Two-state problem

In this section, it was shown how an optimal ADT matrix for an n-electronic-state problem can be obtained. In Section in.D, an application of the method outlined above to a two-state problem for the H3 system is described. 

To gain physical insight into the asymmetric situation let us compare Miller s result (3.81) with that obtained by considering a formal two-state problem with the matrix Hamiltonian,... 

As stated in the introduction, we present the derivation of an extended BO approximate equation for a Hilbert space of arbitary dimensions, for a situation where all the surfaces including the ground-state surface, have a degeneracy along a single line (e.g., a conical intersection) with the excited states. In a two-state problem, this kind of derivation can be done with an arbitary t matrix. On the contrary, such derivation for an N > 2 dimensional case has been performed with some limits to the elements of the r matrix. Hence, in this sence the present derivation is not general but hoped that with some additional assumptions it will be applicable for more general cases. 

We note that the integral over the energetically allowed phase space— that is, the classical level density (97)—was found in Fig. 20 to be in excellent agreement with the quantum-mechanical level density. This finding indicates that there is a valid correspondence between the quantum-mechanical two-state system and its classical mapping representation. A similar conclusion was drawn in a recent smdy of a mapped two-state problem, which focused on the Lyapunov exponents and the energy level statistics of the system [124, 235]. 

The shift of the spectrum has been well discussed previously. The early pulse radiolysis data suggested that the kinetics at 1300 nm [28] matched the kinetics in the blue [16], which led to the description of the kinetics as a two-state problem. However, the results measured by Chase and Hunt, where the kinetics at 1050 nm were considerably slower than the kinetics at 500 and 1300 nm, suggested that the kinetics were more... 

The two-state problem assumes that the two given adiabatic electronic eigenstates d> and quantum treatment is confined to the Hilbert subspace (I>, (l>2 spanned on and , 2. Rotating d>j and [Pg.124]

Such an angle a is called the mixing angle of the adiabatic-to-diabatic transformation (ADT) for the two-state problem. 

Equations (16) and (19) are formally identical. It is suggested that a satisfies equation (17) for the two-state problem and a the similar one for the generalized two-state problem. Applying now the curl operator to both sides of the first equations in equations (16) and (19) and substituting a = a in the last one, one readily obtains that, in the former case, this results in the identity 0 = 0 because the right hand side of equation (11) vanishes for the two-state problem. In the latter case, one finds that the substitution a = a and curl operation, applied to the first equation in equation (19), do not commute, viz., replacing a by a eliminates the right hand side of the first equation in equation (19) and a further application of the curl to the resultant equation yields... 

The result for the E state population thus obtained is depicted in Fig. 6. It should be pointed out that diabatic populations are given here for simplicity (for two-state problems, also adiabatic populations have been computed [47]). In agreement with... 

In particular, in [111] phases and y> entering equation (19) were calculated. As for approximate solutions of two-state problems, one more, differing from equations (29), (30), (31) and (32) has been suggested recently [113]. 

A recent analysis [30a] has shown that the exact solution of the two-state problem (subject only to the MH assumption, p-j = 0 as introduced in connection with Eqs. (74, 81, and 82) yields an expression for A//. entirely in terms of the matrix elements of the dipole moment in the adiabatic basis (see also Eq. 78) ... 

The potential energies of the ground and excited state surfaces, ,are obtained by solving the secular equation for the two-state problem and are ... 

Should the charge carrier become trapped at a surface-state, then the problem becomes virtually identical to the homogenous two-state problem, i.e., the donor and acceptor states are both spatially localised over dimensions comparable to typical molecular dimensions. 

Warshel and coworkers have employed the empirical valence-bond (EVB) method [49] to simulate FERs for PT [50] and other reactions [51]. The PT step between two water molecules in the mechanism of the reaction catalysed by carbonic anhy-drase was described as an effective two-state problem involving reactant-like (H0H)(0H2) and product-like (HO )(HOH2+) VB structures [50a], Diabatic energy curves for these two VB structures were calibrated to reproduce the experimental free energy change for autodissociation in water, and the mixing of the... 

A more complicated situation arises when a reacting species is in equilibrium, or partial equilibrium, with other isomers. This happens in both large neutral molecules (Orchard and Thrush, 1974 Cromwell et al., 1991) as well as in ions (Rosenstock et al., 1982 Baer, 1986 Bunn and Baer, 1986). An example of a two-state problem is shown in figure 7.32 with isomers A and B. Four rates are involved in the formation of products 1 and 2 by the following reaction mechanism ... 

The method is very simple in a two-state problem. If X>, A> and 1>, 2> are adiabatic and diabatic electronic states, respectively, the transformation is... 

Transmission coefficients are determined numerically for a simple two-state problem designed to simulate the passage of a chemical system through a transition state conformation where the electronic nature of the ground state changes substantially. 

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