First-order properties and the Hellmann–Feynman theorem in the case of a limited CI wave function

Two distinct approaches to the calculation of first-order properties with a limited CI wave function are discussed. One is based on the Hellmann–Feynman theorem and the other on the direct evaluation of the total energy derivative at zero perturbation. Corrections to the Hellmann–Feynman expectation value are given for the CI wave function consisting of a single determinant reference state and all single and double replacements of this. These corrections are the extended Brillouin matrix elements and involve interactions between the zeroth-order wave function and triply substituted configurations. The usefulness of these matrix elements for the generation of MC SCF orbitals and for the calculation of cluster corrections to the wave function is briefly discussed. The formulas for the Brillouin matrix elements expressed in terms of one- and two-electron integrals have been automatically generated using the syntax of the algebraic program SCHOONSCHIP.     

On the SCF calculation of excited states: Singlet states in the two-electron problem

The problem of determining SCF wave functions for excited electronic states is examined for singlet states of two-electron systems using a Lowdin natural orbital transformation of the full CI wave function. This analysis facilitates the comparison of various SCF methods with one another. The distribution of the full CI states among the natural orbital MCSCF states is obtained for the S states of helium using a modest Gaussian basis set. For SCF methods that are not equivalent to the full CI wave functions, it is shown that the Hartree-Fock plus all single excitation wave functions are equivalent to that of Hartree-Fock plus one single excitation. It is further shown that these wave functions are equivalent to the perfect pair or TCSCF wave functions in which the CI expansion coefficients are restricted to have opposite signs. The case of the natural orbital MCSCF wave function for two orbitals is examined in greater detail. It is shown that the first excited state must always be found on the lower natural orbital MCSCF CI root, thus precluding the use of the Hylleras-Undeim-MacDonald (HUM) theorem in locating this state. It is finally demonstrated that the solution obtained by applying the HUM theorem (minimizing the upper MCSCF CI root with respect to orbital mixing parameters) is an artifact of the MCSCF method and does not correspond to any of the full CI states.     

Ab initio CI calculations on benzene with an extended basis set

An extended basis set of triple zeta plus polarization quality is employed to carry out configuration interaction (CI ) calculations of the three lowest singlet and triplet excited states of benzene. The CI calculation is carried out by taking into account single and double excitations of π and σ electrons. In the CI , composite natural orbitals (CNO s), which are constructed from the natural orbitals of the ground state of ethylene, are used as virtual orbitals. The aim of using CNOs is to reduce the number of virtual orbitals to be used in constructing configuration-state functions, thus cutting down CI dimensions without losing reasonable accuracy. The excitation energies resulting from the CI are in fairly good agreement with experiment. The root mean square of the deviation is 0.22 eV for the six calculated energies and the largest disagreement is 0.37 eV for the third singlet excited state. To obtain better excitation energies by an ab initio calculation, it seems likely that we need to take into account more electron correlation than in the present calculation. © 1994 John Wiley & Sons, Inc.     

Convergence studies in configuration interaction calculations

A series of CI calculations for the ground state of BH₃ in which various levels of excitation from the ground state are included are compared with each other and with the full-CI (196 configurations) result. The comparisons cover calculations in terms of two different molecular orbital bases—the canonical SCF basis and a simple, arbitrarily chosen, symmetry orbital basis. As expected, single and, to a lesser extent, triple excitations are of little importance in the SCF case but cannot be ignored for the arbitrary basis. However, as soon as all excitations ≦4 are included, there is practically no difference in the results for the two bases, both giving energies quite close to the full-CI value. In fact, the energies for the two bases are in close agreement also in the (0 + 1 + 2 + 3)-excitation calculation and in qualitative agreement in the (0 + 1 + 2) case. Two methods are tested for the selection of the important higher-excitation configurations, and it is found that results very close to full CI can be obtained with substantially fewer functions. Particularly promising is the application of the “unlinked cluster” approach, based on the ideas developed by Sinano?lu and others, to the prediction of the coefficients of evenly excited configurations from those of the doubly-excited functions in a limited-CI calculation.     

