A method of incorporating quadruple correction in the scheme of multi-reference singly and doubly excited configuration interaction — a CSF based coupled pair approximation

Summary We develop an approximate size consistent method within a framework of the multi-reference configuration interaction scheme. The Rayleigh-Schrödinger perturbation theory is employed with a specific selection of the unperturbed part of the electronic Hamiltonian. The second order energy is obtained by a set of equations similar to the quasidegenerate variational perturbation theory of Cave and Davidson. The approximate fourth order energy is obtained by solving a set of equations similar to the coupled electron pair approximation (CEPA). The method has been tested for two simple systems, BeH₂ and N₂, and the results are quite encouraging.     

Approximate inclusion of four-mode couplings in vibrational coupled-cluster theory

The vibrational coupled cluster (VCC) equations are analyzed in terms of vibrational Mo?ller-Plesset perturbation theory aiming specifically at the importance of four-mode couplings. Based on this analysis, new VCC methods are derived for the calculation of anharmonic vibrational energies and vibrational spectra using vibrational coupled cluster response theory. It is shown how the effect of four-mode coupling and excitations can be efficiently and accurately described using approximations for their inclusion. Two closely related approaches are suggested. The computational scaling of the so-called VCC[3pt4F] method is not higher than the fifth power in the number of vibrational degrees of freedom when up to four-mode coupling terms are present in the Hamiltonian and only fourth order when only up to three-mode couplings are present. With a further approximation, one obtains the VCC[3pt4] model which is shown to scale with at most the fourth power in the number of vibrational degrees of freedom for Hamiltonians with both three- and four-mode coupling levels, while sharing the most important characteristics with VCC[3pt4F]. Sample calculations reported for selected tetra-atomic molecules as well as the larger dioxirane and ethylene oxide molecules support that the new models are accurate and useful.     

Electron correlation theories and their application to the study of simple reaction potential surfaces

This paper serves two purposes. The first is to describe an implementation of the coupled cluster theory with double substitutions (CCD ) previously developed by Cizek. The second is to apply this method and closely related fourth-order perturbation methods to some simple molecules and reaction potential surfaces. These studies show that CCD theory gives results close to those of a Møller-Plesset perturbation treatment to fourth order in the space of double and quadruple substitutions MP 4(DQ). Addition of contributions from single substitutions at fourth order makes little change in predicted relative energies. Preliminary results on the potential surfaces for 1,2-hydrogen shifts in C₂H₂, HCN, CH₂O, and N₂H₂ are discussed and compared with previous studies.     

CCSD(T) expectation value calculations of first-order properties

An expectation value approach to calculations of first-order properties using the non-iterative, triple-excitation amplitudes in the coupled cluster wave function is exploited. Three methods are suggested and analysed using the many body perturbation theory (MBPT) expansion arguments. The first method, in which non-iterative triple-excitation amplitudes are used in the expression for the expectation values, makes the wave function accurate through the second order of MBPT. In the second method, which is an extension of the first, effects of triple-excitation amplitudes are coupled with single- and double-excitation amplitudes. The correlated density matrix equivalent through the fourth order to that obtained when CCSDT-la amplitudes are used is employed in the third method. The suggested methods are tested on dipole moment and polarizability calculations for several diatomic closed-shell molecules and are compared to other related approaches. Received: 15 May 1997 / Accepted: 5 June 1997     

New perspectives in multireference perturbation theory: the n-electron valence state approach

The n-electron valence state perturbation theory (NEVPT) is a form of multireference perturbation theory which is based on a zero order reference wavefunction of CAS-CI type (complete active space configuration interaction) and which is characterized by the utilization of correction functions (zero order wavefunctions external to the CAS) of multireference nature, obtained through the diagonalization of a suitable two-electron model Hamiltonian (Dyall’s Hamiltonian) in some well defined determinant spaces. A review of the NEVPT approach is presented, starting from the original second order state-specific formulation, going through the quasidegenerate multi-state extension and arriving at the recent implementations of the third order in the energy and of the internally contracted configuration interaction. The chief properties of NEVPT—size consistence and absence of intruder states—are analyzed. Finally, an application concerning the calculation of the vertical spectrum of the biologically important free base porphin molecule, is presented.     

