Derivatives of orbital function and an extension of Berezin-Gel’fand’s theorem

A generalization of a result of Berezin and Gel’fand in the context of Eaton triples is given. The generalization and its proof are Lie-theoretic free and requires some basic knowledge of nonsmooth analysis. The result is then applied to determine the distance between a point and a G-orbit or its convex hull.We also discuss the derivatives of some orbital functions.

Theory for electronic Raman activity of chlorophyll dimers

We present a modification of a supermolecule model of a general chlorophyll dimer, and calculate the excited state electronic Raman selection rules between the various singlet states. The model utilizes the electronic part of the total wavefunction within the strong intermolecular coupling scheme. Under this condition and the neglect of underlying orbitals, the electronic wave function of the dimer is reduced to a four orbital function with each monomer contributing one HOMO and one LUMO. Using a Hückel type molecular orbital calculation, the one electron orbital energies and coefficients of the ground and excited state wavefunctions of the dimer are obtained and are shown to depend explicitly on the ground and excited state intermolecular interaction terms. The resulting Raman activity calculation between the various excited state configurations yields two types of tensor elements: (1) contains only coordinates of the individual monomer α1 (molecule 1) and α2 (molecule 2) and are called spatially localized elements and (2) labelled spatially delocalized and contains coordinates of both monomer molecules simultaneously (α3 and α4).

Liposarcoma of the Maxilla

A clinically appearing, well-encapsulated lipoma was locally excised from the right posterior upper buccal gingival sulcus of a 24-year-old man and diagnosed histologically as pleomorphic liposarcoma. A right maxillectomy with preservation of orbital function was followed by 5,000 rads of planned postoperative radiation therapy. Four months later the tumor recurred outside of the original site, which has been controlled by further radiation and chemotherapy to these areas.

A quantum-mechanical study of the water molecule

Non-empirical calculations of the electronic structure of the water molecule (treated as a full ten-electron system) are carried out for various types of wave function. These include a ‘bond orbital’ function, a ‘modified electron-pair’ function, and various Roothaan-type self-consistent field (s. c. f.) functions. These approximations are refined by admitting interaction with up to twelve configurational functions of ground-state symmetry. Three bond angles are considered. The results show that the modified electron pair function, which has not hitherto been used in non-empirical calculations, provides the most satisfactory first approximation. This function is constructed from orthonormalized hybrid orbitals in order to overcome non-orthogonality difficulties; and the inclusion of configuration interaction in this particular basis is found to be simple, effective and physically meaningful. The concept of a ‘core’, comprising the oxygen inner shell and lone pairs and providing an effective field for the bond pairs, is found to be remarkably satisfactory. Provided the effective field is properly defined, it is then possible to treat the molecule formally as a four-electron system and to obtain, nevertheless, a total energy 0⋅3 eV better than that given by the best of the s. c. f. calculations. This suggests that the ‘core approximation’ is not a real obstacle to progress. The one-electron density matrices are calculated from the various wave functions and the corresponding dipole moments are evaluated.

The molecular orbital theory of chemical valency IX. The interaction of paired electrons in chemical bonds

In previous parts of this series the molecular orbital theory has been developed in a way which, while dealing satisfactorily with the interaction of electrons in different orbitals, is not adequate to describe the wave theory of two paired electrons in the same orbital. This paper is an attempt to improve this part of the theory. Most of the work is restricted to the problem of two electrons in the bond of a homonuclear diatomic molecule. The method is based on expansions of the wave function and the interelectronic repulsion term of the Hamiltonian over the irreducible representations of the symmetry group. A series of coupled equations are obtained for the terms of the wave-function expansion. These exact equations form a useful background against which to examine some approximate wave functions, including the simple molecular orbital function and the electron-pair function. Some approximate calculations on the hydrogen molecule indicate that inclusion of higher terms in the wave-function expansion substantially reduces the calculated electron repulsion energy.