Adiabatic approximation proper

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. 

Most of the theoretical works concerning dynamical aspects of chemical reactions are treated within the adiabatic approximation, which is based on the assumption that the solvent instantaneously adjusts itself to any change in the solute charge distribution. However, in certain conditions, such as sudden perturbations or long solvent relaxation times, the total polarization of the solvent is no longer equilibrated with the actual solute charge distribution and cannot be properly described by the adiabatic approximation. In such a case, the reacting system is better described by nonequilibrium dynamics. 

The term H e is the electron correlation operator, the term H p corresponds to phonon-phonon interaction and H l corresponds to electron-phonon interaction. If we analyze the last term H l we see that when using crude approximation this corresponds to such phonons that force constant in eq. (17) is given as a second derivative of electron-nuclei interaction with respect to normal coordinates. Because we used crude adiabatic approximation in which minimum of the energy is at the point Rg, this is also reflected by basis set used. Therefore this approximation does not properly describes the physical vibrations i.e. if we move the nuclei, electrons are distributed according to the minimum of energy at point Rg and they do not feel correspondingly the R dependence. The perturbation term H) which corresponds to electron-phonon interaction is too large... 

These considerations make TDDFT a very attractive method in the context of first-principles photodynamics calculations. However, there are problems that need to be resolved. Most current implementations of TDDFT use the adiabatic approximation, which states that the density functional is independent of excitation energy. The severity of this approximation is not known. Furthermore, the TDDFT equations are often linearized for efficient solution. Again, the importance of the higher terms is not well characterized. This linearization leads to a formal similarity of TDDFT to CIS and has made many workers suspect that TDDFT may not properly represent doubly excited states. Althought it has been shown that such an assessment may be overly pessimis-tic,74,8o some amount of caution is nevertheless recommended. 

The possibility of consideration of atoms as elementary subunits of the molecular systems is a consequence of Born-Oppenheimer or adiabatic approximation ( separation of electron and nuclear movements) aU quantum chemistry approaches start from this assumption. Additivity (or linear combination) is a common approach to construction of complex functions for physical description of the systems of various levels of complexity (cf orbital approximation, MO LCAO approximation, basis sets of wave functions, and some other approximations in quantum mechanics). The final justification of the method is good correlation of the results of its applications with the available experimental data and the potential to predict the characteristics of molecular systems before experimental data become available. It can be achieved after careful parameter adjustment and proper use of the force field in the area of its validity. The contributions not considered explicitly in the force field formulae are included implicitly into parameter values of the energy terms considered. 

They acknowledge that they may need to go beyond the adiabatic approximation and work with a frequency dependent coupling matrix. They state that Tn the exact theory it is precisely the complicated pole structure of K in the o>-representation which leads to the proper description of complex excitation processes absent in the adiabatic approximation. ... 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

With proper interchange of heat and proper gas flow, staged adiabatic packed beds become a versatile system, which is able to approximate practically any desired temperature progression. Calculation and design of such a system is simple, and we can expect that real operations will closely follow these predictions. 

The proper evaluation of the quantized energy levels within the SACM requires a separable reaction coordinate and thus numerical implementations have implicitly assumed a center-of-mass separation distance for the reaction coordinate, as in flexible RRKM theory. Under certain reasonable limits the underlying adiabatic channel approximation can be shown to be equivalent to the variational RRKM approximations. Thus, the key difference between flexible RRKM theory and the SACM is in the focus on the underlying potential energy surface in flexible RRKM theory as opposed to empirical interpolation schemes in the SACM. Forst s recent implementation of micro-variational RRKM theory [210], which is based on interpolations of product and reactant canonical partition functions, provides what might be considered as an intermediate between these two theories. 

So far, we have only separated out the HX vibrational motion. Generally, such a solution of the stationary Schiodinger equation is not computationally feasible for clusters with more atoms. Therefore, other approximations have to be employed. At the same time, all phenoniena important for the cluster structure have to be properly included. We have performed an adiabatic separation of the HX libra-tional motion from the motion of the heavy particles, i.e., from the cage modes. Moreover, the cage modes have been calculated within the harmonic approach, i.e., by a diagonalization of the Hessian matrix. Formally, the wavefunction is expressed as... 

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