Atomic orbital basis sets for use with effective core potentials

Basis sets developed for use with effective core potentials describe pseudo‐orbitals rather than orbitals. The primitive Gaussian functions and the contraction coefficients in the basis set must therefore both describe the valence region effectively and allow the pseudo‐orbital to be small in the core region. The latter is particularly difficult using 1s primitive functions, which have their maxima at the nucleus. Several methods of choosing contraction coefficients are tried, and it is found that natural orbitals give the best results. The number and optimization of primitive functions are done following Dunning's correlation‐consistent procedure. Optimization of orbital exponents for larger atoms frequently results in coalescence of adjacent exponents; use of orbitals with higher principal quantum number is one alternative. Actinide atoms or ions provide the most difficult cases in that basis sets must be optimized for valence shells of different radial size simultaneously considering correlation energy and spin‐orbit energy. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 516–520, 2000     

Systematic Sequences of Geometric Relativistic Basis Sets. I:s- and p-Block Elements up to Xe

In this work a scheme for constructing systematic sequences of relativistic SCF basis sets at a reasonable computational cost is presented and applied to atoms of the s- and p-block up to Xe. This scheme, which couples simplex optimization and the use of geometric series given by four-term polynomial expressions for the logarithm of the exponents, allows for the construction of basis sets that exhibit very regular patterns of convergence to the numerical reference values of atomic total energies, spinor energies and radial expectation values. This regularity, together with the broad range of basis set sizes presented, enables these sets to be used as building blocks for basis sets applicable in both routine and benchmark relativistic calculations on atomic and molecular systems.Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.     

Photoionization cross sections of H2(+) and H2 with complex Gaussian-type basis functions optimized for the frequency-dependent polarizabilities

Within the framework of the complex basis function method, the photoionization cross sections of H(2)(+) and H(2) were calculated based on the variational principle for the frequency-dependent polarizabilities. In these calculations, complex orbital exponents of Gaussian-type basis functions for the final state continuum wavefunctions were fully optimized for each photon energy with the numerical Newton-Raphson method. In most cases, the use of only one or two complex Gaussian-type basis functions was enough to obtain excellent agreement with previous high precision calculations and available experimental results. However, there were a few cases, in which the use of complex basis functions having various angular momentum quantum numbers was crucial to obtain the accurate results. The behavior of the complex orbital exponents as a function of photon energy was discussed in relation to the scaling relation and the effective charge for photoelectron. The success of this method implies the effectiveness of the optimization of orbital exponents to reduce the number of basis functions and shows the possibility to calculate photoionization cross sections of general molecules using only Gaussian-type basis functions.     

Systematic Gaussian basis-set limit using completeness-optimized primitive sets. A case for magnetic properties

We discuss the connection between the completeness of a basis set, measured by the completeness profiles introduced by Chong (Can J Chem 1995, 73, 79) at a certain exponent interval, and the possibility of reproducing molecular properties that arise either in the region close to the atomic nuclei or in the valence region. We present a scheme for generating completeness-optimized Gaussian basis sets, in which a preselected range of exponents is covered to an arbitrary accuracy. This is done by requiring Gaussian functions, the exponents of which are selected without reference to the atomic structure, to span the range with completeness profile as close to unity as wanted with as few functions as possible. The initial exponent range can be chosen suitable for calculations of molecular energetics or other valence-like properties. By extending the exponent range, properties requiring augmentation of the basis at a given angular momentum value and/or in a given distance range from the nucleus may be straightforwardly and systematically treated. In this scheme a universal, element-independent exponent set is generated in an automated way. The relation of basis-set completeness and performance in the calculation of magnetizability, nuclear magnetic shielding, and spin-spin coupling is tested with the completeness-optimized primitive sets and literature basis sets.     

Random attractions and bifurcation for the classical Rayleigh–van der Pol equations with small noise

In this paper, a classical Rayleigh–van der Pol equations with small noise is considered. Using the asymptotic expansions of the Lyapunov exponents, invariant measure and Lyapunov’s direct, we investigate the stochastic bifurcation, rotation number, random limits cycle and attractor behavior of the random Rayleigh–van der Pol Equations in detail.     

Ab initio calculation of vibronic levels in the 2πu state of ph2

The results of an ab initio treatment of the Renner effect in the ²π_u state of PH₂ are reported The Born-Oppenheimer potential surfaces calculated by means of the MRD CI method are used A variational method, based on polynomial expansions of the molecular potentials and the kinetic-energy operator, is employed for calculation of vibronic levels.     

Correlation method for variance reduction of Monte Carlo integration in RS-HDMR

The High Dimensional Model Representation (HDMR) technique is a procedure for efficiently representing high-dimensional functions. A practical form of the technique, RS-HDMR, is based on randomly sampling the overall function and utilizing orthonormal polynomial expansions. The determination of expansion coefficients employs Monte Carlo integration, which controls the accuracy of RS-HDMR expansions. In this article, a correlation method is used to reduce the Monte Carlo integration error. The determination of the expansion coefficients becomes an iteration procedure, and the resultant RS-HDMR expansion has much better accuracy than that achieved by direct Monte Carlo integration. For an illustration in four dimensions a few hundred random samples are sufficient to construct an RS-HDMR expansion by the correlation method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.     

