Theor Chem Acc (2015) 134:118
1 3
where
and
Additionally, the matrices D − and E − are of the form
∀ p = q
r = s
B pq,rs = A pq,sr = (n p − n q )(δ ps h qr − δ qr h ps )
+ (c p c s δ I p I s + c q c r δ I q I r )(pq|rs + pq|sr)
+ [n p n s (1 − δ I p I s ) + n q n r (1 − δ I q I r ) − n p n r (1 − δ I p I r ) − n q n s (1 − δ I q I s )]pr||qs
− δ qr c p
t
δ I p I t c t ps|tt − δ ps c q
t
δ I q I t c t qr|tt − δ pr c r
t
δ I r I t c t qs|tt
− δ qs c s
t
δ I s I t c t pr|tt + δ ps
t
n t [n p (1 − δ I p I t ) − n q (1 − δ I q I t )]qt||rt
− δ qr
t
n t [n p (1 − δ I p I t ) − n q (1 − δ I q I t )]pt||st ,
(32)
∀ p =q B pq,rr = B rr,pq = 2c r (c q δ I r I q + c p δ I r I p )rr|pq − 2c r (δ rq + δ rp )
s
c s δ I r I s ss|pq,
(33)
∀ p,q B pp,qq = 4c p c q
δ I q I p qq|pp + δ pq
2
r
n r (1 − δ I q I r )qr||qr + 2h qq − μ I q
.
(34)
∀ p > q
r
(c p − c q )D
−
pq,r = 2c r (δ rp − δ rq )h qp + (δ rp − δ rq )
s
c s δ I r I s ss|pq
+ (c p δ I r I p − c q δ I r I q )rr|pq
+ 2(δ rp − δ rq )c r
s
(1 − δ I r I s )n s qs||ps
+ 2c r [n p (1 − δ I r I p ) − n q (1 − δ I r I q )]qr||pr,
(35)
∀ pq E
−
pq = δ I p I q pp|qq + δ pq
2
r
n r (1 − δ I q I r )qr||qr + 2h qq − μ I q
+ 4c p c q (1 − δ I p I q )pq||pq.
(c p = 0) I p = N/2 + 1 —so there is an additional (N/2 + 1)
th geminal and it includes all the virtual orbitals.
References
1. Shaik S, Hiberty PC (2004) Valence bond theory, its history, fundamentals, and applications: a primer. Wiley, New York
2. Goddard WA III (1967) Phys Rev 157:81
3. Hunt WJ, Hay PJ, Goddard WA III (1972) J Chem Phys 57:738
4. Goddard WA III, Dunning TH Jr, Hunt WJ (1969) Chem Phys
Lett 4:231
5. Hunt WJ, Dunning TH Jr, Goddard WA III (1969) Chem Phys
Lett 3:606
6. Bobrowicz FW, Goddard WA (1977) In: Schaefer HF III (ed)
Modern theoretical chemistry: methods of electronic structure
theory. Plenum, New York, pp 79–127
7. Hurley AC, Lennard-Jones JE, Pople JA (1963) Proc R Soc A 220:446
The two-electron integrals pq|rs and pq||rs are in terms of
the natural orbitals, and they are defi ned in Eqs. ( 11 ) and ( 12 ).
The Lagrange multipliers
μ I p
are obtained from the stationary equations for the ground-state expansion coeffi cients
∀ P ∀ p∈P ∂[E GVB−PP − μ I p
q∈P c 2
q ]/∂c p = 0 reading
Notice that the symbol I p used in Eqs. ( 27 )–( 36 ) in
the context of the GVB-PP for an N -electron system
should be understood as follows: For an occupied orbital
ϕ p (c p = 0) I p is equal to an index of a geminal to which
an orbital is assigned ( I p = P if p ∈ P ), for a virtual orbital
(36)
∀ P=1,...,N/2 ∀ p∈P μ I p c p = 2c p h pp +
q
c q δ I q I p qq|pp
+ 2c p
q
(1 − δ I p I q )n q pq||pq.
