Roothaan's open shell theory from the viewpoint of an orthogonal group

Abstract

Explicit equations for the constants a and b of the energy as the average expectation value for the minimal domain of states giving the Roothaan energy functional are derived in terms of the total spin value and the "seniority" quantum number. The state with off-diagonal long-range order can appear in the framework of the Roothaan scheme for the quasidegenerate system as an alternative to the Hund rule. Admissible many-electron states are established for all configurations of the icosahedral symmetry group systems, and corresponding a and b coefficients are calculated.

Vector coupling coefficients for calculations of transition‐metal atoms and ions by the SCF coupling operator method

“…After this paper was completed, the authors became acquainted with a recent paper [26], several results from which are in sharp contradiction with the above results.…”

“…Particularly, in [26], such states as 'D1 ( d 3 ) , 'S1 (d4), 'D1 (d') and other multiple (double) states are considered as the Roothaan-type ones and the vcc a and b for them are presented (see Table I11 in [26]). 11…”

“…21.11). The same problem holds if one uses the general SCF coupling operator method [S] instead of the atomic Hartree-Fock theory [l-31. Concerning the values of the coefficients a and b, presented in [26] for the state '0 ( d 3 ) and other multiple ones (including those for the systems of the icosahedral symmetry), it should be pointed that they do not correspond to the true energy of a state (which is an eigenvalue of the secular problem). These coefficients correspond to the one of the diagonal matrix element for which c(*) = d4) (see Table A21a-1 in [l]).…”

The electronic transition energies of four alternative C60 molecules

“…The truncated icosahedron has the I h symmetry [6] consisting of 20 hexagons and 12 pentagons. The European football is the best model of this structure.…”

Unitary Group Approach to the Many-Electron Correlation Problem

“…We also see continuously new aspects of group theoretical and algebraic structure of quantum theoretical description of atomic and molecular structure to be discovered (see, e.g., Cizek et al, 1983;Wen et al, 1989Wen et al, , 1990Zhou and Pulay, 1989;Klimko and Mestechkin, 1990;Koutecky and Scheuch, 1990;Nomura, 1990;Planelles and Karwowski, 1990;etc.). In view of the inherent beauty of the representation theory of semisimple Lie groups and algebras (see, e.g., Humphreys;1972), and the potential wealth of new mathematics for hardly explored domain of representations of non-compact topological groups, there is to be expected the presence of a rich algebraic structure in the mathematical description of the submicroscopic world that is yet to be found.…”