Test of an integral approximation scheme based on semiorthogonalized orbitals in LCAO SCF MO CI calculations of naphthalene

Ab initio LCAO SCF MO CI calculations of naphthalene are carried out with a minimal basis set to test an integral approximation scheme proposed in a previous paper. When 71.3 and 53.0% of the two-electron integrals are neglected, the errors in the SCF total energy are only ?0.0534 and ?0.0006 a.u., respectively. In the latter case, the maximum absolute errors of the orbital energy and the gross AO population are 0.007 a.u. and 0.001, respectively. Even in the former case the errors of the excitation energies are less than 0.0004 a.u. Errors of oscillator strengths are also examined and are found to be tolerably small.     

Gaussian approximation of exponential type orbitals based on B functions

This work gives new, highly accurate optimized gaussian series expansions for the B functions used in molecular quantum mechanics. These functions are generally chosen because of their compact Fourier transform, following Shavitt. The inverse Laplace transform in the square root of the variable is used for Gauss quadrature in this work. Two procedures for obtaining accurate gaussian expansions have been compared for the required extended precision arithmetic. The first is based on Gaussian quadratures and the second on direct optimization. Both use the Maple computer algebra system. Numerical results are tabulated and compared with previous work. Special cases are found to agree before pushing the optimization technique further. The optimal gaussian expansions of B functions obtained in this work are available for reference. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Graph theory of molecular orbitals beyond tight-binding approximation

The graph theory of molecular orbitals including second neighbor interactions (η) is considered here. Graphical methods of getting the characteristic polynomial for the π system of a conjugated molecule are given. It is shown that the characteristic polynomial can be factorized if there are symmetries in the molecule.     

Exact analytic computer calculation on a four-center integral with slater orbitals

A method has been developed for reducing an initial sixfold four-center integral to a linear combination of a finite number of terms containing elementary and special functions. The maximum order of integration in the closed expressions is two. The method has been implemented with computers designed to automate handling literal expressions, and the necessary programs have been written. A four-center integral second in complexity has been calculated.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 22, No. 4, pp. 491–494, July–August, 1986.     

Error evaluation of integral methods by consideration on the approximation of temperature integral

Summary In this paper, the integral methods in general use are divided into two types in terms of their different ways to in order to deal with the temperature integral p(x): for Type A the function h(x)=p(x)x²e^x is regarded as constant vs. x, while for Type B h(x) varies vs. x and ln[p(x)] is assumed to have the approximation form of ln[p(x)]=alnx+bx+c (the coefficients a, b, and c are constant). The errors of kinetic parameters calculated by these two types of methods are derived as functions of x and analyzed theoretically. It is found that Type A methods have the common errors of activation energy, while the Coats-Redfern method can lead to more accurate value of frequency factor than others. The accuracy of frequency factor can be further enhanced by adjusting the expression of the Coats-Redfern approximation. Although using quite simple approximation of the temperature integral, the Coats-Redfern method has the best performance among Type A methods, implying that usage of a sophisticated approximation may be unnecessary in kinetic analysis. For Type B, the revised MKN method has a lower error in activation energy and an acceptable error in frequency factor, and thus it can be reliably used. Comparatively, the Doyle method has higher error of activation energy and great error of the frequency factor, and thus it is not recommended to be adopted in kinetic analysis.     

A population analysis based on hybrid orbitals

By a way of partitioning of Mulliken overlap populations, the hybrid orbitals of central atoms and ligands are obtained by the CNDO method. The dipole moments of some hybrids are calculated using the hybrid orbitals obtained, and they are mostly in accordance with experimental values. In addition, overlap integrals may be decomposed into components, and then σ, π, and δ hybrid orbitals can be obtained.     

Localized orbitals based on the fermi hole

The Fermi hole provides a direct (non-iterative) method for tansforming canonical SCF molecular orbitals into localized orbitals. Except for simple overlap integrals required to maintain orthogonality, this method requires no integrals over orbitals or basis functions. This method is demonstrated by application to a furanone (C₄H₄O₂), methylacetylene, and boron trifluoride. The results of these calculations are compared to those determined by the orbital centroid criterion of localization.     

New linear integral kinetic parameters assessment method based on an accurate approximate formula of temperature integral

This work concerns a proposition of a new assessment method to obtain kinetic parameters from nonisothermal solid-state kinetics, based on a new and accurate approximate formula of temperature integral. The new formula was derived numerically by a two-step linearly fitting process without using any further approximating series. The relative error involved in the activation energy has been estimated and found to be less than 0.001% in the practical range of 15 < x < 60. A comparison of the suggested approximations to published approximates has shown significant improvements in terms of accuracy at high and low x values. The validity of the new method has been confirmed by computing activation energy from experimental data. Moreover, two approaches have been proposed to determine the kinetic reaction model and preexponential factor based on the new approximate formula. The comparison of the obtained results arising from the application of the present method to others obtained by the most widely reported methods in the literature shows a remarkable preeminence of the new method.     

Improvement on macroscopic compressibility approximation and beyond

A numerical procedure is proposed to extend the thermodynamic perturbation expansion (TPE) to a higher order. It is shown that the present second order term is superior to that due to a macroscopic compressibility approximation (MCA), a local compressibility approximation, and a superposition approximation by Barker and Henderson [Rev. Mod. Phys. 48, 587 (1976)]. Extensive model calculation and comparison with simulation data available in literature and supplied in the present report indicate that the present third order TPE is superior to a previous second order TPE based on the MCA, two previous perturbation theories, which are respectively based on an analytical mean spherical approximation for an Ornstein-Zernike equation, and an assumed explicit functional form for the Laplace transform of radial distribution function multiplied by radial distance, and a recent generalized van der Waals theory. The present critical temperature for a hard core attractive Yukawa fluid of varying range is in very good agreement with that due to a hierarchical reference theory. The present third order TPE is computationally far more modest than the self-consistent integral equation theory, and therefore is a viable alternative to use of the latter.     

