Single-reference Coupled-cluster Methods Research Articles

Basis set limit electronic excitation energies, ionization potentials, and electron affinities for the 3d transition metal atoms: Coupled cluster and multireference methods

Recently developed correlation consistent basis sets for the first row transition metal elements Sc-Zn have been utilized to determine complete basis set (CBS) scalar relativistic electron affinities, ionization potentials, and 4s(2)3d(n-2)-4s(1)d(n-1) electronic excitation energies with single reference coupled cluster methods [CCSD(T), CCSDT, and CCSDTQ] and multireference configuration interaction with three reference spaces: 3d4s, 3d4s4p, and 3d4s4p3d'. The theoretical values calculated with the highest order coupled cluster techniques at the CBS limit, including extrapolations to full configuration interaction, are well within 1 kcal/mol of the corresponding experimental data. For the early transition metal elements (Sc-Mn) the internally contracted multireference averaged coupled pair functional method yielded excellent agreement with experiment; however, the atomic properties for the late transition metals (Mn-Zn) proved to be much more difficult to describe with this level of theory, even with the largest reference function of the present work.

Critical comparison of various connected quadruple excitation approximations in the coupled-cluster treatment of bond breaking

To assess the limits of single-reference coupled-cluster (CC) methods for potential-energy surfaces, several methods have been considered for the inclusion of connected quadruple excitations. Most are based upon the factorized inclusion of the connected quadruple contribution (Qf) [J. Chem. Phys. 108, 9221 (1998)]. We compare the methods for the treatment of potential-energy curves for small molecules. These include CCSD(TQf), where the initial contributions of triple (T) and factorized quadruple excitations are added to coupled-cluster singles (S) and doubles (D), its generalization to CCSD(TQf), where instead of measuring their first contribution from orders in H, it is measured from orders in H=e(-(T1+T2))He(T1+T2); renormalized approximations of both, and CCSD2 defined in [J. Chem. Phys. 115, 2014 (2001)]. We also consider CCSDT, CCSDT(Qf), CCSDTQ, and CCSDTQP for comparison, where T, Q, and P indicate full triple, quadruple, and pentuple excitations, respectively. Illustrations for F2, the double bond breaking in water, and N2 are shown, including effects of quadruples on equilibrium geometries and vibrational frequencies. Despite the fact that no perturbative approximation, as opposed to an iterative approximation, should be able to separate a molecule correctly for a restricted-Hartree-Fock reference function, some of these higher-order approximations have a role to play in developing new, more robust procedures.

Critical comparison of single-reference and multireference coupled-cluster methods: Geometry, harmonic frequencies, and excitation energies of N2O2

Calculated vertical excitation energies, optimized geometries, and vibrational frequencies of the nitric oxide dimer are reported. The “multireference” (MR) nature of the problem and weak bond between the monomers make a proper description of the system difficult, and standard methods are not as applicable to this system. In this study, recently developed methods such as the double-electron-affinity similarity-transformed equation-of-motion coupled cluster method (DEA-STEOM-CCSD), MR Brillouin–Wigner CCSD (MR-BWCCSD), MR average quadratic CCSD (MR-AQCCSD), and others are used along with a series of basis sets of increasing accuracy. The calculated excitation energies are consistent and convergent with respect to the basis set in DEA-STEOM-CCSD, MR-BWCCSD, and MR-AQCCSD methods. The geometries are highly sensitive to the basis set size and the challenge to obtain the right answers in the basis set limit remains. Nevertheless, we obtain qualitative agreement with the experimental geometry and harmonic vibrational frequencies. The results from the above multireference methods show dramatic improvement over the coupled cluster with singles and doubles and perturbative triples excitation [CCSD(T)] results. Like O3, (NO)2 offers an extremely challenging example in its ground and excited states for single-reference and multireference theory. It deserves to be a standard test molecule as new methods are developed.

