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

Summary 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 theT ₃ andT ₄ 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 forT ₄ terms is critically analyzed by including allT ₃ contributions.     

Multiple solutions of coupled-cluster equations for PPP model of [10]annulene

Multiple (real) solutions of the coupled-cluster (CC) equations (corresponding to the CCD, ACP and ACPQ methods) are studied for the Pariser–Parr–Pople (PPP) model of [10]annulene, C₁₀H₁₀. The long-range electrostatic interactions are represented either by the Mataga–Nishimoto potential, or Pople’s R⁻¹ potential. The multiple solutions are obtained in a quasi-random manner, by generating a pool of starting amplitudes and applying a standard CC iterative procedure combined with Pulay’s DIIS method. Several unexpected features of these solutions are uncovered, including the switching between two CCD solutions when moving between the weakly and strongly correlated regime of the PPP model with Pople’s potential.     

A Hilbert space multi-reference coupled-cluster study of the H4 model system

Summary Employing the Hilbert space ansatz, a fully quadratic coupled-cluster method with a multidimensional reference space is applied to a DZP basis study of the model system, H₄. The reference space is described by two to four configurations at the level of single and double excitations, and single and double excitation operators are included in the expansions for the cluster and wave operator through quadratic terms. The performance of quadratic MRCCSD is investigated for the ground and three excited states of the H₄ system consisting of two stretched hydrogen molecules in a trapezoidal configuration where the degree of quasidegeneracy is varied from a nondegenerate situation to a completely degenerate one. Compared to full CI, in the highly degenerate region, the MRCCSD works quite well. In less degenerate regions, the accuracy is less satisfactory.     

The ground-state potential energy function of a beryllium dimer determined using the single-reference coupled-cluster approach

The accurate ground-state potential energy function of the beryllium dimer, Be(2), has been determined from large-scale ab initio calculations using the single-reference coupled-cluster approach in conjunction with the correlation-consistent core-valence basis sets up to septuple-zeta quality. Results obtained with the conventional and explicitly-correlated coupled-cluster methods were compared. The scalar relativistic and adiabatic (the diagonal correction) effects were also discussed. The vibration-rotation energy levels of Be(2) were predicted and found to be as accurate as those determined from the empirical potential energy function [J. M. Merritt et al., Science, 2009, 324, 1548]. The potential energy function of Be(2) was determined in this study to have a minimum at 2.444 ? and the well depth of 935 cm(-1).     

Application of the valence-universal coupled-cluster method based on various model spaces. II. Nonstandard solutions for Be

The impact of the choices of the complete model space (CMS) and of the orbital basis set on the existence, attainability, and properties of the nonstandard solutions of the valence-universal coupled-cluster (VU-CC) methods has been studied for the case of nonlinear equations corresponding to the atomically oriented form of these methods accounting for one- and two-electron excitations (VU-CCSD/R method) and applied to the Be atom. The results for five ¹S states are discussed. In addition to the previously applied CMS defined by the orbital set (2s, 2p) and (2s, 3s), we have employed the CMSs defined by the (2s, 2p, 3s) set. For each of the CMSs several nonstandard solutions are documented. It is found that the energies of the individual states corresponding to standard and nonstandard solutions differ very little. These energies are also almost independent on the choice of CMS. On the other hand, the energies of excited states disclose a strong dependence on the radial structure of the orbital basis set. It is also demonstrated that the magnitudes of the cluster amplitudes representing a set of states depend both on the choice of the CMS and whether they correspond to the standard or nonstandard solutions. © 1996 John Wiley & Sons, Inc.     

The barrier height of the F+H2 reaction revisited: coupled-cluster and multireference configuration-interaction benchmark calculations

Large scale coupled-cluster benchmark calculations have been carried out to determine the barrier height of the F+H2 reaction as accurately as possible. The best estimates for the barrier height of the linear and bent transition states amount to 2.16 and 1.63 kcal/mol, respectively. These values include corrections for core correlation, scalar-relativistic effects, spin-orbit effects, as well as the diagonal Born-Oppenheimer correction. The CCSD(T) basis-set limits are estimated using extrapolation techniques with augmented quintuple and sextuple-zeta basis sets, and remaining N-electron errors are determined using coupled-cluster singles, doubles, triples, quadruples calculations with up to augmented quintuple-zeta basis sets. The remaining uncertainty is estimated to be less than 0.1 kcal/mol. The coupled-cluster results are used to calibrate multireference configuration-interaction calculations with empirical scaling of the correlation energy.     

