Coupled-cluster Formalism Research Articles

Development of a size-consistent state-specific multireference perturbation theory with relaxed model-space coefficients

We explore the Rayleigh–Schrödinger and the Brillouin–Wigner perturbative counterparts of our recently developed state-specific coupled-cluster formalism with a complete active space. It is size-extensive and designed to avoid intruders. For each reference determinant φ μ , there is a separate cluster operator T μ . The redundancy inherent in such a choice is resolved by postulating suitable sufficiency conditions which at the same time ensure size-extensivity and size-consistency. The combining coefficients c μ for φ μ s are completely relaxed and obtained by diagonalizing an effective operator in the model space, one root of which is the target eigenvalue of the state. We illustrate size-consistency of the perturbative formalisms with an example model problem.

Approximate coupled-cluster methods employing split cluster amplitudes: Implementation of an almost-linear coupled-cluster formalism

A new approximation strategy, split-amplitude strategy, useful within the framework of the coupled-cluster (CC) methodology is proposed. It consists in representing the individual cluster amplitudes as a sum of two components, one of fixed value, which may be obtained from external sources, and the other determined from a set of modified CC equations. This approach provides new possibilities of absorbing information concerning the values of cluster amplitudes from independent calculations. By properly choosing the fixed amplitude components, one may substantially reduce the magnitudes of the most significant amplitudes to be determined for the state considered, which in turn causes that the known approximation procedures are more justifiable when applied to the modified CC equations than to the equations of the standard CC approaches. The split-amplitude strategy has been employed to setting up several almost-linear CC (AL-CC) approaches of a single reference type corresponding to the basic CC methods. These low-cost approaches seem to be useful in theories of processes involving nondynamically correlated (quasidegenerate) states. The AL-CC methods have been applied to the ground states for various geometries of the following molecular systems: H8, H2O, BH, and HF. It is found that the energies obtained for a given AL-CC approach are very close to those for the corresponding basic CC method, which is true even for strongly quasidegenerate states.

A state-specific multi-reference coupled cluster formalism with molecular applications

We present in this paper a state-specific coupled cluster method based on a reference function composed of determinants spanning a complete active space (CAS). The method treats all the reference determinants on the same footing and is hence expected to provide a uniform description over a wide range of molecular geometries. The combining coefficients are determined by diagonalizing an effective operator in the CAS and are thus completely flexible, and not constrained to pre-assigned values. The method uses a separate cluster operator for exciting to virtual functions from each reference determinant. The linear dependence implicit in this choice of cluster operators is eliminated by invoking suitable sufficiency conditions. The choice is dictated to ensure size-extensivity. The use of a CAS also guarantees size-consistency. Illustrative applications to the H8 model system and ground state potential curve for Li2 indicate both the accuracy of the method and its ability to bypass intruders.

A state-specific multi-reference coupled cluster formalism with molecular applications

A state-specific multi-reference coupled cluster formalism with molecular applications

Theoretical Study of the H2 + NO and Related Reactions of [H2NO] Isomers

Mechanisms of the reactions occurring on the multichannel lowest doublet potential energy surface of the [H2NO] system containing H2NO, HNOH, and NOH2 intermediates and including the H2 + NO, NH2 + O, NH + OH, and H + HNO entry channels have been probed using unrestricted coupled-cluster formalism. Also, the transition structure on the doublet energy surface for the direct hydrogen abstraction from H2 + NO yielding H + HNO has been identified. The energetic and molecular parameters derived from coupled cluster singles and doubles with triples correction (CCSD(T)) calculations using the 6-311++G(3df,3pd) basis set, based on their respective optimized geometries obtained with a 6-311++G(d,p) basis set, have then been utilized to compute the apparent rate constants of the different competitive channels in the [H2NO] system within the framework of a quantum version of Rice−Ramsperger−Kassel theory (QRRK). The H2 + NO reaction is found to be a rather slow reaction, and it occurs via bimolecular hydrogen abstraction. The calculated total rate constant for this reaction at 2000 K and 1 atm is 2.35 × 108 cm3 mol-1 s-1. Stabilization of NH2O or NHOH intermediates plays a minor role in the kinetics of NH2 + O reaction which is dominated by the dissociation products, HNO + H, NH + OH and H2 + NO. The H + HNO reaction kinetics has only 10% contribution from addition/isomerization/dissociation paths at 2000 K and 1 atm. H2 + NO and H + HNO are expected to be the competitive products in the reaction of OH with nitrene. The calculated standard heats of formation are ΔHf298(NH2O) = 17.5 ± 2 and ΔHf298(HNOH) = 23.7 ± 2 kcal/mol.

