Complete Model Space Research Articles

Method of moments for the continuous transition between the Brillouin–Wigner-type and Rayleigh–Schrödinger-type multireference coupled cluster theories†

We apply the method of moments to the multireference (MR) coupled cluster (CC) formalism representing the continuous transition between the Brillouin–Wigner-type and Rayleigh–Schrödinger-type theories based on the Jeziorski–Monkhorst wave function ansatz and derive the formula for the noniterative energy corrections to the corresponding MRCC energies that recover the exact, full configuration interaction energies in the general model space case, including complete and incomplete model spaces. We also extend the relationship between the generalised moments of the state-universal (SU) MRCC equations within the Jeziorski–Monkhorst and Kucharski–Bartlett formulations of the SUMRCC theory to the general model space case. Finally, we argue that in the complete model space case, the relationship between moments of the SUMRCC equations corresponding to the Jeziorski–Monkhorst and Kucharski–Bartlett formulations of the SUMRCC theory, derived in this work, implies an equivalence of these two formulations of the SUMRCC approach, provided that the disconnected linked terms are included in the Kucharski–Bartlett formulation, and verify this statement numerically.

Third‐order generalized Van Vleck perturbation theory variant of multireference perturbation theory for electron correlation

AbstractA third‐order approximation to the generalized Van Vleck perturbation theory (GVVPT3) variant of multireference perturbation theory treatment of dynamic electron correlation is described. The approximation builds upon some aspects of the increasingly widely used GVVPT2, which we introduced previously (Khait et al., J Chem Phys 2002, 117, 4133). In addition, using formal relations obtained in connection with GVVPT3, a more thorough understanding of GVVPT2 is obtained, including a key diagnostic that assesses suitability of GVVPT2 for specific tasks. Computational aspects of the implementation of GVVPT3, especially the use of a macroconfiguration description of the model space, are described. The obtained GVVPT3 method can use both complete and incomplete model spaces efficiently. We demonstrate the new method on a series of model problems that demonstrate the applicability of the suggested approach. Among other tests, we show that GVVPT3 is capable of reproducing MRCISD results to within 0.01 eV for the problematic barrier height of the metastable N. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

An exponential multi-reference wavefunction ansatz: connectivity analysis and application to N2

A connectivity analysis for the exponential multi-reference wavefunction ansatz (MRexpT) (J Chem Phys 123:84102, 2005) is carried out. Assuming a complete model space and separating interactions carrying active labels the cluster operator carrying no active labels is fully connected. The valence (active) part of the MRexpT cluster operator, however remains disconnected. Consequently, the MRexpT correlation energy scales linearly with the number of non-active electrons as single reference coupled cluster does while MRexpT additionally can treat multi reference cases. Therefore, MRexpT should be well suited to be applied to a large number of molecular applications. Its applicability to periodic systems with multi-reference unit cells however seems to be limited. An application to the triple bond breaking of the N2 molecule is presented.

Explicitly intruder-free valence-universal multireference coupled cluster theory as applied to ionization spectroscopy

Although it is quite promising to compute the spectroscopic energies [say, ionization potential (IP)] via the traditional valence-universal multireference coupled cluster (VUMRCC) method based on the description of the complete model space being seriously plagued by the perennial intruder state problem, the eigenvalue independent partitioning (EIP) based VUMRCC (coined as EIP-MRCC) method is quite effective to predict the spectroscopic energies in an intruder-free manner. Hence, the EIP-MRCC method is suitable for generating both the principal IPs and the satellite IPs of the inner-valence region. An EIP strategy converts the nonlinear VUMRCC equations for M(m,n) dimensional model space of m hole and n particle to a non-Hermitian eigenproblem of larger dimension whose M(m,n) roots are only physically meaningful. To increase the quality of the computed energy differences in the sense of chemical accuracy and to locate the correct position of it in the spectrum, the inclusion of higher-body cluster operators on top of all the standard singles-doubles is not the only pivotal issue, the effect of the size of the basis set is also equally important. This paper illustrates these issues by calculating the principal and satellite IPs of HF and HCl molecules using various basis sets (viz., Dunning's cc-pVDZ, cc-pVTZ, and cc-pVQZ) via EIP-MRCC method with full inclusion of triples (abbreviated as EIP-MRCCSDT). The results seem quite encouraging in comparison with the experimental values. The controversial 2Pi satellite at 28.67 eV of HCl of Svensson et al. [J. Chem. Phys. 89, 7193 (1988)] is also reported.

