Time-dependent molecular theory electron nuclear dynamics

Time-dependent perturbation theory, electron nuclear dynamics (END), molecular systems, 340-342... 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Electronic structure theory, electron nuclear dynamics (END) structure and properties, 326-327 theoretical background, 324-325 time-dependent variational principle (TDVP), general nuclear dynamics, 334-337 Electronic wave function, permutational symmetry, 680-682 Electron nuclear dynamics (END) degenerate states chemistry, xii-xiii direct molecular dynamics, structure and properties, 327 molecular systems, 337-351 final-state analysis, 342-349 intramolecular electron transfer,... 

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

This level of theory outhned above is implemented in the ENDyne code [18]. The explicit time dependence of the electronic and nuclear dynamics permits illustrative animated representations of trajectories and of the evolution of molecular properties. These animations reveal reaction mechanisms and details of dynamics otherwise difficult to discern, making the approach particularly suitable for the study of the subtleties of contributions to the stopping cross section. 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in molecular electronic structure theory, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

Within the na-qmd approach, electronic (quantum) and nuclear (classical) dof are treated by combining time-dependent (td) density functional theory (dft) with molecular dynamics (md). The basic theorem of tddft [24-26] states that for a system of interacting particles the many-particle state and, thus, any observable are uniquely determined by the time-dependent single-particle density p(r, t) which can be written identically as the density... 

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