1 Stochastic resolution of identity to CC2 for large systems ground -state properties

2025-04-30 0 0 1.01MB 19 页 10玖币
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Stochastic resolution of identity to CC2 for large systems: ground-state
properties
Chongxiao Zhao1,2,* and Wenjie Dou,1,2,*
1Department of Chemistry, School of Science, Westlake University, Hangzhou, Zhejiang 310024, China
2Institute of Natural Sciences, Westlake Institute for Advanced Study, Hangzhou, Zhejiang 310024, China
Email: zhaochongxiao@westlake.edu.cn; douwenjie@westlake.edu.cn
A stochastic resolution of identity approach (sRI) is applied to the second-order coupled cluster
singles and doubles (CC2) model to calculate the ground-state energy. Utilizing a set of stochastic
orbitals to optimize the expensive tensor contraction steps in CC2, we greatly reduce the overall
computational cost. Compared with the RI-CC2 model, the sRI-CC2 achieves scaling reduction from
O(N 5) to O(N 3), where N is a measure for the system size. When applying the sRI-CC2 to a series
of hydrogen dimer chains, we demonstrate that the sRI-CC2 accurately reproduces RI-CC2 results
for the correlation energies and exhibits a scaling of O(
.), with NH being the number of hydrogen
atoms. Our calculations with different systems and basis sets show small changes in standard
deviations, which indicates a broad applicability of our approach to various systems.
I. INTRODUCTION
Developing accurate and affordable electronic structure theory for complex systems is still one of the most
challenging problems in theoretical chemistry. Among the available methodologies, the coupled cluster singles and
doubles model (CCSD) has been proved to be a valuable one, which scales as O(N
6) (with N being a measure of the
system size). The CCSD model was first implemented by Purvis and Bartlett1 in 1982 and became popular in realistic
electronic structure calculations due to further development by Koch et al.2,3 Later in 1995, Christiansen and Koch et
al.4 reported the formulation and implementation of the CC2 model, which scales as O(N5), as an approximation to
CCSD. Significantly, the CC2 model provides the ground state energy as well as the excitation energy from the single
excitation dominant transition, which is correct to the second order in the fluctuation operator. Between 2000 and
2001, Hald et al.5-7 took advantage of integral-direct approach to handle the 4-index electron repulsion integrals (ERIs),
which reduced the computational cost of the CC2 model and extended its applicability to larger molecular systems. In
2
addition to the integral-direct approach, the introduction of the resolution-of-the-identity (RI) approximation8,9 by
Hättig and Weigend10 in 2000 greatly ameliorated the bottlenecks of CPU time and storage and has enabled a
widespread use of the CC2 model.11
There have been many successful cases12-17 of the ground state energy calculations by the RI approach. However,
the RI approximation still scales as O(N 5), consuming huge memory and high disk space. In addition, the CC2
calculation of the ground state properties needs to be achieved through an iterative process, which makes it obviously
inferior to other methods such as MP2 in time consumption. These factors weaken the advantages of CC2 and make
people pay more attention to its performance in the calculation of the electronic excited state energy.18
For these reasons, a stochastic orbital approach is introduced to the RI approximation, abbreviated as sRI
approach, to further reduce the scaling of the CC2 ground state energy calculations. The sRI approximation has been
formulated and implemented for a variety of electronic structure theory, including MP219-21, DFT22,23, GW24 et al. In
the sRI approach, a set of random orbitals are introduced to simplify the achievement of 4-index ERIs and lower the
rank of computational costs from O(N
5) to O(N
3), when basically maintains the original accuracy. In the subsequent
research25-27, this sRI approach has been further applied to the second-order Matsubara Green’s function (sRI-GF2)
theory and turned out to be a practical approach for large weakly-correlated systems. Inspired by its high performance,
we introduce the sRI method to the calculation of the CC2 ground state energy.
In the paper, we develop the sRI-CC2 model to further reduce the scaling. We apply the sRI approximation to
the CC2 theory for the ground state properties and show that the sRI approximation reduces the scaling of CC2 from
O(N
5) to O(N
3). We test the performance of sRI-CC2 for hydrogen dimer chains as well as a list of molecular systems.
We show that sRI-CC2 reproduce the results from RI-CC2 in Q-Chem package28 for intensive properties, with a
stochastic error that does not depend on system size. We further analyze the error to demonstrate the applicability of
the sRI-CC2 to a variety of molecular systems.
This paper is organized as follows: In the section Ⅱ, we briefly review RI and sRI methods. We also demonstrate
the detailed implementation of sRI-CC2 in this section. In the section Ⅲ, a comparison of RI-CC2 and sRI-CC2
approaches for a series of molecules and basis sets is presented, with emphasis on the scaling of CPU time and the
assessment of the correlation energies and standard deviations. Finally, we conclude in the section Ⅳ.
3
II. THEORY
A. Notation
We use the notations in Table I to represent the items used in the main text. In particular, the total number of AO
basis functions, auxiliary basis functions, occupied, and unoccupied sets of MOs are denoted as NAO, Naux, Nocc, and
Nvirt respectively.
TABLE I. Summary of notations in the following equations.
B. Resolution of identity (RI) and stochastic resolution of identity (sRI)
Before introducing the stochastic resolution of identity, we briefly review the resolution of identity approach. In
the RI approximation, the 4-index ERIs are approximated by 3-index and 2-index ERIs using the auxiliary basis {P}:
(|)(|)[]

