The coupled-cluster method with a multiconfiguration reference state

The coupled-cluster approach to obtaining the bond-state wave functions of many-electron systems is extended, with a set of physically reasonable approximations, to admit a multiconfiguration reference state. This extension permits electronic structure calculations to be performed on correlated closed- or open-shell systems with potentially uniform precision for all molecular geometries. Explicit coupled cluster working equations are derived using a multiconfiguration reference state for the case in which the so-called cluster operator is approximated by its one- and two-particle components. The evaluation of the requisite matrix elements is facilitated by use of the unitary group generators which have recently received wide attention and use in the quantum chemistry community.     

Molecular symmetry. IV. The coupled perturbed Hartree–Fock method

Symmetry methods employed in the ab initio polyatomic program HONDO are extended to the coupled perturbed Hartree–Fock (CPHF) formalism, a key step in the analytical computation of energy first derivatives for configuration interaction (CI) wavefunctions, and energy second derivatives for Hartree–Fock (HF) wavefunctions. One possible computational strategy is to construct Fock-like matrices for each nuclear coordinate in which the one- and two-electron integrals of the usual Fock matrix are replaced by the integral first derivatives. “Skeleton” matrices are constructed from the unique blocks of electron-repulsion integral derivatives. The correct matrices are generated by applying a symmetrization operator. The analysis is valid for many wavefunctions, including closed- or open-shell spin-restricted and spin-unrestricted HF wavefunctions. To illustrate the method, we compare the computer time required for setting up the coupled perturbed HF equations for eclipsed ethane using D_3h symmetry point group and various subgroups of D_3h. Computational times are roughly inversely proportional to the order of the point group.     

Parallel implementation of electronic structure energy, gradient, and Hessian calculations

ACES III is a newly written program in which the computationally demanding components of the computational chemistry code ACES II [J. F. Stanton et al., Int. J. Quantum Chem. 526, 879 (1992); [ACES II program system, University of Florida, 1994] have been redesigned and implemented in parallel. The high-level algorithms include Hartree-Fock (HF) self-consistent field (SCF), second-order many-body perturbation theory [MBPT(2)] energy, gradient, and Hessian, and coupled cluster singles, doubles, and perturbative triples [CCSD(T)] energy and gradient. For SCF, MBPT(2), and CCSD(T), both restricted HF and unrestricted HF reference wave functions are available. For MBPT(2) gradients and Hessians, a restricted open-shell HF reference is also supported. The methods are programed in a special language designed for the parallelization project. The language is called super instruction assembly language (SIAL). The design uses an extreme form of object-oriented programing. All compute intensive operations, such as tensor contractions and diagonalizations, all communication operations, and all input-output operations are handled by a parallel program written in C and FORTRAN 77. This parallel program, called the super instruction processor (SIP), interprets and executes the SIAL program. By separating the algorithmic complexity (in SIAL) from the complexities of execution on computer hardware (in SIP), a software system is created that allows for very effective optimization and tuning on different hardware architectures with quite manageable effort.     

Higher-order equation-of-motion coupled-cluster methods for electron attachment

High-order equation-of-motion coupled-cluster methods for electron attachment (EA-EOM-CC) have been implemented with the aid of the symbolic algebra program TCE into parallel computer programs. Two types of size-extensive truncation have been applied to the electron-attachment and cluster excitation operators: (1) the electron-attachment operator truncated after the 2p-1h, 3p-2h, or 4p-3h level in combination with the cluster excitation operator after doubles, triples, or quadruples, respectively, defining EA-EOM-CCSD, EA-EOM-CCSDT, or EA-EOM-CCSDTQ; (2) the combination of up to the 3p-2h electron-attachment operator and up to the double cluster excitation operator [EA-EOM-CCSD(3p-2h)] or up to 4p-3h and triples [EA-EOM-CCSDT(4p-3h)]. These methods, capable of handling electron attachment to open-shell molecules, have been applied to the electron affinities of NH and C2, the excitation energies of CH, and the spectroscopic constants of all these molecules with the errors due to basis sets of finite sizes removed by extrapolation. The differences in the electron affinities or excitation energies between EA-EOM-CCSD and experiment are frequently in excess of 2 eV for these molecules, which have severe multideterminant wave functions. Including higher-order operators, the EA-EOM-CC methods predict these quantities accurate to within 0.01 eV of experimental values. In particular, the 3p-2h electron-attachment and triple cluster excitation operators are significant for achieving this accuracy.     

