Volume 43, Issue 10, 15 November 1965
Index of content:
Self‐Consistent Group Calculations on Polyatomic Molecules. II. Hybridization and Optimum Orbitals in Water43(1965); http://dx.doi.org/10.1063/1.1701471View Description Hide Description
GF calculations for the ground state of the water molecule are reported for three sets of orbital exponents, and optimum valence hybrids, determined by systematic variation of the hybridization parameters until the total electronic energy was a minimum, are compared with hybrids obtained by other criteria. Different methods of constructing localized inner‐shell groups are discussed.
43(1965); http://dx.doi.org/10.1063/1.1701472View Description Hide Description
The historical connection between the molecular orbital and Heitler—London treatments is traced, with particular attention to the contributions of Mulliken. Early discussions of the self‐consistent‐field problem, the relations of Heisenberg's work to the antisymmetry of the wavefunction, and configuration interactions in the two‐electron problem, are reviewed, with references to the heteropolar as well as the homopolar cases. Early discussions of directed valence are mentioned. The Coulson—Fischer, Hurley—Lennard‐Jones—Pople, and alternant molecular orbital approaches to bonding are discussed, with mention of recent work on correlation energy.
43(1965); http://dx.doi.org/10.1063/1.1701473View Description Hide Description
The spin Hamiltonian is derived for two arbitrary nonorthogonal systems, to a certain well‐defined order. It is shown that the coupling constant, which no longer necessarily takes only positive values, is determined by the electron densities and spin densities of the two systems.
43(1965); http://dx.doi.org/10.1063/1.1701474View Description Hide Description
The concept of ``electronegativity'' is discussed and reviewed in the light of recent observed relationships between successive ionization potentials. A differentiation is made between atomic and molecular electronegativity and particular reference is made to the principle of equalization of molecular electronegativities in the process of forming a bond.
The Rydbergequation for atoms is used as a basis for the quantitative calculation of the atomic spectra and electronegativity of systems in any valency state. The resulting equation can be used either to calculate atomic electronegativities or to determine quantitatively the diagonal matrix elements in SCF calculations for molecules involving heteroatoms.
43(1965); http://dx.doi.org/10.1063/1.1701475View Description Hide Description
A general discussion of approximate methods for obtaining self‐consistent molecular orbitals for all valence electrons of large molecules is presented. It is shown that the procedure of neglecting differential overlap in electron‐interaction integrals (familiar in π‐electron theory) without further adjustment may lead to results which are not invariant to simple transformations of the atomic orbital basis set such as rotation of axes or replacement of s, p orbitals by hybrids. The behavior of approximate methods in this context is examined in detail and two schemes are found which are invariant to transformations among atomic orbitals on a given atom. One of these (the simpler but more approximate) involves the complete neglect of differential overlap (CNDO) in all basis sets connected by such transformations. The other involves the neglect of diatomic differential overlap (NDDO) only, that is only products of orbitals on different atoms being neglected in the electron‐repulsion integrals.
Approximate Self‐Consistent Molecular Orbital Theory. II. Calculations with Complete Neglect of Differential Overlap43(1965); http://dx.doi.org/10.1063/1.1701476View Description Hide Description
The approximate self‐consistent molecular orbital method with complete neglect of differential overlap (CNDO), described in Paper I, is used to calculate molecular orbitals for the valence electrons of diatomic and small polyatomic molecules. A small number of bonding parameters (β‐resonance integrals) are chosen semiempirically so that the results are comparable to previous accurate LCAO—SCF wavefunctions for diatomic hydrides using a similar basis set. With this calibration, it is found that calculations on other diatomics and polyatomics lead to molecular orbitals and electron distributions in reasonable agreement with the full calculations where available. Although the new method is not yet successful in predicting bond lengths and dissociation energies, it does lead to the correct geometry, reasonable bond angles and bending force constants for the polyatomic molecules considered. It also gives calculated barriers to internal rotation for ethane, methylamine, and methanol which are in fair agreement with experiment.
