Theor Chem Acc (2015) 134:143
1 3
where the above expansion starts with the energy of the reference function and f
p
q , v 2
pq
sr , v 3
pqr
uvw , etc., are the elements
of the one-, two-, three- and higher-body antisymmetrized
integral lists.
To emphasize the analogy with the single-reference
case, the quasiparticle Fockian denomination will be used
for the one-particle part of the above expansion. Using a
proper unitary transformation of the ordinary molecular
orbitals, the quasiparticle Fockian has a block-diagonal
structure, i.e., (1) for a Φ 0 obtained as a solution of a CAS
problem, its active–active block can be always kept diagonal; (2) using pseudo-canonical orbitals to expand the inactive occupied and inactive virtual subspaces, the inactive
blocks of the quasiparticle Fockian are also diagonal
The fi rst property can be easily understood by considering a general active–inactive matrix element of f
p
q ,
which is zero, as
ˆ
Q
+
A
ˆ
Q
−
I
N
|Φ 0 is orthogonal to the reference state and lies completely in the CAS. As the f AB and
f IJ off-diagonal elements can be eliminated by simple rotations of molecular orbitals, for which the reference function
Φ 0 is invariant, the active–active block of the quasiparticle
Fockian can be transformed into a diagonal form. Investigating the inactive virtual block of the quasiparticle Fockian would lead to similar conclusion.
To investigate the occupied inactive–inactive blocks of
the quasiparticle Fockian we can start from the
relation. After some straightforward manipulation, it leads
to the f
i
j = h ij +
pq ip||jqP pq + ΔE CAS δ ij expression,
where the fi rst and the second terms together at the lefthand side form the occupied block of the generalized Fock
matrix and ΔE CAS is the correlation energy of Φ 0 . When
pseudo-canonical orbitals are used, this block of the generalized Fockian is diagonal, and thus f
i
j is also diagonal.
One can reach a similar conclusion for the virtual inactive
block, where the matrix elements are defi ned by the
relation.
The presented quasiparticle framework offers a simple way to develop the MR versions of the single-reference-based correlation methods, such as the CI or the CC
approaches [ 37 ]. To introduce the MR variant of the MBPT
[ 6 ], fi rst the Hamiltonian should be separated into a zerothoder and a perturbation part, ˆ
H = ˆ
H (0) + ˆ
V . Following the
MBPT, it seems to be a natural choice to use the diagonal
(14)
f
A
I = =Φ 0 | ˆ
H
ˆ
Q
+
A
ˆ
Q
−
I
N
|Φ 0 ,
(15)
f
i
j = −−Φ 0 | ˆ
Q
+
i
ˆ
H N ˆ
Q
−
j |Φ 0
(16)
f
a
b = =Φ 0 | ˆ
Q
−
a
ˆ
H N ˆ
Q
+
b |Φ 0
part of the one-particle term from Eq. ( 13 ) as the zerothorder Hamiltonian,
where ε p = f
p
p . The zeroth- and the fi rst-order perturbation
energies together provide the CAS energy of the reference
function, E (0) + E (1) = =Φ 0 | ˆ
H 0 + ˆ
V |Φ 0 = E CAS . In the
basis of the functions introduced in Eqs. ( 8 ) and ( 9 )—these
are the zeroth-order functions—the zeroth-order Hamiltonian is diagonal,
where, for the sake of brevity, the MR basis elements are
labeled by the serial numbers instead of the one-particle
labels. The second-order energy correction according to the
Rayleigh–Schrödinger PT reads as
where we could replace ˆ
V by ˆ
H utilizing that Φ 0 | ˆ
H 0 |Φ K is
zero for any K .
The above E (2) energy correction directly provides an
approximation to the dynamical electron correlation. For
this correction only those Φ K functions give contributions
where at least one inactive excitation appears, otherwise
Φ K | ˆ
H|Φ 0 = 0 . It means that at least one inactive oneparticle energy shows up in the denominator, which is
a useful property, as it provides a certain amount of protection against the singular behavior of the perturbation
denominator.