Analysis of the multireference state-universal coupled-cluster Ansatz

An algorithm for the multireference (MR) cluster analysis of configuration interaction (CI) wave functions, based on the state-universal (SU) coupled-cluster (CC) Ansatz of Jeziorski and Monkhorst, is formulated. It is illustrated on two often-studied model systems that consist of two (the so-called H4 model) and four (the H8 model), slightly stretched, hydrogen molecules and enable one to vary the degree of quasidegeneracy via a single geometric parameter. The role played by higher-than-two-body clusters is examined and their importance in the MR effective Hamiltonian formalism, particularly in the presence of intruder states, is explored in detail. It is pointed out that in contrast with the single-reference CC methods, the MR SU CC approach with singles and doubles (CCSD) cannot yield exact energies even when exact one- and two-body clusters are available. It is shown that this limitation, which is particularly crucial in the presence of intruder states, can be remedied by employing a small subset of higher-than-pair-cluster amplitudes, extracted via cluster analysis from the MR CISD wave functions that are based on the same model space as the SU CCSD method, not unlike in the reduced MR CCSD method.

To Multireference or not to Multireference: That is the Question?

I present a personal viewpoint on multi-reference coupled-cluster theory, its pros and cons. I also suggest some criteria that should be satisfied by multi-reference CC, not the least of which is to develop a tool that will be (almost!) as easy to apply as today’s powerful array of single reference coupled-cluster methods. Some approaches like the equation of motion CC method offers a multi-reference description of some target states, while being entirely single reference in execution. Perhaps it offers a model for further generalization to a wider array of multi-reference problems.

Can ordinary single-reference coupled-cluster methods describe potential energy surfaces with nearly spectroscopic accuracy? The renormalized coupled-cluster study of the vibrational spectrum of HF

The recently proposed renormalized (R) and completely renormalized (CR) CCSD(T) and CCSD(TQ) methods, which remove the failing of the standard CCSD(T) and CCSD(TQf) approaches at large internuclear separations, have been used to obtain the potential energy function and the vibrational spectrum of the HF molecule. The vibrational term values obtained in the renormalized and completely renormalized CCSD(T) and CCSD(TQ) calculations have been found to be in a better agreement with the experimental [Rydberg–Klein–Rees (RKR)] data than than the results of the expensive full CCSDT calculations. The simple R-CCSD(T) method gives <10 cm−1 errors for the vibrational energies up to ∼41 000 cm−1. The CR-CCSD(T) and CR-CCSD(TQ) methods reduce the ∼300 cm−1 errors in the full CCSDT results for the high-lying states near dissociation to 100–200 cm−1.

Can ordinary single-reference coupled-cluster methods describe the potential energy curve of N 2? The renormalized CCSDT(Q) study

The recently proposed renormalized and completely renormalized CCSDT(Q) methods, which result from the method of moments of coupled-cluster equations, have been implemented and applied to the potential energy curve of N 2. It is shown that the renormalized and completely renormalized CCSDT(Q) methods, employing the restricted Hartree–Fock reference, provide very good description of the potential energy curve of N 2, in spite of the failure of the full CCSDT and perturbative CCSDT(Q f) approaches at large N–N separations.

Single-root multireference Brillouin-Wigner coupled-cluster theory. Rotational barrier of the N2H2 molecule

Recently developed single-root multireference Brillouin–Wigner coupled-cluster (MR BWCC) theory, which deals with one state at a time while employing a multiconfigurational reference wave function, is applied to study the rotational barrier of the N2H2 molecule. The method represents a brand new coupled-cluster (CC) approach to quasi-degenerate problems which combines merits of two approaches: the single-reference CC method in a nondegenerate case and the Hilbert space MR CC method in quasi-degenerate case. The method is able to switch itself from a nondegenerate to a fully degenerate case in a continuous manner, thus providing smooth potential energy surfaces. Moreover, in contrast to the Hilbert space MR CC theory, it does not contain the so-called coupling terms and in a highly nondegenerate case it reduces to a standard single-reference CC method. In order to better judge the abilitiesof our new approach, we study the rotation barrier of the N2H2 molecule at the CCSD level and the results are compared with the single-reference CCSD and Hilbert space MR CCSD methods. The rotation of the N2H2 molecule from a trans- to cis-conformer represents a typical two-state problem in which the weights of reference configurations can change from 0 to 1 in a continuous manner and, in contrast to the H4 models, it represents a real system.

