Cluster Amplitudes and Their Interplay with Self-Consistency in Density Functional Methods Greta Jacobson12Juan M. Marmolejo-Tejada1Mart n A. Mosquera1

2025-04-27 0 0 808.96KB 17 页 10玖币

侵权投诉

Cluster Amplitudes and Their Interplay with Self-Consistency in

Density Functional Methods

Greta Jacobson,1,2Juan M. Marmolejo-Tejada,1Mart´ın A. Mosquera1,∗

1Department of Chemistry and Biochemistry, Montana State University, Bozeman, Montana 59717 USA

2Department of Chemistry, Millikin University, Decatur, Illinois 62522 USA

∗martinmosquera@montana.edu

August, 2022

Abstract

Density functional theory (DFT) provides convenient electronic structure methods for the study

of molecular systems and materials. Regular Kohn-Sham DFT calculations rely on unitary transfor-

mations to determine the ground-state electronic density, ground state energy, and related properties.

However, for dissociation of molecular systems into open-shell fragments, due to the self-interaction

error present in a large number of density functional approximations, the self-consistent procedure

based on the this type of transformation gives rise to the well-known charge delocalization problem.

To avoid this issue, we showed previously that the cluster operator of coupled-cluster theory can be

utilized within the context of DFT to solve in an alternative and approximate fashion the ground-state

self-consistent problem. This work further examines the application of the singles cluster operator to

molecular ground state calculations. Two approximations are derived and explored: i), A linearized

scheme of the quadratic equation used to determine the cluster amplitudes, and, ii), the eﬀect of car-

rying the calculations in a non-self-consistent ﬁeld fashion. These approaches are found to be capable

of improving the energy and density of the system and are quite stable in either case. The theoretical

framework discussed in this work could be used to describe, with an added ﬂexibility, quantum systems

that display challenging features and require expanded theoretical methods.

1 Introduction

Electronic structure methods predict a very large number of measurable quantities that are used to

understand, characterize, and optimize chemical compounds and materials. Quantum mechanics is

the foundation upon which algorithms are designed and applied to compute electronic and structural

properties. From a fundamental standpoint, quantum mechanics states that with a complete knowledge

arXiv:2210.03694v1 [physics.chem-ph] 7 Oct 2022

of the wave function of the system one can thus be able to determine all the information about the

system of interest. For computational eﬃciency, however, density functional theory (DFT) serves as an

alternative to pursue such goal. In DFT one the primary objectives is the calculation of the electronic

density of the system, as opposed to the full wave function of all the electrons. Although it is common

to separate both, wave-function theory (WFT) and DFT, as separate ﬁelds, it can be argued that both

are intrinsically connected, especially from the algorithmic point of view.

DFT methods have been formulated on the basis of physical understanding of model systems and

small molecules. A notable example is the electron gas, which in many ways has led to functional

components that to date still remain an important part of a very large number of density functional

approximations (DFAs). These functionals are available for diﬀerent energy “pieces” such as the

kinetic, exchange, correlation, and van der Waals energies. The kinetic energy is known to be the most

challenging energy to be expressed explicitly as a density-functional. For this reason, Kohn-Sham

(KS) DFT [1] is the most common theory within DFT that is utilized for practical calculations and

to derive concepts.[2, 3] As KS-DFT uses single-electron orbitals to determine a kinetic energy. As is

well known, even though KS-DFT practical calculations perform well for determining properties such

as molecular geometries, and optoelectronic properties of a very large number of compound types, it is

diﬃcult for transition-metal systems [4], bond-breaking [5], and charge-transfer excitations [6], among

others, where erroneous charge delocalization [7–10] is a main manifestation of these adverse eﬀects.

