Piotr Piecuch

Efficient computer implementation of the renormalized coupled-cluster methods: The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approaches

Abstract The recently proposed renormalized (R) and completely renormalized (CR) coupled-cluster (CC) methods of the CCSD[T] and CCSD(T) types have been implemented using recursively generated intermediates and fast matrix multiplication routines. The details of this implementation, including the complete set of equations that have been used in writing efficient computer codes, memory requirements, and typical CPU timings, are discussed. The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) computer codes and similar codes for the standard CC methods, including the LCCD, CCD, CCSD, CCSD[T], and CCSD(T) approaches, have been incorporated into the gamess package. Information about the main features of this new set of CC programs is provided.

The method of moments of coupled-cluster equations and the renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approaches

This paper is the first in a series of papers on the new approach to the many-electron correlation problem, termed the method of moments of coupled-cluster equations (MMCC). A hierarchy of MMCC approximations, including the renormalized and completely renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) methods, which can be viewed as generalizations of the well-known perturbative coupled-cluster CCSD[T], CCSD(T), CCSD(TQf), and CCSDT(Qf) schemes, is introduced. In this initial study, an emphasis is placed on the ability of the MMCC approach to describe bond breaking and large effects due to connected triples and quadruples by modifying the standard noniterative CC approaches, such as the popular CCSD(T) method. The performance of selected MMCC approaches, including the renormalized and completely renormalized CCSD[T], CCSD(T), and CCSD(TQ) schemes, is illustrated by the results of pilot calculations for the HF and H2O molecules.

Renormalized coupled-cluster methods exploiting left eigenstates of the similarity-transformed Hamiltonian.

Completely renormalized (CR) coupled-cluster (CC) approaches, such as CR-CCSD(T), in which one corrects the standard CC singles and doubles (CCSD) energy for the effects of triply (T) and other higher-than-doubly excited clusters [K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18 (2000)], are reformulated in terms of the left eigenstates Phimid R:L of the similarity-transformed Hamiltonian of CC theory. The resulting CR-CCSD(T)(L) or CR-CC(2,3) and other CR-CC(L) methods are derived from the new biorthogonal form of the method of moments of CC equations (MMCC) in which, in analogy to the original MMCC theory, one focuses on the noniterative corrections to standard CC energies that recover the exact, full configuration-interaction energies. One of the advantages of the biorthogonal MMCC theory, which will be further analyzed and extended to excited states in a separate paper, is a rigorous size extensivity of the basic ground-state CR-CC(L) approximations that result from it, which was slightly violated by the original CR-CCSD(T) and CR-CCSD(TQ) approaches. This includes the CR-CCSD(T)(L) or CR-CC(2,3) method discussed in this paper, in which one corrects the CCSD energy by the relatively inexpensive noniterative correction due to triples. Test calculations for bond breaking in HF, F(2), and H(2)O indicate that the noniterative CR-CCSD(T)(L) or CR-CC(2,3) approximation is very competitive with the standard CCSD(T) theory for nondegenerate closed-shell states, while being practically as accurate as the full CC approach with singles, doubles, and triples in the bond-breaking region. Calculations of the activation enthalpy for the thermal isomerizations of cyclopropane involving the trimethylene biradical as a transition state show that the noniterative CR-CCSD(T)(L) approximation is capable of providing activation enthalpies which perfectly agree with experiment.

A state‐selective multireference coupled‐cluster theory employing the single‐reference formalism

A new state‐selective multireference (MR) coupled‐cluster (CC) method exploiting the single‐reference (SR) particle‐hole formalism is described. It is an extension of a simple two‐reference formalism, which we presented in our earlier paper [N. Oliphant and L. Adamowicz, J. Chem. Phys. 94, 1229 (1991)], and a rigorous formulation of another method of ours, which we obtained as an approximation of the SRCC approach truncated at triple excitations (SRCCSDT) [N. Oliphant and L. Adamowicz, J. Chem. Phys. 96, 3739 (1992)]. The size extensivity of the resulting correlation energies is achieved by employing a SRCC‐like ansatz for the multideterminantal wave function. General considerations are supplemented by suggesting a hierarchy of approximate schemes, with the MRCCSD approach (MRCC approach truncated at double excitations from the reference determinants) representing the most important one. Our state‐selective MRCCSD theory emerges through a suitable selection of the most essential cluster components appearing in the full SRCCSDTQ method (SRCC method truncated at quadruple excitations), when the latter is applied to quasidegenerate states. The complete set of equations describing our MRCCSD formalism is presented and the possibility of the recursive intermediate factorization [S. A. Kucharski and R. J. Bartlett, Theor. Chim. Acta 80, 387 (1991)] of our approach, leading to an efficient computer algorithm, is discussed.

New coupled-cluster methods with singles, doubles, and noniterative triples for high accuracy calculations of excited electronic states.

The single-reference ab initio methods for high accuracy calculations of potential energy surfaces (PESs) of excited electronic states, termed the completely renormalized equation-of-motion coupled-cluster approaches with singles, doubles, and noniterative triples [CR-EOMCCSD(T)], are developed. In the CR-EOMCCSD(T) methods, which are based on the formalism of the method of moments of coupled-cluster equations, the suitably designed corrections due to triple excitations are added, in a state-selective manner, to the excited-state energies obtained in the standard equation-of-motion coupled-cluster calculations with singles and doubles (EOMCCSD). It is demonstrated that the CR-EOMCCSD(T) approaches, which can be regarded as the excited-state analogs of the ground-state CR-CCSD(T) theory, provide a highly accurate description of excited states dominated by double excitations, excited states displaying a manifestly multireference character, and PESs of excited states along bond breaking coordinates with the ease of the ground-state CCSD(T) or CR-CCSD(T) calculations. The performance of the CR-EOMCCSD(T) methods is illustrated by the results of calculations for the excited states of CH+, HF, N2, C2, and ozone.