Explicitly correlated coupled-cluster theory using cusp conditions. I. Perturbation analysis of coupled-cluster singles and doubles (CCSD-F12)

Geminal functions based on Slater-type correlation factors and fixed expansion coefficients, determined by cusp conditions, have in recent years been forwarded as an efficient and numerically stable method for introducing explicit electron correlation into coupled-cluster theory. In this work, we analyze the equations of explicitly correlated coupled-cluster singles and doubles (CCSD-F12) theory and introduce an ordering scheme based on perturbation theory which can be used to characterize and understand the various approximations found in the literature. Numerical results for a test set of 29 molecules support our analysis and give additional insight. In particular, our results help rationalize the success of the CCSD(F12) approximation which is based on a very systematic cancellation of the neglected, otherwise individually large third-order geminal-geminal coupling terms. Further approximations to CCSD(F12) can be introduced without sacrificing the accuracy if the entire set of third-order coupling terms between the conventional doubles cluster amplitudes and the geminal doubles amplitudes is retained, leading to the recently proposed CCSD[F12] and CCSD(F12(?)) models, which have negligible overhead compared to conventional CCSD calculations. Particularly, the importance of the ring-term type contribution is pointed out which may be used to improve on other existing approximations such as CCSD-F12b. For small basis sets, it might be advantageous to keep certain higher-order terms leading to CCSD-F12(?), which, for the case of the SP ansatz, merely involves a noniterative correction to CCSD(F12(?)).     

Applicability of quasi-degenerate many-body perturbation theory to quasi-degenerate electronic states: The H4 model revisited

The performance of the quasi-degenerate many-body perturbation theory up to the third order is investigated for the ground state, five excited states, and the first quintet of a simple four-electron H₄ model system consisting of two stretched hydrogen molecules, in which the degree of quasi-degeneracy can be continuously varied from a nondegenerate to a full degenerate situation. We employ a DZP basis set. The effect of intruder states is considered and a comparison with other multireference correlation techniques is also provided. Finally, a criterion for the model space to be quasi-degenerate will be reformulated and generalized. © 1995 John Wiley & Sons, Inc.     

Multiple basis sets in calculations of triples corrections in coupled-cluster theory

Multiple basis sets are used in calculations of perturbational corrections for triples replacements in the framework of single-reference coupled-cluster theory. We investigate a computational procedure, where the triples correction is calculated from a reduced space of virtual orbitals, while the full space is employed for the coupled-cluster singles-and-doubles model. The reduced space is either constructed from a prescribed unitary transformation of the virtual orbitals (for example into natural orbitals) with subsequent truncation, or from a reduced set of atomic basis functions. After the selection of a reduced space of virtual orbitals, the singles and doubles amplitudes obtained from a calculation in the full space are projected onto the reduced space, the remaining set of virtual orbitals is brought into canonical form by diagonalizing the representation of the Fock operator in the reduced space, and the triples corrections are evaluated as usual. The case studies include the determination of the spectroscopic constants of N₂, F₂, and CO, the geometry of O₃, the electric dipole moment of CO, the static dipole polarizability of F⁻, and the Ne⋯Ne interatomic potential. Received: 28 December 1996 / Accepted: 8 April 1997     

Model study of the impact of orbital choice on the accuracy of coupled-cluster energies. I. Single-reference-state formulation

The impact of the choice of molecular orbital sets on the results of single-reference-state coupled-cluster (CC) methods was studied for the H4 model. This model offers a straightforward way of taking into account all possible symmetry-adapted orbitals. Moreover, the degree of quasi-degeneracy of its ground state can be varied over a wide range by changing its geometry. The CCD, CCSD, and CCSDT approaches are considered. Surfaces representing the dependence of the energy on the parameters defining the orbitals are obtained. It is documented that for every method there exist alternative orbital sets which allow one to obtain more accurate energies than the standard (HF, BO, and NO) ones. However, for many of the former orbital sets, one obtains relatively large one-body amplitudes or one may encounter problems with solving the CC equations by conventional methods. An interesting variety of orbitals which might be useful for studies of quasi-degenerate states by the CCD method was found. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 205–219, 1998     

