AbstractsThe total wave function of a system y ( x , y ) is expressed as a product of a marginal amplitude function f ( y ) and a conditional amplitude function $(x I y). The original Sie fuhrt zu einer partiellen Trennung der Variabeln x und y.

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The Schrijdmger equation satisfied by the electron density is derived without approximation from the theory of marginal and conditional amplitudes. The equation arises from a factorization of the total N-electrOn wavefunction defmed by the normalization appropriate to a conditional amplitude. This Schradinger equation is an exact dynamical model for computing effective one-electron potentials from known N-electrOn wavefunctions. Results are presented for several small molecules. They display the shell structure of atoms, and the valence structure of both ionic and covalent molecules.

The exact (nonadiabatic) nuclear and electronic factors of a molecular wave function are expanded in the basis of eigenfunctions of the electronic Hamiltonian according to the Rayleigh-Schrodinger perturbation theory of Born and Huang. Thus it is shown that, with rare exceptions, the exact nuclear factor (a marginal amplitude) is a nodeless function. The nodes in vibrationally excited nuclear wave functions within the Born-Oppenheimer approximation become node-avoiding minima in the exact nuclear wave function. Corresponding to each node-avoiding minimum in the nuclear wave function the exact (nonadiabatic) effective potential for the nuclear motion has a spiky barrier superimposed upon the Born-Oppenheimer (adiabatic) eigenenergy of the electronic Hamiltonian. These barriers are the result of nonadiabatic coupling between electronic states, which is strongest in the vicinity of the nodes in the Born-Oppenheimer-approximation nuclear (vibrational) wave function.

Semianalytic solutions of the Schrödinger equation for the motion of a particle in the static Coulomb field of two other particles are derived in a form suitable for numerical evaluation of the expansion coefficients on an electronic computer. Relevant mathematical aspects and the numerical methods are described. Tables of two-center wavefunctions are made available through the American Documentation Institute. An Appendix describes some ``dynamic scaling'' procedures, designed to facilitate Gaussian elimination and bisection where large-order matrices are involved.

A procedure for deriving ionization potentials is presented based upon the factorization of a wave function into the product of a conditional amplitude and a one-electron marginal amplitude. In the asymptotic limit as one electron is removed the conditional amplitude becomes the wave function of the residual positive ion, so that successive ionization potentials can be generatcd in sequence by the same procedure.

Con-N bond lengths. But, the Co-N stretching frequency of Com(taptacn) is 519 cm"1 (identified by an 8-cm"1 shift upon deuteriation of the primary amines) compared to 479 cm'1 for Coln(tacn)2.45 The greater force constant can then lead to a greater inner-sphere reorganization energy.It has been shown recently that the self-exchange parameters of Co3+/2+ couples may be treated in the framework of the Marcus-Sutin model for outer-sphere electron transfer.38 For complexes of the type under consideration the preequilibrium constant (A0) is 0.04 (/ = 0.1 M) and radii (/•) are 9.0 Á (with reaction thickness = 0.8 Á). From these values, a self-consistent set of parameters for the Marcus-Sutin model has been determined for the Co(taptacn)3+/2+ couple: (/ * = 13.8 kcal/mol; <7 * = 5.2 kcal/mol, and d2-d} = 0.165 A, yielding kmc = 0.04 M"1 s'1 as observed in the present study. The somewhat smaller Ad value calculated for the present complexes is in keeping with the Co(III) species being slightly distorted toward the Co(II) structure (45) Boeyens,

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