
k |vaci
(190)
a+
k |vaci = δik |vaci
(191)
After comparison with Eq.(186) we conclude that
ai = a+
i
†
(192)
i.e. in the case of orthonormal basis functions, the annihilation operator is the adjoint of the
creation operator.
The “bra” corresponding to the “ket”
+ +
|Di = a+
N . . . a2 a1 |vaci
is
hD| = hvac|a1 a2 . . . aN
11.7
Non-orthogonal functions
Overlap matrix is non-diagonal
Sµν = hχµ |χν i = hvac| χ+
µ
Introduce the notation
χ+
µ
†
†
χ+
ν |vaci
= χ−
µ
(193)
(194)
Following anticommutation relation can be derived
−
[χ+
ν , χµ ]+ = Sµν
(195)
P
−1/2
Take an auxiliary Löwdin-orthogonalized basis ψλ =
µ Sλµ χµ . The creation and and annihilation operators can be transformed to this basis as
X −1/2
X 1/2
ψλ+ =
Sλµ χ+
χ+
Sµλ ψλ+
(196)
µ
µ =
µ
ψτ−
=
X
λ
−1/2 −
Sντ
χν
χ+
ν
=
ν
X
−
Sτ1/2
ν ψτ
(197)
τ
Applying these transformations
X 1/2
X 1/2
+
−
−
[χ+
Sµλ Sτ1/2
Sµλ Sτ1/2
ν , χµ ]− =
ν [ψλ , ψτ ]+ =
ν δλτ = Sµν
λτ
(198)
λτ
The operator χ−
µ is not a true annihilation operator, for instance
χ−
µ |χµ χλ i = Sµµ |χλ i − Sµλ |χµ i
27
(199)
11.8
Properties of creation and annihilation operators: Summary
Properties of the vacuum
hvac|vaci = 1
ak |vaci = 0
hvac|a+
k =0
Anticommutation relations (Pauli principle)
+
[a+
i , ak ]+ = 0
[ai , ak ]+ = 0
[a+
i , ak ]+ = δik
Adjoint relation
ak = a+
k
†
Correspondences
|ψi i ⇔ a+
i |vaci
+ +
|Di ⇔ a+
N . . . a2 a1 |vaci
hψi | ⇔ hvac|ai
|Di ⇔ hvac|a1 a2 . . . aN
11.9
Particle number operator
A sequence of annihilation and creation operators acting on the same orbital
n̂i = a+
i ai
“measures” the occupation number of this orbital, i.e. tells if it is occupied (ni = 1) or if it is
empty (ni = 0)
n̂i |Di = a+
i ai |n1 . . . ni . . . nM i = ni |n1 . . . ni . . . nM i = ni |Di
(200)
Determinant wave function is eigenfunction of n̂i with eigenvalue ni .
Operator of the total number of particles
N̂ =
M
X
n̂i =
i=1
M
X
a+
i ai
(201)
i=1
Single determinants are eigenfunctions of N̂
N̂ |Di =
M
X
ni |Di = N |Di
(202)
i=1
as well as multi-determinant wave functions
N̂ |Ψi =
X
K
cK N̂ |DK i =
X
K
cK
M
X
niK |DK i =
i=1
28
X
K
cK N |DK i = N |Ψi
(203)
11.10
Representation of operators
In the first quantization representation an operator ÂS transforms the function Ψ to another one
Φ
ÂS |Ψi = |Φi
We are looking for the second quantized operator that performs the analogous transformation
between wave functions in particle number representation.
ÂS
|Φi
−−−−−−−−→
?
⇓
⇓
|Ψi
⇓
ÂF
|Φ± i −−−−−−−−→ |Ψ± i
The operator ÂS acts on one of the functions φi ∈ {φk }M
k=1 of the orthonormal set. The transformed
function ψi can be developed in the same basis {φk }M
k=1 .
Let us follow step-by-step the correspondences between the first- and second-quantized picture:
FIRST QUANTIZATION
SECOND QUANTIZATION
+
ÂF φ+
i |vaci = ψi |vaci
ÂS |φi i = |ψi i
|ψi i =
ψi+ =
P
k cik |φk i
ÂS |φi i =
P
hφ` |ÂS |φi i =
P
ÂF φ+
i |vaci =
k cik |φk i
k cik hφ` |φk i
+
k cik φk |vaci
P
⇓
ÂF φ+
i =
⇒
A`i = ci`
+
k cik φk
P
P
k
Aki φ+
k
Complete this expression formally by φ−
i
X
−
ÂF φ+
i φi =
−
Aki φ+
k φi
k
Sum on all basis functions φi
ÂF
X
−
φ+
i φi =
X
i
|
−
Aki φ+
k φi
ki
{z
}
N̂
The particle number operator acts as a unit operator on one-electron functions, therefore we can
write
X
−
ÂF =
Aki φ+
(204)
k φi
ki
11.11
One-electron operators
In the Schrödinger representation
ÂS =
N
X
i=1
29
Â(i)
(205)
The action of this operator on a Slater determinant leads to a linear combination of Slater determinants
N X
X
ÂS |Φi = |Ψi =
cik |φ1 . . . φk . . . φN |
(206)
i
k
The second-quantized counterpart of this expression is
+
+
ÂF φ+
N . . . φi . . . φ1 |vaci =
N X
M
X
i
+
+
Aki φ+
N . . . φk . . . φ1 |vaci
(207)
k
We shall prove that the above result is obtained, if we apply the previously “derived” form of the
second-quantized operator, i.e.
M X
M
X
−
ÂF =
Aki φ+
(208)
k φi
i
k
Let us study the expression
M X
M
X
i
− +
+
+
Aki φ+
k φi φN . . . φi . . . φ1 |vaci
(209)
k
the orbital i should be occupied, otherwise φ−
i would give zero ⇒ summation over i is limited to
N instead of M
migrate φ+
i to the first position:
+
+
+
+
±φ+
i φN . . . φi−1 φi+1 . . . φ1 |vaci
using the anticommutation relation and that the spin orbital i has already been moved out of the
string
− +
+
+ −
+
φ+
k φi φi = φk (1 − φi φi ) = φk
| {z }
n̂i
+
migrate φ+
k back to the place of φi . We need the same number of permutations, therefore we
recover the possible sign change
In effect, we retrieve the expression
+
+
ÂF φ+
N . . . φi . . . φ1 |vaci =
N X
M
X
i
+
+
Aki φ+
N . . . φk . . . φ1 |vaci
k
The second-quantized operator does not depend on the number of electrons.
11.12
Two-electron operators
Two-electron operator in the Schrödinger picture
ÂS =
N
X
i<j
30
Â(ij)
(210)
e.g. electron repulsion operator
V̂ee =
N
X
1
r
i<j ij
transforms a pair of orbitals simultaneously
X
Â(ij)|φ1 . . . φi . . . φj . . . φN |
N X
X
i>j
cij,k` |φ1 . . . φk . . . φ` . . . φN |
(211)
i<j k`
with
Z Z
cij,k` =
dx1 dx2 φ∗k (1)φi (1)
1 ∗
φ (2)φj (2) = hk`|iji
r12 `
Following an analogous reasoning, we obtain the following expression for the second-quantized form
of the electron repulsion operator
F
V̂ee
=
1X
+ − −
hij|k`iφ+
i φj φ` φk
2
(212)
ijk`
Note the reversed order of indices k and `!
11.13
Reduced density matrices
The expectation value of a one-electron operator
X
 =
Aµν a+
µ aν
µν
is
hÂi =
X
Aµν ha+
µ aν i
(213)
µν
where the quantity is brackets is the one-particle reduced density matrix (1RDM)
Pνµ = ha+
µ aν i
The expectation value of a one-electron operator can be written as a trace
X
X
hÂi =
Aµν Pνµ =
(AP )µµ = Tr(AP )
µν
(214)
(215)
µ
Analogously, the expectation value of a two-electron operator
hÂi =
1 X
+
Aµνλσ ha+
µ aν aσ aλ i
2
(216)
1 X
Aµνλσ Γλσµν
2
(217)
µνλσ
can be written as a trace
hÂi =
µνλσ
with the two-particle reduced density matrix, Γλσµν
+
Γλσµν = ha+
µ aν aσ aλ i
31
(218)
The symmetry properties of Γ are determined by Hermitian symmetry and anticommutation relations
+ +
+
Γλσµν = ha+
µ aν aσ aλ i = haλ aσ aν aµ i = Γµνλσ
+
= −ha+
µ aν aλ aσ i = −Γσλµν
(219)
+
= −ha+
ν aµ aσ aλ i = −Γλσνµ
+
= ha+
ν aµ aλ aσ i = Γσλνµ
The diagonal elements are zero (Pauli principle)
Γµµλσ = Γµµλλ = 0
(220)
Normalization of the 2RDM
X
X
X
TrΓ =
Γµνµν =
hν + µ+ µ− ν − i = −
hµ+ ν + µ− ν − i
µν
=−
µν
X
µν
− +
+
−
hµ {δµν − ν µ }ν i
µν
=−
X
δµν hµ+ ν − i − hµ+ µ− ν + ν − i
(221)
µν
=−
X
hn̂µ i +
µ
X
hn̂µ n̂ν i
µν
= N2 − N
The 1RDM is derivable from the 2RDM by contraction
X
X
Γνλµλ =
hµ+ λ+ λ− ν − i = (1 − N )hµ+ ν − i = (1 − N )Pνµ
λ
11.14
(222)
λ
Hamiltonian and energy expressions
M
In an orthonormal basis {φµ }µ=1 the electronic Hamiltonian is
Ĥ =
X
hµν a+
µ aν +
µν
1 X
+
hµν|λσia+
µ aν aσ aλ
2
(223)
µνλσ
The SQ Hamiltonian
◦ is not equal to the usual one: it is a projection on a finite (M -dimensional) space of oneelectron basis functions
◦ is independent of the number of electrons - this dependence is shifted to the wave function
The electronic energy – expectation value of the Hamiltonian – can be calculated using the oneand two-particle RDMs
E = hΨ|Ĥ|Ψi =
X
hµν Pνµ +
µν
1 X
hµν|λσiΓλσµν
2
µνλσ
This form of the energy expression is used in current quantum chemical calculations.
32
(224)
An alternative energy expression is based on the reduced two-particle Hamiltonian, defined as
1 X µν + + − −
Kλσ aµ aν aσ aλ
2
(225)
1
(δνσ hµλ + δµλ hνσ ) + hµν|λσi
1−N
(226)
Ĥ =
µνλσ
where the matrix elements are
µν
Kλσ
=
and the energy can be written in terms of the 2RDM
X µν
E=
Kλσ Γλσµν
(227)
µνλσ
(cf. DFT, where the energy is a functional on the one-particle density). Unconstrained variation of Γλσµν leads to much too low energies, since Γ should satisfy additional N-representability
conditions.
Rosina’s theorem: The ground state 2RDM completely determines the exact N-particle ground
state wave function (and p-RDMs) without any specific information about the Hamiltonian.
Contracted Schrödinger equations are of the following structure
K·
11.15
2
D+K ·
3
D+K ·
4
D=E·
2
D
Calculation of matrix elements
Let us consider first operator strings with vacuum:
1. we cannot annihilate on the vacuum
hvac|a+
i ak |vaci = 0
(228)
2. use the anticommutation rule and the previous result
+
hvac|ai a+
k |vaci = hvac|(δik − ak ai )|vaci = δik
(229)
+
hvac|ai aj a+
k a` |vaci =?
3.
+
+
+
+
+
ai aj a+
= ai a+
=
k a` = ai (δkj − ak aj )a`
l δkj − ai ak aj a`
+
+
(δi` − a+
` ai )δkj − (δik − ak ai )(δj` − al aj ) = δi` δkj − δik δjl + zeros
(230)
A graphical evaluation rule can also be deduced
+
hvac|ai aj a+
k a` |vaci = δi` δkj − δik δjl
11.16
Matrix elements between single determinants
Let us calculate
−
−
− + − +
+
hHF|φ+
i φk |HFi = hvac|φ1 . . . φN φi φk φN . . . φ1 |vaci
We proceed by steps
33
(231)
1. if i,k ∈/{1, . . . , N }, the matrix element is zero
2. if i, k ∈ {1, . . . , N }, we order the pairs aa a+
a
transpositions
(a 6= i, k; i 6= k) by an even number of
− + − + + − +
− +
− − + − + +
hvac|φ−
i φk φi φk φi φk φN φN . . . φ1 φ1 |vaci = hvac|φi φk φi φk φi φk |vaci = 0
3. if i, k ∈ {1, . . . , N } and i = k, we order all pairs a 6= i
+ − + − +
− +
− + − +
hvac|φ−
i φi φi φi φN φN . . . φ1 φ1 |vaci = hvac|φi φi φi φi |vaci = 1
Taking the sum of the three cases
−
hHF|φ+
i φk |HFi = ni δik
11.17
(232)
Fermi vacuum
Evaluation of matrix elements involving single determinant (Hartree-Fock, HF) wave functions can
be simplified, since the creation and annihilation operators that generate the hHF| and |HFi states,
respectively, have only a passive role.
−
− −
− +
−
+
+ +
hφ+
i . . . φk iHF = hvac|φ1 φ2 . . . φN [φi . . . φk ] φN . . . φ2 φ1 |vaci
|
{z
}
|
{z
}
hHF|
|HFi
Therefore we define a Fermi-vacuum, which has analogous properties to the true vacuum.
VACUUM
FERMI VACUUM
φ−
i |HFi = 0
ai |vaci = 0
hvac|a+
i =0
if
hHF|φ+
i =0
if
ni = 0
ni = 0
hvac|a+
i ak |vaci = 0
−
hHF|φ+
i φk |HFi = nk δik
hvac|ak a+
i |vaci = δik
+
hHF|φ−
k φi |HFi = (nk − 1)δik
The evaluation of matrix elements consists in simply to find all the combinations of the type
−
φ+
i φk
ak a+
i
and associate to them the factor
δik
nk δik
with a sign depending on the number of permutations.
11.18
Slater rules – an example
One-electron matrix element hDar |Ĥ1 |Di
Using
hDar | = hHF|a+
a ar
and
Ĥ1 =
X
hij a+
i aj
ij
the matrix element is
hDar |Ĥ1 |Di =
X
+
hij hHF|a+
a ar ai aj |HFi
ij
34
(233)
Taking the possible pairings and associating them the factors
a+
i ak ⇒ nk δik
ai a+
k ⇒ (1 − nk )δik
and
we have
hDar |Ĥ1 |Di =
X
=
X
hij na δaj (1 − nr )δri + nr δar nj δij
ij
(234)
hij δaj δri = hra
ij
11.19
Energy of a single determinant
Expectation value of the electronic Hamiltonian with the single-determinant wave function
EHF =
X
=
X
−
hµν hHF|χ+
µ χν |HFi +
µν
µν
1 X
+ − −
(µλ|νσ)hHF|χ+
µ χν χσ χλ |HFi
2
µνλσ
1 X
hµν Pνµ +
(µλ|νσ)(Pσν Pλµ − Pσµ Pλν )
2
(235)
µνλσ
After permutation of the dummy indices
EHF =
X
hµν Pνµ +
µν
1 X
Pνµ Pσλ {(µν|λσ) − (µλ|νσ)}
2
(236)
µνλσ
Remembering the definition of the Fock matrix
X
Fµν = hµν +
Pσλ {(µν|λσ) − (µλ|νσ)}
(237)
µνλσ
the Hartree-Fock energy can be written as a trace
EHF =
12
1
Tr(h + F )P
2
(238)
Hartree-Fock theory
Explicit variation of the energy of a determinant constructed up from a set of orthogonal orbitals
• orthogonality of orbitals should be preserved during the variation process
• modify energy functional by the method of Lagrange multipliers
• mathematically problematic: the orthogonality of orbitals is anticipated in the energy
expression to be varied
Apply the variation principle for an arbitrary variation of the determinant
• take the most general variation of a one-determinant wave function
• a condition is given for stationary character of the energy of a one-determinant wave
function (Brillouin theorem)
• obtain the Hartree-Fock equations from the Brillouin theorem
35
12.1
Brillouin theorem
Take the expression of the variation principle
hδΨ|Ĥ − E|Ψi = 0
(239)
and substitute the general variation of a one-determinant wave function
N X
X
i
p∗ri hΨ(ψi → ψr )|Ĥ − E|Ψi = 0
(240)
r
Since the pri variation parameters are independent, each term should vanish individually, and since
pri → 0 but pri 6= 0, we can simplify as
hΨ(ψi → ψr )|Ĥ|Ψi − EhΨ(ψi → ψr )|Ψi = 0
(241)
The overlap of two determinants differing in one orthogonal spin orbital vanishes
hΨ(ψi → ψr )|Ĥ|Ψi = 0
(242)
Since this condition is trivially satisfied if the spin of ψi and ψr is different, we can constrain the
variation to spin orbitals ψr = φr (r)ηr (σ) having the same spin component as ψi
