Two-state systems

This behaviour is characteristic of any two-state system, and the maximum in the heat capacity is called a Schottky anomaly. 

Figure A2.2.1. Heat capacity of a two-state system as a function of the dimensionless temperature, lc T/([iH). From the partition fimction, one also finds the Helmholtz free energy as...

It is also instructive to start from the expression for entropy S = log(g(A( m)) for a specific energy partition between the two-state system and the reservoir. Using the result for g N, m) in section A2.2.2. and noting that E = one gets (using the Stirling approximation A (2kN)2N e ). 

In tire limit of a small defonnation, a polymer system can be considered as a superjDosition of a two-state system witli different relaxation times. Phenomenologically, tire different relaxation processes are designated by Greek... 

For the special case of a two-state systems, a Flermitean phase operator was proposed, [143], which was said to provide a quantitative measure for phase information. )... 

For a two-state system, the eigenfunctions of the diabatic potential matrix of Eq. (63) in terms of its elements are... 

The concept of two-state systems occupies a central role in quantum mechanics [16,26]. As discussed extensively by Feynmann et al. [16], benzene and ammonia are examples of simple two-state systems Their properties are best described by assuming that the wave function that represents them is a combination of two base states. In the cases of ammonia and benzene, the two base states are equivalent. The two base states necessarily give rise to two independent states, which we named twin states [27,28]. One of them is the ground state, the other an excited states. The twin states are the ones observed experimentally. 

Stabilizing resonances also occur in other systems. Some well-known ones are the allyl radical and square cyclobutadiene. It has been shown that in these cases, the ground-state wave function is constructed from the out-of-phase combination of the two components [24,30]. In Section HI, it is shown that this is also a necessary result of Pauli s principle and the permutational symmetry of the polyelectronic wave function When the number of electron pairs exchanged in a two-state system is even, the ground state is the out-of-phase combination [28]. Three electrons may be considered as two electron pairs, one of which is half-populated. When both electron pahs are fully populated, an antiaromatic system arises ("Section HI). 

The results of the derivation (which is reproduced in Appendix A) are summarized in Figure 7. This figure applies to both reactive and resonance stabilized (such as benzene) systems. The compounds A and B are the reactant and product in a pericyclic reaction, or the two equivalent Kekule structures in an aromatic system. The parameter t, is the reaction coordinate in a pericyclic reaction or the coordinate interchanging two Kekule structures in aromatic (and antiaromatic) systems. The avoided crossing model [26-28] predicts that the two eigenfunctions of the two-state system may be fomred by in-phase and out-of-phase combinations of the noninteracting basic states A) and B). State A) differs from B) by the spin-pairing scheme. 

Ammonia is a two-state system [16], in which the two base states lie at a minimum energy. They are connected by the inversion reaction with a small baiiier. The process proceeds upon the spin re-pairing of four electrons (Fig. 15) and has a very low barrier. The system is analogous to the tetrahedral carbon one... 

The Solution for a Single Conical Intersection The curl equation for a two-state system is given in Eq. (26) ... 

To summarize our findings so far, we may say that if indeed the radial component of a single completely isolated conical intersection can be assumed to be negligible small as compared to the angular component, then we can present, almost fully analytically, the 2D field of the non-adiabatic coupling terras for a two-state system formed by any number of conical intersections. Thus, Eq. (165) can be considered as the non-adiabatic coupling field in the case of two states. 

In this section, we concentrate on a few examples to show the degree of relevance of the theory presented in the previous sections. For this purpose, we analyze the conical intersections of two real two-state systems and one real system resembling a tri-state case. 

The values due to the two separate calculations are of the same quality we usually get from (pure) two-state calculations, that is, veiy close to 1.0 but two comments have to be made in this respect (1) The quality of the numbers are different in the two calculations The reason might be connected with the fact that in the second case the circle surrounds an area about three times larger than in the first case. This fact seems to indicate that the deviations are due noise caused by CIs belonging to neighbor states [e.g., the (1,2) and the (4,5) CIs]. (2) We would like to remind the reader that the diagonal element in case of the two-state system was only (—)0.39 [73] [instead of (—)1.0] so that incorporating the third state led, indeed, to a significant improvement. 

Another subject with important potential application is discussed in Section XIV. There we suggested employing the curl equations (which any Bohr-Oppenheimer-Huang system has to obey for the for the relevant sub-Hilbert space), instead of ab initio calculations, to derive the non-adiabatic coupling terms [113,114]. Whereas these equations yield an analytic solution for any two-state system (the abelian case) they become much more elaborate due to the nonlinear terms that are unavoidable for any realistic system that contains more than two states (the non-abelian case). The solution of these equations is subject to boundary conditions that can be supplied either by ab initio calculations or perturbation theory. 

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