Extended DFAs that are free of incorrect charge delocalization should eliminate the main cause

for such adverse eﬀect, the self-interaction error [11–14]. Additionally, improved methodologies must

also come with relatively low computational costs. Motivated by these considerations, and fueled by

advances in machine learning and the premise of new generation of computing technologies (classical

and quantum), theoretical methods are being advanced by the scientiﬁc community, with the goal of

extending the applicability of DFT methods [15–17]. These extensions include the development of force

ﬁelds, which are creating opportunities for detailed studies of systems at the mesoscopic scale [18]. For

example, artiﬁcial neural network (ANN) algorithms have been used to generate density functional

approximations [19, 20], and have been able to eliminate charge delocalization errors. On the other

hand, ANNs also have led to both transferable and speciﬁc force ﬁelds. This also includes ANNs being

used extensively in materials discovery and properties prediction [21–23]. Machine-learned interactomic

potentials, which are tailored for a particular system of interest demonstrate quite appealing theoretical

prospects for modeling mesoscale phenomena [24–30].

From a foundational perspective, the elimination of charge delocalization still remains a long sought

goal, where theoretical tools are still the subject of continued developments. This problem not only

manifests in DFT development, but also in WFT research. For example, it is known that there are

dynamically correlated post-Hartree-Fock methods that can also cause issues with size-consistency,

whereas the well-known exponential ansatz of WFT, in conjunction with spin-symmetry breaking,

oﬀers a theoretically sound route to restore size-consistency (which implies size-extensivity as well).

We showed previously that this exponential operator, which in turn is determined by what is known

as the “cluster operator” [31–41], can also prevent undesired charge delocalization in DFT calculations

[42]. The cluster operator in the ground-state case is limited in our calculations to single-electron

transitions, as it displays a high degree of accuracy at this level of excitation. The cluster amplitudes

that are used to construct the exponential operator are derived as the solution of a quadratic equation,

which is solved in an iterative fashion. Our proposed method, denoted as “eXp” (due to its relying on

the exponential operator), predicted with physical consistency the binding energy curves of classical

systems such as di-hydrogen, lithium hydride, and hydrogen ﬂuoride, but we also show other cases

where the eXp method functions as an alternative to the standard unitary method of KS-DFAs, and

we suggested they are also compatible with the double-hybrid functional approach [43–45]. These

previous ﬁndings motivate the present work, where we further explore the eXp method under its

linearized version, which simpliﬁes in a very accurate way the determination of the cluster amplitudes

and the exponential operator. We also examine non-self-consistent ﬁeld calculations, where the single-

particle Hamiltonian is determined by the Hartree-Fock density, which is used to estimate directly the

cluster operator and its conjugate, the “lambda” operator. In this study we ﬁnd that the linearized eXp

method performs quite well with excellent agreement with respect to the full quadratic scheme in both

cases, the self-consistent and the non-self-consistent ones. The eXp technique is applied to a couple

of known cases of severe charge delocalization (or strong self-interaction), with the goal of eliminating

it: The positively charged neon dimer, Ne+

2, and lithium-ﬂuoride, LiF. In addition, our methods are

applied to a set of molecules at their minimum-energy geometries, where we show that the linearized

eXp method performs quite similarly as the quadratic version in self-consistent-ﬁeld (SCF) and non-

self-consistent-ﬁeld (NSCF) calculations. However, the NSCF computations, as expected, are less

accurate that the SCF ones, but can be considered for calculations where computational acceleration

is needed. The simulations considered in this work are based on a single-particle Hamiltonian, but

they are also applicable to Hamiltonians that include two-body interactions, such as those used in

double-hybrid approaches.

2 Theory

3 Computational Details

Determining ground-state properties in KS-DFT begins with the calculation of the KS Slater deter-

minant |Φiand subsequently the electronic energy. The wave function |Φiis computed through the

minimization of an auxiliary single-particle energy, which depends on the single-particle Hamilonian,

or KS Fock operator. We denote this density-dependent operator as ˆ

f. The energy function that is

minimized in KS-DFT to obtain the orbitals is then hΦ|ˆ

f|Φi, and it leads to the standard KS equations

where the single particle orbitals are constructed through diagonalization of the KS Fock matrix. The

object ˆ

fis the sum of the kinetic, electron-nucleus, exchange-correlation (XC), and Hartree contribu-

tions.

