Electron nuclear dynamics formalism

V is the derivative with respect to R.) We stress that in this formalism, I and J denote the complete adiabatic electronic state, and not a component thereof. Both /) and y) contain the nuclear coordinates, designated by R, as parameters. The above line integral was used and elaborated in calculations of nuclear dynamics on potential surfaces by several authors [273,283,288-301]. (For an extended discussion of this and related matters the reviews of Sidis [48] and Pacher et al. [49] are especially infonnative.)... 

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)]... 

Transitions between electronic states are formally equivalent to transitions between different vibrational or rotational states which were amply discussed in Chapters 9 11. Computationally, however, they are much more difficult to handle because they arise from the coupling between electronic and nuclear motions. The rigorous description of electronic transitions in polyatomic molecules is probably the most difficult task in the whole field of molecular dynamics (Siebrand 1976 Tully 1976 Child 1979 Rebentrost 1981 Baer 1983 Koppel, Domcke, and Cederbaum 1984 Whetten, Ezra, and Grant 1985 Desouter-Lecomte et al. 1985 Baer 1985b Lefebvre-Brion and Field 1986 Sidis 1989a,b Coalson 1989). The reasons will become apparent below. The two basic approaches, the adiabatic and the diabatic representations, will be outlined in Sections 15.1 and 15.2, respectively. Two examples, the photodissociation of CH3I and of H2S, will be discussed in Section 15.3. 

First attempts to explain the new NMR phenomena invoked electron-nuclear cross-relaxations in intermediate radicals and were based on a formalism similar to that of dynamic nuclear polarization or Overhauser effects ).Accord-... 

A novel data analysis procedure is described, based on a variational solution of the Schrddinger equation, that can be used to analyze gas electron diffraction (GED) data obtained from molecular ensembles in nonequilibrium (non-Boltzmann) vibrational distributions. The method replaces the conventional expression used in GED studies, which is restricted to molecules with small-amplitude vibrations in equilibrium distributions, and is important in time-resolved (stroboscopic) GED, a new tool developed to study the nuclear dynamics of laser-excited molecules. As an example, the new formalism has been used to investigate the structural and vibrational kinetics of C=S, using stroboscopic GED data recorded during the first 120 ns following the 193 nm photodissociation of CS2. Temporal changes of vibrational population are observed, which can... 

These last equations show that the MCTDH method is capable of treating the nuclear dynamics of molecular systems on several coupled electronic states. This formalism has been used, in combination with the vibronic coupling model of Kdppel et al. 

Theoretical aspects related to the above problems have been elaborated and discussed in detail in the Ab initio theory of complex electronic ground state of superconductors , published recently [52a, b]. The main theoretical point is a generalization of the BOA by sequence of canonical - base function transformations. This formalism is equivalent to our original one, based on quasi-particle transformation treatment [53]. The final electronic wave function is explicitly dependent on nuclear coordinates Q and nuclear momenta P. Emerging new quasiparticles, i.e. nonadiabatic fermions, are explicitly dependent on nuclear dynamics. As a result, the effect of nuclear dynamics can be calculated as corrections to the clamped nuclei ground state electronic energy, the one-particle spectrum and the two-particle term, i.e. to the electron correlation energy ... 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

The first topic has an important role in the interpretation and calculation of atomic and molecular structures and properties. It is needless to stress the importance of electronic correlation effects, a central topic of research in quantum chemistry. The relativistic formulations are of great importance not only from a formal viewpoint, but also for the increasing number of studies on atoms with high Z values in molecules and materials. Valence theory deserves special attention since it improves the electronic description of molecular systems and reactions with the point of view used by most laboratory chemists. Nuclear motion constitutes a broad research field of great importance to account for the internal molecular dynamics and spectroscopic properties. 

The formalism developed so far is adequate whenever the motion of the atomic nuclei can be neglected. Then the electron-nucleus interaction only enters as a static contribution to the potential r(r, t) in Eq. (41). This is a good approximation for atoms in strong laser fields above the infrared frequency regime. When the nuclei are allowed to move, the nuclear motion couples dynamically to the electronic motion and the situation becomes more complicated. 

In chemical dynamics, one can distinguish two qualitatively different types of processes electron transfer and reactions involving bond rearrangement the latter involve heavy-particle (proton or heavier) motion in the formal reaction coordinate. The zero-order model for the electron transfer case is pre-organization of the nuclear coordinates (often predominantly the solvent nuclear coordinates) followed by pure electronic motion corresponding to a transition between diabatic electronic states. The zero-order model for the second type of process is transition state theory (or, preferably, variational transition state theory ) in the lowest adiabatic electronic state (i.e., on the lowest-energy Bom-Oppenheimer potential energy surface). 

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