Information theoretic indices for characterization of chemical structures. By Danail Bonchev Wiley,New York, 1983

Wilson, Jankowski, and Paldus have recently applied nondegenerate many-body perturbation theory (MBPT ) to simple models, in which the degree of quasidegeneracy could be varied continuously, and concluded that the nondegenerate theory was applicable even near degeneracy. The error in their results changes, however, considerably with geometry, leading to an incorrect potential surface. An extension of their calculations shows convergence even at exact degeneracy (square planar H₄). It is shown here that the apparently good convergence is due to the suppression of the large (infinite at exact degeneracy) component of the perturbation energy in low order by the way the Hamiltonian is partitioned. This component will, however, resurface at higher orders, leading to slow convergence or even divergence. The low-order sum of the perturbation series is not very meaningful, depends strongly on details of the zero-order Hamiltonian, and yields, in general, incorrect potential surfaces. Multireference MBPT eliminates these problems.     

A coupled-cluster method for quasidegenerate states

A size-extensive, multireferences coupled-cluster method for studies of quasidegenerate states based on the Jeziorski–Monkhorst [16] ansatz for the cluster operator (Ω = ∑eP_j, where the sum is extended over the configurations spanning the model space), is presented and applied in pilot calculations. The method is referred to as multireference coupled electron-pair method (MR CEPM ), because it is assumed that the individual cluster operators can be approximated by their two-body parts, i.e., T_j ≈ T_j(2). The linear version of this method (MR L-CEPM ) is also discussed. Both methods are applied to two simple model systems: (1) a minimum basis set model involving eight hydrogen atoms in various spacial arrangements for which the degree of quasidegeneracy can be continuously varied; (2) a model involving the C_2ν insertion of Be into H₂. For the first time in multireference coupled-cluster calculations, the nonlinear parts of the equations are completely accounted for. The MR CEPM results are very encouraging for strongly quasidegenerate states. The MR L -CEPM results are slightly below the accurate (FCI ) values.     

Influence of molecular geometry on valence space for quasidegenerate many-body perturbation theory

The Be-H₂ insertion reaction is used as a model to study the application of quasidegenerate many-body perturbation theory (QDMBPT) to polyatomic molecules where the pattern of quasidegenerate orbitals varies greatly with geometry. Full active valence space QDMBPT calculations are compared with the exact solutions within the basis and with previous QDMBPT computations that retain only a pair of quasidegenerate valence orbitals.     

Simple recipe for implementing computation of first-order relativistic corrections to electron correlation energies in framework of direct perturbation theory

The computation of the relativistic correction to the first order in 1/c², where c is the velocity of light, is implemented at the levels of coupled cluster and many-body perturbation theory. The relativistic correction is obtained by applying direct perturbation theory through the first order, and it is shown that its implementation is straightforward if analytical energy gradients of the methods under consideration are available. Preliminary results were obtained by a numerical procedure and are reported for some closed-shell atoms (He, Be, Ne, and Ar) and molecules (CuH and SiH₄). © 1997 by John Wiley & Sons, Inc.     

Nonstationary perturbation theory for open-shell molecules

Nonstationary perturbation theory equations have been obtained for open-shell molecules. The equations were formulated in terms of a density matrix in the MO-LCAO method. The first variant is coupled perturbation theory in the framework of the restricted Hartree-Fock method for open shells, and the second variant is variational perturbation theory for ground and excited electronic states of molecules, in which the perturbed wave function of the system is constructed in the form of a superposition of the ground and singly excited configurations composed of the Hartree-Fock orbitals of the open shell. A calculation of the Cauchy moments of the dynamic dipole polarizability of several molecules of conjugated open-shell hydrocarbons, viz., doublet states of odd alternant hydrocarbons, as well as triplet excited states and doublet states of radical ions of even alternant hydrocarbons, has been carried out in the framework of both methods.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 21, No. 1, pp. 18–27, January–February, 1985.     

