Time-dependent nuclear wave function

Determination of the time-dependent nuclear wave function now becomes a straightforward exercise in the locally quadratic theory. We need to propagate Gaussian wave packets of the form... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

Once the nuclear wave function has been collapsed into a product of delta functions that identify the nuclear positions, the time-dependent electronic wave function can be expanded in the adiabatic basis ... 

To deal with the problem of using a superposition of functions, Heller also tried using Gaussian wave packets with a fixed width as a time-dependent basis set for the representation of the evolving nuclear wave function [23]. Each frozen Gaussian function evolves under classical equations of motion, and the phase is provided by the classical action along the path... 

The adiabatic time-dependent equation for the nuclear wave functions 114... 

Note that in Eq. (2.8), the nuclear wave function depends only on the nuclear coordinates, whereas the electronic wave function depends explicitly on the electronic coordinates and implicitly on the nuclear coordinates. This simplification is known as the Born-Oppenheimer approximation [15] and assumes that electrons, much lighter than nuclei, can adapt their position instantaneously to any change of the nuclear coordinates. Under these conditions, the electronic time-independent Schrodinger equation can be written as... 

The total (electronic plus nuclear) wave function is determined by the time-dependent Schrodinger equation (24) with the initial condition (23). In direct analogy to the case of photon absorption and emission [cf. Eq. (39)], the ionization signals defined in Eqs. (18) and (19) are obtained from the second-order wave function. 

It should to be noted that the form chosen here for is somewhat arbitrary. For instance, one could use electronic wave functions in the form of plane waves (exp(iA . )) independent of Q. Similarly one could use nuclear wave functions independent of time (a basis set of steady states) multiplied by a time-dependent scalar factor. In fact, this last form of P might be more convenient for numerical purposes than the one proposed here, that is, eqn (5.4). However, it turns out that eqn (5.4) is convenient for the qualitative discussion on the quantum-to-classical transition described below. 

The nuclear wave functions are the projections of the total wave functions on the electronic PES involved. They depend on the nuclear coordinate Q and on the time t. is the initial vibrational state = 0 of the ground state. The matrix elements of the Hamiltonian (3.12), describing the neutral molecular states, are given by... 

Abedi, A., N. T. Maitra, and E. K. U. Gross. 2010. Exact factorization of the time-dependent electron-nuclear wave function. Phys. Rev. Lett. 105 123002-1-4. 

Finally, the time-dependent wave function has to be analyzed in order to evaluate the spectra, cross-sections, or other measurable quantities. Several methods have been developed to solve this general problem of time-dependent nuclear dynamics. The most efficient ones make (implicit) use of the... 

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... 

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