As an alternative to the standard procedure mentioned above, we stationarize the single-particle

energy with respect to cluster operators, where the reference is a Hartree-Fock (HF) wave function,

which we denote as Ψ0. This wavefunction, as expected, is constructed with occupied orbitals in the

HF molecular orbital basis set. This is a relevant detail, as our calculations rely entirely on such

molecular basis set. The HF wavefunction can either be a restricted or unrestricted reference. We

introduce an auxiliary right-handed wave function of the form |ΥRi= exp(+ˆ

t)|Ψ0i, and the left-ket

hΥL|=hΨ0|(1 + ˆ

Λ) exp(−ˆ

t), where ˆ

tand ˆ

Λ are the cluster operators. The function to stationarize is

hΥL|ˆ

f|ΥRi, so it leads to the auxiliary single-particle energy as:

Es= stat.

t,ˆ

ΛhΨ0|(1 + ˆ

Λ) ¯

f|Ψ0i(1)

where the symbol ¯

fdenotes the transformed operator exp(−ˆ

t)ˆ

fexp(+ˆ

t). We use this notation for

other operators too; so if ˆ

Ω is some arbitrary operator, then ¯

Ω = exp(−ˆ

t)ˆ

Ω exp(+ˆ

t). The cluster

operators that we are interested in have the form ˆ

t=Pai ta

iˆa†ˆ

i, and ˆ

Λ = Pai Λa

iˆ

i†ˆa. The indices i

and adenote occupied and virtual spin-orbitals, respectively. By stationarizing with respect to ˆ

tand

Λ it is then implied that one must ﬁnd, what we regard as vectors computationally, {ta

i}and {Λa

i}.

This demands that the derivatives of the function hΨ0|(1 + ˆ

Λ) ¯

f|Ψ0iwith respect to all the elements

Λa

iand ta

iare all zero.

We denote fpq as the matrix element, hχp|ˆ

f|χqi, where χpis a Hartree-Fock spin-orbital; this

implies that fpq can be non-zero for p6=q. We then have that the t-amplitudes derive from the

equation:

0 = fai +X

ifab −X

jfji −X

fjbtb

ita

j(2)

And the Λ-amplitudes are obtained from the linear system MΛ =−f, where

Mck,ai =Rck,ai −X

jfjc −X

ifkbδac (3)

and

Rck,ai =fcaδik −fkiδca (4)

The symbol frepresents the Fock matrix as a vector, (f)ai =fai. We denote the process of determining

tthrough Eq. 2 as the quadratic eXp scheme, or “Q-eXp”. It, Q-eXp, can be solved using the quasi-

Newton method where an estimate to ta

iis updated according to the equation:

i←ta

i−La

faa −fii

(5)

Where La

irefers to the left-hand side of Equation 2.

By neglecting quadratic terms in Eq. 2, we obtain the approximation:

Rt =−f(6)

We refer to this scheme as “L-eXp”. This approximation requires the solution to a linear system of

equations, so it avoids the need for iterations to ﬁnd t. On the other hand, this linear matrix equation

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ClusterAmplitudesandTheirInterplaywithSelf-ConsistencyinDensityFunctionalMethodsGretaJacobson,1;2JuanM.Marmolejo-Tejada,1MartnA.Mosquera1;1DepartmentofChemistryandBiochemistry,MontanaStateUniversity,Bozeman,Montana59717USA2DepartmentofChemistry,MillikinUniversity,Decatur,Illinois62522USAmartinmo...

展开>> 收起<<

Cluster Amplitudes and Their Interplay with Self-Consistency in Density Functional Methods Greta Jacobson12Juan M. Marmolejo-Tejada1Mart n A. Mosquera1.pdf

共17页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Cluster Amplitudes and Their Interplay with Self-Consistency in Density Functional Methods Greta Jacobson12Juan M. Marmolejo-Tejada1Mart n A. Mosquera1

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: