LCAO Description of Symmetric Molecules by Unified Theory of Finite Graphs

Abstract

LCAO MO descriptions of symmetric molecules have topological and energetic aspects. Topological effects include the point symmetry of the nuclear configuration and the various connections between these points (connectedness). An appropriate mathematical tool for a discussion of molecular topoligies is given by the general theory of graphs. This theory anticipates various properties connected with the topological matrix. Special graphs and their matrix representations which reflect all topological properties of symmetric molecules serve to determine the molecular-orbital-term scheme and the symmetry orbitals for various sorts of molecules. In particular, tetrahedral, square planar, trigonal prismatic, and octahedral molecules are investigated. Finally some remarks are made on the relative choice of parameters in the semiempirical theory as suggested by the consideration of the virial theorem.

Charge-transfer spectra of tetravalent lanthanide ions in oxides

“…= t2u < lt2g < 2tt. < tlg = 2t2g < e. = a2.. [13]. But in the case of a cube, still not considering interactions of the ligand m.o.…”

“…If we follow Schmidtke's topological treatment of an octahedron of six ligands [13], the sequence of ligand m.o. 's is, a~ < I t,, < tz~ < t~,, < tl~ = e~, < 2t,,.…”

Use of symmetry plane fragmentation and graph squaring techniques to express the eigenspectra for some vertex-weighted graphs of linear chains and cycles in analytical form

“…With proper scaling of the energy, graph eigenvalues become exactly equal to the Hu¨ckel molecular orbitals (HMO) of the corresponding molecule [1,2]. There are a large number of graph factorization methods that reduce a large graph into smaller graphs in such a way that the factored graphs contain all the eigenvalues of the large graph and are significant to calculate the eigenvalues of the large graph.…”

Topological aspects of symmetric molecules

“…Such matrices have also been defined in Hiickel theory [3, 41 to determine electron orbital structures of r-electron systems. In an application to general LCAO theory this concept has been extended by using a theory of multigraphs [ 5 ] which allows for a multiple interaction (e.g. by (r and T bonding) of the atoms in the molecule.…”

“…The orbital order of p functions located at the vertices of a tetrahedron is determined from its proper structural matrix [6] in the case of relatively small n-bonding effects as (5) The corresponding s orbital sequence is a, < t, . These sequences are in agreement with an extended Hiickel calculation for the B4H, molecule [7].…”