A simple random‐walk method for obtaining ab initio solutions of the Schrodinger equation is examined in its application to the case of the molecular ion H+3 in the equilateral triangle configuration with side length R=1.66 bohr. The method, which is based on the similarity of the Schrodinger equation and the diffusion equation, involves the random movement of imaginary particles (psips) in electron configuration space subject to a variable chance of multiplication or disappearance. The computation requirements for high accuracy in determining energies of H+3 are greater than those of existing LCAO–MO–SCF–CI methods. For more complex molecular systems the method may be competitive.

A random‐walk simulation of the Schrödinger equation: H+3

The characteristic features of model potentials, effective potentials and pseudopotentials are carefully investigated. Then we justify our choice to work only with hermitian pseudopotential operators, and we develop a general non-empirical method to determine atomic pseudopotentials. In view of their numerical use for molecular calculations, these pseudopotentials are cast into semi-local forms, and their parameters are obtained by a least-squares process; tables of parameter values are given for the two first rows of the periodic system.

A theoretical method to determine atomic pseudopotentials for electronic structure calculations of molecules and solids

A formalism is developed for obtaining ab initio effective core potentials from numerical Hartree–Fock wavefunctions and such potentials are presented for C, N, O, F, Cl, Fe, Br, and I. The effective core potentials enable one to eliminate the core electrons and the associated orthogonality constraints from electronic structure calculations on atoms and molecules. The effective core potentials are angular momentum dependent, basis set independent, and stable against variational collapse of their eigenfunctions to core functions. They are derived from neutral atom wavefunctions using a pseudo‐orbital transformation which is motivated by considerations of the expected accuracy of their use and of basis set economy in molecular calculations. Then the accuracy is demonstrated by multiconfiguration Hartree–Fock calculations of potential energy curves for HF, HCl, HBr, HI, F2, Cl2, Br2, and I2 and one‐electron properties for HF and HBr. The differences between valence‐electron calculations employing the present...

Ab initio effective core potentials: Reduction of all-electron molecular structure calculations to calculations involving only valence electrons

The applications, most of which have been developed in the last decade, of the Hellmann-Feynman (H-F) theorem in molecular quantum mechanics are reviewed. In general, the forces (on the nuclei of molecules) calculated with the use of this theorem provide great qualitative insight into the nature of the phenomena investigated; outstanding examples of these are in the concepts of chemical binding and molecular shapes. However, there are serious limitations in quantitative applications of the H-F theorem with approximate wave functions, since the calculated forces are extremely sensitive to small inaccuracies in the wave functions, especially near the nuclei of interest. Nevertheless, in view of the fact that it is difficult to discern general qualitative features in very accurate or ab initio molecular calculations, the H-F theorem is likely to be a highly useful tool for developing much needed qualitative chemical models which will be based on firm quantum mechanical foundations and will also remain open to quantitative extension, at least in principle.

The Force Concept in Chemistry

Triatomic hydrogen positive ion surface crossing effects in chemical reactions based on potential energy surfaces calculation using diatomics-in- molecules approach

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Effects of surface crossing in chemical reactions - The H3 system

Self‐consistent‐field and CI variational calculations are presented for the ground state of H3+ using a basis of floating 1s Gaussian orbitals. The best SCF calculation, thought to be near the Hartree—Fock limit, gives E=−1.29993 a.u. for Re=1.6405 a.u. in equilateral triangular (D3h) geometry. The CI wavefunction which is the best rigorous variational result presently available for H3+ gives E=−1.33764 a.u. at Re=1.6504 a.u. in D3h geometry. It is estimated to lie 0.004–0.01 a.u. above the exact result. The electronic structure of H3+ is interpreted in terms of three‐center bonds since the electron density is found to be higher within the triangle than along its sides. Enthalpy, entropy, and Gibbs free energy are computed, and the thermodynamics of several reactions of H3+ examined. In particular, the standard molar enthalpy of formation is found to have a rigorous upper bound of 270 kcal. Variational studies are also presented for D3h H3+ + which give E=−0.12373 a.u. at R=1.68 a.u.

Ab Initio Studies of Small Molecules Using 1s Gaussian Basis Functions. II. H3

Exploratory studies of the ground states of H2+, H2, He2+ +, and HeH+ using a basis of 1s Gaussian orbitals are presented. The emphasis is on the direct calculation of molecular wavefunctions with complete optimization of all parameters including the position and size of all orbitals. For the two‐electron systems both self‐consistent‐field molecular‐orbital and many‐configuration studies are done. Results for total energies, MO orbital energies, and equilibrium bond lengths are satisfactory when compared to other variational studies of various degrees of accuracy, including some nearly exact studies. As expected, the larger nuclear charge in He2+ + and the lower symmetry of HeH+ make convergence slower than for H2+ and H2. Further examination of the H2 results is made by calculation of the averages of the nonenergy quantities x2, z2, and 1/r12.

