Quantum numbers effective

The symbols in the second column represent the electronic state in particular the first number is the total quantum number of the excited electron. We shall see later that in one case at least the symbol is probably incorrect. The third column gives the wave-number of the lowest oscillational-rotational level, the fourth the effective quantum number, the fifth and sixth the oscillational wave-number and the average intemuclear distance for the lowest oscillational-rotational level. The data for H2+ were obtained by extrapolation, except rQ, which is Burrau s theoretical value (Section Via). 

For three-dimensional motion an effective quantum number may be defined as... 

Fig. 7.5 Plot of observed ionizing fields for Na m = 0,1, and 2 states for n = 8-20 plotted vs ns, the effective quantum number in the strong field. The fine indicates the classical ionization threshold (from ref. 8).

Fig. 11.12 Plot of the depopulation cross sections of the Rb n( states by ground state Rb nf( ) (ref. 32) ns (O), nd3/2(x), nd ( ) (ref. 57) np ( ) (ref. 26). The cross sections are plotted vs effective quantum number n. For reference the geometric cross section...

Fig. 11.27 Individual Penning rate coefficients for Rb n + Rb 5p, kP[(nf), against the effective quantum number n (from ref. 108).

All the energy dependence in the cross section arises from the factors A2 and (vb B v2 ) 2. These are, respectively, the spectral density of the channel 2 autoionizing states, and the overlap integral between the bound and continuum ni states with effective quantum numbers vb and v2. We have already seen that A is simply given by -dvl/dv2, the derivative of the quantum defect surface, and repeats modulo 1 in v2. The overlap integral is given is closed form by5... 

Fig. 21.6 Calculated cross section o from a bound state of effective quantum number vb = 12.35 to an autoionizing Rydberg series with width T = O.llr-3 and quantum defect d2 = 0.15 so that the autoionizing states are found at v2 = 0.85(mod 1) (from ref. 6).

Fig. 21.12 The three dimensional, quantum defect surface for the 6pnd states below the 6p1/2 limit. Here v2 and v3 are the effective quantum numbers relative to the 6p1/2 and 6p3/2 limits, respectively. is the continuum phase. The direction of the normal to the surface at any point indicates the amounts of 6p1/2nd5/2,6p3/2ndJ-, and continuum character in the...

Actual calculations of compressed-atom densities, performed with suitably modified SCF software, show that the increased pressure raises all electronic energy levels, at different rates that depend on the shell structure. The effect is more pronounced on those levels of highest effective quantum number l and it is not uncommon for levels of different l to cross during compression. The interpretation of photoelectron spectra in terms of free-atom electron configurations may therefore be misleading in the study of surface chemistry and catalytic effects, for which they are routinely used. 

Electronic bound-state levels are inversely proportional to the square of an effective quantum number, E oc — 1/n2, as shown on an arbitrary scale in the diagram. 

C. Klixbull J0rgensen Effective Quantum Numbers in d and /-Shells, J. Inorg. and Nuclear Chem>, 1, 301 (1955). 

Here n denotes "effective quantum number", exponent 5C, is an arbitrary positive number, r, t, y) are polar coordinates for a point with respect to the origin A in which the function (2,3) is centered. Apart from the first two terms that represent a normalizing factor, the function (2,3) is closely related to hydrogen-like orbitals. For the hydrogen Is orbital the function I q q 0 identical with Q q, if we assume Z Z/n, However, it should be recalled that in contrast to hydrogen-like orbitals STO s are not mutually orthogonal. Another essential difference is in the number of nodes. Hydrogen functions have (n -i - 1) nodes, whereas STO s are nodeless in their radial part. Alternatively, the STO may be expressed by means of Cartesian coordinates as follows... 

With increase in the collision energy, more and more partial cross sections contribute to the total VR cross section, so that one has to calculate the transition probabilities Pj for J>0. If the suppression-enhancement effects are ignored for J>0, one can use the so-called effective quantum number approximation [21J.This amounts to the following ... 

The convergence limits were obtained from the series data using the criteria that the quantum defect (n-n ) is a constant (.27) for the correct limit. To do this we determine the effective quantum number n of each observed Rydberg level using the relation... 

Table 2.1. Effective quantum numbers n ni for alkali atoms.

To demonstrate how useful the alkali model is for d subshell excitation in Zn, Cd and Hg, consider the effective quantum numbers n listed in table 2.2 for np and nf orbitals. The behaviour of these numbers is closely similar to that for alkali spectra, with the most hydrogenic states (near integral n ) being those of greatest , while the n values indicate that the outermost np electron, in its lowest available state, occurs at a binding energy (referred to a core-excited threshold) intermediate between the values for n = 1 and n = 2 of H. 

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