Nuclear coordinates, dynamics

If now the nuclear coordinates are regarded as dynamical variables, rather than parameters, then in the vicinity of the intersection point, the energy eigenfunction, which is a combined electronic-nuclear wave function, will contain a superposition of the two adiabatic, superposition states, with nuclear... 

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.)... 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

Effect of off-diagonal dynamic disorder (off-DDD). The interaction of the electron with the fluctuations of the polarization and local vibrations near the other center leads to new terms VeP - V P, Vev - Vev and VeAp - VAPd, VA - VAd in the perturbation operators V°d and Vfd [see Eqs. (14)]. A part of these interactions corresponding to the equilibrium values of the polarization P0l and Po/ results in the renormalization of the electron interactions with ions A and B, due to their partial screening by the dielectric medium. However, at arbitrary values of the polarization P, there is another part of these interactions which is due to the fluctuating electric fields. This part of the interaction depends on the nuclear coordinates and may exceed the renormalized interactions of the electron with the donor and the acceptor. The interaction of the electron with these fluctuations plays an important role in processes involving solvated, trapped, and weakly bound electrons. 

The dynamics of the nuclear coordinates (Rj) and that of the expansion coefficients (CL) is governed by a generalized steepest-descent procedure that involves solving two sets of dynamical equations ... 

An alternative approach was introduced by Car and Parrinello [12], who developed a DFT-MD method to study periodic systems using a planewave expansion in which the electronic parameters, as well as the nuclear coordinates, are treated as dynamical variables. Following the Car and Parrinello... 

We wanted to extend this approach to include dynamical effects on line shapes. As discussed earlier, for this approach one needs a trajectory co t) for the transition frequency for a single chromophore. One could extract a water cluster around the HOD molecule at every time step in an MD simulation and then perform an ab initio calculation, but this would entail millions of such calculations, which is not feasible. Within the Born Oppenheimer approximation the OH stretch potential is a functional of the nuclear coordinates of all the bath atoms, as is the OH transition frequency. Of course we do not know the functional. Suppose that the transition frequency is (approximately) a function of a one or more collective coordinates of these nuclear positions. A priori we do not know which collective coordinates to choose, or what the function is. We explored several such possibilities, and one collective coordinate that worked reasonably well was simply the electric field from all the bath atoms (assuming the point charges as assigned in the simulation potential) on the H atom of the HOD molecule, in the direction of the OH bond. 

The surface hopping study was rather expensive in terms of CPU time, and consequently large numbers of trajectories could not be run. This is important to obtain statistically converged dynamical properties. The main goal of the surface hopping study was thus not to obtain such information but to provide mechanistic insight into the photodissociation and subsequent relaxation processes. The semi-classical work in the full space of nuclear coordinates provides the important vibrational degrees of freedom that one needs to include in any quantum model of the nuclear motion. This will now be described. 

The acetylene A <- X electronic transition is a bent <- linear transition that would be electronically forbidden ( - ) at the linear structure. The usual approximation is to ignore the possibility that the electronic part of the transition moment depends on nuclear configuration and to calculate the relative strengths of vibrational bands as the square of the vibrational overlap between the initial and final vibrational states (Franck-Condon factor). A slightly more accurate picture would be to express the electronic transition moment as a linear function of Q l (the fra/w-bending normal coordinate on the linear X1 state) in such a treatment, the transition moment would be zero at the linear structure and the vibrational overlap factors would be replaced by matrix elements of Qfl- Nevertheless, as long as one makes use of low vibrational levels of the A state, neglect of the nuclear coordinate dependence of the electronic excitation function is unlikely to affect the predicted dynamics or to complicate any proposed control scheme. 

The RAS concept combines the features of the CAS wave functions with those of more advanced Cl wave functions, where dynamical correlation effects are included. It is thus able to give a more accurate treatment of correlation effects in molecules. The fact that orbital optimization is included makes this method especially attractive for studies of energy surfaces, when there is a need to compute the energy gradient and Hessian with respect to the nuclear coordinates. 

Molecular dynamics involves the calculation of the time-dependent movement of each atom in a molecule11121. This is achieved by solving Newton s equations of motion. For this process the energy surface and the derivative of the energy in terms of the nuclear coordinates are required (Eqs. 4.1 and 4.2 mass m, acceleration a, potential energy E, coordinates r, time t). 

The equality Eenfajrx0 ) = Een(A) is not obvious as R stands for nuclear coordinates appearing in the dynamics of such species, while a represents a space which can be used to define gauge fields [8] for either classical or quantum mechanical models. This latter issue is not examined here. For the time being, let us take the equality and use it by relaxing symmetry constraints that may be related to an but not necessarily to a. Then, eq.(10) follows trivially from eq.(2) after... 

A systematic route to achieve a mixed quantum classical description of EET may start with the partial Wigner representation p(R,P t) of the total density operator referring to the CC solvent system. R and P represent the set of all involved nuclear coordinates and momenta, respectively. However, p(R,P t) remains an operator in the space of electronic CC states (here 4>o and the different first order of the -expansion one can change to electronic matrix elements. Focusing on singly excited state dynamics we have to consider pmn( It, / /,) = 4>m p R, P t) 4>n) which obeys the following equation... 

Intra chromophore vibrations, i.e. the relative motion of all atoms of a particular chromophore, of course, are included in the MD simulations. But, it is less easy to account for their influence on the EET dynamics. They enter via the single chromophore PES of the ground and the excited state Umg(Rm) and Ume(Rm), respectively. If the nuclear coordinate dependence... 

---