Scaled internuclear distance

Figure 6 plots the zeros of the fnam polynomials as a function of the scaled internuclear distance, for n = 4 a nd m = 0. The state labeled 4s(t and associated with the united atom quantum numbers n = 4, / = 0,m = 0 must have three A nodes and no ft nodes for the entire range of internuclear separations, since nx = n — I — 1 = 3, and... 

Figure 6. Zeros of the fnam polynomials for n = 4, m = 0 as a function of scaled internuclear distance = States are labelled by the corresponding united atom quantum numbers.

We can apply a somewhat anedogous argument to in which case we use the distance scaling r — /r with / oc D(D — 1), which leads to the scaled internuclear distance R = R/f. R is by assumption independent of dimension. Therefore, our dimensional continuation of the Schrodinger equation corresponds to a dimension dependent internuclear distance... 

For the sake of simplicity, we will consider a diatomic molecule with the internuclear distance R, but the result is directly general-izable to a system with several internuclear distances Rv R2,. In addition to the trial function = q>(rlt r2,. . ., rN, R), we will now also consider the scaled function ... 

Our derivation of Eq. 11.33 based on the use of the variation principle is different from Slater s original treatment, but so far follows Hirschfelder and Kincaid. Here we will now show that it also permits such a scaling of an arbitrary trial function internuclear distance according to Eq. 11.29. 

By multiplying Eq. 11.32 by 77 and by using Eq. 11.30 and Eq. 11.31, it is then easily checked that the virial theorem (Eq. 11.33) is satisfied for the scaled function internuclear distance R — rj xp. The distance R is here a simple function of p, and, after establishing the relationship in the form of a graph or a table, we can also solve the reverse problem of finding the properly scaled func-... 

Fig. 1-7.—Curves showing the energy of interaction of two normal hydrogen atoms. The scale for the internuclear distance Tab is based on the unit a0 = 0.530 A.

Figure 21-11 Comparison of calculated exact" ab initio energies of H2 as a function of internuclear distance, r, with the energies calculated for simple MO and simple VB methods. The dissociation energy calculated by the ab initio procedure is in close agreement with the experimental value of 102 kcal. The zero of the energy scale in this figure is the energy of widely separated hydrogen atoms.

The results, for several values of the masses and force constants, are given in Table 1, where we have chosen the masses to correspond roughly to H, C, and N atoms and the force constants to be approximately those of single, double, and triple bonds. In the drawings of the normal modes the scale of the displacements is not the same as that of the internuclear distances, nor is the scale of displacements the same in all modes. Only... 

The position of the peak on a distance scale gives information of the internuclear distance, and the shape of the peak itself is a representation of the weighted sum of... 

Other traditional scales, such as that of Goldschmidt, are more empirically based and rely upon experimental internuclear distances found in oxides and fluorides (which are expected to be particularly ionic) assumed values for the radii of F and O2- are needed. A very complete tabulation, based on a large amount of modern data for oxides and fluorides, both binary and complex, has been published by Shannon and Prewitt using values of 133 pm for F and 140 pm for O2-. One of the merits of this collection is that it recognises the dependence of ionic radii oh the coordination number. For example, in the case of Cd2+ radii are quoted for coordination numbers 4, 5, 6, 7, 8 and 12 as shown below ... 

The various scales of ionic radii are in fact about equally good, within the criterion that, when added together, they should reproduce observed internuclear distances in crystals. The realistic radii derived from... 

Recently, two basic questions of chemical dynamics have attracted much attention first, is it possible to detect ( film ) the nuclear dynamics directly on the femtosecond time scale and second, is it possible to direct (control) the nuclear dynamics directly as it unfolds These efforts of real-time detection and control of molecular dynamics are also known as femtosecond chemistry. Most of the work on the detection and control of chemical dynamics has focused on unimolecular reactions where the internuclear distances of the initial state are well defined within, of course, the quantum mechanical uncertainty of the initial vibrational state. The discussion in the following builds on Section 7.2.2, and we will in particular focus on the real-time control of chemical dynamics. It should be emphasized that the general concepts discussed in the present section are not limited to reactions in the gas phase. 

For heavier triatomic systems, Flartree-Fock potential surfaces had earlier been calculated for FI + NO- FfNO [120] andO + NO [121]. Recently Pipano and Kaufman [122] performed an ab initio large-scale configuration-interaction calculation for the reaction 0+ + Na - NO+ + N that included all single and double excitations relative to the lowest configurations of the 2n(, 2 +, 4n, states of the intermediate NNO+ as a function of internuclear distance. These results, as an example of the melding together of the way the theoretical techniques outlined earlier complement each other and experimental observations, will be discussed in more detail in Section IV of this chapter. 

The cross-section for dissociative ionization process (19.56) is experimentally known for v = 0 [83] (see also [14]). The cross-sections for v > 1 can be roughly estimated from the cross-section for v = 0 by using the I 1 scaling of ionization cross-section, where Iv is the transition energy from H (X v) energy level to the (H+ + H+) interaction potential at the internuclear distance equal to the equilibrium distance of (X v). 

Figure 4 shows the known RKR turning points for both the X and A states of the alkali hydrides. The internuclear distance is scaled by the equilibrium bond distance in the state. 

Fig. 5. Density difference contour maps Ap for the H2 system at several internuclear distances. The maps for R = 8.0 and 6.0 are drawn one-half scale relative to the others. (Reproduced from Bader and Chandra, 1968.)...

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