Vibronic matrix elements

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. 

In dealing with the MO-LCAO wave function no additional assumptions concerning the vibronic matrix elements are necessary. The evaluation of the total molecular energy exactly copies the lower sheet of the adiabatic potential. This is a consequence of the well-known fact that the Hartree-Fock equations are equivalent to the statement of the Brillouin theorem the matrix elements of the electronic Hamiltonian between the ground-state and... 

We have used equation (8.31) in deriving the above in equation (8.293). The distribution operator in the vibronic matrix element is simply R 3, since 6 = 0 in this problem. Hence our final result is... 

In Eqs. 31, 36, and 37 we have employed the Condon approximation [66], factoring Tjf out of the full vibronic matrix element, with the understanding that 7)y is to be evaluated for values of the nuclear coordinates pertinent to the configuration or range of configurations of the system in which the ET process occurs. The validity of the Condon factorization depends on the extent to which 7)y varies with the coordinates Q, a topic to which we return below. The coordinates of interest in this connection include the reaction coordinate [t] in Figure 3), as well as others such as conformational modes of the DBA system. The influence of fluctuations in these coordinates (and hence in the magnitude of 7)y) on the overall kinetics depends in detail on the relationship between the timescale for such fluctuations and the time-scales of the other dynamic processes [97]. 

The apparent non-Hermitian relationship (i.e., T// 7 Tf, even though Hu = //n and Sif = Sfi), is an artifact of the Condon factorization of the full vibronic matrix element, and, in a complete vibronic model, the vibrational manifold would compensate the mismatch in //, and //// [6, 60], In using the Condon-factorized form (Eqs. 72 and 73), one may simply bypass the problem by using a mean value (//) for Ha and Hp, thereby reverting to the Hermitian form of Eq. 69. 

This calculated matrix element of d2/ dR2, acting on the electronic wavefunc-tions for the E, F and G, K states of H2 (Fig. 3.8), is displayed in Fig. 3.9 and is seen not to deviate appreciably from the derivative of a Lorentzian curve. Its contribution to the Hi,Vm-,2,Vn vibronic matrix element [Eq. (3.3.11)] is generally smaller than the contribution due to the d/dR operator acting on the electronic functions, but it is in no case negligible. 

The We(R) function, resulting from a simple two-state interaction, has a Lorentzian form [Eq. (3.3.14)]. For the Nj B C interaction, We(R) has its maximum in an H-region of numerous constructive and destructive interferences between the vibrational wavefunctions. Thus, slight changes in the vibrational functions resulting from isotopic substitution can drastically alter the vibronic matrix element,... 

The linear vibronic matrix element is non-zero for any point group the direct product of the irreducible representations of the trial wave functions is a reducible representation which necessarily contains the irreducible representation of a symmetry coordinate... 

Here PQ is the reduced mass corresponding to the nuclear coordinate Q, g and h are the vibronic matrix elements... 

Table 4 summarizes the expressions for the vibronic matrix elements Vy, which all correspond to the [2/2]-type of the adiabatic potential. 

This Appendix is concerned with an appropriate expression for the vibronic matrix elements Vu (41) and Vg in the basis set of degenerate electronic wave functions k) and 1). The operator parts... 

It follows that the only possible values for la + Ip are S A and the computation of vibronic levels can be carried out for each K block separately. Matrix elements of the electronic operator diagonal with respect to the electronic basis [first of Eqs. (60)], and the matrix elements of T are diagonal with respect to the quantum number I = la + Ip. The off-diagonal elements of [second and third of Eqs. (60)] connect the basis functions with I — la + Ip and I — l + l — l 2A. 

In the lowest optieally excited state of the molecule, we have one eleetron (ti ) and one hole (/i ), each with spin 1/2 which couple through the Coulomb interaetion and can either form a singlet 5 state (5 = 0), or a triplet T state (S = 1). Since the electric dipole matrix element for optical transitions — ep A)/(me) does not depend on spin, there is a strong spin seleetion rule (AS = 0) for optical electric dipole transitions. This strong spin seleetion rule arises from the very weak spin-orbit interaction for carbon. Thus, to turn on electric dipole transitions, appropriate odd-parity vibrational modes must be admixed with the initial and (or) final electronic states, so that the w eak absorption below 2.5 eV involves optical transitions between appropriate vibronic levels. These vibronic levels are energetically favored by virtue... 

The effects of spin-orbit coupling on geometric phase may be illustrated by imagining the vibronic coupling between the two Kramers doublets arising from a 2E state, spin-orbit coupled to one of symmetry 2A. The formulation given below follows Stone [24]. The four 2E components are denoted by e, a), e a), e+ 3), c p), and those of 2A by coa), cop). The spin-orbit coupling operator has nonzero matrix elements... 

Terms involving Majorana operators are nondiagonal, but their matrix elements can be simply constructed using the formulas discussed in the preceding sections. The total number of parameters to this order is 15 in addition to the vibron numbers, N and N2- This has to be compared with 4 for the first-order Hamiltonian (4.91). For XY2 molecules, some of the parameters are equal, Xi,i = X2,2 XU2 = X2,12, Y112 = Y2 U, A] = A2, reducing the total number to 11 plus the vibron number N = Aj = N2. Calculation of vibrational spectra of linear triatomic molecules with second-order Hamiltonians produce results with accuracies of the order of 1-5 cm-1. An example is shown in Table 4.8. 

The parameterized, analytical representations of fi, ., fiy, fifi determined in the fitting are in a form suitable for the calculation of the vibronic transition moments V fi V") (a—O, +1), that enter into the expression for the line strength in equation (21). These matrix elements are computed in a manner analogous to that employed for the matrix elements of the potential energy function in Ref. [1]. 

It is seen from Equation 19 that the electronic transitions take place without changing the equilibrium positions of the nuclei, and the electronic component of the dipole transition moment is non-zero only if there is no change of the vibronic state during this transition. Dg is non-zero only if the transitions occur between the vibronic states within one electronic state, and the selection rules of Equation 16 are derived from the conditions for a non-vanishing matrix element in Dg. ... 

Ch. Jungen The vibronic coupling is included through the R dependence of the diagonal and off-diagonal quantum defect matrix elements. The effective principal quantum number, or more precisely the quantum defect, gives a handle on the electronic wave function. The variation with R then contains the information concerning the derivative with respect to R of the electronic wave function. 

To consider the nature of this approximation one should notice that the nuclear kinetic energy operator acts both on the electronic and the nuclear parts of the BO wavefunction. Hence, the deviations from the adiabatic approximation will be measured by the matrix elements of the nuclear kinetic energy, T(Q), and of the nuclear momentum. The approximate adiabatic wavefunctions have the following off-diagonal matrix elements between different vibronic states ... 

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