Fock Reference Research Articles

Linear and cyclic isomers of C4. A theoretical study with coupled-cluster methods and large basis sets

The ground electronic states of linear and rhombic C4 have been studied by high level ab initio quantum chemical techniques. Geometries, harmonic vibrational frequencies, infrared intensities, and other quantities have been determined using 4s3p2d1f correlation consistent basis sets and coupled-cluster methods including triple excitations. The linear–rhombic isomer energy difference has been investigated with a range of basis sets, including a 5s4p3d2f1g correlation consistent set. The linear–rhombic energy difference is influenced significantly by basis set, presence of triple excitations, and the choice of reference function for the open-shell linear isomer. The effect of basis set variation is complex, but once a reasonable quality of basis set has been achieved, further extensions favor the rhombic isomer. The inclusion of triple excitations also favors the rhombic isomer. The use of a restricted Hartree–Fock reference function for the linear isomer yields higher energies at the coupled-cluster level than if an unrestricted Hartree–Fock reference function is used, thereby again favoring the rhombic isomer. The most complete calculations of this study [coupled-cluster singles and doubles with noniterative triples (CCSD(T)) with a 5s4p3d2f1g basis set] indicate that the rhombic isomer is preferred by about 1 kcal mol−1. The coupled-cluster vibrational frequencies of the linear isomer are all real, in agreement with previous work, indicating that this isomer is not bent in the gas phase. The infrared intensities of linear C4 obtained in this work differ significantly from those obtained previously with smaller basis sets and either self-consistent field theory or second-order perturbation theory. The present calculations give a dissociation energy of C4 of 433 kcal mol−1, which is close to a previous value obtained with the aid of an empirical correction, and implies that several experimental estimates of the heat of formation of C4 are unreliable. Electron detachment energies of linear C4− and electron affinities of C4 are computed with larger basis sets than previously and are in very good agreement with recent anion photoelectron data.

Analytic ROHF–MBPT(2) second derivatives

The theory and a first implementation of analytic second derivatives for the second-order energy within many-body perturbation theory [MBPT(2)] based on a restricted open-shell Hartree–Fock (ROHF) reference function are presented. The applicability of the method is demonstrated by calculating harmonic frequency and infrared intensities for the 2B1 state of the CH2N molecule, which suffers from appreciable spin contamination.

The form of spin orbitals for open-shell restricted Hartree—Fock reference functions

A new set of spin orbitals is proposed for single-reference correlation procedures based on an open-shell restricted Hartree—Fock (RHF) reference function. The spatial parts of the new spin orbitals are from an open-shell RHF calculation. The spin part of the doubly occupied (or unoccupied) orbitals is the usual α or β spin function, but the spin part of the singly occupied and singly virtual orbitals takes the form σ + = ( 1 √2 )(α+β) and σ −=( 1 √2 )(α−β), respectively. These spin orbitals form a complete, orthonormal set. The RHF single determinant based on these spin orbitals is an S 2 eigenfunction, but a linear combination of S z eigenfunctions. The advantage of these spin orbitals is that equations for perturbation theory or coupled-cluster theory are symmetric in the spin indices and hence the number of independent parameters to be determined is dramatically reduced.

Restricted open-shell Hartree–Fock-based many-body perturbation theory: Theory and application of energy and gradient calculations

A new approach for many-body perturbation theory (MBPT) built upon a restricted open-shell Hartree–Fock (ROHF) reference function is presented. ROHF-MBPT is shown to give much improved results compared to unrestricted Hartree–Fock (UHF) MBPT in cases where there is large spin contamination of the UHF reference function, and to converge much more rapidly to the infinite-order coupled-cluster result. Equations for analytical gradients at the MBPT(2) level are described and implemented. ROHF-MBPT and restricted open-shell Hartree–Fock single- and double-excitation coupled cluster (ROHF-CCSD) applications are presented for several difficult cases. These include the structure and electron affinity of the CN radical; structure, binding energy, and vibrational frequencies of Li3; the structure and vibrational frequencies for the unobserved FCS molecule; and the multiplet structure of the Ni atom.

