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.     

Calculation of transition density matrices by nonunitary orbital transformations

An efficient scheme for calculating one- and two-electron transition density matrices for two wave functions is described. The method applies to CAS (complete active space) wave functions and certain multireference CI expansions. The orbital sets of the two wave functions are not assumed to be equal. They are transformed to a biorthonormal basis, and the corresponding transformation of the CI coefficients is carried out directly, using the one-electron coupling coefficients.     

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     

Geometrical energy derivative evaluation with MRCI wave functions

The theory of MCSCF and CI energy derivatives with respect to geometrical variations is briefly reviewed with special attention given to the MCSCF and MRCI energy gradients. A computational procedure is proposed for MRCI energy gradients that does not require the solution to any “coupled-perturbed MCSCF ” equations, it does not require any expensive direct-CI matrix-vector products involving derivative integrals, and it does not require any derivative integrals to be transformed from the AO basis to the MO basis. An additional feature is that it does not require any changes to existing MCSCF gradient evaluation programs in order to compute MRCI gradients. The only difference in the two cases is the exact nature of the data passed to the gradient evaluation program from the previous steps in the computational procedure. The additional effort required to compute the entire MRCI energy gradient vector is approximately that required for one additional iteration of the MRCI diagonalization procedure and for one additional MCSCF iteration. For large scale MRCI wave functions, the MRCI energy gradient evaluation should only require about 10% of the effort of computing the wave function itself. This computational procedure removes a major computational botleneck of potential energy surface evaluation.     

Orbital symmetry,orbital stability,and orbital pairing rules for organic reactions in the ground state

A generalization of the Hartree–Fock molecular orbital (MO) theory for treating diradical intermediates was explained pictorially by drawing molecular orbitals of diradical species such as ring-opened trimethylene. The generalized MO theory applied to elucidate electronic mechanisms of concerted, ionic, radical, and ion-radical reactions of organic reactants in the ground state. Generalized MO computations revealed the most essential characteristics of these reactions and mutal relationships between the worlds of Woodward–Hoffmann and Hughes–Ingold. Generalized MO studies supported our orbital symmetry, stability and pairing rules for concerted, ionic and radical reactions in the ground state, respectively. An extension of MO treatments to excited states reactions was briefly pointed out in relation to the density and spin correlation functions by the multireference CI wave functions.     

LCAO–MO similarity measures and taxonomy

Similarity measures between pairs of molecular wave functions are described. They are based on the geometrical structure of the LCAO–MO framework and upon multivariate analysis ideas. The theoretical framework is presented, and formulae for some integrals needed are given. Two main measures, distance and correlation coefficients, are used. Distance and correlation matrices induce relationships in the whole MO set, which can be depicted through minimal spanning tree techniques. Furthermore, principal component analysis allows a two-dimensional visualization of the Mo manifold geometrical relationships. Various examples are given in order to obtain information on how basis set, environment, excitation, bending, stretching, and electronegativity affect the induced order. For this purpose “ab initio” SCF–LCAO–MO calculations with double- and single-zeta quality basis sets have been used for various simple molecular structures: H₂O, NH₃, CH₄, N₂, O₂, C₂, NO, CN, and CO. The results obtained can open the way to LCAO–MO taxonomy. Using this information, other areas of interest are connected with similarity measures (SCF and CI , localization procedures, etc.), proving in this manner their potential utility.     

A study on the electronic spectra of vinylpyridines and 1,2-(dipyridyl)ethylenes. Molecular orbital calculations

The electronic absorption spectra of 2-, 3-, and 4-vinylpyridines and 1,2-(2,3-dipyridyl), 1,2-(2,4-dipyridyl), 1,2-(3,4-dipyridyl), and 1,2-(4,4-dipyridyl) ethylenes have been investigated in polar and nonpolar solvents. A correlation has been made between the geometry of the molecule and the observed spectrum. Molecular orbital calculations have been carried out using the INDO/S? CI procedure and a limited geometry optimization. The solvent effect at the MO level has been calculated. MO calculations predicted the existence of nπ* transitions that were not observed experimentally. The wave functions of the different CI states were calculated. The experimental transition energy as well as oscillator strength corresponded satisfactorily with the calculated ones. The observed transitions were assigned according to the results of MO calculations.     

Cluster analysis of the full configuration interaction wave functions of cyclic polyene models

The first two members of the cyclic polyene homologous series are studied over a wide range of the coupling constant using the Hubbard and Pariser–Parr–Pople model Hamiltonians. The full and various limited configuration interaction (CI ) correlation energies and wave functions are calculated exploiting the unitary group approach. The formalism for the cluster analysis of the exact wave function expressed through the unitary group formalism electronic Gelfand states is developed and applied to the full CI wave functions of the cyclic polyene models studied. It is shown that the connected tetraexcited clusters become essential in the fully correlated limit and that their contribution also significantly increases with electron number even for the coupling constant corresponding to the spectroscopic parametrization of the model Hamiltonians used.     

