Evaluation of integrals over STOs on different centers and the complementary convergence characteristics of ellipsoidal‐coordinate and zeta‐function expansions

One‐electron integrals over three centers and two‐electron integrals over two centers, involving Slater‐type orbitals (STOs), can be evaluated using either an infinite expansion for 1/r₁₂ within an ellipsoidal‐coordinate system or by employing a one‐center expansion in spherical‐harmonic and zeta‐function products. It is shown that the convergence characteristics of both methods are complimentary and that they must both be used if STOs are to be used as basis functions in ab initio calculations. To date, reports dealing with STO integration strategies have dealt exclusively with one method or the other. While the ellipsoidal method is faster, it does not always converge to a satisfactory degree of precision. The zeta‐function method, however, offers reliability at the expense of speed. Both procedures are described and the results of some sample calculation presented. Possible applications for the procedures are also discussed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 1–13, 1999     

High‐order symplectic integration in quasi‐classical trajectory simulation: Case study for O(1D) + H2

Classical trajectory calculations for the O(¹D) + H₂ reaction system are employed to assess the effectiveness of the symplectic integrators. The sixth‐order symplectic integrator has been found to be the most suitable method for the quasi‐classical trajectory calculation of a long‐lived complex‐forming reaction system. In comparison with the traditional fourth‐order Runge–Kutta initialized fourth‐order Admas–Moulton–Hamming predictor‐corrector integrator (RK4‐AMH4), the sixth‐order symplectic integrator is six times less computationally expensive and exhibits better energy conservation. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

A new nonempirical tuning scheme with single self‐consistent field calculation: Comparison with global and IP‐tuned range‐separated functional

System‐dependent nonempirical tuning of range‐separated functional provides a way to minimize the delocalization error of the system. However, existing nonempirical tuning method requires the computation of several ΔSCF calculations to determine the optimal μ value. In this article, we have defined a scheme to evaluate the optimal μ value with single self‐consistent field calculation. Our method is based on the evaluation of the spherically symmetric average Electron localization function (ELF) region. According to this scheme, the radius of the spherically symmetric average ELF region gives is a measure of the distance at which the long‐range part of the range‐separated functional becomes dominant. Numerical results indicate that our method improves the reproduction of HOMO energies and HOMO‐LUMO gap in comparison to global and IP‐tuned range‐separated functional. Moreover, in case of HOMO energies, maximum error of the ELF‐tuned functional is considerably smaller than the global and IP‐tuned functional. Furthermore, our method gives considerably smaller deviation of HOMO energies from ΔSCF IP than global range‐separated functional. © 2017 Wiley Periodicals, Inc.     

Non-integer Slater orbital calculations

In order to calculate the one- and two-electron, two-center integrals over non-integer n Slater type orbitals, use is made of elliptical coordinates for the monoelectronic, hybrid, and Coulomb integrals. For the exchange integrals, the atomic orbitals are translated to a common center. The final integration is performed by Gaussian quadrature.As an example, an SCF ab initio calculation is performed for the LiH molecule, both with integer and non-integer principal quantum number.     

Comparison of two simulation methods to compute solvation free energies and partition coefficients

The thermodynamic integration (TI) and expanded ensemble (EE) methods are used here to calculate the hydration free energy in water, the solvation free energy in 1‐octanol, and the octanol‐water partition coefficient for a six compounds of varying functionality using the optimized potentials for liquid simulations (OPLS) all‐atom (AA) force field parameters and atomic charges. Both methods use the molecular dynamics algorithm as a primary component of the simulation protocol, and both have found wide applications in fields such as the calculation of activity coefficients, phase behavior, and partition coefficients. Both methods result in solvation free energies and 1‐octanol/water partition coefficients with average absolute deviations (AAD) from experimental data to within 4 kJ/mol and 0.5 log units, respectively. Here, we find that in simulations the OPLS‐AA force field parameters (with fixed charges) can reproduce solvation free energies of solutes in 1‐octanol with AAD of about half that for the solute hydration free energies using a extended simple point charge (SPC/E) model of water. The computational efficiency of the two simulation methods are compared based on the time (in nanoseconds) required to obtain similar standard deviations in the solvation free energies and 1‐octanol/water partition coefficients. By this analysis, the EE method is found to be a factor of nine more efficient than the TI algorithm. For both methods, solvation free energy calculations in 1‐octanol consume roughly an order of magnitude more CPU hours than the hydration free energy calculations. © 2012 Wiley Periodicals, Inc.     

