Model semiclassical

For this reason, we will restrict our subsequent approach to planar configurations of the two electrons and of the nucleus, with the polarization axis within this plane. This presents the most accurate quantum treatment of the driven three body Coulomb problem to date, valid in the entire nonrelativistic parameter range, without any adjustable parameter, and with no further approximation beyond the confinement of the accessible configuration space to two dimensions. Whilst this latter approximation certainly does restrict the generality of our model, semiclassical scaling arguments suggest that the unperturbed three... 

Dissociative attachment can be divided into resonant and nonresonant cases. The resonant case is fairly amenable to theoretical treatment (Bardsley et ai, 1964 O Malley, 1966). In that case, the dissociation process can often be well modeled semiclassically in terms of the lifetime of the temporary anion and the survival probability for it to move fl om the geometry at which attachment occurs to the point beyond which the anion is more stable than the neutral. While a fully detailed theoretical treatment can be complex (O Malley, 1966), the minimal ingredients to form a useful estimate of the cross section are an anion potential energy surface and a resonance lifetime or width, each of which can be computed in a fairly straightforward manner. 

H2, a harmonic model semiclassical adiabatic (SCAD) calculation for H + H2 and D + H2, a rotationally averaged lOS (RATOS) calculationl O for D + H2> ICVT calculations using a least action (LA) tunnelling... 

Miller W H 1974 Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants J. Chem. Phys. 61 1823-34... 

To calculate N (E-Eq), the non-torsional transitional modes have been treated as vibrations as well as rotations [26]. The fomier approach is invalid when the transitional mode s barrier for rotation is low, while the latter is inappropriate when the transitional mode is a vibration. Hamionic frequencies for the transitional modes may be obtained from a semi-empirical model [23] or by perfomiing an appropriate nomial mode analysis as a fiinction of the reaction path for the reaction s potential energy surface [26]. Semiclassical quantization may be used to detemiine anliamionic energy levels for die transitional modes [27]. 

The present paper is organized as follows In a first step, the derivation of QCMD and related models is reviewed in the framework of the semiclassical approach, 2. This approach, however, does not reveal the close connection between the QCMD and BO models. For establishing this connection, the BO model is shown to be the adiabatic limit of both, QD and QCMD, 3. Since the BO model is well-known to fail at energy level crossings, we have to discuss the influence of such crossings on QCMD-like models, too. This is done by the means of a relatively simple test system for a specific type of such a crossing where non-adiabatic excitations take place, 4. Here, all models so far discussed fail. Finally, we suggest a modification of the QCMD system to overcome this failure. 

Approximation Properties and Limits of the QCMD Model 383 2.2 Semiclassical Ansatz and QCMD... 

We will refer to this model as to the semiclassical QCMD bundle. Eqs. (7) and (8) would suggest certain initial conditions for /,. However, those would not include any momentum uncertainty, resulting in a wrong disintegration of the probability distribution in g as compared to the full QD. Eor including an initial momentum uncertainty, a Gaussian distribution in position space is used... 

Fig. 2. The BO model is the adiabatic limit of full QD if energy level crossings do not appear. QCMD is connected to QD by the semiclassical approach if no caustics are present. Its adiabatic limit is again the BO solution, this time if the Hamiltonian H is smoothly diagonalizable. Thus, QCMD may be justified indirectly by the adiabatic limit excluding energy level crossings and other discontinuities of the spectral decomposition.

A calculation of tunneling splitting in formic acid dimer has been undertaken by Makri and Miller [1989] for a model two-dimensional polynomial potential with antisymmetric coupling. The semiclassical approximation exploiting a version of the sudden approximation has given A = 0.9cm" while the numerically exact result is 1.8cm" Since this comparison was the main goal pursued by this model calculation, the asymmetry caused by the crystalline environment has not been taken into account. 

When Va varied within the interval 1-8 cm the tunneling splitting was found to depend nearly linearly on Fj, in agreement with the semiclassical model of section 3.5 [see eq. (3.92)], and the prefactor AjA ranged from 0.1 to 0.3, indicating nonadiabatic tunneling. Since this model is one-dimensional, it fails to explain the difference between splittings in the states with the [Pg.127]

Explicit forms for the potential energy in the terms Hi and Hf have been proposed by Saveant [1993], who has developed a semiclassical version, along the lines of the Marcus theory, and applied it successfully to several reactions. In his model, the potential curve for the reactants is a Morse curve, and that for the products is the repulsive branch of a Morse curve ... 

Theoretical models of the electron impact ionization process have focused on the calculation of the ionization cross section and its energy dependence they are divided into quantum, semiclassical and semiempirical. Methods for the calculation of the ionization cross section and experimental techniques developed for the measurement of absolute ionization cross sections will be described in more detail below. Cross sections calculated using the semiempirical additivity method developed by Deutsch and Mark (DM) and their coworkers,12-14 the binary-encounter-Bethe (BEB) method of Kim and Rudd,15 16 and the electrostatic model (EM) developed by Vallance, Harland, and Maclagan17,18 are compared to each other and to experimental data. 

A multitude of semiempirical and semiclassical theories have been developed to calculate electron impact ionization cross sections of atoms and atomic ions, with relatively few for the more complicated case of molecular electron impact ionization cross sections. One of the earlier treatments of molecular targets was that of Jain and Khare.38 Two of the more successful recent approaches are the method proposed by Deutsch and Mark and coworkers12-14 and the binary-encounter Bethe method developed by Kim and Rudd.15,16 The observation of a strong correlation between the maximum in the ionization efficiency curve and the polarizability of the target resulted in the semiempirical polarizability model which depends only on the polarizability, ionization potential, and maximum electron impact ionization cross section of the target molecule.39,40 These and other methods will be considered in detail below. 

Stratt, R. M., Semiclassical statistical mechanics of fluids nonperturbative incorporation of quantum effects in classical many body models, J. Chem. Phys. 1979, 70, 3630-3638... 

This remark, which shows that the deep nature of the semiclassical RY model is to lead to a collapse of the broadened structure at zero temperature [that was not obvious in the RY paper, because of the use of the quantum statistical expression (138) in place of the classical one (139)], will later appear to be of some interest. 

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