The use of improved atomic orbitals in the evaluation of zero-field splitting integrals

The effect of improving the 2p _z-atomic orbital representation on the values of molecular zero-field splitting integrals is assessed on the example of the two-center Coulomb integral involving the (r ²–3z²)/r⁵ operator in the cases of nitrogen and carbon. The results suggest that the use of the classical Slater orbital or its Gaussian equivalent may be misleading.

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Zusammenfassung Ein spezielles, bei der Berechnung der molekularen Nullfeldaufspaltung von Tripletts auftretendes Zweizentrenintegral wird in AbhÄngigkeit von der Güte der 2p _z-Orbitale des Kohlenstoff- bzw. Stickstoffatoms untersucht. Offenbar sind die üblichen Slaterorbitale und deren Approximation durch Gau\funktionen ungeeignet.

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Résumé On a étudié l'effet de l'amélioration de la base d'orbitales atomiques 2p _z sur les valeurs des intégrales qui interviennent dans le calcul de la séparation des niveaux d'un triplet moléculaire en l'absence de champ magnétique. L'utilisation de l'orbitale de Slater classique ou de son équivalent en Gaussienne peut conduire à des conclusions appréciablement erronées.

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This work was supported by grant GM 12289-02 of the United States Public Health Service (National Institute of General Medical Sciences).     

The np Rydberg series of boron monohydride: l-uncoupling and its evolution for intermediate principal quantum numbers n = 4 to n = 11

Using optical-optical-optical triple-resonance spectroscopy, we assign rotational levels with N = 0-5 in the vibrationless, lower-n, p Rydberg states of (11)BH. We apply the Hill and Van Vleck formulation for energy levels with l = 1 in a Hund's case intermediate between (b) and (d) to gauge the energy separating (1)Π and (1)Σ(+) states with zero rotation for n = 4-11. This energy difference, A(l, ξ), represents the strength of the coupling, ξ, between the electron orbital angular momentum, l, and the internuclear axis, which determines the Λ-splitting constant, q(0). The np series exhibits a large q(0) that increases monotonically with n to reach a magnitude similar to the rotational constant, B(0), by n = 9. For higher principal quantum numbers, Λ ceases to be a good quantum number, and l-uncoupling becomes virtually complete for n > 10.     

The atomic number revolution in chemistry: a Kuhnian analysis

This paper argues that the field of chemistry underwent a significant change of theory in the early twentieth century, when atomic number replaced atomic weight as the principle for ordering and identifying the chemical elements. It is a classic case of a Kuhnian revolution. In the process of addressing anomalies, chemists who were trained to see elements as defined by their atomic weight discovered that their theoretical assumptions were impediments to understanding the chemical world. The only way to normalize the anomalies was to introduce new concepts, and a new conceptual understanding of what it is to be an element. In the process of making these changes, a new scientific lexicon emerged, one that took atomic number to be the defining feature of a chemical element.     

Unified treatment for the evaluation of arbitrary multielectron multicenter molecular integrals over Slater‐type orbitals with noninteger principal quantum numbers

The expansion formula has been presented for Slater‐type orbitals with noninteger principal quantum numbers (noninteger n‐STOs), which involves conventional STOs (integer n‐STOs) with the same center. By the use of this expansion formula, arbitrary multielectron multicenter molecular integrals over noninteger n‐STOs are expressed in terms of counterpart integrals over integer n‐STOs with a combined infinite series formula. The convergence of the method is tested for two‐center overlap, nuclear attraction, and two‐electron one‐center integrals, due to the scarcity of the literature, and fair uniform convergence and great numerical stability under wide changes in molecular parameters is achieved. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Surface Green's function calculations: a nonrecursive scheme with an infinite number of principal layers

A novel computational method for a surface Green's function matrix is introduced for the calculation of electrical current in molecular wires. The proposed nonrecursive approach includes an infinite number of principal layers and yields the second-order matrix equation for the transformed Green's function matrix. The solution is found by the direct diagonalization of the auxiliary matrix without any iteration process. As soon as complex roots of the auxiliary matrix (approximately GS) are calculated, the gaps and the bands in the surface electronic structure are found. It is shown that the solution of a second-order matrix equation determines the spectral density matrix, that is, the density of states for noninteracting electrons. Single and double principal layer models are studied both analytically and numerically. The energy interval for nonvanishing spectral matrices is determined. This method is applicable to matrices of any rank.     