Multi-reference Brillouin–Wigner coupled-cluster method with a general model space

We describe a multi-reference Brillouin–Wigner coupled-cluster (BWCC) method that employs a general model space (GMS) by relying on the recently introduced C-conditions [X. Li, J. Paldus. J. chem. Phys., 119, 5320 (2003).] This generalization preserves all the essential features of the standard complete model space (CMS) BWCC method. This makes it possible to substantially reduce the dimension of the model space employed by keeping only important configurations and thus to minimize the computational cost. We also present the results of a preliminary numerical test using the cc-pVDZ model of the LiH molecule.

A constrained variational approach for energy derivatives in Fock-space multireference coupled-cluster theory.

In this paper, we present a formulation based on constrained variational approach to enable efficient computation of energy derivatives using Fock-space multireference coupled-cluster theory. Adopting conventional normal ordered exponential with Bloch projection approach, we present a method of deriving equations when general incomplete model spaces are used. Essential simplifications arise when effective Hamiltonian definition becomes explicit as in the case of complete model spaces or some special quasicomplete model spaces. We apply the method to derive explicit generic expressions upto third-order energy derivatives for [0,1], [1,0], and [1,1] Fock-space sectors. Specific diagrammatic expressions for zeroth-order Lagrange multiplier equations for [0,1], [1,0], and [1,1] sectors are presented.

A general state-selective multireference coupled-cluster algorithm

A state-selective multireference coupled-cluster algorithm is presented which is capable of describing single, double (or higher) excitations from an arbitrary complete model space. One of the active space determinants is chosen as a formal Fermi-vacuum and single, double (or higher) excitations from the other reference functions are considered as higher excitations from this determinant as it has been previously proposed by Oliphant and Adamowicz [J. Chem. Phys. 94, 1229 (1991)]. Coupled-cluster equations are generated in terms of antisymmetrized diagrams and restrictions are imposed on these diagrams to eliminate those cluster amplitudes which carry undesirable number of inactive indices. The corresponding algebraic expressions are factorized and contractions between cluster amplitudes and intermediates are evaluated by our recent string-based algorithm [J. Chem. Phys. 115, 2945 (2001)]. The method can be easily modified to solve multireference configuration interaction problems. Performance of the method is demonstrated by several test calculations on systems which require a multireference description. The problem related to the choice of the Fermi-vacuum has also been investigated.

Application of the intermediate Hamiltonian valence-universal coupled-cluster method to the magnesium atom

The atomically oriented valence-universal coupled-clusters method in the intermediate Hamiltonian formulation with one- and two-body part(s) included in the cluster operator (VU-CCSD/R) is applied to describe the ground state and some excited states of the Mg atom. Two Slater-type orbitals (STO) basis sets are employed to see the effect of the basis set on the results. In addition to that, several complete model spaces are used in the calculation starting with (3s,3p) and ending with (3s,3p,3d,4s,4p) orbitals as active. It has been found that the quality of the description of the ground- and excited states depends on the STO basis set used in the calculation. In order to obtain reliable excitation energies, special attention has been paid to the construction of well-balanced basis sets in which functions necessary for adequate description of both the ground state and excited states under consideration are included. Therefore, in spite of the moderate size of the orbital basis, our results compare quite well in terms of the total energy as well as in energy differences with results of other methods which, in some cases, use significantly larger basis sets.

Potential energy surface and ro‐vibrational energies of Ne in the ground electronic state

AbstractA global potential energy surface of Ne in its ground electronic state is evaluated using a Rayleigh–Schrödinger version of the third‐order multireference many‐body perturbation theory (MR‐MBPT) in a complete model space. The ab initio data are fitted quantitatively to a model potential energy function expressed in the Jacobi coordinates, and the fitted function is used to evaluate the ro‐vibrational energies of Ne using the (bond length–bond angle) Sutcliffe–Tennyson (ST) and (hyperspherical) Yamashita–Morokuma–Leforestier (YML) approaches. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Four-Reference State-Specific Brillouin-Wigner Coupled-Cluster Method: Study of the IBr Molecule

We implemented the state-specific Brillouin–Wigner coupled-cluster method for the complete model space spanned by four reference configurations generated by two electrons in two active orbitals. We applied the method (together with the previously suggested a posteriori size-extensivity correction) to the calculation of spectroscopic constants of the IBr molecule, using averaged relativistic effective core potential.

