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.     

Direct minimization of the energy in density functional theory

The minimization of the energy functional of the first-order density matrix γ( r , r ') is achieved using unitary transformations applied to γ. Equivalently, such transformations can be carried out also on one-electron orbitals (natural orbitals) and their occupation (integer or non-integer) numbers. The conventional local density approximation based on the electron density p( r ) is then considered as a special case. The direct minimization of the energy functional of p with respect to the parameters of the unitary transformation leads to stationary conditions that are all equivalent to the Kohn–Sham equations. Preliminary numerical tests show that the proposed algorithms for the direct minimization of the energy work in a satisfactory manner. © John Wiley & Sons, Inc.     

A semiempirical SCF–LCAO–MO treatment of some oxygen- and nitrogen-containing heterocyclic molecules

The π-electronic properties of furan, oxazole, benzofuran, benzoxazole, anthranil, and dibenzofuran are calculated by the semiempirical self-consistent-field molecular orbital method. A single set of parameters is found which satisfactorily reproduces the π → π* electronic transition energies and other π-electronic properties.     

Direct energy functional minimization under orthogonality constraints

The direct energy functional minimization problem in electronic structure theory, where the single-particle orbitals are optimized under the constraint of orthogonality, is explored. We present an orbital transformation based on an efficient expansion of the inverse factorization of the overlap matrix that keeps orbitals orthonormal. The orbital transformation maps the orthogonality constrained energy functional to an approximate unconstrained functional, which is correct to some order in a neighborhood of an orthogonal but approximate solution. A conjugate gradient scheme can then be used to find the ground state orbitals from the minimization of a sequence of transformed unconstrained electronic energy functionals. The technique provides an efficient, robust, and numerically stable approach to direct total energy minimization in first principles electronic structure theory based on tight-binding, Hartree-Fock, or density functional theory. For sparse problems, where both the orbitals and the effective single-particle Hamiltonians have sparse matrix representations, the effort scales linearly with the number of basis functions N in each iteration. For problems where only the overlap and Hamiltonian matrices are sparse the computational cost scales as O(M2N), where M is the number of occupied orbitals. We report a single point density functional energy calculation of a DNA decamer hydrated with 4003 water molecules under periodic boundary conditions. The DNA fragment containing a cis-syn thymine dimer is composed of 634 atoms and the whole system contains a total of 12,661 atoms and 103,333 spherical Gaussian basis functions.     

Direct minimization of the energy functional in LCAO-MO density matrix formalism

The general multiconfiguration self-consistent-field method is presented along the density matrix formalism. The proposed optimization procedure for orbitals makes use of an orthogonal transformation in the space spanned by the fixed basis set. Acting on the unconstrained parameters of the transformation a direct minimization of the energy expression is performed using a gradient approach. A similar method may also be applied to the optimization of the expansion coefficients. The method works not only for the ground state of a given system, but also for any excited state, yielding an upper bound to the true energy of the considered state.     

Direct minimization of the energy functional in LCAO-MO density matrix formalism

A general theory is presented for the optimization of the coefficients of orbitals and configuration interaction expansion in the case of multiconfiguration wavefunctions containing all single excitations. The orbital coefficients are optimized by suitable orthogonal transformations of the atomic basis; the Cl coefficients are determined solving the usual secular problem. The energy minimization is performed directly by a gradient approach. The method works both for ground and excited states and no convergence difficulties are met. Computational examples are given for H₂O and H₂S molecules.     

Direct minimization of the energy functional in the LCAO-MO density matrix formalism

A previously proposed method of energy minimization is developed for MC SCF wavefunctions formed by all-pair excitations for a closed-shell system. The orbital coefficients are optimized by a gradient approach using a suitable orthogonal transformation of the atomic basis, while optimum CI coefficients are determined solving the usual secular problem for the lowest eigenvalue, after each optimization of the orbitals. Applications to LiH and NH₃ molecules show that the method is numerically well stable, and is capable of accounting for a large part of the correlation energy giving results which compare well with those of the conventional CI method.     

