Basis Set Of Slater-type Orbitals Research Articles

On the calculation of the R-matrix by arbitrary basis sets for one-channel elastic electron-helium scattering

From the calculations of electron scattering by the helium atom, the authors present a rule showing how to choose an arbitrary basis set of Slater-type orbitals for the scattering electron such that the R-matrix becomes convergent.

Pair-correlation energies in sodium hydride with many-body perturbation theory

Many-body perturbation theory (MBPT) is applied to the electronic structure of the sodium hydride molecule using a basis set of Slater-type orbitals. Both a ${V}^{N}$ and a ${V}^{N\ensuremath{-}1}$ potential are considered for determining the virtual orbitals, but these are shown to give nearly equivalent results when all pairlike diagrams are summed through third order in the MBPT expansion using shifted denominators. A value of -0.2544 hartrees is obtained for the correlation energy, which is \ensuremath{\sim}59% of the experimental value. A clear separation into intrashell and intershell correlation energy effects is observed.

Low-lying valence states of the PO molecule according to configuration-interaction calculations

Ab initio configuration-interaction (CI) calculations were carried out for low-lying 2Π, 4Π, 2Σ+, 2Σ−, 4Σ−, 2Δ, and 2Φ valence states of the PO molecule at 11 internuclear distances from 2.45 to 6.5 a.u. All configurations that can be formed from a minimal basis set of Slater-type orbitals, with the restriction of a frozen core, were included. Of the 57 states that dissociate up to 2P(P) + 1S(O), seven 2Π, two 4Π, three 2Σ+, three 2Σ−, one 4Σ−, two 2Δ, and two 2Φ states show a minimum in the potential energy curve. This is to be compared with seven observed valence states.

Long-range interaction between hydrogen atoms: estimate of third-order contributions

Third-order long-range coefficients C 11, C 13, C 15 for two hydrogen atoms in their ground states are evaluated by perturbation-variation methods in terms of finite basis sets of Slater-type orbitals. Some tests for the accuracy of the variational solutions are proposed and discussed.

Perturbation theories for the calculation of molecular interaction energies. II. Application to H2+

The many perturbation expansions reviewed in Paper I [D. M. Chipman, J. D. Bowman, and J. O. Hirschfelder, J. Chem. Phys. 59, 0000 (1973)] are applied to the interaction of a ground state hydrogen atom with a proton to form the 1s σg and 2p σu electronic states of H2+. The calculations were made with high precision for the range of separations R = 0.2–15ao using a large basis set of Slater-type orbitals. For the polarization and symmetrized polarization methods the energies were obtained analytically, these results providing a check on the completeness of the basis set. Also, long range and short range asymptotic formulas are given for the first order wavefunctions and the perturbation energies. The HS and MSMA methods gave the best second order energies. The EL-HAV second order energy is good at intermediate separations but becomes bad at large separations. The expectation value of the Hamiltonian using the wavefunction truncated after the first order (where the coefficient of the zero order function is energy optimized) gave very good results in the OPT, COR, CH, HAV, EL, HS, DEM-I, AM, and SYM-P methods in the order listed; the MSMA and P results were poor due to the use of unsymmetrized wavefunctions. The OPT treatment, which energy optimizes the coefficients of three separate first order functions, has an error of less than 0.04 kcal/mole for the 2p σu state, over the full range of separations, and for the 1s σg state has an error of only 1.4 kcal/mole at the equilibrium separation R = 2ao, becoming much better than this at larger separations. Reasonably good interaction energies can be calculated from the polarization function φ p(1) without using the auxiliary functions θ and ω. In the limit of large separations, we find that the EL-HAV second order energy approaches 41/54 = 0.759 ··· times the exact Coulombic energy, rather than the previously estimated factor of one-half. Also, at large R, the SYM-P, HS, and DEM-I exchange energy agrees with Herring's exact asymptotic expression to within 0.1%.

AB initio SCF calculation of the force constants and the dipole moment derivatives in methanol

Using a gaussian lobe basis set nearly equivalent to the double-zeta Slater type orbital basis set, the principal force constants and the derivatives of the electric dipole moment are calculated. The force constants are found to be in satisfactory agreement with experimental values, and the dipole moment derivatives are in qualitative agreement with preliminary experimental relative intensities.