Evaluation of two‐center overlap and nuclear attraction integrals over Slater‐type orbitals with integer and noninteger principal quantum numbers

A general formula has been established for the expansion of the product of two normalized associated Legendre functions centered on the nuclei a and b. This formula has been utilized for the evaluation of two‐center overlap and nuclear attraction integrals over Slater‐type orbitals (STOs) with integer and noninteger principal quantum numbers. The formulas given in this study for the evaluation of two‐center overlap and nuclear attraction integrals show good rate of convergence and great numerical stability under wide range of quantum numbers, orbital exponents, and internuclear distances. © 2001 Wiley Periodicals, Inc. Int J Quantum Chem, 2001     

The topology of the Ehrenfest force density revisited. A different perspective based on Slater‐type orbitals

The topology of the Ehrenfest force density was studied with Slater‐type orbitals (STO). At larger distances from the nuclei, STOs generate similar artefacts as noticed before with Gaussian‐type orbitals. The topology of the Ehrenfest force density was found to be mainly homeomorphic with the topology of the electron density. For the first time, reliable integrations of several properties over force density atomic basins were performed successfully. Integration of the electron density of a number of hydrides, fluorides, and chlorides of first row elements over force density basins indicate substantial differences between the partial charges of the atoms as compared with those obtained from electron density basins. Calculations on saturated hydrocarbons confirm that the electronegativity of carbon atoms increases with increasing geometrical strain. Atomic interaction lines are observed to exist in the Ehrenfest force density between the hydrogen atoms of several so‐called “congested” molecules, and also in some inclusion complexes of alkanes with helium. However, interaction lines are lacking in several other controversial cases. © 2015 Wiley Periodicals, Inc.     

The ground state tunneling splitting of malonaldehyde: accurate full dimensional quantum dynamics calculations

Benchmark calculations of the tunneling splitting in malonaldehyde using the full dimensional potential proposed by Yagi et al. are reported. Two exact quantum dynamics methods are used: the multiconfigurational time-dependent Hartree (MCTDH) approach and the diffusion Monte Carlo based projection operator imaginary time spectral evolution (POITSE) method. A ground state tunneling splitting of 25.7+/-0.3 cm(-1) is calculated using POITSE. The MCTDH computation yields 25 cm(-1) converged to about 10% accuracy. These rigorous results are used to evaluate the accuracy of approximate dynamical approaches, e.g., the instanton theory.     

The ground state tunneling splitting and the zero point energy of malonaldehyde: a quantum Monte Carlo determination

Quantum dynamics calculations of the ground state tunneling splitting and of the zero point energy of malonaldehyde on the full dimensional potential energy surface proposed by Yagi et al. [J. Chem. Phys. 1154, 10647 (2001)] are reported. The exact diffusion Monte Carlo and the projection operator imaginary time spectral evolution methods are used to compute accurate benchmark results for this 21-dimensional ab initio potential energy surface. A tunneling splitting of 25.7+/-0.3 cm-1 is obtained, and the vibrational ground state energy is found to be 15 122+/-4 cm-1. Isotopic substitution of the tunneling hydrogen modifies the tunneling splitting down to 3.21+/-0.09 cm-1 and the vibrational ground state energy to 14 385+/-2 cm-1. The computed tunneling splittings are slightly higher than the experimental values as expected from the potential energy surface which slightly underestimates the barrier height, and they are slightly lower than the results from the instanton theory obtained using the same potential energy surface.     

