Ab initio calculations on large molecules using molecular fragments. Initial studies on open shell systems

An ab initio procedure, designed for investigation of large molecules and based upon studies of molecular fragments, is extended to open shell systems using the unrestricted Hartree-Fock method. Investigated initially are the ethyl and vinyl radicals, and the ethylene triplet state.     

Convergence problems in the solution of SCF equations

Some problems connected with the convergence of iterative solutions of the Hartree-Fock equations for the open shell systems are discussed. The nonuniqueness of the Hartree-Fock operator form is used. The method of iterating by the operator's power is developed for obtaining solutions of the Hartree-Fock equations. Some particular results for the molecules Li, Li are presented.     

H2 solved by the finite element method

We report on the solution of the Hartree-Fock equations for the ground state of the H₂ molecule using the finite element method. Both the Hartree-Fock and the Poisson equations are solved with this method to an accuracy of 10⁻⁸ using only 26 × 11 grid points in two dimensions. A 41 × 16 grid gives a new Hartree-Fock benchmark to ten-figure accuracy.     

On open shell Hartree-Fock calculations

The errors in open shell Hartree-Fock calculations are discussed. various hamiltonians for single eigenvalue methods are considered, and it is shown that a better energy is obtained with those hamiltonians that correctly account for the open shell/closed shell interactions.

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Zusammenfassung Die Fehler des Hartree-Fock-Verfahrens bei offenen Schalen werden diskutiert, indem verschiedene Hamilton-Operatoren für das Gesamtsystem betrachtet werden. Dabei zeigt sich, daß diejenigen Hamilton-Operatoren, die die Wechselwirkung zwischen offenen und abgeschlossenen Schalen am besten wiedergeben, auch die besten Energien liefern.

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Résumé Discussion des erreurs dans les calculs Hartree-Fock à couche ouverte. On considère divers hamiltoniens pour les méthodes à valeur propre unique, et l'on montre qu'une meilleure énergie est obtenue avec les hamiltoniens qui rendent compte correctement de l'interaction couche ouvertecouche fermée.

     

Sources of error in open shell Hartree-Fock calculations

It is demonstrated that single eigenvalue methods for solving open shell Hartree-Fock equations can give an energy that is not invariant to a unitary transformation of the orbitals. This is discussed with reference to potential curves for polyatomic molecules.     

Variational grand-canonical electronic structure method for open systems

An ab initio method is developed for variational grand-canonical molecular electronic structure of open systems based on the Gibbs-Peierls-Boguliobov inequality. We describe the theory and a practical method for performing the calculations within standard quantum chemistry codes using Gaussian basis sets. The computational effort scales similarly to the ground-state Hartree-Fock method. The quality of the approximation is studied on a hydrogen molecule by comparing to the exact Gibbs free energy, computed using full configuration-interaction calculations. We find the approximation quite accurate, with errors similar to those of the Hartree-Fock method for ground-state (zero-temperature) calculations. A further demonstration is given of the temperature effects on the bending potential curve for water. Some future directions and applications of the method are discussed. Several appendices give the mathematical and algorithmic details of the method.     

Geometry optimization of molecular structures in solution by the polarizable continuum model

A new implementation of analytical gradients for the polarizable continuum model is presented, which allows Hartree-Fock and density functional calculations taking into account both electrostatic and nonelectrostatic contributions to energies and gradients for closed and open shell systems. Simplified procedures neglecting the derivatives of the cavity surface and/or using single spheres for XH_n groups have also been implemented and tested. The solvent-induced geometry relaxation has been studied for a number of representative systems in order to test the efficiency of the procedure and to investigate the role of different contributions. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 404–417, 1998     

Nonstationary perturbation theory for open-shell molecules

Nonstationary perturbation theory equations have been obtained for open-shell molecules. The equations were formulated in terms of a density matrix in the MO-LCAO method. The first variant is coupled perturbation theory in the framework of the restricted Hartree-Fock method for open shells, and the second variant is variational perturbation theory for ground and excited electronic states of molecules, in which the perturbed wave function of the system is constructed in the form of a superposition of the ground and singly excited configurations composed of the Hartree-Fock orbitals of the open shell. A calculation of the Cauchy moments of the dynamic dipole polarizability of several molecules of conjugated open-shell hydrocarbons, viz., doublet states of odd alternant hydrocarbons, as well as triplet excited states and doublet states of radical ions of even alternant hydrocarbons, has been carried out in the framework of both methods.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 21, No. 1, pp. 18–27, January–February, 1985.     

A family of intracules, a conjecture and the electron correlation problem

We present a radical approach to the calculation of electron correlation energies. Unlike conventional methods based on Hartree-Fock or density functional theory, it is based on the two-electron phase-space information in the Omega intracule, a three-dimensional function derived from the Wigner distribution. Our formula for the correlation energy is isomorphic to the Hartree-Fock energy expression but requires a new type of four-index integral. Preliminary results, obtained using a model that is based on the known correlation energies of small atoms, are encouraging.     

