Theor Chem Acc (2015) 134:107
1 3
should be neglected in practical calculations. Hirata and
He [ 32 ] recently proposed the renormalized formalism
for the FT many-body perturbation theory that changes
the divergence rate to be consistent to the zero-temperature counterpart:
This formula comes from the thermal Wick’s theorem [ 34 ,
35 ]. Note that this renormalized FT-MP2 energy as well
as the conventional FT-MP2 energy diverges for metallic
system. The conventional FT-MP2 energy, however, may
diverge even for insulators. In this subsection, we clarify
the correspondence of the above two formalisms with two
E MP2 [D] functionals.
According to the standard AO-based formalism [ 17 ], the
Laplace transformation of E C
MP2 and E R
MP2 leads to
The integrand in Eq. ( 27 ) is given by
For the conventional formalism, the energy-weighted density matrices are given by
and for the renormalized formalism, they are
The expressions for the conventional formalism are different
from Eqs. ( 11 ) and ( 12 ) only in the existence of the occupation number before C p C T
p . For the renormalized formalism,
the occupation number also appears on the exponential.
If the HF equation of Eq. ( 4 ) is satisfi ed, one can derive
the following equation:
(25)
E
R
MP2 =
pqrs
pq|rs
2˜ t
R
pq,rs − ˜ t
R
pq,sr
,
(26)
˜ t
R
pq,rs = −
f pq,rs rs|pq
¯
f r ε r + ¯
f s ε s − f p ε p − f q ε q
.
(27)
C/R
MP2 = −
∞
0
e
C/R
2 (t)dt.
(28)
e
C/R
2 (t) =
γ δκε
μνσ
X
C/R
μγ (t)Y
C/R
νδ (t)X
C/R
κ (t)Y
C/R
σ ε (t)
γ κ|δε[2νσ |μ − −νσ |μ].
(29)
X
C
(t) =
p
e
ε p t f p C p C
T
p ,
(30)
Y
C
(t) =
p
e
−ε p t ¯
f p C p C
T
p ,
(31)
X
R
(t) =
p
e
ε p f p t f p C p C
T
p ,
(32)
Y
R
(t) =
p
e
−ε p ¯
f p t ¯
f p C p C
T
p .
(33)
X
C
(t) = e
tS −1 F D,
where ¯
D is the complement of the FT-HF density matrix,
These are formally the same as the S −1 F formulas of
Eqs. ( 13 ) and ( 14 ). However, it cannot be further transformed to the DF formulas because the density matrix is no
longer idempotent at fi nite temperature but has the following relationship:
As for the renormalized formulas, on the contrary, using
the relationships of FDS = SDF and Eq. ( 36 ) leads to
These are formally the same as the DF formulas of
Eqs. ( 16 ) and ( 17 ).
Here, we summarize the important point in this Section.
As derived in the previous studies [ 14 , 18 ], two DM-MP2
formalisms (i.e., the S −1 F and DF formulas) are equivalent at zero temperature (see Eqs. 13 and 16 , for example).
At fi nite temperature, however, these two formulas are no
longer equivalent and further coincide with the conventional and renormalized FT-MP2 formulas (see Eqs. 33 and
37 , for example).
When Kobayashi and Nakai [ 18 ] calculated the MP2
correlation energy using Eq. ( 10 ) with the density matrix
having random noise, they obtained a reasonable MP2
correlation energy with Eqs. ( 16 ) and ( 17 ), although they
could not with Eqs. ( 13 ) and ( 14 ). The noise-introduced
HF density matrix can be represented as Eq. ( 18 ) having
noise in the occupation number, f p , if the basis set spans
the complete space. Therefore, this result can now be interpreted that the amplitude of Eq. ( 22 ) for p = r and q = s,
for instance, diverges when using a density matrix for a
mixed state.
2.3 Finite-temperature DC-MP2 method
Kobayashi and coworkers [ 20 , 21 , 26 , 41 ] have proposed a
linear-scaling DC-MP2 method. This method utilizes MOs
determined in the subsystem α, {ψ α
p }, which are constructed as
(34)
Y
C
(t) = e
−tS −1 F ¯
D,
(35)
¯
D =
p
¯
f p C p C
T
p = S
−1
− D.
(36)
(DS)
n
=
p
f
n
p C p C
T
p S.
(37)
X
R
(t) = e
tDF D,
(38)
Y
R
(t) = e
−t ¯
DF ¯
D.