Theoretical Evidence for a Stable form of Cyclic Ozone, and its Chemical Consequences

The total energy of the symmetrical, bent O3 molecule is studied as a function of internuclear angle and internuclear distance. The method used is an ab initio molecular orbital calculation with a minimum basis set of Slater-type orbitals, plus limited configuration interaction. Best single configuration energies show the existence of two stable minima: a cyclic (60°) structure and a bent (115°) structure. The cyclic structure is preferred by 48 kcal/mol, contrary to experiment (116° 48′) Configuration interaction results with two low lying orbitals show the cyclic form still preferred by 6 kcal/mol. A transition state for the bent-cyclic pathway lies 26 kcal/mol (or less) above the bent form. Based on these results, a possible low energy path via the cyclic form is proposed for the thermal decomposition of ozone. It is also proposed that cyclic ozone may have been observed in the pulse radiolysis of oxygen.

Two-center calculations for x-ray scattering

An analytical procedure for the evaluation of two-center integrals in x-ray scattering calculations has been developed. The scattering potential is represented in prolate spheroidal coordinates. The plane wave operator is expanded in spheroidal wave-functions. All integrals are evaluated analytically with the aid of stable recursion relations. It is found that the several infinite sums involved are rapidly convergent, so that a practical computational method is at hand. The method is illustrated with a calculation of the molecular scattering factor for C-H from a near Hartree-Fock wavefunction comprised of an extended basis set of Slater-type orbitals. The coherent x-ray scattering intensity of HZ has been computed from a natural spin orbital expansion of the exact electronic wavefunction for the 1 Σ g + state of H 2.

Calculation of Hyperfine Coupling Constants in Inorganic Radicals

Hyperfine coupling constants have been calculated for the well-known radicals HCO, NO2, NF2, ClO2, and CF3 as well as a large number of related species. An SCF Hartree-Fock-type approach was used with a minimum basis set of Slater-type orbitals to obtain MO coefficients of the singly occupied orbital. Some of the coupling constants were obtained directly from the ab initio calculations; others were estimated by use of semiempirical relations. Comparison with experimental data shows fair agreement. In addition, the structures of NF2, O3+,O3−, O3, and CF3 were computed.

A comparison of the resonance, perfect pairing, and molecular orbital wavefunctions for BeH2

Series of wavefunctions for the ground state of BeH2 have been compared in the frameworks of the valence bond and the molecular orbital theories for Be-H distances between 1·5 a.u. and 6·0 a.u. using minimum basis sets of Slater-type orbitals with exponents optimized at each internuclear distance. It is confirmed that the model based on the resonance valence state can be useful, particularly when ionic configurations are incorporated with it, although the model emphasizing perfect pairing and including an additional parameter to determine ionic character provides significantly the lowest energies for cases where there is a single degree of freedom in the mixing of valence bond structures. Both molecular orbital and valence bond calculations indicate it is a very good approximation to consider BeH2 as involving localized electron-pair bonds.

A Semi-empirical Analysis of the Electronic Spectra and Molecular Structures of Hydrogen Cyanide and Carbon Dioxide

All valence electron molecular orbital calculations have been carried out on hydrogen cyanide and carbon dioxide using a minimum basis set of Slater-type orbitals by a semi-empirical SCF-LCAO-MO method described earlier for formaldehyde. Since many of the excited states of these molecules are bent, a series of calculations is carried out on hydrogen cyanide in the linear configuration and in bent configurations with H—C—N angles of 150°, 120°, and 90°, and on carbon dioxide in the linear configuration and bent configuration with O—C—O angle of 90°. The binding energy changes in various molecular orbitals on bending are used to interpret electronic spectra and are also compared with Walsh's curves.

Potential Curves for the Valence-Excited States of Silicon Monoxide. A Theoretical Study

Ab initio quantum mechanical calculations have been carried out at 10 internuclear separations for those 72 molecular states of silicon monoxide which dissociate to a Si atom in 3P, 1D, 1S, or 5S state plus an oxygen atom in 3P, 1D, or 1S state. Full configuration interaction calculations were made from a minimal basis set of Slater-type orbitals, with the restriction that the core orbitals (1s, 2s, 2p Si and 1s O) were fully occupied. The results were strikingly similar to those obtained for CO in an analogous theoretical study. Ten bound states were found to dissociate to ground state Si plus ground state O; five of these states have been observed experimentally in SiO, but eight have experimentally known counterparts in CO. The predicted ordering of states is X 1Σ+, a 3II, a′ 3Σ+, 3Δ, e 3Σ−, I 1Σ−, A 1II, 1Δ, 5Σ+, and 5II. Molecular orbital configurations are assigned to each of these 10 states. Among 10 higher predicted bound states, the 1II III state, with calculated De=1.32 eV, is perhaps the most likely to be observed experimentally. The tentative assignment of the I 1Σ− state by Verma and Mulliken is considered unlikely on the basis of the present calculations. The A 1II potential curve is very flat as it approaches the dissociation limit, but, unlike CO, there is no maximum in this curve for SiO.