Improved perturbed uncoupled Hartree-Fock (IUCHF) methods for second-order physical properties

Two different kinds of improved uncoupled Hartree-Fock methods (IUCHF-I and IUCHF-II) are proposed for the calculation of second-order physical properties. These methods provide a better approximation to the coupled Hartree-Fock (CHF) second-order energy than the ordinary uncoupled Hartree-Fock (UCHF) theory.     

A new energy decomposition scheme for molecular interactions within the Hartree-Fock approximation

A new method is proposed for the analysis of components of molecular interaction energy within the Hartree-Fock approximation. The Hartree-Fock molecular orbitals of the isolated molecules are used as the basis for the construction of Fock matrix of the supermolecule. Then certain blocks of this matrix are set to zero subject to specify boundary conditions of the supermolecule molecular orbitals, and the resultant matrix is diagonalized iteratively to obtain the desired energy components. This method can be considered as an extension of our previous method, but has an advantage in the explicit definition of the charge transfer energy, placing it on an equal footing with the exchange and polarization terms. The new method is compared with existing perturbation methods, and is also applied to the energy and electron density decomposition of (H₂O)₂.     

Coupled-cluster calculations of vibrational Raman optical activity spectra

We present the first calculations of Raman optical activity spectra at the coupled-cluster level of theory. Calculations are presented for (S)-methyloxirane and compared to recent experimental gas-phase measurements as well as the results obtained at the Hartree-Fock and density functional level of theory using the popular B3LYP functional. For the experimentally relevant frequency region of 400-1600 cm(-1), the Hartree-Fock, B3LYP and coupled-cluster spectra are very similar when the same force field is used, and the results also agree well with experiment. For high-frequency vibrational modes, differences in the ROA difference parameters are observed and are analyzed. The new coupled-cluster ROA code will allow for critical benchmarking of the accuracy of modern exchange-correlation functionals in the calculation of ROA spectra.     

A reformulation of the coupled perturbed self-consistent field equations entirely within a local atomic orbital density matrix-based scheme

To exploit the exponential decay found in numerical studies for the density matrix and its derivative with respect to nuclear displacements, we reformulate the coupled perturbed self-consistent field (CPSCF) equations and a quadratically convergent SCF (QCSCF) method for Hartree-Fock and density functional theory within a local density matrix-based scheme. Our D-CPSCF (density matrix-based CPSCF) and D-QCSCF schemes open the way for exploiting sparsity and to achieve asymptotically linear scaling of computational complexity with molecular size (M), in case of D-CPSCF for all (M) derivative densities. Furthermore, these methods are even for small molecules strongly competitive to conventional algorithms.     

Two-dimensional fully numerical solutions of molecular Hartree-Fock equations: LiH and BH

A fully numerical Hartree-Fock approach is developed for diatomic molecules. The exchange potential is solved relaxing a local, Poisson-like equation. Improved Hartree-Fock limits are reported for LiH and BH.     

Alternative wavefunction ansatz for including explicit electron-proton correlation in the nuclear-electronic orbital approach

The nuclear-electronic orbital (NEO) approach treats specified nuclei quantum mechanically on the same level as the electrons with molecular orbital techniques. The explicitly correlated Hartree-Fock (NEO-XCHF) approach was developed to incorporate electron-nucleus dynamical correlation directly into the variational optimization of the nuclear-electronic wavefunction. In the original version of this approach, the Hartree-Fock wavefunction is multiplied by (1+G?), where G? is a geminal operator expressed as a sum of Gaussian type geminal functions that depend on the electron-proton distance. Herein, a new wavefunction ansatz is proposed to avoid the computation of five- and six-particle integrals and to simplify the computation of the lower dimensional integrals involving the geminal functions. In the new ansatz, denoted NEO-XCHF2, the Hartree-Fock wavefunction is multiplied by √(1+G?) rather than (1+G?). Although the NEO-XCHF2 ansatz eliminates the integrals that are quadratic in the geminal functions, it introduces terms in the kinetic energy integrals with no known analytical solution. A truncated expansion scheme is devised to approximate these problematic terms. An alternative hybrid approach, in which the kinetic energy terms are calculated with the original NEO-XCHF ansatz and the potential energy terms are calculated with the NEO-XCHF2 ansatz, is also implemented. Applications to a series of model systems with up to four electrons provide validation for the NEO-XCHF2 approach and the treatments of the kinetic energy terms.     

Hartree-fock calculations for excited rydberg states

A method is introduced which allows to compute self-consistent restricted Hartree-Fock wave functions for excited Rydberg configurations. The concepts of reorganization and electron correlation of Rydberg states are discussed. As an illustration Hartree-Fock calculations for the (ls)(nl) Rydberg series of He are presented.