Molecular properties from perturbation theory: A Unified treatment of energy derivatives

The definition of a molecular property as a derivative of the electronic energy with respect to one or more applied perturbations is reviewed. The explicit enumeration of terms entering the derivative formulas is performed by considering in turn the various parameter spaces on which the energy and wave function depend. After deriving general expressions for first, second, and third derivatives for different types of perturbation, the parameter spaces involved in MCSCF and CI cases are identified and used to obtain expressions for the first and second derivatives. An example of an MCSCF third derivative is also given. In addition, the various equation systems defining the perturbed wave functions in each order are derived. Some attention is given to the efficient computer implementation of derivative calculations, and the present work is compared with that of other authors.     

Extended Koopmans' theorem ionization potentials for beryllium atom shake-up transitions

The ionization potentials were calculated for Be using the extended Koopmans' theorem (EKT ) using several full configuration interaction (CI ) and multiconfigurational-self-consistent-field (MCSCF ) wave functions as reference wave functions. The wave functions used account for 89.7–96.7% of the correlation energy. Comparisons are made with experimental values and with δCI values calculated as the difference in energy obtained from CI wave functions for Be and Be⁺. The best EKT IP differed from the δCI value by 0.0003 eV for the lowest IP and by 0.0006 eV for ionization into the lowest ²P state of Be⁺. A calculation of ionization into the second ²P state of Be⁺ requires diffuse orbitals that are unimportant in the wave function for the ground state of Be. This results in small natural orbital occupation numbers for natural orbitals needed in the EKT calculation. © 1994 John Wiley & Sons, Inc.     

Ground-state correlation energy of Ne

A series of basis sets and configuration interaction (CI ) wave functions, both of which were constructed so as to systematically approach to the complete set limit and the full CI limit, were used for the ground state of Ne. These calculations yielded an estimated correlation energy of ?0.3891 au, which is 99.6% of a recent theoretical estimate of ?0.3905 au. The CI value, ?0.3821 au, was obtained by SDCI calculation with seven reference configurations by using Slater-type orbitals (STO s) from s to h functions. © 1994 John Wiley & Sons, Inc.     

The Valence-Bond (VB) Model and Its Intimate Relationship to the Symmetric or Permutation Group

VB and molecular orbital (MO) models are normally distinguished by the fact the first looks at molecules as a collection of atoms held together by chemical bonds while the latter adopts the view that each molecule should be regarded as an independent entity built up of electrons and nuclei and characterized by its molecular structure. Nevertheless, there is a much more fundamental difference between these two models which is only revealed when the symmetries of the many-electron Hamiltonian are fully taken into account: while the VB and MO wave functions exhibit the point-group symmetry, whenever present in the many-electron Hamiltonian, only VB wave functions exhibit the permutation symmetry, which is always present in the many-electron Hamiltonian. Practically all the conflicts among the practitioners of the two models can be traced down to the lack of permutation symmetry in the MO wave functions. Moreover, when examined from the permutation group perspective, it becomes clear that the concepts introduced by Pauling to deal with molecules can be equally applied to the study of the atomic structure. In other words, as strange as it may sound, VB can be extended to the study of atoms and, therefore, is a much more general model than MO.     

Spin-projected Extended Hartree-Fock using a Valence Bond approach

Summary We describe spin-projected Extended Hartree-Fock calculations, performed with a Valence Bond Self-Consistent Field program. Potential energy curves are given for BH, BeH, and N₂. For BH the EHF function ranks well with the corresponding Spin-coupled and full CI wave functions. For BeH, the EHF function introduces spin contamination in the separated Be atom due to the rigidity of the wave function. This results in an inferior potential energy curve compared to Spin-coupled and full CI. The triple bond breaking in N₂ is again nicely described by EHF. The Extended Hartree-Fock method as suggested by Löwdin can be a feasible tool in describing bond breaking.     

Optical, tsvetometric, and acid-base characteristics of methyl orange

The optical and tsvetometric characteristics of 4-dimethylaminoazobenzen-4′-sulphoacid (methyl orange, MO) in water solutions at pH 2–9 are studied. The equations of calibration plots of the dependence of colorimetric functions on concentration are obtained and calculated as molar coefficients of colorimetric functions. The method determines The constants of MO dissociation at an ionic force of 0.01–0.2 are determined by spectrophotometry; they are given to an ionic force of 0 and are compared to the literature data. The advantages of tsvetometry over spectrophotometry are shown in determining the MO concentration.     