Symmetry-Adapted Pair Correlations for Open-Shell Systems: Correlation Energy and Hyperfine Structure of the Nitrogen Atom

Using two different extended basis sets of Slater-type orbitals, we have carried out a study of the correlation energy and hyperfine structure of the $^{4}S$ ground state of the nitrogen atom. Symmetry-adapted pair correlations were evaluated with respect to a multiconfiguration reference state, the first-order wave function. It was found that each symmetry-adapted pair correlation energy could be rather accurately (to about 3%) approximated as a sum of effects due to symmetry-determined classes of configurations. Using the larger of the two basis sets, the effects of 1633 configurations, constructed from 19 530 distinct Slater determinants, were considered. With this basis set the correlation energy obtained was 0.181 75 hartree, compared to the estimated correlation energy 0.186 hartree. It was found that pair correlations had large canceling effects on the hyperfine structure, the net result of which was a spin density 77% of experiment, and only slightly larger than the first-order spin density, which includes core polarization effects. Several points in common are found in a comparison with the many-body perturbation theory results of Dutta, Matsubara, Pu, and Das. The most useful and interesting result of the present work is a clear picture of the different effects which contribute to the correlation energy and hyperfine structure of $^{4}S$ N.

Molecular SCF calculations for the ground state of some three-membered ring molecules: Cis and trans diaziridine, oxaziridine and the corresponding imminium ions

LCAO SCF MO calculations with minimal basis sets of Slater-type orbitals are performed for some three-membered heterocycles. A reinterpretation of the resulting wavefunctions in terms of localized (exclusive) orbitals is employed to show the bending of the ring bonds and to measure the degree of transferability of some groups among chemically different yet related molecules. The electrostatic potentials produced in the neighbouring space by the nuclear and electronic charge distributions are evaluated and employed to indicate the molecular sites most likely subject to electrophilic attack. Some features of the electrostatic potential may be easily related to the entities employed in the usual intuitive description of chemical binding (lone pairs, etc.).

Potential energy surfaces of CH5+by non-empirical calculations

Potential energy surfaces have been calculated for various configurations of the CH5 + ion using a non-empirical LCAO SCF MO method with a minimum basis set of Slater-type orbitals. Emphasis has been placed on a wide coverage of the surface rather than on accurate values for the extrema. The approach of H2 along the C 3 axis of CH3 + leads to a stable Cs configuration and no activation energy is required. The D 3h and C 4v configurations can be considered as limiting forms of a C2v molecule in which H2 approaches the CH3 + along a C 2 axis. We have shown that the formation of CH5 + by this route requires a high activation energy.

Valence-Excited States of Carbon Monoxide

Precise quantum mechanical calculations have been performed on the 72 states of carbon monoxide which dissociate to a 3P, 1D, 1S, or 5S carbon atom plus a 3P, 1D, or 1S oxygen atom. A minimal basis set of Slater-type orbitals (optimized for the C and O atoms) was used, and holding the atomic 1s orbitals doubly occupied, full configuration interaction calculations were carried out for all molecular states at no fewer than nine internuclear separations. Seventeen bound states (De ≥ 0.27 eV) were obtained, eight of which have been observed experimentally. The theoretical ordering of known bound states agrees with experiment except for the a 3Π and A 1Π states. This fact is rationalized by noting that these two states have much smaller re values than do the other excited states. The most interesting of the unobserved predicted bound states are the 5Σ+ and 5Π states, which dissociate to 3PC+3PO, and the third 1Π state, with a relatively large calculated dissociation energy. A dominant molecular orbital configuration is associated with each of the first eleven bound states. Calculated spectroscopic constants are compared with available experimental data. Potential curves are presented and discussed for both bound and repulsive molecular states. The A 1Π state has a calculated maximum of 1135 cm−1, in good agreement with the experimentally based estimate (950 ± 150 cm−1) of Simmons, Bass, and Tilford. Calculations were also carried out on the X 1Σ+, a 3Π, and a′ 3Σ+ states using 1s atomic Hartree–Fock orbitals and 2s and 2p STO's optimized for the X state at the experimental re. These calculations correctly placed the a 3Π state below the a′ 3Σ+ and resulted in improved De and re values for all three states.

