The adiabatic calculations

For each n, S and Ms, we allow the 64 (6x4 Gg plus 8x5 //,) phonon coordinates to relax, and determine the optimal distortion, by full minimization of the lowest adiabatic potential sheet Vad(Q) in the space of all the phonons coordinates Q. We leave the Ag modes out, since they contribute a trivial 

In Table 4 we summarize some global properties of the obtained JT minima for all charge and spin states. In these multi-mode JT systems, the local symmetry of an optimal distortion is described in terms of the subgroup GiOCai of symmetry operations which leave that minimum invariant. We remind that the minima in the 

In the fifth column the number of neighbors of all orders are listed for a given minimum. The last column gives the total amount of dimensionless JT distortion at each minimum 

For the other cases of lower symmetry, the number of neighbors of any given order must be complemented by some extra connectivity information. First, we observe that the minima for n = 2. 5=1 and for n = 3, S = are exactly the same. Indeed, these two cases are related by a particle-hole symmetry applied only to one spin flavor. For all nonequivalent cases, the complete topological information about the wells is contained in the connectivity matrix C(n,S), whose matrix elements 

In Table 6 we report the range of vertical excitation energies AE for all final spin symmetries S, in the frozen minimum configurations Qmin(/i. 5), for all values of n and 5. The complete spectmm (available upon request from the authors) is very dense and not much informative. The listed energies give a quantitative prevision of the spectral range where a fast (optical) spectroscopy is likely to locate the intraband HOMO excitations of the C )0 ions. For the experimentally most accessible case n = 2, 5 = 1, here follows the complete list of the triplet-triplet excitation energies 127, 149, 150, 178, 182, 218, 326, 337, and 346 meV. 

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Both vapor-liquid flash calculations are implemented by the FORTRAN IV subroutine FLASH, which is described and listed in Appendix F. This subroutine can accept vapor and liquid feed streams simultaneously. It provides for input of estimates of vaporization, vapor and liquid compositions, and, for the adiabatic calculation, temperature, but makes its own initial estimates as specified above in the absence (0 values) of the external estimates. No cases have been encountered in which convergence is not achieved from internal initial estimates. 

The nodal planes used qualitatively to discuss cis/trans isomerization clearly appear in the ELF profiles. For AA and BB at the crossing point, the picture reveals the numerical artifact of the adiabatic calculations. At the respective attractors, AA and BB show the expected nodal plane distribution. The method is not capable to distinguish between singlet and triplet spin-state ELF. This issue was discussed during the meeting. The reader may find appropriate discussions in this volume. 

For a given basis set, one has to ascertain that the VBCMD energy ordering preserves the ordering of the structures, as reflected from their relative weights in the adiabatic calculations. 

There are significant deviations between the quantum and adiabatic predictions for the triplet soliton structures. The soliton width in the adiabatic calculation (ca. 10 bond lengths) is much shorter than the corresponding quantum calculation. 

For both flat flames and shielded Meker-type flames, knowledge of the unburned flow velocity, the change in number of moles of gas, and the flame temperature permit the calculation of linear flow velocity in the flame, so that measurements as a function of distance become measurements in reaction time. Fortunately, because ionization usually peaks after the peak in flame temperature, the adiabatically calculated flame temperature is often adequate however, at 1-10 Torr, the flame temperature is often 500°K lower than the adiabatic temperature. Actual measurements of ion or other species profiles, together with temperature profiles, are difficult to make in the reaction zone because the insertion of a probe in the flame front, where chemical and thermodynamic conditions are changing very rapidly, perturbs the natural environment. 

On the addition of fours waters the adiabatic ionization potentials of the purines in base pairs slightly increase (ca. 0.3eV), but to a lesser extent than the increase for the Koopmans ionization potentials after water addition. This is expected as relaxation of the nuclear framework in the adiabatic calculation allows the hydration shell to adjust to the formation of the cation radical. To investigate the effects of additional solvation layers on the adiabatic ionization... 

The two adiabatic potential energy surfaces that we will use in the present calculations, are called a reactive double-slit model (RDSM) [59] where the first surface is the lower and the second is the upper surface, respectively,... 

Single surface calculations with a vector potential in the adiabatic representation and two surface calculations in the diabatic representation with or without shifting the conical intersection from the origin are performed using Cartesian coordinates. As in the asymptotic region the two coordinates of the model represent a translational and a vibrational mode, respectively, the initial wave function for the ground state can be represented as. 

Single surface calculations with proper phase treatment in the adiabatic representation with shifted conical intersection has been performed in polai coordinates. For this calculation, the initial adiabatic wave function tad(9, 4 > o) is obtained by mapping t, to) ittlo polai space using the relations,... 

The numerical calculations have been done on a two-coordinate system with q being a radial coordinate and <() the polar coordinate. We consider a 3 x 3 non-adiabatic (vector) mabix t in which and T4, aie two components. If we assume = 0, takes the following form,... 

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. 

The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

The Hamiltonian provides a suitable analytic form that can be fitted to the adiabatic surfaces obtained from quantum chemical calculations. As a simple example we take the butatriene molecule. In its neutral ground state it is a planar molecule with D2/1 symmetry. The lowest two states of the radical cation, responsible for the first two bands in the photoelectron spectrum, are and... 

Now, we discuss briefly the situation when one or both of the adiabatic electronic states has/have nonlinear equilibrium geometry. In Figures 6 and 7 we show two characteristic examples, the state of BH2 and NH2, respectively. The BH2 potential curves are the result of ab initio calculations of the present authors [33,34], and those for NH2 are taken from [25]. 

The approach developed by Jungen and Merer (JM) [24] is of a similar level of sophistication. The main difference is that IM prefer to remove the coupling between the electronic states by a transformation of the Hamiltonian matrix (i.e., vibronic energy matrix), rather that of the Hamiltonian itself. They first calculate the large amplitude bending functions for one of the adiabatic potentials, as if it belonged to a E electronic state. These functions are used as... 

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