Method of moments of coupled-cluster equations: a new formalism for designing accurate electronic structure methods for ground and excited states

The method of moments of coupled-cluster equations (MMCC), which provides a systematic way of improving the results of the standard coupled-cluster (CC) and equation-of-motion CC (EOMCC) calculations for the ground- and excited-state energies of atomic and molecular systems, is described. The MMCC theory and its generalized MMCC (GMMCC) extension that enables one to use the cluster operators resulting from the standard as well as nonstandard CC calculations, including those obtained with the extended CC (ECC) approaches, are based on rigorous mathematical relationships that define the many-body structure of the differences between the full configuration interaction (CI) and CC or EOMCC energies. These relationships can be used to design the noniterative corrections to the CC/EOMCC energies that work for chemical bond breaking and potential energy surfaces of excited electronic states, including excited states dominated by double excitations, where the standard single-reference CC/EOMCC methods fail. Several MMCC and GMMCC approximations are discussed, including the renormalized and completely renormalized CC/EOMCC methods for closed- and open-shell states, the quadratic MMCC approaches, the CI-corrected MMCC methods, and the GMMCC approaches for multiple bond breaking based on the ECC cluster amplitudes.     

Making More Extensive Use of the Coupled-cluster Wave Function: from the Standard Energy Expression to the Energy Expectation Value

The standard coupled-cluster (CC) scheme with single and double excitations in the cluster operator (CCSD) includes only up to quadruple excitations in the equations. The CCSD exponential expansion generates, however, all possible excitations out of the reference function through products of the cluster operators. Clearly, in all standard approximate CC approaches only a part of the CC wave function is used in the equations. If the standard CCSD wave function is inserted into the energy expectation value expression then the complete CCSD wave function contributes to the energy. Such an energy expectation value expression can be presented as a sum of the standard CCSD energy formula plus correction terms. The correction terms provide an information about the quality of the total CC function. Contributions associated with the presence of higher than double excitations in the bra CCSD wave function supplement the CCSD energy obtained within the standard scheme. These contributions can be generated in a sequential way by considering intermediate excitation levels for the bra CCSD wave function in the expectation value expression before reaching the highest excitation level. In this way the importance of particular components differing in the standard and expectation value CCSD energies can be investigated. Some of the contributions can be recognized as close to or identical with the so-called renormalized noniterative corrections to the CC methods. We try to see to what an extent the nonstandard energy expressions, like the energy expectation value or the asymmetric energy formula, can be used to extend the applicability of the CCSD method illustrating our considerations with some numerical examples.Dedicated to Professor Jean-Paul Malrieu to honor his contribution to quantum chemistry and physics     

Model study of the impact of orbital choice on the accuracy of coupled-cluster energies. II. Valence-universal coupled-cluster method

The impact of the choice of molecular orbital sets on the results of the valence-universal coupled cluster method involving up to three-body amplitudes (VU-CCSDT) was studied for the H4 model. This model offers a straightforward way of representing all possible symmetry-adapted orbitals. Moreover, the degree of quasi-degeneracy of its lowest ¹A₁ states can be varied over a wide range by changing its geometry. Calculations were performed both for 13 sets of standard quantum chemical orbitals and for a vast variety of nonstandard orbital sets defined by nodes of a two-dimensional orbital grid. The performance of various standard orbital sets in VU-CCSDT calculations is compared. It is also documented that for every quasi-degeneracy region there exist nonstandard orbital sets which allow one to obtain more accurate VU-CCSDT energies than the standard orbital sets. In an attempt to provide a general interpretation for some of the alternative orbital sets, we defined a set of orbitals which maximize the proximity of the model and target spaces—maximum proximity orbitals (MPO). It is demonstrated that outside the strong quasi-degeneracy region the energies obtained for the VU-CCSDT approach based on the MPOs are more accurate than for the standard orbital sets. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 221–237, 1998     

Model study of the impact of orbital choice on the accuracy of coupled-cluster energies. I. Single-reference-state formulation

The impact of the choice of molecular orbital sets on the results of single-reference-state coupled-cluster (CC) methods was studied for the H4 model. This model offers a straightforward way of taking into account all possible symmetry-adapted orbitals. Moreover, the degree of quasi-degeneracy of its ground state can be varied over a wide range by changing its geometry. The CCD, CCSD, and CCSDT approaches are considered. Surfaces representing the dependence of the energy on the parameters defining the orbitals are obtained. It is documented that for every method there exist alternative orbital sets which allow one to obtain more accurate energies than the standard (HF, BO, and NO) ones. However, for many of the former orbital sets, one obtains relatively large one-body amplitudes or one may encounter problems with solving the CC equations by conventional methods. An interesting variety of orbitals which might be useful for studies of quasi-degenerate states by the CCD method was found. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 205–219, 1998     

