Effective interactions matrix elements

A common case is that the matrix element ( coupling, rq f) is non-zero between donor and bridge at a single atomic orbital which we give index 1. The acceptor is similarly in contact with the last orbital of the bridge, with index n (coupling=9). Most matrices of eq.(27) are thus sparse and it is easy to derive the effective interaction matrix element [19], The energy difference A between and a may be written as ... 

A numerical deperturbation of the NO B2n, L2n, and C2n states has been performed by Gallusser and Dressier (1982) in which many vibrational levels were simultaneously deperturbed (see Section 6.2.2). Another deperturbation, which included the D2S+ state, has been performed (Lefebvre-Brion, unpublished calculation). In this four-state deperturbation, only one vibrational level from each state was treated. The deperturbed levels are represented by the straight lines on Fig. 4.2 (solid lines for 2n states, dashed lines for D2E). The effective interaction matrix elements obtained from the four-state deperturba-... 

In the same way, we finally arrive at the following expression for the effective interaction matrix element between the donor and the acceptor ... 

The stabilizing effect of the intermolecular nF—ctcf interaction (Fig. 5.53(b)) can also be assessed by deleting the nF F cTcF ) interaction-matrix element and recalculating the potential-energy surface E s) in the absence of this interaction. 

We can use an alternative scheme in order to predict the effect of the nature of the atoms A and X on the preferred geometry of AX2 molecules. Thus, for example, consider the MO s of linear H20 which are shown in Fig. 43. Upon bending, the key stabilizing interaction introduced is the interaction between the original lone pair HOMO and the original sigma LUMO. This will increase as the HOMO-LUMO gap in the linear molecule decreases and the corresponding interaction matrix element increases or remains constant. 

Consider the two systems CH2F—SH and CH2F—OH. According to our approach both are predicted to exist in a preferred gauche conformation. However, the extent to which the nx-o F interaction obtains in the two molecules may be subject to matrix element control simply because ns is a better donor than no but yields a smaller interaction matrix element with a F- The variation of these two effects may conceivably be comparable and subject to matrix element control due to the fact that the n—o orbital interaction involves well separated energy levels. Hence, one... 

Our studies of electronic energy transfer reactions are consistent with a strong decrease in the electronic interaction matrix element in accordance with eq 2. However, these studies also indicate that at least two potentially distinguishable effects can result in constants larger than one might naively predict using eq 2 ... 

Rates for nonradiative spin-forbidden transitions depend on the electronic spin-orbit interaction matrix element as well as on the overlap between the vibrational wave functions of the molecule. Close to intersections between potential energy surfaces of different space or spin symmetries, the overlap requirement is mostly fulfilled, and the intersystem crossing is effective. Interaction with vibrationally unbound states may lead to predissociation. 

We have now formulated the approximation and presented one way of actually calculating it, namely by means of Eqs. (42), (43) and Fig. 21. Another way of including much of the higher order effects in Eq. (42) is to calculate the wave functions for the excited electron m in a potential which directly includes a certain selection of the interaction matrix elements in Fig. 21. This is going to be our normal mode of operation, and the zeroth-order basis set will be chosen in the following way ... 

The second term in Eq. (12) has the form characteristic for atomic polarization, the summation in this term being performed over the vibronic states of the ground electronic term. The occurrence of this term is due to the Jahn-Teller effect, since in the absence of the effect the matrix elements < o o- ne) are identically equal to zero. Its contribution to the atomic polarization is determined by the magnitude of vibronic interaction and, generally speaking, it is not small. Visually, the vibronic contribution to the atomic polarization can be explained by the increase of the mobility and hence the polarizability of the vibrational system due to the Jahn-Teller effect. 

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