Development of a relaxation-inducing cluster expansion formalism for treating strong relaxation and correlation effects

We present in this paper a comprehensive account of an explicitly spin-free coupled cluster theory for treating energy differences of open-shell states relative to a closed-shell ground state, where the open-shell states of interest are dominated by a few simple configuration state functions. We develop a valence-universal coupled cluster formalism to achieve this via a novel cluster expansion ansatz for the valence part of the wave operator, where the orbital relaxation and the correlation relaxation accompanying ionization/excitation from the ground state are taken care of to all orders in compact, efficient, and explicitly spin-free manner. The essential difference of our proposed ansatz from the ordinary and the normal-ordered cluster ansatz in vogue is that (a) we allow the valence cluster operators to be connected among themselves with spectator valence lines only and (b) we use suitable combinatoric factors accompanying powers of cluster operators thus connected, which are equal to the number of ways the operators can be joined, leading to the same excitation (the automorphic factor). We emphasize that such an ansatz does not generate terms (diagrams) with chains of cluster operators joined among themselves via spectator lines only. Barring only a few, almost all the terms in the working equations determining the cluster amplitudes involve contraction of the Hamiltonian with the cluster operators via at least one nonspectator line, leading to what we call a "strongly connected" series. The structure of the working equation is remarkably similar to the single-reference closed-shell equation, with a few additional terms. The presence of contractions among cluster operators via spectator lines introduces the additional physical effects of orbital and correlation relaxation using low-body cluster operators. As an illustrative application of the new multireference coupled cluster (CC) theory, we consider in this paper computation of ionization potentials (IPs) of one-valence problem with only one active orbital. The numerical applications are made for both the core- and the inner- and outer-valence IPs for several molecular systems. The numerical values demonstrate the superiority of the relaxation-inducing CC theory, as compared to the normal-ordered ansatz.     

Formulation and implementation of a unitary group adapted state universal multi-reference coupled cluster (UGA-SUMRCC) theory: Excited and ionized state energies

The traditional state universal multi-reference coupled cluster (SUMRCC) theory uses the Jeziorski-Monkhorst (JM) based Ansatz of the wave operator: Ω = ∑(μ)Ω(μ)∣φ(μ)??φ(μ)∣, where Ω(μ) = exp?(T(μ)) is the cluster representation of the component of Ω inducing virtual excitations from the model function φ(μ). In the first formulations, φ(μ)s were chosen to be single determinants and T(μ)s were defined in terms of spinorbitals. This leads to spin-contamination for the non-singlet cases. In this paper, we propose and implement an explicitly spin-free realization of the SUMRCC theory. This method uses spin-free unitary generators in defining the cluster operators, {T(μ)}, which even at singles-doubles truncation, generates non-commuting cluster operators. We propose the use of normal-ordered exponential parameterization for Ω:∑(μ){exp?(T(μ))}∣φ(μ)??φ(μ)∣, where {} denotes the normal ordering with respect to a common closed shell vacuum which makes the "direct term" of the SUMRCC equations terminate at the quartic power. We choose our model functions {φ(μ)} as unitary group adapted (UGA) Gel'fand states which is why we call our theory UGA-SUMRCC. In the spirit of the original SUMRCC, we choose exactly the right number of linearly independent cluster operators in {T(μ)} such that no redundancies in the virtual functions {χ(μ) (l)} are involved. Using example applications for electron detached/attached and h-p excited states relative to a closed shell ground state we discuss how to choose the most compact and non-redundant cluster operators. Although there exists a more elaborate spin-adapted JM-like ansatz of Datta and Mukherjee (known as combinatoric open-shell CC (COS-CC), its working equations are more complex. Results are compared with those from COS-CC, equation of motion coupled cluster methods, restricted open-shell Hartree-Fock coupled cluster, and full configuration interaction. We observe that our results are more accurate with respect to most other theories as a result of the use of the cluster expansion structure for our wave operator. Our results are comparable to those from the more involved COS-CC, indicating that our theory captures the most important aspects of physics with a considerably simpler scheme.     

