A. C. Hurley

Potential Curves for Doubly Positive Diatomic Ions

An integral form of the quantum‐mechanical virial theorem, applicable to ionic as well as neutral systems, is established. This theorem is used to derive an approximate expression for the potential curve for a doubly positive diatomic ion in terms of the corresponding curve for a related neutral molecule. The appearance potentials of a number of these ions are calculated and compared with values obtained from electron‐impact measurements. Satisfactory agreement is found, but a crucial test of the theory is vitiated by uncertainties in the experimental value of either the appearance potential of the doubly charged ion or the dissociation energy of the corresponding neutral molecule. The accuracy of the approximations underlying the theory is estimated by a series of calculations on two‐electron systems.

The coupled-pair approximation in a basis of independent-pair natural orbitals

Abstract A computational method based on a rapidly convergent form of the unlinked cluster expansion is presented. Ciźeks coupled-pair approximation (CPA) is derived in a basis of partially non-orthogonal orbitals which transform each pair function to diagonal form; this produces a simple (non-variational) set of equations from which may be extracted the energy and coefficients of a wavefunction constructed from the Hartree-Fock function, all double excitations and all unlinked clusters of these. The relationship of the CPA to simpler treatments is developed using the results of Hurley, and numerical results of a simple illustrative study of the BH 3 molecule are given.

Unlinked cluster effects in molecular electronic structure. I. The HCN and HNC molecules

Extensive calculations on the molecules HCN and HNC have been performed using our recently proposed ’’coupled‐pair approximation’’ (CPA) in a basis of nonorthogonal independent‐pair natural orbitals. A number of linear geometries are used for both systems, allowing prediction of equilibrium geometry, rotational constants and force constants for the stretching vibrational modes. The CPA values are in substantially better agreement with experimental results (where available) than those obtained from variational CI calculations including all double excitations, and can be generated with little extra computational effort. In addition, several approximate coupled‐pair techniques, which require no more effort than a CI calculation, are investigated in order to estimate their accuracy relative to the full coupled‐pair method. Using the bond‐stretching potentials, we have calculated vibrational energy levels and transition energies. Again, the values obtained by the CPA method are in better agreement with experim...

Integrated and Integral Hellmann—Feynman Formulas

For an isoelectronic molecular process X→Y, the energy change ΔW = WY—WX may be computed from any one of three equivalent formulas if the exact molecular wavefunctions, ψX and ψY, are known: the expectation‐value difference, the integrated Hellmann—Feynman formula, and the integral Hellmann—Feynman formula. Should only approximate wavefunctions be available, ΨX and ΨY, these formulas give different estimates of the energy change. If the Hamiltonians for X and Y differ in values of some parameter or parameters λ, say λ = 0 for X and λ = 1 for Y, one has HX=H(0), HY=H(1), H=H(λ), ΨX=Ψ(0), ΨY=Ψ(1), Ψ=Ψ(λ), and the three formulas are as follows: ΔWed=〈Ψ(1) |H(1) |Ψ(1)〉/〈Ψ(1) |Ψ(1)〉−〈Ψ(0) |H(0) |Ψ(0)〉/〈Ψ(0) |Ψ(0)〉,ΔWd= ∫ 01dλ〈Ψ(λ) |∂H(λ)/∂λ |Ψ(λ)〉〈Ψ(λ) |Ψ(λ)〉,ΔWl=〈Ψ(0) |H(1)−H(0) |Ψ(1)〉/〈Ψ(0) |Ψ(1)〉. Relative advantages and disadvantages of these formulas are discussed, and illustrations are given of their use. Conditions for the equivalence of the formulas are established. It is shown...

Unlinked cluster effects in molecular electronic structure. III. Potential curve for the CN− ion and the adiabatic electron affinity of CN

A ground state potential curve (around equilibrium) is constructed for the CN− ion, using CI and coupled‐pair methods. Significant changes arise in the values of the spectroscopic constants when the effects of unlinked cluster excitations are included; a set of predicted CN− spectroscopic constants is obtained using the uncertainties observed in a recent calculation on HCN. Evidence of the great similarity in the HCN, HNC, and CN− electronic structures is obtained by analyzing the CN− valence‐shell correlation energy into pair contributions, thus giving a very simple picture of the correlation effects. Using the direct CI method extended to include approximate coupled pair techniques, the ground state energy for the CN radical is calculated, obtaining the adiabatic electron affinity of CN to form CN−; agreement with the very accurate experimental data is satisfactory.

