Gary Simons

New alternative to the Dunham potential for diatomic molecules

A new systematic procedure for constructing potential curves for diatomic molecules is developed. The procedure is similar to the well‐known Dunham method, except that the expansion parameter is (R‐Re)/R instead of (R‐Re)/Re. The new expansion, which has a formal theoretical basis, is shown to be superior in terms of both rate of convergence and region of convergence. It is shown how the expansion coefficients may be obtained from spectroscopic data, and the proper behavior of the potential at large R is shown to allow one to determine additional coefficients and to determine dissociation energies. To illustrate the method, the ground states of hydrogen flouride and carbon monoxide are treated. Possible extensions to polyatomic molecules are briefly discussed.

New Model Potential for Pseudopotential Calculations

The use of model potentials in pseudopotential calculations is discussed, and a set of desirable criteria for model potentials is suggested. The Hellmann potential and the Abarenkov and Heine potential are examined, and it is shown that both potentials are useful, but neither fully satisfies the suggested desiderata. A new potential of the form, V(r) = − Z / r + ΣlBlPl / r2, where Pl is the projection operator over the subspace of spherical harmonics of a given l, is proposed. A physical interpretation of the potential in terms of a “Pauli force” and a polarized core is given. Ionization energies for excited S, P, D, and F states for one‐valence‐electron atoms are calculated. The new potential is also applied to two‐valence‐electron atoms, and their calculated valence‐state energies are shown to be in good agreement with experimental values.

New procedure for generating valence and Rydberg orbitals. I. Atomic oscillator strengths

A new procedure for generating approximate valence and Rydberg orbitals is proposed. The method involves the analytic determination of the eigenfunctions of a model Hamiltonian and leads to orbitals which are functions of quantum numbers and quantum defects. The new functions are shown to be similar to previously obtained orbitals and are used to derive a general formula for transition integrals. Illustrative results for Li and Be+ are excellent; results for Na, K, and Mg are satisfactory.

New procedure for generating valence and Rydberg orbitals. II. Atomic photoionization cross sections

A previously described procedure for generating approximate valence and Rydberg orbitals from spectral data is extended to continuum states. Formulas for photoionization cross sections are derived, and cross sections are reported for the lithium sequence ions (Li I to Si XII), sodium, and helium. The predicted values are shown to be in good agreement with experimental and polarized‐Hartree–Fock values, and results are generally superior to cross sections obtained from the same spectral data by the quantum defect method.

Simple Bond‐Charge Model for Potential‐Energy Curves of Heteronuclear Diatomic Molecules

For a homonuclear diatomic molecule near its equilibrium internuclear distance Re, in some bound electronic state, a potential‐energy function W(R) of the form W = W0 + W1 / R + W2 / R2 has previously been shown to be a good approximation to the true potential. From this equation and the molecular virial theorem, there follow expressions for the total electronic potential energy V(R) and the total electronic kinetic energy T(R), V = 2W0 + W1 / R, T = −W0 + W2 / R2. The R‐dependent, Coulombic part of V is modeled by locating a positive charge Ze at each nucleus and a negative charge −qe at the bond center, with q = 2Z. The Rdependent, free‐electron‐like part of T is modeled by assuming that the charge q moves freely in a one‐dimensional box of length νR. Thus W1 / R = e2(Z2 − 4Zq) / R, W2 / R2 = h2q / 8mν2R2, and W = W0 + (e2 / R) (Z2 − 4Zq) + (1 / R2)[(h2 / 8m) (q / ν2)]. For 17 molecules in 63 different electronic states, parameters q and ν are given that reproduce exactly the experimental equilibrium di...

Expansion variables for general quartic force fields of triatomic molecules

A framework for general quartic force fields of polyatomic molecules, employing expansion coordinates of the form ρ=(r − re)/r, is developed. The new approach is shown to be superior in terms of both rate and region of convergence to traditional expansions, and a procedure for obtaining the necessary coefficients is outlined. Results are presented for CO2, CS2, HCN, and N2O, and the chemical utility of the new force fields is assessed.

Atomic and Molecular Pseudopotential Studies Using Gaussian Orbitals

The Hellmann model potential is employed for the first time in atomic and molecular calculations with a Gaussian basis set. The use of Gaussian functions allows for the simple application of the pseudopotential method to large molecular systems while maintaining the accuracy exhibited by Slater‐type‐orbital calculations. A new model potential specifically designed for Gaussians, V(r) = C1r2 + C2, is proposed and used to examine a series of two‐valence‐electron atoms. A number of diatomic molecules are studied with both the Hellmann and Gaussian model potentials. Bond lengths, vibrational frequencies, dipole moments, and dissociation energies are calculated, and the results are generally in good accord with experimental data. The Phillips–Kleinmann pseudopotential formalism is used to examine the model potentials and analyze the atomic and molecular results.

New procedure for generating valence and Rydberg orbitals

A previously proposed procedure for generating approximate valence, Rydberg, and continuum orbitals from spectral data is extended to two-valence electron systems. Oscillator strengths and photoionization cross sections are reported for the respective 1s22s2 →1s22s np and 1s22s2→1s22s ep transitions of Be, B+, C++, N+++, 0+4, Ne+6, Mg+8, Al+9, and Si+10. Comparisons show a high level of agreement with Hartree-Fock oscillator strengths, but the accuracy of the cross sections is uncertain.

Regional stationary principles and regional virial theorems

If the Born‐Oppenheimer Hamiltonian operator for a molecular system is averaged over all space for electrons 2,3, ··· , N, but for electron 1 is averaged only over the volume A, with surface S, it is shown that the regional expectation value so defined is stationary with respect to change of a parameter ζ in the exact wavefunction, provided that ∫S[(∂/∂ n1) Pζ(1,1′)]1=1′ dS1=0, where, for nodeless ψ, Pζ(1,1′)=∫ ··· ∫ψ*(1′,2,··· , N) ψ (1,2, ··· , N) × ∂ ln[ψ(1, 2, ··· , N)/ ψ*(1′, 2, ··· , N)]∂ ζ |ζ=1 dτ2 ··· dτN. Under this condition with ζ a scale parameter, a regional virial theorem is shown to hold for the volume A, in the sense previously described by Bader and co‐workers [J. Am. Chem. Soc. 93, 3095 (1971): Chem. Phys. Lett. 8, 29 (1971), J. Chem. Phys. 56, 3320 (1972); 58, 557 (1973)]. This condition differs from the corresponding condition suggested by Bader (∂ / ∂ n1)ρ (r1)=0 on S, but in most cases the surfaces determined by it should be similar to those determined by Baders condition. Correspon...

Capabilities and limitations of an analytic potential expansion for diatomic molecules

Abstract A recently proposed alternative to the Dunham potential for diatomic molecules is examined in detail. This new potential is a series in powers of the variable z = (R − R e ) R . Constants derived from spectroscopic data are used to construct potential curves for CO, N 2 , NO, C 2 , CO + , N 2 + , O 2 + , NO + , and O 2 . The accuracy of the potential expansion is tested in three ways by calculating: (a) the potential at the vibrational turning points, (b) the turning points from input values of the potential, and (c) numerical eigenvalues and rotational constants. The computed potentials are in excellent agreement with RKR curves for the lower portions of the potential well and are in semiquantitative agreement for the upper portions.