Theor Chem Acc (2015) 134:143
1 3
where Φ K = Φ
a A ...C A ...a B ...C B ...
i A ...J A ...i B ...J B ...
and thus the QMBPT2
energy of the supersystem is the sum of the subsystem
energies.
We can also check the extensivity of the method [ 36 ],
i.e., the connectedness of the QMBPT2 energy. To that
end it is practical to reformulate the matrix elements in the
numerator of Eq. ( 19 ), Φ 0 | ˆ
H|Φ K = =0| ˆ
U † ˆ
H ˆ
U|K , where
|K is the determinant assigned to |Φ K . Using the exponential parameterization of ˆ
U it is easy to show that ˆ
U † ˆ
H ˆ
U is
a connected quantity (see Ref. [ 37 ] for more details) and in
this way the QMBPT2 energy is also connected.
3 Computational considerations
To investigate the scaling of the computation cost of
QMBPT2, it is advantageous to use the orbital labeling
instead of the composite label K in Eq. ( 19 ). For a small
CAS problem the most expensive terms contain four inactive indices,
where the above term has the same form that is known
from the SR theory (see later) and its scaling is n 2
v n 2
c , where
n v and n c are the number of inactive virtual and inactive
occupied orbitals, respectively. The most expensive contributions coming from the MR nature of QMBPT2 have
three inactive indices,
Supposing that the Φ
abA...C
iJK...L | ˆ
H|Φ 0 matrix element is given,
the computation cost of this term scales as n 2
v n c n ao N act ,
where n ao is the number of active occupied orbitals and N act
is the number of active confi gurations.
There is another component of the calculations which
has a signifi cant cost, namely the evaluation of the
Φ
abA...C
iJK...L | ˆ
H|Φ 0 type of matrix elements. As the effi cient
implementation of these terms are not that straightforward
the rest of this section is dedicated to this problem.
The matrix element in Eq. ( 19 ) can be expanded in the
basis of determinants,
(21)
E
(2)
=
abij
Φ 0 | ˆ
H|Φ ab
ij Φ ab
ij | ˆ
H|Φ 0
ε i + ε j − ε a − ε b
+ · · · ,
(22)
E
(2) = . . .
+
abA...CiJK...L
Φ 0 | ˆ
H|Φ
abA...C
iJK...L Φ
abA...C
iJK...L | ˆ
H|Φ 0
ε i + ε J + ε K + . . . + ε L − ε a − ε b − ε A + . . . + ε B
(23)
Φ K | ˆ
H|Φ 0 =
LM
[U K ]
L [U 0 ]
M
L| ˆ
H|M
=
LM
[U K ]
L [U O ]
M
L, k| ˆ
H|M, o,
where in the last term we separated the active and inactive parts of the determinants. Here calligraphic letters are
used for the active, and roman letters stand for the inactive
part of the determinants, |L = |L, l . As ˆ
U transforms only
the active labels, the coeffi cient matrix [U K ] L holds exclusively calligraphic letters.
Since the coeffi cient matrix belongs to a unitary transformation, those components of the above expression
where no active orbitals appear in the Hamiltonian can be
trivially calculated. For example,
where o ab
ij is a double inactive excitation with respect to the
core part ( o ) of Φ 0 . Those terms where the active part of the
Hamiltonian is involved are more complex. The most timeconsuming component from this class of terms is the one
where one active occupied orbital appears,
where M J is the inactive confi guration obtained by eliminating J from confi guration M . According to the above
expression the overall computation cost of these terms
scales as n 2
v n c n ao N 2
act . To reduce the computation cost one
can try to factorize this expression into active and inactive
parts,
At this point we can realize that though formally the matrix
element containing the inactive second-quantized expression still M J dependent, its value (1 or -1) is completely
determined by the number of active electrons ( N ae ) in M J ,
i.e.,
(24)
Φ
ab
ij | ˆ
H|Φ 0 =
LM
[U K ]
L [U O ]
M
×
pqrs
1
4
pq||rsL, o
ab
ij |ˆ p
+
ˆ
q
+
ˆ
s
−
ˆ
r
−
|M, o = =ab||ij,
(25)
Φ
abA...C
iI...K | ˆ
H|Φ 0
=
LM
U
A...C
I...K
L
[U O ]
M
pqrs
1
4
pq||rs
× ×L, o
ab
i |ˆ p
+
ˆ
q
+
ˆ
s
−
ˆ
r
−
|M, o
=
M
J
U
A...C
I...K
M J
[U O ]
M
ab||iJ
× ×M J , o
ab
i | ˆ
b
+
ˆ
a
+ ˆ i
− ˆ
J
−
|M, o,
(26)
J
ab||iJ
M
U
A...C
I...K
M J
[U O ]
M
M J , o
ab
i |ˆ a
+ ˆ
b
+ ˆ i
−
|M J , oM J , o| ˆ
J
−
|M, o.
(27)
sign(a, b, i, N ae − 1) = =M J , o
ab
i |ˆ a
+ ˆ
b
+ ˆ i
−
|M J , o.