Coupled-cluster singles and doubles for extended systems

Coupled-cluster theory with connected single and double excitation operators (CCSD) and related approximations, such as linearized CCSD, quadratic configuration interaction with single and double excitation operators, coupled-cluster with connected double excitation operator (CCD), linearized CCD, approximate CCD, and second- and third-order many-body perturbation theories, are formulated and implemented for infinitely extended one-dimensional systems (polymers), on the basis of the periodic boundary conditions and distance-based screening of integrals, density matrix elements, and excitation amplitudes. The variation of correlation energies with the truncation radii of short- and long-range lattice sums and with the number of wave vector sampling points in the first Brillouin zone is examined for polyethylene, polyacetylene, and polyyne, and is shown to be a function of the degree of pi-electron conjugation or the fundamental band gaps. The t2 and t1 amplitudes in the atomic orbital (AO) basis are obtained by first computing the t amplitudes in the Bloch-orbital basis and subsequently back-transforming them into the AO basis. The plot of these AO-based t amplitudes as a function of unit cells also indicates that the t2 amplitudes of polyacetylene and polyyne exhibit appreciably slower decay than those of polyethylene, although the asymptotic decay behavior is invariably 1/r3. The AO-based t1 amplitudes appear to correlate strongly with the electronic structure, and they decay seemingly exponentially for polyethylene whereas they stay at a constant magnitude across the seventh nearest neighbors of polyacetylene and polyyne, which attests to far reaching effects of nondynamical electron correlation mediated by orbital rotation. Nonetheless, the unit cell contributions to the correlation energies taper below 10(-6) hartree after 15 A for all three polymers. The basis set dependence of the decay behavior of t2 amplitudes is also examined for linear hydrogen fluoride polymer (HF)infinity and linear beryllium polymer (Be)infinity employing the STO-3G, 6-31G, and 6-31G* basis sets, and proves to be rather small.     

Accuracy of energy extrapolation in multireference configuration interaction calculations

The accuracy of extrapolation procedures in conjunction with energy-based configuration selection in CI calculations is examined. The normally high accuracy of such extrapolation can deteriorate in multireference CI calculations when configuration functions of low weight are included in the root (reference) set. This is due to the inadequacy of second-order energy contribution estimates for the very large number of discarded low-contribution functions generated as single and double excitations from the minor members of the root set. The problem may be overcome by increasing the number of configurations included in the zero-order function used for the energy contribution estimation process. Illustrative results are presented for excited states of the H₂O molecule and the H₂O⁺ ion.     

Dissociation energies within selected configuration interaction and perturbation theory

Selected configuration interaction (CI) calculations and second-order perturbational theory are used to truncate systematically multireference single and double excitation CI (MRCI) expansions in the calculation of the bond dissociation energies of several systems like the single-bonded LiF molecule or the multiple-bonded N₂, NO and O₂ diatomic systems. The method is extended to compute the CH bond dissociation energy ofethene C₂H₄. It is shown how the proposed scheme (perturbation-selected MRCI (MRCI-PS)) is able to reproduce the accuracy of complete MRCI expansions with only a small number of configurations variationally evaluated.     

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.     

Quantum Monte Carlo study of ground and first excited state of C,N, O,F, and Ne atoms using Slater-Jastrow-Backflow wave function

All-electron fixed-node diffusion quantum Monte Carlo energies of the two lowest-lying states of C, N, O, F, and Ne atoms are reported. The Slater-Jastrow form is used as the trial wave function. We will use single- and multideterminant wave functions as the Slater part. The single-determinant wave function has been computed by the Hartree-Fock method and the multideterminant wave functions have been computed by the complete active space self-consistent field, configuration interaction with single and double excitation, configuration interaction with single, double, triple, and quadruple excitation and second-order configuration interaction. For the ground- and first excited states, the multideterminant wave functions have computed more than 99% of the correlation energy. Significant improvements have been achieved using the backflow transformations and up to 99.8% of the correlation energy has been recovered. A very good agreement with the experimental data has been obtained for the excitation energies.     