Second-order shakeup terms in electron propagator calculations on F2 and H2O2

A new approximation of the electron propagator includes second-order terms in the superoperator Hamiltonian matrix that are not present in the nondiagonal, renormalized method (NR2) of J. Chem. Phys. 108, 1008 (1998). These terms are not difficult to calculate and resemble expressions obtained with various propagator and coupled-cluster formalisms. Their inclusion produces larger, more accurate vertical ionization energies of F2 and H2O2. Cancellations between the second-order terms considered here and the first-order terms present in the NR2 method reduce the importance of shakeup operators in describing the first few cationic states of these molecules. Dyson orbitals for 2A final states of H2O2 exhibit qualitatively important mixings between canonical orbitals. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 175–182, 1998

Reduced multireference CCSD method: An effective approach to quasidegenerate states

Standard multireference (MR) coupled cluster (CC) approaches are based on the effective Hamiltonian formalism and generalized Bloch equation. Their implementation, relying on the valence universal or state universal cluster Ansatz, is very demanding and their practical exploitation is often plagued with intruder state and multiple solution problems. These problems are avoided in the so-called state selective or state specific (SS) MR approaches that concentrate on one state at a time. To preserve as much as possible the flexibility and generality offered by the general MR CC approaches, yet obtaining a reliable and manageable algorithm, we propose a novel SS strategy providing a size-extensive CC formalism, while exploiting the MR model space and the corresponding excited state manifold. This strategy involves three steps: (i) The construction of a variational configuration interaction (CI) wave function within the singly (S) and doubly (D) excited state manifold, (ii) the cluster analysis of this CI wave function providing the information about the higher than pair cluster amplitudes, and (iii) the exploitation of these amplitudes in the so-called externally corrected CCSD procedure. This approach is referred to as the reduced MR (RMR) SS CCSD method and is implemented at the ab initio level and applied to several model systems for which the exact full CI results are available. These include two four electron H4 systems (usually referred to as the H4 and S4 models), an eight electron H8 model and the singlet-triplet separation problem in CH2. It is shown that the RMR CCSD approach produces highly accurate results, is free from intruder state problems, is very general and effective and applicable to both closed and open shell systems.

Electric properties of the water molecule in 1A1, 1B1, and 3B1 electronic states: Refined CASSCF and CASPT2 calculations

The dipole moments and dipole polarizabilities of the 1A1, 1B1, and 3B1 electronic states of the water molecule have been calculated by using the CASSCF approach followed by the evaluation of the dynamic electron correlation contribution by the second-order perturbation scheme CASPT2. All calculations have been carried out in a specifically extended ANO basis set which accounts for the Rydberg character of the two excited states. In order to estimate the correctness and accuracy of the present data a scan over a variety of different active spaces for the CASSCF wave function has been made. The present results are superior to earlier CASSCF calculations, although their qualitative features remain essentially the same. The dipole moments in 1B1 and 3B1 states are predicted to be about 0.49 a.u. and 0.33 a.u., respectively, and have the opposite orientation with respect to the ground state dipole moment. The dipole polarizability tensors of the excited states are characterized by high anisotropy and are dominated by the in-plane component perpendicular to the symmetry axis. All their components are found to be about an order of magnitude larger than those of the ground state polarizability tensor. The excitation energy dependence on the choice of the active orbital space in the CASSCF reference function is also considered and the analysis of the present data concludes in the concept of what is called the mutually compatible active spaces for the two states involved in excitation. All CASPT2 results are in good agreement with the results of recent calculations carried out in the framework of the open-shell coupled cluster formalism. This agreement confirms the high efficiency of the CASSCF/CASPT2 approach to the treatment of the electron correlation effects.

Size-consistent approach and density analysis of hyperpolarizability: Second hyperpolarizabilities of polymeric systems with and without defects

Various size-consistent approaches to the calculation of molecular hyperpolarizabilities are analyzed based on the double perturbation theory. General equations for the nth-order response property with respect to an external time-independent field are derived on the basis of the Rayleigh–Schrödinger perturbation theory (RSPT) and the coupled-cluster (CC) theory. The corresponding equations for the time-dependent case are also derived by the CC formalism, which is referred to as the time-dependent CC (TDCC) method. In order to clarify the spatial characteristics of polarizability and hyperpolarizability, we present an analysis method using a new concept ‘‘the polarizability and hyperpolarizability densities.’’ As an application of the size-consistent methods, the static second hyperpolarizabilities (γ) of π-conjugated polymeric systems are calculated by the use of the uncoupled (UCHF), and coupled-Hartree–Fock (CHF) methods combined with the semiempirical INDO approximation. Characteristics of γ values calculated for regular polyenes, solitonlike polyenes and donor(D)–acceptor(A) disubstituted polyenes are investigated, particularly in relation to the chain-length effect. Further, we employ γ density analysis method, in which the third derivatives of the Mulliken charge densities against applied electric fields are plotted for exploring the local contributions of the constituent atoms to γ values. Results for the finite polyenes are also extrapolated to an infinity of the chain length to predict the intrinsic γ values per unit carbon–carbon (CC) bond of polymeric chains. Furthermore, the CHF+Mo/ller–Plesset second-order perturbation (MP2) method in the PPP approximation is applied to polymeric systems with larger chain length in order to elucidate the electron correlation effects on the chain-length dependencies of γ values.

