1 3
Theor Chem Acc (2015) 134:107
DOI 10.1007/s00214-015-1710-y
REGULAR ARTICLE
Second-order Møller–Plesset perturbation (MP2) theory at fi nite
temperature: relation with Surján’s density matrix MP2 and its
application to linear-scaling divide-and-conquer method
Masato Kobayashi
1,2 · Tetsuya Taketsugu
1,2
Received: 24 June 2015 / Accepted: 28 July 2015 / Published online: 15 August 2015
© Springer-Verlag Berlin Heidelberg 2015
Keywords Fractional occupation number · Many-body
perturbation theory · Laplace-transformed Møller–Plesset
perturbation · Linear-scaling electronic structure method
1 Introduction
The perturbation theory has widely been used in quantum
chemistry to account for the dynamical electron correlation
in single Slater [i.e., Hartree–Fock (HF)] and multideterminantal states [ 1 ]. Surján has worked on the perturbation
theories for both HF and non-HF references. For non-HF
reference functions, he and his coworkers proposed a series
of multiconfi guration perturbation (MCPT) theories [ 2 – 6 ].
Because the MCPT theories are applicable to any reference functions, they have occasionally been applied to the
antisymmetric product of strongly orthogonal geminals
wave functions [ 7 – 9 ].
Although many of Surján’s famous works are related
to the multideterminantal framework, he has also worked
on the Møller–Plesset perturbation (MP) theory [ 10 , 11 ],
where the HF wave function is adopted as the reference
function. The importance of the MP theory, especially the
second-order MP (MP2) method, remains unchanged as
the simplest non-empirical way to consider the electron
correlation, even though the practical density functional
theory (DFT) is getting mature [ 12 , 13 ]. The computational
time for the canonical MP2 calculation, however, scales as
O(N 5 ) , where N is the number of basis functions, which is
signifi cantly longer than those for the DFT and HF calculations of O(N 3 ) . There have been proposed several techniques to reduce the computational time for the MP2 calculation. Surján [ 14 ] proposed a method to evaluate the
MP2 correlation energy as the functional of the HF density
matrix (DM), D , based on the Laplace-transformed MP2
Abstract In 2005, Surján showed two explicit formulas
for evaluating the second-order Møller–Plesset perturbation (MP2) energy as a functional of the Hartree–Fock density matrix D (Chem Phys Lett 406:318, 2005 ), which are
referred to as the E MP2 [D] functionals. In this paper, we
present the fi nite-temperature (FT) MP2 energy functionals of the FT Hartree–Fock density matrix. There are also
two formulas for the FT-MP2, namely the conventional and
renormalized ones; the latter of which has recently been
formulated by Hirata and He (J Chem Phys 138:204112,
2013 ). We proved that there exists one-to-one correspondence between the formulas of two FT-MP2 and the
E MP2 [D] functionals. This fact can explain the different
behavior of two E MP2 [D] functionals when an approximate Hartree–Fock density matrix is applied, which was
previously investigated by Kobayashi and Nakai (Chem
Phys Lett 420:250, 2006 ). We also applied the FT-MP2 formalisms to the linear-scaling divide-and-conquer method
for improving the accuracy with tiny addition of the computational efforts.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Masato Kobayashi
k-masato@mail.sci.hokudai.ac.jp
Tetsuya Taketsugu
take@sci.hokudai.ac.jp
1
Department of Chemistry, Faculty of Science , Hokkaido
University , Sapporo 060-0810 , Japan
2
Elements Strategy Initiative for Catalysts and Batteries
(ESICB) , Kyoto University , Kyoto 615-8245 , Japan
257
Reprinted from the journal
Theor Chem Acc (2015) 134:107
DOI 10.1007/s00214-015-1710-y
REGULAR ARTICLE
Second-order Møller–Plesset perturbation (MP2) theory at fi nite
temperature: relation with Surján’s density matrix MP2 and its
application to linear-scaling divide-and-conquer method
Masato Kobayashi
1,2 · Tetsuya Taketsugu
1,2
Received: 24 June 2015 / Accepted: 28 July 2015 / Published online: 15 August 2015
© Springer-Verlag Berlin Heidelberg 2015
Keywords Fractional occupation number · Many-body
perturbation theory · Laplace-transformed Møller–Plesset
perturbation · Linear-scaling electronic structure method
1 Introduction
The perturbation theory has widely been used in quantum
chemistry to account for the dynamical electron correlation
in single Slater [i.e., Hartree–Fock (HF)] and multideterminantal states [ 1 ]. Surján has worked on the perturbation
theories for both HF and non-HF references. For non-HF
reference functions, he and his coworkers proposed a series
of multiconfi guration perturbation (MCPT) theories [ 2 – 6 ].
Because the MCPT theories are applicable to any reference functions, they have occasionally been applied to the
antisymmetric product of strongly orthogonal geminals
wave functions [ 7 – 9 ].
Although many of Surján’s famous works are related
to the multideterminantal framework, he has also worked
on the Møller–Plesset perturbation (MP) theory [ 10 , 11 ],
where the HF wave function is adopted as the reference
function. The importance of the MP theory, especially the
second-order MP (MP2) method, remains unchanged as
the simplest non-empirical way to consider the electron
correlation, even though the practical density functional
theory (DFT) is getting mature [ 12 , 13 ]. The computational
time for the canonical MP2 calculation, however, scales as
O(N 5 ) , where N is the number of basis functions, which is
signifi cantly longer than those for the DFT and HF calculations of O(N 3 ) . There have been proposed several techniques to reduce the computational time for the MP2 calculation. Surján [ 14 ] proposed a method to evaluate the
MP2 correlation energy as the functional of the HF density
matrix (DM), D , based on the Laplace-transformed MP2
Abstract In 2005, Surján showed two explicit formulas
for evaluating the second-order Møller–Plesset perturbation (MP2) energy as a functional of the Hartree–Fock density matrix D (Chem Phys Lett 406:318, 2005 ), which are
referred to as the E MP2 [D] functionals. In this paper, we
present the fi nite-temperature (FT) MP2 energy functionals of the FT Hartree–Fock density matrix. There are also
two formulas for the FT-MP2, namely the conventional and
renormalized ones; the latter of which has recently been
formulated by Hirata and He (J Chem Phys 138:204112,
2013 ). We proved that there exists one-to-one correspondence between the formulas of two FT-MP2 and the
E MP2 [D] functionals. This fact can explain the different
behavior of two E MP2 [D] functionals when an approximate Hartree–Fock density matrix is applied, which was
previously investigated by Kobayashi and Nakai (Chem
Phys Lett 420:250, 2006 ). We also applied the FT-MP2 formalisms to the linear-scaling divide-and-conquer method
for improving the accuracy with tiny addition of the computational efforts.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Masato Kobayashi
k-masato@mail.sci.hokudai.ac.jp
Tetsuya Taketsugu
take@sci.hokudai.ac.jp
1
Department of Chemistry, Faculty of Science , Hokkaido
University , Sapporo 060-0810 , Japan
2
Elements Strategy Initiative for Catalysts and Batteries
(ESICB) , Kyoto University , Kyoto 615-8245 , Japan
257
Reprinted from the journal