Interstate electronic couplings

The polaron transformation, executed on the Hamiltonian (12.8)-( 12.10) was seen to yield a new Hamiltonian, Eq. (12.15), in which the interstate coupling is renormalized or dressed by an operator that shifts the position coordinates associated with the boson field. This transformation is well known in the solid-state physics literature, however in much of the chemical literature a similar end is achieved via a different route based on the Bom-Oppenheimer (BO) theory of molecular vibronic stmcture (Section 2.5). In the BO approximation, molecular vibronic states are of the form (/) (r,R)x ,v(R) where r and R denote electronic and nuclear coordinates, respectively, R) are eigenfunctions of the electronic Hamiltonian (with corresponding eigenvalues E r ) ) obtained at fixed nuclear coordinates R and... 

The intersection between the two states (Cl-S/Sg) was optimized together with the TS between the two minima, which lies along the interstate coupling vector for CI-S/S2. (The fourth critical point of the three-electron model discussed... 

A high symmetry of the molecule does not only help to (sometimes dramatically) reduce the number of parameters, it also provides a solid basis for the vibronic coupling model Hamiltonian. When the two interacting electronic states are of different symmetry (as assumed here), the interstate coupling must be an odd function of the couphng coordinate. Hence, there can be no constant or quadratic terms, only linear or bilinear ones are allowed. The vibronic coupling Hamiltonian was first derived by Cederbamn et and is more fully described in a review article by Kbppel et and in Chapter 7 of this book. 

In practical applications of the theory the computational problem is simplified by restricting the electronic Hilbert space to just two or three electronic states of interest, e.g., the ground state and two excited states. A further significant simplification arises from symmetry selection rules. It follows from the definition of equation (15) that the first-order intra-state coupling constants kI" can be nonzero only for totally symmetric modes. V (0) is zero if transform according to different irreducible representations. The first-order interstate coupling constant xj"" is nonzero only for modes which transform according to the irreducible representation Fg which fulfils... 

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