It can be easily seen that the above generalization of the
MBPT2 keeps the size-consistency of the original theory.
Supposing that we have two noninteracting subsystems, A
and B , separately described by ˆ
H A and ˆ
H B , and the reference function has a product form, Φ 0 = Φ A
0 Φ B
0 , where Φ A
0
and Φ B
0 are solutions of the CAS problems of the subsystems, and then the unitary transformation at Eq. ( 3 ) is also
product separable, ˆ
U = ˆ
U A ˆ
U B . Considering the defi nition
of quasiparticles at Eqs. ( 5 ) and ( 6 ) it is obvious now that
these are localized either on subsystem A or on subsystem
B, and the zeroth-order functions defi ned by the quasiparticles at Eq. ( 9 ) are product separable, i.e., Φ K = Φ K A Φ K B .
Using these results the contribution of any Φ K to the energy
is a sum of the subsystem contributions,
(17)
ˆ
H 0 =
p
ε p
ˆ
Q
+
p
ˆ
Q
−
p
N
,
(18)
ˆ
H 0 |Φ K = E K |Φ K , K = 0, 1, 2, . . . ,
(19)
E
(2)
=
K,K =0
Φ 0 | ˆ
H|Φ K Φ K | ˆ
H|Φ 0
E 0 − E K
,
(20)
Φ 0 | ˆ
H A |Φ
a A ...C A ...
i A ...J A ... Φ
a A ...C A ...
i A ...J A ... | ˆ
H A |Φ 0
ε i A + · · · + ε J A + · · · − ε a A − · · · − ε C A − · · ·
+
Φ 0 | ˆ
H B |Φ
a B ...C B ...
i B ...J B ... Φ
a B ...C B ...
i B ...J B ... | ˆ
H B |Φ 0
ε i B + · · · + ε J B + · · · − ε a B − · · · − ε C B − · · ·
,
252
Reprinted from the journal
1 3
where the above expansion starts with the energy of the reference function and f
p
q , v 2
pq
sr , v 3
pqr
uvw , etc., are the elements
of the one-, two-, three- and higher-body antisymmetrized
integral lists.
To emphasize the analogy with the single-reference
case, the quasiparticle Fockian denomination will be used
for the one-particle part of the above expansion. Using a
proper unitary transformation of the ordinary molecular
orbitals, the quasiparticle Fockian has a block-diagonal
structure, i.e., (1) for a Φ 0 obtained as a solution of a CAS
problem, its active–active block can be always kept diagonal; (2) using pseudo-canonical orbitals to expand the inactive occupied and inactive virtual subspaces, the inactive
blocks of the quasiparticle Fockian are also diagonal
The fi rst property can be easily understood by considering a general active–inactive matrix element of f
p
q ,
which is zero, as
ˆ
Q
+
A
ˆ
Q
−
I
N
|Φ 0 is orthogonal to the reference state and lies completely in the CAS. As the f AB and
f IJ off-diagonal elements can be eliminated by simple rotations of molecular orbitals, for which the reference function
Φ 0 is invariant, the active–active block of the quasiparticle
Fockian can be transformed into a diagonal form. Investigating the inactive virtual block of the quasiparticle Fockian would lead to similar conclusion.
To investigate the occupied inactive–inactive blocks of
the quasiparticle Fockian we can start from the
relation. After some straightforward manipulation, it leads
to the f
i
j = h ij +
pq ip||jqP pq + ΔE CAS δ ij expression,
where the fi rst and the second terms together at the lefthand side form the occupied block of the generalized Fock
matrix and ΔE CAS is the correlation energy of Φ 0 . When
pseudo-canonical orbitals are used, this block of the generalized Fockian is diagonal, and thus f
i
j is also diagonal.
One can reach a similar conclusion for the virtual inactive
block, where the matrix elements are defi ned by the
relation.
