Quasi-adiabatic states

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... 

The Quasi-variational method (Method II) An alternative approach, which we recommend, consists of optimizing each diabatic state separately, in an independent calculation. Consequently, the resulting orbitals of the diabatic states are different from those of the adiabatic states, and each diabatic state possesses its best possible set of orbitals. The diabatic energies are obviously lower compared with those obtained by the previous method, and are therefore quasi-variational. The diabatic energies of the covalent and ionic structures of F2, calculated with Methods I and II in the L-BOVB framework, are shown in Table 10.4. It is seen that the ionic structures have much lower energies in the quasi-variational procedure, and as such, the procedure can serve a basis for deriving quantities such as resonance energies (see below). 

The choice of the saturation gas is critical. When Ar and Kr were sparged in water irradiated at 513 kHz, an enhancement in the production of OH radicals of between 10% and 20%, respectively, was observed, compared with 02-saturated solutions [22]. The higher temperatures achieved with the noble gases upon bubble collapse under quasi-adiabatic conditions account for the observed difference. Because the rate of sonochemical degradation is directly linked to the steady state concentration of OH radicals, the acceleration of those reactions is expected in the presence of such background gases. The use of ozone as saturation gas (in mixtures with 02) provided new reaction pathways in the gas phase inside the bubbles, which also increase the measured reaction rates (see Sect. IV.G.l). 

Figure 15 Porosity structure of a high-resolution single-channel calculation for an upwelling system undergoing melting by both adiabatic decompression and reactive flow (see Spiegelman and Kelemen, 2003). Colors show the porosity field at late times in the run where the porosity is quasi steady-state. The maximum porosity at the top of the column (red) is 0.8% while the minimum porosity at the bottom (dark blue) is 10 times smaller. Axis ticks are height and width relative to the overall height of the box. In the absence of channels this problem is identical to the equilibrium one-porosity transport model of Spiegelman and Elliott (1993). Introduction of channels, however, produces interesting new chemical effects similar to the two porosity models.

Operator equations have been employed by George and Ross (1971) to analyse symmetry in chemical reactions. In order to preserve the identity of electronic states of reactants and products, these authors worked within a quasi-adiabatic representation of electronic motions. By introducing a chain of approximations, going from separate conservation of total electronic spin to complete neglect of dynamics, they discussed the Wigner-Witmer angular momentum correlation rules, Shuler s rules for linear molecular conformations and the Woodward-Hoffmann rules. 

The optimal catalyst distribution problem was studied in an adiabatic reactor (Ogunye and Ray, 197la,b). The optimal initial distribution of catalyst activity along the axis of a tubular fixed-bed reactor was examined for a class of reactivation-deactivation problems by Gryaert and Crowe (1976). A general set of simultaneous reactions was considered, quasi steady state approximation was used, and the decay of the catalyst expressed as a function of temperature, concentration and catalyst activity. The influence of various initial catalyst activity distributions upon the reactor performance was also considered. 

A system which approaches the adiabatic state. (See Quasi-adiabatic .)... 

Figure 21.15 Schematic representation of classical and vibrationally adiabatic potentials, showing the presence of a potential well along the reaction coordinate for v = 1. This well can support quasi-bound states or resonances. Adapted from Schatz, permission AAAS...

It is also important to note how, even in the case that a well on the PES is not present, the (vibrationally) adiabatic curves can show wells and barriers, as the PES perpendicular to the translational coordinate widens and narrows respectively. These wells can support quasi-bound states similar to shape resonances. Therefore, reactive scattering through this temporarily bound state can give rise to reactive resonances. 

The operating conditions of calorimeters are defined first and foremost with regard to an ideal state. For this reason, the designations isothermal and adiabatic as used here are not in strict accordance with the concepts of thermodynamics. In calorimetric practice, it would be more appropriate to use the terms quasi-isothermal and quasi-adiabatic. There is a common tendency to use thermodynamic concepts even when the ideal conditions required by them are not complied with. This fact must be kept in mind, in particular in the uncertainty analysis. 

Presumably the most straightforward approach to chemical dynamics in intense laser fields is to use the time-independent or time-dependent adiabatic states [352], which are the eigenstates of field-free or field-dependent Hamiltonian at given time points respectively, and solve the Schrodinger equation in a stepwise manner. However, when the laser field is approximately periodic, one can also use a set of field-dressed periodic states as an expansion basis. The set of quasi-static states in a periodic Hamiltonian is derived by a Floquet type analysis and is often referred to as the Floquet states [370]. Provided that the laser field is approximately periodic, advantages of using the latter basis set include (1) analysis and interpretation of the electron dynamics is clearer since the Floquet state population often vary slowly with the timescale of the pulse envelope and each Floquet state is characterized as a field-dressed quasi-stationary state, (2) under some moderate conditions, the nuclear dynamics can be approximated by mixed quantum-classical (MQC) nonadiabatic dynamics on the field-dressed PES. The latter point not only provides a powerful clue for interpretation of nuclear dynamics but also implies possible MQC formulation of intense field molecular dynamics. 

Most procedures, but not all of them, imply the preliminary determination of the adiabatic wavefuactions, which is.fe trivial way of achieving the block diagonalization of //ei by means of the standard methods of ab initio theoretical chemistry then, a rotation or projection of the adiabatic states yields the quasi-diabatic ones. In all cases, the central point is to obtain the 5 block H of the electronic Hamiltonian matrix. The diagonalization of H returns the adiabatic energies Ek and the eigenvectors Cg, which express the adiabatic states in the quasi-diabatic basis ... 

Several methods to determine quasi-diabatic states are based on the assumption that single determinants or configurations are quasi-diabatic wavefunctions in other words, their physical content should not change too much with the nuclear geometry and the matrix elements of d/dQa in this basis should be small. When this is true, the main source of variation of the adiabatic wavefunctions, and the dominant contribution to is the change of the Cl coefficients as functions of the nuclear coordinates keeping the Cl coefficients as constant as... 

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