Analytic evaluation of second derivatives using second-order many-body perturbation theory and unrestricted Hartree—Fock reference functions

Explicit spin-orbital equations are presented for the second derivative of the second-order many-body perturbation theory energy based on unrestricted Hartree—Fock reference functions, and their implementation is briefly discussed. Results are reported for the electric polarizabilities, harmonic vibrational frequencies and infrared intensities of the 3B 1 state of CF 2.

Restricted Møller—Plesset theory for open-shell molecules

Møller—Plesset perturbation theory calculations for open-shell molecules based on a spin-restricted Hartree—Fock reference wavefunction are examined through the development of a new prescription for obtaining unique semi-canonical orbitals. These orbitals, which are different for α and β spins whilst avoiding the spin contamination present in UHF reference functions, satisfy the criteria on which Koopmans' theorem is based, lending justification to their use in defining a zeroth-order Hamiltonian for perturbation theory. This new and straightforward Møller—Plesset theory for open-shell molecules is called RMP theory. The convergence of the RMP series is examined to high order, and shows the greatly improved convergence characteristics also found with our alternative ROMP theory. For the molecules NH 2 ( r e)1.5 r e, 2 r e) and CN, RMP2 energies are substantially lower than UMP2 energies.

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.

Many-body perturbation theory with a restricted open-shell Hartree—Fock reference

A new, efficient ROHF based MBPT method is presented. The method, which is non-iterative, invariant to transformations among occupied or virtual orbitals, and generalizable to any order, is illustrated by application to the UHF spin contaminated CN radical and the H+OCH 2 transition state.

Analytic evaluation of energy gradients at the coupled-cluster singles and doubles level using quasi-restricted Hartree–Fock open-shell reference functions

Analytic formulas for the evaluation of energy gradients at the coupled-cluster singles and doubles (CCSD) level for a special class of open-shell quasi-restricted Hartree–Fock (QRHF) reference functions are presented and their computational implementation is discussed. The reference functions considered here are those which are formed from a closed-shell determinant for the molecule in a different ionization state by either adding or removing a single electron. Due to the approximate invariance of the CCSD model with respect to the choice of orbitals, results obtained with QRHF-CCSD closely approximate those from CCSD calculations using unrestricted Hartree–Fock (UHF) or restricted open-shell Hartree–Fock (ROHF) reference functions in cases where the latter references are well behaved. However, in cases where spin contamination (UHF) or symmetry breaking (UHF and ROHF) effects are important, QRHF-CCSD offers an attractive alternative since the resulting wave function is a projected eigenfunction of the spin operator Ŝ2 and transforms as a pure irreducible representation of the molecular point group.

Coupled-cluster open-shell analytic gradients: Implementation of the direct product decomposition approach in energy gradient calculations

Analytic energy gradients for the coupled-cluster singles and doubles (CCSD) method have been implemented for closed-shell systems using restricted Hartree–Fock (RHF) and open-shell systems using unrestricted Hartree–Fock (UHF) reference functions. To achieve maximum computational efficiency, the basic theory has been reformulated in terms of intermediates, thus reducing the number of required floating-point operations, and all computational steps are given in terms of matrix products in order to exploit the vector capabilities of modern supercomputers. Furthermore, the implementation has been designed to take full advantage of Abelian symmetry operations. To illustrate the computational efficiency of our implementation and in particular to demonstrate the possible savings due to the exploitation of symmetry, computer timings and hardware requirements are given for several representative chemical systems. In addition, the newly developed analytic CCSD gradient methods are applied to calculate the equilibrium geometry and energy splitting of the lowest singlet and triplet states of the C4O2 molecule.

Analytic energy gradients for open-shell coupled-cluster singles and doubles (CCSD) calculations using restricted open-shell Hartree—Fock (ROHF) reference functions

The theory of analytic energy gradients for the open-shell single and double excitation coupled-cluster (CCSD) method based on restricted open-shell Hartree—Fock (ROHF) reference functions is presented. The new CCSD gradient method is applied to the dissociation of the 3A″ state of formaldehyde (CH 2O) to H and HCO. Complete geometry optimization is carried out for the initial reactant, the transition state of the reaction, and for the dissociation products. Non-iterative triples appropriate to ROHF are introduced and are used to define a new ROHF-CCSD(T) method. Using this approach and zero-point corrections, the activation energy is calculated to be 21.0 kcal/mol and the exit barrier height is predicted to be 6.1 kcal/mol, both in excellent agreement with experiment.