hΨ(φi ηi → φr ηi )|Ĥ|Ψi = 0
(243)
Brillouin theorem: Stationary one-determinant wave function does not interact with the singly
excited configurations.
Equivalent to the variation principle applied to a one-determinant wave function.
12.2
Unrestricted Hartree-Fock equations
Consider the determinant wave function
Ψ = det[ψ1 (1)ψ2 (2) . . . ψN (N )]
(244)
If the {ψi } spin orbitals were not orthogonal, they can always be orthogonalized (c.f. invariance
property of determinants).
Apply the Brillouin theorem
hΨ(φi ηi → φr ηi )|Ĥ|Ψi = 0
(245)
Use the Slater rules
hΨri |Ĥ|Ψi = hψr |ĥ|ψi i +
N
X
hψr ψj ||ψi ψj i
(246)
j=1
j6=i
and write the expression in spatial orbitals for the case of ηi = ηr we obtain
(φr |ĥ|φi ) +
N X
(φr φi |φj φj ) − (φr φj |φj φi )δηi ηj = 0
(247)
j=1
j6=i
Introduce (local) the CoulombJˆj φ(r 1 ) =
Z
dr 2
φj (r 2 )∗ φj (r 2 )
φ(r 1 )
r12
36
(248)
and (nonlocal) exchange operators
Z
K̂j φ(r 1 ) =
dr 2
φ∗j (r 2 )φ(r 2 )
φj (r 1 )
r12
(249)
The orbital form of the Brillouin-conditions
(φr |ĥ +
N
X
Jˆj − K̂j δηi ηj |φi ) = 0
(250)
j=1
j6=i
Since φr lies in the orthogonal complement of the occupied orbitals, it is sufficient to require that
N
N
X
X
ˆ
ĥ +
Jj − K̂j δηi ηj φi =
λji φj δηi ηj
j=1
j6=i
(251)
j=1
i.e. the function obtained after the action of the operator in curly brackets Fockian lies entirely in
the space of the occupied orbitals of the same spin.
na
Let us apply this result to an unrestricted determinant made of na α and nb β orbitals, {φα
i }i=1
β nb
and {φi }i=1 . Use the Coulomb- and exchange operators defined separately for orbitals with α
and β spin, to obtain the following two sets of equations
nb
na
na
X
X
X
a
a
b
ˆ
ˆ
ĥ +
Jj − K̂j +
Jj φα
aji φα
i =
j
j=1
j6=i
j=1
(252)
j=1
nb
nb
na
X
X
X
ĥ +
Jˆja +
Jˆjb − K̂ja φβi =
bji φβj
j=1
j=1
j6=i
(253)
j=1
Since Jˆja φα
j =
which are coupled by the Coulomb-interaction with electrons of antiparallel spin.
K̂ja φα
j , we can remove the summation restriction by adding the
a α
Jˆja φα
j − K̂j φj = 0
self-repulsion term
given set (α or β)
(254)
and define the Fock-operators, which are now identical for all orbitals in a
F̂ a = ĥ +
na X
nb
X
Jˆja − K̂ja +
Jˆjb
j=1
F̂ b = ĥ +
j=1
na
X
j=1
Jˆja +
nb X
Jˆjb − K̂jb
(255)
j=1
The UHF equations are obtained in the following form
F̂ a |φα
i i=
na
X
F̂ b |φβi i =
aji |φα
ji
j=1
nb
X
bji |φβj i
(256)
j=1
It is quite impractical to solve directly these coupled integro-differential equations. Before bringing
them to a more convenient form, check some properties of the Fockian.
The Fock operator does not change if constructed from a set of orbitals obtained by a unitary
transformation of the orbitals of the same spin
37
Unitary transformation
φi =
na
X
Uji φ0j
φ0i =
na
X
j
(U † )ji φj
(257)
j
Consider, for instance, the Coulomb operator
na
X
Jˆia =
i
na Z
X
n
dr 2
i
n
a
a X
φ∗i (r 2 )φi (r 2 ) X
=
r12
i
Z
dr 2
j,k
∗ 0∗
Uji
φj (r 2 )Uki φ0k (r 2 )
r12
(258)
∗
Using the unitary character of the transformation, Uji
= (U † )ij = (U −1 )ij we obtain
na X
na
X
i
†
na
0
X
φ0∗
j (r 2 )φk (r 2 )
dr 2
=
r12
j
Z
Uki (U )ij
j,k
Z
na
0
X
φ∗0
0
j (r 2 )φj (r 2 )
dr 2
=
Jˆja
r12
j
The Fock operator is hermitian
Exchange operator
Z
Z
φ∗i (r 2 )ψ(r 2 )
∗
φi (r 1 )
h φ | K̂i ψ i = dr 1 φ (r 1 )
dr 2
r12
Z
Z
φ∗ (r 1 )φ∗i (r 2 )φi (r 1 )ψ(r 2 )
= dr 1 dr 2
= hφφi |φi ψi
r12
∗
φ∗i (r 2 )φ(r 2 )
dr 2
φi (r 1 ) ψ(r 1 )
h K̂i φ | ψ i = dr 1
r12
Z
Z
φ∗ (r 1 )φ∗ (r 2 )ψ(r 1 )φi (r 2 )
= dr 1 dr 2 i
= hφi φ|ψφi i
r12
Z
(259)
(260)
Z
(261)
After an interchange of the integration variables
h φ | K̂i ψ i = hφφi |φi ψi = hφi φ|ψφi i = h K̂i φ | ψ i
Q.E.D.
(262)
We can show analogously that the same holds for the Coulomb operator. The Fock operator, being
itself the sum of hermitian operators, is hermitian too.
12.3
Canonical UHF equations
ji are the elements of a hermitian Fock-operator matrix
F̂
a
|φα
i i
=
na
X
ji |φα
ji
(263)
j
a α
hφα
k |F̂ |φi i =
na
X
j
α
ji hφα
k |φj i = ki
| {z }
δkj
(264)
Since F̂ a can be constructed from any set of orbitals that is obtained by a unitary transformation
of the orbitals of the same spin, we can take U that diagonalizes the hermitian matrix
α0
U † U ik = 0ik = φi δik
(265)
38
In fact, this transformation applied to the orbitals leads to a diagonal ε matrix
X
X
X
a
α
∗
a α
0ik =
Uji φα
|
F̂
|
U
φ
=
Uji
hφα
`k
j
`
j |F̂ |φ` i U`k =
|
{z
}
j
`
j,`
aj`
X
0
∗ a
=
Uji
j` U`k = U † U ik = ai δik
(266)
j,`
There exists a set of spin orbitals for which the Fockian is diagonal. Using this basis set the
canonical unrestricted Hartree-Fock equations are obtained
a α
F̂ a φα
i = i φi
i = 1, 2, . . . , ∞
F̂ b φβi
i = 1, 2, . . . , ∞
=
bi φβi
(267)
that are coupled pseudo-eigenvalue equations. The first na and nb solutions are the occupied
orbitals while the remaining solutions are the virtual orbitals.
• Although the eigenfunctions of the HF-equations have no physical meaning (the wave
function is invariant to their unitary transformation) the eigenvalues (orbitals energies)
can be approximately related to the ionization potentials (Koopmans theorem).
• An infinite number of equivalent orbital sets can be obtained by unitary transformations
of the canonical orbitals. Chosen according to some well-defined criteria, these unitary
transformations may lead to orbitals with specific properties, i.e. the localized orbitals,
closer to our feeling about chemical bonding.
12.4
Unrestricted Hartree-Fock energy expression
Let us apply the general result for the expectation value of the Hamiltonian for a single determinant
built from N = na + nb spin orbitals
E=
nX
a +nb
hi|ĥ|ii +
i
na +nb nX
a +nb
1 X
hij||iji
2 i
j
(268)
Separate sums over orbitals of spin α and spin β and write integrals in terms of space orbitals
E=
na
X
i
+
nb
X
i
(i|ĥ|i) +
na nX
na
X
a +nb
1X
(ij|ij) +
(ii|jj) −
2 i
j
j
nb nX
nb
X
a +nb
1X
hi|ĥ|ii +
(ii|jj) −
(ij|ij)
2 i
j
j
(269)
Use the notations
α
α
hαα
ii = (i |ĥ|i )
αα
Jij
= (iα iα |j α j α )
αβ
Jij
= (iα iα |j β j β )
αα
Kij
= (iα j α |iα j α )
39
(270)
and the energy expression can also be written as
na
X
E=
hαα
ii +
i
+
nb
X
hββ
ii
i
n
n
n
n
n
n
a X
a
a X
b
αα
1X
1X
αα
Jij − Kij
+
J αβ +
2 i j
2 i j ij
n
n
b X
b
b X
a
ββ
1X
1X
ββ +
Jij − Kij
+
J βα
2 i j
2 i j ij
(271)
Orbital energies are the diagonal matrix elements of the α- and β-spin Fock operators in a canonical molecular orbital basis, e.g.
α
i
=
α α
(φα
i |F̂ |φi )
α
= (i |ĥ +
na
X
(Jˆja − K̂ja ) +
j
=
hαα
ii
+
na
X
αα
Jij
αα
− Kij
nb
X
Jˆb |iα )
j
+
j
nb
X
(272)
αβ
Jij
j
The sum of orbital energies
nX
a +nb
i =
i
na
X
hαα
ii +
i
+
nb
X
na X
na
X
i
hββ
ii
+
i
j
i
nb X
nb
X
i
nb
na X
X
αβ
αα
αα
Jij
− Kij
+
Jij
ββ
Jij
−
ββ Kij
j
+
j
nb X
na
X
i
(273)
βα
Jij
6= E
j
is not equal to the total energy, due to the double counting of electron-electron interactions.
12.5
Spin properties of the UHF wave function
Eigenfunction of the Ŝz operator
Ŝz |Di =
1 α
(n − nβ )|Di
2
(274)
but not eigenfunction of the Ŝ 2 = Ŝ− Ŝ+ + Ŝz + Ŝz2 operator!
β
For example, a two-electron DODS (different orbitals different spin) system |Di = |Â[φα φ ]i
β
β
β
α
β
Ŝ 2 |Di = Â[Ŝ− Ŝ+ φα φ + Ŝz φα φ + Ŝz2 φα φ ]
where we use that [Ŝ 2 , Â] = 0.
Last two terms are zero, since (nα − nβ ) = 0.
β
Ŝ− Ŝ+ {φα φ } = Ŝ− {φα φβ } = φ φβ + φα φ
(275)
The two-electron UHF wave function is not eigenfunction of the Ŝ 2 operator
α
Ŝ 2 |Di = |Di + Â[φ φβ ] 6= (S(S + 1)|Di
(276)
Expectation value of Ŝ 2 is
β
α
hD|Ŝ 2 |Di = hD|Di + hÂφα φ |Âφ φβ i
α β α α
hφ |φ i hφ |φ i
=1+ α β
= 1 − |S αβ |2
α
hφ |φ i hφ |φα i
40
(277)
12.6
Symmetry dilemma
While the exact solutions of the molecular Schrödinger equation are eigenfunctions of all the
operators that commute with the molecular Hamiltonian, this is not necessarily the case for an
approximate (e.g. UHF) solution. This can be the case for the symmetry operation of the molecular
point group and it is the case for the Ŝ 2 operator, which has the expectation value for a UHF wave
function:
X X αβ
2
n α − nβ n α − n β
Ŝ UHF =
+ 1 +nβ −
|Sij |2
(278)
2
2
|
{z
}
i
j
Ŝ 2
exact
If we require that the solutions have the correct symmetry, it appears as a constraint (restriction)
in the variation problem, and leads necessarily to a higher energy than the solution of the unconstrained (unrestricted) problem. We should decide whether we give up the symmetry requirements
to get a better energy or we restrict the solutions on physical grounds and obtain a worse energy.
This is the Löwdin symmetry dilemma.
12.7
Restricted Hartree-Fock equations
A determinant built up from pairs of spin orbitals composed of the same space orbital multiplied
by either α(σ) or β(σ) is a restricted determinant. The case of closed shells, i.e. N = 2n is
particularly important
(279)
Ψ = A φ1 (1)φ1 (2)φ2 (3)φ2 (4) . . . φn (2n − 1)φn (2n)
Instead of repeating the derivation from the Brillouin theorem, we can simply consider this problem
as a special case of the UHF equations with na = nb = n and Jˆia = Jˆib = Jˆi , etc. We obtain
F̂ = ĥ +
n
X
2Jˆj − K̂j
(280)
j=1
where the summation is over the n space orbitals.
as eigenfunctions of the canonical RHF equations
The canonical molecular orbitals are obtained
F̂ |φi i = i |φi i
i = 1, 2, . . . , ∞
(281)
with orbital energies as eigenvalues.
12.8
RHF energy expression
Closed shell RHF electronic energy in terms of n space orbitals
E=2
n
X
(i|ĥ|i) +
n X
n
X
i
i
2(ii|jj) − (ij|ji) = 2
n
X
j
i
hii +
n X
n
X
i
2Jij − Kij
(282)
j
Orbital energies are the diagonal elements (and eigenvalues) of the RHF Fock operator and using
integrals in space orbitals in the chemists’ (charge density) convention
i = hii +
n
X
2Jij − Kij = (i|ĥ|i) +
j
n
X
2(ii|jj) − (ij|ji)
(283)
j
The sum of the energies of the n doubly occupied orbitals
2n
X
i
εi = 2
n
X
hii + 2
n
X
i
j
is not equal to the total energy.
41
2Jij − Kij 6= E
(284)
Orbital energies
12.9
hψi |F̂ |ψj i = εj hψi |ψj i = εj δij
εi = hi|ĥ|ii +
N
X
hib||ibi
(285)
(286)
b
• Occupied orbitals
εa = ha|ĥ|ai +
N
X
hab||abi
(287)
b=1
b6=a
where haa||aai = 0 is used. The energy of an occupied orbital corresponds to a particle
moving in a N − 1-electron potential.
• Virtual orbitals
εr = hr|ĥ|ri +
N
X
hrb||rbi
(288)
b=1
describe electrons in the potential of all the N electrons of the Hartree-Fock ground state.
12.10
Koopmans theorem
Let us group together those terms of the electronic energy, which depend on a given molecular
orbital, e.g. k
EN =
N
X
N
hi|ĥ|ii + hk|ĥ|ki +
i6=k
N
N
X
1 XX
1
hij||iji +
hik||iki + hkk||kki
2
2
i6=k j6=k
(289)
i6=k
Since the self-interaction hkk||kki = 0, the last term is 0 and we can drop the summation restriction
EN =
N
X
i6=k
N
hi|ĥ|ii +
N
N
X
1 XX
hij||iji + hk|ĥ|ki +
hik||iki = EN −1 (k) + k
2
i
(290)
i6=k j6=k
The negative of the orbital energy is equal to the energy difference of (N − 1)-electron and the
N -electron determinants, the ionization potential,
IP = EN −1 (k) − EN = −k
(291)
provided we neglect the orbital relaxation effects (spin orbitals are supposed to be frozen in the
(N − 1)-electron determinant.
Similarly, we can examine the energy of an (N + 1)-electron determinant, obtained by adding a
virtual spin orbital ψr to the N -electron determinant.
EN +1 = EN + hr|ĥ|ri +
N
X
hri||rii = EN + r
(292)
i
The electron affinity, EA, can be approximated as the negative of the virtual orbital energy:
EA = EN − EN +1 (r) = −r
(293)
Koopmans theorem: Provided that the spin-orbitals satisfy the canonical HF equations, and
the same set of orbitals are used to describe the neutral system and its ions, their energy difference
is stationary against spin-orbital variations.
42
12.11
Excitation operators - Generalities
Define an excitation operator by
Ôn† |0i = |ni
(294)
Ôn† = |nih0|
(295)
Ôn = |0ihn|
(296)
In bra-ket representation
Th corresponding de-excitation operator
Apply the commutator [Ĥ, Ôn† ] to the exact ground state wave function
[Ĥ, Ôn† ]|0i = Ĥ Ôn† |0i − Ôn† Ĥ|0i = Ĥ|ni − E0 Ôn† |0i
= En Ôn† |0i − E0 Ôn† |0i = ∆E0n Ôn† |0i
(297)
This leads to the (super)operator equation
[Ĥ, Ôn† ] = ∆E0n Ôn†
(298)
Excitation energy as Rayleigh quotient
∆E0n =
h0|Ôn [Ĥ, Ôn† ]|0i
h0|Ôn Ôn† |0i
(299)
These are the basic relations of the EOM (equation of motion) approach.
12.12
Generalized Koopmans theorem
An ionization process can be described by the operator
X
Ôn† =
c∗r ar = Ô+
(300)
r
which creates a positive ion by removing one electron. The ionization potential, ∆E + is
P ∗
c cs h0| a+
c† Kc
s [Ĥ, ar ]|0i
∆E + = rs
= †
