Two-state system


This behaviour is characteristic of any two-state system, and the maximum in the heat capacity is called a Schottky anomaly.  [c.403]

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 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 ).  [c.403]

Consider a two-state system, where  [c.1156]

For a two-state system, the eigenfunctions of the diabatic potential matrix of Eq. (63) in terms of its elements are  [c.281]

In principle, the Landau-Zener formula could be used to calculate a hop probability for a trajectory, but this is often not practical as it requires knowledge about the position of the crossing point. Studies [32,211] indicate instead that the best method for accuracy and simplicity is the fewest switches algorithm [203]. The aim is that the percentage of trajectories in each state equals the state populations with a minimum number of transitions occurring to maintain this. The state populations are provided by integrating the equation for state amplitudes Eq. (94). Changes in the populations over a time step then mean that for a two-state system the probability of a trajectory changing out of state 2 into state 1 is  [c.293]

A, A Chemical Reaction as a Two-State System  [c.327]

The phase change of the total polyelectronic wave function in a chemical reaction [22-25], which is more extensively discussed in Section in, is central to the approach presented in this chapter. It is shown that some reactions may be classified as phase preserving (p) on the ground-state surface, while others are phase inverting (i). The distinction between the two can be made by checking the change in the spin pairing of the electrons that are exchanged in the reaction. A complete loop around a point in configuration space may be constmeted using a number of consecutive elementaiy reactions, starting and ending with the reactant A. The smallest possible loop typically requires at least three reactions two other molecules must be involved in order to complete a loop they are the desired product B and another one C, so that the complete loop is A B C A. Two types of phase inverting loops may be constructed those in which each reaction is phase inverting (an i loop) and those in which one reaction is phase inverting, and the other two phase preserving (an ip loop). At least one reaction must be phase inverting for the complete loop to be phase inverting and thus to encircle a conical intersection and lead to a photochemical reaction. It follows, that if a conical intersection is crossed during a photochemical reaction, in general at least two products are expected, B and C. A single product requires the existence of a two-component loop. This is possible if one of the molecules may be viewed as the out-of-phase combination of a two-state system. The allyl radical (Section IV, cf. Fig. 12) and the triplet state are examples of such systems. We restrict the discussion in this chapter to singlet states only.  [c.329]

A. A Chemical Reaction as a Two-State System  [c.330]

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).  [c.330]

There are two nuclear configurations on the ground-state surface that are of special interest to the chemist One is the energy minimum for the in-phase combination of H(I)) and H(II)), which is the equilibrium geometry of H(III). The second is also a stationary point on the ground-state surface, but for the out-of-phase combination of H(I)) and H(II))—it is the TS between H(I) and H(II). Clearly, the geometries (nuclear configuration) of these two species are quite different. Each of these structures is constructed from two base functions, and is therefore a two-state system. As for any two-state system, each has a twin state on the electronic excited surface. Thus, the in-phase combination of the two electronic wave functions H(I)) and H(II)) at the nuclear configuration of the transition state is found on the excited-state potential surface. Likewise, the out-of-phase combination at the nuclear geometry of the minimum energy of H(III)) also lies on the excited-state potential. Thus a given spin-paired scheme of the H4 system is seen to support very different nuclear geometries on the each potential surfaces.  [c.335]

A. The Treatment of the Two-State System in a Plane  [c.635]

In 1992, Baer and Englman [55] suggested that Berry s topological phase [60-62], as derived for molecular systems, and likewise the Longuet-Higgins phase [14-17], should be related to the adiabatic-to-diabatic transformation angle as calculated for a two-state system [56] (see also [63]). Whereas the Baer-Englman suggestion was based on a study of the Jahn-Teller conical intersection model, it was later supported by other studies [11,12,64—75]. In particular, it can be shown that these two angles are related by comparing the extended Bom-Oppenheimer approximation, once expressed in terms of the gradient of the Longuet-Higgins phase (see Appendix A) and once in terms of the two-state non-adiabatic coupling term [75].  [c.637]

