Theor Chem Acc (2015) 134:107
1 3
where the coeffi cient C α
μp is obtained by solving the HF equation in subsystem α :
In Eq. ( 39 ), L(α) is the set of AOs used in the expansion of
MOs in subsystem α, which consists of two parts:
S(α) is the AOs corresponding to the central region of the
subsystem α, which is mutually exclusive:
and B(α) is the AOs corresponding to the neighboring
region of the central region of the subsystem α, called
the buffer region . F α and S α are the subsystem Fock and
overlap matrices, respectively, of which the elements are
expressed by
Obviously, the matrix elements of Eqs. ( 43 ) and ( 44 )
are the same as Eqs. ( 5 ) and ( 6 ). Here in Eq. ( 43 ), one
can adopt any density matrix, D, which is appropriately
determined. A possible candidate is the DC-HF density
matrix:
where
f α
p is the Fermi-distributed occupation number:
and P is the partition matrix defi ned by
Fermi level, ε F , in Eq. ( 47 ) is determined from the number
of electrons in the entire system, n e , by solving the following
equation:
As shown above, the Fermi level and the FT formalism are
introduced in the DC-HF method. Recently, Yoshikawa and
Nakai proposed an alternative DC procedure, where the number of electrons in each subsystem is fi xed after a couple of
iterations for improving the parallel effi ciency of the DC-HF
(40)
F
α C
α
p = ε
α
p S
α C
α
p .
(41)
L(α) = S(α) B(α).
(42)
S(α) ∩ S(β) = ∅, ∀α = β,
(43)
F
α
μν = H
core
μν +
σ
D σ [2μσ |ν − −μσ |ν],
(44)
S
α
μν = =φ μ |φ ν .
(45)
D
DC
μν =
α
P
α
μν D
α
μν ,
(46)
D
α
=
MO(α)
p
f
α
p C
α
p C
αT
p .
(47)
f
α
p = f β (ε F − ε
α
p ),
(48)
P
α
μν =
⎧
⎨
⎩
1 (μ ∈ S(α) ∧ ν ∈ S(α))
1/2 (μ ∈ S(α) ∧ ν ∈ B(α), or vice versa)
0 (otherwise).
(49)
n e = tr(D
DC S).
calculations [ 42 ]. The Fermi level is no longer constant
through the entire system in this scheme, which is also related
to the adjustable density matrix assembly method [ 43 , 44 ].
In the DC-MP2 method, the total correlation energy
is estimated by summing up the subsystem correlation
energies:
The correlation energy of the subsystem α, E α
MP2 , has
been estimated from the energy density analysis [ 45 ] with
the subsystem MOs as follows:
˜ t α
ij,ab is the effective two-electron excitation amplitude for
subsystem α, which is expressed in MP2 case with the subsystem MOs as follows:
In the previous DC-MP2 method with integer occupation
numbers, the subsystem MOs should clearly be separated
into occupied and virtual ones by the Fermi level. In the
DC-HF method, however, fractional occupations of MOs
are allowed around Fermi level because the FT ensemble is
formally treated (see Eq. 46 ).
It is also possible to adopt the FT-MP2 formalisms of
Eqs. ( 21 ) or ( 25 ) in the DC-MP2 method. On the analogy
to Eq. ( 51 ), the subsystem correlation energy at fi nite temperature can be evaluated as
˜ t C/Rα
pq,rs is
or
where
(50)
E DC-MP2 =
α
E
α
MP2 .
(51)
E
α
MP2 =
occ(α)
ij
vir(α)
ab
μ∈S(α)
C
α
μi
μj
α
|a
α b
α
×
2 ˜ t
α
ij,ab − ˜ t
α
ij,ba
.
(52)
˜ t
α
ij,ab = −
a α b α |i α j α
ε α
a + ε α
b − ε α
i − ε α
j
.
(53)
E
C/Rα
MP2 =
MO(α)
pqrs
μ∈S(α)
C
α
μp
μq
α
|r
α s
α
×
2˜ t
C/Rα
pq,rs − ˜ t
C/Rα
pq,rs
.
