Theor Chem Acc (2016) 135:3
1 3
within the finite-dimensional subspace spanned by the
chosen basis set. The last term allows this subspace
to be rotated within the Hilbert space of one-particle
states to attain the deeper minimum with respect to the
total energy.
Substituting Eq. ( 21 ) into Eq. ( 19 ) and taking into
account the independence of the variations and their arbitrariness, we obtain the following equations for orbitals
(see [ 10 , 11 ] for more details):
and equations for basis set optimization
If the K th excited state is considered then, as mentioned, the projector P α
u takes the following form
P α
u =
K−1
k=0
ϕ α
kn
ϕ α
kn
. Equations ( 22 ) represent the
orthogonality-constrained HF method in its general form.
According to the AP methodology, the orthogonality
constraint of Eqs. ( 12 ) and ( 18 ) is satisfi ed in the limit
o → ∞ and s → ∞, , respectively. By setting s = 0 , we
can relax the spin-purity constraint ( 18 ) and go back from
ROHF to UHF solutions. By setting o = 0 , we fall back to
the ground state. The corresponding orbitals form an optimal set which satisfy the generalized Brillouin’s theorem
(see [ 40 ] for more details) and lead to the same total energy
that the one obtained by the Roothaan procedure. In our
method, the only additional computation required, beyond
that arising in the standard UHF scheme, is the evaluation
of the overlap matrix element ϕ 0n |ϕ kn .
Left side of Eq. ( 23 ) represent a gradient of the total
energy with respect to nonlinear basis set parameters {μ a } .
This expression allows these parameters to be determined
variationally and can be also used to construct an algorithm for optimization based on the gradient-like methods.
Since neither λ s nor λ o can be infi nity in practical calculations, one has to settle on some large fi nite values. The
recommended values are λ s = 100 hartrees for the spinpurity constraint and λ o = 1000 hartrees for the orthogonality constraint. They provide target accuracy close to 10
−6 .
In concluding this section, it is also worth noting that in
our method all excited confi gurations based on the excited
Slater determinant Φ 1 , viz., Φ a
i , Φ ab
ij etc., where i and j refer
(22)
P 1
F α − s P
β
1 + o P α
u − α
i
P 1
ϕ α
1i
= 0 , s , o → ∞
P 1
F β + s Q α
1 −
β
i
P 1
ϕ
β
1i
= 0,
i = 1, 2, . . . , M
(23)
n α
i
ϕ
α
1i
(∂ a P 1 )F
α
ϕ
α
1i
+
n β
i
ϕ
β
1i
(∂ a P 1 )F
β
ϕ
β
1i
= 0, a = 1, 2, . . . A
to occupied orbitals and a and b to virtual ones, are orthogonal both to Φ 0 and among themselves. Therefore, these
functions form the orthonormal basis set in the many-body
space and can be used, unlike other SCF methods which do
not satisfy the orthogonality of states in the explicit form,
to develop many-body methods incorporating the correlation effects, in particular, a many-body Møller–Plesset-like
perturbation theory (see next Section).
4 Second-order correction to the energy
for excited states
It is known that within the framework of the Roothaan coupling operator approach, there is no unique way of choosing a reference Hamiltonian, H (0) , with respect to which
a perturbation expansion for correlation effects can be
developed. Several proposals have been made for openshell many-body perturbation theory expansions based
on a reference from the ROHF formalism [ 41 , 42 ] or the
unrestricted Hartree–Fock formalism [ 43 , 44 ]. We follow
our papers [ 10 , 45 ] where an alternative technique for the
open-shell systems has been developed. In our method, the
second-order correction to the ground state energy can be
presented by [ 45 ]:
The summations are over spin-orbitals. Subscripts i , j and
a , b correspond to occupied and virtual orbitals of the
ground state determinant, respectively. Unlike the formalism developed in Refs. [ 43 , 44 ], single excitations do
not contribute because our orbitals satisfy the generalized
Brillouin theorem.
An optimum set of MOs obtained by means of
Eq. ( 22 ) allows us to construct a well-defi ned open-shell
perturbation theory for excited states which is a natural
extension of the popular closed-shell MP2. For example,
the zeroth-order Hamiltonian for the fi rst excited state is
as follows:
with Fockians
The summation up to M − 1 means that the vector |ϕ α
0n is
excluded from the subspace of virtual orbitals. Remind
M is the dimension of the basis set for the fi rst ES. We
(24)
E
(2)
0 =
occ
i>j
virt
a>b
ϕ 0a ϕ 0i |ϕ 0b ϕ 0j
−
ϕ 0a ϕ 0j |ϕ 0b ϕ 0i
2
0i + 0j − 0a − 0b
H
(0)
=
n α
m=electrons
F
α
(m) +
n β
m=electrons
F
β
(m)
F
σ
=
M−1
i
ϕ
σ
i
σ
i
ϕ
σ
i
, σ = α, β
193
Reprinted from the journal
1 3
within the finite-dimensional subspace spanned by the
chosen basis set. The last term allows this subspace
to be rotated within the Hilbert space of one-particle
states to attain the deeper minimum with respect to the
total energy.
