Diabatization, coupled electronic/nuclear

In Chapter VI, Ohm and Deumens present their electron nuclear dynamics (END) time-dependent, nonadiabatic, theoretical, and computational approach to the study of molecular processes. This approach stresses the analysis of such processes in terms of dynamical, time-evolving states rather than stationary molecular states. Thus, rovibrational and scattering states are reduced to less prominent roles as is the case in most modem wavepacket treatments of molecular reaction dynamics. Unlike most theoretical methods, END also relegates electronic stationary states, potential energy surfaces, adiabatic and diabatic descriptions, and nonadiabatic coupling terms to the background in favor of a dynamic, time-evolving description of all electrons. 

Density functional theory, direct molecular dynamics, complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 404-411 Density operator, direct molecular dynamics, adiabatic systems, 375-377 Derivative couplings conical intersections, 569-570 direct molecular dynamics, vibronic coupling, conical intersections, 386-389 Determinantal wave function, electron nuclear dynamics (END), molecular systems, final-state analysis, 342-349 Diabatic representation ... 

Figure 32. Vibronic periodic orbits of a coupled electronic two-state system with a single vibrational mode (Model IVa). All orbits are displayed as a function of the nuclear position x and the electronic population N, where N = Aidia (left) and N = (right), respectively. As a further illustration, the three shortest orbits have been drawn as curves in between the diabatic potentials Vi and V2 (left) as well as in between the corresponding adiabatic potentials Wi and W2 (right). The shaded Gaussians schematically indicate that orbits A and C are responsible for the short-time dynamics following impulsive excitation of V2 at (xo,po) = (3,0), while orbit B and its symmetric partner determine the short-time dynamics after excitation of Vi at (xo,po) = (3, —2.45).

When the field is absent and the nuclei are viewed as quantum particles, a complete description requires inclusion of electron-phonon and spin-orbit coupling operators. Now, the By( )-coefficients couple electronic states whose diabatic potential energy surfaces jrield degenerate functions for the nuclear dynamics. Finally, these geometry-dependent coefficients couple all diabatic electronic states whenever a fuUy quantum-mechanical molecular system is embedded in an external field. 

The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

Neumann boundary conditions, electronic states, adiabatic-to-diabatic transformation, two-state system, 304-309 Newton-Raphson equation, conical intersection location locations, 565 orthogonal coordinates, 567 Non-Abelian theory, molecular systems, Yang-Mills fields nuclear Lagrangean, 250 pure vs. tensorial gauge fields, 250-253 Non-adiabatic coupling ... 

To apply the mapping formalism to vibronically coupled systems, we identify the / ) with electronic states and the h m with operators of the nuclear dynamics. Hereby, the adiabatic as well as a diabatic electronic representation may be employed. In a diabatic representation, we have [cf. Eq. (1)]... 

The import of diabatic electronic states for dynamical treatments of conical intersecting BO potential energy surfaces is well acknowledged. This intersection is characterized by the non-existence of symmetry element determining its location in nuclear space [25]. This problem is absent in the GED approach. Because the symmetries of the cis and trans conformer are irreducible to each other, a regularization method without a correct reaction coordinate does not make sense. The slope at the (conic) intersection is well defined in the GED scheme. Observe, however, that for closed shell structures, the direct coupling of both states is zero. A configuration interaction is necessary to obtain an appropriate description in other words, correlation states such as diradical ones and the full excited BB state in the AA local minimum cannot be left out the scheme. 

The non-adiabatic character of the process under study is included in the present approach in the evaluation of the initial wavepacket in Eq.(7). In an electronic diabatic representation, the electronic wavefunctions are considered to do not depend on the nuclear coordinates, so that the coupling between different states is only given by the electronic Hamiltonian, being of potential-type character. 

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