Nuclear-electronic orbital nonorthogonal configuration interaction approach

The nuclear-electronic orbital nonorthogonal configuration interaction (NEO-NOCI) approach is presented. In this framework, the hydrogen nuclei are treated quantum mechanically on the same level as the electrons, and a mixed nuclear-electronic time-independent Schrodinger equation is solved with molecular orbital techniques. For hydrogen transfer systems, the transferring hydrogen is represented by two basis function centers to allow delocalization of the nuclear wave function. In the two-state NEO-NOCI approach, the ground and excited state delocalized nuclear-electronic wave functions are expressed as linear combinations of two nonorthogonal localized nuclear-electronic wave functions obtained at the NEO-Hartree-Fock level. The advantages of the NEO-NOCI approach are the removal of the adiabatic separation between the electrons and the quantum nuclei, the computational efficiency, the potential for systematic improvement by enhancing the basis sets and number of configurations, and the applicability to a broad range of chemical systems. The tunneling splitting is determined by the energy difference between the two delocalized vibronic states. The hydrogen tunneling splittings calculated with the NEO-NOCI approach for the [He-H-He]+ model system with a range of fixed He-He distances are in excellent agreement with NEO-full CI and Fourier grid calculations. These benchmarking calculations indicate that NEO-NOCI is a promising approach for the calculation of delocalized, bilobal hydrogen wave functions and the corresponding hydrogen tunneling splittings.     

Multi-component molecular orbital study of isotope effects on lithium hydride molecules with the configuration interaction scheme

The multi-component molecular orbital method, which can take account of the quantum effect of the electrons and nuclei, is applied to the calculation of lithium hydride isotope species with the configuration interaction (CI) scheme. The optimum basis set functions for quantum nuclei are proposed by the fully variational procedure under single electronic–single nuclear excitation CI level. The average internuclear distances and dipole moments for isotopic lithium hydride molecules calculated with small basis functions are reasonable agreement with the corresponding experimental values.     

A direct CI method with a multiconfigurational reference state

A method for direct configuration–interaction (CI ) calculations with a multiconfigurational reference function is described. The reference state can contain several closed-shell electronic configurations and the CI expansion comprises all single and double replacements out of all these configurations. The resulting secular problem is solved using a variation-perturbation method. A number of examples are given showing the efficiency of the method. The largest CI expansion used in calculations with this program so far contains 76,471 spin- and space-symmetrized configurations.     

On calculations of correlated wave functions with heavy configurational mixing

We discuss aspects of the theory and computation of wave functions and energies of discrete states of polyelectronic atoms that are represented in zero order by configurations with holes in subshells below the valence subshell. Both in zero order and in the remaining correlation components, such wave functions have particularities stemming from the state‐specific self‐consistent field and the heavy configurational mixing associated with near‐degeneracies and hole‐filling correlations. By referring to a variety of examples from small‐ and large‐scale calculations, it is noted that appropriate penetration into the many‐body problem can provide, in an economic and physically transparent way, reliable interpretations and semi‐ and fully quantitative understanding of issues related to states with inner holes and to cases of near‐degeneracies that result in strongly correlated wave functions. Whenever hole‐filling correlations are allowed, multiple correlations (i.e., beyond single‐ and double‐orbital substitutions in the single reference configuration) acquire increased importance relative to that in ordinary electronic structures. This is demonstrated via large‐scale multiconfigurational Hartree–Fock (MCHF) plus configuration interaction (CI) calculations on the Cl KL3s3p^{6 2}S discrete state, which is the lowest of its symmetry. The calculations incorporated correlations up to selected sextuple orbital excitations from the M shell. MCHF plus CI calculations at the level of quadruple orbital substitutions were also carried out for the Cl KL3s²3p^{5 2}P^o ground state and the excitation energy at this level of calculation was found to be 85,364 cm^?1, in excellent agreement with the experimental value of the fine‐structure‐weighted average, 85,385 cm^?1 (10.59 eV). Within the approximations of the calculation, the hole‐filling triple and quadruple orbital correlations, which, of course, are absent from the ²P^o state, contribute about 1 eV, which is significant. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

An ab initio study of the ground and excited states of p-benzoquinone

The results of anab initio SCF calculation for the ground state and CI calculations for the excited states of p-benzoquinone are presented and discussed. A minimum basis set of Slater type orbitals was employed and the CI calculations were performed by considering single excitations from valence to virtual SCF molecular orbitals. The convergence of the calculated excitation energies is studied as a function of the number of orbitals used in the CI calculations. These calculations explain quite well the experimental results.     

