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.     

Effects of global orbital cutoff value and numerical basis set size on accuracies of theoretical atomization energies

Numerical basis sets are known for their rapid convergence in density functional theory calculations. The selections of global orbital cutoff values and numerical basis set sizes are important to the computational accuracies and efficiencies. In this study, the effects of global orbital cutoff values and numerical basis set sizes on the theoretical atomization energies (D ₀) were investigated using density functional theory with the generalized gradient approximation. Our results on the total energies of seven atoms and D ₀ of a set of 44 molecules demonstrate that the numerical orbital cutoff value should be larger than 6.5 Å to get the converged energetic properties. Through comparing the D ₀ of these 44 molecules obtained by using four kinds of different numerical basis sets, DN, DND, DNP, and TNP, it demonstrates that the DNP basis set is good enough to predict accurate D ₀ with affordable computational cost.     

Sturmian basis matrix solution of vibrational potentials

An efficient basis set for use with the Harris, Engerholm and Gwinn solution of one-dimensional vibrational potentials has been derived from centrosymmetric sturmian Laguerre functions using the l = 0 condition. Six-figure accuracy is achieved for n bound levels of the Morse and Fues potentials using 4n sturmian functions. With simple formulas and sparse arrays, convergence to all bound levels is achieved.     

Physical nature of the chemical bond

The effect of such variables as size of basis set, number of S and P type functions in the basis set, size of initial exponent in the basis set, size of multiplicative factor in the geometrical progression of exponents, and interatomic distance has been examined by a self-consistent-field calculation with Gaussian orbitals on HeH⁺. A quasi-optimized wave function has been obtained by allowing only the initial exponent and multiplicative factor to vary in the optimization process.     

Optimum Gaussian basis set for the Bromine atom. Ab initio calculations on the HBr molecule

An optimum (15 ^s 12 ^p 2 ^d ) Gaussian basis set was obtained for the Bromine atom by minimizing the open shell energy functional. In the minimization procedure the method of conjugate gradients was applied. The optimum (15 ^s 12 ^p 2 ^d ) basis set was contracted to an [8 ^s 6 ^p 2 ^d ] “double zeta” quality basis set and this contracted set was tested on the HBr molecule.     

Investigations with the finite element method. II. The collinear F + H2 reaction

We are investigating systematically the use of the finite element method (FEM) for solving the Schrödinger equation. The present work is devoted to the calculation of vibrational transition probabilities for the collinear reactive system F + H₂. The calculations are fully two-dimensional and the results are compared with the conventional basis set expansion methods using the R-matrix or S-matrix propagation. Extensive analysis of FEM on the vector computer Cyber 205 was made and a vector code for the efficient use in two dimensions was developed, so that in the near future applications even in three dimensions will be possible. The details of our FEM calculations are the following: The integration area was discretized into triangles where quadratic polynomials for the local wavefunction were defined. Convergent results can be reached with this simple ansatz with roughly 10000 grid points.     

Some aspects of parallelization of two-electron integrals in molecular orbital programs

Evaluation of two-electron integrals forms a substantial part of the CPU time for any ab initio molecular orbital program. This part of the package, “MICROMOL”, is parallelized. However, this parallelization leads to only sublinear speedups (typically 3 on a 4-node machine). In view of these results, the task of development of an efficient program for two-electron integrals suitable for the parallel environment has been taken up. The program is written in FORTRAN considering specific symmetry features and application of rigorous bounds. This program is further parallelized with a good load balancing strategy. The molecules used as the test cases are: trans-butadiene, benzene, nitrobenzene, naphtalene and cytosine, with 3G and 4–31G basis sets. The results indicate that the parallel version of this program gives a typical speedup of 3.6 for a 3G basis set and approximately 3.4 for a 4–31G basis set for all the molecules tested. The sequential version of this program is ～1.2 times faster than the sequential version of MICROMOL, whereas the parallel version is ～1.4 times faster than the parallelized MICROMOL.     