Theory of the Hyperfine Splittings of Pi-Electron Free Radicals. III. Methyl Radical in a Pyramidal Configuration: Temperature Dependence of the Hyperfine Splittings

Nonempirical calculations for methyl radical (CH3·) in a pyramidal configuration (C3υ) were performed using two minimum basis sets of Slater-type orbitals, one in which orbital exponents were chosen according to Slater's rules (unoptimized) and the other in which they were optimized by minimization of the SCF energy. The spin-restricted SCF plus configuration-interaction method, including all spin-adapted configurations with single and double excitations of space orbitals, was employed. The temperature dependence of the contact hyperfine splittings was computed assuming that the variation with temperature arises from the out-of-plane bending motion. Agreement with experiment at various temperatures is good. The optimized-basis values of the temperature coefficient of the proton splitting, daH / dT, are about 20%–50% too large and the unoptimized-basis results are about 10%–25% too small. Temperature coefficients of the carbon-13 splitting calculated using the optimized and unoptimized basis sets also bracket the experimental values, from which they deviate by less than 10%, which is within the experimental uncertainty. The variation of the spin densities at the magnetic nuclei with the out-of-plane angle is analyzed in detail. Other results of the calculation are discussed, namely, the equilibrium molecular geometry, an approximate frequency for the bending motion, and the incomplete orbital following.

Molecular SCF Calculations for the Ground State of Some Three-Membered Ring Molecules: (CH2)3, (CH2)2NH, (CH2)2NH2+, (CH2)2O, (CH2)2S, (CH)2CH2, and N2CH2

LCAO SCF MO calculations with minimum basis sets of Slater-type orbitals are performed for some three-membered ring compounds. A reinterpretation of the resulting wavefunctions in terms of localized (exclusive) orbitals is presented and discussed: The bent bonds in the rings are closely evidenced. The electrostatic potentials produced in the neighboring space by the nuclear and electronic charge distributions are evaluated and employed to evidence the molecular sites more likely subject to electrophilic attacks.

Determination of Accurate Single-Center Expansions with Slater-Type Orbitals: The H3 System

Electronic wavefunctions are determined for the linear symmetric H3 system. The basis functions are Slater-type orbitals centered on the central proton. For separations of 1.6, 1.7, and 1.8 bohr between adjacent protons, total energies of −1.62648, −1.63074, and −1.62897 a.u. are found using 70-term configuration-interaction wavefunctions. These energies are each in error by about 15 kcal/mole. Detailed analysis is made of the convergence. It is concluded that the single-center Slater-type orbital basis set is not particularly suited for the simple configuration interaction method for determining many-electron wavefunctions. It is suggested that higher accuracy might be achieved by using groups of Slater-type orbitals, taken from off-center hydrogen-atom expansions.

Nonempirical calculations of the isotropic hyperfine splitting constants of 13-C, 14-N, and 17-O in simple radicals.

The isotropic hyperfine splitting constants in ESR spectra for simple radicals 13C1H3, 14N1H2, and 17O1H have been calculated nonempirically by using a configuration-interaction method correct to the second order in the energy matrix elements. Included in the treatment are all the singly excited configurations plus certain types of doubly excited configurations. Each configuration is expressed in terms of the self-consistent-field molecular orbitals and virtual orbitals derived for the ground state. The molecular orbitals are expanded about a single center with an extended basis set of Slater-type orbitals. The calculated hyperfine splitting constants for 13C, 14N, and 17O are found to be 2.7, 0.9, and −0.5 G, respectively, when all the virtual orbitals with symmetry species A1′ or A1 are used; the values are improved to 40.6, 19.0, and −35.4 G, respectively, when certain virtual orbitals with undesirable characteristics are excluded. Though not to be given serious consideration because of the nature of the wavefunctions employed, calculations for 1H are carried out and the values are found to range from −4 to −7 G. Attention is called to the possible danger in using virtual orbitals indiscriminately for calculations on ground-state properties other than energy. Based on the experience gained from this work we suggest that only the orbitals that are better approximations to the excited-state molecular orbitals be used.