(31)
226
Reprinted from the journal
1 3
where
and
Additionally, the matrices D − and E − are of the form
∀ p = q
r = s
B pq,rs = A pq,sr = (n p − n q )(δ ps h qr − δ qr h ps )
+ (c p c s δ I p I s + c q c r δ I q I r )(pq|rs + pq|sr)
+ [n p n s (1 − δ I p I s ) + n q n r (1 − δ I q I r ) − n p n r (1 − δ I p I r ) − n q n s (1 − δ I q I s )]pr||qs
− δ qr c p
t
δ I p I t c t ps|tt − δ ps c q
t
δ I q I t c t qr|tt − δ pr c r
t
δ I r I t c t qs|tt
− δ qs c s
t
δ I s I t c t pr|tt + δ ps
t
n t [n p (1 − δ I p I t ) − n q (1 − δ I q I t )]qt||rt
− δ qr
t
n t [n p (1 − δ I p I t ) − n q (1 − δ I q I t )]pt||st ,
(32)
∀ p =q B pq,rr = B rr,pq = 2c r (c q δ I r I q + c p δ I r I p )rr|pq − 2c r (δ rq + δ rp )
s
c s δ I r I s ss|pq,
(33)
∀ p,q B pp,qq = 4c p c q
δ I q I p qq|pp + δ pq
2
r
n r (1 − δ I q I r )qr||qr + 2h qq − μ I q
.
(34)
∀ p > q
r
(c p − c q )D
−
pq,r = 2c r (δ rp − δ rq )h qp + (δ rp − δ rq )
s
c s δ I r I s ss|pq
+ (c p δ I r I p − c q δ I r I q )rr|pq
+ 2(δ rp − δ rq )c r
s
(1 − δ I r I s )n s qs||ps
+ 2c r [n p (1 − δ I r I p ) − n q (1 − δ I r I q )]qr||pr,
(35)
∀ pq E
−
pq = δ I p I q pp|qq + δ pq
2
r
n r (1 − δ I q I r )qr||qr + 2h qq − μ I q
+ 4c p c q (1 − δ I p I q )pq||pq.
(c p = 0) I p = N/2 + 1 —so there is an additional (N/2 + 1)
th geminal and it includes all the virtual orbitals.
References
1. Shaik S, Hiberty PC (2004) Valence bond theory, its history, fundamentals, and applications: a primer. Wiley, New York
2. Goddard WA III (1967) Phys Rev 157:81
3. Hunt WJ, Hay PJ, Goddard WA III (1972) J Chem Phys 57:738
4. Goddard WA III, Dunning TH Jr, Hunt WJ (1969) Chem Phys
Lett 4:231
5. Hunt WJ, Dunning TH Jr, Goddard WA III (1969) Chem Phys
Lett 3:606
6. Bobrowicz FW, Goddard WA (1977) In: Schaefer HF III (ed)
Modern theoretical chemistry: methods of electronic structure
theory. Plenum, New York, pp 79–127
7. Hurley AC, Lennard-Jones JE, Pople JA (1963) Proc R Soc A 220:446
The two-electron integrals pq|rs and pq||rs are in terms of
the natural orbitals, and they are defi ned in Eqs. ( 11 ) and ( 12 ).
The Lagrange multipliers
μ I p
are obtained from the stationary equations for the ground-state expansion coeffi cients
∀ P ∀ p∈P ∂[E GVB−PP − μ I p
q∈P c 2
q ]/∂c p = 0 reading
Notice that the symbol I p used in Eqs. ( 27 )–( 36 ) in
the context of the GVB-PP for an N -electron system
should be understood as follows: For an occupied orbital
ϕ p (c p = 0) I p is equal to an index of a geminal to which
an orbital is assigned ( I p = P if p ∈ P ), for a virtual orbital
(36)
∀ P=1,...,N/2 ∀ p∈P μ I p c p = 2c p h pp +
q
c q δ I q I p qq|pp
+ 2c p
q
(1 − δ I p I q )n q pq||pq.
(31)
226
Reprinted from the journal