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.     

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.     

Exclusion principle and the symmetry adapted wave-functions

Wave-functions of various spin-dependent and spin-free methods are examined from the point of view of the requirements of the exclusion principle and the spin projection. It is shown that the “two-rowed” or “two-columned” requirements of the standard Young tableaux are necessary but not sufficient to replace the “antisymmetry” requirement of a wave-function and to be regarded as exclusion principle. The symmetry adapted wave-functions which are constructed from the matric basis e may not satisfy the exclusion principle and, hence, their usages are open to question. The appropriate symmetry adapted wave-functions which satisfied the exclusion principle are given for any pure spin state. We have also shown that the structure operators NP for bond functions are spin projectors in the Waller-Hartree double-antisymmetrized space only, and should not be used in the Hartree product space. Furthermore, if the corresponding matric operators PNP are used in the Hartree product space, then the wave-functions thus constructed may not be antisymmetric with respect to the permutations of indistinguishable electrons.     

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.     

Reply to comment on an unexpected symmetry of the Coulomb potential in the Debye-Smoluchowski equation

A properly modified inner boundary condition is used in an analysis of the conditional reaction probability, φ^*_R (r₀, t), using the Debye-Smoluchowski equation. It is shown that the Coulomb potential modifies the reaction velocity at the boundary by the addition of a drift term. φ^*_R (r₀, t) is identifical for attractive and repulsive cases if a constant term is added to the boundary velocity for the attractive potential.     

The molecular basis of vision: a reply to a comment

Electrical polarizability tensors and dipole noments for closed-shell molecules are calculated using a self-consistent field perturbation theory in which each molecular orbital is written as a linear combination of electric-field-dependent atomic functions Values calculated using different sets of atomic functions are compared with experiment and with previous theoretical work.     

Bases for the irreducible representations of O(3) symmetry adapted to a crystallographic point group

We construct bases for the irreducible representations of the rotation group O(3) which are symmetry adapted to a Crystallographic point group. We obtain explicit expressions for the cubic groups, which are valid for arbitrary values of the angular momentum quantum number l. Our method yields an efficient algorithm for both analytical and numerical work. An explicit formula for the multiplicities of an irreducible representation for the cubic groups in an arbitrary angular momentum term l is also derived.     

Molecule‐adapted basis sets optimized with a quantum Monte Carlo method

The Monte Carlo simulated annealing method is adapted to optimize correlated Gaussian‐type functions in nonrelativistic molecular environments. Starting from an atom‐centered atomic Gaussian basis set, the uncontracted functions are reoptimized in the molecular environments corresponding to the H₂O, CN^?, N₂, CO, BF, NO⁺, CO₂, and CS systems. These new molecular adapted basis sets are used to calculate total energies, harmonic vibrational frequencies, and equilibrium geometries at a correlated level of theory. The present methodology is a simple and effective way to improve molecular correlated wave functions, without the need to enlarge the molecular basis set. Additionally, this methodology can be used to generate hierarchical sequences of molecular basis sets with increasing size, which are relevant to establish complete basis set limits. © 2014 Wiley Periodicals, Inc.     

A fundamental connection between symmetry and spatial localization properties of basis sets

The problem of the compatibility between symmetry and localization properties of basis sets is addressed here. It is shown that both concepts are closely related from a fundamental point of view through the notion of invariance extent. This quantity is a functional that depends on the symmetry group and the basis set choices, and it is shown that all basis sets adapted in a general way to symmetry, i.e. induced from irreducible bases of the subgroups, are stationary points of it. In particular, the usual irreducible bases of the full group display a maximal invariance extent, while those symmetry-adapted basis sets that display a minimal value of this quantity feature in most cases the same symmetry properties as localized functions obtained by means of the Boys scheme. The most relevant conclusions are illustrated by means of simple molecular and periodic examples.     

Spectroscopy of sodium atom in liquid helium cluster: a symmetry adapted cluster-configuration interaction (SAC-CI) study

Excitation line shift of the principal resonance line of sodium atom embedded in liquid helium is estimated using symmetry adapted cluster-configuration interaction (SAC-CI) method with a reasonably good choice of basis set. The effect of the liquid surroundings is considered by taking only the nearest neighbor interaction using several model clusters of helium atoms with a centrally located isolated sodium atom. The transition wavelength of the ²P ← ²S line of Na is estimated both for the free Na and for the Na atom centrally located inside a model helium cluster. The shift depends obviously on the model cluster and the distance of the central Na atom from the surrounding He atom. With a physically realizable structure, SAC-CI calculation correctly predicts the theoretical results obtained previously on such estimates.     

On the evaluation of spin–orbit coupling matrix elements in a spin‐adapted basis

In the present article, we outline a simple scheme for generating configuration interaction matrix elements for spin–orbit interactions in molecules. The procedure leads to a close parallelism with spin‐free permutation‐group approaches. Unitary shift operators were successfully used on the orbital space to generate the matching permutations necessary to evaluate the required matrix elements. The procedure is adequately illustrated using examples. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 23–27, 1999     

A comment on a theory of electron affinities

Simons and Smith have calculated electron affinities and ionization potentials through third order in the electronic interaction by invoking the equations of motion method. An alternative analysis of the order is offered here by expanding the denominator in their pseudoeigenvalue equation to generate a Rayleigh-Schrödinger-like perturbation expansion. It is shown that considering the off-diagonal terms in the self-energy denominator leads to terms not considered by Simons and Smith which are third order in the electronic interaction as counted in our expansion. The connection with the diagram methods of Cederbaum is made.     

Fast assembly of Fock matrices utilising symmetry properties of the basis set

A method of assembling the elements of the Fock matrix is described which is a modification of that due to Dacre. Lists of symmetry equivalent one-elec