Approximations , Adiabatic diabatic

There are three ways of implementing the GP boundary condition. These are (1) to expand the wave function in terms of basis functions that themselves satisfy the GP boundary condition [16] (2) to use the vector-potential approach of Mead and Truhlar [6,64] and (3) to convert to an approximately diabatic representation [3, 52, 65, 66], where the effect of the GP is included exactly through the adiabatic-diabatic mixing angle. Of these, (1) is probably the most... 

The coupling of electronic and vibrational motions is studied by two canonical transformations, namely, normal coordinate transformation and momentum transformation on molecular Hamiltonian. It is shown that by these transformations we can pass from crude approximation to adiabatic approximation and then to non-adiabatic (diabatic) Hamiltonian. This leads to renormalized fermions and renormalized diabatic phonons. Simple calculations on H2, HD, and D2 systems are performed and compared with previous approaches. Finally, the problem of reducing diabatic Hamiltonian to adiabatic and crude adiabatic is discussed in the broader context of electronic quasi-degeneracy. 

BORN-OPPENHEIMER APPROXiMATION ADIABATIC PHOTOREACTION DIABATIC PHOTOREACTION BORNYL PYROPHOSPHATE SYNTHASE BOROHYDRIDE REDUCTION ACYL-PHOSPHATE INTERMEDIATE BOSE-EINSTEIN CONDENSATE Bound water,... 

The sudden changes in the adiabatic wavefunctions near avoided crossings make it more convenient to use diabatic potential energy surfaces when simulating photodissociation dynamics. The adiabatic potentials, usually constructed from electronic structure calculation data, should therefore be transformed to diabatic potentials. The adiabatic-diabatic transformation yields diabatic states for which the derivative couplings above approximately vanish. The diabatic potential energy surfaces are obtained from the adiabatic ones by a unitary orthogonal transformation [22,23]... 

This matrix was introduced by F. T. Smith [25] for the treatment of non-adiabatic (diabatic) couplings in atomic collisions. It is now familiar also in molecular structure problems, to indicate local breakdowns of the Born-Oppenheimer approximation. Within the hyperspherical formalism, it has been introduced in the three-body Coulomb problem [20] and in chemical reactions [21-24], see also Section 3. Also, from equation (A4)... 

Adiabatic energy transfer occurs when relative collision velocities are small. In this case the relative motion may be considered a perturbation on adiabatic states defined at each intermolecular position. Perturbed rotational states have been introduced for T-R transfer at low collision energies and for systems of interest in astrophysics.A rotational-orbital adiabatic basis expansion has also been employed in T-R transfer,as a way of decreasing the size of the bases required in close-coupling calculations. In T-V transfer, adiabatic-diabatic transformations, similar to the one in electronic structure studies, have been implemented for collinear models.Two contributions on T-(R,V) transfer have developed an adiabatical semiclassical perturbation theory and an adiabatic exponential distorted-wave approximation. Finally, an adiabati-cally corrected sudden approximation has been applied to RA-T-Rg transfer in diatom-diatom collisions. 

In this section, we prove that the non-adiabatic matiices have to be quantized ( similar to Bohr-Sommerfeld quantization of the angulai momentum) in order to yield a continous, uniquely defined, diabatic potential matrix W(i). In another way, the extended BO approximation will be applied only to those cases that fulfill these quantization rules. The ADT matrix A(s,so) transforms a given adiabatic potential matiix u(i) to a diabatic matiix W(s, so)... 

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. 

Obviously, the fact that the solution of the adiabatic-to-diabatic transformation matrix is only perturbed to second order makes the present approach rather attractive. It not only results in a very efficient approximation but also yields an estimate for the error made in applying the approximation. 

It is to be emphasized that, despite the formal similarity, the physical problems are different. Moreover, in general, diabatic coupling is not small, unlike the tunneling matrix element, and this circumstance does not allow one to apply the noninteracting blip approximation. So even having been formulated in the standard spin-boson form, the problem still remains rather sophisticated. In particular, it is difficult to explore the intermediate region between nonadiabatic and adiabatic transition. 

Suppose at(w,0) kl > only the hP1 state of the system is present initially. The first term on the right of Eq. (7-73) can then contribute nothing to any state l that differs from k it will induce no transitions from the initial state. Retaining it alone constitutes the adiabatic approximation. The second term contributes to at(w,t) provided (il)uc is finite. It is the first diabatic term in the expansion. 

The plot of the pressure drop depending on the bulk velocity in adiabatic and diabatic flows is shown in Fig. 3.6a,b. The data related to the adiabatic flow correspond to constant temperature of the fluids Tjn = 25 °C, whereas in the diabatic flow the fluid temperature increased along micro-channel approximately from 40 to 60 °C. It is seen that in both cases the pressure drop for Habon G increases compared to clear water. The difference between pressure drop corresponding to flows of a surfactant solution and solvent increases with increasing bulk velocity. 

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... 

Note that since the profile of the lower adiabatic potential energy surface for the proton depends on the coordinates of the medium molecules, the zeroth-order states and the diabatic potential energy surfaces depend also on the coordinates of the medium molecules. The double adiabatic approximation is essentially used here the electrons adiabatically follow the motion of all nuclei, while the proton zeroth-order states adiabatically follow the change of the positions of the medium molecules. 

Now, recall that for weak hydrogen bonds the high-frequency mode is much faster than the slow mode because 0 m 20 00. As a consequence, the quantum adiabatic approximation may be assumed to be verified when the anharmonic coupling parameter aG is not too strong. Thus, neglecting the diabatic part of the Hamiltonian (22) and using Eqs. (18) to (20), one obtains... 

In the previous section, we discussed the calculation of the PESs needed in Eq. (2.16a) as well as the nonadiabatic coupling terms of Eqs. (2.16b) and (2.16c). We have noted that in the diabatic representation the off-diagonal elements of Eq. (2.16a) are responsible for the coupling between electronic states while Dp and Gp vanish. In the adiabatic representation the opposite is true The off-diagonal elements of Eq. (2.16a) vanish while Du and Gp do not. In this representation, our calculation of the nonadiabatic coupling is approximate because we assume that Gp is negligible and we make an approximation in the calculation of Dp. (See end of Section n.A for more details.)... 

Manifold approximation, non-adiabatic coupling, line integral conditions, adiabatic-to-diabatic transformation... 

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