Substituting Eq. ( 21 ) into Eq. ( 19 ) and taking into
account the independence of the variations and their arbitrariness, we obtain the following equations for orbitals
(see [ 10 , 11 ] for more details):
and equations for basis set optimization
If the K th excited state is considered then, as mentioned, the projector P α
u takes the following form
P α
u =
K−1
k=0
ϕ α
kn
ϕ α
kn
. Equations ( 22 ) represent the
orthogonality-constrained HF method in its general form.
According to the AP methodology, the orthogonality
constraint of Eqs. ( 12 ) and ( 18 ) is satisfi ed in the limit
o → ∞ and s → ∞, , respectively. By setting s = 0 , we
can relax the spin-purity constraint ( 18 ) and go back from
ROHF to UHF solutions. By setting o = 0 , we fall back to
the ground state. The corresponding orbitals form an optimal set which satisfy the generalized Brillouin’s theorem
(see [ 40 ] for more details) and lead to the same total energy
that the one obtained by the Roothaan procedure. In our
method, the only additional computation required, beyond
that arising in the standard UHF scheme, is the evaluation
of the overlap matrix element ϕ 0n |ϕ kn .
Left side of Eq. ( 23 ) represent a gradient of the total
energy with respect to nonlinear basis set parameters {μ a } .
This expression allows these parameters to be determined
variationally and can be also used to construct an algorithm for optimization based on the gradient-like methods.
Since neither λ s nor λ o can be infi nity in practical calculations, one has to settle on some large fi nite values. The
recommended values are λ s = 100 hartrees for the spinpurity constraint and λ o = 1000 hartrees for the orthogonality constraint. They provide target accuracy close to 10
−6 .
In concluding this section, it is also worth noting that in
our method all excited confi gurations based on the excited
Slater determinant Φ 1 , viz., Φ a
i , Φ ab
ij etc., where i and j refer
(22)
P 1
F α − s P
β
1 + o P α
u − α
i
P 1
ϕ α
1i
= 0 , s , o → ∞
P 1
F β + s Q α
1 −
β
i
P 1
ϕ
β
1i
= 0,
i = 1, 2, . . . , M
(23)
n α
i
ϕ
α
1i
(∂ a P 1 )F
α
ϕ
α
1i
+
n β
i
ϕ
β
1i
(∂ a P 1 )F
β
ϕ
β
1i
= 0, a = 1, 2, . . . A
to occupied orbitals and a and b to virtual ones, are orthogonal both to Φ 0 and among themselves. Therefore, these
functions form the orthonormal basis set in the many-body
space and can be used, unlike other SCF methods which do
not satisfy the orthogonality of states in the explicit form,
to develop many-body methods incorporating the correlation effects, in particular, a many-body Møller–Plesset-like
perturbation theory (see next Section).
4 Second-order correction to the energy
for excited states
It is known that within the framework of the Roothaan coupling operator approach, there is no unique way of choosing a reference Hamiltonian, H (0) , with respect to which
a perturbation expansion for correlation effects can be
developed. Several proposals have been made for openshell many-body perturbation theory expansions based
on a reference from the ROHF formalism [ 41 , 42 ] or the
unrestricted Hartree–Fock formalism [ 43 , 44 ]. We follow
our papers [ 10 , 45 ] where an alternative technique for the
open-shell systems has been developed. In our method, the
second-order correction to the ground state energy can be
presented by [ 45 ]:
The summations are over spin-orbitals. Subscripts i , j and
a , b correspond to occupied and virtual orbitals of the
ground state determinant, respectively. Unlike the formalism developed in Refs. [ 43 , 44 ], single excitations do
not contribute because our orbitals satisfy the generalized
Brillouin theorem.
An optimum set of MOs obtained by means of
Eq. ( 22 ) allows us to construct a well-defi ned open-shell
perturbation theory for excited states which is a natural
extension of the popular closed-shell MP2. For example,
the zeroth-order Hamiltonian for the fi rst excited state is
as follows:
with Fockians
The summation up to M − 1 means that the vector |ϕ α
0n is
excluded from the subspace of virtual orbitals. Remind
M is the dimension of the basis set for the fi rst ES. We
(24)
E
(2)
0 =
occ
i>j
virt
a>b
ϕ 0a ϕ 0i |ϕ 0b ϕ 0j
−
ϕ 0a ϕ 0j |ϕ 0b ϕ 0i
2
0i + 0j − 0a − 0b
H
(0)
=
n α
m=electrons
F
α
(m) +
n β
m=electrons
F
β
(m)
F
σ
=
M−1
i
ϕ
σ
i
σ
i
ϕ
σ
i
, σ = α, β
193
Reprinted from the journal