Applicability of non-degenerate many-body perturbation theory to quasidegenerate electronic states: A model study

The nondegenerate finite-order many-body perturbation theory is applied to simple model systems in which the degree of quasidegeneracy can be continuously varied over a wide range. Three ab initio minimum basis set models involving four hydrogen atoms in various spatial arrangements are considered. The results are compared with the exact full configuration-interaction approach, double-excitation configuration-interaction and the coupled-pair many-electron theory.     

Unitary transformation on the model space. Comparison of Brandow's and Kirtman's versions of quasidegenerate many body perturbation theory

In this paper we study the unitary transformations on the model space in quasidegenerate many body perturbation theory (QDMBPT ). From this point of view we compare Brandow's and Kirtman's versions of QDMBPT up to the fifth order of perturbation expansion. We show that, starting from Brandow's QDMBPT , we can derive various versions of effective Hamiltonians order by order, of which the Kirtman's version is the simplest one. The operator of unitary transformation we express through the correlation operator.     

A constant denominator perturbation theory for molecular energy

A constant denominator perturbation theory is developed based on a zeroth order Hamiltonian characterized by degenerate subsets of orbitals. Such a formulation allows for a decoupling of the numerators of the perturbation sequence, allowing for much more rapid evaluation of the resultant sums. For example, the full fourth order theory can be evaluated as an N ⁶ step rather than N ⁷, where N is proportional to the basis set.Although the theory is general, a constant denominator is chosen for this study as the difference between the average occupied and average virtual orbital energies scaled so that the first order wavefunction yields the lowest possible variational bound. The third order correction then appears naturally as a scaled Langhoff-Davidson correction. The full fourth order with this partitioning is developed. Results are presented within the localized bond model utilizing both the Pariser-Parr-Pople and CNDO/2 model Hamiltonians. The second order theory presents a useful bound, usually containing a good deal of the basis set correlation. In all cases examined the fourth order theory shows remarkable stability, even in those cases in which the Nesbet-Epstein partitioning seems poorly convergent, and the Moller-Plesset theory uncertain.     

Potential energy curve for isomerization of N2H2 and C2H4 using the improved virtual orbital multireference Møller-Plesset perturbation theory

Multireference M?ller-Plesset (MRMP) perturbation theory [K. Hirao, Chem. Phys. Lett. 190, 374 (1992)] is modified to use improved virtual orbitals (IVOs) and is applied to study ground state potential energy curves for isomerization and dissociation of the N2H2 and C2H4 molecules. In contrast to traditional MRMP or multistate multiconfiguration quasidegenerate perturbation theory where the reference functions are obtained from (often difficult to converge) state averaged multiconfiguration self-consistent field methods, our reference functions are represented in terms of computationally efficient IVOs. For convenience in comparisons with other methods, a first order complete active space configuration interaction (CASCI) calculation with the IVOs is followed by the use of the IVOs in MRMP to incorporate residual electron correlation effects. The potential energy curves calculated from the IVO-MRMP method are compared with computations using state-of-the-art coupled cluster singles and doubles (CCSD) methods and variants thereof to assess the efficacy of the IVO-MRMP scheme. The present study clearly demonstrates that unlike the CCSD and its variants, the IVO-MRMP approach provides smooth and reliable ground state potential energy curves for isomerization of these systems. Although the rigorously size-extensive completely renormalized CC theory with noniterative triples corrections (CR-CC(2,3)) likewise provides relatively smooth curves, the CR-CC(2,3) calculations overestimate the cis-trans barrier height for N2H2. The ground state spectroscopic constants predicted by the IVO-CASCI method agree well with experiment and with other highly correlated ab initio methods.     

