Correlation quantum numbers

Until recently, another problem with the so-called correlation quantum numbers was that they did not enable the most accurate calculations of doubly-excited spectra to be performed. With the work of Tang and Shi-mamura [330] on the hyperspherical close-coupling method, this situation has changed, and there is renewed interest in this approach since the situation is still evolving, we merely summarise developments in this area in section 7.11. 

In principle, the hyperspherical method corresponds rather well to the strategy advocated by Langmuir (see [308] and section 7.5) namely that one should seek to quantise the motion of more than just one electron in a many-electron system. The coordinates R and a describe the combined motion of an electron pair, and so the quantum numbers which arise in the solution are radial correlation quantum numbers. 

In the simplified summary given here, only the radial part of the problem has been mentioned. Clearly, there are also angular equations and angular correlation quantum numbers to consider. Another, related, approach is to use group theoretical methods to classify doubly-excited states, and this has been pursued mainly by Herrick [328]. 

Fig. 49. Correlation between the energy levels of (1) free rotation of the symmetric top, and (2) torsion vibrations in the potential with symmetry Cj. Quantum numbers J and K enumerate rotational levels, n vibrational levels. Relative positions of A and E levels are shown on the right.

Summarizing, the principal quantum number (n) can have any positive integral value. It indexes the energy of the electron and is correlated with orbital size. As U increases, the energy of the electron increases, its orbital gets bigger, and the electron is less tightly bound to the atom. 

The value of / correlates with the number of preferred axes in a particular orbital and thereby identifies the orbital shape. According to quantum theoiy, orbital shapes are highly restricted. These restrictions are linked to energy, so the value of the principal quantum number ( ) limits the possible values of /. The smaller U is, the more compact the orbital and the more restricted its possible shapes ... 

Armed with these conditions, we can correlate the rows and columns of the periodic table with values of the quantum numbers it and /. This correlation appears in the periodic table shown in Figure 8. Remember that the elements are arranged so that Z increases one unit at a time from left to right across a row. At the end of each row, we move down one row, to the next higher value of It, and return to the left side to the next higher Z value. Inspection of Figure reveals that the ribbon of elements is cut after elements 2, 10, 18, 36, 54, and 86. 

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

Kier and Hall noticed that the quantity (S -S) jn, where n is the principal quantum number and 5 is computed with Eq. (2), correlates with the Mulliken-Jaffe electronegativities [19, 20]. This correlation suggested an application of the valence delta index to the computation of the electronic state of an atom. The index (5 -5)/n defines the Kier-Hall electronegativity KHE and it is used also to define the hydrogen E-state (HE-state) index. 

Figure 10. Low-energy vibronic levels in the X2II state of HCCS computed in various approximations [152]. Hq zeroth-order approximation (both vibronic and spin-orbit couplings neglected). Hi. vibronic coupling taken into account, spin-orbit interaction neglected. Hi + Hs0 both vibronic and spin-orbit couplings taken into account. Solid horizontal lines K = 0 vibronic levels dashed line K — 1 dash-dotted lines K = 2 dotted lines K — 3. Values of the quantum numbers V4, N of the basis functions dominating the vibronic wave function of the level in question are indicated. Approximate correlation of vibronic states computed in various approximations is indicated by thin lines. In all cases the stretching quantum numbers are assumed to be zero.

F. Hund, "Zur Deutung der Molekulspektren. IV," ZP 51 (1928) 759795 R. S. Mulliken, "The Assignment of Quantum Numbers for Electrons in Molecules. II. Correlation of Molecular and Atomic Electron States," Physical Review 32 (1928) 761772 E. Hiickel, "Zur Quantentheorie der Doppelbindung,"... 

With this definition, due to Child and Halonen (1984), local-mode molecules are near to the = 0 limit, normal mode molecules have —> 1. The correlation diagram for the spectrum is shown in Figure 4.3, for the multiplet P = va + vb = 4. It has become customary to denote the local basis not by the quantum numbers va, vh, but by the combinations... 

Inamoto and co-workers (97,98) introduced a new inductive parameter i (iota) based on atomic properties of X, namely the effective nuclear charge in the valence shell and the effective principal quantum number, as well as E(X) (97). They thereby established a reasonable correlation between the a-SCSs in substituted methanes and ethanes and the t. parameters for a series of substituents not including X = CN and I (97). 

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