Pr ∗
+
c Pc
rs cr cs h0|as ar |0i
(301)
In the denominator we recognize the density matrix elements, and in the numerator we have the
matrix elements of the Koopmans operator
Krs = h0| a+
s [Ĥ, ar ]|0i
(302)
After expansion of the SQ Hamiltonian operator
X
Krs =
htu s+ t+ u− r− − s+ r− t+ u−
tu
1 X
+ + + − − −
+ − + + − −
+
htu|vwi s t u w v r − s r t u w v
2 tuvw
(303)
Applying the anticommutation relations we obtain the 1RDM operator
s+ t+ u− r− − s+ r− t+ u− = s+ t+ u− r− − s+ (δrt − t+ r− )u− = −δrt s+ u−
43
(304)
and the 2RDM operator
s+ t+ u+ w− v − r− − s+ r− t+ u+ w− v − =
s+ t+ u+ w− v − r− − s+ (δrt − t+ r− )u+ w− v − =
s+ t+ u+ w− v − r− − δrt s+ u+ w− v − + s+ t+ (δru − u+ r− )w− v − =
(305)
− δrt s+ u+ w− v − − δru s+ t+ w− v − = −δrt s+ u+ w− v − + δru s+ t+ w− v −
Krs
X
1 X
+ −
+ + − −
+ + − −
=−
htu δrt s u +
htu|vwi δrt s u w v − δru s t w v
2 tuvw
tu
X
X
1 X
=−
hru s+ u− +
hru|vwis+ u+ w− v − −
htr|vwis+ t+ w− v −
2 uvw
u
tvw
X
X
=−
hrt Pts +
hrt|uviΓuv,st
t
(306)
tuv
The condition for a stationary value of ∆E + is
†
δc† Kc
c Kc δc† P c
+
δ∆E = †
−
+ c.c. = 0
c Pc
c† P c c† P c
(307)
leading to the matrix equation
Kc = ∆E + P c
For a HF wave function P = I and the Koopmans matrix is
X
X
Krs = −
hrt δts +
hrt|uvinu nv (δtu δrv − δtv δru ) = −Frs
t
(308)
(309)
tuv
equal to the matrix of the Fock operator, which has as eigenvalues the orbital energies. It can
be concluded that for a HF wave function the variational ionization potential is just the orbital
energy ∆E + = −εi and the corresponding excitation operator is ai , which removes an occupied
spin orbital from the HF determinant.
12.13
How good is the Koopmans theorem?
• orbital relaxation correction - lowers the energy of the ions
• correlation energy correction - depends on the number of electrons
44
E(N-1)
E(N+1)
E(N)
6
−εr
?
6
6
? EA
6
−εi
IP
?
?
Two errors compensate each other for the ionization potential, enhance each other for the electron
affinity.
12.14
Excited states
Excite an electron from the i-th occupied to the r-th virtual level of a RHF wave function. Two
possible excited determinants
Ψri
Ψr̄ī
(310)
Their linear combinations form pure singlet and triplet states
1 r
Ψi + Ψr̄ī
2
1
3
Ψ = Ψri − Ψr̄ī
2
1
Ψ=
(311)
(312)
The excitation energies ∆ES,T in this single transition approximation, (STA) depend on the orbital
energies and the electron-electron interactions.
ES,T =
1
hΨri |Ĥ|Ψri i + hΨr̄ī |Ĥ|Ψr̄ī i ± hΨr̄ī |Ĥ|Ψri i ± hΨri |Ĥ|Ψr̄ī i
2
(313)
The first two matrix elements are equal (just α and β are interchanged) and the last two are
complex conjugates, which are also equal for real wave functions.
The energy of the excited determinant is obtained from the ground state energy by removing εi and
replacing it by εr and correcting for the missing e− -repulsion term between electrons of parallel
spin
hΨri |Ĥ|Ψri i = E0 + εr − εi − hφr |Jˆi − K̂i |φr i
(314)
Transition matrix element between determinants that differ in two spin orbitals
hΨri |Ĥ|Ψr̄ī i = hφ1 φ1 . . . φr φi . . . |Ĥ|φ1 φ1 . . . φi φr . . .i
= hrī||ir̄i = [rī|ir̄] − [rī|r̄i] = (ri|ir) − 0 = Kir
45
Summarizing these results
ES,T =
1
2(E0 + εr − εi − Jir + Kir ) ± 2Kir
2
(315)
Excitation energies
∆ES = εr − εi − Jir + 2Kir
∆ET = εr − εi − Jir
(316)
(317)
Relative magnitude of singlet/triplet splittings can sometimes be estimated from the shape of the
occupied and virtual orbitals, as the self-energy of the transition density φ∗i (r)φr (r).
13
EOM for excited states
Let us use the equation of motion method for electronic excited states. Define the singlet excitation
operator
X
1X ∗ + −
−
∗
†
Ôn† =
Xar (arα aaα + a+
Xar
Êar
(318)
rβ aaβ ) =
2 ar
ar
and its adjoint
Ôn =
X
1X
+ −
−
Xar (a+
Xar Êar
aα arα + aaβ arβ ) =
2 ar
ar
(319)
The excitation energy can be obtained from
∆En =
h0|Ôn [Ĥ, Ôn† ]|0i
(320)
h0|Ôn Ôn† ]|0i
Substitution of the excitation operators
P P
∗
†
ar
bs Xar Xbs h0|Êbs [Ĥ, Êar ]|0i
∆En = P
P
†
∗
ar
bs Xar Xbs h0|Êbs Êar |0i
(321)
The matrix elements in the denominator
†
h0|Êbs Êar
|0i =
1
1
+
(h0|a+
bα aaα + abβ aaβ |0iδrs = ρab
4
2
is the half of the spinless one-particle reduced density matrix.
(322)
Define Q as
†
Qbs,ar = 2h0|Êbs [Ĥ, Êar
]|0i
(323)
and the excitation energy expression in matrix form
∆En =
X † QX
X † ρX
(324)
The stationary condition leads to the matrix equation
QX = ∆EρX
(325)
In the special case of a one-determinant HF wave function as ground state ρ = I and
Qbs,ar = 2(h1 Dbs |Ĥ|1 Dar i − δab δsr hD|Ĥ|Di)
46
(326)
The Q matrix is the single-CI matrix (TDA – Tamm-Dancoff approximation). The diagonal
elements are the STA excitation energies that we have already derived previously
Qar,ar = εa − εb − (aa|rr) + 2(ar|ar)
(327)
while the off-diagonal elements are
Qbs,ar = −(ab|sr) + 2(ar|bs)
(328)
The use of TDA is mandatory when the STA gives qualitatively wrong results. For instance, the
benzene excited states
• 4 possible single excitations between doubly degenerate HOMO to doubly degenerate
LUMO
• four-fold degenerate i.r. with 4 identical STA excitation energies
• TDA leads to the correct symmetry-adapted combinations and splitting of the excitation
energies
TDA is often used with semi-empirical model Hamiltonians (PPP, CNDO/S) and sometimes also
with ab initio (e.g. Gaussian program).
13.1
Hartree-Fock density matrix
The operator of the one-particle reduced density matrix in SQ is
X
ρ̂(x1 ; x01 ) =
ψ` (x1 )ψk∗ (x01 )k + `−
(329)
k`
Since the 1RDM can be expanded as
ρ(x1 ; x01 ) =
X
ψ`∗ (x1 )P`k ψk (x01 )
(330)
k`
the matrix elements P`k = hk + `− i form a discrete representation of the 1RDM.
In the case of a single determinant HF wave function the 1RDM on the basis the MOs
Pki = hHF|ψi+ ψk− |HFi = nk δik
is diagonal. The trace of Pki is equal to N =
PM
i
(331)
ni , the number of electrons.
The density matrix of a HF wave function is idempotent. This property follows simply from the
fact that ni = 0 or ni = 1, therefore
X
X
(P 2 )k` =
Pki Pil =
nk δik · ni δ`i = n2i δk` = Pk`
(332)
i
i
The idempotency of the one-particle reduced density matrix is a necessary and sufficient condition
for the one-determinant character of the wave function.
47
13.2
Natural orbitals
For multi-configurational (e.g. CI) wave functions the 1RDM is not necessarily diagonal. Nevertheless, since P is hermitian, one can find a unitary matrix U that U † P U be diagonal. Let U
such a matrix. Then the transformation from the original (MO) and the new basis is
X
X
X
∗
ηi (U † )ik =
ηi Uki
ηi =
ψk Uki
ψk =
(333)
i
k
i
Substituting this transformation to the expansion of the 1RDM
XX
∗
ρ̂(x1 ; x01 ) =
U`i
ηi (x1 )P`k Ukj ηj∗ (x01 )
k`
(334)
ij
Since U has been chosen such that U † P U ij = λi δij
X
ρ̂(x1 ; x01 ) =
λi ηi (x1 )ηi∗ (x01 )
(335)
i
The orbitals, that diagonalize the 1RDM are called natural orbitals. Let us write the wave
function as a linear combination of determinants built from the natural orbitals. In this case the
diagonal elements of the density matrix
XX
Pii = λi =
c∗K cL hDK |ηi+ ηi− |DL i
K
=
L
XX
K
c∗K cL nL
i hDK |DL i =
L
X
|cK |2 nK
i
(336)
K
are nK
i is the occupation number of the i-th NO in determinant K. The diagonal elements, λi , can
be any number between 0 and 1. The matrix representation of the 1RDM for a multi-configurational
wave function is not idempotent.
13.3
Two-particle density matrix for a HF wave function
The matrix representation of the two-particle reduced density matrix is obtained from the following
expectation value
+ + − −
ΓHF
lk,ij = hHF|ψi ψj ψk ψl |HFi = ni nj (δjk δil − δjl δik )
(337)
in the basis of molecular orbitals.
The two-particle density matrix of a one-determinant wave function can be expressed by the
elements of the 1RDM matrix elements as
ΓHF
lk,ij = Pli Pjk − Plj Pki
(338)
The first part corresponds to the Coulomb-component
ΓHF,Coul
(x1 , x2 ; x1 0, x02 ) = P (x1 , x01 )P (x2 , x02 )
lk,ij
(339)
and the second part first part is the exchange-component
ΓHF,exch
(x1 , x2 ; x1 0, x02 ) = −P (x1 , x02 )P (x2 , x01 )
lk,ij
48
(340)
14
Hartree-Fock-Hall-Roothan equations
The UHF and RHF equations can be brought to a more practical form, by writing the molecular
orbitals as linear combination of some fixed functions. In quantum chemistry, these are usually
some approximations to atomic orbitals (AO). In this LCAO approximation
φi =
m
X
cµi χµ
(341)
µ=1
where cµi = (ci )µ is an element of the column vector of the expansion coefficients. We suppose
that the m basis functions (AOs) are linearly independent and in general, form a non-orthogonal
set. The cµi MO-coefficients are not true linear variation parameters, since the HF equations are
only pseudo-eigenvalue equations.
We can proceed again from the Brillouin condition
hΨ(φi → φ0i )|Ĥ|Ψi = 0
(342)
and we suppose that both φi and φ0i can be expanded in the same AO basis
φi =
m
X
φ0i =
cµi χµ
µ=1
m
X
qµ χµ
(343)
µ=1
Introduce the projection operator to the subspace of occupied orbitals
P̂ =
occ
X
|φj ihφj |
(344)
j=1
P
Any function P̂ |ψi = P̂ | ν pν χν i lies in the occupied subspace and any function (1 − P̂ )|ψi =
P
(1− P̂ )| ν pν χν i is orthogonal to this subspace. Since the pν coefficients are arbitrary, we should
require the fulfillment of the Brillouin theorem for all functions of the form |φ0 i = (1 − P̂ )|χν i.
Using the Slater rules
hφ0 |F̂ |φi i = 0
(345)
and substituting the definition hφ0 |
hχν |(1 − P̂ )F̂ |φi i = hχν |F̂ |φi i − hχν |
occ
X
j=1
φj i hφj |F̂ |φi i = 0
| {z }
(346)
εji
Putting the LCAO expansion of |φi i and |φj i
hχν |F̂ |
m
X
cµi χµ i =
µ=1
occ
X
hχν |
j=1
m
X
cµi χµ iεji
(347)
µ=1
Supposing that we have already performed the unitary transformation that diagonalizes the ε
matrix, and using obvious matrix element notations
m
X
Fνµ cµi = εi
µ=1
m
X
Sνµ cµi
(348)
µ=1
The result is a generalized pseudo-eigenvalue equation
F ci = εi Sci
i = 1, 2, . . . , m
49
(349)
which can be written in matrix form by constructing the m × m MO coefficient matrix, C, from
the column vectors of the expansion coefficients
F C = εSC
(350)
Solutions
N (in closed shell RHF case N/2) occupied orbitals
m − N virtual orbitals
The Hartree-Fock-Roothaan equations are not equivalent with the Hartree-Fock equations. By
increasing the basis set we can only approach the “Hartree-Fock limit”.
14.1
P-matrix
In the LCAO-theory we use frequently the projector to the subspace of occupied orbitals, expressed
by the LCAO coefficients
P̂ =
occ
X
|φi ihφi | =
m X
occ
X
ciµ c∗iν |χµ ihχν |
=
µν j=1
i=1
m
X
Pµν |χµ ihχν |
(351)
µν
In components, and also in matrix notation using the column vector of the LCAO coefficients, ci
Pµν =
occ
X
ciµ c∗iν
P =
i=1
occ
X
ci c†i
(352)
i=1
In the RHF case it is used multiplied with a factor of 2
D = 2P
The P-matrix is often called density matrix. The electronic density has the expression
%(r) =
occ
X
φi (r)φ∗i (r) =
m X
m
X
µ
i
Pµν χ∗ν (r)χµ (r)
(353)
ν
The P-matrix appears in the expectation values of one-electron operators.
The operator P̂ is a projector, i.e. P̂ 2 = P̂ . The idempotency of its matrix representation in the
MO basis and in any orthonormal basis is a criterion for the one-determinant character of the
underlying wave function.
However, this property does not hold for its matrices in a non-orthogonal basis, like the AO basis
hχµ |P̂ |χν i =
occ
X
hχµ |φi ihφi |χν i =
i
occ
X
(SP S)µν
(354)
i
which is obviously not idempotent.
Let us calculate that component of an arbitrary orbital ψ =
subspace. Using the P̂ projection operator
P̂ |ψi =
occ
X
|φi ihφi |ψi =
m
X
µ
i=1
50
P
τ
dτ χτ which lies in the occupied
pµ |χµ i
(355)
Using the LCAO expansions of P̂ and ψ
XX
P̂ |ψi =
Pµν dτ |χµ ihχν |χτ i =
µν
XX
τ
µν
Pµν Sντ dτ |χµ i =
τ
X
(P S)µτ dτ |χµ i
µν
The matrix of the projection, P S can be deduced from
X
(P S)µν dν
pµ =
(356)
ν
The PS matrix is idempotent. To see that, consider the expression of the overlap of two molecular
orbitals
XX
hφi |φj i =
c∗µi cνj hχµ |χν i = δij
(357)
µ
ν
in matrix notation
c†i Scj = δij
(358)
to calculate the square of the P S matrix
P SP S =
occ X
occ
X
i
ci c†i Scj c†j S =
j
occ X
occ
X
i
ci δij c†j S =
j
occ
X
ci c†i S = P S
The trace of the P S matrix is equal to the number of electrons.
Z
Z
XX
XX
X
N = dr%(r) =
Pµν drχ∗ν (r)χµ (r) =
Pµν Sνµ =
(P S)µµ
µ
ν
(359)
i
µ
ν
(360)
µ
The P S matrix can be considered as the correct representation of the one-particle reduced density
matrix (1RDM).
14.2
Population analysis
Total number of electrons (AO basis)
XX
X
N=