Orthogonally spin-adapted single-reference coupled-cluster formalism: Linear response calculation of static properties

An orthogonally spin-adapted coupled-cluster (CC) linear response (LR) method, to calculate first-order static properties of closed-shell systems, is formulated and implemented. Cluster operators are truncated at single and double excitations (CCSD-LR approach) and the explicit first-order CCSD-LR equations, in terms of molecular integrals and zeroth- and first-order cluster amplitudes, are given. Various aspects of the computer implementation of the CCSD-LR approach are discussed. In particular, the efficiency of the CCSD-LR approach is compared with corresponding finite field calculations. Advantages of using the LR approach are illustrated by calculating electrostatic multipole moments for CH2 and HF at equilibrium and displaced geometries.

Displaced and squeezed parity operator: Its role in classical mappings of quantum theories.

The displaced parity operators are shown to have properties that bear a deep relationship with those of the Wigner functions. By exploiting these properties we show that these operators play an important role in linking together many of the aspects of the various exact phase-space mappings of quantum mechanics. These include the Wigner and Weyl representations, coherent states and the Bargmann representation, the P and Q representations, the Weyl correspondence, and the Moyal star product formalism. We also introduce corresponding displaced Fourier operators and show that their squares are just the displaced parity operators. The formalism is extended to squeezed and displaced parity operators, and their corresponding central role in the theory of squeezed coherent states and general squeezing is explained. We also elucidate the part played by the displaced parity operators in the Moyal star product and its extensions, as a first step towards a potential application of these operators in such modern developments as deformation theory and quantum groups. Finally, we indicate how the apparatus developed might also find applications in other recent exact classical mappings of many-particle quantum mechanics or quantum field theory, which are not special cases of deformation theory. Prime examples here include the powerful so-called independent-cluster method techniques, which incorporate the coupled-cluster method formalism with its inbuilt supercoherent states. Throughout the work we stress the central and unifying role played by the displaced parity operator and its generalizations.

Supermolecular approach to many-body dispersion interactions in weak van der Waals complexes: He, Ne, and Ar trimers

Using the diagrammatic many-body perturbation theory, various three-body dispersion terms that appear in the intermolecular Mo/ller–Plesset perturbation theory (MPPT) are identified and classified with regard to the effects of intramonomer electron correlation on the dispersion term. Via the connection with the supermolecular MPPT, it is demonstrated how the leading dispersion nonadditivities arise within supermolecular calculations that employ MPPT or coupled cluster formalisms. The numerical calculations for He3, Ne3, and Ar3 in triangular geometries fully confirm theoretical predictions. The calculated values of dispersion nonadditivity clearly show that the coupled cluster theory with single, double, and noniterative triple excitations provides the proper framework for the efficient inclusion of the intramonomer correlation effects in dispersion nonadditivity. The convergence of the two-body and three-body terms is shown to be very similar if we compare the three-body terms of an order higher than the two-body terms. This pattern is used to provide the estimates of the total nonadditivities in the three trimers within a few percent accuracy.

Orthogonally spin-adapted state-universal coupled-cluster formalism: Implementation of the complete two-reference theory including cubic and quartic coupling terms

The complete orthogonally spin-adapted Hilbert-space (or state-universal) coupled-cluster (CC) theory involving singly and doubly excited clusters for a model space spanned by two closed-shell configurations is considered. Explicit expressions for the previously neglected cubic and quartic coupling terms are derived using diagrammatic methods of many-body perturbation theory and graphical methods of spin algebras. The resulting formalism has been implemented and the general purpose ab initio program has been written using newly developed procedure for improving the convergence of the reduced linear equation method. Results of the pilot calculations for the two lowest singlet states of the minimum basis set and double zeta plus polarization basis H4 models as well as for the CH2 molecule at equilibrium and displaced geometries are presented and compared with the available single-reference CC and configuration interaction data. They indicate negligible role of the cubic and quartic coupling terms, which justifies the validity of approximations considered so far.

