Semiclassical expression

The semiclassical expressions in the ZN theory are given below for the Stokes constant and other imporatant physical quantities. 

In the quantum mechanical formulation of electron transfer (Atkins, 1984 Closs et al, 1986) as a radiationless transition, the rate of ET is described as the product of the electronic coupling term J2 and the Frank-Condon factor FC, which is weighted with the Boltzmann population of the vibrational energy levels. But Marcus and Sutin (1985) have pointed out that, in the high-temperature limit, this treatment yields the semiclassical expression (9). 

The evaluation of the semiclassical Van Vleck-Gutzwiller propagator (106) amounts to the solution of a boundary-value problem. That is, given a trajectory characterized by the position q(f) = q, and momentum p(f) = p, we need to hnd the roots of the equation = q floiPo)- To circumvent this cumbersome root search, one may rewrite the semiclassical expression for the transition amplitude (105) as an initial-value problem [104-111]... 

The absorbing potential factor is the only nonstandard feature in Eq. (3.1). Fortunately, it is much simpler to deal with the absorbing potential semiclassically than it is quantum mechanically it cannot cause any unwanted reflections in the above semiclassical expression because we have implicitly made an infinitesimal approximation for it. Thus, it does not affect the dynamics and only causes absorption see also the disscussion by Seide-man et al. [3b]. 

The remainder of this paper is organized as follows the global optimization procedure used in the formulation is discussed in Sect. 6.2. The semiclassical expression of the correlation function is derived in Sect. 6.3, and the properties of the semiclassical correlation function are discussed in Sect. 6.4. In Sect. 6.5 we introduce the idea of guided optimal control. The full control algorithm is provided. In Sect. 6.6 we provide three numerical examples i) the control of wavepacket motion where a two-dimensional model of H2O is used as an example, ii) the control of the H + OD —> HO + D reaction using a two-dimensional model of HOD, and iii) the control of the 4-D model of HCN-CNH isomerization (i.e., isomerization in a plane). Future perspectives from the authors point of view are summarized in Sect. 6.7. 

By applying the saddle point approximation to its formula, the semiclassical expression of the wave matrix is given as follows [22] ... 

Electron transfer in proteins generally involves redox centers separated by long distances. The electronic interaction between redox sites is relatively weak and the transition state for the ET reaction must be formed many times before there is a successhil conversion from reactants to products the process is electronically nonadiabatic. A Eandau-Zener treatment of the reactant-product transition probability produces the familiar semiclassical expression for the rate of nonadiabatic electron transfer between a donor (D) and acceptor (A) held at fixed distance (equation 1). Biological electron flow over long distances with a relatively small release of free energy is possible because the protein fold creates a suitable balance between AG° and k as well as adequate electronic coupling between distant redox centers. 

The modified form of the semiclassical expression for the total number of states with energy avi in a collection of iV oscillators of frequencies Vj is... 

This is true provided that D( ) increases with E much more slowly than D(E) depends on E like p,( ). Using the semiclassical expression p ( ) = (Elhaif lh[Pg.551]

The semiclassic expression for the two-body configurational integral follows from Eqs. (2.110) - (2.112) as... 

In Section 2.5.3 we derived the semiclassic expression for the ceinonical partition function [see Eq. (2.110)] based on the assumption that at sufficiently high temperatures we may replace the Hamiltonian operator by its classic analog, the Hamiltonian function [see Elq. (2.100)]. In this section we will sketch a more refined treatment of the semiclassic theory developed in Section 2.5 originally due to Hill and presented in detail in his classical work on stati.stical mechanics [326]. Because of Hill s clear and detailed exposition and because we need the final result mainly as a justification to treat confined fluids by means of classic statistical thermodynamics, we will just briefly outhne the key ideas of Hill s treatment for reasons of completeness of the current work. 

In this case the matrix elements of H can also be expressed in terms of single coordinate matrix elements of FjOfr). The semiclassical expression for the matrix elements of F/q,) (between nonorthogonal SCF states) is33... 

In neither the collinear nor the coplanar cases, however, is the agreement as quantitative as for the non-reactive applications discussed above. The difficulty lies in applying the appropriate uniform semiclassical expression, for the small differences between action integrals of the various trajectories which contribute to the 0 - 0 transition makes the primitive semiclassical expressions inaccurate. For the collinear case, for example, Fig. 3 shows the... 

Briefly, the Bessel function uniform expression is generated from the primitive semiclassical expression... 

See Ref. 37 for a derivation. Where the Airy function expression is valid, the two uniform semiclassical expressions are essentially equivalent. For the case of present interest, n, = n2 = 0, (59) and (60) become... 

In highly quantum-like situations such as these, therefore, it is necessary to use the appropriate uniform semiclassical expressions to obtain quantitative results for transitions between individual quantum states. These will, too, undoubtedly be cases for which the quantum number function is too highly structured for any semiclassical treatment to be quantitatively useful. 

The semiclassical expression for the dipole matrix element is therefore given by... 

Sometimes the proper inclusion of interference terms means that an appropriate uniform semiclassical expression must be utilized rather than the primitive semi-classical expressions of (31) and (51). 

We first consider the case where the reaction probabilities are computed for the adiabatic model with the reaction-path curvature neglected, the so-called vibrationally adiabatic zero-curvature approximation [36]. We approximate the quantum mechanical ground-state probabilities P (E) for the one-dimensional scattering problem by a uniform semiclassical expression [48], which for E < is given by... 

The applicability conditions for this high energy semiclassical expression for 5—matrix are as follows... 

Recently, Zhu and Nakamura carefully analyzed the Stokes phenomenon of Eq. (60) and succeeded in deriving not only quantum mechanically exact, but also new semiclassical expressions of SR. The quantum mechanically exact solutions in the LZ case (20-23) are... 

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