LCAO–MO studies on hydrogen bonding: The interaction between carbonyl and hydroxyl groups

Potential curves that show the energy dependence of hydrogen bonds between carbonyl and hydroxyl groups on the O? H bond length, on the distance between the molecules, and on the angle between the functional groups have been calculated with the CNDO /2 method. The results are presented for a small model system–formaldehyde/water—and for the dimer of formic acid. Good agreement is obtained with the available experimental data. The influence of the molecular geometry on the calculated results is discussed.     

Small,simultaneous adjustments of orbital exponents in LCAO–MO–SCF calculations using self-consistent perturbation theory

Self-consistent perturbation theory is introduced to facilitate making small, simultaneous variations in orbital exponents. This is accomplished by interpreting these variations as perturbations on the quantum mechanical system. The minimum-energy condition yields a set of linear equations for the desired exponential corrections.     

A multi-configuration LCAO–MO study for complex unsaturated molecules. II. Application to the benzene cation

The method of the MC –LCAO –MO approach, described in the preceding paper, is further applied to the benzene cation. Through the iteration process the π-electron energies and the molecular shapes are computed for the ground and two lowest excited states of the cation in both D_6h and D_2h geometries. A remarkable fact obtained is that a comparatively small variation of the geometrical structure (c. 0.010 – 0.013 Å bond length difference) brings about a considerable change of the energy value (c. 0.85 – 1.25 eV). The π-electronic excitation energies obtained from the iteration process are compared with the transition energies calculated from the usual method in which the structures of the excited states are assumed to be the same as the corresponding ground state structures. The difference in the excitation energy between the cation and the anion, and the CI effect on the excited states, are discussed. It is found that the doubly excited configurations play an important role in CI , which is somewhat different from that of the singly excited configurations. The stabilization energy due to the Jahn–Teller distortion is estimated for the ground state of the cation.     

SCF CI MO calculations on the base–base interactions in dinucleoside monophosphates

In order to elucidate further details of RNA conformations we have studied base stacking in dinucleoside monophosphates (DNP's) using UV difference spectra and the hypochromic effect. SCF CI MO calculations according to PPP and MIM approximations were carried out for six pyrimidine-containing DNP's in each of several stacking geometries. The calculated and plotted difference spectra were fitted to the experimental spectra. Different DNP's showed distinct geometries in the range of the helix angle of 35°. From our results we conclude that there is a microstructure in the helix. This would imply an additional content of information in this macromolecule, a higher precision in nucleic acid interactions, and make possible the prediction of the conformation of polynucleotides.     

X–NO2 rotational energy barriers: Local density functional calculations

The X–NO₂ rotational energy barriers of nitromethane, nitroethylene, nitrobenzene, and a group of nitramines have been computed using a local density functional (LDF ) procedure, using ab initio Hartree–Fock (HF )-optimized structures of the ground and rotational transition states. The results have been discussed in relation to HF and some correlated ab initio values and the available experimental data. Our LDF barriers are overall quite reasonable, in generally satisfactory agreement with the experimental and correlated ab initio results. © 1993 John Wiley & Sons, Inc.     

Scaled one-electron hamiltonian model for open-shell LCAO–MO–SCF calculations: Comparison with the restricted open-shell method of roothaan

The performance of a recently proposed scaled one-electron Hamiltonian (SOEH ) model is tested against parallel sets of restricted open-shell calculations by the method of Roothaan. It is found that the energy calculated by SOEH model, in general, lies slightly higher than the energy computed by the restricted open-shell method of Roothaan lending credibility to the application of variational argument to the scaled pseudoenergy functional (E^av) for deriving the SOEH model. The numerical stability of the converged SOEH energy with respect to changes in trial vectors indicates the reliability of the method. The SOEH model is shown to perform well in the calculation of geometries of radicals and ions. The convergence behavior of the SOEH model is compared with that of the restricted open-shell method of Roothaan.     

