Three-state systems

VIII. An Analytical Derivation for the Possible Sign Rips in a Three-State System... 

In Section XIV.B, this derivation is extended to a three-state system. 

In Section IV, we introduced the topological matrix D [see Eq. (38)] and showed that for a sub-Hilbert space this matrix is diagonal with (-1-1) and (—1) terms a feature that was defined as quantization of the non-adiabatic coupling matrix. If the present three-state system forms a sub-Hilbert space the resulting D matrix has to be a diagonal matrix as just mentioned. From Eq. (38) it is noticed that the D matrix is calculated along contours, F, that surround conical intersections. Our task in this section is to calculate the D matrix and we do this, again, for circular contours. 

Because this is a three-state system, it has two independent rate equations and two relaxation times. It is somewhat easier to write the rate equations when Scheme IV is presented in the form of Scheme V. 

The usual EVB procedure involves diagonalizing this 3x3 Hamiltonian. However, here we wish to use a very simple model for our reaction and represent the potential surface and wavefunction of the reacting system using only two electronic states. Using a two-state system will preserve most of the important features of the potential energy surface while at the same time provide a simple model that will be more amenable to discussion than the three-state system. For the two-state system we define the following states as the reactant and product wavefunctions ... 

With mechanisms such as these, it is often possible to simplify the analysis of the action of a channel blocker by assuming that agonist binding is much faster than channel opening and closing and then combining several closed states together so that the mechanism approximates a three-state system ... 

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