Development of Experiment and Theory to Detect and Predict Ligand Phase Separation on Silver Nanoparticles

MALDI mass‐spectrometry measurements are combined with self‐consistent mean‐field (SCF) calculations to detect and predict ligand phase separation on Ag nanoparticles. The experimental and theoretical techniques complement each other by enabling quantification of the nearest‐neighbor distribution of a ligand mixture in a monolayer shell. By tracking a characteristic metallic fragment family, analysis of a MALDI spectrum produces a frequency distribution corresponding to specific ligand patterning. Inherent to the SCF calculation is the enumeration of local interactions that dictate ligand assembly. Interweaving MALDI and SCF facilitates a comparison between the experimentally and theoretically derived frequency distributions as well as their deviation from a well‐mixed state. Thus, we combine these techniques to detect and predict phase separation in monolayers that mix uniformly or experience varying degrees of de‐mixing, including microphase separation and stripe formation. Definition of MALDI removed as this is a commonly recognized technique.     

Fractional occupation numbers and spin density functional calculations of degenerate systems

We implemented ab initio self‐consistent field (SCF) fractional occupation numbers (FON) calculation with Dunlap's interpolation scheme for the twisted ethylene, which is a prototype molecule of a σ–π biradical system. The calculational results are compared with those of complete‐active‐space (CAS) SCF and spin‐unrestricted Kohn–Sham (UKS) calculations on potential surfaces, occupation numbers of natural orbitals, and correlation entropies. It was found that the UKS methods gave similar results to CASSCF, while the FON solutions appeared in only the nearly complete degenerate region. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 93: 317–323, 2003     

Atomic and molecular cubature grids

This article presents cubature grids of the Gaussian type that are adapted for the purpose of wave function calculation on atoms and molecules. The problems of the singularity at the nucleus, the derivatives in the kinetic energy, and the presence of two‐electron integrals are shown to be resolved. Each grid has a definite degree of accuracy so that it reproduces the exact values of all the integrals in a defined class. Seventh‐degree accuracy can be obtained from a grid of 143 nodes. The grids are applied, as simple illustrations, to well‐known self‐consistent field (SCF) calculations on helium. Grids for homonuclear diatomics are also discussed and an illustrative application given to a homonuclear diatomic molecule. A comparison between a molecular grid and the union of two unmodified atomic grids shows that overlaps and distortions in weights can occur to the extent that this is not practical. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Evaluation of atomic pressure in the multiple time‐step integration algorithm

In molecular dynamics (MD) calculations, reduction in calculation time per MD loop is essential. A multiple time‐step (MTS) integration algorithm, the RESPA (Tuckerman and Berne, J. Chem. Phys. 1992, 97, 1990–2001), enables reductions in calculation time by decreasing the frequency of time‐consuming long‐range interaction calculations. However, the RESPA MTS algorithm involves uncertainties in evaluating the atomic interaction‐based pressure (i.e., atomic pressure) of systems with and without holonomic constraints. It is not clear which intermediate forces and constraint forces in the MTS integration procedure should be used to calculate the atomic pressure. In this article, we propose a series of equations to evaluate the atomic pressure in the RESPA MTS integration procedure on the basis of its equivalence to the Velocity‐Verlet integration procedure with a single time step (STS). The equations guarantee time‐reversibility even for the system with holonomic constrants. Furthermore, we generalize the equations to both (i) arbitrary number of inner time steps and (ii) arbitrary number of force components (RESPA levels). The atomic pressure calculated by our equations with the MTS integration shows excellent agreement with the reference value with the STS, whereas pressures calculated using the conventional ad hoc equations deviated from it. Our equations can be extended straightforwardly to the MTS integration algorithm for the isothermal NVT and isothermal–isobaric NPT ensembles. © 2017 Wiley Periodicals, Inc.     

Efficient evaluation of Coulomb integrals in a mixed Gaussian and plane‐wave basis

An efficient way of calculation is presented for matrix elements between two plane waves interacting with a molecular Coulombic field. In concurrence with the absolute value of the momentum transfer vector, K = k₁ ? k ₂ , the most effective method of calculation is selected. The case of K = 0 requires special treatment. For 0 < |K| ≤ 0.3, it is profitable to evaluate the integrals by means of the multipole expansion, and for |K| > 0.3 the density fitting can be applied. For the large |K| the electronic part of the integral is much smaller than the nuclear part and the integral may be approximated by the nuclear contribution only. Some examples for testing the accuracy and time saved are presented. The primary purpose of this paper is to accelerate electron scattering calculations, but it also may be profitable for the electronic structure theory in attempts to use mixed Gaussian and plane‐wave basis sets. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Performance of the elongation method with larger basis sets

