Exact non-adiabatic theory

The total wave function that describes both electrons and nuclei can be proposed in the following form  

The second term is non-zero only for the heteronuclear case and contains the mixed product of nablas V V,- with V/ = + j- Rx,Ry,R as the components of the vector R. 

If the problem were solved exactly, then the solution of the Schrodinger equation could be sought, e.g., by using the Ritz method (p. 202). Then we have to decide what kind of basis set to use. We could use two auxiliary complete basis sets one that depended on the electronic coordinates i/ffc (/ ), and the second on the nuclear coordinates j i(R). The complete basis set for the Hilbert space of our system could be constructed as a Cartesian product ipk(r) x ), i.e. all possible product-like functions ipklr) l(R). Thus, the wave function could be expanded in a series 

However, we are unable to manage the complete sets, instead, we are able to take only a few terms in this expansion. We would like them to describe the molecule reasonably well, and at the same time to 

let us write down the Schiddinger equation with the Hamiltonian (6.6) and the wave function as in (6.7) 

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The rectangular approximation (7.6) of dependence E(r) implies that ts = 0. This simplification being valid only for non-adiabatic interaction, exact knowledge of the time-dependence V(t) is not obligatory. Random walk approximation is quite acceptable. The value Ro/R is a free parameter of the model ( Ro/R < 1) and makes it possible to vary the ratio of times 0 < tc/to < oo. This interval falls into two regions one of them corresponds to impact theory (0 < tc/to < 1), and the other (1 <, tc/t0 < oo) to the fluctuating liquid cage. In the first case non-adiabaticity of the process is provided by the condition... 

The Bom-Oppenheimer Principle in the adiabatic approximation is one among many approximations that are made in molecular theory. The question of how precise is the approximation is not often asked and is difficult to answer. It is often brushed aside. One way to proceed is to compare exact solutions of the full molecular Hamiltonian with solutions found in the adiabatic approximation. It should be possible to test whether the latter solutions approach the former as non-adiabatic corrections are added. It is necessary to work with a Hamiltonian and associated Schrodinger equation that is capable of exact solution. 

Garrett and Truhlar circumvent the latter problem by making separable Taylor expansions around the bottom of the well of the internal Hamiltonian. For transition state theory, at low temperatures, this is a reasonable prescription, however the coordinate system remains arbitrary. Miller and coworkershave derived exact expressions for the Hamiltonian of the system as a function of a reaction path. Their theory incorporates non-adiabatic and rotational coupling between the internal degrees of freedom, however, it too is based on harmonic expansions and an arbitrary reaction coordinate. 

As the transition probabilities depend not only on the Massey parameter but also on the value of the matrix element of the interaction causing non-adiabatic transitions, an important role in the non-adiabatic transition theory is allotted to selection rules which establish the general connection between the type of the non-adiabatic interaction and the symmetries of states between which the transition occurs. The use of these selection rules and also a specific feature of the non-adiabatic interaction, namely the localization over relatively small regions, allows to approximate in these regions the adiabatic terms and the matrix elements of non-adiabatic coupling by simple functions which permits an exact solution of equations for non-adiabatic coupling. 

With eqn (7) the time-dependent Kohn-Sham scheme is an exact many-body theory. But, as in the time-independent case, the exchange-correlation action functional is not known and has to be approximated. The most common approximation is the adiabatic local density approximation (ALDA). Here, the non-local (in time) exchange-correlation kernel, i.e., the action functional, is approximated by a time-independent kernel that is local in time. Thus, it is assumed that the variation of the total electron density in time is slow, and as a consequence it is possible to use a time-independent exchange-correlation potential from a ground-state calculation. Therefore, the functional is written as the integral over time of the exchange-... 

For comparison, in the BCS theory this ratio is 3,52. In relative values both the BCS and our dependence of the energy gap on the temperature are exactly the same (i.e. the dependences of A/Aq on T/Tc). The study of other physical properties, such as specific heat, is published in our previous paper [19]. Let us note that the Eq. 28.93 was derived without any specific requirements for the detailed mechanism of superconductivity in comparison with the BCS theory. It reflects the thermodynamical properties of non-adiabatic systems in a more general form, solely as a consequence of the solution of the extended Born-Handy formula. 

We have presented some of the most recent developments in the computation and modeling of quantum phenomena in condensed phased systems in terms of the quantum-classical Liouville equation. In this approach we consider situations where the dynamics of the environment can be treated as if it were almost classical. This description introduces certain non-classical features into the dynamics, such as classical evolution on the mean of two adiabatic surfaces. Decoherence is naturally incorporated into the description of the dynamics. Although the theory involves several levels of approximation, QCL dynamics performs extremely well when compared to exact quantum calculations for some important benchmark tests such as the spin-boson system. Consequently, QCL dynamics is an accurate theory to explore the dynamics of many quantum condensed phase systems. 

Virtually all non-trivial collision theories are based on the impact-parameter method and on the independent-electron model, where one active electron moves under the influence of the combined field of the nuclei and the remaining electrons frozen in their initial state. Most theories additionally rely on much more serious assumptions as, e.g., adiabatic or sudden electronic transitions, perturbative or even classical projectile/electron interactions. All these assumptions are circumvented in this work by solving the time-dependent Schrodinger equation numerically exact using the atomic-orbital coupled-channel (AO) method. This non-perturbative method provides full information of the basic single-electron mechanisms such as target excitation and ionization, electron capture and projectile excitation and ionization. Since the complex populations amplitudes are available for all important states as a function of time at any given impact parameter, practically all experimentally observable quantities may be computed. 

Reaction conditions (i) and (ii) resemble the Semenov classifications of stable and unstable behaviour. For (iii), the reaction conditions are called parametrically sensitive. With absolute control of system parameters, any degree of self-heating can be produced and a complete range of maximum temperature excesses attained up to the adiabatic flame temperature. Physically such exact control is impossible and althou in the laboratory we should expect to see occasional temperature rises of the order of 100 K, repeatable non-explosive temperatures will be practically bounded by the steady-state limits. For simple systems, therefore, stationary-state treatments are still of great value. First, they impose a stability bound, inasmuch as conditions stable under stationary-state theory always remain stable under... 

If the treatment was limited to these two determinants with a common set of 3sci3pci atomic orbitals (AO) taken from either the Cl- or Cl Hartree-Fock (HF) calculation, the treatment would be quite incorrect and would predict an erroneous curve crossing distance and avoidance. But one may use these two determinants to define a 2 x 2 model space and apply the theory of effective Hamiltonians, as suggested by Levy " (with a slightly non-orthodox definition of the effective Hamiltonian). One may use either the Bloch or des Cloizeaux definition of /f " as a 2 x 2 matrix, the eigenvalues of which are the exact adiabatic eigenvalues... 

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