Electric Nuclear Born-Oppenheimer

Equation (28) is the set of exact coupled differential equations that must be solved for the nuclear wave functions in the presence of the time-varying electric field. In the spirit of the Born-Oppenheimer approximation, the ENBO approximation assumes that the electronic wave functions can respond immediately to changes in the nuclear geometry and to changes in the electric field and that we can consequently ignore the coupling terms containing... 

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. 

To deduce whether a transition is allowed between two stationary states, we investigate the matrix element of the electric dipole-moment operator between those states (Section 3.2). We will use the Born-Oppenheimer approximation of writing the stationary-state molecular wave functions as products of electronic and nuclear wave functions ... 

Solid State Physical Methods. - The theoretical treatment of a molecule or a polymer in the presence of an electric field or more generally of a laser beam presents a formidable problem. Here we shall remain first within the framework of the Born-Oppenheimer approximation and shall not consider the change of the phonons in the presence of an electric field because we shall work in a fixed nuclear (framework). Further, first we shall not take into account the effect of the interaction between the linear polymers on their polarizabilities and hyperpolarizabilities either although both effects are non-neglible.110-1,2 They will be treated subsequently. 

In the previous sections, the Born-Oppenheimer approximation was assumed, i.e. nuclear motion on a single PES was considered. However, in many simations, the dynamics of the nuclei need to be treated on several PESs corresponding to coupled electronic state. The coupling between the electronic states can be due to the presence of an external electric field or to internal vibronic interactions. There exists two different ways of treating several coupled electronic states with the MCTDH method [60], the single-set formulation and the multi-set formulation. 

Let us look first at the transition of the original definitions as integrals over the charge density, Eqs. (4.5), (4.6) and (4.8), to quantum mechanics that we will illustrate for the example of the electric dipole moment. In the Born-Oppenheimer approximation, Section 2.2, the electrons in a molecule form a continuous charge distribution whereas the discrete nuclear charges are located at fixed points Rk- The expression, Eq. (4.5) for the a-component of the electric dipole moment can therefore be rewritten as... 

The ]V-electron operators p. Ro) and Q Ro) will in the following often be called the electric dipole operator and the electric quadrupole operator, respectively. Although we are working within the Born-Oppenheimer approximation we have included the interaction of the electric field and field gradient with the nuclear charges in the molecular Hamiltonian in Eq. (2.101). This interaction then leads to nuclear contributions to the perturbation Hamiltonian operators. The operators and... 

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