Channel orbital

For an atom, or outside a sphere that completely encloses a target molecule, channel orbital functions are of the form... 

Using the projection-operator formalism of Feshbach [ 115,116], an implicit variational solution for the coefficients cIJiS in can be incorporated into an equivalent partitioned equation for the channel orbital functions. This is a multichannel variant of the logic used to derive the correlation potential operator vc in orbital-functional theory. Define a projection operator Q such that... 

Effective one-electron equations for the channel orbital functions can be obtained either by evaluating orbital functional derivatives of the variational functional S or more directly by projecting Eq. (8.3) onto the individual target states p. With appropriate normalizing factors, ((")/ TV) = if/ps. Equations for the radial channel functions fps(r) are obtained by projecting onto spherical harmonics and elementary spin functions. The matrix operator acting on channel orbitals is... 

The asymptotic form of radial open-channel orbitals fps(r) is given by Eq. (8.2). Functions of this form can be represented as linear combinations of two independent continuum basis functions for each open channel. These basis functions must be regular at the coordinate origin, but have the asymptotic forms... 

The last term here does not vanish even if ma = 0. The nonhermitian part of mfj1 comes from the free-free matrix of H — E, which is characteristic of scattering theory. When channel orbital functions are normalized to unit flux, with the asymptotic forms specified above, this nonhermitian term is given by... 

This formula expresses the surface integral obtained by integrating the kinetic energy integral by parts for open-channel orbital functions of the specified asymptotic form. In matrix notation,... 

The Wigner-Eisenbud [428] 7 -matrix, or derivative matrix, is defined by the relationship between radial channel orbitals fps(r) and their derivatives on some sphere of radius r that surrounds a target system [214,33], Assuming spherical geometry, the dimensionless radial f -matrix Rpq at r is defined by... 

The theory of the / -matrix was developed in nuclear physics. As usually presented, the theory makes use of a Green function to relate value and slope of the radial channel orbitals at r, expanding these functions for r < r as linear combinations of basis functions that satisfy fixed boundary conditions at r. The true logarithmic derivative (or reciprocal of the / -matrix in multichannel formalism)... 

The A-matrix can be matched at r to external channel orbitals, solutions in principle of external close-coupling equations, to determine scattering matrices. Radial channel orbital vectors, of standard asymptotic form for the A -matrix,... 

As written, these equations refer to open channels only. When external closed channels are considered, an external closed channel orbital that vanishes as r -> oo must be included for each such channel. The indices p,q,s run over all channels,... 

In the case of a scattering resonance, bound-free correlation is modified by a transient bound state of fV+1 electrons. In a finite matrix representation, the projected (fV+l)-electron Hamiltonian H has positive energy eigenvalues, which define possible scattering resonances if they interact sufficiently weakly with the scattering continuum. In resonance theory [270], this transient discrete state is multiplied by an energy-dependent coefficient whose magnitude is determined by that of the channel orbital in the resonant channel. Thus the normalization of the channel orbital establishes the absolute amplitude of the transient discrete state, and arbitrary normalization of the channel orbital cannot lead to an inconsistency. 

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

In this example of Hg-H2 photoinitiated reactions, it was shown that selectivity in entrance-channel orbital orientation results in significant differences for both reactivity and product state distributions. 

Keywords Bond information probes Bond localization Chemical bonds Chemical reactivity Contra-gradience criterion Covalent/ionic bond components Direct/indirect bond multiplicities Entropic bond indices Fisher information Information theory Molecular information channels Orbital... 

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