Comparison of a quantum random number generator with pseudorandom number generators for their use in molecular Monte Carlo simulations

Four pseudorandom number generators were compared with a physical, quantum‐based random number generator using the NIST suite of statistical tests, which only the quantum‐based random number generator could successfully pass. We then measured the effect of the five random number generators on various calculated properties in different Markov‐chain Monte Carlo simulations. Two types of systems were tested: conformational sampling of a small molecule in aqueous solution and liquid methanol under constant temperature and pressure. The results show that poor quality pseudorandom number generators produce results that deviate significantly from those obtained with the quantum‐based random number generator, particularly in the case of the small molecule in aqueous solution setup. In contrast, the widely used Mersenne Twister pseudorandom generator and a 64‐bit Linear Congruential Generator with a scrambler produce results that are statistically indistinguishable from those obtained with the quantum‐based random number generator. © 2017 Wiley Periodicals, Inc.     

Implementation of screened hybrid density functional for periodic systems with numerical atomic orbitals: basis function fitting and integral screening

We present an efficient O(N) implementation of screened hybrid density functional for periodic systems with numerical atomic orbitals (NAOs). NAOs of valence electrons are fitted with gaussian-type orbitals, which is convenient for the calculation of electron repulsion integrals and the construction of Hartree-Fock exchange matrix elements. All other parts of Hamiltonian matrix elements are constructed directly with NAOs. The strict locality of NAOs is adopted as an efficient two-electron integral screening technique to speed up calculations.     

A simple model of atomic collision in the presence of an intense radiation field: A comparison test for gauge invariant vs. conventional quantum transition probabilities

A simple model mimicking the collision between two atomic partners in the presence of an intense radiation field is investigated according to a two-state, non perturbative approach, to put in quantitative evidence inadequacies in both transition probability and cross section arising from the use of transition amplitudes which are not gauge-invariant.     

Gaussian vs. Slater representations of d orbitals: An information theoretic appraisal based on both position and momentum space properties

A detailed appraisal of Gaussian-type orbital (GTO) and Slater-type orbital (STO) expansions of 3d orbitals is carried out for the ²S state of copper—a case that should be maximally unfavorable for STOs. The appraisal is based on a wide variety of both position and momentum space properties and utilizes an information theoretic quality assessment technique. It is found that GTO expansions are not as useful as STO expansions for the prediction of 〈p⁸〉, 〈p⁷〉, and 〈r^?6〉 because these properties probe the functional deficiencies of GTOs at small r and large p. On the other hand, GTO expansions can predict accurate values of large r properties like 〈r⁸〉 despite the fact that their position space asymptotic decay is too fast. Unlike the case of s orbitals in helium, there does not seem to be any consistent ordering between accuracy in position space and accuracy in momentum space. The quality measures are found to be very useful for pinpointing the deficiencies of various expansions. This information enables us to construct easily a new GTO and a new STO expansion that are more accurate than any of the others in the literature. It is suggested that one STO is worth no more than two GTOs in the case of d orbitals.     

Unbiased expectation values from diffusion quantum Monte Carlo simulations with a fixed number of walkers

We append forward walking to a diffusion Monte Carlo algorithm which maintains a fixed number of walkers. This removes the importance sampling bias of expectation values of operators which do not commute with the Hamiltonian. We demonstrate the effectiveness of this approach by employing three importance sampling functions for the hydrogen atom ground state, two very crude. We estimate moments of the electron-nuclear distance, static polarizabilities, and high-order hyperpolarizabilites up to the fourth power in the electric field, where no use is made of the finite field approximation. The results agree with the analytical values, with a statistical error which increases substantially with decreasing overlap of the guiding function with the exact wave function.     

Energy decomposition in the topological theory of atoms in molecules and in the linear combination of atomic orbitals formalism: a note

 It is shown that the molecular energy calculated at the self-consistent-field level can be strictly expressed as a sum of one- and two-atom energy components in the framework of Bader's topological theory of atoms in molecules (AIM). The expressions of our recent “chemical energy component analysis” can be obtained from the AIM ones as some linear combination of atomic orbitals mappings of the integrations over the atomic basins. Received: 15 June 2000 / Accepted: 4 October 2000 / Published online: 19 January 2001