Coupling matrices, electronic state adiabatic representation

Coupling matrices, electronic state adiabatic representation ... 

Single-valued potential, adiabatic-to-diabatic transformation matrix, non-adiabatic coupling, 49-50 topological matrix, 50-53 Skew symmetric matrix, electronic states adiabatic representation, 290-291 adiabatic-to-diabatic transformation, two-state system, 302-309 Slater determinants ... 

The remaining combinations vanish for symmetry reasons [the operator transforms according to B (A") hreducible representation]. The nonvanishing of the off-diagonal matrix element fl+ is responsible for the coupling of the adiabatic electronic states. 

In an electronic adiabatic representation, however, the electronic Hamiltonian becomes diagonal,i.e. ( a 77e C/3) = da,0Va, where the adiabatic Va potentials for initial (A,B,B ) and final (X) electronic states were described in Ref.[31]. The couplings between different electronic states arises from the matrix elements of the nuclear kinetic operator Tn, giving rise to the so-called non-adiabatic coupling matrix elements (NACME) and are due to the dependence of the electronic functions on the nuclear coordinates. The actual form of these matrix elements depends on the choice of the coordinates. 

In order to better understand the similarity between the coupled- and the uncoupled-surface results, we show the time evolution of the electronic populations in Fig. 4. The initial WP corresponds to H2(< = 0,j = 0) and is again prepared in the asymptotic reactant channel of the lower adiabatic sheet. [It is transformed to the diabatic representation using the S-matrix of Eq. (7) prior to propagation]. As can be seen from Fig. 4, we obtain a 0.71/0.29 population of the two component diabatic electronic states (shown by the dashed and solid lines) at t = 0. Therefore, the diabatic potentials do not approach the asymptotic adiabatic states of H - - H2 but represent a mixture of them. 

When a group of adiabatic electronic states have (almost) no couplings with all the other adiabatic electronic states, we may keep only the equations for the group of states in the matrix of Eq. (5.4), and make a diabatic representation from the group. This is different from making diabatic states from the complete set of adiabatic states and then keeping only a group of them, but it is in fact a better approximation, and is used more frequently. 

In Eq. [7], the electronic wavefunctions are taken as the eigenfunctions of the electronic Hamiltonian. In this case, all the coupling matrix elements H[j = ( // ff /y) are zero, and the coupling between different electronic states occurs through the nuclear kinetic energy terms this formulation is called the adiabatic representation. ... 

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