Adiabatic approximation functions

This means that the electronic and nuclear wave functions cannot be separated anymore, and therefore the adiabatic approximation cannot be applied beyond the second-order perturbation. 

Thus, the neglect of the off-diagonal matrix elements allows the change from mixed states of the nuclear subsystem to pure ones. The motion of the nuclei leads only to the deformation of the electronic distribution and not to transitions between different electronic states. In other words, a stationary distribution of electrons is obtained for each instantaneous position of the nuclei, that is, the elechons follow the motion of the nuclei adiabatically. The distribution of the nuclei is described by the wave function x (R i) in the potential V + Cn , known as the proper adiabatic approximation [41]. The off-diagonal operators C n in the matrix C, which lead to transitions between the states v / and t / are called operators of nonadiabaticity and the potential V = (R) due to the mean field of all the electrons of the system is called the adiabatic potential. 

Now we make the usual assumption in nonadiabatic transition theory that non-adiabaticity is essential only in the vicinity of the crossing point where e(Qc) = 0- Therefore, if the trajectory does not cross the dividing surface Q = Qc, its contribution to the path integral is to a good accuracy described by adiabatic approximation, i.e., e = ad Hence the real part of partition function, Zq is the same as in the adiabatic approximation. Then the rate constant may be written as... 

Then fie may be thought of as the phase accumulated by the function Ci(t) during the period p. To find B we should compare the phase 4>(P -I- t t) to that calculated in the adiabatic approximation... 

In the adiabatic approximation the form of the total wave function is restricted to one electronic surface, i.e. all coupling elements in eq. (3.12) are neglected (only the terms with i = i survive). Except for spatially degenerate wave functions, the diagonal first-order non-adiabatic coupling element is zero. 

In the Bom-Oppenheimer picture the nuclei move on a potential energy surface (PES) which is a solution to the electronic Schrodinger equation. The PES is independent of the nuclear masses (i.e. it is the same for isotopic molecules), this is not the case when working in the adiabatic approximation since the diagonal correction (and mass polarization) depends on the nuclear masses. Solution of (3.16) for the nuclear wave function leads to energy levels for molecular vibrations (Section 13.1) and rotations, which in turn are the fundamentals for many forms of spectroscopy, such as IR, Raman, microwave etc. 

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. 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

Bauernschmitt, R., Ahlrichs, R., 1996b, Treatment of Electronic Excitations Within the Adiabatic Approximation of Time Dependent Density Functional Theory , Chem. Phys. Lett., 256, 454. 

For molecular system, the Bom-Oppenheimer (B-O) adiabatic approximation is often used. In this case the molecular wave function can be expressed as... 

On the other hand, the undamped autocorrelation function (17) we have obtained within the standard approach avoiding the adiabatic approximation must lead after Fourier transform to spectral densities involving very puzzling Dirac delta peaks given by... 

Figure 19. The mean decay time as a function of frequency of the driving signal for different values of noise intensity, kT — 0.5,0.1,0.05, A — 1. The phase is equal to zero. Solid lines represent results of computer simulation, and dashed lines represent an adiabatic approximation (6.15).

In this framework, we have developed an analytical model based on a self-consistent solution of the Poisson equation using an adiabatic approximation for laser generated fast electrons [75], This model, briefly outlined in the following, allows the determination of the optimal target thickness to optimize the maximum proton (and ion) energies, as well as the particle number as a function of given UHC laser pulse parameters. 

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