Implementation of molecular symmetry in valence bond calculation

A novel strategy for the construction of many-electron symmetry-adapted wave function is proposed for ab initio valence bond (VB) calculations and is implemented for valence bond self-consistent filed (VBSCF) and breathing orbital valence bond (BOVB) methods with various orbital optimization algorithms. Symmetry-adapted VB functions are constructed by the projection operator of symmetry group. The many-electron symmetry-adapted wave function is expressed in terms of symmetry-adapted VB functions, and thus the VB calculations can be performed with the molecular symmetry restriction. Test results show that molecular symmetry reduces the computational cost of both the iteration numbers and CPU time. Furthermore, excited states with specific symmetry can be conveniently obtained in VB calculations by using symmetry-adapted VB functions.     

Simple method to obtain symmetry harmonics of point groups

We propose a simple, self-consistent method to obtain basis functions of irreducible representations of a finite point group. Our method is based on eigenproblem formulation of a projection operator represented as a nonhomogeneous polynomial of angular momentum L. The method is shown to be more efficient than the usual numerical methods when applied to the analysis of high-order symmetry harmonics in cubic and icosahedral groups. For low-order symmetry harmonics the method provides rational coefficients of expansion in the Y(L,M) basis.     

Treatment of symmetry in MO calculations. II. Numerical projection operators for molecular orbitals

This article describes the numerical application of projection operators to restore the symmetry of molecular orbitals in self-consistent field (SCF) calculations when the symmetry is lost because of degeneracy or near degeneracy. The application of projection operators is particularly useful in cases of near degeneracies of three or more molecular orbitals, where it is difficult to find an effective algorithm for restoring the symmetry of molecular orbitals by orthonormal transformations.     

Continuous symmetry measures for complex symmetry group

Symmetry is a fundamental property of nature, used extensively in physics, chemistry, and biology. The Continuous symmetry measures (CSM) is a method for estimating the deviation of a given system from having a certain perfect symmetry, which enables us to formulate quantitative relation between symmetry and other physical properties. Analytical procedures for calculating the CSM of all simple cyclic point groups are available for several years. Here, we present a methodology for calculating the CSM of any complex point group, including the dihedral, tetrahedral, octahedral, and icosahedral symmetry groups. We present the method and analyze its performances and errors. We also introduce an analytical method for calculating the CSM of the linear symmetry groups. As an example, we apply these methods for examining the symmetry of water, the symmetry maps of AB₄ complexes, and the symmetry of several Lennard‐Jones clusters. © 2014 Wiley Periodicals, Inc.     

The symmetrized random matrix approach, an efficient method to obtain relativistic molecular symmetry adapted functions

In relativistic quantum chemical calculation of molecules, where the spin-orbit interaction is included, the electron orbitals possess both the double point group symmetry and the time-reversal symmetry. If symmetry adapted functions are employed as the basis functions of electron orbitals, it would allow a significant reduction of the computational expense. The point group symmetry adapted functions can be obtained by the group projection operators via its actions on the atomic orbital functions. We have proposed an efficient and simple method to obtain all irreducible representation matrices, which are the basis of the group projection operators, of any finite double point group. Both double point group symmetry and time-reversal symmetry are automatically imposed on the representation matrices. This is achieved by the symmetrized random matrix (SRM) approach, where the SRM is constructed in the regular representation space of a finite group and the eigenfunctions of SRM provide all irreducible representation matrices of the given point group.     

New N-representability results with symmetry in molecular quantum chemistry

We prove a new type of N-representability result: given a totally symmetric density function ρ, we construct a wavefunction Ψ such that the totally symmetric part of $\rho \Psi $ (its projection over the totally symmetric functions) be equal to ρ, and, furthermore, such that Ψ belongs to a given class of symmetry associated to the symmetry group of a molecule. Our proof uses deformations of density functions and which are solutions of a “Jacobian problem”. This allows us to formalize rigorously an idea of A. Görling (Phys. Rev. A 47 (1993) 2783), for Density-Functional Theory in molecular quantum chemistry, by defining a density functional that takes into account the symmetry of the molecule under study.     