Lifetimes of superdeformed rotational states in36Ar

Lifetimes have been measured in a superdeformed rotational band recently identified in the N=Z nucleus {sup 36}Ar. A large low-spin quadrupole deformation ({beta}{sub 2}=0.46{+-}0.03) is confirmed and a decrease in the collectivity is observed as the high-spin band termination at I{sup {pi}}=16{sup +} is approached. Detailed comparisons of the experimental B(E2) values with the results of cranked Nilsson-Strutinsky and large-scale (s{sub 1/2}d{sub 3/2})-pf spherical shell model calculations indicate the need for a more refined treatment of transition matrix elements close to termination in the former, and the inclusion of the complete sd-pf model space in the latter description of this highly-collective rotational band.

Spin-adapted multipartitioning perturbation theory

The multireference generalization of the Mller-Plesset perturbation theory employing multiple partitionings of the total Hamiltonian is revised in order to avoid spin contaminations and non-physical dependence of the results on the z projection of the total spin. The novel formulation retains the main advantages of the original approach (computational simplicity, numerical stability, uncontracted treatment of the reference-space part of wavefunctions, size consistency of second-order results for complete model spaces). The results of pilot calculations on the transitions between electronic states with different spin multiplicities in CH2, SiH2, ScH, Sc2 and Cu2 are reported.

On the origin of size inconsistency of the second-order state-specific effective Hamiltonian method

The deviations of the second-order state-specific effective Hamiltonian method from the strict size consistency are analyzed. Provided that complete or separable model spaces are used, these deviations can be suppressed by a proper choice of nonuniform shifts of energy denominators.

Temperature dependence of pair correlations in nuclei in the iron region

We use the shell-model Monte Carlo approach to study thermal properties and pair correlations in 54,56,58Fe and in 56Cr. The calculations are performed with the modified Kuo-Brown interaction in the complete 1p0f model space. We find generally that the proton-proton and neutron-neutron J = 0 pairing correlations, which dominate the ground-state properties of even-even nuclei, vanish at temperatures around 1 MeV. This pairing phase transition is accompanied by a rapid increase in the moment of inertia and a partial unquenching of the M1 strength. We find that the M1 strength totally unquenches at higher temperatures, related to the vanishing of isoscalar proton-neutron correlations, which persist to higher temperatures than the pairing between like nucleons. The Gamow-Teller strength is also correlated to the isoscalar proton-neutron pairing and hence also unquenches at a temperature larger than that of the pairing phase transition.

Application of the valence‐universal coupled‐cluster method based on various model spaces to 1S states of Be

AbstractThe recently formulated atomically oriented valence‐universal theory (VU–CCSD/R) based on various complete model spaces is applied to the accurate description of the 2s2 1S, 2s3s 1S, 2p2 1S, and 3s2 1S states of Be. Two extensive 9s9p7d7f5g STO basis sets that differ by one s‐type function are employed. Both standard and nonstandard solutions of the VU–CC equations are considered. The eight values of the ground‐state energies obtained when considering various model spaces and basis sets were found to differ very little. The energies of the individual excited states disclosed a relatively weak dependence on the choice of model spaces defining the method (or, equivalently, on the fact whether they correspond to standard or nonstandard solutions). In turn, the energies of the excited states strongly depend on the STO basis set used in the calculations. To obtain a reliable description of both the ground state and excited states, well‐balanced STO basis sets have to be employed. © 1995 John Wiley & Sons, Inc.

Impact of the choice of model spaces and basis sets on the performance of the valence-universal coupled-cluster method: Energies for Be and C2+

To study the impact of the choices of complete model spaces (CMS's) and of orbital basis sets on the performance of valence-universal (VU) coupled-cluster (CC) methods, extensive calculations have been performed for Be and ${\mathrm{C}}^{2+}$ by means of the atomically oriented form of these methods accounting for one(S)- and two(D)-electron radial (R) excitations (VU-CCSD/R method). In addition to the commonly used complete model space defined by the orbital set (2s,2p) we have employed the CMS's defined by the (2s,3s) and (2s,2p,3s) sets. Despite the anticipated increase of intruder state difficulties, for the new CMS's, no problems have been encountered when solving the nonlinear VU-CCSD/R equations. The results for the total energies and energy differences disclosed a weak dependence on the CMS chosen. We have used four Slater-type orbitals basis sets partially balanced for an accurate description of the ground and excited states. It turned out that the energies of excited states strongly depend on the basis set used, which emphasizes the importance of developing methods for constructing well balanced basis sets. It has been demonstrated that the VU-CCSD/R method can be used for the accurate localization of the position of the 2${\mathit{p}}^{2}$ $^{1}$S resonance with respect to the ionization limit. Some attention is also paid to a singular behavior of the energy obtained with the linearized version of the VU-CCSD/R method.