253
Reprinted from the journal
1 3
where Φ K = Φ
a A ...C A ...a B ...C B ...
i A ...J A ...i B ...J B ...
and thus the QMBPT2
energy of the supersystem is the sum of the subsystem
energies.
We can also check the extensivity of the method [ 36 ],
i.e., the connectedness of the QMBPT2 energy. To that
end it is practical to reformulate the matrix elements in the
numerator of Eq. ( 19 ), Φ 0 | ˆ
H|Φ K = =0| ˆ
U † ˆ
H ˆ
U|K , where
|K is the determinant assigned to |Φ K . Using the exponential parameterization of ˆ
U it is easy to show that ˆ
U † ˆ
H ˆ
U is
a connected quantity (see Ref. [ 37 ] for more details) and in
this way the QMBPT2 energy is also connected.
3 Computational considerations
To investigate the scaling of the computation cost of
QMBPT2, it is advantageous to use the orbital labeling
instead of the composite label K in Eq. ( 19 ). For a small
CAS problem the most expensive terms contain four inactive indices,
where the above term has the same form that is known
from the SR theory (see later) and its scaling is n 2
v n 2
c , where
n v and n c are the number of inactive virtual and inactive
occupied orbitals, respectively. The most expensive contributions coming from the MR nature of QMBPT2 have
three inactive indices,
Supposing that the Φ
abA...C
iJK...L | ˆ
H|Φ 0 matrix element is given,
the computation cost of this term scales as n 2
v n c n ao N act ,
where n ao is the number of active occupied orbitals and N act
is the number of active confi gurations.
There is another component of the calculations which
has a signifi cant cost, namely the evaluation of the
Φ
abA...C
iJK...L | ˆ
H|Φ 0 type of matrix elements. As the effi cient
implementation of these terms are not that straightforward
the rest of this section is dedicated to this problem.
The matrix element in Eq. ( 19 ) can be expanded in the
basis of determinants,
(21)
E
(2)
=
abij
Φ 0 | ˆ
H|Φ ab
ij Φ ab
ij | ˆ
H|Φ 0
ε i + ε j − ε a − ε b
+ · · · ,
(22)
E
(2) = . . .
+
abA...CiJK...L
Φ 0 | ˆ
H|Φ
abA...C
iJK...L Φ
abA...C
iJK...L | ˆ
H|Φ 0
ε i + ε J + ε K + . . . + ε L − ε a − ε b − ε A + . . . + ε B
(23)
Φ K | ˆ
H|Φ 0 =
LM
[U K ]
L [U 0 ]
M
L| ˆ
H|M
=
LM
[U K ]
L [U O ]
M
L, k| ˆ
H|M, o,
where in the last term we separated the active and inactive parts of the determinants. Here calligraphic letters are
used for the active, and roman letters stand for the inactive
part of the determinants, |L = |L, l . As ˆ
U transforms only
the active labels, the coeffi cient matrix [U K ] L holds exclusively calligraphic letters.
Since the coeffi cient matrix belongs to a unitary transformation, those components of the above expression
where no active orbitals appear in the Hamiltonian can be
trivially calculated. For example,
where o ab
ij is a double inactive excitation with respect to the
core part ( o ) of Φ 0 . Those terms where the active part of the
Hamiltonian is involved are more complex. The most timeconsuming component from this class of terms is the one
where one active occupied orbital appears,
where M J is the inactive confi guration obtained by eliminating J from confi guration M . According to the above
expression the overall computation cost of these terms
scales as n 2
v n c n ao N 2
act . To reduce the computation cost one
can try to factorize this expression into active and inactive
parts,
At this point we can realize that though formally the matrix
element containing the inactive second-quantized expression still M J dependent, its value (1 or -1) is completely
determined by the number of active electrons ( N ae ) in M J ,
i.e.,
(24)
Φ
ab
ij | ˆ
H|Φ 0 =
LM
[U K ]
L [U O ]
M
×
pqrs
1
4
pq||rsL, o
ab
ij |ˆ p
+
ˆ
q
+
ˆ
s
−
ˆ
r
−
|M, o = =ab||ij,
(25)
Φ
abA...C
iI...K | ˆ
H|Φ 0
=
LM
U
A...C
I...K
L
[U O ]
M
pqrs
1
4
pq||rs
× ×L, o
ab
i |ˆ p
+
ˆ
q
+
ˆ
s
−
ˆ
r
−
|M, o
=
M
J
U
A...C
I...K
M J
[U O ]
M
ab||iJ
× ×M J , o
ab
i | ˆ
b
+
ˆ
a
+ ˆ i
− ˆ
J
−
|M, o,
(26)
J
ab||iJ
M
U
A...C
I...K
M J
[U O ]
M
M J , o
ab
i |ˆ a
+ ˆ
b
+ ˆ i
−
|M J , oM J , o| ˆ
J
−
|M, o.
(27)
sign(a, b, i, N ae − 1) = =M J , o
ab
i |ˆ a
+ ˆ
b
+ ˆ i
−
|M J , o.
253
Reprinted from the journal