Pµν Sνµ =
(P S)µµ
µ
ν
(361)
µ
Each term can be interpreted as the number of electrons associated to the orbital χµ . Since the
AO-s are assigned to atoms, we can group together the net charge (electronic + nuclear) associated
with an atom A
X
QA (Mulliken) = ZA −
(P S)µµ
(362)
µ∈A
This is the Mulliken population analysis. It corresponds to the halving of the electron density
associated with a pair of AOs.
This definition of the atomic charge is not unique. Since TrAB = TrBA
X
N=
(S α P S 1−α )µµ
(363)
µ
with a value of α ≤ 1.
In particular
QA (Löwdin) = ZA −
X
µ∈A
This is the Löwdin-population analysis.
51
(S 1/2 P S 1/2 )µµ
(364)
14.3
Potential-fitted charges
The electrostatic properties (multipole moments, electrostatic potential, etc.) cannot be described
by Mulliken (or Löwdin) charges correctly. Take the molecular dipole moment.
X
XXX
µ=
ZA RA −
Pµν hχµ |r̂|χν i
A µ∈A ν
A
=
X
=
X
=
X
A
XXX
A
XXX
A
X
ZA R −
Pµν hχµ |RA + r̂ A |χν i
A µ∈A ν
A
ZA R −
Pµν Sµν RA −
A µ∈A ν
A
A
Q R −
A
XXX
Pµν hχµ |r̂ A |χν i
(365)
A µ∈A ν
A
µ = µ(Mulliken) +
A
X
µA
A
If the molecular multipoles (dipole, etc.) are wrong, asymptotic behaviour of the electrostatic
potential calculated from them
X
QA
(366)
V (r k ) =
|r k − RA |
A
is also incorrect.
Approximately correct electrostatics can be obtained from fitted charges. Take the expression of
the potential on a grid of points
N
V (r k ) =
X
a
a
X
qa
=
T ka q a
|r k − Ra |
a
k = 1, . . . , Ngrid
(367)
and look for the best set of charges that reproduce the quantum chemically calculated electrostatic
potential in the same points,
Vk =
X
A
X
ZA
1
−
Pµν hχµ |
|χν i
|r k − RA |
|r k − r|
µν
(368)
This leads to an overdetermined system of equations (Ngrid Na )
Vk =
Na
X
T ka q a
(369)
a
In matrix form, the formal solution (in the least squares sense)
v = Tq
q = (T † T )−1 v
(370)
Depend to some extent on the choice of sampling grid.
14.4
Covalent bond orders
Correlation of fluctuation of the number of electrons associated to two atoms
BAB = −2(hNA NB i − hNA ihNB i)
is considered as a measure of the covalent bond order between two atoms.
52
(371)
It can be shown that it is related to the change of correlation content during bond formation in
the diatomic fragment. The deviation of the pair density from the simple product of one-particle
densities is the correlation function, f (r 1 , r 2 )
1
%(r 1 )%(r 2 )[1 + f (r 1 , r 2 )]
2
γ(r 1 , r 2 ) =
Integrated on a given region (e.g. atomic region), it gives the total correlation content
Z
Z
F (ΩA , ΩA ) =
dr 1
dr 2 %(r 1 )%(r 2 )f (r 1 , r 2 )
ΩA
(372)
(373)
ΩA
The correlation content is dominated by the exchange correlation component, therefore the bond
order is often related to directly with the exchange part of the above quantity
Z
Z
BAB = 2
dr1
dr2 [%α (r2 ; r1 ) %α (r1 ; r2 ) + %β (r2 ; r1 ) %β (r1 ; r2 )]
(374)
ΩA
ΩB
Using Mulliken’s definition of the atomic population, we get the following bond order definition
X
BAB =
(P α S)µν (P α S)νµ + (P β S)µν (P β S)νµ
(375)
µν
Gives 1 for single, 2 for double and 3 for triple bonds.
14.5
Implementation of the HFR equations: SCF procedure
We suppose that a reasonable initial guess of the P-matrix, P (0) , is available. For a closed shell
system
• Calculate the one-electron integrals to form the core-Hamiltonian matrix
core
nucl
Hµν
= Tµν + Vµν
(376)
• Calculate the list of two-electron integrals
Z
Z
(µν|λσ) =
dr 1
−1 ∗
dr 2 χ∗µ (1)χν (1)r12
χλ (2)χσ (2)
(377)
There are O(K 4 /8) unique integrals, therefore this step constitutes the major part of the
computational effort.
• Form the Fock matrix of the n-th iteration
(n)
core
Fµν
= Hµν
+
X
(n−1) Pλσ
core
2(µν|λσ) − (µλ|σν) = Hµν
+ Gµν (P (n−1) )
(378)
λσ
• Solve the matrix equation
F C = SCε
(379)
• Introduce the canonical orthogonalization (avoiding problems with linearly depen-
dent basis sets) by the following equations
U † SU = s
X = U s−1/2
(380)
• Transform the Roothaan equations directly by substituting C = XC 0
F XC 0 = SXC 0 ε
(X † F X)C 0 = (X † SX)C 0 ε
53
(381)
• Diagonalize F 0 = (X † F X) to get C 0 and ε
• Transform back the coefficient matrix to the original basis
C = XC 0
(382)
• Form a new P-matrix
P n+1 = CC †
(383)
• Check convergence by comparing P (n) and P (n+1) ; if necessary start a new cycle
14.6
Direct SCF procedure
After a few SCF cycles, most of the P-matrix elements do not change appreciably. Consider the
Fockian update formula
F (n) = H + G(P (n−1) )
(384)
The density matrix of the (n − 1)th iteration is P (n−1) = P (n−2) + ∆P (n−1) . Since
F (n−1) = H + G(P (n−2) )
(385)
the update formula can be written also as
F (n) = F (n−1) + G(∆P (n−1) )
(386)
The matrix of two-electron update matrix
X
(n−1) G(∆P (n−1) )µν =
∆Pλσ
2(µµ|λσ) − (µλ|σν)
(387)
λσ
is sparse and we need only those two-electron integrals, which are multiplied by |∆P (n−1)
| > .
µν
These integrals can be calculated directly, at each iteration, thus reducing huge data storage and
disk access overheads.
14.7
Basis sets
• Slater type orbitals (STO) or exponential orbitals (ETO)
O
n−1 −ζra
χST
e
· Y`m (θa , φa )
n`m (r a ) = N ra
(388)
• correct R → 0 and R → ∞ behaviour
• atomic HF orbitals are linear combinations of a few STOs
• a molecular basis set may contain one, two (double ζ), three (triple ζ), four
(quadruple ζ), etc. sets of STOs per atom
• Electron repulsion integrals (ERI) are difficult to calculate for more than two
centers
• Gaussian type orbitals (GTO)
2
guvw (r a ) = N xu y v z w · e−αra
gn`m (r a ) = N e
2
−αra
(Cartesian)
rn−1 · Y`m (θa , φa )
(389)
(spherical)
• incorrect behaviour at R → 0 (no cusp) and at R → ∞ (too fast decay)
• much more (5–10 times) GTOs are needed to describe an atomic HF orbital
• basis functions are fixed combination of primitive Gaussians: contracted Gaussian
functions
• easy calculation of ERIs for three and four centers too (product theorem)
54
14.8
Product theorem for Gaussian functions
The ERIs involve the product of two primitive Gaussian functions
Y
exp (−αra2 ) exp (−βrb2 ) =
exp −α(ri − Ai )2 − β(ri − Bi )2
(390)
i=x,y,z
We can transform the exponent, e.g. for the component x
−α(x − Ax )2 − β(x − Bx )2 = −α(x2 − 2Ax x + A2x ) − β(x2 − 2Bx x + Bx2 )
αA2x + βBx2 αAx + βBx
= −(α + β) x2 − 2x
+
α+β
α+β
αβ
αAx + βBx 2
−
(Ax − Bx )2
= −(α + β) x −
α+β
α+β
(391)
leading to a new Gaussian function
exp −
αβ
(Ax − Bx )2 · exp −(α + β)(x − XAB )2
α+β
(392)
αAx + βBx
α+β
(393)
with (α + β) exponent and
XAB =
center. It means that even four-center integrals can be handled as easily as two-center ones.
14.9
Contracted Gaussian functions
Reduce the number of variational parameters in the HFR equations by taking fixed linear combination of primitive functions
χCGF
(r − RA ) =
µ
L
X
dpµ gp (αpµ , r − RA )
(394)
p=1
• STO-nG basis sets
obtained as least square fits to STO basis function with ζ = 1.0
• generalized contraction
MOs from atomic GTO calculations
• ANO basis
atomic natural orbitals from correlated atomic calculations
• split-valence basis sets
minimal contracted basis for core orbitals, more flexibility for valence shell: valence double
zeta (VDZ), valence triple zeta (VTZ)
• polarization functions
have higher angular quantum number than the highest occupied atomic orbital
• diffuse functions
to describe excited states, negative ions
55
14.10
Second Quantized Hamiltonian in AO basis
In non-orthogonal basis, like the AO basis, creation and annihilation operators follow the anticommutation relation
−
[χ+
(395)
ν , χµ ]+ = Sµν
and the operators χ−
µ do not act as true annihilation operators. Try to find the “true” annihilation
operators χ̃−
associated
with χ+
µ
ν , by requiring to satisfy the usual anticommutation rule
−
[χ+
µ , χ̃ν ]+ = δµν
(396)
X
(397)
−
Expand χ̃−
ν as linear combination of χν
χ̃−
ν =
Lνλ χ−
λ
λ
Substitution into Eq.(396) gives
−
[χ+
µ , χ̃ν ]+ =
X
−
Lνλ [χ+
µ , χλ ]+ =
λ
X
Lνλ Sλµ = δµν
(398)
λ
The above relationship is satisfied if L = S −1
χ̃−
ν =
X
−1 −
Sνλ
χλ
(399)
λ
In general, a set of functions {χ̃i } form the bi-orthogonal set related to the non-orthogonal set
{χi } if they satisfy the hχ̃i |χj i = δij relation. The χ̃−
λ operators form a bi-orthogonal set related
to χ−
.
µ
Let us write the Hamiltonian with the symmetrically orthogonalized auxiliary AO basis set ψi =
P
−1/2
χµ . In this orthogonal basis the Hamiltonian has the usual form
µ Siµ
Ĥ =
X
hij ψi+ ψj− +
ij
1X
hij|kliψi+ ψj+ ψl− ψk−
2
(400)
ijkl
After transformation of the operators to the original overlapping AO basis
XX
−1/2 −1/2
−
Ĥ =
hij Sµi Sjν χ+
µ χν
ij
+
µν
1X X
2
−1/2
hij|kliSµi
−1/2
Sνj
−1/2
Slσ
−1/2 + + − −
χµ χν χσ χλ
(401)
Skλ
ijkl µνλσ
Transform back the one-electron integrals too into the original basis
X −1/2 −1/2
hij =
Siα Sβj hαβ
(402)
αβ
and similarly for the two-electron integrals. Summation over i, k and i, j, k, l leads to the following
relatively simple form
X
1 X X −1 −1
−1 + −
−1
−1 −1 + + − −
Ĥ =
Sµα
hαβ Sβν
χµ χν +
Sµα Sνβ hαβ|γδiSγσ
Sδλ χµ χν χσ χλ
(403)
2
µναβ
µνλσ αβγδ
−
We can take advantage of the simple anticommutation relations between χ+
µ and χ̃λ , if we use the
above Hamiltonian in the following form
X
1 X
+ − −
−
hµ̃λ̃|νσiχ+
Ĥ =
hµ̃ν χ+
(404)
µ χ̃ν +
µ χν χ̃σ χ̃λ
2
µν
µνλσ
56
where the half-transformed integrals are defined as
X
−1
hµ̃ν =
Sµλ
hλν
(405)
λ
and
(µ̃λ̃|νσ) =
X
−1 −1
Sµτ
Sλ (τ |νσ)
(406)
τ
In order to calculate the expectation values of the operator strings in this energy expression, let
us consider the AO/MO transformations in both original and bi-orthogonal bases.
φ = C †χ
χ = (C † )−1 φ
(407)
(C † )−1 = SC
(408)
using that the MOs are orthonormal
C † SC = 1
and the definition of the bi-orthogonal orbitals
χ̃ = χ† S −1 = φ† C †
(409)
we get the relationships
χ+
µ =
X
(C † S)iµ φ+
i
χ̃−
µ =
i
X
cµi φ−
i
(410)
i
Substitution of these expressions and taking the expectation value
−
hχ+
µ χ̃ν i = (P S)νµ
X
+ − −
hχ+
Γσλητ Sηµ Sτ ν
µ χν χ̃σ χ̃λ i =
(411)
ητ
In the special case of HF wave functions we obtain for the two-particle string
X
+ − −
hHF|χ+
Pστ Pλη − Pση Pλτ Sηµ Sτ ν
µ χν χ̃σ χ̃λ |HFi =
(412)
ητ
= (P S)σν (P S)λµ − (P S)σµ (P S)λν
14.11
Hellmann-Feynman theorem
Consider a Hamiltonian, which depends on one or more parameters, αi
X
Ĥ(α) = Ĥ(0) +
αi Âi
(413)
i
The first derivative of the total energy for a (normalized) Hartree-Fock determinant wave function,
D is
∂
∂D
∂ Ĥ
∂D
∂E
=
hD|Ĥ|Di = h
|Ĥ|Di + hD|
|Di + hD|Ĥ|
i
(414)
∂αi
∂αi
∂αi
∂αi
∂αi
Since the Hartree-Fock wave function is stationary with respect to any variation in the class of
one-determinant wave functions, we remain with
∂E
∂ Ĥ
= hD|
|Di = hD|Âi |Di
∂α
∂α
The exact Hartree-Fock wave function satisfies the Hellmann-Feynman theorem.
57
(415)
Let us apply this result to the calculation of the molecular dipole moment. The Hamiltonian in
the presence of an electric field
Ĥ(F) = Ĥ(0) + Fα µ̂α
where µ̂α is the α = x, y, z component of the multipole moment operator. Expanding the total
energy in Taylor series
∂E(F)
1 ∂ 2 E(F)
E(F) = E(0) +
Fα +
Fα Fβ + . . .
(416)
∂Fα F=0
2 ∂Fα ∂Fβ F=0
The dipole moment is defined as
∂E(F)
µα = −
∂Fα F=0
(417)
According the previous result, the dipole moment can be calculated from a HF wave function as
the expectation value
µα = −hD|µ̂α |Di
(418)
In the case of non-variational wave functions the molecular properties must be calculated from the
energy derivatives.
14.12
Hartree-Fock gradients
Exact solutions of the molecular Hamiltonian require infinite basis sets. The advantage of the
SQ formalism is that eigenfunctions of the SQ Hamiltonian in a given finite basis set are exact
solutions of this model Hamiltonian. Therefore, the Hellmann-Feynman theorem applies to its
exact eigenfunctions, as well as variational solutions, like the Hartree-Fock wave function,in the
given finite basis.
The derivative of the energy can be calculated as
δE = hΨ|δ Ĥ|Ψi
(419)
The derivative of the SQ Hamiltonian
δ Ĥ =
X
+
δhµ̃ν χ+
µ χ̃ν +
µν
1 X
+ − −
δhµ̃λ̃|νσiχ+
µ χν χ̃σ χ̃λ
2
(420)
µνλσ
Use the expectation values of the operator strings and the following relationships to calculate the
derivative of the half-transformed AO-integrals, e.g.
δ(S −1 h) = δS −1 · h + S −1 · δh
δS −1 = S −1 δSS −1
(421)
We obtain for the special case HF wave function
δE HF =
X
µν
δhµν Pµν +
X
1 X
−1
δhµν|λσi Pλµ Pσν − Pλν Pσµ −
Fµν Pνλ δSλσ Sσµ
2
µνλσ
(422)
µνλσ
Using the definition of the P-matrix and the SCF condition
term can be rewritten as
occ
XX
−
εi cµi c∗νi δSνµ
µν
i
|
{z
Wµν
