Electronic coefficient

In order to generate the equations of motion, one needs the gradients of the energy with respect to the nuclear coordinates and the variational parameters. The expression for the nuclear gradients are reported in Ref. [18c], and the derivation of the gradients with respect to the electronic coefficients is straightforward ... 

The nuclear equations of motion are given by Eq. (12) and the corresponding ones for the electronic coefficients are ... 

In a mixed quantum-classical simulation such as a mean-field-trajectory or a surface-hopping calculation, the population probability of the diabatic state v[/ t) is given as the quasiclassical average over the squared modulus of the diabatic electronic coefficients dk t) defined in Eq. (27). This yields... 

Figure 2. Diabatic (left) and adiabatic (right) population probabiUties of the C (fuU line), B (dotted line), and X (dashed line) electronic states as obtained for Model II, which represents a three-state five-mode model of the benzene cation. Shown are (A) exact quantum calculations of Ref. 180, as well as mean-field-trajectory results [(B), (E)] and surface-hopping results [(C),(D),(F),(G)]. The latter are obtained either directly from the electronic coefficients [(C),(F)] or from binned coefficients [(D),(G)].

For example, the vibrational initial state may be represented by a Wigner distribution (17), while the initial electronic coefficients may be determined by <4(0) = q being an arbitrary phase [200]. 

A further important property of a MQC description is the ability to correctly describe the time evolution of the electronic coefficients. A proper description of the electronic phase coherence is expected to be particularly important in the case of multiple curve-crossings that are frequently encountered in bound-state relaxation dynamics [163]. Within the limits of the classical-path approximation, the MPT method naturally accounts for the coherent time evolution of the electronic coefficients (see Fig. 5). This conclusion is also supported by the numerical results for the transient oscillations of the electronic population, which were reproduced quite well by the MFT method. Similarly, it has been shown that the MFT method in general does a good job in reproducing coherent nuclear motion on coupled potential-energy surfaces. 

As a consistency test of the stochastic model, one can check whether the percentage Nk t) of trajectories propagating on the adiabatic PES Wk is equal to the corresponding adiabatic population probability Pf t). In a SH calculation, the latter quantity may be evaluated by an ensemble average over the squared modulus of the adiabatic electronic coefficients [cf. Eq. (22)], that is. 

Figure 11. Time-dependent population probability of the upper (a) adiabatic and (b) diabatic electronic state of Model 1. The quantum-mechanical results (thick lines) are compared to SH results obtained directly from the electronic coefficients (dashed lines) and to SH results obtained from binned coefficients (thin solid lines), reflecting the percentage N2(t) of trajectories propagating on the upper adiabatic surface. Panel (c) shows the absolute number of successful (thick hue) and rejected (thin line) surface hops occurring in the SH calculation.

However, a much coarser approach can yield significant results. Several authors, for example, have attempted to relate the kinetic-secondary-electron coefficient to the stopping-power relationships used for the analysis of atomic ion ranges in solids. A very lucid paper on this approach has been presented by Beuhler and Friedman and their perspective will be summarized in the following paragraphs. 

Secondary-electron coefficients are strongly dependent upon the condition of the surface. The presence of adsorbed gas or surface roughness can significantly alter the number of secondary electrons. Moreover, much of the work in this field predates ultra-high-vacuum technology and the associated surface-characterization tools (for reviews see Refs. 144-146). In addition, surfaces exposed to a plasma are not well characterized. Therefore, crude, estimates of the magnitude of the secondary-electron coefficients seem to be the most useful type of data in the present context. 

Part 2 is devoted to the foundations of the mathematical apparatus of the angular momentum and graphical methods, which, as it has turned out, are very efficient in the theory of complex atoms. Part 3 considers the non-relativistic and relativistic cases of complex electronic configurations (one and several open shells of equivalent electrons, coefficients of fractional parentage and optimization of coupling schemes). Part 4 deals with the second-quantization in a coupled tensorial form, quasispin and isospin techniques in atomic spectroscopy, leading to new very efficient versions of the Racah algebra. 

ZT Y r A A A A A AC dimensionless thermoelectric figure of merit electronic coefficient of heat capacity (1+ZT)F2 crystal field singlet non-Kramers doublet (crystal field state) crystal field triplet crystal field triplet hybridization gap jump in heat capacity at Tc K KL -min P 6>d X JCO total thermal conductivity of solid thermal conductivity of electrons or holes thermal conductivity of lattice minimum lattice thermal conductivity electrical resistivity Debye temperature magnetic susceptibility magnetic susceptibility at T = 0... 

For the ls2s state, the theoretical uncertainty of 6 MHz is considerably less due to Pachucki s calculation of AE Its contribution of 3.00(1) MHz to the ionization energy reduces the difference between theory and experiment to 1.7 6 MHz. The 6 MHz theoretical uncertainty is conservatively taken to be the entire amount of the ln(o ) a.u. term in Table 8. Since the n-dependence of this term is more complicated than a simple 1/n dependence, it is at best an approximation to replace a factor of Z /(Trn ) by (< (ri) + < (r2)) to form the two-electron generalization [see Eq. (14)]. For each term of O(o ) a.u. in Table 8, the two-electron coefficient C(lsns) multiplying (< (ri) (5(f2)) is calculated from the corresponding one-electron coefficients according to... 

Until recently, photoionization cross sections and recombination (radiative and di-electronic) coefficient sets used in photoionization computations were not obtained self-consistently. Photoionization and recombination calculations are presently being carried out using the same set of eigenfunctions as in the IRON project (Nahar Pradhan 1997, Nahar et al. 2000). The expected overall uncertainty is 10 - 20%. Experimental checks on a few species (see e.g. Savin 1999) can provide benchmarks for confrontation with numerical computations. 

The original idea of Paldus was to generate the two-electron coefficients as a matrix product of the one-electron coefficients... 

Analytical technique. Secondary electrons are produced by inelastic collisions between high-energy beam electrons and atoms within the specimen (Goldstein et al., 1992). The number of secondary electrons emitted, and the resulting secondary electron coefficient, is... 

The values given in Table 12 have been determined for the electronic coefficient (y) and the limiting Debye temperature (0d)-... 

The electronic coefficient is notably lower than the value of 13.7 mJ/ (mol K) determined by Lounasmaa and Veuro (1972) for the even-even isotope Sm which has no nuclear contribution. Although these measurements were carried out over the temperature range of 0.45-6 K and generally represent an improvement over the measurements of Lounasmaa (1962a), unfortunately, the results were only given in the form of a small graph. 

The values given in Table 115 were obtained for the electronic coefficient (7) and the limiting Debye temperature (0d). In each case, the nuclear contribution to the heat capacity has been subtracted. In the case of the measurements of Lounasmaa and Roach (1962) and HiU et al. (1974), no magnetic contribution was considered, but both Wells et al. (1976) and Dceda et al. (1985) included a magnetic term 9.57 exp(—14.8/T). 

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