The presented quasiparticle framework offers a simple way to develop the MR versions of the single-reference-based correlation methods, such as the CI or the CC
approaches [ 37 ]. To introduce the MR variant of the MBPT
[ 6 ], fi rst the Hamiltonian should be separated into a zerothoder and a perturbation part, ˆ
H = ˆ
H (0) + ˆ
V . Following the
MBPT, it seems to be a natural choice to use the diagonal
(14)
f
A
I = =Φ 0 | ˆ
H
ˆ
Q
+
A
ˆ
Q
−
I
N
|Φ 0 ,
(15)
f
i
j = −−Φ 0 | ˆ
Q
+
i
ˆ
H N ˆ
Q
−
j |Φ 0
(16)
f
a
b = =Φ 0 | ˆ
Q
−
a
ˆ
H N ˆ
Q
+
b |Φ 0
part of the one-particle term from Eq. ( 13 ) as the zerothorder Hamiltonian,
where ε p = f
p
p . The zeroth- and the fi rst-order perturbation
energies together provide the CAS energy of the reference
function, E (0) + E (1) = =Φ 0 | ˆ
H 0 + ˆ
V |Φ 0 = E CAS . In the
basis of the functions introduced in Eqs. ( 8 ) and ( 9 )—these
are the zeroth-order functions—the zeroth-order Hamiltonian is diagonal,
where, for the sake of brevity, the MR basis elements are
labeled by the serial numbers instead of the one-particle
labels. The second-order energy correction according to the
Rayleigh–Schrödinger PT reads as
where we could replace ˆ
V by ˆ
H utilizing that Φ 0 | ˆ
H 0 |Φ K is
zero for any K .
The above E (2) energy correction directly provides an
approximation to the dynamical electron correlation. For
this correction only those Φ K functions give contributions
where at least one inactive excitation appears, otherwise
Φ K | ˆ
H|Φ 0 = 0 . It means that at least one inactive oneparticle energy shows up in the denominator, which is
a useful property, as it provides a certain amount of protection against the singular behavior of the perturbation
denominator.
It can be easily seen that the above generalization of the
MBPT2 keeps the size-consistency of the original theory.
Supposing that we have two noninteracting subsystems, A
and B , separately described by ˆ
H A and ˆ
H B , and the reference function has a product form, Φ 0 = Φ A
0 Φ B
0 , where Φ A
0
and Φ B
0 are solutions of the CAS problems of the subsystems, and then the unitary transformation at Eq. ( 3 ) is also
product separable, ˆ
U = ˆ
U A ˆ
U B . Considering the defi nition
of quasiparticles at Eqs. ( 5 ) and ( 6 ) it is obvious now that
these are localized either on subsystem A or on subsystem
B, and the zeroth-order functions defi ned by the quasiparticles at Eq. ( 9 ) are product separable, i.e., Φ K = Φ K A Φ K B .
Using these results the contribution of any Φ K to the energy
is a sum of the subsystem contributions,
(17)
ˆ
H 0 =
p
ε p
ˆ
Q
+
p
ˆ
Q
−
p
N
,
(18)
ˆ
H 0 |Φ K = E K |Φ K , K = 0, 1, 2, . . . ,
(19)
E
(2)
=
K,K =0
Φ 0 | ˆ
H|Φ K Φ K | ˆ
H|Φ 0
E 0 − E K
,
(20)
Φ 0 | ˆ
H A |Φ
a A ...C A ...
i A ...J A ... Φ
a A ...C A ...
i A ...J A ... | ˆ
H A |Φ 0
ε i A + · · · + ε J A + · · · − ε a A − · · · − ε C A − · · ·
+
Φ 0 | ˆ
H B |Φ
a B ...C B ...
i B ...J B ... Φ
a B ...C B ...
i B ...J B ... | ˆ
H B |Φ 0
ε i B + · · · + ε J B + · · · − ε a B − · · · − ε C B − · · ·
,
252
Reprinted from the journal