Reduced interatomic distances

This is the simplest possible mechanistic model of the PES, derived from an approximate treatment of energy according to eq. (3.69). The FA type of treatment implies that the geminal amplitude-related ES V eqs. (2.78) and (2.81) are fixed at their invariant values eq. (3.7). This corresponds clearly to a simplified situation where all bonds are single ones. Within such a picture, the dependence of the energy on the interatomic distance reduces to that of the matrix elements of the underlying QM (MINDO/3 or NDDO) semiempirical Hamiltonian. 

The second step concerns distance selection and metrization. Bound smoothing only reduces the possible intervals for interatomic distances from the original bounds. However, the embedding algorithm demands a specific distance for every atom pair in the molecule. These distances are chosen randomly within the interval, from either a uniform or an estimated distribution [48,49], to generate a trial distance matrix. Unifonn distance distributions seem to provide better sampling for very sparse data sets [48]. 

If the electrode potential is further reduced to h-350 mV, a hexagonal superstructure with a periodicity of 2.4 0.2 nm is observed. With respect to the interatomic distances in the Au(lll) structure at the surface, this corresponds - within the error limits - to an 8 X 8 superstructure (Figure 6.2-9). 

A single particle of (reduced) mass p in an orbit of radius r = rq + r2 (= interatomic distance) therefore has the same moment of inertia as the diatomic molecule. The classical energy for such a particle is E = p2/2m and the angular momentum L = pr. In terms of the moment of inertia I = mr2, it follows that L2 = 2mEr2 = 2EI. The length of arc that corresponds to particle motion is s = rep, where ip is the angle of rotation. The Schrodinger equation is1... 

For mi = m2, the expression reduces to that obtained for a monoatomic chain (eq. 8.18). When q approaches zero, the amplitudes of the two types of atom become equal and the two types of atom vibrate in phase, as depicted in the upper part of Figure 8.10. Two neighbouring atoms vibrate together without an appreciable variation in their interatomic distance. The waves are termed acoustic vibrations, acoustic vibrational modes or acoustic phonons. When q is increased, the unit cell, which consists of one atom of each type, becomes increasingly deformed. At < max the heavier atoms vibrate in phase while the lighter atoms are stationary. 

On the basis of Eq. (1), NOEs are usually treated as upper bounds on interatomic distances rather than as precise distance constraints, because the presence of internal motions and, possibly, chemical exchange may diminish the strength of an NOE [23]. In fact, much of the robustness of the NMR structure determination method is due to the use of upper distance bounds instead of exact distance constraints in conjunction with the observation that internal motions and exchange effects usually reduce rather than increase the NOEs [5]. For the same reason, the absence of an NOE is in general not interpreted as a lower bound on the distance between the two interacting spins. 

As mentioned in the previous paragraphs, to define an atomic environment they used the maximum gap rule. The Brunner-Schwarzenbach method was considered, in which all interatomic distances between an atom and its neighbours are plotted in a histogram such as those shown in Fig. 3.17. The height of the bars is proportional to the number of neighbours, and all distances are expressed as reduced values relative to the shortest distance. In the specific case of CsCl, having a = 411.3 pm,... 

Figure 3.17. Interatomic distances in CsCI. The distances are given for the CsCI compound (cubic, cP2-CsCl type, a = 411.3 pm) with Cs and Cl in the representative positions 0, 0, 0, and A, A, A respectively, white and black atoms in Fig. 3.8. In the tables the first two groups of distances (in pm) are given as positions of each atom around the reference atom. Notice that not only atoms in the reference cell but also those in the adjacent cells must be considered (see Figs. 3.8 (d)-(f)). At the right side, the corresponding histograms using the reduced distances d/dmm are shown the first two bars summarize the data contained in the table.

Reduced interatomic distances (dv = d/dmin) may be defined as the ratios of the actual distance values to the minimum value. 

In conclusion, the repulsive interactions arise from both a screened coulomb repulsion between nuclei, and from the overlap of closed inner shells. The former interaction can be effectively described by a bare coulomb repulsion multiplied by a screening function. The Moliere function, Eq. (5), with an adjustable screening length provides an adequate representation for most situations. The latter interaction is well described by an exponential decay of the form of a Bom-Mayer function. Furthermore, due to the spherical nature of the closed atomic orbitals and the coulomb interaction, the repulsive forces can often be well described by pair-additive potentials. Both interactions may be combined either by using functions which reduce to each interaction in the correct limits, or by splining the two forms at an appropriate interatomic distance . 

A recent recalculation [43], utilizing the interatomic distance obtained from EXAFS [39,40] and the spring constant corrected for the higher binding energy extracted from the DSC results [57] reduced the ditference between the calculated and measured s to about 2%. 

The vibrational frequency of a diatomic molecule (a one-dimensional system) is proportional to the square root of force constant (the second derivative of the energy with respect to the interatomic distance) divided by the reduced mass (which depends on the masses of the two atoms). 

Fig. 2. Illustration to the JT and PJT effect in diatomic molecule formation. In the homonuclear case (a) the two atomic states at large interatomic distances <2m = l/R = 0 form a double degenerate term which at larger QM (smaller R) splits due to bonding, thus reducing the energy and symmetry quite similar to any other JT E bi problem (cf. Fig. la at Q > 0) for heteronuclear diatomics (b) the bonding picture is that of the pseudo JT effect (cf. Fig. lc at Q > 0).

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