Eckart

We find it convenient to reverse the historical ordering and to stait with (neatly) exact nonrelativistic vibration-rotation Hamiltonians for triatomic molecules. From the point of view of molecular spectroscopy, the optimal Hamiltonian is that which maximally decouples from each other vibrational and rotational motions (as well different vibrational modes from one another). It is obtained by employing a molecule-bound frame that takes over the rotations of the complete molecule as much as possible. Ideally, the only remaining motion observable in this system would be displacements of the nuclei with respect to one another, that is, molecular vibrations. It is well known, however, that such a program can be realized only approximately by introducing the Eckart conditions [38]. 

Figure 4 demonstrates that in order to variationally describe a realistic barrier shape (Eckart potential) by an effective parabolic one, the frequency of the latter, should drop with decreasing temperature. At high temperatures, T > T, transitions near the barrier top dominate, and the parabolic approximation with roeff = is accurate. 

Fig. 4. Variationally determined effective parabolic barrier frequency co ff for the Eckart barrier in units of 2n/hfi [Voth et al. 1989b], The dotted line is the high-temperature limit co = co. ...

While being very attractive in view of their similarity to CLTST, on closer inspection (3.61)-(3.63) reveal their deficiency at low temperatures. When P -rcc, the characteristic length Ax from (3.60b) becomes large, and the expansion (3.58) as well as the gaussian approximation for the centroid density breaks down. In the test of ref. [Voth et al. 1989b], which has displayed the success of the centroid approximation for the Eckart barrier at T> T, the low-temperature limit has not been reached, so there is no ground to trust eq. (3.62) as an estimate for kc ... 

Strictly speaking, the concept of itself makes no sense for a potential like the Eckart one, unless one artificially introduces Zo as the partition function of a bound initial state, which is not described by this potential. That is to say, it is reasonable to consider the combination kZo, not k alone. 

Such calculations have been performed by Takayanagi et al. [1987] and Hancock et al. [1989]. The minimum energy of the linear H3 complex is only 0.055 kcal/mol lower than that of the isolated H and H2. The intermolecular vibration frequency is smaller than 50cm L The height of the vibrational-adiabatic barrier is 9.4 kcal/mol, the H-H distance 0.82 A. The barrier was approximated by an Eckart potential with width 1.5-1.8 A. The rate constant has been calculated from eq. (2.1), using the barrier height as an adjustable parameter. This led to a value of Vq similar to that of the gas-phase reaction H -I- H2. 

It seems as if an energy value of sufficiently high accuracy has now been found for the helium problem, but we still do not know the actual form of the corresponding exact eigenfunction. In this connection, the mean square deviation e = J — W 2 (dx) and criteria of the Eckart type (Eq. III.27) are not very informative, since s may turn out to be exceedingly small, even if trial function... 

For two-electron systems the basic idea of using different orbitals for different electrons goes back to Hylleraas (1929) and to Eckart (1930) who both used it in treating He. The method was thoroughly discussed at the Shelter Island Conference 1951 in treatments of He and H2 (Kotani 1951, Taylor and Parr 1952, Mulliken 1952), but the circumstances are here exceptionally simple because of the possibility of separating space and spin according to Eq. III. 1. 

Eckart, C, Phys. Rev. 36, 878, "Theory and calculation of screening constants." Open shell idea discussed, b. 

Eckart, criteria, 264, 298 procedure, 267 Effective charge, 274, 276 Effective Hamiltonian, 226 Elastic model, excess entropy calculation from, 141 of a solid solution, 140 Electric correlation, 248 Electric field gradient, 188, 189 Electron (s), 200... 

In the latter expression the matrix element of operator dq> is transformed according to the Wigner-Eckart theorem and the definition used is... 

The matrix element of operator is written in terms of the Wigner-Eckart theorem, and the integral part is denoted as... 

Owing to liquid isotropy, the averaged matrix elements of X are expressed in the Wigner-Eckart form (omitting further on the overbar denoting averaging)... 

Weisskopf generalization, adiabatic broadening theory 132, 136 Wigner D-fiinotions 86, 275 Wigner-Eckart theory 232, 244, 253... 

S., Ott, T., Davies, R., Eckart, A., 1999, in Astronomy with Adaptive Optics -Present Results and Future Programs, ESO and OSA, 333... 

STUART C. ALTHORPE, JUAN CARLOS JUANES-MARCOS, AND ECKART WREDE... 

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