The elongation method proposed by Imamura serves as a theoretical model for polymerization processes. It can now be used together with larger basis sets, Hartree–Fock and density functional methods from the Gaussian 94 package with direct self‐consistent field (SCF). This allows electronic structure calculation of elongating clusters with an efficiency superior to full cluster calculations and a precision superior to previous versions of our elongation method. Performance and accuracy compared with full cluster calculations on a regular polymer using the BLYP/6‐31G(d, p) method. Interaction energies of water and hydrogen fluoride polymers of increasing length are compared between HF, BLYP methods and 4‐31G, 6‐31G(d, p) basis sets: Diffuse and polarization functions have a large influence on the interaction energy on both polymers. Local density of states are calculated for different cluster lengths. They are in good agreement with full cluster calculations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 35–47, 1999     

CAS–SCF and density functional calculations of second hyperpolarizabilities for a nitronyl nitroxide radical

In this study, we investigated a multireference‐based nondynamical electron correlation dependence of the second hyperpolarizability, γ. We performed complete active space self‐consistent‐field (CAS–SCF) calculations including the effect of π electrons. Although the sign of γ obtained by CAS–SCF calculation was found to be negative, the magnitude of γ was found to be much larger than that at the CCSD(T) level. We also applied density functional (DF) methods with different exchange–correlation functionals (BLYP and B3LYP) to the calculation of γ. Although the B3LYP method is found to provide negative γ, its magnitude is much smaller than that at the CCSD(T) level. This feature is discussed in detail by using γ density plots. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 329–336, 1999     

A full‐pivoting algorithm for the Cholesky decomposition of two‐electron repulsion and spin‐orbit coupling integrals

A significant reduction in the computational effort for the evaluation of the electronic repulsion integrals (ERI) in ab initio quantum chemistry calculations is obtained by using Cholesky decomposition (CD), a numerical procedure that can remove the zero or small eigenvalues of the ERI positive (semi)definite matrix, while avoiding the calculation of the entire matrix. Conversely, due to its antisymmetric character, CD cannot be directly applied to the matrix representation of the spatial part of the two‐electron spin‐orbit coupling (2e‐SOC) integrals. Here, we present a computational strategy to achieve a Cholesky representation of the spatial part of the 2e‐SOC integrals, and propose a new efficient CD algorithm for both ERI and 2e‐SOC integrals. The proposed algorithm differs from previous CD implementations by the extensive use of a full‐pivoting design, which allows a univocal definition of the Cholesky basis, once the CD δ threshold is made explicit. We show that is the upper limit for the errors affecting the reconstructed 2e‐SOC integrals. The proposed strategy was implemented in the ab initio program Computational Emulator of Rare Earth Systems (CERES), and tested for computational performance on both the ERI and 2e‐SOC integrals evaluation. © 2017 Wiley Periodicals, Inc.     

Implementation of the diagonalization‐free algorithm in the self‐consistent field procedure within the four‐component relativistic scheme

A recently developed Thouless‐expansion‐based diagonalization‐free approach for improving the efficiency of self‐consistent field (SCF) methods (Noga and ?imunek, J. Chem. Theory Comput. 2010, 6, 2706) has been adapted to the four‐component relativistic scheme and implemented within the program package ReSpect. In addition to the implementation, the method has been thoroughly analyzed, particularly with respect to cases for which it is difficult or computationally expensive to find a good initial guess. Based on this analysis, several modifications of the original algorithm, refining its stability and efficiency, are proposed. To demonstrate the robustness and efficiency of the improved algorithm, we present the results of four‐component diagonalization‐free SCF calculations on several heavy‐metal complexes, the largest of which contains more than 80 atoms (about 6000 4‐spinor basis functions). The diagonalization‐free procedure is about twice as fast as the corresponding diagonalization. © 2014 Wiley Periodicals, Inc.     

Single molecule conductivity: the role of junction-orbital degeneracy in the artificially high currents predicted by ab initio approaches

A priori evaluations, using Hartree-Fock self-consistent-field (SCF) theory or density-functional theory (DFT), of the current passing between two electrodes through a single bridging molecule result in predicted conductivities that may be up to one to two orders of magnitude larger than observed ones. We demonstrate that this is, in part, often due to the improper application of the computational methods. Conductivity is shown to arise from tunneling between junction states of the electrodes through the molecule; these states are inherently either quasi two-fold or four-fold degenerate and always comprise the (highest occupied molecular orbital) HOMO band at the Fermi energy of the system. Frequently, in previous cluster based molecular conduction calculations, closed-shell SCF or Kohn-Sham DFT methods have been applied to systems that we demonstrate to be intrinsically open shell in nature. Such calculations are shown to induce artificial HOMO-LUMO (LUMO-lowest unoccupied molecular orbital) band splittings that Landauer-based formalisms for steady-state conduction interpret as arising from extremely rapid through-molecule tunneling at the Fermi energy, hence, overestimating the low-voltage conductivity. It is demonstrated that these shortcomings can be eliminated, dramatically reducing calculated current magnitudes, through the alternate use of electronic-structure calculations based on the spin-restricted open-shell formalism and related multiconfigurational SCF of DFT approaches. Further, we demonstrate that most anomalies arising in DFT implementations arise through the use of hybrid density functionals such as B3LYP. While the enhanced band-gap properties of these functionals have made them the defacto standard in molecular conductivity calculations, we demonstrate that it also makes them particularly susceptible to open-shell anomalies.     