58
}
P
ν
Fµν cnui = εi
P
ν
Sµν cνi the last
(423)
and we obtain the final result
X
X
1 X
δE HF =
δhµν Pµν +
δhµν|λσi Pλµ Pσν − Pλν Pσµ −
Wµν δSνµ
2
µν
µν
(424)
µνλσ
This expression is completely general. In the special case of forces the derivatives are taken with
respect to the nuclear coordinates. Since the atomic orbitals follow the nuclei (orbital following)
the geometrical derivatives of the integrals enter in the force expression (wave function forces).
14.13
BSSE
S
Take a complex A B and trace the energy as a function of their separation, EA S B (R, Ω). As a
consequence of the orbital-following principle, we have a different basis set at each distance and
relative orientation. In particular, for the interaction energy
∆E = EA∪B (R, Ω) − EA∪B (R = ∞, Ω) = EA∪B (R, Ω) − EA − EB
(425)
At small separations the partner basis functions improve the monomer description (nothing to do
with the physical interaction) and lead to an extra stabilization of the complex.
Remedy: Boys-Bernardi counterpoise correction
Calculate monomer energies in the dimer basis at each geometry
∆E = EA∪B (R, Ω) − EA (A ∪ B|R, Ω) − EB (A ∪ B|R, Ω)
(426)
As an example, the dissociation curve of the Ar dimer. An artificial minimum appears in the RHF
calculations; after counterpoise correction the potential curve has no minimum (electron correlation
would be needed to describe the van der Waals minimum in this system).
14.14
Response properties
In the Taylor-expansion of the total energy of a system in the presence of an external perturbation
∂E(F)
1 ∂ 2 E(F)
E(F) = E(0) +
Fα +
Fα Fβ + . . .
(427)
∂Fα F=0
2 ∂Fα ∂Fβ F=0
the second-order terms are related to the polarizability, i.e. linear response of the system to the
perturbation
2
∂ E(F)
ααβ =
(428)
∂Fα ∂Fβ F=0
We can apply the general Taylor-expansion of the energy by remarking that in the presence of an
external perturbation
Ĥ = Ĥ0 + Ĥ 0
a = ∇H = ∇H0 + ∇H 0
(429)
If the unperturbed energy is variational, ∇H0 = 0, we obtain the following equation for the
parameter variations, d that make the perturbed system energy stationary
∇H 0
M
Q
d
=0
(430)
+
∗
∗
H 0∇
Q
M
d∗
which simplifies in case of real wave function and perturbation as
∇H 0 + (M + Q)d = 0
59
(431)
The first order parameter change, d is proportional to the derivative of the Hamiltonian a = ∇H 0
d = (M + Q)−1 a
(432)
P
In the case of an external field perturbation Ĥ 0 = α Fα µ̂α the energy can be expressed as
XX
Fα (∇µα )† (M + Q)−1 (µβ ∇)Fβ
(433)
E = E0 − 2
α
β
Comparison with the definition of the electric dipole polarizability yields immediately
ααβ = (∇µα )† (M + Q)−1 (µβ ∇)
14.15
(434)
General wave function variation
We have seen that the first order variation of a single determinant wave function can be parameterized as
X
|Ψi = |Ψ0 i +
par |Ψra i
(435)
ar
A more general approach, valid to any kind of wave function and also to higher order variations,
can be developed by SQ.
P
Let us describe the variation of orbitals as resulting from a unitary transformation, ψi0 = k ψk Uki ,
which means that the corresponding creation and annihilation operators transform according to
X †
†
a0i =
ak Uki
k
a0i
=
X
(436)
∗
ak Uki
k
• The unitary transformation can be written as
+
a0i = exp (−iΛ̂)a+
i exp (iΛ̂)
(437)
where Λ̂ is a hermitian operator
Λ̂ =
XX
i
Λij a+
i aj
(438)
j
Using the expansion of the exponential operator
1 +
+
+
exp (−iΛ̂)a+
i exp (iΛ̂) = ai + [ai , iΛ̂] + [[ai , iΛ̂], iΛ̂] + . . .
2
(439)
The commutator expressions can be simplified using the anticommutator relationships,
e.g.
+
+ +
+
+
[a+
i , ak al ] = ai ak al − ak al ai
+
+
+
= −a+
k ai al − ak al ai
=
−a+
k (δil
−
al a+
i )
−
+
a+
k al ai
(440)
=
−a+
k δil
Applied to the first commutator
X
X
X
+
[a+
iΛkl [a+
iΛkl (−a+
a+
i , iΛ̂] =
i , ak al ] =
k δil ) = −
k iΛki
kl
kl
60
k
(441)
The second-order commutator yields
[[a+
i , iΛ̂], iΛ̂] =
X
a+
k (iΛiΛ)ki
(442)
k
and finally one can show that
exp (−iΛ̂)a+
i exp (iΛ̂) =
X
k
X
1
a+
a+
k (1 − iΛ̂ + (iΛ̂iΛ̂) + . . .)ki =
k exp (−iΛ̂)ki
2
(443)
k
This is a unitary transformation, provided that Λ is hermitian, i.e. we can parameterize
the transformation as
U = exp (−iΛ)
(444)
• The effect of orbital transformations is identical to a unitary (norm conserving!) transfor-
mation of the wave function. Consider a transformed determinant Ψ
+
+
+
|Ψ0 i = a01 a02 . . . a0N |vaci
(iΛ̂) (−iΛ̂) + (iΛ̂)
(iΛ̂)
= e(−iΛ̂) a+
e
a2 e
. . . e(−iΛ̂) a+
|vaci
1e
Ne
(445)
+
+
= exp (−iΛ̂)a+
1 a2 . . . aN |vaci = exp (−iΛ̂)|Ψi
and the result holds for any linear combination of determinants too.
• Since the hermitian matrix Λ can be expressed with the real symmetric and antisymmetric
matrices λ and K as Λ = λ + iK and the purely imaginary part of the transformation
concerns only an uninteresting phase factor exp (iλ), we can write
Û = exp (−iΛ̂) = exp (K̂)
(446)
• For small values of the transformation parameters, Λkl , we can expand the exponential
operator as
|Ψ0 i = exp (K̂)|Ψi = |Ψi+
X
Kkl a+
k al |Ψi−
1 XX
+
Kkl Kmn a+
k al am an |Ψi+. . .
2
mn
(447)
kl
kl
Bring the operator product of last term in the normal order
+
+
+
+
k +
a+
k al am an = ak (δlm − am al )an = δlm ak an + a am an al
(448)
and the transformed wave function is written as a linear combination of singly-, doubly-,
etc. excited configurations
|Ψ0 i = |Ψi +
X
kl
14.16
Kkl |Ψkl i −
1 XX
Kkl Kmn δlm |Ψkl i + |Ψkm
ln i + . . .
2
mn
(449)
kl
Coupled Hartree-Fock equations
Having a general expression of the parameterization of the wave function, we can write the energy
as
E = hΨ|e−K̂ ĤeK̂ |Ψi
1
1
= hΨ|(1 − K̂ + K̂ 2 − . . .)Ĥ(1 + K̂ + K 2 + . . .)|Ψi
2!
2!
1
= hΨ|Ĥ|Ψi + hΨ|[Ĥ, K̂]|Ψi + hΨ| [Ĥ, K̂], K̂ |Ψi + . . .
2!
61
(450)
The operator K̂ can be written in term of the one-particle excitation operators Êij as
X
X
K̂ =
pij a+
pij Êij
i aj =
ij
(451)
ij
and the Taylor expansion of the energy (for real wave function and variations)
E = hΨ|Ĥ|Ψi +
X
pij hΨ|[Ĥ, Êij ]|Ψi +
ij
1 XX
pij pkl hΨ| [Ĥ, Êij ], Êkl |Ψi + . . .
2! ij
(452)
kl
The M and Q matrices can be identified as
Qij,kl = hΨ|Êij Ĥ Êkl + Êkl Ĥ Êij |Ψi
(453)
Mij,kl = −hΨ|Ĥ Êij Êkl + Êkl Êij Ĥ|Ψi
The matrix elements in the second derivative of the energy for the special case of a Hartree-Fock
wave function are obtained after expansion of the SQ operators.
15
Comparison of RHF and UHF solutions
Example of the Li atom
β
|2 ΨRHF i = |φ1s φ1s φ2s i
α
|2 ΨUHF i = |φα
1s φ1s φ2s i
φα
2s
6
φ2s
6
φβ1s
φ1s
6
?
φα
1s
6
?
1sα and 1sβ electrons experience different effective potentials (with and without exchange interaction with 2sα). The 2sα electron “polarizes” the 1s shell, and the corresponding spatial functions
tend to be different.
The unrestricted doublet state can be expanded on the basis of exact doublet, quadruplet, etc.
states:
|2 ΨUHF i = c2 |2i + c4 |4i + c6 |6i + . . .
(454)
Since the contamination comes always from higher multiplicity components, hS 2 iUHF is always too
large, as can be seen from its expression.
15.1
RHF solution of minimal basis H2
The molecular orbitals are fully determined by symmetry
1
φ1 = p
2(1 + SAB )
1
φ2 = p
2(1 − SAB )
χa + χb
(455)
χb − χb
where χa and χb are 1s atomic orbitals on the two H atoms.
62
As the form of orbitals does not depend on the intermolecular separation, we can analyze the case
Rab → ∞, when Sab = hχa |χb i → 0. The one-determinant RHF wave function
ΨRHF
R→∞ = Â φ1 (1)α(1) φ1 (2)β(2)
1 = Â {χa (1) + χb (1)}α(1) {χa (2) + χb (2)}β(2)
2
1
=
Â[χa χa ] + Â[χb χb ]
ionic terms
2
+ Â[χa χb ] + Â[χb χa ]
covalent terms
(456)
The RHF wave function has an equal weight of “ionic” and “covalent” terms, instead of the correct
dissociation limit
1 ΨR→∞ = √ Â[χa χb ] + Â[χb χa ]
(457)
2
i.e. the symmetrized combination of two H-atom wave functions.
15.2
UHF orbitals of minimal basis H2
The unrestricted orbitals are not symmetry-constrained. The one degree of freedom can be incorporated easily if we write the unrestricted orbitals as linear combination of the restricted ones
φα,β
= cos θ φ1 ± sin θ φ2
1
φα,β
= ∓ sin θ φ1 + cos θ φ2
2
(458)
These are the unitary transformations which bring the orbitals from the original RHF set {φ1 , φ2 }
to the UHF ones.
cos θ ± sin θ
0 ±θ
U α,β = ∓ sin θ
=
exp
(459)
cos θ
∓θ
0
The two occupied orbitals of interest can be written in terms of the AOs as
φβ1 = c2 χa + c1 χb
(460)
1
sin θ
cos θ ± p
2(1 − Sab )
(461)
φα
1 = c1 χa + c2 χb
with the coefficients
1
c1,2 = p
2(1 + Sab )
63
θ=0
0 < θ < π/4
θ = π/4
α-spin MO
1.4
1.4
1.4
1.2
1.2
1.2
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
-2
-1
0
1
0.2
-2
2
-1
0
1
2
-2
-1
0
1
2
-2
-1
0
1
2
β-spin MO
1.4
1.4
1.4
1.2
1.2
1.2
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
-2
-1
0
1
2
0.2
-2
c1 = c2
15.3
-1
0
1
2
c1 > c2
c1 = 1.0 c2 = 0.0
UHF solutions for minimal basis H2
UHF energy
β
β
α
α α β β
E0 = (φα
1 |ĥ|φ1 ) + (φ1 |ĥ|φ1 ) + (φ1 φ1 |φ1 φ1 )
(462)
can be written after expanding the UHF orbitals in terms of the integrals over RHF MOs as
E0 (θ) = 2 cos2 θ h11 + 2 sin2 θ h22
+ cos4 θ J11 + sin4 θ J22 + 2 cos2 θ sin2 θ (J12 − 2K12 )
(463)
The energy is extremum if
dE0 (θ)/dθ = 4 cos θ sin θ
× h22 − h11 + sin2 θ J22 − cos2 θ J11
+ (cos2 θ − sin2 θ)(J12 − 2K12 ) = 0
(464)
• RHF solution exists at any interatomic distance with θ = 0, i.e.
4 cos θ sin θ = 0
(465)
with energy
E0 (0) = 2h11 + J11 = 2ε1 − J11
(466)
• UHF solution exists, if at a given R the values of integrals are such that
h22 − h11 + sin2 θ J22 − cos2 θ J11 + (cos2 θ − sin2 θ)(J12 − 2K12 ) = 0
(467)
i.e. the equation
cos2 θ =
h22 − h11 + J22 − J12 + 2K12
J11 + J22 − 2J12 + 4K12
can be satisfied.
64
(468)
Stability of the RHF/UHF solutions
15.4
The nature of the restricted solution can be studied through the second derivative of the energy
d2 E(θ)/dθ2 |θ=0 = E 00 (0) = 4(h22 − h11 + J22 − J12 + 2K12 )
= 4(ε2 − ε1 − J12 − K12 )
(469)
At the “saddle point” (E 00 (0) = 0) the UHF and RHF solutions become separated.
-0.6
R=1.4
-0.6
R=2.5
-0.6
R=4.0
-0.7
-0.7
-0.8
-0.8
-0.8
E(θ)
-0.7
-0.8
E(θ)
-0.7
E(θ)
E(θ)
-0.6
-0.9
-0.9
-0.9
-0.9
-1
-1
-1
-1
-1.1
-1.1
-1
-0.5
0
0.5
1
-1.1
-1
-0.5
0
0.5
1
R=10.0
-1.1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
θ
θ
θ
θ
E 00 (0) > 0
E 00 (0) ≈ 0
E 00 (0) < 0
E 00 (0) < 0
1
0.1
0.05
E (a.u.)
0
RHF
UHF
-0.05
-0.1
-0.15
CI
-0.2
1
1.5
2
2.5
3
3.5
R (a.u.)
15.5
Behaviour at dissociation
The UHF wave function tends to the correct dissociation limit, while the RHF leads to a “dissociation catastrophe”.
65
0.4
0.3
E (a.u.)
0.2
0.1
RHF
0
UHF
-0.1
CI
-0.2
0
1
2
3
4
5
6
7
R (a.u.)
The UHF wave function in the R → ∞ limit is
ΨUHF
R→∞ = Â[χa (1)χb (2)]
(470)
Instead of the correct (symmetrical) form
1 ΨR→∞ = √ Â[χa (1)χb (2)] + Â[χb (1)χa (2)]
2
Although the orbitals are correct, the wave function is not!
Expand the UHF wave function using the unitary transformation of the orbitals
β
|ΨUHF i = |φα
1 φ1 i = | cos θ φ1 + sin θ φ2 cos θ φ1 − sin θ φ2
= cos2 θ|φ1 φ1 i − sin2 θ|φ2 φ2 i − cos θ sin θ |φ1 φ2 i − φ2 φ1 i
|
{z
}
√
(471)
(472)
2|3 Ψ21 i
There is a triplet contaminant, with growing weight as R increases. At the dissociation limit
(θ = π/4)
1
1
|ΨUHF
|φ1 φ1 i − |φ2 φ2 i − |3 Ψ21 i
(473)
R→∞ i =
2
2
The triplet contaminant is the price to pay for having a one-determinant wave function.
16
Configuration interaction method
The multi-configurational wave function
1 2 X r r
1 2 X rs rs
ca |Ψa i +
cab |Ψab i
|Φi = c0 |Ψi +
1!
2!
ar
abrs
1 2 X rst rst
1 2 X rstu rstu
+
cabc |Ψabc i +
cabcd |Ψabcd i + . . .
3!
4!
abc
rst
abcd
rstu
66
(474)
where the coefficients are the eigenvectors of the CI-matrix.
The CI wave function is not normalized in the sense that
X
1 4 X rs 2
hΦ|Φi = 1 +
(cba )2 +
(cab ) + . . .
2!
ab
(475)
abrs
However, it satisfies the intermediate normalization condition
hΨ|Φi = 1
(476)
Schrödinger equation
Ĥ|Φi = E0 |Φi
(477)
By subtracting E0 |Φi
(Ĥ − E0 )|Φi = (E0 − E0 )|Φi = Ecorr |Φi
(478)
multiplication and integration by hΨ|
hΨ|Ĥ − E0 |Φi = Ecorr hΨ|Φi = Ecorr
(479)
lets appear explicitly the correlation energy, Ecorr .
Substitute the CI-expansion of the wave function
X
X
t
t
tu
tu
hΨ|Ĥ − E0 |Φi = hΨ|Ĥ − E0 |Ψi +
cc |Ψc i +
ccd |Ψcd i + . . .
ct
=
X
c<d
t<u
(480)
tu
ctu
cd hΨ|Ĥ|Ψcd i
c<d
t<u
The (unnormalized) coefficients of the double excited configurations determine the correlation
energy
X
tu
Ecorr =
ctu
(481)
cd hΨ|Ĥ|Ψcd i
c<d
t<u
In order to know these coefficients exactly, we must solve the full CI problem!
17
Structure of the CI matrix
The CI matrix