A coupled cluster study of As2

Theoretical calculations at the self-consistent field (SCF) and coupled-cluster method including all single and double excitations (CCSD) have been carried out for As2. A variety of basis sets employing both segmented contractions and atomic natural orbitals have been developed and investigated. The predicted equilibrium bond length (re=2.107 Å), harmonic vibrational frequency (ωe=447 cm−1), and ionization potentials (9.75, 10.08, and 14.78 eV) are in excellent agreement with experimental values 2.103 Å, 430 cm−1, and 9.89 eV, 10.22 eV, and 15.32 eV; respectively. The best theoretical estimates for the dissociation energy (71.6 kcal/mol) and for the electron affinity (0.56 eV) were obtained at the CCSD level of theory employing large basis sets. However, given the importance of correlation effects on these two properties (the corresponding SCF values are 12.4 kcal/mol and 0.42 eV), the theoretical predictions could substantially be affected by the inclusion of connected triple excitations not considered in this work. The open-shell atomic As(4S) and diatomic calculations for As+2 and As−2 were carried out using a recently developed CCSD program that is based on restricted Hartree–Fock references and does not include any spin contamination for the correlation energies. The importance of correlating 3d electrons and the basis-set superposition error for De are analyzed. The effect of including d- and f-type polarization functions in the basis set is also discussed.

The effect of electron correlation on the electron density distributions of molecules: Comparison of perturbation and configuration interaction methods

Two methods for including the effects of electron correlation on the electron density distributions of molecules are compared. Second-order Mo/ller–Plesset perturbation theory (MP2) is shown to give electron density distributions in good qualitative agreement with much more time-consuming configuration interaction calculations. The latter include all single and double substitutions from the Hartree–Fock reference configuration. The two methods are compared by means of difference density maps calculated with large basis sets for a few representative molecules. From an analysis of the relative importance of single and double substitutions, it is shown that double substitutions are dominant near nuclei, but in other regions of space the single substitutions are more important. The MP2 method is shown to provide a practical method for calculating the correlation corrections to the electron density distributions of molecules of intermediate size.

Multiplicity of many-body wavefunctions using unrestricted Hartree-Fock reference functions

Spin multiplicity formulae for the correlated Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC) methods based upon an unrestricted Hartree Fock (UHF) reference function are presented and applied to evaluate the multiplicity in calculations of first row atoms. Spin projection in CC theory is discussed.

Analytic evaluation of energy gradients for the single and double excitation coupled cluster (CCSD) wave function: Theory and application

The theory for the analytic evaluation of energy gradients for coupled cluster (CC) wave functions is presented. In particular, explicit expressions for the analytic energy gradient of the CC singles and doubles (CCSD) wave function for a closed-shell restricted Hartree–Fock reference determinant are presented and shown to scale as N6 where N is the one-electron number of atomic basis functions for the molecular system. Thus analytic CCSD gradients are found to be of the same magnitude in computational cost as is the evaluation of analytic gradients for the configuration interaction singles and doubles (CISD) wave function. Applications of this method are presented for the water molecule and the formaldehyde molecule using a double-ζ plus polarization (DZ+P) basis set. The CCSD equilibrium geometries, dipole moments, and, via finite differences of gradients, CCSD harmonic vibrational frequencies and infrared intensities are reported. For H2O these results are compared to analogous CISD, CISDT, CISDTQ, and experimental results, and it is found that the CCSD predictions are most comparable to those of CISDTQ for this particular system. For the case of H2CO, the CCSD results are compared to CISD and experimental predictions. In general, the CCSD results and timings are encouraging.