Semidirect parallel self-consistent field: the load balancing problem in the input/output intensive self-consistent field iterations

The full capacity of contemporary parallel computers can, in the context of iterative ab initio procedures like, for example, self-consistent field (SCF) and multiconfigurational SCF, only be utilized if the disk and input/output (I/O) capacity are fully exploited before the implementation turns to an integral direct strategy. In a recent report on parallel semidirect SCF http://www.tc.cornell.edu/er/media/1996/collabrate.html, http://www.fp.mcs.anl.gd/grand-challenges/chem/nondirect/index.html it was demonstrated that super-linear speedups are achievable for algorithms that exploit scalable parallel I/O. In the I/O-intensive SCF iterations of this implementation a static load balancing, however, was employed, dictated by the initial iteration in which integral evaluation dominates the central processing unit activity and thus determines the load balancing. In the present paper we present the first implementation in which load balancing is achieved throughout the whole SCF procedure, i.e. also in subsequent iterations. The improved scalability of our new algorithm is demonstrated in some test calculations, for example, for 63-node calculation a speedup of 104 was observed in the computation of the two-electron integral contribution to the Fock matrix.Contribution to the Björn Roos Honorary Issue Acknowledgement.We thank J. Nieplocha for valuable help and making the toolkit (including ChemIO) available to us. R.L. acknowledges the Intelligent Modeling Laboratory and the University of Tokyo for financial support during his stay in Japan.     

Solvatochromic shifts of Br2 and I2 in water cages of type 512, 51262, 51263, and 51264

Despite the relatively small size of molecular bromine and iodine, the physicochemical behavior in different solvents is not yet fully understood, in particular when excited‐state properties are sought. In this work, we investigate isolated halogen molecules trapped in clathrate hydrate cages. Relativistic supermolecular calculations reveal that the environment shift to the excitation energies of the (nondegenerate) states and lie within a spread of 0.05 eV, respectively, suggesting that environment shifts can be estimated with scalar‐relativistic treatments. As even scalar‐relativistic calculations are problematic for excited‐state calculations for clathrates with growing size and basis sets, we have applied the subsystem‐based scheme frozen‐density embedding, which avoids a supermolecular treatment. This allows for the calculation of excited states for extended clusters with coupled‐cluster methods and basis sets of triple‐zeta quality with additional diffuse functions mandatory for excited‐state properties, as well as a facile treatment at scalar‐relativistic exact two‐component level of theory for the heavy atoms bromine and iodine. This simple approach yields scalar‐relativistic estimates for solvatochromic shifts introduced by the clathrate cages. © 2015 Wiley Periodicals, Inc.     

Energy-conformation studies of HCN,HNC and CN−: A comparison of results from EH-SCC and SCF molecular orbital calculations

We have made an Extended Hückel Self Consistent Charge (EH-SCC) molecular orbital calculation for hydrogen cyanide, hydrogen isocyanide and cyanide ion. The main purpose of this calculation was to compare the EH-SCC and the more accurate SCF MO calculations for HCN in order to evaluate the method we used here for future use. Specifically, we have calculated and compared the following properties of HCN: total energy, binding energy, variation of ground state energy with geometric conformation, ionization potential and dipole moment. In addition, we have extended previous calculations of HCN by also considering its energy variation with bond angle for two excited state configurations and deducing some of the characteristics of its electronic spectra. Finally we have also made an MO calculation of the isocyanide isomer HNC and CN^– ion to compare with and add to the known characterization of the H, C, N, system.

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Zusammenfassung Rechnungen nach der erweiterten Hückeltheorie werden für HCN, HNC und CN^– durchgeführt und mit ab initio Resultaten verglichen. Im einzelnen wurden Gesamtenergie, Bindungsenergie in Abhängigkeit von der geometrischen Struktur, Ionisierungspotential und Dipolmoment von HCN berechnet und außerdem die Energie für zwei doppelt angeregte Konfigurationen in Abhängigkeit vom Bindungswinkel bestimmt. Darüber hinaus sind MO-Rechnungen für HNC und CN^– gemacht worden.