(39)
ψ
α
p =
μ∈L(α)
C
α
μp φ μ ,
260
Reprinted from the journal
1 3
should be neglected in practical calculations. Hirata and
He [ 32 ] recently proposed the renormalized formalism
for the FT many-body perturbation theory that changes
the divergence rate to be consistent to the zero-temperature counterpart:
This formula comes from the thermal Wick’s theorem [ 34 ,
35 ]. Note that this renormalized FT-MP2 energy as well
as the conventional FT-MP2 energy diverges for metallic
system. The conventional FT-MP2 energy, however, may
diverge even for insulators. In this subsection, we clarify
the correspondence of the above two formalisms with two
E MP2 [D] functionals.
According to the standard AO-based formalism [ 17 ], the
Laplace transformation of E C
MP2 and E R
MP2 leads to
The integrand in Eq. ( 27 ) is given by
For the conventional formalism, the energy-weighted density matrices are given by
and for the renormalized formalism, they are
The expressions for the conventional formalism are different
from Eqs. ( 11 ) and ( 12 ) only in the existence of the occupation number before C p C T
p . For the renormalized formalism,
the occupation number also appears on the exponential.
If the HF equation of Eq. ( 4 ) is satisfi ed, one can derive
the following equation:
(25)
E
R
MP2 =
pqrs
pq|rs
2˜ t
R
pq,rs − ˜ t
R
pq,sr
,
(26)
˜ t
R
pq,rs = −
f pq,rs rs|pq
¯
f r ε r + ¯
f s ε s − f p ε p − f q ε q
.
(27)
C/R
MP2 = −
∞
0
e
C/R
2 (t)dt.
(28)
e
C/R
2 (t) =
γ δκε
μνσ
X
C/R
μγ (t)Y
C/R
νδ (t)X
C/R
κ (t)Y
C/R
σ ε (t)
γ κ|δε[2νσ |μ − −νσ |μ].
(29)
X
C
(t) =
p
e
ε p t f p C p C
T
p ,
(30)
Y
C
(t) =
p
e
−ε p t ¯
f p C p C
T
p ,
(31)
X
R
(t) =
p
e
ε p f p t f p C p C
T
p ,
(32)
Y
R
(t) =
p
e
−ε p ¯
f p t ¯
f p C p C
T
p .
(33)
X
C
(t) = e
tS −1 F D,
where ¯
D is the complement of the FT-HF density matrix,
These are formally the same as the S −1 F formulas of
Eqs. ( 13 ) and ( 14 ). However, it cannot be further transformed to the DF formulas because the density matrix is no
longer idempotent at fi nite temperature but has the following relationship:
As for the renormalized formulas, on the contrary, using
the relationships of FDS = SDF and Eq. ( 36 ) leads to
These are formally the same as the DF formulas of
Eqs. ( 16 ) and ( 17 ).
Here, we summarize the important point in this Section.
As derived in the previous studies [ 14 , 18 ], two DM-MP2
formalisms (i.e., the S −1 F and DF formulas) are equivalent at zero temperature (see Eqs. 13 and 16 , for example).
At fi nite temperature, however, these two formulas are no
longer equivalent and further coincide with the conventional and renormalized FT-MP2 formulas (see Eqs. 33 and
37 , for example).
When Kobayashi and Nakai [ 18 ] calculated the MP2
correlation energy using Eq. ( 10 ) with the density matrix
having random noise, they obtained a reasonable MP2
correlation energy with Eqs. ( 16 ) and ( 17 ), although they
could not with Eqs. ( 13 ) and ( 14 ). The noise-introduced
HF density matrix can be represented as Eq. ( 18 ) having
noise in the occupation number, f p , if the basis set spans
the complete space. Therefore, this result can now be interpreted that the amplitude of Eq. ( 22 ) for p = r and q = s,
for instance, diverges when using a density matrix for a
mixed state.
2.3 Finite-temperature DC-MP2 method
Kobayashi and coworkers [ 20 , 21 , 26 , 41 ] have proposed a
linear-scaling DC-MP2 method. This method utilizes MOs
determined in the subsystem α, {ψ α
p }, which are constructed as
(34)
Y
C
(t) = e
−tS −1 F ¯
D,
(35)
¯
D =
p
¯
f p C p C
T
p = S
−1
− D.
(36)
(DS)
n
=
p
f
n
p C p C
T
p S.
(37)
X
R
(t) = e
tDF D,
(38)
Y
R
(t) = e
−t ¯
DF ¯
D.
(39)
ψ
α
p =
μ∈L(α)
C
α
μp φ μ ,
260
Reprinted from the journal