On the microscopic derivation of a mass formula

To analyze the steps leading to exact solutions of the Schrödinger equation, we proceed as if we were to do shell-model calculations in valence spaces defined by carefully specified doubly-magic cores. First we sketch the coupled-cluster formalism as a general framework. Then we separate the hamiltonian H = H m + H M (m = monopole, M = multipole or residual) by constructing explicitly H m, entirely responsible for self-consistency aspects. Next we construct renormalized hamiltonians H = H m + H M H m describes the core and single-particle energies. It also contains quadratic effects. The core energies are incorporated in a smooth function E (SLD)of liquid-drop (LD) form. Shell effects emerge as a residual part of H m plus contributions due to configuration mixing through H M. Deformed nuclei are treated phenomenologically, by invoking the extraordinary simplicity of their energy patterns, and elementary arguments about their structure. A general mass formula is presented. It is shown that a minimal version of its bring down the r.m.s. deviation (in a fit to N, Z > 20 nuclei) from 2.5–3 MeV for an LD form to 900 keV (800 keV) by the adjunction of only two (three) parameters. The work of Duflo (preceding paper) is examined and to a large extent vindicated. Study of the data provides firm evidence that the full specification of H m contains the clue to further progress.

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 the full CI (FCI) method. Along with the ground-state energies, we also analyze the resulting wave functions by examining some selected cluster components. This analysis enables us to assess the quality of the resulting wave functions. Our SS CC theory truncated at double excitations, which emerges through selection of the most essential clusters appearing in the full SR CCSDTQ formalism [SS CCSD (TQ) method] provides equally good results in nondegenerate and quasidegenerate regions. The difference between the ground-state energy obtained with the SS CCSD(TQ) approach and the FCI energy does not exceed 1.1 mhartree over all the geometries considered. This value compares favorably with the maximum difference of 2.8 mhartree between the SU CCSD energies and the FCI energies obtained for the same range of geometries. The SS CCSD(T) method, emerging from the SR CCSDT theory through selection of the most essential clusters, is less stable, since it neglects very important semi-internal quadruple excitations. Unlike the genuine multideterminantal SU CC formalism, our SS CC approach is not affected by the intruder state problem and its convergence remains satisfactory in nondegenerate and quasidegenerate regimes.

Approximate account of connected quadruply excited clusters in multi-reference Hilbert space coupled-cluster theory. Application to planar H 4 models

An approximate account of the connected quadruply excited cluster components that is achieved by considering only those pair cluster interaction diagrams which are separable over the hole lines is examined in the context of the multi-reference Hilbert space coupled-cluster formalism. The results for two simple model systems involving two interacting, slightly stretched hydrogen molecules are most encouraging.

Coupled-cluster method for quantum lattices: Application to square S=1/2 Heisenberg antiferromagnets.

A coupled-cluster formalism similar to that used in electronic-structure theory is developed for quantum-mechanical systems localized to sites on a lattice. It is shown how diagrammatic methods may be used to represent the cluster contributions in a convenient and systematic manner. The formalism is applied to the square S=1/2 Heisenberg antiferromagnet, yielding with minor computational effort results of quality comparable to those from computationally intensive quantum Monte Carlo calculations.

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.

Analysis of atomic electric dipole moment in thallium by all-order calculations in many-body perturbation theory.

A procedure is presented for calculations of double-perturbation problems in the coupled-cluster formalism. With use of this method to include to all orders the correlation effects of single and pair excitations, the enhancement factor R=-585 is obtained for the electric dipole moment (EDM) in atomic thallium. Equations for EDM perturbed single-excitation amplitudes are solved by diagonalization to take into account mixing between excited core states, and perturbed pair functions are introduced to incorporate correlation effects. Considering contributions from higher-order perturbation terms, we estimate the accuracy of the obtained enhancement factor is approximately at the 5--10 % level.

Stochastic averaging in quantum dynamics driven by colored noise: a nonperturbative cluster cumulant approach

We present here a nonperturbative cluster cumulant method for generating the stochastically averaged evolution operator U for quantum systems driven by Ornstein—Uhlenbeck colored noise (real and complex). The method induces a boson mapping of the stochastic variable, and interprets the averaging as an expectation value with respect to the boson vacuum |0 B〉. The evolution of the mapped operator U B, written in a factorized cluster expansion, is governed by a deterministic Hamiltonian H B defined in an expanded Fock space. The apparatus of our recently developed multireference coupled cluster formalism is invoked to solve the resultant cluster cumulant equations. Numerical applications are presented to illustrate the formalism.