Optimized linear combinations of simple exponentials for atomic systems

Hartree-Fock wave functions for the He and Be isoelectronic sequences of ions are calculated using orbitals which are linear combinations of simple exponential functions. By a full optimization of the exponents and coefficients close approximations to the HartreeFock energies were obtained. To the same order of accuracy the resulting Hartree–Fock orbitals require fewer basis functions than used previously. A number of difficulties which arise in the numerical procedures as the size of the basis set is increased are analysed in detail. Similar results are obtained for the Li sequence using the Unrestricted HartreeFock method with and without projection.     

Calculation of excitation and photoionization spectra by quasi-degenerate perturbation theory

The ability of the QDPT CI method, in which the quasi-degenerate perturbation theory is applied within the configuration interaction (CI ) approach, in dealing with the calculation of excitation and photoionization spectra is shown through an overview of comparisons between the QDPT CI values and the full CI ones in the same basis set. A direct comparison with the experimental data is given for the core photoelectron spectrum of Ne and the valence photoelectron spectra of Ar and HCl. The quality of the results obtained is very satisfactory, both as regards the energies and the wave functions, indicating the validity and the flexibility of the method that can confidently be applied also in cases where strong correlation effects are present. © 1994 John Wiley & Sons, Inc.     

Examination of the Hund rule in closed-shell systems: Investigation of spin correlation effects

The validity of the Hund rule in atomic orbitals (AO s) of the carbon atoms inside closed-shell molecules, such as acetylene, ethylene, and ethane, is examined. Electron-pair populations and contributions of the two-electron covalent structures with parallel (?) and antiparallel (↑↓) spins are calculated by multielectron population analysis of MO + CI wave functions. Such an analysis, which allows the visualization of various cooperative electronic effects in some target AO s, is extended on the basis of (strictly orthogonal) hybrid orbitals. Although the HF level shows, incorrectly, that the Hund rule is not satisfied, the CI clearly shows a preference for (?) spins to those of (↑↓): This holds for both the electron-pair populations [those of (↑↓) spins diminish with the CI more drastically than those of (?) spins], as well as for contributions of the two-electron covalent structures [those of (?) spins increase with the CI more drastically than do those of (?) spins]. The calculation of the correlation functions (or dependent functions) in AO space allows the comparison of repulsive or “attractive” behaviour of (?) and (↑↓) spins in various AO couples. Mutual dependence of the two electrons inside the sigma system increases in the series ethane < ethylene < acetylene. Also found is that parallel spins in (pure) AO s of sigma systems are preferred to the antiparallel spins when going from ethane to acetylene. The preference parallel–antiparallel spins in AO s belonging to two different atoms, including hydrogens, is also examined. © 1993 John Wiley & Sons, Inc.     

Configuration interaction spaces with arbitrary restrictions on orbital occupancies

Configuration interaction (CI) spaces obtained from the full CI space by imposing arbitrary restrictions on the occupancies of molecular orbital (MO) groups are studied. It is proved that such restricted spaces are in a certain sense “closed.” Namely, in the course of the Hamiltonian matrix construction the excitations out of the chosen restricted CI space may be easily replaced by the excitations within this space. © 1996 John Wiley & Sons, Inc.     

Dependence of valency on geometry for configuration interaction wave functions

Earlier definitions of valencies of atoms, molecules, and molecular orbitals are extended to configuration interaction (CI ) wave functions. Using these definitions, valencies both at equilibrium and nonequilibrium geometries of molecules are calculated at the CI level and compared with non-CI results. CI valency correlation diagrams are obtained. Valency variation with bond length using correlated wave functions is found to behave properly unlike in the case of SCF wave functions.     

Matrix‐covariant representation of high‐order configuration interaction and coupled cluster theories

We present the closed form of the reduced density matrices (RDMs) of arbitrary order for configuration interaction (CI) wave functions at any excitation level, up to the full CI. A special operator technique due to Bogoliubov is applied and extended. It focuses on constructions of matrix‐covariant expressions independent of the basis set used. The corresponding variational CI equations are given in an explicit form containing the matrices related to conventional excitation operators. A subsequent transformation of the latter to an irreducible form makes it possible to generate the matrix‐covariant representation for coupled cluster (CC) models. Here this transformation is performed for a simplified high‐order CC scheme somewhat reminiscent of the quadratic CI model. A generalized spin‐flip approximation closely related to high‐order CI and CC models is presented, stressing on a possible inclusion of nondynamical and dynamical correlation effects for multiple bond breaking. A derivation of the full CI and simple CC models for systems involving effective three‐electron interactions is also given, thereby demonstrating the capability of the proposed method to deal with complicated many‐body problem. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008