Semirigorous LCAO—MO—SCF Methods for Three‐Dimensional Molecular Calculations Including Electron Repulsion43(1965); http://dx.doi.org/10.1063/1.1701477View Description Hide Description
A semirigorous LCAO—MO—SCF method including electron repulsion, for three‐dimensional closed‐shell molecular calculations has been derived. The method starts with the complete many‐electron Hamiltonian (in which interelectronic interactions are included explicitly) and the self‐consistent molecular orbital equations of Roothaan, then makes a series of systematic approximations for the integrals involved. There are several different levels of approximation which evolve depending on how restrictive one makes the conditions for neglecting a′(1)a″(1), where a′ and a″ are two orbitals on Atom A; and a′(1)b″(1), where a is an orbital on Atom A and b is an orbital on Atom B, A ≠ B, and for neglecting (a′c′′′| G | b″d′v), where a′ is an orbital on Atom A, b″ is an orbital on Atom B, c′′′ is an orbital on Atom C, d′v is an orbital on Atom D, and A, B, C, D are, in general, different atoms. The approximations for the core operators are defined and the core Hamiltonian matrix elements are presented in detail for several of the levels of approximation. The complete expressions for Fa′a′, Fa′a″ , and Fa′b″ for closed‐shell systems in the two simplest of the approximative schemes are included in the body of this paper. The salient details of the more complicated schemes (corresponding to lesser and lesser neglect of the interaction of charge distributions) are indicated.
These semirigorous LCAO—MO—SCF schemes outlined in this paper hold promise for less‐than‐rigorous yet better‐than‐empirical calculations for the whole gamut of three‐dimensional molecules from simple inorganics through complicated organics. An analogous treatment with the same approximations for the neglect of certain integrals can also be applied to an open‐shell LCAO—MO—SCF procedure.
Semiempirical Molecular Orbitals for General Polyatomic Molecules. II. One‐Electron Model Prediction of the H–O–H Angle43(1965); http://dx.doi.org/10.1063/1.1701478View Description Hide Description
A consistent one‐electron semiempirical molecular orbital method based on the approximationhas been used with atomic data to predict the equilibrium geometry of the water molecule. A variety of plausible parameters predict an equilibrium H–O–H angle in the range 95°—105°. Success and limitations of the method are attributed to the fidelity with which it reproduces important one‐electron effects. In less critical cases, this model resembles the Wolfsberg—Helmholz and other non‐Hückel semiempirical MO treatments.
43(1965); http://dx.doi.org/10.1063/1.1701479View Description Hide Description
A simple internally consistent recipe for obtaining atomic orbital matrix elements (Hii ) from atomic spectral data agrees with values obtained from empirical molecular ionization potentials via molecular orbital calculation. Regularities observed lead to very simple expressions useful for adjustment of parameters for charge transfer. Various assumed relations between atomic valence‐state quantities and orbital matrix elements are compared with the observed data for selected atoms. In the cases studied, valence‐state ionization potentials depend quadratically on the degree of ionization, but can be satisfactorily approximated by linear functions for net charges between −1 and +1. In the linear approximation, the second term is the same for both occupancies of both s and p orbitals.
43(1965); http://dx.doi.org/10.1063/1.1701480View Description Hide Description
Convenient and accurate formulas have been found which implement the spherical‐harmonic expansion of a Slater‐type orbital about a point displaced from the orbital's center. These formulas include, as a special case, the expansion of spherically symmetric orbitals which has been reported previously. The results are presented in algebraic form, suitable for machine computation, and have been tested for the transformation of any Slater‐type orbital with n≤7, l≤3, | m |≤l to an arbitrary space‐fixed point.