Molecular symmetry perception

An algorithm for molecular symmetry perception is presented. The method identifies the full set of molecular symmetry elements (proper and improper) and determines their coordinates. The algorithm eliminates the necessity to explore the entire graph automorphism group; as a result its computer application is extremely effective. Application to several dendrimers and fullerenes with high topological symmetry is presented.     

Tables of subgroups,site groups and interchange groups of symmetry point groups and the construction of SALCs

A table listing subgroups and supergroups of 43 chemically important symmetry point groups is presented. Other tables give the site groups and the corresponding interchange groups for those point groups which are of finite order. It is shown how the tables may be used to facilitate the structure determination of metal-ligand and related compounds from spectroscopic data and how the construction of symmetry adapted linear combinations. SALCs, may be simplified. The tables also may be used in factor group analysis.     

Simplification of point group projection operators and its application to symmetry adaptation of multishell electron configurations

A practical method for generating irreducible matrix reps of point groups and a concise formula about projection operators are proposed. By using this formula as well as versatile classification schemes, the symmetry adaptation of a many-electron system is simplified. A unified algorithm and program of symmetry adaptation of spin-free space have been developed.     

Full non-rigid group theory and symmetry of melamine

The non-rigid molecule group (NRG) theory in which the dynamic symmetry operations are defined as physical operations is a new field in chemistry. Smeyers, in a series of papers, applied this notion to determine the character table of restricted NRG of some molecules. For example, Smeyers and Villa computed the r-NRG of the triple equivalent methyl rotation in pyramidal trimethylamine with inversion and proved that the r-NRG of this molecule is a group of order 648, containing two subgroups of order 324 without inversion [5]. In this work, a simple method is described, through which it is possible to calculate character tables for the symmetry group of molecules. We study the full NRG of melamine, and prove that it is a groups of order 48, with 27 and 10 conjugacy classes. Also, we compute the symmetry of melamine and prove that it is a non-abelian groups of order 6. The method can be generalized to apply to other non-rigid molecules.     

Continuous symmetry measures of irreducible representations: application to molecular orbitals

The formalism of continuous symmetry measures is extended to describe the extent to which a function, such as a molecular orbital, transforms under the symmetry operations of a given point group according to each irreducible representation of this group. For distorted molecules we are able to calculate the degree to which any molecular orbital transforms with respect to the irreducible representations of the pseudosymmetry group that is valid for a higher symmetry, idealized geometry, showing which irreducible representations participate in each molecular orbital upon symmetry loss in the geometry of the nuclear framework.     

SASS: a symmetry adapted stochastic search algorithm exploiting site symmetry

A simple symmetry adapted search algorithm (SASS) exploiting point group symmetry increases the efficiency of systematic explorations of complex quantum mechanical potential energy surfaces. In contrast to previously described stochastic approaches, which do not employ symmetry, candidate structures are generated within simple point groups, such as C2, Cs, and C2v. This facilitates efficient sampling of the 3N-6 Pople's dimensional configuration space and increases the speed and effectiveness of quantum chemical geometry optimizations. Pople's concept of framework groups [J. Am. Chem. Soc. 102, 4615 (1980)] is used to partition the configuration space into structures spanning all possible distributions of sets of symmetry equivalent atoms. This provides an efficient means of computing all structures of a given symmetry with minimum redundancy. This approach also is advantageous for generating initial structures for global optimizations via genetic algorithm and other stochastic global search techniques. Application of the SASS method is illustrated by locating 14 low-lying stationary points on the cc-pwCVDZ ROCCSD(T) potential energy surface of Li5H2. The global minimum structure is identified, along with many unique, nonintuitive, energetically favorable isomers.     

