Exponential Potential Model Research Articles

Non-adiabatic transitions in a two-state exponential potential model

A general two-state exponential potential model is investigated and the corresponding two-channel scattering problem is solved by means of semiclassical theory. The analytical expression for the non-adiabatic transition matrix yields a unified expression in the repulsive and previously studied attractive case. The final formulae are expressed in terms of model-independent quantities, i.e. the contour integrals of adiabatic local momenta. Oscillations of the overall transition probability below the crossing of diabatic potentials are observed in the case of strong coupling. The theory is demonstrated to work very well even at energies lower than the diabatic crossing region. Based on our results the unified theory of non-adiabatic transitions, covering the Landau-Zener-Stueckelberg and Rozen-Zener-Demkov models in such an energy range, is possible.

Semiclassical theory of nonadiabatic transitions in a two-state exponential model

A general two-state exponential potential model is solved with use of the Bessel transformation and the WKB (Wentzel-Kramers-Brillouin) type semiclassical approximation. Accurate expressions are obtained for the nonadiabatic transition probability for one passage of the transition point and for the two dynamical phases. Functionalities of these quantities in terms of two basic parameters are the same as those obtained before by Nikitin. The two basic parameters are, however, expressed in more general and accurate forms. Accuracies of these expressions are numerically confirmed. The three quantities, the nonadiabatic transition probability and the two dynamical phases, constitute the nonadiabatic transition matrix and can be used to describe various (spectroscopic as well as scattering) processes not only for a two-state but also for a multichannel system. A possible generalization of the present theory is also briefly discussed to formulate a unified theory that can cover both Landau-Zener-Stueckelberg and Rosen-Zener-Demkov cases within the adiabatic state representation.

Unified semiclassical theory for the two-state system: Analytical solutions for scattering matrices

Unified semiclassical theory is established for general two-state system by employing an exactly analytical quantum solution [C. Zhu, J. Phys. A29, 1293 (1996)] for the Nikitin exponential–potential model which contains the two-state curve crossing and noncrossing cases as a whole. Analytical solutions for scattering matrices are found for both three- and two-channel cases within the time-independent treatment. This is made possible by introducing a very important parameter d(R0)=√)/[V22(R0)−V11(R0)]2 (V11(R), V22(R) and V12(R) are diabatic potentials and coupling, R0 is real part of complex crossing point between two adiabatic potentials) which represents a type of nonadiabatic transition for the two-state system. For instance, d=∞ represents the Landau–Zener type and d=√ represents Rosen–Zener type. Since d(R0) runs from unity to infinity, this parameter provides a quantitative description of nonadiabatic transition. The idea used here is the parameter comparison method which makes a unique link between the model and general potential system at the complex crossing point. This method is testified not only by numerical examples, but also by agreement of the present semiclassical formulas with all existing semiclassical formulas.

Quantum mechanically exact analytical solutions of a two-state exponential model

A certain two-state exponential potential model is solved quantum mechanically exactly. Compact expressions for nonadibatic transition matrices are obtained. Interesting quantum mechanical threshold effects are found. Simple very accurate expressions are found from a semiclassical viewpoint for the nonadiabatic transition probabilities, indicating that the exponential model may present a third important basic model in addition to the Landau–Zener–Stueckelberg and the Rosen–Zener–Demkov models. Extension to general cases is also briefly discussed.

On the equation of state of a classical fluid

A rigorous cumulant theory is developed for deriving the Van der Waals equation of state in a direct way in the Van der Waals limit for a system with a hard-sphere potential and a long-range attractive potential; thereby the role played by the Van der Waals limit is cleared out. A linear- potential model is examined in comparison with the exponential potential model. The cases with the Van der Waals attractive potential and other potentials are also investigated. From the rigorous cumulant formula for the free energy, upper and lower free-energy bounds are derived. These bounds agree with what have been derived by the author by a different method. They coincide with each other in the Van der Waals limit to yield the free energy corresponding to the Van der Waals equation of state. This conclusion agrees with Lebowitz and Penrose's, but for certain potentials the radial distribution function of the reference system is needed in the free- energy expression. Considerations are given of the virial type equation of state and the pair distribution function. The uniqueness of the Van der Waals limits in the derivation of a Van der Waals type equation in general and the role played by the limit in the random phase approximation are pointed out.

Consideration of a phase transition in a one-dimensional gas

A phase transition in a one-dimensional gas is treated based on a linear-potential model. In this model, the molecules interact with each other with a long-range attractive potential plus a hard-sphere repulsive potential. Because of the simple potential, the phase-space integral can be evaluated rigorously in an elementary way. The role of the Van der Waals limit is discussed, and the Van der Waals equation is obtained for the pressure in this limit. When 1 > 8 aβ p, where a is the hard-sphere diameter, β = 1/ kT and p is the pressure, and with the Maxwell rule of equal areas, a first-order phase transition appears as in the case of Kac, Uhlenbeck and Hemmer. The scaling law and other details of the equation of state are discussed. Different from the exponential-potential model, the linear-potential model is applicable to a wider class of potentials that can be expanded in a Taylor series. In fact, it will be shown that the higher-order terms in the potential constant γ in the expansion of the Kac exponential potential do not contribute to the Van der Waals equation of state.