By the use of this analysis, in conjunction with the properties of the rotation group, it is now possible to evaluate directly, the general multicenter electron‐repulsion integral expressed in terms of Slater‐type orbitals. The convenience of this direct approach suggests that previous analyses in terms of Gaussian orbitals or Gaussian approximations to Slater‐type orbitals, replete with their poor radial dependence, may no longer offer any computational advantages.
43(1965); http://dx.doi.org/10.1063/1.1701481View Description Hide Description
43(1965); http://dx.doi.org/10.1063/1.1701482View Description Hide Description
Various methods for the calculation of lower bounds for eigenvalues are examined, including those of Weinstein, Temple, Bazley and Fox, Gay, and Miller. It is shown how all of these can be derived in a unified manner by the projection technique. The alternate forms obtained for the Gay formula show how a considerably improved method can be readily obtained. Applied to the ground state of the helium atom with a simple screened hydrogenic trial function, this new method gives a lower bound closer to the true energy than the best upper bound obtained with this form of trial function. Possible routes to further improved methods are suggested.
Studies in Perturbation Theory. XI. Lower Bounds to Energy Eigenvalues, Ground State, and Excited States43(1965); http://dx.doi.org/10.1063/1.1701483View Description Hide Description
The bracketing theorem in the partitioning technique for solving the Schrödinger equation may be used in principle to determine upper and lower bounds to energy eigenvalues. Practical lower bounds of any accuracy desired may be evaluated by utilizing the properties of ``inner projections'' on finite manifolds in the Hilbert space. The method is here applied to the ground state and excited states of a Hamiltonian H=H 0+V having a positive definite perturbationV. Even if inspiration is derived from the method of intermediate Hamiltonians, the final results are of bracketing type and independent of this approach. The method is numerically illustrated in some accompanying papers.
43(1965); http://dx.doi.org/10.1063/1.1701484View Description Hide Description
The lower‐bounds method presented by Löwdin in the preceding paper has been applied to oscillators perturbed by third‐ and fourth‐power terms in the potential‐energy expression. For favorable cases agreement between upper and lower bounds is easily carried to many more figures than are likely to be physically significant. In many cases the lower bounds agreed more closely to the true eigenvalue than did the corresponding upper bounds. For a given basis set, this method gives closer bounds than that of Bazley and Fox, except for energy levels too high to be satisfactorily treated in the given basis. The only disadvantage found was that for close bounds double precision proved necessary, indicating more than ordinary loss of computational accuracy.
43(1965); http://dx.doi.org/10.1063/1.1701485View Description Hide Description
Upper and lower bounds have been calculated for the energy levels of a rigid rotator in an electric field, in order to study the problems associated with the use of the partitioning method for bracketing an eigenvalue of the Schrödinger equation. Results of arbitrarily high accuracy are possible in this example.
43(1965); http://dx.doi.org/10.1063/1.1701486View Description Hide Description
A brief discussion is given of the early contributions of R. S. Mulliken and his school to the study of the electronic spectra of polyatomic molecules. An up‐to‐date summary of the polyatomic molecules and free radicals for which high‐resolution studies of their band spectra have been carried out is included. Some points of likely interest to the theoretical worker are discussed, including the Renner effect in AH2 and BAC molecules, predissociation in HAB molecules, and some recent experimental results on the allyl and vinoxy radicals.
43(1965); http://dx.doi.org/10.1063/1.1701487View Description Hide Description
Several methods of obtaining upper and lower bounds to the eigenvalues of self‐adjoint operators bounded from below have recently been developed by Löwdin. All these procedures are based on a bracketing theorem. We mention them and discuss one of the variants that makes use of intermediate Hamiltonians. To illustrate the power of the method, an application is made to various forms of double‐minimum (D.M.) potentials, both symmetric and asymmetric, which have already been studied by Somorjai and Hornig. The new results show the importance of obtaining these lower bounds in connection with the resonance interaction between accidentally coincident ``left'' and ``right'' levels of weakly asymmetric D.M. potentials. An agreement between upper and lower bounds of 12–15 significant figures is obtained.