Coupled-cluster methods with internal and semi-internal triply and quadruply excited clusters: CCSDt and CCSDtq approaches

Extension of the closed-shell coupled-cluster (CC) theory to studies of bond breaking and general quasidegenerate situations requires the inclusion of the connected triply and quadruply excited clusters, T3 and T4, respectively. Since the complete inclusion of these clusters is expensive, we explore the possibility of incorporating dominant T3 and T4 contributions by limiting them to active orbitals. We restrict T3 and T4 clusters to internal or internal and semi-internal components using arguments originating from the multireference formalism. A hierarchy of approximations to standard CCSDT (CC singles, doubles, and triples) and CCSDTQ (CC singles, doubles, triples, and quadruples) schemes, designated as the CCSDt and CCSDtq approaches, is proposed and tested using the H2O and HF molecules at displaced nuclear geometries and C2 at the equilibrium geometry. It is demonstrated that the CCSDt and CCSDtq methods provide an excellent description of bond breaking and nondynamic correlation effects. Unlike pert...

The active-space equation-of-motion coupled-cluster methods for excited electronic states: Full EOMCCSDt

The full version of the equation-of-motion coupled-cluster (EOMCC) method with all singles and doubles, and a selected set of triples defined through active orbitals (EOMCCSDt) has been implemented and tested using the H8, H2O, N2, C2, and CH+ systems. It is demonstrated that the full EOMCCSDt method provides the results of the full EOMCCSDT (EOMCC singles, doubles, and triples) quality at the fraction of the computer effort associated with the EOMCCSDT calculations. This includes excited states that are dominated by doubles and states that have large triexcited components. The excellent performance of the EOMCCSDt approach is observed even when the ground electronic state has a quasidegenerate character, which means that we can apply the EOMCCSDt formalism to excited states that cannot be adequately described by the perturbative triples models. The EOMCCSDt method is equivalent to the EOMCCSDT approach if all orbitals used in the EOMCCSDt calculations are active.

Renormalized CCSD(T) and CCSD(TQ) approaches: Dissociation of the N2 triple bond

The recently proposed renormalized and completely renormalized CCSD(T) and CCSD(TQ) methods, which can be viewed as generalizations of the noniterative perturbative CCSD(T) and CCSD(TQf) schemes and which result from the more general method of moments of coupled-cluster equations, are applied to the dissociation of the ground-state N2 molecule. It is shown that the renormalized and completely renormalized CCSD(T) and CCSD(TQ) methods provide significantly better results for large N–N separations than their unrenormalized CCSD(T) and CCSD(TQf) counterparts.

State‐selective multireference coupled‐cluster theory employing the single‐reference formalism: Implementation and application to the H8 model system

The new state‐selective (SS) multireference (MR) coupled‐cluster (CC) method exploiting the single‐reference (SR) particle‐hole formalism, which we have introduced in our recent paper [P. Piecuch, N. Oliphant, and L. Adamowicz, J. Chem. Phys. 99, 1875 (1993)], has been implemented and the results of the pilot calculations for the minimum basis‐set (MBS) model composed of eight hydrogen atoms in various geometrical arrangements are presented. This model enables a continuous transition between degenerate and nondegenerate regimes. Comparison is made with the results of SR CC calculations involving double (CCD), single and double (CCSD), single, double, and triple (CCSDT), and single, double, triple, and quadruple (CCSDTQ) excitations. Our SS CC energies are also compared with the results of the Hilbert space, state‐universal (SU) MR CC(S)D calculations, as well as with the MR configuration interaction (CI) results (with and without Davidson‐type corrections) and the exact correlation energies obtained using...

Combined coupled-cluster and many-body perturbation theories.

Various approximations combining coupled-cluster (CC) and many-body perturbation theories have been derived and implemented into the parallel execution programs that take into account the spin, spatial (real Abelian), and permutation symmetries and that are applicable to closed- and open-shell molecules. The implemented models range from the CCSD(T), CCSD[T], CCSD(2)(T), CCSD(2)(TQ), and CCSDT(2)(Q) methods to the completely renormalized (CR) CCSD(T) and CCSD[T] approaches, where CCSD (CCSDT) stands for the CC method with connected single and double (single, double, and triple) cluster operators, and subscripted or parenthesized 2, T, and Q indicate the perturbation order or the excitation ranks of the cluster operators included in the corrections. The derivation and computer implementation have been automated by the algebraic and symbolic manipulation program TENSOR CONTRACTION ENGINE (TCE). The TCE-synthesized subroutines generate the tensors with the highest excitation rank in a blockwise manner so that they need not be stored in their entirety, while enabling the efficient reuse of other precalculated intermediate tensors defined by prioritizing the memory optimization as well as operation minimization. Consequently, the overall storage requirements for the corrections due to connected triple and quadruple cluster operators scale as O(n(4)) and O(n(6)), respectively (n being a measure of the system size). For systems with modest multireference character of their wave functions, we found that the order of accuracy is CCSD<CR-CCSD(T) approximately CCSD(2)(T) approximately CCSD(T)<CCSDT approximately CCSD(2)(TQ)<CCSDT(2)(Q), whereas CR-CCSD(T) is more effective in cases of larger quasidegeneracy. The operation costs of the TCE-generated CCSD(2)(TQ) and CCSDT(2)(Q) codes scale as rather steep O(n(9)), while the TCE-generated CCSD(T), CCSD(2)(T), and CR-CCSD(T) codes are near operation minimum [a noniterative O(n(7))]. The perturbative correction part of the CCSD(T)/cc-pVDZ calculations for azulene exhibited a 45-fold speedup upon a 64-fold increase in the number of processors from 8 to 512.