Multi-partitioning quasidegenerate perturbation theory. A new approach to multireference Møller-Plesset perturbation theory

We propose a new approach to the construction of perturbative expansions for effective Hamiltonians, implying the use of multiple partitioning of the total Hamiltonian. This approach offers the possibility to formulate the state-universal multireference many-body perturbation theory retaining the most important features of conventional Møller-Plesset expansions. Restricted to the second order, the novel theory gives rise to a simple method of electronic structure calculations which combines a decontracted treatment of model-space components of wavefunctions with the strict size-consistency for complete model spaces and the ability to avoid, or at least attenuate, the intruder-state problem.

A valence-universal coupled-cluster single- and double-excitations method for atoms. III. Solvability problems in the presence of intruder states

For pt. II see ibid., vol. 27, p. 829-42 (1994). To better understand the problems met when solving the equations of VU-CC approaches in the presence of intruder states, the authors are concerned with the following aspects of the solvability problem for sets of non-linear equations: the existence and properties of multiple solutions and the attainability of these solutions by means of various numerical methods. The study is concentrated on the equations obtained for Be within the framework of the recently formulated atomically oriented form of the valence-universal coupled-cluster theory accounting for one- and two-electron excitations (VU-CCSD/R) and based on the complete model space (2s2, 2p2). Six pairs of multiple solutions representing four 1S states are found and discussed. Three of these solutions provide amplitudes describing the 2p2 1S state for which the intruder state problem has been considered as extremely serious. Several known numerical methods have been applied to solve the same set of non-linear equations for the two-valence cluster amplitudes. It is shown that these methods perform quite differently in the presence of intruder states, which seems to indicate that the intruder state problem for VU-CC methods is partly caused by the commonly used methods of solving the non-linear equations.

Perturbative and complete model space linked diagrammatic expansions for the canonical effective operator

Effective Hamiltonians and effective operators produce, respectively, exact energies and matrix elements of a time-independent operator A for a finite number of eigenstates of a time-independent Hamiltonian H. We obtain degenerate and quasidegenerate perturbative expressions for the particularly useful canonical effective operator ÂC through second order in perturbation theory. The corresponding ÂC diagrammatic expressions are derived for the case where ÂC acts in a complete finite space. Our first order results have been used previously for ab initio computations of dipole and transition dipole moments in diatomic hydrides and for testing the assumptions in semiempirical methods for dipole properties. A brief discussion is also provided on the computational labors required by first and second order ÂC many-body calculations, the derivation of ÂC diagrams when ÂC acts in an incomplete finite space, and on the derivation of diagrammatic rules for ÂC in arbitrary perturbation order.

A valence-universal coupled-duster single- and double-excitations method for atoms. II. Application to Be

For pt. I see Malinowski and Jankowski, ibid., vol. 26, p. 3035 (1993). The recently formulated atomically oriented form of the valence-universal coupled-cluster theory accounting for one- and two-electron excitations (VU-CCSD/R) is applied to several states or Be. The complete model space (CMS) is employed. Various approximate forms of this method including the lowest-order perturbation theory are also considered. The standard single-reference (SR) formulation of the CCSD, method, treated as a special case of the VU-CCSD/R approach, is applied to the ground state. The calculations for the 9s9p7d7f5g STO basis set are the most accurate ones performed so far for any SR- and VU-CCSD method formulated in terms of analytical orbitals. Proceeding from the SR- to the VU-formulation increased the magnitude of the ground-state correlation energy by 1.1%. Special attention is paid to the well known intruder state problem for the pair of 2s2 1S and 2p2 1S states. It is demonstrated that, despite earlier findings, the CMS based VU-CCSD, methods can be safely used in the presence of intruder states. The impact of the electronic structure of the series of systems considered in VU-CCSD calculations on the accuracy of results is also addressed.