Nonadiabatic calculations

There have been a few studies that do not seem to fit into any of the categories thus far identified. Moss [70] has made a number of high-precision nonadiabatic calculations on hJ both by variational methods that specifically include vibrational wavefunctions and by methods based on scattering theory. The same system has also been studied by Ackermann and Shertzer [71] using finite-element methods. Finally, we mention the very precise variational Monte Carlo studies of Ps2 and its generalization to four-charge systems of type M" m M m by Bressanini et al. [72]. 

Fig. 5. photoelectron band in furan (a) experiment, (b) nonadiabatic calculation,... 

Figure 6, Equi-nonadiabatic coupling lines for the terms ri2(r,v) and r23(r,y) as calculated for the C2H molecule for a fixed C—C distauce, that is, rcc — 1-35 A. (a) Equi-non-adiabaric coupling term lines for the ri2(r,y). (b) Equi-non-adiabatic coupling term lines for X23(r,y). The Cartesian coordinates (x,y) are related to (q, 9) as follows x — qcosO] y —, sin9, where q and 9 are measured with respect to the midpoint between the two carbons.

The problem of nonadiabatic tunneling in the Landau-Zener approximation has been solved by Ovchinnikova [1965]. For further refinements of the theory beyond this approximation see Laing et al. [1977], Holstein [1978], Coveney et al. [1985], Nakamura [1987]. The nonadiabatic transition probability for a more general case of dissipative tunneling is derived in appendix B. We quote here only the result for the dissipationless case obtained in the Landau-Zener limit. When < F (Xe), the total transition probability is the product of the adiabatic tunneling rate, calculated in the previous sections, and the Landau-Zener-Stueckelberg-like factor... 

It has been shown that there is a two-dimensional cut of the PES such that the MEP lies completely within it. The coordinates in this cut are 4, and a linear combination of qs-q-j. This cut is presented in fig. 64, along with the MEP. Motion along the reaction path is adiabatic with respect to the fast coordinates q -q and nonadiabatic in the space of the slow coordinates q -qi-Nevertheless, since the MEP has a small curvature, the deviation of the extremal trajectory from it is small. This small curvature approximation has been intensively used earlier [Skodje et al. 1981 Truhlar et al. 1982], in particular for calculating tunneling splittings in (HF)2- The rate constant of reaction (6.45a) found in this way is characterized by the values T<. = 20-25 K, = 10 -10 s , = 1-4 kcal/mol above T, which compare well with the experiment. 

The heat transfer problem which must be solved in order to calculate the temperature profiles has been posed by Lee and Macosko(lO) as a coupled unsteady state heat conduction problem in the adjoining domains of the reaction mixture and of the nonadiabatic, nonisothermal mold wall. Figure 5 shows the geometry of interest. The following assumptions were made 1) no flow in the reaction mixture (typical molds fill in <2 sec.) ... 

Procedures enabling the calculation of bifurcation and limit points for systems of nonlinear equations have been discussed, for example, by Keller (13) Heinemann et al. (14-15) and Chan (16). In particular, in the work of Heineman et al., a version of Keller s pseudo-arclength continuation method was used to calculate the multiple steady-states of a model one-step, nonadiabatic, premixed laminar flame (Heinemann et al., (14)) a premixed, nonadiabatic, hydrogen-air system (Heinemann et al., (15)). 

The method is composed of the following algorithms (1) transition position is detected along each classical trajectory, (2) direction of transition is determined there and the ID cut of the potential energy surfaces is made along that direction, (3) judgment is made whether the transition is LZ type or nonadiabatic tunneling type, and (4) the transition probability is calculated by the appropriate ZN formula. The transition position can be simply found by... 

As discussed by Miller and co-workers [52,53], it is worthwhile to develop theories that enable us to evaluate thermal reaction rate constants directly and not to rely on the calculations of the most detailed scattering matrix or the state-to-state reaction probabihty. Here, our formulation of the nonadiabatic transition state theory is briefly described for the simplest case in which the transition state is created by potential surface crossing [27]. 

The factor k takes into acount the effects of nonadiabatic transition and tunneling properly. Also note that the electronic coupling //ad is assumed to be constant in the Marcus formula, but this is not necessary in the present formulation. The coupling Had cancels out in k of Eq. (126) and the ZN probability can be calculated from the information of adiabatic potentials. 

The author would like to thank all the group members in the past and present who carried out all the researches discussed in this chapter Drs. C. Zhu, G. V. Mil nikov, Y. Teranishi, K. Nagaya, A. Kondorskiy, H. Fujisaki, S. Zou, H. Tamura, and P. Oloyede. He is indebted to Professors S. Nanbu and T. Ishida for their contributions, especially on molecular functions and electronic structure calculations. He also thanks Professor Y. Zhao for his work on the nonadiabatic transition state theory and electron transfer. The work was supported by a Grant-in-Aid for Specially Promoted Research on Studies of Nonadiabatic Chemical Dynamics based on the Zhu-Nakamura Theory from MEXT of Japan. 

The C matrix, the columns ofwhich, Cj(, are the eigenvectors of H, is normally not too different from the matrix defined above. However, the QDPT treatment, applied either to an adiabatic or to a diabatic zeroth-order basis, is necessary in order to prevent serious artefacts, especially in the case of avoided crossings [27]. The preliminary diabatisation makes it easier to interpolate the matrix elements of the hamiltonian and of other operators as functions of the nuclear coordinates and to calculate the nonadiabatic coupling matrix elements ... 

Approximate calculation of the integral over 8 in Eq. (34.27) shows that the ejfective electron transmission coefficient for nonadiabatic reactions is equal to... 

A quantum-mechanical description of spin-state equilibria has been proposed on the basis of a radiationless nonadiabatic multiphonon process [117]. Calculated rate constants of, e.g., k 10 s for iron(II) and iron(III) are in reasonable agreement with the observed values between 10 and 10 s . Here again the quantity of largest influence is the metal-ligand bond length change AR and the consequent variation of stretching vibrations. 

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