Configuration interactions with molecular orbitals from the SCF Xα scattered-wave approximation

A method has been devised for calculating the state energies and wave functions for molecules, with computer algorithms and implementation programs. Molecular integrals can be determined in an orbital basis by the X-SW method and can be used in programs based on nonempirical methods, which serve to extend the X. method. Estimates have been made of the state energies with correction for electron correlation for titanium and zirconium monofluorides.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 26, No. 2, pp. 137–142, March–April, 1990.     

ACE algorithm for the rapid evaluation of the electron-repulsion integral over Gaussian-type orbitals

A new series of general formulas to evaluate the electron-repulsion integral (ERI) can be derived from modifying the Gauss-Rys quadrature formula. These named as “accompanying coordinate expansion (ACE) formulas” are capable of evaluating very fast ERIs, especially for contracted Gaussian-type orbitals (GTOs). According to the degree of the contraction of GTOs, the optimum formula can be selected among these ACEs. Numerical examples are shown for (ps|ps) and (pp|pp) ERIs as typical examples. It is found that the present ACE algorithm is numerically stable and is most efficient among all algorithms in the literature in the floating-point-operation (FLOP) count for all varieties of the degree of contraction. © 1996 John Wiley & Sons, Inc.     

Ab initio calculations on large molecules: The multiplicative integral approximation

In the multiplicative integral approximation (MIA), two-electron integrals are evaluated using an expansion of a product of two Gaussians in terms of auxiliary functions. An estimator of the error introduced by the approximation is incorporated in the self-consistent field (SCF) calculations and the integrals for which the error estimate is larger than a preset value are systematically corrected. In this way the results of a MIA-assisted calculation have the same accuracy as a conventional calculation. The full exploitation of the expansion technique while constructing the Fock-matrix allows important time savings. Results are presented for a number of test cases.     

Molecular orbital calculations based on linear combinations of fragment orbitals

A semiempirical MO method based on localized fragment orbitals has been developed, which is particularly suited for the construction of orbital correlation diagrams for the discussion of the electronic structure of complex molecules in terms of fragments and their interactions. The method allows for the inclusion of experimental ionization potentials and electron affinities of the fragments within the calculation of the Fock matrix elements and may thus form the basis of an interpretation of photoelectron spectra, comparable to the interpretation of UV spectra by means of the MIM method of Longuet-Higgins and Murrell. Several levels of approximation are discussed using the acrolein molecule as an example.     

A multi-configuration reference CEPA method based on pair natural orbitals

Summary A multi-reference CI scheme is proposed which is aiming at a considerable reduction of the generally very large number of configurations of CI expansions in multi-configuration reference cases. This reduction is achieved by combining the idea of internal contraction, the concept of pair natural orbitals (PNO's) and CEPA (coupled electron pair) type approximations for the contributions of higher than double excitations. This latter estimate leads to size consistent results and also permits to employ reference wavefunctions that contain only the dominantly occupied configurations of the considered system. Applications to two test cases, the lowest states (³ P,¹ D and¹ S) of the carbon atom and the symmetry forbiddenC _2v insertion reaction of Be and H₂, show that our method is able to truncate CI expansions to lengths of no more than 10³–10⁴ without losing more than 1–2% of the correlation energy. The calculated excitation energies and energy barriers agree with the full CI results in the respective basis within about 1 kcal/mol. Thus the MC-CEPA-PNO method presents a very efficient way to obtain chemical accuracy in CI-calculations for molecular systems.Dedicated to Prof. Dr. W. Kutzelnigg on the occasion of his 60th birthday     

On a definition of bond energies based on localized molecular orbitals

A theoretical model is presented for defining bond energies based on localized molecular Orbitals. These bond energies are obtained by rearranging the total SCF energy including the nuclear repulsion term to a sum over orbital and orbital interaction terms and then to total orbital terms, which can be interpreted as the energies of localized orbitals in a molecule. A scaling procedure is used to obtain a direct connection with experimental bond dissociation energies. Two scale parameters are employed, the C-C and the C-H bond dissociation energy in C₂H₆ for A-B and C-H type bonds, respectively. The implications of this scaling procedure are discussed. Numerical applications to a number of organic molecules containing no conjugated bonds gives in general a very satisfactory agreement between experimental and theoretical bond energies.     

Spatially ioslated redox orbitals - an update

The concept of the “spatially isolated orbital” was introduced to rationalize the unusual optical and magnetic properties determined for metal chelate complexes, ML₃ⁿ⁺, in which a chelate ligand, L, as 2,2′-bipyridine possesses low energy π^* orbitals. Optical excitation or electrochemical reduction produces a species whose effective symmetry is that of the ligand alone. This review discusses the results associated with the redox orbital emphasizing anomalies that occur.     

Evaluation of the charge penetration energy between non-orthogonal molecular orbitals using the Spherical Gaussian Overlap approximation

An overlap dependent formula for evaluating the charge penetration energy between non-orthogonal molecular orbitals is derived using the Spherical Gaussian Overlap approximation. When combined with an accurate multipole representation of the electrostatic energy, such as in the effective fragment potential method, ab initio electrostatic energies are generally reproduced to within 0.2 kcal/mol for a variety of molecular dimers and basis sets. The only larger error is for the DMSO dimer, where the electrostatic energy is overestimated by 0.7 kcal/mol.