Adiabatic approximation accuracy

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... 

Figure 4. Accuracy of the lineshapes with respect to the adiabatic approximation. When unspecified, the parameters are co0 = 3000cm-1, co00 = 150cm-1, y = 30cm-1, and T = 300 K.

The adiabatic approximation is one of the keystones on which the theory of electron tunneling is based (see Sect. 2). In particular, the matrix element for the transition between the initial and the final electron states contain the adiabatic wave functions of the donor and acceptor. Adiabatic approximation is known [25] to have a very high degree of accuracy. Because of this the non-adiabatic effects have been neglected until recently in the theory of electron tunneling without detailed analysis of whether this can actually be done. In the present section we shall try to fill in this blank and to discuss to what extent the non-adiabatic effects can influence the process of electron tunneling. 

The following ideas have been used in these studies. If the electron is located near the donor (say at a distance less than rQ = 15 A from it), then the wave function of the system comprising "the core of the donor + electron is described with good accuracy in terms of the adiabatic approximation... 

Nonadiabaticity—the additional "friction" between electrons and nuclei (ions)—neglected in obtaining the BO approximation is describable by an operator H. VJe now evaluate H (72,73) and estimate the accuracy of the BO adiabatic approximation. The operator H can be found in the following way (29). The total Hamiltonian H is written as a sum of two terms,... 

We shall employ the adiabatic approximation that was shown to provide a rather good accuracy in the calculation of the magnetic susceptibility for the vibronic mixed valence systems exhibiting PJT in a wide range of parameters [9]. At the same time this approximation allows us to gain a descriptive comprehension of the physical role of the JT interaction. The full Hamiltonian of the system in the adiabatic approximation can be written as follows ... 

Kolos W (1970) Adiabatic approximation and its accuracy. Adv Quantum Chem 5 99-133... 

In order to reveal the effects of the JT vibronic interaction [74]-[76] one can employ the adiabatic approximation that was proved to provide a quite good accuracy in the description of the magnetic properties of MV clusters [77] and allowed to avoid numerical solutions of the dynamic problem. According to the adiabatic approach the magnetization can be obtained by averaging the derivatives —dUi(p, H)/dHa over the vibrational coordinates. In the case of an arbitrary p 7 0 the gap between... 

H. We understood H to be complete and including electronic as well as nuclear degrees of freedom, and in which case the states are the true nonadiabatlc vlbronic eigenstates of the system and hence the properties are the exact ones. Nothing prevents us, however, to introduce the adiabatic approximation and to assume the wave functions to be products of electronic and nuclear (vibrational) parts. In this case, the Born-Oppenheimer electronic plus vibrational properties will appear. We can even reduce the accuracy to the extent that we adopt the electronic Hamiltonian, work with the spectrum of electronic states, and thus extract the electronic part of the properties. In all these cases, the SOS property expressions remain unchanged. 

The degree of accuracy of the adiabatic approximation can be inferred from Eq. (4.27). The nonadiabatic terms are proportional to the adiabatic channel index n and to h2/m. For fixed E, the electronic quantum index is proportional to Emtl2/h, hence the nonadiabatic terms are proportional to Eh/mtl2. 

W. Kolos, Adiabatic approximation and its accuracy f Advan. Quantum Chem., 5, 99 (1970). 

There are certain situations in which the accuracy of time-dependent DFT applications may be unsatisfactory, due to the fact that the currently used xc approximations are not satisfactory in these cases. Multiple excitations are not yet well described, for example. It is generally believed that treatment of such excitations requires an xc kernel /xc which is frequency-dependent and significantly more complicated than the currently popular adiabatic approximation which is almost universally employed. In practice it turns out that excitations, which in theories based on HF have a significant multiple-excitation character, may still be surprisingly well described at the TDDFT level. If, however, an excitation is almost purely a multiple excitation, the current TDDFT descriptions are likely to miss it completely. 

W. Kolos, Adiabatic Approximation and its Accuracy , Advaw. Quantum Chem. 5 (1970)... 

Equation (2.13) constitutes the Born-Oppenheimer approximation. When the electronic adiabatic wavefunctions are chosen real, the non-adiabatic term simply reads A = G = ( 6 T ). Usually, A is very small and Eq.(2.13) is only used if very high accuracy is sought. Neglecting A leads to the so-called adiabatic approximation... 

For more than two decades, density functional theory (DFT) has been used as a reliable tool to describe the electronic stmcture, total energy, and associated characteristics of molecules and solid-state materials, in the adiabatic approximation [1, 2]. The level of accuracy achieved by DFT is acceptable for a range of purposes in quantum chemistry with a better balance between accuracy and computational cost than more sophisticated approaches based on interacting electronic wave function theory (WFT) [3]. 

For further discussion, we must first consider some aspects of quantum mechanics, associated with the transitions of systems from one state to another. Complex polyatomic systems consist of nuclei and electrons. An absolutely accurate description of such systems even in the stationary state is quite complicated. For this reason, an approximate method is employed, which gives a fairly high accuracy in a majority of cases (we are speaking about the Born-Oppenheimer adiabatic approximation). The physical meaning of this method was in fact explained in the previous section. Since the characteristic velocities of motion of nuclei and electrons differ considerably, we can consider the motion of electrons in the field created by fixed nuclei. Incidentally, it is clear from this that the Franck-Condon principle is a direct corollary of the adiabatic approximation. 

Physically the cutting-corner trajectory implies that the particle crosses the barrier suddenly on the time scale of the slow -vibration period. In the literature this approximation is usually called sudden , frozen bath and fast flip approximation, or large curvature case. In the opposite case of small curvature (also called adiabatic and slow flip approximation), coj/coo < sin tp, which is relevant for transfer of fairly heavy masses, the MEP may be taken to a good accuracy to be the reaction path. 

The aim of this work is to obtain the four lowest E curves and wavefunctions of BH at the same level of accuracy and to bring out the interplay of ionic, Rydberg and valence states at energies and internuclear distances which were not previously investigated. We have therefore made use of a method, already put forward by us [16,17] to determine at once quasi-diabatic and adiabatic states, potential energy cnrves and approximate nonadiabatic couplings. We have analogously determined the first three E+ states, of which only the lowest had been theoretically studied... 

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