A new approach to solve the Schrodinger equation with an anharmonic sextic potential

In this study, the N-dimensional radial Schrodinger equation with an anharmonic sextic potential is solved by the extended Nikirov-Uranov method. We prove that the radial function can be factorised as the product between an exponential function and a polynomial function solution of the biconfluent Heun equation. The approach investigated in this article aims to be an alternative to other known methods of solving, as it has the advantage of dealing with simple, first-order differential and algebraic equations and avoiding numerous and laborious coordinate transformations and series expansions.

     

Plutonium valence state distributions

A method for ascertaining equilibrium valence state distributions of plutonium in acid solutions as a function of the plutonium oxidation number and the solution acidity is illustrated with an example. The method may be more practical for manual use than methods based upon polynomial equations. Mound Laboratory is operated by Monsanto Research Corporation for the U.S. Atomic Energy Commission under Contract No. AT-33-1-GEN-53.     

Polynomial Forms of Typical Interatomic Potential Functions

The use of polynomial functionals for describing two-body interactions in computational chemistry softwares has been surveyed and found to be prevalent. In this paper, Binomial and Maclaurin series expansions are used for expressing typical interatomic potential functions – such as Lennard-Jones, Morse, Rydberg and Buckingham potential – in a generic polynomial function, with the coefficients presented in a tabular format. Theoretical plots of these potential functions and their corresponding polynomial forms show increasing correlation with the order of polynomial, thereby validating the obtained polynomial’s coefficients. Conversely, a polynomial functional obtained by curve-fitting of experimental data can be converted into Morse, Rydberg and Buckingham potentials by using the generated table.     

Basis Set Orbital Relaxation in Atomic and Molecular Hydrogen Systems

Orbital relaxation (OR) amounts to variation of the orbital exponents in hydrogen molecules and ions relative to the exponents of the isolated atom; it is represented as the sum of the one- and two-center contributions depending on the effective atomic charge and on the presence of other atoms in the molecule. The procedure for isolating the contributions of the exponent includes treatment of the OR of hydrogen in a special set of neutral and charged atoms and molecules with certain multiplicities of their electronic states. Within the framework of the spin-unrestricted Hartree-Fock method, we found and discussed the optimal values of the exponents of the basis orbitals of hydrogen atoms and molecules using the minimal split valence-shell basis set, the basis set that includes the polarization function, and the expanded set of grouped natural orbitals. A simple energy model is suggested for OR. Expressions are derived for evaluating the exponents of the relaxed orbitals in hydrogen-containing systems.Original Russian Text Copyright © 2004 by A. I. Ermakov, A. E. Merkulov, A. A. Svechnikova, and V. V. Belousov__________Translated from Zhurnal Strukturnoi Khimii, Vol. 45, No. 6, pp. 973–978, November–December, 2004.     

Double even tempering of orbital exponents: Application to Roothaan-Hartree-Fock calculations for He through Xe in Slater-type basis sets

Summary Double even tempering (DET) of orbital exponents is proposed as a useful generalization of even tempering (ET). The DET scheme uses two sets of basis functions for each angular momentum. The two sets have different principal quantum numbers and their exponents are generated by two different geometric sequences. Roothaan-Hartree-Fock (RHF) calculations on the atoms from He through Xe using both ET and DET Slater-type basis sets of the same size are carried out to demonstrate the substantial improvement offered by the DET scheme. The DET scheme reduces the maximum deviation of the RHF energies relative to the Hartree-Fock limit from 1.4 to 0.3 millihartrees.     

Method of approximation of dependence of isobaric heat capacity on temperature

A method of approximating temperature dependence of heat capacity has been developed on the basis of a power polynomial in variables with different exponents in combination with the Nelder-Mead simplex method. For especially complicated dependences, the method makes it possible to match polynomials at junction points with infinitesimal deviation. The method provides high reproducibility and correct interpolation of empirical and semiempirical data.     

用改进的基向量法重建光谱反射比

如何从已知物体表面颜色的三刺激值及其照明条件和观察条件准确地重建其光谱反射比,是彩色图像研究领域尚待解决的一个重要课题.本文首先介绍了两种前人提出的方法,即“伪逆矩阵法”和“Wiener方法”,然后,分析了这两种方法的优缺点,指出了共同存在的问题,即所重建的光谱反射系数有可能超出实际可能的范围(0,1),进而提出了一种新的改进的基向量法.最后,文中还给出此改进的算法和上述两种算法的特性模拟仿真.从仿真的结果中可以看出,改进的算法不仅在精度上较上述两种算法更精确,而且能够保证所重建的光谱反射比能够满足其实际可能的范围(0,1).因而,用本文中所给出的改进的基向量法重建物体表面的光谱反射比,较上述两种前人提出的算法更能满足工业实际需要.     

Angular dependence of Gaussian-Lobe orbitals. I. Analysis of standard p- and d-orbitals

Multipole expansions of Gaussian-lobe atomic orbitals around their centers are theoretically investigated in order to study the exact angular dependence of such functions. Analytical expressions of the multipole coefficients are derived for standard lobe orbitals. It is shown that the average-square values of multipole components are related to a unique orbital parameter λ. The numerical values of p- and d-components are given for selected λ and the choice of this parameter is discussed on the basis of symmetry and computational arguments. The transferability of optimized atomic exponents from harmonic (or Cartesian) functions to lobe functions is established so that the possibility of applying the Gaussian-lobe orbital approach in chemical studies is greatly extended.