For a two-state system Eq. (25) simplifies significantly to become  [c.644]

In [7,8,80], it was shown that in a two-state system the nonadiabatic coupling term, Ti2, has to be quantized in the following way  [c.667]

A. The Treatment of the Two-State System in a Plane  [c.692]

The Solution for a Single Conical Intersection The curl equation for a two-state system is given in Eq. (26)  [c.692]

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.  [c.696]

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.  [c.711]

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.  [c.714]

We consider a two state system, state A and state B. A state is defined as a domain in phase space that is (at least) in local equilibrium since thermodynamic variables are assigned to it. We assume that A or B are described by a local canonical ensemble. There are no dark or hidden states and the probability of the system to be in either A or in B is one. A phenomenological rate equation that describes the transitions between A and B is  [c.276]

Injection pressures for hot-chamber machines are between 6.9 and 20.7 MPa (1000—3000 psi). The velocity with which the metal enters the die is controlled by the pressure and die design. The optimum may differ from each casting and in the past it has been determined primarily by trial and error. In recent years, techniques have been developed for the use of instmmentation to caUbrate the shot system and die design programs are available for optimizing gate and mnner systems in dies (100,101). The P—technique (relationship between pressure and flow), developed by the Commonwealth Scientific and Industrial Research Organization, is utilized to characterize the rate with which the die-casting machine-shot system deflvers zinc to the die. Using this information, metal flow during die filling can be controlled for castings of various sizes (102). Computer programs are available to aid the die caster in applying these rules and guidelines (103).  [c.412]

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  [c.2531]

Already in 1938, Evans and Warhurst [17] suggested that the Diels-AIder addition reaction of a diene with an olefin proceeds via a concerted mechanism. They pointed out the analogy between the delocalized electrons in the tiansition states for the reaction between butadiene and ethylene and the tt electron system of benzene. They calculated the resonance stabilization of this transition state by the VB method earlier used by Pauling to calculate the resonance energy of benzene. They concluded that the extra aromatic stabilization of this transition state made the concerted route more favorable then a two-step process. In a subsequent paper [18], Evans used the Hilckel MO theory to calculate the transition state energy of the same reaction and some others. These ideas essentially introduce a chemical reacting complex (reactants and products) as a two-state system. Dewar [42] later formulated a general principle for all pericyclic reactions (Evans principle) Thermal pericyclic reactions take place preferentially via aromatic transition states. Aromaticity was defined by the amount of resonance stabilization. Evans principle connects the problem of themial pericyclic reactions with that of aromaticity Any theory of aromaticity is also a theory of pericyclic reactions [43]. Evans approach was more recently used to aid in finding conical intersections [44], (cf. Section Vni).  [c.341]

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.  [c.342]

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  [c.350]

On this occasion, we want also to refer to an incorrect statement that we made more than once [72], namely, that the (1,2) conical intersection results indicate that for any value of ri and r2 the two states under consideration form an isolated two-state sub-Hilbert space. We now know that in fact they do not form an isolated system because the second state is coupled to the thud state via a conical intersection as will be discussed next. Still, the fact that the series of topological angles, as calculated for the various values of r and r2, are either multiples of it or zero indicates that we can form, for this adiabatic two-state system, single-valued diabatic potentials. Thus if for some numerical heatment only the two lowest adiabatic states are required, the results obtained here suggest that it is possible to foiTn from these two adiabatic surfaces singlevalued diabatic potentials employing the line-integral approach. Indeed, recently Billing et al. [104] carried out such a photodissociation study based on the two lowest adiabatic states as obtained from ab initio calculations. The complete justification for such a study was presented in Section XI.  [c.706]


See pages that mention the term Two-state system : [c.329]    [c.650]    [c.692]    [c.28]   
The role of degenerate states in chemistry (0) -- [ c.0 ]