 (|)
=[[(|)/][

/


(|)] (1)
Where we have defined 4-, 3- and 2-index ERIs as
(|) = ()()()()
 (2)
(|) = () () ()
 (3)
item function or indices
AO Gaussian basis functions
χ
α
(r1), χ
β
(r1), χ
γ
(r1), χ
δ
(r1), …
auxiliary basis functions P, Q, R, S, …
general sets of AOs
α
,
β
,
γ
,
δ
, …
general sets of MOs p, q, r, s, …
occupied (active) MOs i, j, k, l, …
unoccupied (virtual) MOs a, b, c, d, …
4
= (|) = () ()
 (4)
Defining

(|) 
/

(5)
we can rewrite the RI approximation as
(|)


(6)
Note that the formation of the RI matrix 
in Eq. (5) scales as O(

). Furthermore, the transformation of
this 3-index matrix from AO basis to MO basis can be done in two steps,

=


(7)

= 

(8)
which scale as O(
). Here and are the usual SCF MO coefficients. Since both  and  scale
linearly with the system size N, the formal scaling of the above transformation is O().
The stochastic realization of the RI approximation utilizes another set of stochastic orbitals {
θ
ξ
},
ξ
= 1, 2, …,
Ns (with Ns being the number of stochastic orbitals). All these stochastic orbitals are column arrays of length Naux with
random elements ±1, (i.e.
θ
ξ
= ±1). Thus, due to the central limit theorem, we have the following identity:
θ
⊗
θ
ξ
=1
θ
ξ
⊗ (
θ
ξ
)
ξ
 =
θ
θ
ξ
θ
θ
ξ
θ
θ
ξ
θ
θ
ξ
θ
θ

ξ
θ
θ

ξ
 
θ

θ
ξ
θ

θ
ξ
 
θ

θ

ξ
(9)
In Eq. (9), since
θ
ξ
(and
θ
) is a random choice of ±1, the diagonal matrix element denoted by 〈
θ
θ
〉
ξ
always
equals 1; the off-diagonal element denoted by 〈
θ
θ
〉
ξ
, however, converges to 0 when averaging over Ns stochastic
orbitals.
With the introduction of the stochastic resolution of identity, we can approximate 4-index ERIs using the
stochastic orbitals in the following form:
摘要：

1StochasticresolutionofidentitytoCC2forlargesystems:ground-statepropertiesChongxiaoZhao1,2,*andWenjieDou,1,2,*1DepartmentofChemistry,SchoolofScience,WestlakeUniversity,Hangzhou,Zhejiang310024,China2InstituteofNaturalSciences,WestlakeInstituteforAdvancedStudy,Hangzhou,Zhejiang310024,ChinaEmail:zhaoch...

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