Adiabatic approximation generalized

THE CRUDE BORN-OPPENHEIMER ADIABATIC APPROXIMATION 555 which is a more general foiin of Eq. (131). The modification is simple ... 

The general potential U (8) has not been used before 1999 [52] because its numerical matrix representation requires huge basis sets, incompatible with the common computers. In order to avoid this situation, an approximation has been undertaken in previous studies the adiabatic approximation [54,55], Following an idea of Stepanov [56], Marechal and Witkowski assumed that the fast mode follows adiabatically the slow intermonomer motions, just as the electrons are assumed to follow adiabatically the motions of the nuclei in a molecule. It has been shown [57] that the adiabatic approximation is only suitable for very weak hydrogen bonds, as discussed in the next section. 

We must stress that the use of a single damping parameter y supposes that the relaxations of the fast and bending modes have the same magnitude. A more general treatment of damping has been proposed [22,23,71,72] however, this treatment (discussed in Section IV.D) requires the use of the adiabatic approximation, so that its application is limited to very weak hydrogen bonds. 

As shown in Section IV.D, it is possible, within the adiabatic approximation, to account for the general situation where the relaxation parameters of the fast mode y0 and of the bending mode y are not supposed to be equal. 

These conclusions must be considered keeping in mind that the general theoretical spectral density used for the computations, in the absence of the fast mode damping, reduces [8] to the Boulil et al. spectral density and, in the absence of the slow mode damping, reduces to that obtained by Rosch and Ratner one must also rember that these two last spectral densities, in the absence of both dampings [8], reduce to the Franck-Condon progression involving Dirac delta peaks that are the result of the fundamental work of Marechal and Witkowski. Besides, the adiabatic approximation at the basis of the Marechal... 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

In Table II we also compare our total variational energies with the energies obtained by Wolniewicz. In his calculations Wolniewicz employed an approach wherein the zeroth order the adiabatic approximation for the wave function was used (i.e., the wave function is a product of the ground-state electronic wave function and a vibrational wave function) and he calculated the nonadiabatic effects as corrections [107, 108]. In general the agreement between our results... 

In order to separate the electronic and nuclear coordinates in an eigenvalue problem for the Hamiltonian defined by Equation 1, the adiabatic approximation in the version of a Bom-Oppenheimer model is used. In general, the eigenfunction defined within the adiabatic approximation is defined as a linear combination. 

The direct variational solution of the Schrddinger equation after separation of the center of mass motion is in general possible and can be performed very accurately for three- and four- body systems such as (Kolos, 1969) and H2 (Kolos and Wolniewicz, 1963 Bishop and Cheung, 1978). For larger systems it is unlikely to perform such calculations in the near future. Therefore the usual way in quantum chemistry is to introduce the adiabatic approximation. The nonrelativistic hamiltonian for a diatomic N-electron molecule in the center of mass system has the following form (in atomic units). 

Here, the pt are the permanent dipoles of molecules i = 1 and 2, and the ptj( r, i 2, Rij) are the dipoles induced by molecule i in molecule j the are the vectors pointing from the center of molecule i to the center of molecule j and the r, are the (intramolecular) vibrational coordinates. In general, these dipoles are given in the adiabatic approximation where electronic and nuclear wavefunctions appear as factors of the total wavefunction, 0(rf r) ( ). Dipole operators pop are defined as usual so that their expectation values shown above can be computed from the wavefunctions. For the induced dipole component, the dipole operator is defined with respect to the center of mass of the pair so that the induced dipole moments py do not depend on the center of mass coordinates. For bigger systems the total dipole moment may be expressed in the form of a simple generalization of Eq. 4.4. In general, the molecules will be assumed to be in a electronic ground state which is chemically inert. 

It is necessary next to relate these small displacements to the collective coordinates g, from a Taylor expansion of the general potential energy U(Q). In the adiabatic approximation, the nuclear coordinates Q are free parameters and can be used as a basis for the Taylor expansion. Thus we write IJ(Q) in the general form (Ref. [2], Chapter 3) ... 

Periodic orbits also explain the long-lived resonances in the photodissociation of CH.30N0(S i), for example, which we amply discussed in Chapter 7. But the existence of periodic orbits in such cases really does not come as a surprise because the potential barrier, independent of its height, stabilizes the periodic motion. If the adiabatic approximation is reasonably trustworthy the periodic orbits do not reveal any additional or new information. Finally, it is important to realize that, in general, the periodic orbits do not provide an assignment in the usual sense, i.e., labeling each peak in the spectrum by a set of quantum numbers. Because of the short lifetime of the excited complex, the stationary wavefunctions do not exhibit a distinct nodal structure as they do in truly indirect processes (see Figure 7.11 for examples). 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

It should be noted that the condition of the adiabatic approximation, although inherent to the classical treatment of the nuclei, may contradict the condition for the classical limit of quantum mechanics [generally, the condition of slow nuclear motion, P 0, and the condition, Eq. (8), may not be fulfilled simultaneously]. 

Approximations made in the XC potential generally also affect the quality of the XC response kernel if it is derived from the potential. In addition, in essentially all applications of TDDFT to computations of molecular response properties, the XC kernel is adiabatic (not frequency-dependent), even though it should be a function of frequency. One of the better known consequences of the adiabatic approximation is the inability of TDDFT to describe simultaneous excitations of more than one electron. Due to the sometimes very pronounced effects from the approximations under points 1-3, along with effects from limited basis set flexibility, it is not clear how strongly the adiabatic approximation affects present-day computations of molecular chiroptical response properties in terms of its ability to predict ECD and ORD in the UV-Vis range of frequencies. 

While the one-dimensional case may seem too simple, even trivial, it presents a good opportunity to put forward some very general concepts. These concepts, like the existence of barriers in phase space and the stable/unstable manifolds theorem, are best introduced here, having in mind that most interesting applications will come later on. Also, the one-dimensional case has been employed in less trivial ways, by reducing all rapid DOFs to some adiabatic approximation allowing nonlinear one-dimensional TST to be applied [34]. 

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