Unitary group approach to spin-adapted open-shell coupled cluster theory

We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open-shell, and virtual orbitals. When acting on the closed-shell reference state with n_c doubly occupied and n_v unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(n_c) ? U(n_V) generate all Gelfand-Tsetlin (GT) states corresponding to appropriate irreducible representation of U(n_c + n_v). The tensor operators generating the M-tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group S_M. This provides an alternative to the Nagel-Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(n_c) ? U(n_s) tensors, can be shown to be connected, the irreducible tensor operators of U(n_c) ? U(n_v) represent a convenient basis for a spin-adapted full coupled cluster calculation for closed-shell systems. For a high-spin reference determinant with n, singly occupied open-shell orbitals, the corresponding representation of U(n), n=n_c + n_v + n_s is not simply reducible under the group U(n_c) ? U(n_s) ? U(n_v). The multiplicity problem is resolved using the group chain U(n) ? U(n_c + n_v) ? U(n_s) ? U(n_c) ?U(n_s)? U(n_v) ? U(n_v). The labeling of the resulting configuration-state functions (which, in general, are not GT states when n_c > 1) by the irreducible representations of the intermediate group U(n_c + n_v) ?U(n_s) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first-, second-, and third-order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin-adapted open-shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin-restricted open-shell many-body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.     

Towards the Development and Applications of Manifestly Spin-free Multi-reference Coupled Electron-pair Approximation-like Methods: A State Specific Approach

As a practical tool of being applicable to bigger molecules, a full-blown state-specific multi-reference coupled cluster formalism developed by us (Mahapatra et al. in J Chem Phys 110:6171, 1999) would be rather demanding computationally, and it is worthwhile to look for physically motivated approximation schemes which capture a substantial portion of the correlation of the full-blown theory. In this spirit, we have recently proposed coupled electron-pair approximation (CEPA)-like various approximants to the parent spin-adapted state-specific multi-reference coupled cluster (SS-MRCC) theory which depend on the inclusion of EPV terms to various degree. Here, the space of excitations is confined to the first order interactive virtual space generated by the cluster operator, but the EPV terms are included exactly. We call them spin-free state specific multi-reference CERA (SS-MRCEPA) theories. They work within the complete active space (CAS) and have been found to be very effective in bypassing the intruders, similar in performance to that of the parent SS-MRCC theory. The spin-adaptation of the working equations of both the SS-MRCC and the CEPA-like approximants is a non-trivial exercise. In this paper, we delineate briefly the essentials of a spin-free formulation of the SS-MRCC and SS-MRCEPA theories. This allows us to include open-shell configuration state functions (CSF) in the CAS. We consider three variants of SS-MRCEPA method. Two are explicitly orbital invariant: (1) SS-MRCEPA(0), a purely lineralized version of the SS-MRCC theory, (2) SS-MRCEPA(I), which includes all the EPV terms explicitly and exactly in an orbital invariant manner and (3) the SS-MRCEPA(D), which emerges when we keep only the diagonal terms of a set of dressed operators in the working equations. Unlike the first two, the third version is not invariant under the orbital transformation within the set of doubly occupied core, valence and virtual orbitals. The SS-MRCEPA methods produce very encouraging results as was evidenced in the applications on the computation of potential energy surfaces for the ground states of LiH and HF molecules.     

Ground‐State and Excitation Spectra of a Strongly Correlated Lattice by the Coupled Cluster Method

The coupled cluster method is applied to a strongly correlated lattice Hamiltonian, and the coupled cluster linear response method is extended to the calculation of electronic spectra by finding an approximation to a resolvent operator which describes the spectral response of the coupled cluster solution to excitation operators. In this spectral coupled cluster method, the ground and excited states appear as resonances in the spectra, and the resolvent can be iteratively improved in selected spectral regions. The method is applied to a MnO₂ plane model which corresponds to previous experimental works.     

A coupled cluster approach with a hybrid treatment of connected triple excitations: implementation and applications for open-shell systems

An implementation of the coupled cluster (CC) singles, doubles, and a hybrid treatment of connected triples [denoted as CCSD(T)-h], based on the unrestricted Hartree-Fock (UHF) reference, is presented. Based on the spin-integrated formulation, we have developed a computer program to achieve the automatic derivation and implementation of the CCSD(T)-h approach. The CCSD(T)-h approach computationally scales as the seventh power of the system size, and is affordable for many medium-sized systems. The present approach has been applied to study the equilibrium geometries and harmonic vibrational frequencies in a number of open-shell diatomic molecules and bond breaking potential energy profiles in several open-shell molecules, including CH(3), NH(2), and SiH(2). For all systems under study, the overall performance of the UHF-based CCSD(T)-h approach is very close to that of the corresponding CCSDT (CC singles, doubles, and triples), and much better than that of the UHF-based CCSD(T) (CC singles, doubles, and perturbative triples).     