Analytic energy derivatives for the equation-of-motion coupled-cluster method: Algebraic expressions,implementation and application to theS 1 state of HFCO

Summary The general theory of analytic derivatives for the equation-of-motion coupled cluster (EOM-CC) method is reviewed. Special attention is paid to the EOM-CC singles and doubles (EOM-CCSD) approximation, which has the same computational scaling properties as the coupled-cluster singles doubles (CCSD) ground state method and is therefore applicable to a wide range of molecular systems. The detailed spin orbital equations that must be solved in EOM-CCSD gradient calculations are presented for the first time, and some guidelines are discussed regarding their computational implementation. Finally, use of the EOM-CCSD gradient method is illustrated by determining the structure, dipole moment components, harmonic frequencies and infrared intensities of formyl fluoride (HFCO) in its singlet excited (n, *) state.     

Electron correlation: the many-body problem at the heart of chemistry

The physical interactions among electrons and nuclei, responsible for the chemistry of atoms and molecules, is well described by quantum mechanics and chemistry is therefore fully described by the solutions of the Schr?dinger equation. In all but the simplest systems we must be content with approximate solutions, the principal difficulty being the treatment of the correlation between the motions of the many electrons, arising from their mutual repulsion. This article aims to provide a clear understanding of the physical concept of electron correlation and the modern methods used for its approximation. Using helium as a simple case study and beginning with an uncorrelated orbital picture of electronic motion, we first introduce Fermi correlation, arising from the symmetry requirements of the exact wave function, and then consider the Coulomb correlation arising from the mutual Coulomb repulsion between the electrons. Finally, we briefly discuss the general treatment of electron correlation in modern electronic-structure theory, focussing on the Hartree-Fock and coupled-cluster methods and addressing static and dynamical Coulomb correlation.     

A modified multi-reference second order perturbation theory

A new scheme with extended model space is proposed to improve the calculation of multi-reference second order perturbation theory (MRPT2). The new scheme preserves the concise code structure of the original program, and avoids intruder states in constructions of the potential energy surface, which is confirmed by a series of comparable calculations. The new MRPT2 program is an available tool for the research of molecular excited states and electronic spectrum.     

Configuration-based multi-reference second order perturbation theory

Various configuration-based multi-reference second order perturbation approaches were investigated and a new scheme averting intruder states was suggested. The codes based on these schemes were tested by example calculations.     

多参考态二级微扰方法的改进

本文提出一个新的扩大模型空间的方案用于改进多参考态二级微扰理论(MRPT2)计算. 新方案保持了原方案中扩大模型空间之前的简单程序结构, 理论上完全可以避免势能面计算中入侵态的出现, 并在一系列比较计算中得到证实. 新MRPT2程序是研究分子激发态和电子光谱的有用工具.     

AuX (X=O,S)低电子态的理论研究

通常要用多参考态方法才能合理处理需考虑旋轨耦合（SOC）效应的开壳层分子如AuO和AuS的低电子态. 事实上,通过选取合适的参考态,采用运动方程耦合簇方法（EOM-CC）也能计算这些分子的一些低电子态,而且EOM-CC方法是单参考态方法,使用起来比多参考态方法更加简单. 本文采用最近发展的含旋轨耦合的EOM-CC 计算电离能的方法（EOMIP-CC）,选取对应的负离子为参考态,在CCSD 级别上计算了AuO 和AuS低电子态的性质. 在不考虑旋轨耦合时,通过比较EOMIP-CCSD和EOMIP-CCSDT的结果考察EOMIPCCSD的精度. 此外,与EOMIP-CCSDT结果相比,如果自旋污染较为显著而且T₁的模较大时,UCCSD（T）方法对能量最低的某一特定对称性的电子态的所对应的电离能误差约为0.1-0.15 eV. 在考虑了旋轨耦合效应后,我们的方法得到的键长和振动频率与实验值吻合较好. 另一方面,虽然EOMIP-SOC-CCSD高估了能量较高的²Δ_3/2态、²Σ_1/2⁺态和²Π_1/2态的能量,但是对于其它能量更低的电子态,它们的能量与已有实验值误差在0.2 eV 左右. 这显示我们所用的含SOC的EOMIP-CCSD方法对原本需要用多参考态方法才能处理的AuO和AuS低电子态能给出可靠的结果.