Study of influences of various excitation classes onab initio calculated isotropic hyperfine coupling constants

Summary Reliable prediction of the isotropic hyperfine coupling constantA _iso is still a difficult task forab initio calculations. Strong dependence on the method employed for its calculation has been found. Within a CI ansatzA _iso is considerably affected by the excitation classes taken into account within the CI calculation. In the present work the influence of various excitation classes onA _iso is examined. Calculations including all single, double, triple and a large part of the quadruple excitations are performed and the individual effects of the excitation classes are studied. It is found that the surprisingly good agreement found for S-CI treatments is due to large error cancellations. The importance of higher than double excitations arises from their indirect influence on the single excitations.     

Quantum dynamical treatment of inelastic scattering of atoms at a surface at finite temperature: the random phase thermal wave function approach

We present quantum dynamical calculations for the inelastic scattering of atoms at a nonrigid surface at finite temperature. The surface degrees of freedom are discretized and treated in a multiconfigurational wave function picture. The thermal averaging is carried out with the random phase thermal wave function approach. We show that it is sufficient to restrict the random phases to the intermediate basis of single particle functions, discuss the convergence of the method with the number of configurations and realizations, and analyze the flow of energy between different parts of the system for a range of temperatures between 4 and 500 K.     

Calculation of transition matrix elements by nonsingular orbital transformations

A general strategy is described for the evaluation of transition matrix elements between pairs of full class CI wave functions built up from mutually nonorthogonal molecular orbitals. A new method is proposed for the counter‐transformation of the linear expansion coefficients of a full CI wave function under a nonsingular transformation of the molecular‐orbital basis. The method, which consists in a straightforward application of the Cauchy–Binet formula to the definition of a Slater determinant, is shown to be simple and suitable for efficient implementation on current high‐performance computers. The new method appears mainly beneficial to the calculation of miscellaneous transition matrix elements among individually optimized CASSCF states and to the re‐evaluation of the CASCI expansion coefficients in Slater‐determinant bases formed from arbitrarily rotated (e.g., localized or, conversely, delocalized) active molecular orbitals. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Efficient configuration selection scheme for vibrational second-order perturbation theory

A fast algorithm of vibrational second-order Moller-Plesset perturbation theory is proposed, enabling a substantial reduction in the number of vibrational self-consistent-field (VSCF) configurations that need to be summed in the calculations. Important configurations are identified a priori by assuming that a reference VSCF wave function is approximated well by harmonic oscillator wave functions and that fifth- and higher-order anharmonicities are negligible. The proposed scheme has reduced the number of VSCF configurations by more than 100 times for formaldehyde, ethylene, and furazan with an error in computed frequencies being not more than a few cm(-1).     

Floating functions satisfying the hellmann—feynman theorem: Single floating scheme

The electrostatic calculation for molecules using approximated variational wave functions leads to well known difficulties connected with the application of the Hellmann-Feynman (H? F) theorem. This is due to the basis set inadequacies in the underlying calculations. This defect can easily be remedied by floating functions, whose centers are optimized in space. We can keep almost everything of the traditional wave function with a nuclear-fixed basis set, but we apply single floating to ensure the H? F theorem. Then, one can obtain a wave function obeying the H? F theorem. This provides a great conceptual simplification and may lead to practical advantages. The single floating scheme, which retains one expansion center per nucleus, is successfully applied to a series of small molecules using SCF and CASSCF wave functions with sufficiently polarized basis sets.     

Configuration interaction calculations on the 2P ground state of boron atom and C+ using Slater orbitals

Configuration Interaction (CI) calculations on the ground ²P state of boron atom are presented using a wave function expansion constructed with L‐S eigenfunction configurations of s‐, p‐, and d‐Slater orbitals. Two procedures of optimization of the orbital exponents have been investigated. First, CI(SD) calculations including few types of configurations and full optimization of the orbital exponents led to the energy ?24.63704575 a.u. Second, full‐CI (FCI) calculations including a large number of configuration types using a fixed set of orbital exponents for all configurations gave ?24.63405222 a.u. using the basis [4s3p2d] and 2157 configurations, and to an improved result of ?24.64013999 a.u. for 3957 configurations and a [5s4p3d] basis. This last result is better than earlier calculations of Schaefer and Harris (Phys Rev 1968, 167, 67), and compares well with the recent ones from Froese Fischer and Bunge (personal communication). In addition, using the same wave functions, CI calculations of the boron isoelectronic ion C⁺ have been performed obtaining an energy of ?37.41027598 a.u. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011