Role of Atomic Valence States in Molecular Energy Calculations

Ab initio calculations of molecular binding energies are examined critically in a basis of antisymmetrized products of approximate atomic state functions (the approximate composite functions of Moffitt). It is shown that much more accurate results may be expected if the binding energy is calculated relative to a suitably defined generalized valence state V of the dissociation products. The energy of this state V relative to the ground state of the dissociation products is estimated from atomic spectral data. This procedure leads directly to the intra‐atomic correlation correction previously proposed by the author as a necessary modification of Moffitts method of atoms in molecules.Detailed calculations are carried out for the ground states of the molecules LiH and BH, and for the ground and excited states of the benzene molecule (in the usual π‐electron approximation). In each case the intra‐atomic correlation correction leads to a substantial improvement in the accuracy of the results.

Thermochemistry in the Hartree-Fock Approximation

Publisher Summary The aim of quantum chemistry is to predict the enthalpies of reaction of molecules from an ab initio calculations based directly on the Schrodinger equation. A more modest aim is the Hartree–Fock limit that for a closed-shell ground state is defined as the lowest electronic energy obtainable with a variational trial wave function expressed as a single Slater determinant. For a range of small molecules, recent self-consistent field molecular orbital (SCF-MO) calculations help to estimate the Hartree–Fock limit to within near chemical accuracy. The total energy obtained is then used to investigate the utility of the Hartree–Fock approximation in thermochemical calculations. In the expansion method the unknown SCF-MOs are expressed linearly in terms of a certain set of basis functions. As a consequence the integro–differential equations of Hartree–Fock theory are reduced to nonlinear matrix eigenvalue equations that are solved iteratively. The SCF-MOs and the total energy converge to accurate Hartree–Fock results as the basis set is extended to comprise an infinite complete set of functions. The Hartree–Fock binding energy was obtained from extrapolated Hartree–Fock total energy and atomic Hartree–Fock energy. The correlation energy should remain invariant in any change that preserves the number of electron pairs and their local spatial relationship to each other. The Hartree–Fock approximation should then provide an accurate estimate of the change in total energy. Any compilation of enthalpies of formation depends on some definition of reference states, both for the molecules concerned and for the elements. Accurate Hartree–Fock total energy provide a good account of the thermochemistry of those chemically stable small molecules for which current SCF calculations are adequate to estimate the Hartree–Fock limit. Better HFR calculations for a wider variety of molecules are needed to assess the true role and limitations of Hartree–Fock theory in thermochemistry.

Anharmonic Vibration in Fluorite Structures

The high-temperature neutron diffraction measurements of UO2 and CaF2 made recently by Willis are examined by means of the generalized formulation of the structure factor which recognizes the antisymmetric properties of atoms occupying non-centrosymmetric lattice sites. Numerical analysis of the neutron data in terms of an anharmonic vibration formalism developed for effective one-particle potential fields of the type Vj(r) = V0,j + ½αj(x2 + y2 + z2) + βjxyz + quartic terms shows that certain unusual features of the observations are readily accounted for by the antisymmetric (β) components associated with the tetrahedral point symmetry of the anionic sites. Values are derived for the αj and βj parameters in both UO2 and CaF2, and their ability to reproduce the experimental observations over a wide range of elevated temperature is demonstrated. The parameters derived for CaF2 are compared with theoretical estimates obtained from an Einstein-type model of the vibrating lattice, using the various expressions for the interionic interaction in fluorite that have been proposed by Benson & Dempsey. Despite evident limitations in this model, it is possible to conclude that the Pauling radii for Ca2+ and F- cannot reproduce the physical behaviour of this system.

On the Method of Atoms in Molecules

The method of atoms in molecules is investigated in detail for the case of covalent-ionic resonance in the ground state of the hydrogen molecule. It is shown that the method is reliable if, and only if, different orbitals are used to approximate atomic and ionic states of the same molecule.

Unlinked cluster effects in molecular electronic structure. II. Pair correlations in the molecules HCN and HNC

Valence‐shell correlation energies for the HCN and HNC molecules are analyzed into pair contributions at the independent‐pair, CI, and coupled‐pair level. The correlation effects for each system are very similar, and are easily related to the spatial extension and interpenetration of the localized SCF orbitals which form the reference configuration. In addition to the recently proposed ’’coupled‐pair approximation’’ (CPA), two approximate methods, including the widely‐used CEPA technique, are used, and the results are compared with those obtained using the full treatment. The CPA″ method proposed earlier is found to give energy results in agreement with the full CPA to ’’chemical accuracy’’ (1kcal/mole); the agreement obtained with CEPA is not as good, although there is still a substantial improvement over the CI values.