Vibronic -matrix theory

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, p)xv(q), multiplied by a 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. 

An electronic R-matrix radius a is chosen such that exchange can be neglected for r a. An upper limit qd for the intemuclear coordinate q is chosen so that a dissociating electronic state t ,/ is bound for q qd- This defines a vibronic hypercylinder [284] with two distinct surface regions an electronic wall with r = a for 0 q qd and a dissociation cap defined by the enclosed volume of 

A matrix of operators Rpp is defined by projection of R into the multichannel representation indexed by A-electron target states p. This defines the vibrational excitation submatrix of the A-matrix as 

This matrix can be computed from the general variational formula derived in Chapter 8, using a complete set of vibronic basis functions 

It is assumed that target states p are indexed for each value of q such that a smooth diabatic energy function Ep(q) is defined. This requires careful analysis of avoided crossings. The functions should be a complete set of vibrational functions for the target potential Vp = Ep, including functions that represent the vibrational continuum. All vibrational basis functions are truncated at q = qd, without restricting their boundary values. The radial functions fra should be complete for r a. 

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In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch.(7) In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Bom-Oppenheimer states. Henry and Kasha(8) and Jortner and co-workers(9-12) have pointed out that the Bom-Oppenheimer (BO) approximation is only valid if the energy difference between the BO states is large relative to the vibronic matrix element connecting these states. When there are near-degenerate or degenerate zeroth-order vibronic states belonging to different configurations the BO approximation fails. 

The vibronic integrals Vs and Vst contain a radial part and an angular part. The angular part can be determined with the help of the group theory and the remainder (the reduced matrix element) is taken as a parameter depending only on the symmetry type (Xe and Xee). Considering the quadratic approximation to the E-e vibronic coupling the vibronic matrix becomes expressed as follows [88-90] ... 

A parameterization method of the Hamiltonian for two electronic states which couple via nuclear distortions (vibronic coupling), based on density functional theory (DFT) and Slaters transition state method, is presented and applied to the pseudo-Jahn-Teller coupling problem in molecules with an s2-lone pair. The diagonal and off-diagonal energies of the 2X2 Hamiltonian matrix have been calculated as a function of the symmetry breaking angular distortion modes and r (Td)] of molecules with the coordination number CN = 3... 

For the off-diagonal contributions to the vibronic potential matrix, application of group theory gives ... 

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