Single-root multireference Brillouin-Wigner coupled-cluster theory: Applicability to the F2 molecule

Recently developed single-root multireference Brillouin-Wigner coupled-cluster (MR BWCC) theory, which deals with one state at a time while employing a multiconfigurational reference wave function, is applied to the ground state of the F2 molecule using a two-determinant reference space at the level of the CCSD approximation. The method represents a brand-new coupled-cluster (CC) approach to quasidegenerate problems which combines merits of two theories: the single-reference CC method in a nondegenerate case and the Hilbert space MR CC method in quasidegenerate case. The method is able to switch itself from a nondegenerate to a fully degenerate case in a continuous manner, providing thus smooth potential energy surfaces. Moreover, in contrast to the Hilbert space MR CC approaches, it does not contain the so-called coupling terms and completely reduces to the standard single-reference CC method in a highly nondegenerate region. Using a [4s,3p,1d] and [4s,3p,2d,1f ] basis sets, the calculated potential energy curves are smooth, dissociate correctly and the results are compared with other available multireference techniques as well as experiment.

Single-Root Multireference Brillouin-Wigner Coupled-Cluster Theory. Rotational Barrier of the Ethylene Molecule

Recently developed single-root multireference Brillouin-Wigner coupled cluster (MR BWCC) theory is applied to study the rotational barrier of the ethylene molecule. The method belongs to a broad family of state-selective coupled-cluster (CC) methods and may be considered as a bridge connecting the single-reference and multireference CC theories. In a highly nondegenerate case, it becomes identical with the standard single-reference CC method while in a quasi-degenerate case gives results very close to the Hilbert space MR CC method. The method switches between the two cases in a continuous way, providing thus smooth potential energy surfaces, not plagued by intruder states. The rotation about the double bond in ethylene represents a typical two-state problem. To demonstrate abilities of our approach, we study the rotation barrier using the single-root MR BWCC theory at the CCSD level of approximation and the results are compared with other CC methods.

A Comparison of Variational and Coupled-Cluster Calculations of Molecular Properties: The Polarizabilities of BeO, 1Σg+, and C2, 1Σg+, 3Πu, and 3Σg-

The parallel and perpendicular components of the static dipole polarizabilities of the title molecules in different spectroscopic states were calculated using a large variety of high-level electron correlation methods. Excitation energies between different states for C2 are presented as well. We used both single-reference coupled-cluster methods at different level of sophistication and complete active space SCF, multireference configuration interaction, and averaged quadratic coupled-cluster methods with various active spaces. The reliability of calculated properties for BeO and C2, known as difficult to calculate accurately, was deduced from the pattern of results observed by a systematic improvement of the level of sophistication and from a comparison of coupled-cluster and multireference variational methods.

Valence bond corrected single reference coupled cluster approach

An extension of the single reference coupled cluster method truncated to 1- and 2-body cluster components (CCSD) to quasidegenerate systems, where 3-and 4-body connected cluster components play an important role, is proposed. The basic idea is to extract the information concerning the 3- and 4-body clusters from some independent source, similarly as was implicitly done in the so-called ACPQ or ACC(S)D methods, and correct accordingly the absolute term in the CCSD equations. As a source of these approximate 3- and 4-body clusters, simple valence bond (VB) type wave functions are employed, since they are capable of describing electronic structure of various molecular systems for a wide range of nuclear conformations including their dissociation. The cluster analysis of these VB wave functions, that provides the desired information concerning the connected 3- and 4-body cluster components, is outlined and the explicit form of required correction terms to the CCSD equations is given.

Applicability of single-reference coupled-cluster methods to excited states. A model study

It is demonstrated that standard single-reference coupled-cluster (SR-CC) theories can be successfully applied to the studies of some excited states not belonging to the lowest energy in their symmetry class. The prerequisite of such applications is that a proper reference state configuration and orbital set are employed for the parametrization of the SR-CC method. The 2 1A 1 and 5 1A 1 states of the simple H4 model system are considered for geometries related to various degrees of quasidegeneracy of these states. It is found that the efficiency of various versions of the SR-CC method (except the linearized versions) is similar to that of standard applications to the closed-shell ground state of H4.

Solving the single-reference coupled-cluster equations involving highly excited clusters in quasidegenerate situations

A new procedure for improving the convergence of the reduced linear equation method for solving the single-reference (SR) coupled-cluster (CC) equations in highly degenerate cases, where the convergence of the iterative process is often destabilized by ‘‘dangerous’’ denominators, is discussed and implemented. Emphasis is placed on the SR CC methods involving highly excited clusters, such as the CC method with singles, doubles, triples, and quadruples (CCSDTQ). The new algorithm is based on rearranging and quasilinearizing the nonlinear terms and correcting the energy denominators by the quasilinearized terms. The method is tested in the SR CCSDTQ calculations for a dissociating single bond.

