The Semiclassical Approach

A different approach to simplifying RP spin evolution for calculation is the semiclassical approach developed by Schulten and coworkers.In this approximation. 

MAGNETIC FIELD EFFECTS ON RADICAL PAIRS IN HOMOGENEOUS SOLUTION 

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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. 

The semiclassical approach to QCMD, as introduced in [10], derives the QCMD equations within two steps. First, a separation step makes a tensor ansatz for the full wavefunction separating the coordinates x and q ... 

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.

On the other hand, in the semiclassical approaches [83-85,92-95], one uses Eqs. (126) and (127) and assumes that the fast mode obeys quantum mechanics whereas the slow one obeys classical mechanics. 

We want to mention here that the application of the near-nuclear corrections could have been performed with the complete relativistic functional, and we have utilized the semi-relativistic expressions just for simplicity and for testing them. For not large Z, the remaining errors above mentioned should be addressed to limitations of the semiclassical approach and of the procedure utilized for the near-nuclear corrections, rather than to the truncation of the expansion in powers of... 

Error bars of the experimental data were not specified. However, Hunt provided measurements at two different ortho- to para-H2 concentration ratios, 3 1 (solid squares and dots) and 1 1 (open squares and circles). According to the theory developed above, variation of this ratio should not affect the results. While symmetry considerations of the interacting H2 molecules are important at low temperatures (T < 40 K), the semiclassical approach does not distinguish para- and ortho-H2 we think that the differences of the data taken at different ortho-para ratios may, in essence, reflect the uncertainties of the measurement. We note that earlier works by Chisholm and Welsh [121] and by Hare and Welsh [175] gave values... 

As a first test [22] of the semiclassical approach described above we have computed the transmission probability through the lie kart potential barrier,... 

On the contrary, the semiclassical approach in the problem of the optical absorption is restricted to a great extent and the adequate description of the phonon-assisted optical bands with a complicated structure caused by the dynamic JTE cannot be done in the framework of this approach [13]. An expressive example is represented by the two-humped absorption band of A —> E <8> e transition. The dip of absorption curve for A —> E <8> e transition to zero has no physical meaning because of the invalidity of the semiclassical approximation for this spectral range due to essentially quantum nature of the density of the vibronic states in the conical intersection of the adiabatic surface. This result is peculiar for the resonance (optical) phenomena in JT systems full discussion of the condition of the applicability of the adiabatic approximation is given in Ref. [13]. 

Equation (153) is the semiclassical limit of the quantum approach of indirect damping. Now, the question may arise as to how Eq. (153) may be viewed from the classical theory of relaxation in order to make a connection with the semiclassical approach of Robertson and Yarwood, which used the classical theory of Brownian motion. 

As it appears, the classical spectral density (174) is very similar to the semiclassical approached (144) obtained above and it is equivalent to that given by Eq. (151). 

Within the semiclassical approach, each wavepacket is propagated by running a large number of classical trajectories. In order to calculate the correlation function, a double summation with respect to these large number of trajectories should be performed at each time step, which is, unfortunately, also very demanding computationally. 

Let us consider an isolated molecule perturbed by an electromagnetic field. According to the semiclassical approach, the external radiation is described as a plane monochromatic wave traveling with velocity c and obeying the Maxwell equations [21] (i.e., the fields are not quantized). 

A. This is the Bohr radius of the semiclassical approach (1913) and is often adopted as the atomic unit of length. The electron has a maximum probability of being at this distance from the nucleus but a good chance of being anywhere within a very considerable volume. 

The equations of motion (14a) and (14b) contain time-dependent coefficients. Nevertheless, surprisingly, it is possible to find their analytical solution in the semiclassical approach for an arbitrary but real modulation amplitude... 

The basic semiclassical idea is that one uses a quantum mechanical description of the process of interest but then invokes classical mechanics to determine all dynamical relationships. A transition from initial state i to final state f, for example, is thus described by a transition amplitude, or S-matrix element Sfi, the square modulus of which is the transition probability Pf = Sfi 2. The semiclassical approach uses classical mechanics to construct the classical-limit approximation for the transition amplitude, i.e. the classical S-matrix the fact that classical mechanics is used to construct an amplitude means that the quantum principle of superposition is incorporated in the description, and this is the only element of quantum mechanics in the model. The completely classical approach would be to use classical mechanics to construct the transition probability directly, never alluding to an amplitude. 

In many cases, too, the semiclassical model provides a quantitative description of the quantum effects in molecular systems, although there will surely be situations for which it fails quantitatively or is at best awkward to apply. From the numerical examples which have been carried out thus far— and more are needed before a definitive conclusion can be reached—it appears that the most practically useful contribution of classical S-matrix theory is the ability to describe classically forbidden processes i.e. although completely classical (e.g. Monte Carlo) methods seem to be adequate for treating classically allowed processes, they are not meaningful for classically forbidden ones. (Purely classical treatments will not of course describe quantum interference effects which are present in classically allowed processes, but under most practical conditions these are quenched.) The semiclassical approach thus widens the class of phenomena to which classical trajectory methods can be applied. 

The accuracy of the semiclassical approach to vi vj (typically better than 10%) is not limited to the case EiThis approach is particularly valuable for displaying the general qualitative form of versus E (see... 

Fig. 5.5 for an example) and for relating certain crucial features of the experimentally well-known oscillatory behavior of HitVi 2,Vj (Lagerqvist and Miescher, 1958, Fig. 16) to Ec, Re, (dV2/dR) R=Rc, and a,2 - Rc or 62 - Rc, where V2 is the unknown potential. However, the semiclassical approach has not been widely applied to bound bound interactions, its major use having been bound free interactions (Section 7.6). 

As in the previous studies, we apply the semiclassical approach to the problem. This means that the angular momenta are considered to be large and the teriris l//ri in the matrix of E(p(103) can be neglected. Then the Hamiltonian matrix can be reduced to the form... 

To avoid the account of the edge effects let us consider rather long structures (L > 50 nm), i.e. we will consider the armchair single-wall carbon nanotubes with the length greater than electron mean free path [2-6]. To describe the electron-phonon transport in nanotubes like that the semiclassical approach and the kinetic Boltzmann equation for one-dimensional electron-phonon gas can be used [4,6]. In this connection the purpose of the present study is to develop a model of electron transport based on a numerical solution of the Boltzmann transport equation. 

In the past few years, we have proposed a model for correcting the wrong description of the electron density near the nucleus of the semiclassical approaches [4], This provides values of average atomic properties at the accuracy of the Hartree-Fock procedure with a very simple approach [5], allows relativistic extensions [6], and can describe negative ions [7],... 

At n 1, such a state, corresponding to the circular electron orbit perpendicular to the z axis, minimizes the uncertainty relations of the radial and transverse (to the orbit plane) components of p and r. Moreover, the quantum fluctuations of the radius of the orbit and the angle of inclination of the orbit plane to the z axis decrease as This makes the applicability of the semiclassical approach... 

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