Stanisław A. Kucharski

Non-iterative fifth-order triple and quadruple excitation energy corrections in correlated methods

Abstract In critical cases, single-reference correlated methods like coupled-cluster theory or its quadratic CI approximations fail because of the importance of additional highly excited excitations that cannot usually be included, like connected triple and quadruple excitations. Here we present the first, non-iterative method to evaluate the full set of fifth-order corrections to CCSD and QCISD and assess their accuracy compared to full CI for the very sensitive Be 2 curve and other cases.

A coupled cluster approach with triple excitations

The coupled‐cluster model for electron correlation is generalized to include the effects of connected triple excitation contributions. The detailed equations for triple excitation amplitudes are presented, and a simplified version implemented that retains the dominant terms. The model presented, CCSDT‐1, provides the energy correct through fourth order and the wave function through second order. The CCSDT‐1 model is illustrated by comparing with full CI results for HF, BH, and H2O, the latter at several geometries.

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 coupled‐cluster single, double, triple, and quadruple excitation method

A general implementation of the coupled‐cluster (CC) single, double, triple, and quadruple excitation (CCSDTQ) method is presented and applied to several molecules, including BH, HF, H2O, and CO with DZP basis sets. Comparisons with full CI show average errors to be 14 μhartree at equilibrium and 26 μhartree at twice Re. CCSDTQ is exact for four electrons and is the first CC method correct through sixth order in perturbation theory.

Fifth-Order Many-Body Perturbation Theory and Its Relationship to Various Coupled-Cluster Approaches

Publisher Summary This chapter presents the fifth-order terms generated by several different coupled-cluster (CC) approaches and compares them with all fifth-order MBPT diagrams. This will enable to analyze the steps required to compute these terms. Several simplifications are identified. In particular, all the fifth-order diagrams arising from T 4 may be obtained with only an n 6 algorithm. To enumerate the fifth-order terms, a modified type of diagram is introduced that reduces the overall number of fifth-order diagrams. There are several different methods of calculating fifth-order diagrams. One of the most straightforward would be to collect all the different fifth-order diagrams, to factorize them if possible, and to compute them one by one, several of them requiring multiplication by two to account for their Hermitian conjugates demonstrated by comparing, for instance, groups of diagrams. Later, the results discussed in the chapter exhibit that the calculation of the fifth-order energy is feasible and when carefully implemented may be applied to moderate-sized systems. The number of terms that must be considered may be substantially reduced by taking into account the fact that many of the diagrams have identical values and many are amendable to factorization.

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...

Recursive intermediate factorization and complete computational linearization of the coupled-cluster single, double, triple, and quadruple excitation equations

SummaryThe nonlinear CCSDTQ equations are written in a fully linearized form, via the introduction of computationally convenient intermediates. An efficient formulation of the coupled cluster method is proposed. Due to a recursive method for the calculation of intermediates, all computational steps involve the multiplication of an intermediate with aT vertex. This property makes it possible to express the CC equations exclusively in terms of matrix products which can be directly transformed into a highly vectorized program.

Coupled-cluster theory for excited electronic states: The full equation-of-motion coupled-cluster single, double, and triple excitation method

The equation-of-motion coupled-cluster method with the full inclusion of the single, double, and triple excitations (EOM-CCSDT) has been formulated and implemented. The proper factorization procedure ensures that the method scales as n8, i.e., in the same manner as the standard CCSDT method for ground states. The method has been tested on the vertical excitation energies of the N2 and CO molecules for several basis sets up to 92 basis functions. The full inclusion of the triple excitations improves the EOM-CCSD results by up to 0.2 eV for considered systems.

Noniterative energy corrections through fifth-order to the coupled cluster singles and doubles method

Perturbation corrections through fifth order in the many-body perturbation theory energy with respect to a coupled cluster singles and doubles reference have been derived and analyzed. The formulas employ the T1 and T2 amplitudes obtained as a solution of the coupled cluster singles and doubles equations. Four different energy functionals have been considered as a starting point in the derivation: the regular coupled cluster energy expression, the coupled cluster functional incorporating Λ amplitudes, the one constructed via an expectation value coupled cluster method, and that obtained on the basis of the extended coupled cluster method. The proposed corrections have been applied to several small molecules to test their performance compared to full configuration interaction. The fourth-order Λ-based formulas improve upon CCSD(T), (coupled cluster singles and doubles with noniterative triples), while the best fifth-order formulas reduce the fourth-order error by about two-thirds. We also introduce a facto...

A multireference coupled-cluster method for special classes of incomplete model spaces

A multireference coupled‐cluster (MRCC) method for special but general classes of incomplete model spaces is developed within a Hilbert space framework. The formulation avoids the valence universality requirement and related Fock space considerations that require hierarchical solutions for different number of electrons. Consequently, the Hilbert space approach has fewer amplitudes to determine. It is shown that diagonalization of the effective Hamiltonian leads to purely extensive energies, providing the formal basis for a general MR‐CC methods for potential energy surfaces.