hΨ|Ĥ|Ψi
0
hΨ|Ĥ|Di

hS|Ĥ|Si hS|Ĥ|Di


hD|Ĥ|Di





0
hS|Ĥ|T i
hD|Ĥ|T i
hT |Ĥ|T i
0
0
hD|Ĥ|Qi
hT |Ĥ|Qi
hQ|Ĥ|Qi

...
. . .


. . .

. . .

. . .
• Brillouin theorem hΨ|Ĥ|Si = 0
• Slater rules hΨ|Ĥ|T i = hΨ|Ĥ|Qi = hS|Ĥ|Qi = 0
• nonzero blocks are sparse
• double excitations play a predominant role
• by indirect coupling all types of excitations contribute to the correclation energy
67
(482)
17.1
Doubly excited CI
Neglect all configurations except the doubly excited ones
X
tu
|ΦDCI i = |Ψi +
ctu
cd |Ψcd i
(483)
c<d
t<u
After substitution into
(Ĥ − E0 )|ΦDCI i = Ecorr |ΦDCI i
(484)
and successive multiplication by hΨ| and by hΨrs
ab | we obtain the system of equations
X
tu
ctu
cd hΨ|Ĥ|Ψcd i = Ecorr
c<d
t<u
hΨrs
ab |Ĥ|Ψi +
X
rs
tu
rs
ctu
cd hΨab |Ĥ − E0 |Ψcd i = cab Ecorr
(485)
c<d
t<u
Introduce the matrix notations
(B)rasb = hΨrs
ab |Ĥ|Ψi
tu
(D)rasb,tcud = hΨrs
ab |Ĥ − E0 |Ψcd i
rs
(c)rasb = cab
(486)
and the DCI matrix equations are
17.2
0
B
B†
D
1
1
= Ecorr
c
c
(487)
Approximate solutions of the DCI equations
The exact solution of the DCI problem is the lowest eigenvalue of the CI matrix
0 B†
B D
(488)
We can solve this problem by the matrix partitioning technique by considering the corresponding
partitioned system of equations
B † c = Ecorr
B + Dc = cEcorr
(489)
(490)
Solve the second equation for c
c = −(D − 1Ecorr )−1 B
(491)
and
Ecorr = −B † (D − 1Ecorr )−1 B
(492)
As a first approximation we can set Ecorr = 0 in the denominator and get
Ecorr ≈ −B † D −1 B
(493)
This expression is not variational (not upper bound to the energy) but size-extensive (unlike
truncated CI).
68
As a further simplification, we neglect off-diagonal elements of D, i.e.
(D −1 )rasb,tucd =
δac δbd δrt δsu
rs
hΨab |Ĥ − E0 |Ψrs
ab i
(494)
and the correlation energy becomes approximately
Ecorr ≈ −
X hΨ|Ĥ|Ψrs ihΨrs |Ĥ|Ψi
ab
a<b
r<s
ab
rs
hΨrs
ab |Ĥ − E0 |Ψab i
=
X
a<b
r<s
Ecorr
rs
ab
(495)
sum of double-excitation contributions.
17.3
Natural orbitals
Natural orbitals diagonalize the 1RDM for any (including multi-configurational) wave function.
Take a two-electron singlet system with the following CI expansion
1
| Φi = c0 |11̄i +
K
X
K
cr1 |1 Ψr1 i
r=2
K
1 X X rs 1 rs
c | Ψ11 i
+
2 r=2 s=2 11
(496)
where the spin-adapted singlet configuration functions (CF) are
1
|1 Ψr1 i = √ (|1r̄i + |r1̄i)
2
1
|1 Ψrs
11 i = √ (|rs̄i + |sr̄i)
2
(497)
Substitute these CFs
K
K K
1 X X rs 1 X r
c1 |1r̄i + |r1̄i + √
c11 |rs̄i + |sr̄i
|1 Φi = c0 |11̄i + √
2 r=2
2 r=2 s=2
=
K X
K
X
(498)
Cij |ij̄i
i=1 j=1
where the Cij coefficients form a symmetric matrix. The CI expansion contains K 2 terms.
The 1RDM can be calculated as
Z
XX
∗
0
∗
P (x1 , x1 ) =
Cij Ckl [i∗ (1)j̄ ∗ (2) − i∗ (2)j̄ ∗ (1)][k ∗ (10 )¯l∗ (2) − k ∗ (2)l¯0 (1)]d2
ij
=
X
kl
(499)
(CC † )ij [i(1)j ∗ (10 ) + ī(1)j̄ ∗ (10 )]
ij
Let us choose a transformation U such that
U † CU = d
(500)
where d is a diagonal matrix.
The orbitals ηk obtained from the transformation
X
X
ηk =
ψi Uik
and
ψi =
ηm Uim
(501)
m
i
diagonalize the 1RDM
P (x1 , x01 ) =
X
d2i [ηi (1)ηi∗ (10 ) + η̄i (1)η̄i∗ (10 )]
i
69
(502)
Substitution in the CI expansion leads to
K
X
Cij |ij̄i =
K X
K
X
ij
ij
=
K
X
Uim Cij Ujn |ηm η̄n i
mn
(U † CU )mn |ηm η̄n i =
K
X
(503)
dm |ηm η̄m i
m
mn
which contains only K terms instead of K 2 !
17.4
Minimal basis H2 molecule
Out of the six possible determinants four symmetry adapted configuration functions (CFs) can be
formed
X 1 Σ+
g
|Ψi
1
B Σ+
u
E 1 Σ+
g
b3 Σ +
u
=|11̄i
1
| Ψ21 i
|1 Ψ22
11 i
=|12̄i + |21̄i
=|22̄i


|12i
|3 Ψ21 i = |12̄i − |21̄i


|1̄2̄i
Taking into account the spin multiplicity (singlet and triplet) and space symmetry (g or u), the
CI matrix has the following structure

h1 Ψ|Ĥ|1 Ψi



h1 Ψ |Ĥ|1 Ψ22
11 i
1 22
|
Ĥ|
Ψ
h1 Ψ22
11
11 i
0
0
1 2
h Ψ1 |Ĥ|1 Ψ21 i

0

0


0
3 2
3 2
h Ψ1 |Ĥ| Ψ1 i
(504)
The energies of the B 1 Σ+
u state is given by
1
h12̄|Ĥ|12̄i − h21̄|Ĥ|12̄i − h12̄|Ĥ|21̄i + h21̄|Ĥ|21̄i
2
= h11 + h22 + J12 + K12
h1 Ψ21 |Ĥ|1 Ψ21 i =
(505)
and of the b3 Σ+
u state
h3 Ψ21 |Ĥ|3 Ψ21 i = h12|Ĥ|12i = h11 + h22 + J12 − K12
(506)
which lies below the corresponding singlet state (Hund’s rule) and tends to the limit of E(H) at
R → ∞.
The MO energies of the two 1 Σ+
g states are
E MO (X 1 Σ+
g ) = 2h11 + J11
E MO (E 1 Σ+
g ) = 2h22 + J22
which dissociate to the same limit (in the MO approximation!).
70
(507)
1
0.8
E (a.u.)
0.6
B 1 Σ+
u
0.4
E1 Σ+
g (MO)
b3 Σ +
u
0.2
X1 Σ+
g (MO)
0
-0.2
0
1
2
3
4
5
6
7
R (a.u.)
17.5
DCI of the minimal basis H2
To solve the 2 × 2 CI problem we need the matrix elements
(B)212̄1̄ = hΨ212̄1̄ |Ĥ|Ψi = h1̄||22̄i = [12|1̄2̄] − [12̄|1̄2] = (12|12) − 0 = K12
(508)
and
(D)212̄1̄,212̄1̄ = hΨ212̄1̄ |Ĥ − E0 |Ψ212̄1̄ i = 2h22 + J22 − 2h11 − J11
(509)
Using the orbital energies
ε1 = h11 + J11
ε2 = h22 + 2J12 − K12
(510)
we can write
(D)212̄1̄,212̄1̄ = 2∆ = 2(ε2 − ε1 ) + J11 + J22 − 4J12 + 2K12
The DCI matrix takes the simple form
0
K12
1
1
= Ecorr
K12 2∆
c
c
(511)
(512)
The solution can be found from
K12 + 2∆c = Ecorr c
K12
Ecorr − 2∆
2
K12
=
Ecorr − 2∆
c=
K12 c = Ecorr
Ecorr
71
(513)
(514)
The exact (in minimal basis) correlation energy is
q
2
Ecorr = ∆ − ∆2 + K12
(515)
At dissociation ε1 = ε2 , h11 = h22 = E(H). Since all two-electron integrals J11 = J22 = J12 =
K12 = (χa χa |χa χa ) tend to the same value, ∆ = 0 and the total energy
q
2
E CI (X 1 Σ+
)
=
2h
+
J
+
∆
−
∆2 + K12
(516)
11
11
g
tends to 2E(H).
In this limit the CI coefficient
c=
−K12
→ −1
2 )1/2
∆ + (∆2 + K12
(517)
which means that the CI wave function in the dissociation limit
22
|ΨCI
R→∞ i = |Ψi + |Ψ11 i = |χa χb i + |χb χa i
(518)
is the linear combination of two ground state H atoms.
1
0.8
E 1 Σ+
g (CI)
E (a.u.)
0.6
B 1 Σ+
u
0.4
E1 Σ+
g (MO)
b3 Σ +
u
0.2
X1 Σ+
g (MO)
0
X 1 Σ+
g (CI)
-0.2
0
1
2
3
4
5
6
7
R (a.u.)
17.6
Approximate CI solution of minimal basis H2
Use the approximate correlation energy expression
Ecorr ≈ −B † D −1 B
72
(519)
0.04
1
0.035
0.9
0.03
0.8
n1
-Ecorr
0.025
0.02
0.7
0.015
Since B = K12 and D † =
2∆ we get immediately
the estimation
Ecorr ≈ −
0.01
0.6
K12
2∆
0.005
0
0.5
1
1.5
0.5
2.5
2
R (a.u.)
In this special case, the one-particle density matrix is diagonal in the original MO basis (NO=MO).
Using the normalized CI expansion coefficients
C11 = √
1
1 + c2
the P-matrix
C22 = √
2
C11
0
0
2
C22
(520)
(521)
0.5
1
n1
0.45
0.4
0.9
Ecorr
0.35
0.8
n1
0.3
-Ecorr
c
1 + c2
0.25
0.2
0.7
0.15
0.1
0.6
Eappr
corr
0.05
0
The “perturbational” correlation energy is a good
approximation as far as
n1 ≈ 1 and it deteriorates when the wave
function becomes multiconfigurational (n1 < 1).
0.5
0
2
4
6
8
10
R (a.u.)
17.7
Size-consistency problem
The energy of a non-interacting dimer (N-mer) should be the sum of monomer energies.
The Hartee-Fock wave function of an infinitely separated dimer of H2 molecules is
|Ψ0 i = |11 1̄1 12 1̄2 i
73
(522)
with the energy
E0 = 4h11 + h11 1̄1 ||11 1̄1 i + h12 1̄2 ||12 1̄2 i + 2h11 1̄1 ||12 1̄2 i = 2(2h11 + J11 )
(523)
sum of the monomer energies.
The DCI wave function involves only “local” excitations
|Φi = |Ψ0 i +
2
X
ci |Φ21ii 2̄1̄ii i
(524)
i=1
Matrix elements
hΨ|Ĥ|Φ2111 2̄1̄11 i = h11 1̄1 ||21 2̄1 i = (11 21 |11 21 ) = K12
and the DCI matrix is