Quasidegenerate coupled-cluster approach with hermitian model hamiltonian

The quasidegenerate coupled-cluster approach with a hermitian model Hamiltonian is constructed. This theory provides an efficient method for simultaneous construction and evaluation of the hermitian diagrammatic model Hamiltonian up to an arbitrary high order of perturbation theory and prescribed form of topology.     

Valence bond corrected single reference coupled cluster approach

Summary 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.     

Exploiting regularity in systematic sequences of wavefunctions which approach the full CI limit

Summary The ground state total energy and related 1-electron properties are computed for three small molecules (N₂, H₂O, and H₂CN) using several systematic sequences of wavefunctions which approach the full CI. These sequences include multireference CI, averaged coupled pair functional and quasidegenerate variational perturbation theory wavefunctions. It is demonstrated that sufficient regularity exists in the sequence of variationally computed energies to permit extrapolation to the full CI limit using simple analytic expressions. It is furthermore demonstrated that a subset of the original list of configurations employed in the normal singles and doubles CI procedure can be selected using second order perturbation theory without adversely affecting the extrapolation to the full CI limit. This significantly broadens the range of applicability of the method. Along these lines, a scheme is proposed for the extrapolation of the selected CI results to the zero threshold (i.e. unselected) values in cases where the numbers of configurations associated with the latter would render the calculations intractable. Due to the vast reduction in the number of configurations which are handled variationally, the proposed scheme makes it possible to derive estimates of the full CI limit in cases where explicit full CI is either very difficult or currently impossible.Dedicated to Prof. Klaus RuedenbergThe Pacific Northwest Laboratory is operated for the U.S. Department of Energy by Battelle Memorial Institute under contract DE-AC06-76RLO 1830     

Structure, force fields, and vibrational spectra of cerium tetrahalides

The geometrical structure, force fields, and vibrational spectra of CeX₄ (X = F, Cl, Br, I) were investigated by second, third, and fourth order Möller-Plesset perturbation theory, CISD+Q configuration interaction method, and the CCSD(T) coupled cluster method. Calculations on CeF₄ were also performed by multiconfiguration second order perturbation theory MCQDPT2/CASSCF. The wave function of the ground state of CeX₄ molecules was found to be appreciably non-one-configurational; this property increases from cerium fluorides to iodides and leads to the divergence of the series of Möller-Plesset perturbation theory. The calculated data point to a tetrahedral equilibrium nuclear configuration in CeX₄ molecules. The energy barriers to the inversion of the tetrahedral CeX₄ molecules via the square configurations are high enough, 74–89 kJ/mol. The calculated vibration frequencies, effective internuclear distances, and mean amplitudes of nuclear vibrations in CeF₄ agree with IR and Raman spectroscopic and high-temperature gas-phase electron diffraction data.     

Photophysics of Schiff Bases: Theoretical Study of Salicylidene Methylamine

The proton‐transfer reaction in a model aromatic Schiff base, salicylidene methylamine (SMA), in the ground and in the lowest electronically‐excited singlet states, is theoretically analyzed with the aid of second‐order approximate coupled‐cluster model CC2, time‐dependent density functional theory (TD‐DFT) using the Becke, three‐parameter Lee–Yang–Parr (B3LYP) functional, and complete active space perturbation theory CASPT2 electronic structure methods. Computed vertical‐absorption spectra for the stable ground‐state isomers of SMA fully confirm the photochromism of SMA. The potential‐energy profiles of the ground and the lowest excited singlet state are calculated and four photophysically relevant isomeric forms of SMA; α, β, γ, and δ are discussed. The calculations indicate two S₁/S₀ conical intersections which provide non‐adiabatic gates for a radiationless decay to the ground state. The photophysical scheme which emerges from the theoretical study is related to recent experimental results obtained for SMA and its derivatives in the low‐temperature argon matrices (J. Grzegorzek, A. Filarowski, Z. Mielke, Phys. Chem. Chem. Phys. 2011 , 13, 16596–16605). Our results suggest that aromatic Schiff bases are potential candidates for optically driven molecular switches.