Local symmetry and chirality of molecular faces

The concept of local symmetry has been applied to faces of planar sites such as carbon–carbon double bonds and aromatic rings with the principal results being as follows. The two faces of a planar site must have the same local symmetry group. This local symmetry group is limited to the polar point groups. For cyclic compounds, directed cycles must have chirotopic faces although the reverse is not necessarily true: chirotopic faces are possible for both directed and undirected cycles. A number of examples are provided to illustrate these results. Received: 30 June 1999 / Accepted: 4 October 1999 / Published online: 19 April 2000     

Use of molecular symmetry in SCF calculations

A method is described whereby molecular symmetry properties may be used to reduce the numbers of one- and two-electron integrals that need to be calculated and stored in the course of a molecular SCF calculation. The method is a generalization of a previously reported procedure, extending the earlier work to cover those molecules belonging to point groups which have complex representations. The practical application of the method is discussed and an illustrative example given. The quite extensive tables of molecular symmetry properties which the method uses may be computer generated in a straightforward manner. A procedure for doing this using a minimum amount of input data is presented.     

Optimization of molecular electronic structure calculations. 1. Local symmetry and local symmetricized orbitals

A formalism is suggested of the so-called local symmetricized orbitals to be used for the construction of a symmetricized basis in molecular electronic structure calculations. The local symmetricized orbitals are defined as additive contributions to the symmetry orbitals of a molecule that arise from symmetry operations of a corresponding atom. The local symmetricized orbitals are transformed according to the irreducible representations of the molecular symmetry group. This approach appears to be most suitable for the optimization of quantum mechanical calculations accounting for the spatial symmetry of compounds under consideration. This fact is due to the formalism of the local symmetricized orbitals that explicitly accounts for the local symmetry of basis function centers, which is essential for such optimization.     

General method for symmetry orbitals and tensors in electronic structure calculations

The symmetry orbital tensor (SOT) method, which makes full use of symmetries in all point groups and can be applied to the self-consistent field (SCF) and post-SCF calculations, is introduced. The principal feature of this method is the definition of the symmetry orbitals (SOs). Any element in a molecular point group will transform one SO to another equivalent SO or simply to itself, and no mixture among SOs exists. Thus, although the SOs for non-Abelian point groups may adapt to reducible representations, their transformation properties are much simpler than in conventional treatments. This article also presents a general scheme to generate SOs for all point groups. The direct products of N SOs form an Nth-rank SOT group, and each matrix element between SOTs is the product of a physical factor and a geometric factor. Compared with the canonical molecular orbitals, the use of SOs can noticeably reduce the computation efforts by decreasing the number of integrals needed in the SCF calculations or the number of configurations needed in the configuration interaction (CI) calculations. The SOT-SCF and SOT-CI approaches are formulated and a preliminary SOT-SCF program is written. Pilot calculations demonstrate the value of the SOT approach, at least at the closed-shell Hartree–Fock level. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 305–321, 1999     

The symmetry of molecular polarization tensors

Polarization tensors are discussed in terms of their intrinsic symmetry group which is a direct product of the point group and the subgroup of the permutation group relevant to the experiment. The study of these latter groups is simplified by use of the isomorphism with certain point groups and permutations of suffixes can be visualized by rotations and reflections of the vertices of various objects in space. The approach unites the previous treatments and provides a means of constructing the bases for the irreducible tensor components. The difficulties introduced by Laplace's equation are explained and the information obtainable from induced birefringence experiments (Kerr and Cotton–Mouton effects) discussed for various systems.     

Construction of linearly independent relativistic symmetry orbitals for finite double-point groups including time reversal symmetry

A formalism is developed for the construction of relativistic symmetry-adapted molecular basis functions under consideration of time reversal invariance. The theory is applicable to the finite double point groups C_n, C_nh, S_n, C_nv, D_n, D_nd, D_nh, T, T_h, T_d, O, and O_h. It is based on the LCAO method. A projection operator technique is employed to construct molecular symmetry orbitals from atomic orbitals. The search for linearly independent basis function is simplified by means of group theoretical considerations.