Calculation of the Transition Probability

Let us consider an electron transfer system, whose Hamiltonian may be written (r,Q)=H(r,Q)-t-TN, 

The system is assumed to be initially prepared in a vibronic state in which the donor center is reduced and the acceptor center oxidized, and we intend to find the transition probability to a vibronic state in which the donor is oxidized and the acceptor reduced. These two states, which of course are not stationary states of, are written as Xav(Q) (r= Q) and Xbw (Q) kb (r. Q) respectively, where ia and j/b are normalized with respect to r for any value of Q  

It is convenient to seek a solution v /(r, Q, t) of the time-dependent Schrodinger equation  

In the left-hand side of Eq. (4), second-order cross terms have been neglected. 

This formulation emphasizes the importance of the residual interactions H , in the electron transfer process. 

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The occupation of energy level 8 depends on its position with respect to the Fermi level and should be taken into account in calculation of the transition probability. The number of electrons within a given energy interval de is equal to p(8)/(8)d8. 

Note that in the reference model all the interactions of the electron with the medium polarization VeP are included in Eqs. (8) determining the electron states. The dependence of A and B on the polarization and intramolecular vibrations was entirely neglected in most calculations of the transition probability [the approximation of constant electron density (ACED)]. This approximation, together with Eqs. (4)-(7), resulted in the parabolic shape of the diabatic PES Ut and Uf. The latter differed only by the shift... 

This new approach enables us to consider all the physical effects due to the interaction of the electron with the medium polarization and local vibrations and to take them into account in the calculation of the transition probability. These physical effects are as follows ... 

A general method for the calculation of the transition probability in the harmonic approximation developed in Ref. 44 enabled us to take into account, in a rigorous way, both the dependence of the tunneling of the quantum particles on the coordinates of other degrees of freedom of the system and the effects of the inertia and nonadiabaticity of the tunneling particle, taking into account the mixing of the normal coordinates of the system in the initial and... 

A formalism similar to that used for partially adiabatic proton transfer reactions was applied in the calculation of the transition probability. This model of the diffusion jump is similar to the model of the diffusion of light defects in solids which was first considered in Ref. 62. 

Recently, much attention has been paid to the investigation of the role of this interaction in relation to the calculations for adiabatic reactions. For steady-state nonadiabatic reactions where the initial thermal equilibrium is not disturbed by the reaction, the coupling constants describing the interaction with the thermal bath do not enter explicitly into the expressions for the transition probabilities. The role of the thermal bath in this case is reduced to that the activation factor is determined by the free energy in the transitional configuration, and for the calculation of the transition probabilities, it is sufficient to know the free energy surfaces of the system as functions of the coordinates of the reactive modes. 

The 14N hf and quadrupole parameters observed in Co(acacen) by Rudin et al.59 are the first magnetic data reported on equatorial nitrogen ligand nuclei in a low-spin Co(II) complex. Only two of the four predicted AmN = 1 ENDOR transitions (3.9) were observed for each nitrogen nucleus. A numerical calculation of the transition probabilities shows that the corresponding transitions in the other ms-state are at least ten times less intense (hyperfine enhancement). 

We assume again that the atoms follow straight line trajectories, and we calculate the transition probability, P(b), from the initial to the final state in a collision with a given impact parameter, b. We then compute the cross section by integrating over impact parameter, and, if necessary, angle of v relative to E to obtain the cross section. The central problem is the calculation of the transition probability P(b). The Schroedinger equation for this problem has the Hamiltonian... 

Fig. 14.6 Calculations of the transition probability for the Na 17s + 17s —> 17p + 16p, (0,0) transition for (a) v E, and (b) vlE. Solid lines are for calculations which include the effects of the permanent dipole moments dashed lines indicate that the permanent dipole moments have been neglected (from ref. 16).

Calculation of the transition probability by the Landau method is based on the analytical continuation of classical dynamical variables into the classically forbidden region of the potential [5]. Alternatively, the Landau transition probability can be recovered from the Fourier components of certain classical quantities, related to the transitions in question [10,11]. This allows one to write the Landau VR probability as... 

The above theoretical scheme has been implemented by first obtaining the forms of the wavefunction for initial and final states as predicted by FOTOS and then carrying out a Cl so that the coefficients are obtained to all orders within the finite expansion. Obviously, in special cases, adjustments can be made, without, however, demanding much additional work, for example. Ref. [22]. The final calculation of the transition probability amplitude takes into account the overall NON between initial and final sets of orbitals. 

As soon as bound states are considered there are only discrete energy levels. Nevertheless it was shown by Bell [77] that it is possible to employ approximately a continuum of energy levels for the calculations of the tunnel rates, which is adequate for the description of many experimental systems. In the simplest form (see Fig. 21.5) of the Bell model, the potential barrier is an inverted parabola. This allows the use of the known solution of the quantum mechanical harmonic oscillator for the calculation of the transition probability through the barrier. The corresponding Schrodinger equation is... 

For a single crystal at 55 K with the c axis parallel to the axis of observation, the spectrum consists of two lines only because all the spin moments lie along this axis and the other three lines for the 0+ -> 2+ transition have zero probability. At 4-2 K the spins form a spiral arrangement leading to a finite probability for all five lines. Both types of ordering are complex, but a statistical calculation of the transition probabilities agreed well with the experimental data. 