A multi-configuration LCAO–MO study for complex unsaturated molecules. I. General theory and its application to the benzene anion

A multi-configuration LCAO –MO approach using a π-bond order–bond length linear relation is introduced to predict the geometrical structures for the electronic ground and excited states of unsaturated hydrocarbons. The procedure is designed to include configuration interaction in each iterative computation where the π-electron approximation is employed under the Pariser–Parr type semi-empirical treatment. The π-bond order–bond length relation is determined as r_pq = 1.523 – 0.193P_pq, when the bond lengths of ethylene, benzene and naphthalene are used and the groundstate functions including the singly and doubly excited configurations are taken into account to obtain the bond orders P_pq. The iterative calculation is applied to the ground state and the two lowest excited states of the benzene anion in both D_6h and D_2h molecular geometries. The geometrical structures and the π-electron energies are computed for the ground and excited states of the anion; for the latter, two types of configuration species are used. It is found that the first lowest excited state is not subjected to the Jahn–Teller effect and the calculated excited state energies do not agree with the observed values (c. 1.0 ～ 2.5 eV higher than the observed values). The latter point is discussed in detail. It is also found that the resultant ground state energy depression due to configuration mixing is not very large and the two types of configuration species used give different CI effects on the energy levels of the two lowest excited states of the anion. Finally, the stabilization energy due to the Jahn–Teller distortion is estimated for the ground state of the anion.     

B-spline method for energy minimization in grid-based molecular mechanics calculations

A method is described for molecular mechanics calculations based on a cubic B-spline approximation of the potential energy. This method is useful when parts of the system are allowed to remain fixed in position so that a potential energy grid can be precalculated and used to approximate the interaction energy between parts of a molecule or between molecules. We adapted and modified the conventional B-spline method to provide an approximation of the Empirical Conformational Energy Program for Peptides (ECEPP) potential energy function. The advantage of the B-spline method over simpler approximations is that the resulting B-spline function is C2 continuous, which allows minimization of the potential energy by any local minimization algorithm. The standard B-spline method provides a good approximation of the electrostatic energy; but in order to reproduce the Lennard–Jones and hydrogen-bonding functional forms accurately, it was necessary to modify the standard B-spline method. This modification of the B-spline method can also be used to improve the accuracy of trilinear interpolation for simulations that do not require continuous derivatives. As an example, we apply the B-spline method to rigid-body docking energy calculations using the ECEPP potential energy function. Energies are calculated for the complex of Phe-Pro-Arg with thrombin. For this system, we compare the performance of the B-spline method to that of the standard pairwise summation in terms of speed, accuracy, and overhead costs for a variety of grid spacings. In our rigid-body docking calculations, the B-spline method provided an accurate approximation of the total energy of the system, and it resulted in an 180-fold reduction in the time required for a single energy and gradient calculation for this system. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 71–85, 1998     

Direct minimization of the energy functional in LCAO-MO density matrix formalism I. Closed and open shell systems

A direct method of minimization of the energy expression for closed and open shell systems in LCAO-MO density matrix formalism is presented. The method makes use of a unitary transformation acting directly on the density matrices. Expressions of the gradient and second energy derivatives are worked out. Some preliminary calculations to test the rate of minimization using a variable metric method have been made on H₂S and SO molecules and have given satisfactory results.[/p]     

A comparison of accelerators for direct energy minimization in electronic structure calculations

We compare three different methods for direct energy minimization in electronic structure calculations where the gradient of the energy functional with respect to the molecular orbitals is available. These methods make use of the preconditioned gradient to increase robustness. An orbital transformation is used to ensure that the orthogonality constraint on the orbitals remains satisfied when using standard minimization methods. In addition, we propose an adaptive scheme for estimating the curvature of the energy functional to increase the performance of a line search free quasi-Newton method. We show that the performance of all methods is similar when robustness of the methods is ensured.