0
K
K
K
2∆
0
 
 
K
1
1
0   c  = Ecorr  c 
2∆
c
c
(525)
(526)
leading to the equations
K12 + 2∆c = Ecorr c
K12
Ecorr − 2∆
2
2K12
=
Ecorr − 2∆
c=
2Kc = Ecorr
Ecorr
(527)
(528)
DCI correlation energy of the dimer
(2)
Ecorr = ∆ −
q
2
∆2 + 2K12
(529)
is not equal to the sum of monomer correlation energies
(2)
q
2
Ecorr 6= 2(1) Ecorr = 2∆ − 2 ∆2 + K12
(530)
Similar result holds for an N-mer:
(N )
Ecorr = ∆ −
q
2
∆2 + N K12
(531)
which means that the DCI (truncated CI) correlation energy per monomer tends to 0 with increasing system size.
If we take the approximate correlation energy expression, applied to an N-mer
(N )
Ecorr ≈ −
N
X
B†Bi
i
Di
i
=−
NK
2∆
(532)
i.e. N times the approximate monomer correlation energy.
17.8
Full CI of a non-interacting H2 dimer
The wave function includes a quadruple excitation too
|Φi = |Ψ0 i + c1 |21 2̄1 12 1̄2 i + c2 |11 1̄1 22 2̄2 i + c3 |21 2̄1 22 2̄2 i
The FCI matrix is augmented with respect to the DCI one

 
 
0 K K
0
1
1
K 2∆ 0
 c 
c
K

 
 
K 0 2∆ K   c  = Ecorr  c 
0 K K 4∆
c3
c3
74
(533)
(534)
Solve the equations by taking first the fourth and first rows
2Kc + 4∆c3 = Ecorr c3
2Kc
Ecorr − 4∆
Ecorr
=
Ecorr − 4∆
c3 =
2Kc = Ecorr
Ecorr
(535)
(536)
Using the second row
K + 2∆ c + Kc3 = Ecorr c
(537)
and substitution of c3
2K
Ecorr − 4∆
which leads to the equation for the correlation energy
c=
Ecorr =
2K 2
Ecorr − 4∆
and the solution is twice the monomer correlation energy
p
Ecorr = 2(∆ − ∆2 + K 2 )
(538)
(539)
(540)
Remark that the coefficient of the quadruple excitation
c3 =
18
18.1
2Kc
=c×c
Ecorr − 4∆
(541)
Rayleigh-Schroedinger perturbation theory
Schroedinger equation with perturbation
We are looking for the ground state solution of the problem
Ĥψ = (Ĥ0 + V̂ )ψ = Eψ
(542)
and we already know the (exact) solution of the zero-order problem
Ĥ0 ϕ0 = E0 ϕ0
(543)
Let us write ∆E = E − E0 , the energy correction
(Ĥ0 − E0 )ψ = (∆E − V̂ )ψ
(544)
and in order to fix the phase of ψ, impose the intermediate normalization
hϕ0 |ψi = 1
(545)
Introduce the reduced resolvent operator as
R̂0 = (1 − |ϕ0 ihϕ0 |)(Ĥ0 − E0 )−1
(546)
which can be regarded as the inverse of the operator Ĥ0 − E0 in the space of functions orthogonal
to ϕ0
R̂0 (Ĥ0 − E0 ) = 1 − |ϕ0 ihϕ0 |
(547)
Multiplying the Schrödinger equation by R̂0
R̂0 (Ĥ0 − E0 )ψ = R̂0 (∆E − V̂ )ψ
75
(548)
using the definition of the resolvent and the intermediate normalization an equation is obtained
for the wave function
ψ = ϕ0 + R̂0 (∆E − V̂ )ψ
(549)
After multiplication of the Schrödinger equation by hϕ0 |
hϕ0 |(Ĥ0 − E0 )|ψi = hϕ0 |(∆E − V̂ )|ψi
(550)
we get the energy correction
∆E = hϕ0 |V̂ |ψi
18.2
(551)
Iterative solution of the perturbed Schroedinger equation
These equations can be solved iteratively
∆En = hϕ0 |V̂ |ψn−1 i
ψn = ϕ0 + R̂0 (∆En − V̂ )ψn
(552)
To the lowest orders of iteration we find by using ψ0 = ϕ0 and R̂0 ϕ0 = 0
∆E1 = hϕ0 |V̂ |ψ0 i
ψ1 = ϕ0 − R̂0 V̂ ψ0
∆E2 = hϕ0 |V̂ |ψ1 i = hϕ0 |V̂ |ψ0 i − hϕ0 |V̂ R̂0 V̂ |ψ0 i = ∆E1 − hϕ0 |V̂ R̂0 V̂ |ψ0 i
ψ2 = ϕ0 − R̂0 (∆E2 − V̂ )ψ1
(553)
= ϕ0 − R̂0 (hϕ0 |V̂ |ψ0 i − hϕ0 |V̂ R̂0 V̂ |ψ0 i − V̂ )(ϕ0 − R̂0 V̂ ψ0 )
= ϕ(1) − R̂0 (V̂ − ∆E2 )R̂0 V̂ ϕ0
18.3
H-atom in electric field
Hamiltonian of the H-atom in an electric field, Fz
Ĥ = Ĥ0 + ẑFz
1
1
Ĥ0 = − ∆ −
2
r
where
The Hamiltonian of the isolated H-atom has
the lowest eigenvalue E0 = − 12 and eigenfunction ϕ0 =
First iteration in the energy yields zero
(554)
√1 e−r .
π
∆E1 = Fz hϕ0 |ẑ|ψ0 i = µz · Fz = 0
(555)
First iteration in the wave function leads to the equation
ψ1 = ϕ0 − R̂0 V̂ ϕ0
(Ĥ0 − E0 )ψ1 = − V̂ − ∆E1 ϕ0
1
1 1
1
− ∆− +
ψ1 = − √ Fz z e−r
2
r
2
π
which has the solution
Fz r
ψ1 = − √ z
− 1 e−r
2
π
76
(556)
(557)
Second energy iteration
9
∆E2 = Fz hϕ0 |ẑ|ψ1 i = − · Fz2
4
This leads to a development of the energy in the powers of Fz
(558)
9
· F2
4 z
(559)
9
a.u.
2
(560)
E = E0 + ∆E2 = E0 −
the dipole polarizability is
α=−
18.4
∂2E
∂Fz2
=
Fz =0
Rayleigh-Schroedinger perturbation expansion
Assuming that the iteration process converges, the exact wave function and energy can be expanded
in power series of a perturbation parameter λ
(Ĥ0 + λV̂ )ψ = ∆Eψ
(561)
as
∆E =
∞
X
λn ∆E (n)
and
∞
X
ψ=
n=1
λn ψ (n)
(562)
n=0
Substitute the series expansions in
∆E = hϕ0 |V̂ |ψi
ψ = ϕ0 + R̂0 (∆E − V̂ )ψ
and
leading to
X
λn ∆E (n) =
n=1
X
λm+1 hϕ0 |V̂ |ψ (m) i
m=0
(563)
!
X
n
λ ψ
(n)
= ϕ0 + R̂0
n=1
X
m
λ ∆E
(m)
− λV̂
m=1
X
λk ψ (k)
k=1
and collect terms of the same power to obtain the general recursion formulae
∆E (n) = hϕ0 |V̂ |ψ (n−1) i
ψ (n) = −R̂0 V̂ ψ (n−1) −
n−1
X
∆E (k) R̂0 ψ (n−k)
(564)
k=1
18.5
Explicit formulae at low orders
The reduced resolvent has the spectral resolution
R̂0 =
X |ϕ0 ihϕ0 |
Ek − E 0
(565)
k6=0
◦ First order
∆E (1) = hϕ0 |V̂ |ϕ0 i
ψ (1) = −R̂0 V̂ ϕ0 = −
X hϕk |V̂ |ϕ0 i
k6=0
77
Ek − E 0
(566)
ϕk
(567)
(n)
The wave function corrections are often expressed in terms of the expansion coefficients ck
hϕk |ψ (n) i on the basis of the eigenfunctions of the zero order Hamiltonian
(1)
ck = −
X hϕk |V̂ |ϕ0 i
k6=0
E k − E0
=
(568)
◦ Second order
Energy
E (2) =hϕ0 |V̂ |ψ (1) i = −hϕ0 |V̂ R̂0 V̂ |ϕ0 i
X hϕ0 |V̂ |ϕk ihϕk |V̂ |ϕ0 i
=−
Ek − E 0
(569)
k6=0
Wave function
ϕ(2) = R̂0 V̂ R̂0 V̂ ψ (1) + R̂0 E (1) R̂0 V̂ ψ (0) =
= R̂0 V̂ R̂0 V̂ ϕ0 − R̂0 hV̂ iR̂0 V̂ ϕ0 = R̂0 V R̂0 V̂ ϕ0
(570)
where we used the definitions
hV̂ i = hϕ0 |V̂ |ϕ0 i
and
V = V̂ − hV̂ i
(571)
◦ Third order
∆E (3) = hϕ0 |V̂ |ψ (2) i = −hϕ0 |V̂ R̂0 V R̂0 V̂ |ϕ0 i
X X hϕ0 |V̂ |ϕk ihϕk |V |ϕl ihϕl |V̂ |ϕ0 i
=
(Ek − E0 )(El − E0 )
(572)
k6=0 l6=0
Summary of energy corrections
∆E (1) = hV̂ i
∆E
(2)
(573)
= hV̂ R̂0 V̂ i
(574)
∆E (3) = hV̂ R̂0 V R̂0 V̂ i
∆E (4) = hV̂ R̂0 V R̂0 V − hV R̂0 V i R̂0 V̂ i
(575)
(576)
∆E (5) = hV̂ R̂0 (V R̂0 V R̂0 V − hV R̂0 V R̂0 V i
− V R̂0 hV R̂0 V i − hV R̂0 V iR̂0 V )R̂0 V̂ i
18.6
(577)
Energy with the first-order wave function
The energy (Rayleigh quotient)
E=
hψ|Ĥ + λV̂ |ψi
hψ|ψi
(578)
with the first order wave function, ψ = ϕ0 + λψ (1) = ϕ0 − λV̂ R̂0 ϕ0
E=
hϕ0 − λϕ0 V̂ R̂0 |Ĥ + λV̂ |ϕ0 − λV̂ R̂0 ϕ0 i
1 + λ2 hϕ0 V̂ R̂0 |R̂0 V̂ ϕ0 i
78
(579)
After expanding the denominator and using that R̂0 Ĥ0 ϕ0 = 0
E = E0 + λE (1) − 2λ2 hϕ0 |V̂ R̂0 V̂ |ϕ0 i+
+ λ2 hϕ0 |V̂ R̂0 (Ĥ + λV̂ )R̂0 V̂ |ϕ0 i ×
× 1 − λ2 hϕ0 V̂ R̂0 |R̂0 V̂ ϕ0 i
(580)
and collecting terms of the same order
E = E0 + λE (1) −
− λ2 2hϕ0 |V̂ R̂0 V̂ |ϕ0 i−
(581)
− λ2 hϕ0 |V̂ R̂0 (Ĥ − E0 )R̂0 V̂ |ϕ0 i
+ λ3 hϕ0 |V̂ R̂0 (V̂ − E (1) )R̂0 V̂ |ϕ0 i
Use that E (1) = hV̂ i and R̂0 (Ĥ − E0 ) = 1 − |ϕ0 ihϕ0 | and obtain the previously derived 3rd order
energy expression
E = E0 + hϕ0 |V̂ |ϕ0 i − hϕ0 |V̂ R̂0 V̂ |ϕ0 i + hϕ0 |V̂ R̂0 (V̂ − hV̂ i)R̂0 V̂ |ϕ0 i
(582)
This result can be generalized: the nth order wave function determines the (2n-1)th order energy
expression, provided the normalization is taken into account.
The perturbational energy is not a upper bound to the exact energy, but the Rayleigh-quotient is
an upper bound.
18.7
Deformation energy and perturbation energy
Up to second order the Rayleigh quotient can be written as a sum of two terms, the expectation
value of the zero order Hamiltonian
hϕ − λϕV̂ R̂0 |Ĥ|ϕ − λV̂ R̂0 ϕi
1 + λ2 hϕV̂ R̂0 |R̂0 V̂ ϕi
(2)
= E0 + λ2 hϕ|V̂ R̂0 V̂ |ϕi = E0 + ∆Edef
(583)
and the expectation value of the perturbation operator
hϕ − λϕV̂ R̂0 |λV̂ |ϕ − λV̂ R̂0 ϕi
1+
λ2 hϕV̂
R̂0 |R̂0 V̂ ϕi
(2)
= λhϕ|V̂ |ϕi − λ2 2hϕ|V̂ R̂0 V̂ |ϕi = E (1) + ∆Estab
(584)
The second order correction to the expectation value of the perturbation is twice the second order
energy correction and it is twice the energy raise of the wave function due to the deformation of
the wave function.
(2)
(2)
∆Estab = −2∆Edef
(585)
18.8
Dalgarno’s (2n+1) theorem
Although the recursion formulae would suggest that we need the (n − 1) order wave function
correction to calculate the n-th order energy correction, as the previous example showed, there is
a much stronger statement:
In order to obtain the (2n + 1) order energy correction, it is sufficient to know the n-th
order wave function correction.
Wave function
n
X
ψ=
ψ (k) + O(λn+1 )
(586)
k
79
Energy
E=
n
n X
X
k
hψ (k) |Ĥ|ψ (l) i + O(λ2n+2 )
(587)
l
Transformation formulae (Löwdin)
∆E (2n) = hϕ0 |V̂ |ψ (2n−1) i
= hψ (n) |V̂ |ψ (n−1) i −
n X
n
X
∆E (2n−k−l) hψ (k) |ψ (l) i
(588)
k=1 l=1
∆E (2n+1) = hϕ0 |V̂ |ψ (2n) i
= hψ (n) |V̂ |ψ (n) i −
n X
n
X
∆E (2n+1−k−l) hψ (k) |ψ (l) i
(589)
k=1 l=1
Computational difficulties increase significantly at 2n orders.
18.9
Pertubational correction of expectation values
Expectation value of an arbitrary Hermitian operator, B̂ with a first order wave function ψ ≈
(1 + R̂0 V̂ )ϕ
hϕ − λϕV̂ R̂0 |B̂ + λV̂ |ϕ − λV̂ R̂0 ϕi
B=
(590)
1 + λ2 hϕV̂ R̂0 |R̂0 V̂ ϕi
Expanding up to first order in the perturbation
hϕ|B̂ + B̂ R̂0 V̂ + V̂ R̂0 B̂ + V̂ R̂0 (B̂ − B0 )R̂0 V̂ |ϕi = B0 + ∆B (1)
(591)
Suppose that the perturbation is of the form V̂ = a = Âa† , then
∆B (1) = a† hÂR̂0 B̂ + B̂ R̂0 Âia = a† K(ÂB̂)a
(592)
where K(ÂB̂) = hÂR̂0 B̂ + B̂ R̂0 Âi is the linear response function.
18.10
Partitioning method
Let us consider an orthonormal basis {ϕk } divided into two subsets, A and B, containing nA and
nB functions, respectively. The result, obtained by using only nA functions can be improved by
adding the nB extra functions, i.e. the secular equations are partitioned as
AA
A
A
H
H AB
c
c
=
E
(593)
cB
cB
H BA H BB
which is equivalent to the system of equations
H AA cA + H AB cB =EcA
H BA cA + H BB cB =EcB
(594)
A formal solution can be obtained by expressing cB from the second equation
cB = (E1BB − H BB )−1 H BA cA
80
(595)
and inserting it into the first one leading to
H eff cA = EcA
(596)
The effective Hamiltonian is an nA × nA matrix
H eff = H AA + H AB (E1BB − H BB )−1 H BA
(597)
including implicitly the effect of the nB other functions. The effective Hamiltonian depends on the
yet unknown energy, E, therefore the solution should be obtained by iteration.
Take the case nA = 1, cA = c1 = 1, then H eff is just a single element
E = H eff = f (E)
(598)
Insert as a first approximation E = H11 and expand the inverse matrix, using (I + ∆)−1 =
I − ∆ + ∆2 leading to second order in the off-diagonal elements
E = H11 +
X H1κ Hk1
H11 − Hκκ
κ>1
(κ > 1)
(599)
The expansion coefficients in the cB vector (c.f. above)
cB = (E1BB − H BB )−1 H BA cA
(600)
can be approximated similarly as
cκ =
Hκ1
H11 − Hκκ
(601)
Analogous to RSPT, but
◦ basis is finite
◦ no a priori separation of the Hamiltonian is necessary
◦ complete set of eigenfunctions not needed
◦ analogies with RSPT to handled with caution
To make the connection with RSPT clear, choose the eigenfunctions of Ĥ0 as basis
(0)
Ĥ0 ϕk = Ek ϕk
(602)
and the relevant matrix elements are
(0)
H11 = hϕ1 |Ĥ0 + V̂ |ϕ1 i = E1 + hϕ1 |V̂ |ϕ1 i
(603)
H1k = hϕ1 |Ĥ0 + V̂ |ϕk i = hϕ1 |V̂ |ϕk i
(604)
By application of the previously derived results we obtain for the perturbational corrections to the
i-th state
X hϕi |V̂ |ϕk ihϕk |V̂ |ϕi i
(0)
Ei = Ei + hϕi |V̂ |ϕi i +
(605)
(0)
(0)
Ei − E k
k(6=i)
ψ i = ϕi +
X
cik ϕk = ϕi −
k(6=i)
X hϕk |V̂ |ϕi i
(0)
k(6=i) Ei
(Quasi)-degenerate cases can be handled by taking(nA > 1).
81
(0)
− Ek
ϕk
(606)
In non-orthogonal basis sets the matrix equations are
AA
A
AA
H
H AB
c
M
=
E
cB
H BA H BB
M BA
M AB
M BB
cA
cB
(607)
where M is the overlap (metric) matrix of the basis. The effective equation is
H eff cA = EM AA cA
(608)
H AA + (H AB − EM AB )(E1BB − H BB )−1 (H BA − EM BA )
(609)
with the effective Hamiltonian
and the energy to second order is
2
E=
H00 X [Hk0 − Mk0 (H00 /M00 )]
+
M00
[H00 /M00 − (Hkk /Mkk )]
(610)
k
19
RSPT treatment of intramolecular correlation
Let us separate the N-electron Hamiltonian into two parts
Ĥ = Ĥ0 + Ŵ
(611)
where Ĥ0 is the Hartree-Fock Hamiltonian
X
Ĥ0 =
F̂ (i)
(612)
i
and the perturbation is the difference between the HF and total Hamiltonians
Ŵ = Ĥ −
X
X 1
X
ˆ − K̂(i)
−
J(i)
r
i
i<j ij
F̂ (i) =
i
(613)
which often called the fluctuation potential.
The zero order wave function is the Hartree-Fock determinant.
The zero order energy is the sum of occupied orbital energies
E (0) = hΨ|
X
F̂ (i)|Ψi =
N
X
εa
(614)
a
i
The first order energy correction is just the “double counting correction” we discussed in the
Hartree-Fock theory
X
XX
X
E (1) = hΨ|Ŵ |Ψi =
hab||abi −
(Jab − Kab ) = −
hab||abi
(615)
a
a<b
b
a<b
The Hartree-Fock energy is
E0 = E (0) + E (1) =
N
X
N
εa −
a
N
1 XX
hab||abi
2 a
(616)
b
In order to apply the second order energy and first order wave function correction formulae, the
“intermediate states”, ϕk should be identified ⇒ excited determinants formed from the complete
set of eigenfunctions of F̂ .
82
◦ singly excited determinants
hΨ|Ŵ |Ψra i = hΨ|Ĥ − Ĥ0 |Ψra i = hΨ|Ĥ|Ψra i − Far = 0
(617)
they cannot contribute by the virtue of the Brillouin theorem and HF equations.
◦ triply (and higher) excited determinants do not contribute – Slater rules
◦ double excitations, |Ψrs
ab i – yes
Application of the general formula leads to
E (2) = −
X hΨ|Ŵ |Ψrs ihΨrs |Ŵ |Ψi
ab
a<b
r<s
ab
(0) |Ψrs i
hΨrs
ab |Ĥ0 − E
ab
(618)
Let us use that
(0)
hΨrs
|Ψrs
ab |Ĥ0 − E
ab i = −(εa + εb − εr − εs )
and the Slater rules
hΨ|Ŵ |Ψrs
ab i = hΨ|
X 1
|Ψrs
ab i = hab||rsi
r
ij
i<j
(619)
(620)
we end up the second order Møller-Plesset (or MBPT) correction
E (2) =
X
a<b
r<s
|hab||rsi|2
1X
|hab||rsi|2
=
εa + εb − εr − εs
4
εa + εb − εr − εs
(621)
abrs
In terms of the regular integrals the second order energy
E (2) =
1 X hab|rsihrs|bai
1 X hab|rsihrs|abi
−
2
ε a + ε b − ε r − εs
2
εa + εb − εr − εs
abrs
(622)
abrs
and for closed shells in terms of spatial orbitals
N/2
E
(2)
=2
X
abrs
19.1
N/2
X hab|rsihrs|bai
hab|rsihrs|abi
−
ε a + ε b − εr − εs
εa + εb − εr − εs
(623)
abrs
Third order MBPT energy
The third order energy expression is more laborious
E (3) =
1 X
hab||rsihcd||abihrs||cdi
8
(εa + εb − εr − εs )(εc + εd − εr − εs )
abcdrs
1 X
hab||rsihrs||tuihtu||abi
+
8
(εa + εb − εr − εs )(εa + εb − εt − εu )
abrstu
X
hab||rsihcs||tbihrt||aci
+
(εa + εb − εr − εs )(εa + εc − εr − εt )
abcrst
83
(624)
19.2
Comparison with the partitioning method
We have already seen (c.f. discussion of the CI method) that the DCI correlation energy can be
obtained from the following matrix