In Section II the object was primarily the description of the states in which separated molecules are found, and in Section III we discussed the interaction between separated molecules. In this section we are concerned with the relationship of the interaction to the relative motion of the molecules, i.e., the transition probability on collision, or the collision cross-section. Naturally, we would want the best interaction potential for any computation of a specific transition probability. However, as we have seen, the potentials are not known as exactly as we would desire and perhaps this is fortunate because the correct potential may complicate the calculation of the transition probability to such an extent that the work would most likely be abandoned, although with machines we may see more rigorous treatments in the future. Indeed, evaluations of collision cross-sections will become more and more important as molecular beam techniques are perfected. The main difficulty in such experiments is detection, but there has been some recent development and application. ... 

A misunderstanding of this situation may lead to serious errors, as can be demonstrated by numerous examples in the literature in which one "tunneling" factor is substituted for another. Thus, for instance, making exact numerical calculations of the transition probabilities knni colinear H2+ H reaction, TRUHLAR and KUPPERMANN /19d/... 

The al ini. o potential energy surface of SHAVITT, STEVENS, MINN, and KARPLUS (SSMK), as modified by TRUHLAR and KUPPERMANN / 4/ has been used by these authors for exact quantal calculations of the transition probabilities for the colinear H2 + H and D2 + D reactions. On the basis of these data, CHRISTOV and PARLAPANSKI /132/ directly computed the values of the factor le. in the collision theory expression (23.IV) in the temperature range 300-1000 K. The corresponding values of the factor in the statistical expression (67,... 

Theoretical work has focused on two aspects of the problem perturbation theory and other calculations of the transition probabilities in the material, and the effect on the transition probability of the statistical nature of the radiation field. Makinson and Buckingham [7,29] were the first to predict the second-order effect and calculate its magnitude based on a surface model of photoemission this work was expanded by Smith [7.30], Bowers [7.31], and Adawi [7.32], The analogous volume calculation was performed by Bloch [7.33] and later corrected by Teich and Wolga [7.24, 25],... 

In the equation the quantity df/dQ is the so-called differential oscillation strength. dfldQ is proportional to the optical absorption coefficient. The logarithmic term varies only slowly with Q, therefore the probability of resonant transition is roughly proportional to the dfldQ)/Q ratio, i.e., it is proportional to the ratio of optical absorption coefficient and the transition energy. If the optical absorption spectrum is accurately known, this equation gives a possibility for the calculation of the transition probabilities and also the yields of the different excited molecules (Hatano 1968, 1999 Makarov and Polak 1970 Kouchi and Hatano 2004). 

The solution of this problem was given by Dogonadze and Kuznetsov. The general scheme of the calculation of the transition probability was derived, based on the density matrix method (see Section 3). It was shown that Vif has to be taken out of the integral sign over the nuclei coordinates at the point at which the product of the density matrices of the initial and final states for the heavy particles is maximal. 

In 1976 Schmickler used an interesting method for the calculation of the transition probability starting from the description of the solvent as a set... 

However, it should be noted that from the very beginning the dipole moment of the molecule was divided into two parts, p, = m, + aEy where the first term, m is the permanent part of the dipole moment and the second is the dipole moment induced by the electric field, E, in the medium (a is the electronic polarizability of the molecule). If m, in this expression is meant to be an operator over all the degrees of freedom, including the electrons, the second term, aE, has to be omitted. However, if some high frequency modes (e.g., electronic or vibrational) are excluded from m the second term, aE, has to be retained, but in this case, m, is the operator with respect only to the remaining degrees of freedom (i.e., orientational and vibrational, or solely orientational ones). Thus, such separation is essentially equivalent to the use of the concepts of the inertia and inertialess polarizations. In addition, for the calculation of the transition probability, it was necessary to calculate the average quantities of the type These were expressed in Ref. 81... 

For the calculation of the transition probabilities P the scattering problem has to be solved in two... 

The calculation of the transition probability B is standard. The perturbation due to the interaction of radiation with matter can be written... 

If em adsorbed molecule is moving in electronically adiabatic potential well the model of two-dimensional anharmonic oscillator can be used for the description of its vibrational spectrum. This spectrum E(,n) (n = (ni,Ti2)) can be calculated by the numerical or analytical integration of the two-dimensional Schrodinger equation. The calculations of the transition probabilities e E, E) (or e(n, n)) for such oscillator have been performed with the help of the pertiu-bation theory or more sophisticated approaches. Provided these transition probabiUties e(n, n) are known for dense enough energy spectrum (AE ksT ), a diffusion model of energy relaxation may be used... 

Let us now suppose the state s) to be represented by the approximate Hamiltonian Ho of Equation Al. If, for example, the state s) and another state i>) of the manifold (Al) between the transitions are isolated from the remaining states of Ho, the problem is relatively simple. However, if the states are not isolated, the calculation of the transition probability is not simple. To reduce the complexity and to make clear the states of the system to be included in the calculation of the matrix elements of the Green s function, we use the Feshbach projection operator formalism. In this formalism, a projection operator associated with a model function or a set of states, the so-called dose coupled states, is introduced... 

Transition probabilities of forbidden lines. Accurate theoretical calculations of the transition probabilities of forbidden lines can be made without great difficulty for those lines which satisfy all the selection rules in Table 7.1. However, there are many cases of magnetic dipole or electric quadrupole lines where one or more of the approximate rules is broken. In these cases the atomic wave-functions must be calculated in intermediate coupling and the resulting transition probabilities are consequently less reliable. 

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