The error due to neglecting the core spin–orbit splitting in valence ZORA calculations with frozen core approximation and its elimination

In valence zeroth-order regular approximation (ZORA) calculations with frozen core approximation, when the basis set optimized to the related scalar relativistic ZORA calculations is used, neglecting the core spin–orbit splitting may result in additional basis set truncation errors. It is found that the error is negligible for most elements except the 6p-block elements. When the basis set is extended by a p-type STO function put on the 6p element atoms with the ζ value proper to 5p_1/2 orbitals, the error can be reduced to be negligible. The calculated atomic properties related to valence orbitals can be improved greatly by use of this extended basis set. The frozen core approximation calculations of some molecules containing Tl, Pb and Bi with closed shells show that neglecting the core spin–orbit splitting only slightly affects the calculated bond lengths and bond energies, and the calculated molecular property can also be improved slightly by use of the extended basis sets.     

Analysis of approximations and errors in equations of motion method calculations

We analyze a number of fundamental questions associated with the use of a finite one-particle orbital basis in equations of motion (EOM) method calculations of excitation energies etc., of atomic and molecular systems. This approximation yields an approximate n_e-electron ground state and say, N excited states, while there are (N + 1)² different possible basis operators for EOM calculations. We show that sets of at most 2N basis operators can contribute to the EOM calculations. Any set of 2N basis operators, satisfying certain conditions, provides the exact EOM energies which are equivalent to complete configuration interaction results within the same orbital basis. We investigate the use of particle-particle shifting operators which are not employed in EOM calculations in model calculations on He with operator bases smaller than the complete 2V to consider the convergence of the expansion. The dependence of EOM calculations on the quality of the approximate ground state wavefunction is studied through calculations for Be where additional support is provided for the frequent need for multiconfigurational zeroth order reference functions (as corrected perturbatively). Excited state EOM wavefunctions from EOM calculations are shown to not necessarily be orthogonal to either the exact or approximate ground state wavefunction, suggesting implications in the use of EOM methods to evaluate excited state properties. The He and Be examples and a simple two-level problem are also utilized to illustrate questions concerning the use of the EOM equations to obtain an iteratively improved ground state wavefunction.     

Ab initio SCF calculations on hydrogen bonded cresol isomers

Ab initio GAUSSIAN 80 calculations with two different basis sets (STO-3G and 4–31 G*) were performed on hydrogen bonded cresol isomers for comparison with experimental data from free jet fluorescence excitation spectroscopy. Form-cresol, the calculated barriers for hindered internal rotation of the OH-group and the CH₃-group are in good agreement with experiment. The calculations show the trans-linear configuration ofp-cresol·B-clusters (B = H₂O, CH₃OH) to be more stable than the all-planar configuration. This agrees with CI calculations and microwave spectroscopic investigations of the water dimer. Calculations of both the intermolecular stretch and bend frequencies ofp-cresol·B-clusters show little dependence on the all-planar or trans-linear configuration but a strong dependence on the choice of the basis set. With the minimal basis set STO-3G, the vibrational energies are generally too high. The agreement between the calculated vibrational frequencies from the 4–31 G* basis set and the experimental values is fair.     

The coupling algebra of trigonally subduced crystal field eigenvectors

The general theory of subduction of eigenvectors between infinite groups is used to derive a finite group subduction operator and define the corresponding subduction coefficients. The coupling behaviour of these subduced eigenvectors can then be described in terms of 3 Γ symbols. These symbols, defined only in relation to complex basis sets are all fully real and have all phases fixed by the subduction operator. They differ from V coefficients in two phase relationships and have the advantage, unlike V coefficients, of retaining all the symmetry properties and selection rules of Wigner 3-j symbols. Appropriate label systems which render these properties in terms of simple algebras are given for all quantizing axes available in O _h . The specific set of 3Γ symbols for each quantization is determined by the orientation of the coordinate axes in the Hamiltonian. The four possible orientations for trigonal quantization are examined and the operator chosen which produces eigenvectors with conventional conjugate phases and a fully real set of 3Γ symbols.     