Complete active space coupled-cluster method. Extension of single-reference coupled-cluster method using the CASSCF wavefunction

It is proposed that the calculation of non-dynamical electron correlation effects by the complete active space self-consistent field (CASSCF) method can be extended by the inclusion of dynamical correlation effects through the use of a suitably modified single-reference coupled-cluster (CC) procedure. The CASSCF method supplies the reference configuration and the complete set of the CC amplitudes corresponding to the space of active orbitals. These amplitudes are then fed into the CC procedure to determine the CC amplitudes of low rank, but corresponding to the full orbital space. The proposed CASCC method combines the formal simplicity of the single-reference CC method with the flexibility of the CASSCF approach, but is limited to cases where there is a single configuration with the highest weight in the CASSCF wavefunction.

Multiple solutions of the single-reference coupled-cluster method

The nonlinear coupled-cluster (CC) equations possess several solutions. They describe various excited states as long as they contain a contribution from the reference function. We study the additional solutions of the single-reference CC equations and demonstrate that unlike the ground state, the approximate solution for excited states in general does not satisfy the cluster condition nor the standard proof of extensivity. For such states that makes the CC approximation poorer than that for the analogous CI.

Multiconfigurational spin-adapted single-reference coupled cluster formalism

We exploit the unitary group formalism in formulating a multiconfigurational single-reference coupled cluster method for cases involving one or two electrons in open shells. The linear version of CCSD theory for the simple open shell case and for low lying singlet states of closed shell systems are considered in detail. The entire formalism is related to the unitary group based CISD method, and explicit expressions for size-extensivity corrections, leading to the L-CCSD formalism, are given. An illustrative example of the minimum basis set model of the BeH radical is also presented. © 1993 John Wiley & Sons, Inc.

A coupled cluster study of the equilibrium bond distance in methane

The equilibrium C-H bond distance of methane has been predicted using the single-reference coupled cluster method including all single, double and perturbative triple configuration excitations [CCSD(T)] in conjunction with large atomic natural orbital basis sets originating from Almof and Taylor's and Dunning's contractions. The best theoretical prediction for the r e of methane being 1·087 A, obtained with a [5 + 1s 4 + 1p 3d 2f 1g/5s 3p 1d] basis set at the CCSD(T) level of theory, is in agreement with the experimental value of 1·086 ± 0·001 A.

A coupled-cluster study of inversion symmetry breaking in the F+2 molecular ion

Several coupled-cluster methods have been used to calculate equilibrium properties (re, ωe, and De ) of the ground state (2Πg) of the F+2 molecular ion. Two unrestricted Hartree–Fock reference determinants have been used. The first was made up of inversion symmetry constrained D2h orbitals, the second of inversion symmetry broken C2v orbitals. The results of the coupled-cluster calculations are rather insensitive to the choice of reference determinant, in contrast to what is observed for finite-order perturbation theory. It follows that for certain symmetry breaking problems, single reference coupled-cluster methods are sufficiently powerful to overcome a symmetry broken reference function, thus in principle obviating the need for either a multireference starting point or a symmetry constrained single reference starting point. Some extended basis set coupled-cluster calculations of equilibrium properties of F+2 and F2 were performed. Very good agreement with experiment was obtained for F2, suggesting that the results for F+2 are also reliable.

Performance of single-reference coupled-cluster methods for quasidegenerate problems: The H4 model

Several coupled-cluster methods based on a single-determinantal reference function have been applied to the model system composed of four hydrogen atoms in a trapezoidal arrangement. For nondegenerate regions all methods with the exception of CCD provide results within 1 mhartree of the exact (FCI) value. For degenerate regions such an accuracy can be achieved with the inclusion of theT3 andT4 clusters, in an iterative and in a noniterative manner. We report results for CCSDT, CCSDTQ-1, CCSD+TQ*(CCSD), CCSD+Q(CCSDT) plus other methods. In particular, the ACP method which has been proposed to indirectly account forT4 terms is critically analyzed by including allT3 contributions.