Nuclear dynamics energy functional form

The T =0 time-dependent mean-field theory currently provides the best description of nuclear dynamics at low energies [5,6]. We consider two single-particle operators, Q, P interpreted as a collective coordinate and a collective momentum. Their nature depends on the kind of motion that we want to focus upon. We require that Q Q and P-r — P under time reversal and that IP, Q] 0. We then form a constrained Hartree-Fock (CHF) calculation on the many-body Hamiltonian H by minimizing the functional... 

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 ... 

It should be pointed out that the use of a thermostat affects the energy conservation in MD. Namely, in thermostatted dynamics the conserved energy (kinetic and potential energy of nuclei plus the fictitious kinetic energy of the wave function) discussed in Section 2.1 is no longer conserved. Instead, the energy that includes additional terms due to the thermostats (nuclear and electronic ) is constant. For example, for a system thermostatted by a chain of n nuclear thermostats, controlled by variables J and QJ, the conserved energy takes the form ... 

As reference data for comparison, the full quantum mechanical wave-functions of the coherent-type Gaussian form as in Eq. (6.106) are numerically propagated. The wavefunction is chosen to start at a point xi,X2) = (5.5,0.0) with zero initial momentum on the first diabatic state. The point of conical intersection is located at (xi,X2) = (2.5,0.0), the potential energy of which is exactly the same as the value at (xi, X2) = (5.5,0.0). This symmetric dynamics is performed to purely illustrate how the quantum wave functions are branched behind the conical intersection. The contour plots of the density of nuclear wavefunctions at selected times are given in Fig. 6.12 (solid curves). 

The existence of the biradicals and the multipHcity of the surfaces on which these are formed have not been demonstrated directly however, experimental results (stereochemistry of the reaction, CIDNP [chemically induced dynamic nuclear polarization], radical trapping experiments, and quantum yield measurements) support their existence. Recently, the mechanism of 1,3-migration and oxa-di-Jt-methane reactions in terms of potential energy surface and decay funnels has been described this also supports the aforementioned mechanistic impHcations. The detailed mechanism, however, depends, in a very subtle way, on the structure of the chromophoric system and the presence of the functional groups. 

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