Hypervirial theorems and restrictions on wave functions

The conditions that ensure that an optimal variational wave function ø under general restricting requirements satisfies the hypervirial theorem are analysed. Application is made to a system where thez -component of angular momentum is a constant of motion and results are discussed in connection with those obtained via symmetry considerations.     

Permutation representations in molecular symmetry

The reducible representations of the point groups are generally studied because of their relevance to molecular orbital and vibration theory. Triple correlations within the polyhedra are described by group-theoretical invariants that are related to the permutation representations and termed polyhedral isoscalar factors. These invariants are applied in theorems on matrix elements referring to the symmetry-adapted bases at different centres. Further invariants or geometrical weight factors inter-relate different types of reduced matrix elements of irreducible tensors (generalization of the Wigner-Eckart theorem to the polycentric case). As a demonstration a complete tabulation is given for the point group C ₄.     

Hypervirial theorems for some simple quantum chemistry systems

A very general form of the hypervirial theorems, which can be applied to several quantum systems of great interest for the theoretical quantum chemists is presented.     

The Hiller-Sucher-Feinberg density is not integrable

Summary The large-distance asymptotic behavior of the electron density obtained from the Hiller-Sucher-Feinberg identity is analyzed. It is shown that, unless the generating wavefunction is exact, such electron density decays like a polynomial in inverse powers of the distance. Therefore, the Hiller-Sucher-Feinberg density is not integrable in general. The sufficient conditions to be imposed upon the generating wavefunctions in order to assure integrability are spelled out.Dedicated to Prof. Klaus Ruedenberg     

The use of site symmetry in constructing symmetry adapted functions

It is shown that the application of a projection operator from a given group to a function is equivalent to the successive application of projection operators from factor groups of the starting group to that function. When used with the factor groups representing the site symmetry of a position and the simplest group of interchanges of positions, this concept provides a very simple method for obtaining symmetry adapted linear combinations of basis functions.     

Invariance criteria and symmetry conservation rules for geometry optimizations

The objective of this paper is to analyze the behaviour of some minimization methods such as steepest descent method, generalized Newton and quasi-Newton methods under transformations of the variables of the function to be minimized. Energy and molecular coordinates are the function and the variables, respectively, in the case of geometry optimizations. Invariant levels are shown to be decisive for the area where the minimization methods can be successfully employed without rescaling of the coordinates. Specific conditions for symmetry conservation are worked out in context of invariant levels. Symmetry making, breaking and conservation are shown with working examples of geometry optimizations and calculation of energy minimum paths on the basis of certain kinds of molecular coordinates.     

Bonding in C2 and Be2: Broken symmetry and correlation in DFT solutions

Summary The solution of both Hartree-Fock (HF) and Kohn-Sham (KS) equations is based on the variational principle. Exact wavefunctions would obey the same symmetry restrictions contained in the total hamiltonian. However, the variational principle does not guarantee these symmetry restrictions and the HF and KS solutions are not necessarily symmetric in spin and space. Spatial and spin symmetry broken solutions with lower energies than their restricted analogues are examined for C₂ and Be₂, in the context of the KS formalism. Comparison with UHF solutions shows that KS instabilities are far less pronounced. The main differences between HF and KS solutions are related to effects of electron correlation.     

Integral and regional virial theorems in density functional theory

Integral forms of the virial theorem in the density functional theory are derived. A relationship between the orbital energy sum and the integral of the pressure is studied. A new form of the regional virial theorem and the regional analogue of the Levy-Perdew relation are obtained. Local density approximation to the regional virial theorem is presented.     

Fixed-nuclei and laboratory-frame formalisms for electron scattering by a spherical top,with full incorporation of symmetry

We discuss molecule-frame and laboratory-frame symmetry-adapted formalisms for electron scattering by a spherical top. The molecule-frame formalism is based on the fixed-nuclear-orientation approximation, both for electronically elastic scattering by a vibrationally rigid molecule and also for the more general case where electronic excitation and vibrational degrees of freedom are included. The laboratory-frame formalism is based on the exact symmetries of the problem, which are carefully related to the approximate symmetries of the molecule-frame treatment. We present both the forward and backward transformations between the two representations.     