hΨ|Ĥ0 + Ŵ |Ψi
hΨ|Ŵ |Ψrs
...
hΨ|Ŵ |Ψtu
ab i
cd i
 hΨrs |Ŵ |Ψi

rs
tu
hΨrs
hΨrs


ab
ab |Ĥ0 + Ŵ |Ψab i . . .
ab |Ŵ |Ψcd i
(625)


...
...
...
...
rs
tu
hΨtu
hΨtu
. . . hΨtu
cd |Ŵ |Ψi
cd |Ŵ |Ψab i
cd |Ĥ0 + Ŵ |Ψcd i
where we used the notations
hΨ|Ĥ0 + Ŵ |Ψi
B abrs
B †abrs
D abrs,cdtu
(626)
and the correlation energy is
∆E =
XX
−1
B †abrs (D abrs,cdtu )
B abrs
(627)
abrs cdtu
Decompose the D abrs,cdtu matrix D = K + W , i.e. as a sum of a diagonal (K) and non-diagonal
(W ) contribution and expand the inverse matrix
D−1 =(K + W )−1 = [K(1 + K −1 W )]−1
=(1 + K −1 W )−1 K −1 = K −1 − K −1 W K −1 + K −1 W K −1 W K −1 . . .
(628)
rs
Since the inverse of the diagonal matrix Kab
is trivial, the approximate correlation energy has the
form
X †
−1
∆E =
B abrs (K rs
B abrs
(629)
ab )
abrs
Two possible choices for the partition
◦ Møeller-Plesset partition
rs
rs
(0)
K rs
− (εa + εb − εr − εs )
ab (MP) = hΨab |Ĥ0 |Ψab i = E
(630)
leading to the MP2 result
∆E(MP2) =
1 X B †abrs B abrs
4
εa + εb − εr − εs
(631)
abrs
◦ Epstein-Nesbet partitioning
rs
rs
rs
rs
K rs
ab (EN) = hΨab |Ĥ0 + Ŵ |Ψab i = K ab (MP) − dab
(632)
leading to
∆E(EN) =
B †abrs B abrs
1XX
4
(εa + εb − εr − εs ) − drs
ab
(633)
abrs cdtu
which is the result obtained previously as an approximate correlation energy expression.
84
19.3
Partial summation of MBPT series
Second order correction
E(2) =
1X
rs −1
hab||rsihrs||abi(Kab
)
4
(634)
abrs
A component of the third order correction
1X
rs −1
hab||rsidrs
ab hrs||abi(Kab )
4
E(3) =
(635)
abrs
where
drs
ab = hab||abi + hrs||rsi − har||ari − hbr||bri − has||asi − hbs||bsi
(636)
Similar terms come in fourth-, fifth-, etc. order
"
#
2
1 X −1
d
d
E(2) + E(3) + . . . =
K hab||rsihrs||abi 1 +
+
+ ...
4
K
K
abrs
1 X hab||rsihrs||abi
=
4
K(1 − d/K)
(637)
abrs
We obtain by partial summation the Epstein-Nesbet result.
19.4
MBPT correlation energy of minimal basis H2
The exact correlation energy was found for this system as
2 1/2
Ecorr = ∆ − (∆2 + K12
)
(638)
with
∆ = 2(ε2 − ε1 ) + J11 + J22 − 4J12 + 2K12
(639)
We can apply directly the formulae by considering that a = b = 1 and r = s = 2. Second order
E (2) = 2
2
h11|22ih22|11i h11|22ih22|11i
K12
−
=
2(ε1 − ε2 )
2(ε1 − ε2 )
2(ε1 − ε2 )
(640)
The third order result can be obtained from the general formula by taking into account that the
first order energy correction
E (1) = hΨ|Ŵ |Ψi = −J11
(641)
and the perturbation matrix element
hΨ|Ŵ |Ψ212̄1̄ i = h11̄||22̄i = h11|22i = K12
(642)
Third order energy
E (3) =
2
K12
(J11 + J22 − 4J12 + 2K12 )
4(ε1 − ε2 )
85
(643)
0.4
0.1
0.05
0.3
MP2
RHF
0.2
MP3
E (a.u.)
E (a.u.)
0
-0.05
-0.1
0.1
RHF
MP2
0
MP3
-0.15
-0.1
CI
-0.2
CI
-0.2
1
1.5
2
2.5
3
3.5
0
1
2
3
R (a.u.)
19.5
4
5
6
7
R (a.u.)
Size-consistency
N-mer of non-interacting H2 molecules
Second order energy
N
2
X
|hΨ|Ŵ |Ψ21ii 2̄1̄ii i|2
N K12
=
E (2) =
2(ε1 − ε2 )
2(ε1 − ε2 )
i=1
(644)
is N times the monomer MP2 energy.
Third order energy correction
E (3) =
2
N K12
(J11 + J22 − 4J12 + 2K12 )
4(ε1 − ε2 )2
(645)
The general result is true: the MPn energy correction is size-consistent.
19.6
Size-consistent methods
The size consistency can be qualitatively discussed by invoking the one- and two-particle excitation
operators
X
X
rs + +
T̂1 =
Tar a+
T̂2 =
Tab
ar as ab aa
(646)
r aa
ar
abrs
◦ SDCI wave function of a complex
(1 + T̂1A + T̂2A )|ΨA i(1 + T̂1B + T̂2B )|ΨB i
= |ΨAB (SDCI)i + (T̂1A T̂2B + . . .)|ΨA ΨB i
which is not of SCDI form
◦ coupled cluster wave functions (CCSD) of a complex
exp (T̂1A + T̂2A )|ΨA i exp (T̂1B + T̂2B )|ΨB i
is still of the CCSD form.
86
=
exp (T̂1A + T̂1B + T̂2A + T̂2B )|ΨA ΨB i
19.7
Local MP2
One can use localized (Wannier-) orbitals instead of canonical ones in the MPn theory. Problem:
Fock operator is not diagonal.
First order wave function correction satisfies ψ (1) = ϕ0 − R̂0 Ŵ ϕ0 , i.e.
(Ĥ0 − E (0) )ψ (1) = −Ŵ ϕ0
(647)
Let ϕ0 = Ψ0 , the Hartree-Fock wave function written in localized orbitals and the first order wave
function correction is
XX
ab rs
ψ (1) =
Trs
Ψab
(648)
rs
ab
with r, s localized virtual orbitals.
X
ab
(Ĥ0 − E (0) )|Ψrs
ab iTrs = −Ŵ |Ψ0 i
(649)
abrs
Multiplication from left by hΨtu
cd | leads to the linear equations
X
(0)
ab
tu
hΨtu
|Ψrs
cd |Ĥ0 − E
ab iTrs = −hΨcd |Ŵ |Ψ0 i
(650)
abrs
ab
which can be solved directly for the unknown amplitudes, Trs
, and the MP2 energy is
X
ab
E (LMP2) =
Trs
hΨ0 |Ŵ |Ψrs
ab i
(651)
abrs
Advantage: linear scaling correlation method!
Of course, this is a generalization of the canonical MP2 result, where the linear equations can be
solved analytically, since
(0)
hΨtu
|Ψrs
cd |Ĥ0 − E
ab i = δac δbd δrt δsu (εt + εu − εb − εc )
is a diagonal matrix.
87
(652)