Nonadiabatic functions

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

The approximations defining minimal END, that is, direct nonadiabatic dynamics with classical nuclei and quantum electrons described by a single complex determinantal wave function constructed from nonoithogonal spin... 

Thus, the neglect of the off-diagonal matrix elements allows the change from mixed states of the nuclear subsystem to pure ones. The motion of the nuclei leads only to the deformation of the electronic distribution and not to transitions between different electronic states. In other words, a stationary distribution of electrons is obtained for each instantaneous position of the nuclei, that is, the elechons follow the motion of the nuclei adiabatically. The distribution of the nuclei is described by the wave function x (R i) in the potential V + Cn , known as the proper adiabatic approximation [41]. The off-diagonal operators C n in the matrix C, which lead to transitions between the states v / and t / are called operators of nonadiabaticity and the potential V = (R) due to the mean field of all the electrons of the system is called the adiabatic potential. 

Now we make the usual assumption in nonadiabatic transition theory that non-adiabaticity is essential only in the vicinity of the crossing point where e(Qc) = 0- Therefore, if the trajectory does not cross the dividing surface Q = Qc, its contribution to the path integral is to a good accuracy described by adiabatic approximation, i.e., e = ad Hence the real part of partition function, Zq is the same as in the adiabatic approximation. Then the rate constant may be written as... 

NONADIABATIC CHEMICAL DYNAMICS COMPREHENSION AND CONTROL OF DYNAMICS, AND MANIFESTATION OF MOLECULAR FUNCTIONS... 

Figure 10. Arrhenius plot of the thermal rate constants for the 2D model system. Circles-full quantum results. Thick solid (dashed) curve present nonadiabatic transition state theory by using the seam surface [the minimum energy crossing point (MECP)] approximation. Thin solid and dashed curves are the same as the thick ones except that the classical partition functions are used. Taken from Ref. [27].

The laser parameters should be chosen so that a and p can make the nonadiabatic transition probability V as close to unity as possible. Figure 34 depicts the probability P 2 as a function of a and p. There are some areas in which the probabilty is larger than 0.9, such as those around (ot= 1.20, p = 0.85), (ot = 0.53, p = 2.40), (a = 0.38, p = 3.31), and so on. Due to the coordinate dependence of the potential difference A(x) and the transition dipole moment p(x), it is generally impossible to achieve perfect excitation of the wave packet by a single quadratically chirped laser pulse. However, a very high efficiency of the population transfer is possible without significant deformation of the shape of the wave packet, if we locate the wave packet parameters inside one of these islands. The biggest, thus the most useful island, is around ot = 1.20, p = 0.85. The transition probability P 2 is > 0.9, if... 

Figure 34. Contour map of the nonadiabatic transition probability Pn induced by quadratically chirped pulse as a function of the two basic parameters a and p. Taken from Ref. [37]-...

Nonadiabatic transitions definitely play crucial roles for molecules to manifest various functions. The theory of nonadiabatic transition is very helpful not only to comprehend the mechanisms, but also to design new molecular functions and enhance their efficiencies. The photochromism that is expected to be applicable to molecular switches and memories is a good example [130]. Photoisomerization of retinal is well known to be a basic mechanism of vision. In these processes, the NT type of nonadiabatic transitions play essential roles. There must be many other similar examples. Utilization of the complete reflection phenomenon can also be another candidate, as discussed in Section V.C. In this section, the following two examples are cosidered (1) photochromism due to photoisomerization between cyclohexadiene (CHD) and hexatriene (HT) as an example of photoswitching molecular functions, and (2) hydrogen transmission through a five-membered carbon ring. 

The parameters c and 8 are defined in Section A.3 The nonadiabatic transition amplitude that connects the wave function just before and right after the transition at the avoided crossing is given by... 

Tj and are the turning points on E2 R) (see Fig. Al). The nonadiabatic transition amplitude to connect the wave functions between the right and the left sides of the crossing point is given by... 

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. 

Nonadiabatic Chemical Dynamics Comprehension and Control OF Dynamics, and Manifestation of Molecular Functions By Hiroki Nakamura... 

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 ... 

Fig. 8 shows the nonadiabatic coupling functions between states, which are completely dominated by the double crossing feature. The coefficient mixing term is a very good approximation of the total matrix elements, thus confirming the considerations already put forward with regard to the singlets. 

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