Molecular van der Waals symmetry affecting bulk properties of condensed phases: melting and boiling points

Qualitative analysis of the trends in melting temperatures T _m and boiling temperatures T _b of organic compounds and related substances supports a concept of van der Waals symmetry suggested earlier by the authors. Data for substituted methanes, ethanes, ethylenes, benzenes and cyclohexanes, reveal linear T _b versus M ^α patterns but usually non-uniform T _m versus M ^α patterns (where M is the molecular mass, α varies from 0.5 to 1). In the same set of isomers, T _m and T _b often change in different succession. To explain the observed trends, a local van der Waals field U that acts on a molecule in a solid or liquid phase, was represented by a sum of an averaged long-range field U ₀ from all molecules, and a short-range “contact” term W determined by a molecular shape. Variations of T _m and T _b among closely related substances correspond to different strength of perturbation of U ₀ ^(C) (in a crystal) and U ₀ ^(L) (in a liquid) by W. Higher T _b of isomers with medium-symmetry molecules (like 1,2-disubstituted benzenes and cyclohexanes) reflect the better fitting of their molecules to a local molecular environment in a liquid (that contains vacancies). Highly symmetrical quasispherical molecules (e.g., C₂Cl₆ or C₆F₁₂) produce a stabilised solid and destabilised liquid state, hence their molecular crystals easily sublime rather than melt at ambient pressure. Dedicated in memory of Professor Petr M. Zorky     

Chemical applications of topology and group theory. 17. An information theoretical approach to polyhedral symmetry [1]

Information theoretic parameters are described which measure the asymmetry of polyhedra based on partitions of their vertices, faces, and edges into orbits under action of their symmetry point groups. Such asymmetry parameters are all zero only for the five regular polyhedra and are all unity for polyhedra having no symmetry at all, i.e. belonging to the C ₁ symmetry point group. In all other cases such asymmetry parameters have values between zero and unity. Values for such asymmetry parameters are given for all topologically distinct polyhedra having five, six, and seven vertices; all topologically distinct eight-vertex polyhedra having at least six symmetry elements; and selected polyhedra having from nine to twelve vertices. Effects of polyhedral distortions on these asymmetry parameters are examined for the tetrahedron, trigonal bipyramid, square pyramid, and octahedron. Such information theoretic asymmetry parameters can be used to order site partitions which are incomparable by the chirality algebra methods of Ruch and co-workers.     

Application of coset representations to the construction of symmetry adapted functions

Summary A coset representation (G(/G _i )), which is defined algebraically by a coset decomposition of a finite groupG by its subgroupG _i , is shown to be a method for the decomposition of a regular body into its point group orbits. This proof also shows that each member of theG(/G _i ) orbit belongs to theG _i site-symmetry. In addition, a general equation concerning the multiplicities of such coset representations is derived and shown to involve Brester's equations and thek-value equations of framework groups as special cases. The relationship of the coset representation and the site-symmetry affords a general procedure for obtaining symmetry adapted functions.     

The combinatorics of symmetry adaptation

A method is developed for obtaining the generating functions for the equivalence classes of orbitals wherein only orbitals within an equivalence class participate in symmetry adaptation. It is shown that using Williamson's combinatorial theorem the generating functions for the symmetry species contained in each equivalence class can be obtained. The method is illustrated with Porphindianion.     

分子轨道图形理论中对称轴定理

本文试图以浅显明确的形式描述对称轴定理，使之能与对称面定理相媲美，并对对称轴定理作一定的扩展，给出分子轨道构造定理。     

The method of ascending symmetry for irreducible characters of finite groups

The irreducible characters of a finite group are determined uniquely by those of a minimal set of maximal subgroups. The method is based on the construction of all class functions which are irreducible characters on every maximal subgroup. These are generalized characters by a theorem of Brauer, so that the irreducible characters are obtained by checking the norm. An alternative characterization of irreducible characters, the Maximum Mixing Rule, works for all point symmetry groups, and its physical significance is discussed. As an example, the character tables for all point symmetry groups and crystal double-groups are constructed in this way.Bat-Sheva de Rothschild fellow for 1985     

Hypervirial Theorems and SCF Wavefunctions for Coupled Oscillators

The SCF formalism is analysed in connection with hypervirial theorems. Coupled oscillator models are chosen in order to illustrate the formal results.     

Spin density in first-row atoms from the Hiller-Sucher-Feinberg identity

Summary The performance of delta function and Hiller-Sucher-Feinberg (HSF) operators is compared for calculations of the electronic spin density at the nucleus, which determines the observed Fermi contact hyperfine splitting. Calculations are performed on the ground states of the first-row open-shell atoms boron through fluorine. The wavefunctions include low order spin polarization effects calculated through the multiconfigurational self-consistent-field procedure. It is shown that while delta function and HSF operators give nearly the same results when essentially exact numerical grid methods are used, the HSF operator gives a significant advantage when contracted Gaussian type basis sets are utilized.     

Extension of the quantum virial and Hellmann–Feynman theorems to the quantum statistical averages

In the present article, we extended the quantum virial and Hellmann–Feynman theorems to the quantum statistical averages, that is, to the thermal states. We obtained some new formulas which make possible expressing the thermodynamical observables of the system as functions of the moments of coordinates, as we see in a short example relating to the pseudoharmonical oscillator. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 159–165, 1998