Benchmark studies of variational, unitary and extended coupled cluster methods

Comparative benchmark calculations are presented for coupled cluster theory in its standard formulation, as well as variational, extended, and unitary coupled cluster methods. The systems studied include HF, N(2), and CN, and with cluster operators that for the first time include up to quadruple excitations. In cases where static correlation effects are weak, the differences between the predictions of molecular properties from each theory are negligible. When, however, static correlation is strong, it is demonstrated that variational coupled cluster theory can be significantly more robust than the traditional ansatz and offers a starting point on which to base single-determinant reference methods that can be used beyond the normal domain of applicability. These conclusions hold at all levels of truncation of the cluster operator, with the variational approach showing significantly smaller errors.     

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

Summary The 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.This work has been supported by the U.S. Air Force Office of Scientific Research, Grant No. 90-0079     

Higher-order equation-of-motion coupled-cluster methods

The equation-of-motion coupled-cluster (EOM-CC) methods truncated after double, triple, or quadruple cluster and linear excitation operators (EOM-CCSD, EOM-CCSDT, and EOM-CCSDTQ) have been derived and implemented into parallel execution programs. They compute excitation energies, excited-state dipole moments, and transition moments of closed- and open-shell systems, taking advantage of spin, spatial (real Abelian), and permutation symmetries simultaneously and fully (within the spin-orbital formalisms). The related Lambda equation solvers for coupled-cluster (CC) methods through and up to connected quadruple excitation (CCSD, CCSDT, and CCSDTQ) have also been developed. These developments have been achieved, by virtue of the algebraic and symbolic manipulation program that automated the formula derivation and implementation altogether. The EOM-CC methods and CC Lambda equations introduce a class of second quantized ansatz with a de-excitation operator (Y), a number of excitation operators (X), and a physical (e.g., the Hamiltonian) operator (A), leading to the tensor contraction expressions that can be performed in the order of ((...((yx)x)...)x)a or ((...((ax)x)...)x)y at the minimal peak operation cost, where x, y, and a are basis-set representations (i.e., tensors) of the respective operators X, Y, and A. Any intermediate tensor resulting from either contraction order is shown to have at most six groups of permutable indices, and this knowledge is used to guide the computer-synthesized programs to fully exploit the permutation symmetry of any tensor to minimize the arithmetic and memory costs.     

Cluster Amplitudes and Their Interplay with Self-Consistency in Density Functional Methods

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 transformations 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. ii) The effect of carrying 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.     

Coupled cluster methods including triple excitations for excited states of radicals

We report an extension of the coupled cluster iterative-triples model, CC3, to excited states of open-shell molecules, including radicals. We define the method for both spin-unrestricted Hartree-Fock (UHF) and spin-restricted open-shell Hartree-Fock (ROHF) reference determinants and discuss its efficient implementation in the PSI3 program package. The program is streamlined to use at most O(N(7)) computational steps and avoids storage of the triple-excitation amplitudes for both the ground- and excited-state calculations. The excitation-energy program makes use of a Lowdin projection formalism (comparable to that of earlier implementations) that allows computational reduction of the Davidson algorithm to only the single- and double-excitation space, but limits the calculation to only one excited state at a time. However, a root-following algorithm may be used to compute energies for multiple states of the same symmetry. Benchmark applications of the new methods to the lowest valence (2)B(1) state of the allyl radical, low-lying states of the CH and CO(+) diatomics, and the nitromethyl radical show substantial improvement over ROHF- and UHF-based CCSD excitation energies for states with strong double-excitation character or cases suffering from significant spin contamination. For the allyl radical, CC3 adiabatic excitation energies differ from experiment by less than 0.02 eV, while for the (2)Sigma(+) state of CH, significant errors of more than 0.4 eV remain.     

Spin-Projected EHF equations

Extended Hartree-Fock (EHF ) equations are developed for the general open-shell case using a modified pair-orthogonality-constrained variation (POCV ) method. The EHF energy is expressed in terms of corresponding orbitals that are required to remain orthogonal and paired for all arbitrary infinitesimal variations. The Euler equations for each set of orbitals are reduced to unique pseudosecular equations, the LCAO form of which may easily be derived. The Euler equations and the expressions obtained for the off-diagonal elements of the ?^γδ (γ, δ = a or b) matrices for the closed-shell case are identical to those obtained by Mayer, who used the generalized Brillouin theorem method. However, the present method yields equations for both closed- and open-shell cases and for any spin state.