Ab Initio Studies of Small Molecules Using 1s Gaussian Basis Functions. I. Exploratory Calculations

A new model potential for one‐valence‐electron atoms is introduced. It consists [Eq. (6)] of a core Coulomb potential modified by the addition of a Gaussian screened Coulomb potential. This potential has the desired features outlined for a model potential, and it has particularly convenient and simple mathematical properties when used with Gaussian basis functions. Since the smooth valence orbitals are sought, Gaussian functions are a good basis set because their deficiencies in nuclear regions do not enter the problem. The potential is calibrated to experimental energies for the 2s and 2p states of Li and 3s and 3p states of Na, using extended basis sets. The model Hamiltonians so defined are used with a variety of more modest basis sets to determine pseudowavefunctions. Based on comparisons with ab initio orbitals, energy analyses, radial density calculations, and overlap and expectation value calculations, the conclusion is that good valence pseudowavefunctions can be obtained by this approach with relatively small basis sets. The approach looks promising for valence‐only molecular studies.

Valence Electron Studies with Gaussian‐Based Model Potentials and Gaussian Basis Functions. I. General Discussion and Applications to the Lowest s and p States of Li and Na

A previously developed simple valence‐only electronic structure theory based on atomic core model potentials and using flexible Gaussian valence basis functions is applied to the two‐valence electron systems Li2, Na2, NaLi, LiH, NaH, Li3+, Na3+, Li2Na+, LiNa2+, LiH2+, Li2H+, NaH2+, Na2H+, and H2, within the SCF MO model. Results for calculated equilibrium geometries and energy changes for certain chemical reactions are compared to the corresponding quantities from analogous all‐electron ab initio calculations. The model potential results are quite similar to those from the ab initio ones and are generally satisfactory for such a simple theory. There is a tendency toward slightly long internuclear distances (average deviation of all unique independent distances is +0.06 bohr) and slightly high energy of complex relative to separate constituents (average deviation of dissociation reaction energies is −2.1 kcal/mole) in these calculations relative to the all‐electron ones, the explanation of which will requi...

Valence electron studies with Gaussian‐based model potentials and Gaussian basis functions. III Applications to two‐valence‐electron systems composed of combinations of Li, Na, H, or their unipositive ions

The ground states of the three‐ and four‐electron systems He2+, He2, and linear symmetric H3, H4+, and H4 are investigated using the linear combinations of Gaussian‐type orbitals (LCGTO) self‐consistent‐field molecular‐orbital (SCF MO) approach The basis functions are 1s GTO's, not necessarily located at nuclear positions Optimized, near‐Hartree–Fock (HF) wavefunctions are presented for the isoelectronic H3 and He2+ systems, for which such results were not previously available They are found to have similar correlation energies With simpler wavefunctions, E‐vs‐R plots for H3 and He2+ give good bond lengths and shapes of the energy curve It is found that previous SCF MO studies have not closely approximated the true HF function for He2 The H4+ calculations are the first SCF MO results available for the species, and indicate its instability with respect to H3+ + H, in agreement with recent experimental data The GTO studies for H4 are the only rigorously executed ab initio calculations yet done for th

Ab Initio Studies of Small Molecules Using 1s Gaussian Basis Functions. III. LCGTO SCF MO Wavefunctions of the Three‐ and Four‐Electron Systems He2+, He2, and Linear H3, H4+, H4

Maurice E. Schwartz

Papers

Ab Initio Studies of Small Molecules Using 1s Gaussian Basis Functions. II. H3

Ab Initio Studies of Small Molecules Using 1s Gaussian Basis Functions. I. Exploratory Calculations

Valence Electron Studies with Gaussian‐Based Model Potentials and Gaussian Basis Functions. I. General Discussion and Applications to the Lowest s and p States of Li and Na

Valence electron studies with Gaussian‐based model potentials and Gaussian basis functions. III Applications to two‐valence‐electron systems composed of combinations of Li, Na, H, or their unipositive ions

Ab Initio Studies of Small Molecules Using 1s Gaussian Basis Functions. III. LCGTO SCF MO Wavefunctions of the Three‐ and Four‐Electron Systems He2+, He2, and Linear H3, H4+, H4