The efficient evaluation of configuration interaction analytic energy second derivatives: Application to hydrogen thioperoxide, HSOH

We present an efficient reformulation of the analytic configuration interaction (CI) energy second derivative. Specifically, the Z-vector method of Handy and Schaefer is used to avoid solving the second order coupled perturbed Hartree–Fock (CPHF) equations. We have incorporated translational–rotational invariance into the new method. We present a more efficient method for the evaluation of the Y matrix contribution. The procedure which has been implemented can accommodate very large basis sets and CI expansions for any general restricted Hartree–Fock (RHF) reference wave function. As a test case, we apply the new procedure to the HSOH molecule using a double zeta plus polarization basis set. This leads to 50 contracted Gaussian basis functions and 116 403 configurations in the CI expansion. Harmonic vibrational frequencies and infrared intensities are predicted for HSOH and its deuterated isotopomers. The analytic method described herein requires only 56% of the central processor unit time used by a numerical method.

A modified coupled pair functional approach

A modified coupled pair functional (MCPF) method is presented that dramatically improves properties for cases where the Hartree–Fock reference configuration is not a good zeroth-order description of the wave function. This new methodology is compared to singles-plus-doubles configuration interaction and the CPF method of Ahlrichs et al. for the ground states of NiH, CuH, and ZnH.

Exponentially generated wave functions

We consider several generalizations of the exponential ansatz in a rather formal way, giving several new wave functions which we call exponentially generated (EG) wave functions. There are three distinct ways of the exponential-type generations of the wave functions, two of which are new. They are named ESAC (extended symmetry-adapted-cluster) wave function and exponentially generated CI (EGCI) wave function. The ESAC wave function is a simple extension of the SAC wave function and is applicable even when the Hartree–Fock reference configuration is not dominant. The EGCI wave function is a CI wave function constructed in the spirit of the cluster expansion theory. Formally, it has the merits of both the CI theory and the cluster expansion theory; for example, the upper bound nature, size consistency, and the applicability to quasidegenerate states and excited states. We then introduce several new wave functions by a multiple and mixed use of the exponential-type operators. We call such a class of wave functions multiexponentially generated (MEG) wave functions. There are many possibilities for the MEG wave functions, and the MR-SAC wave function proposed previously is one of them. When the system involves several classes of electron correlations, the MEG wave function permits an optimal (physically and practically) use of the exponential-type operators for the distinct classes of electron correlations. We described the method of solution of the EG and MEG wave functions and examined size consistency and some other properties.

Electronic symmetry breaking in polyatomic molecules. Multiconfiguration self-consistent field study of the cyclopropenyl radical C3H3

For equilateral triangle geometries (point group D3h), the C3H3 radical has a degenerate 2E″ electronic ground state. Although the 2A2 and 2B1 components separate in energy for C2v geometries, these two components should have identical energies for equilateral triangle structures. In fact, when approximate wave functions are used and the orbitals not required to transform according to the D3h irreducible representations, an energy separation between the 2A2 and 2B1 components is observed. At the single configuration self-consistent field (SCF) level of theory this separation is 2.8 kcal with a double-zeta basis set and 2.4 kcal with double-zeta plus polarization. It has been demonstrated that this spurious separation may be greatly reduced using multiconfiguration self-consistent field (up to 7474 variationally optimum configurations) and configuration interaction (up to 60 685 space and spin adapted configurations) techniques. Configurations differing by three and four electrons from the Hartree–Fock reference function are found necessary to reduce the 2A2-2B1 separation to below 0.5 kcal.

Generalization of analytic configuration interaction (CI) gradient techniques for potential energy hypersurfaces, including a solution to the coupled perturbed Hartree–Fock equations for multiconfiguration SCF molecular wave functions

Analytic configuration interaction gradient techniques for multiconfiguration self-consistent field (MCSCF) wave functions have been developed by solving the coupled perturbed equations for the generalized Hartree–Fock operator. The method has been implemented in conjunction with the loop-driven graphical unitary group approach for the determination of correlated wave functions. This theoretical approach is in principle applicable to any type of configuration interaction (CI) wave function constructed from orbitals determined from a rigorous SCF or MCSCF procedure. Initial applications have been to CI wave functions involving all single and double excitations relative to an open-shell singlet restricted Hartree–Fock reference function. The specific example chosen was the first excited singlet state (b 1B1) of CH2, for which the equilibrium geometry and harmonic vibrational frequencies have been predicted with a double zeta plus polarization basis set.