One-dimensional model for water and aqueous solutions. I. Pure liquid water

Two simplified one-dimensional models for waterlike particles are studied. One is referred to as the primitive model which is a simplified version of a model introduced by Ben-Naim in 1992 [Statistical Thermodynamics for Chemists and Biochemists (Plenum, New York, 1992)]. The second, referred to as the primitive cluster model, is a simplified version of the model used by Lovett and Ben-Naim in 1969 [J. Chem. Phys. 51, 3108 (1969)]. The two models are shown to be nearly equivalent and both exhibit some of the most characteristic behavior of liquid water. It is argued that a key feature of the molecular interactions--the correlation between the strong binding energy and low local density--is essential for the manifestation of the anomalous behavior of liquid water. It is also essential for the understanding of the outstanding behavior of liquid water.     

Model study of the impact of orbital choice on the accuracy of coupled-cluster energies. II. Valence-universal coupled-cluster method

The impact of the choice of molecular orbital sets on the results of the valence-universal coupled cluster method involving up to three-body amplitudes (VU-CCSDT) was studied for the H4 model. This model offers a straightforward way of representing all possible symmetry-adapted orbitals. Moreover, the degree of quasi-degeneracy of its lowest ¹A₁ states can be varied over a wide range by changing its geometry. Calculations were performed both for 13 sets of standard quantum chemical orbitals and for a vast variety of nonstandard orbital sets defined by nodes of a two-dimensional orbital grid. The performance of various standard orbital sets in VU-CCSDT calculations is compared. It is also documented that for every quasi-degeneracy region there exist nonstandard orbital sets which allow one to obtain more accurate VU-CCSDT energies than the standard orbital sets. In an attempt to provide a general interpretation for some of the alternative orbital sets, we defined a set of orbitals which maximize the proximity of the model and target spaces—maximum proximity orbitals (MPO). It is demonstrated that outside the strong quasi-degeneracy region the energies obtained for the VU-CCSDT approach based on the MPOs are more accurate than for the standard orbital sets. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 221–237, 1998     

Multiple solutions of unrestricted Hartree-Fock equations: The SNH+ radical as an example

A case of appearance of multiple unrestricted Hartree-Fock (UHF) solutions in the SNH⁺ (²A') radical is reported. This kind of solution does not arise from different electron configurations or symmetry breaking effects and seem to be exclusive of the UHF method. Three solutions are found and studied on the basis of wave function stability tests and several conclusions about their origin are given.     

Orthogonally spin-adapted coupled-cluster equations involving singly and doubly excited clusters. Comparison of different procedures for spin-adaptation

A particularly compact form of the orthogonally spin-adapted coupled-cluster equations involving all singly and doubly excited clusters is derived for the general case of a non-Hartree–Fock closed-shell reference determinant. The diagrammatic approach based on the graphical methods of spin algebras is applied. The relationship of different diagrammatic procedures for spin-adaptation, employing both bare and spin-adapted two-electron interaction vertices, is discussed. A comparison with the results obtained with algebraic spin-adaption approaches is also given.     

Carbodications. I. The structures and energies of C4H isomers

At high levels of ab initio theory (6-31G*//4-31G), the most stable C₄H isomer is indicated to be the nonplanar cyclobutadiene dication ( 1a ); the planar form, 1b , is indicated to be 7.5 kcal/mol less stable. The second most stable C₄H isomer, the methylenecyclopropene dication, is indicated to prefer the perpendicular ( 2a ) over the planar ( 2b ) arrangement by 7 kcal/mol. The “anti van't Hoff” cyclo-(HB)₂C?CH₂ system ( 4 ), isoelectronic with 2 , also prefers the perpendicular conformation ( 4a ), and retains the C?C double bond. The linear butatriene dication ( 3 ) is the least stable C₄H species investigated. The perpendicular (D_2d) arrangement ( 3a ), permitting double allyl cationlike conjugation, is preferred over the planar D_2h form ( 3b ) by 26 kcal/mol. The heat of formation of the most stable form of C₄H, 1a , is estimated to be 623–640 kcal/mol. This species should be thermodynamically stable toward dissociation into smaller charged fragments.