(54)
˜ t
Cα
pq,rs = −
f α
pq,rs r α s α |p α q α
ε α
r + ε α
s − ε α
p − ε α
q
,
(55)
˜ t
Rα
pq,rs = −
f α
pq,rs r α s α |p α q α
¯
f α
r ε α
r + ¯
f α
s ε α
s − f α
p ε α
p − f α
q ε α
q
,
(56)
¯
f
α
p =1 − f
α
p ,
261
Reprinted from the journal
1 3
where the coeffi cient C α
μp is obtained by solving the HF equation in subsystem α :
In Eq. ( 39 ), L(α) is the set of AOs used in the expansion of
MOs in subsystem α, which consists of two parts:
S(α) is the AOs corresponding to the central region of the
subsystem α, which is mutually exclusive:
and B(α) is the AOs corresponding to the neighboring
region of the central region of the subsystem α, called
the buffer region . F α and S α are the subsystem Fock and
overlap matrices, respectively, of which the elements are
expressed by
Obviously, the matrix elements of Eqs. ( 43 ) and ( 44 )
are the same as Eqs. ( 5 ) and ( 6 ). Here in Eq. ( 43 ), one
can adopt any density matrix, D, which is appropriately
determined. A possible candidate is the DC-HF density
matrix:
where
f α
p is the Fermi-distributed occupation number:
and P is the partition matrix defi ned by
Fermi level, ε F , in Eq. ( 47 ) is determined from the number
of electrons in the entire system, n e , by solving the following
equation:
As shown above, the Fermi level and the FT formalism are
introduced in the DC-HF method. Recently, Yoshikawa and
Nakai proposed an alternative DC procedure, where the number of electrons in each subsystem is fi xed after a couple of
iterations for improving the parallel effi ciency of the DC-HF
(40)
F
α C
α
p = ε
α
p S
α C
α
p .
(41)
L(α) = S(α) B(α).
(42)
S(α) ∩ S(β) = ∅, ∀α = β,
(43)
F
α
μν = H
core
μν +
σ
D σ [2μσ |ν − −μσ |ν],
(44)
S
α
μν = =φ μ |φ ν .
(45)
D
DC
μν =
α
P
α
μν D
α
μν ,
(46)
D
α
=
MO(α)
p
f
α
p C
α
p C
αT
p .
(47)
f
α
p = f β (ε F − ε
α
p ),
(48)
P
α
μν =
⎧
⎨
⎩
1 (μ ∈ S(α) ∧ ν ∈ S(α))
1/2 (μ ∈ S(α) ∧ ν ∈ B(α), or vice versa)
0 (otherwise).
(49)
n e = tr(D
DC S).
calculations [ 42 ]. The Fermi level is no longer constant
through the entire system in this scheme, which is also related
to the adjustable density matrix assembly method [ 43 , 44 ].
In the DC-MP2 method, the total correlation energy
is estimated by summing up the subsystem correlation
energies:
The correlation energy of the subsystem α, E α
MP2 , has
been estimated from the energy density analysis [ 45 ] with
the subsystem MOs as follows:
˜ t α
ij,ab is the effective two-electron excitation amplitude for
subsystem α, which is expressed in MP2 case with the subsystem MOs as follows:
In the previous DC-MP2 method with integer occupation
numbers, the subsystem MOs should clearly be separated
into occupied and virtual ones by the Fermi level. In the
DC-HF method, however, fractional occupations of MOs
are allowed around Fermi level because the FT ensemble is
formally treated (see Eq. 46 ).
It is also possible to adopt the FT-MP2 formalisms of
Eqs. ( 21 ) or ( 25 ) in the DC-MP2 method. On the analogy
to Eq. ( 51 ), the subsystem correlation energy at fi nite temperature can be evaluated as
˜ t C/Rα
pq,rs is
or
where
(50)
E DC-MP2 =
α
E
α
MP2 .
(51)
E
α
MP2 =
occ(α)
ij
vir(α)
ab
μ∈S(α)
C
α
μi
μj
α
|a
α b
α
×
2 ˜ t
α
ij,ab − ˜ t
α
ij,ba
.
(52)
˜ t
α
ij,ab = −
a α b α |i α j α
ε α
a + ε α
b − ε α
i − ε α
j
.
(53)
E
C/Rα
MP2 =
MO(α)
pqrs
μ∈S(α)
C
α
μp
μq
α
|r
α s
α
×
2˜ t
C/Rα
pq,rs − ˜ t
C/Rα
pq,rs
.
(54)
˜ t
Cα
pq,rs = −
f α
pq,rs r α s α |p α q α
ε α
r + ε α
s − ε α
p − ε α
q
,
(55)
˜ t
Rα
pq,rs = −
f α
pq,rs r α s α |p α q α
¯
f α
r ε α
r + ¯
f α
s ε α
s − f α
p ε α
p − f α
q ε α
q
,
(56)
¯
f
α
p =1 − f
α
p ,
261
Reprinted from the journal