Linear model for estimating the entropy of formation of aqueous anions

Recently, two of us published a series of articles in which the entropies of formation, TΔ_f S ^o, of different classes of aqueous species were estimated. A set of equations were presented wherein only simple parameters were derived from the ionic stoichiometry. In the current paper new equations of comparable precision are presented using a smaller set of simple parameters.     

Ground state geometry of cinso

The ground state geometry for ClNSO is investigated using ab initio calculations in a moderately large basis set. Variation of the dihedral angle yields a planar syn geometry. The geometry is discussed on the basis of a population analysis.     

Ab initio calculations of the equilibrium geometry of cis and trans methylglyoxal using the force method

Ab initio calculations of the equilibrium ground-state conformation of trans-methylglyoxal, CH₃COCHO, have been carried out using the gradient method and a 4-3 IG basis set. The results are compared with experiment and with a theoretical geometry computed using an STO-3G basis set, and the more usual stepwise optimization of the parameters. The molecular orbital energies, dipole moment and ionization energy are calulated. Calculations are also carried out on the cis-isomer and the barrier to cis-trans isomerization is obtained.     

CI method for the study of general molecular potentials

A method of configuration interaction designed for general molecular potentials is outlined. The technique employed to arrive at a symmetrized multideterminantal basis for such calculations relies heavily on certain properties of Abelian groups; in particular the M _L quantum number commonly employed in atomic structure calculations is replaced in the general molecular case by the index p which labels irreducible representations of some appropriate Abelian group. Formation of the desired symmetrized linear combinations of determinants is thereupon accomplished solely by means of a series of simple diagonalizations, a procedure which insures both the linear independence and the orthonormality of the resultant basis set. CI treatments involving the C₄H₄ isomer tetrahedrane (T _d point group) and the linear nitrous oxide NNO molecule are considered in some detail with a view toward illustrating the use of the techniques described herein. Finally, a survey of a series of recent calculations utilizing the CI method is made and it is concluded from these results that the effects of such treatment vary strongly from one system to another, depending in a very specific manner upon the individual characteristics of a given molecule.     

Atomic core-ionization energies; approximately piecewise-linear and linear relationships

In the generalized Sturmian method, solutions to the many-particle Schrödinger equation are built up from isoenergetic sets of solutions to an approximate Schrödinger equation with a weighted potential \({\beta_\nu \mathsf{V}_0({\bf x})}\). The weighting factors β _ν are chosen in such a way as to make all of the members of the basis set correspond to the energy of the state being represented. In this paper we apply the method to core ionization in atoms and atomic ions, using a basis where \({\mathsf{V}_0({\bf x})}\) is chosen to be the nuclear attraction potential. We make use of a large-Z approximation, which leads to extremely simple closed-form expressions not only for energies, but also for values of the electronic potential at the nucleus. The method predicts approximately piecewise linear dependence of the core-ionization energies on the number of electrons N for isonuclear series, and an approximately linear dependence of ΔE ? Z ²/2 on the nuclear charge Z for isoelectronic series.     

Application of the Seth-Paul-Van Duyse equation—I : A simple relation defining new X+(R) substituent constants on the basis of electrophilic σ+ constants

The Seth-Paul-Van Duyse equation (SPVDE) correlating the CO stretching frequencies of a great number of R₁COR₂ molecules with X (R) substituent constants has been reinvestigated. A simple empirical relation defining new X⁺(R) constants on the basis of electrophilic σ constants has been derived: X⁺ (R) = 0·238 σ⁺ + 1·077. The three earlier reported relationships between the X(R) constants and Hammett σ values have been replaced by this single relationships and a set of new X⁺(R) constants has been calculated. The new X⁺(R) constants applied to 287 R₁COR₂ compounds containing the para- or meta-substituted benzene rings fit very well the SPVDE. It has been possible to extend the SPVDE to various aromatic systems.