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Blech A, Ebeling RMM, Heger M, Koch CP, Reich DM. Numerical evaluation of orientation averages and its application to molecular physics. J Chem Phys 2024; 161:131501. [PMID: 39365019 DOI: 10.1063/5.0230569] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2024] [Accepted: 09/16/2024] [Indexed: 10/05/2024] Open
Abstract
In molecular physics, it is often necessary to average over the orientation of molecules when calculating observables, in particular when modeling experiments in the liquid or gas phase. Evaluated in terms of Euler angles, this is closely related to integration over two- or three-dimensional unit spheres, a common problem discussed in numerical analysis. The computational cost of the integration depends significantly on the quadrature method, making the selection of an appropriate method crucial for the feasibility of simulations. After reviewing several classes of spherical quadrature methods in terms of their efficiency and error distribution, we derive guidelines for choosing the best quadrature method for orientation averages and illustrate these with three examples from chiral molecule physics. While Gauss quadratures allow for achieving numerically exact integration for a wide range of applications, other methods offer advantages in specific circumstances. Our guidelines can also be applied to higher-dimensional spherical domains and other geometries. We also present a Python package providing a flexible interface to a variety of quadrature methods.
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Affiliation(s)
- Alexander Blech
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany
| | - Raoul M M Ebeling
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany
| | - Marec Heger
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany
| | - Christiane P Koch
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany
| | - Daniel M Reich
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany
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Yagi T, Sato H. Density functional theory for molecular liquids based on interaction site model and self-consistent integral equations for site–site pair correlation functions. J Chem Phys 2020; 153:164102. [DOI: 10.1063/5.0022568] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Tomoaki Yagi
- Department of Molecular Engineering, Graduate School of Engineering, Kyoto University, Kyoto 615-8510, Japan
| | - Hirofumi Sato
- Department of Molecular Engineering, Graduate School of Engineering, Kyoto University, Kyoto 615-8510, Japan
- Elements Strategy Initiative for Catalysts and Batteries (ESICB), Kyoto University, Kyoto 615-8520, Japan
- Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 606-8103, Japan
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Miyata T. Sigma enlarging bridge function for heteronuclear Lennard-Jones diatomic solute solvated in a Lennard-Jones monatomic solvent in terms of the parameter transferability. Chem Phys Lett 2020. [DOI: 10.1016/j.cplett.2020.137777] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Tanimoto S, Yoshida N, Yamaguchi T, Ten-no SL, Nakano H. Effect of Molecular Orientational Correlations on Solvation Free Energy Computed by Reference Interaction Site Model Theory. J Chem Inf Model 2019; 59:3770-3781. [DOI: 10.1021/acs.jcim.9b00330] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022]
Affiliation(s)
- Shoichi Tanimoto
- Department of Chemistry, Graduate School of Science, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
| | - Norio Yoshida
- Department of Chemistry, Graduate School of Science, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
| | - Tsuyoshi Yamaguchi
- Graduate School of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan
| | - Seiichiro L. Ten-no
- Graduate School of Science, Technology, and Innovation, Kobe University, Nada-ku, Kobe 657-8501, Japan
| | - Haruyuki Nakano
- Department of Chemistry, Graduate School of Science, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
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Yokogawa D. Toward Accurate Solvation Free Energy Calculation with the Reference Interaction Site Model Self-Consistent Field: Introduction of a New Bridge Function. J Chem Theory Comput 2018; 14:3272-3278. [DOI: 10.1021/acs.jctc.8b00314] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- D. Yokogawa
- Department of Chemistry, Graduate School of Science, Nagoya University, Chikusa, Nagoya 464-8602, Japan
- Institute of Transformative Bio-Molecules (WPI-ITbM), Nagoya UniversityChikusa, Nagoya 464-8602, Japan
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Zhao S, Liu Y, Liu H, Wu J. Site-site direct correlation functions for three popular molecular models of liquid water. J Chem Phys 2013; 139:064509. [DOI: 10.1063/1.4817784] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/19/2023] Open
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Sato H. A modern solvation theory: quantum chemistry and statistical chemistry. Phys Chem Chem Phys 2013; 15:7450-65. [DOI: 10.1039/c3cp50247c] [Citation(s) in RCA: 52] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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Yokogawa D, Sato H, Sakaki S. An integral equation theory for structural fluctuation in molecular liquid. Chem Phys Lett 2010. [DOI: 10.1016/j.cplett.2010.01.061] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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Ten-no S, Jung J, Chuman H, Kawashima Y. Assessment of free energy expressions in RISM integral equation theory: theoretical prediction of partition coefficients revisited. Mol Phys 2010. [DOI: 10.1080/00268970903451848] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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10
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Yokogawa D, Sato H, Sakaki S. An integral equation theory for 3D solvation structure: A new procedure free from 3D Fourier transform. Chem Phys Lett 2006. [DOI: 10.1016/j.cplett.2006.10.093] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Sumi T, Sekino H. An interaction site model integral equation study of molecular fluids explicitly considering the molecular orientation. J Chem Phys 2006; 125:34509. [PMID: 16863364 DOI: 10.1063/1.2215603] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We implemented an interaction site model integral equation for rigid molecules based on a density-functional theory where the molecular orientation is explicitly considered. In this implementation of the integral equation, multiple integral of the degree of freedom of the molecular orientation is performed using efficient quadrature methods, so that the site-site pair correlation functions are evaluated exactly in the limit of low density. We apply this method to Cl(2), HCl, and H(2)O molecular fluids that have been investigated by several integral equation studies using various models. The site-site pair correlation functions obtained from the integral equation are in good agreement with the one from a simulation of these molecules. Rotational invariant coefficients, which characterize the microscopic structure of molecular fluids, are determined from the integral equation and the simulation in order to investigate the accuracy of the integral equation.
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Affiliation(s)
- Tomonari Sumi
- Department of Knowledge-Based Information Engineering, Toyohashi University of Technology, Tempaku-cho, Toyohashi 441-8580, Japan.
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Stevensson B, Edén M. Efficient orientational averaging by the extension of Lebedev grids via regularized octahedral symmetry expansion. JOURNAL OF MAGNETIC RESONANCE (SAN DIEGO, CALIF. : 1997) 2006; 181:162-76. [PMID: 16697671 DOI: 10.1016/j.jmr.2006.04.008] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2006] [Revised: 04/14/2006] [Accepted: 04/19/2006] [Indexed: 05/09/2023]
Abstract
Gaussian spherical quadrature methods in the guise of the Lebedev sampling grids are highly efficient for some orientational ("powder") averaging problems in solid state NMR. However, their applicability is currently restricted, as the sets of orientations are derived analytically and because they are not well adapted to simulate the broad peakshapes encountered, for example, in the NMR on static powders or on half-integer quadrupolar spins subject to second order quadrupolar interactions under magic-angle spinning conditions. We remedy these problems by (i) introducing the recursive procedure regularized octahedral symmetry expansion (ROSE), to which any existing Lebedev set may be subjected. Each recursive step gives a 9-fold enlarged set of orientations. (ii) We demonstrate that ROSE-expanded grids, in conjunction with spectral interpolation, is well suited for calculating broad peakshapes. These advances combine into the apparently most efficient general-purpose two-angle orientational averaging technique proposed to date for solid state NMR applications.
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Affiliation(s)
- Baltzar Stevensson
- Physical Chemistry Division, Arrhenius Laboratory, Stockholm University, SE-106 91 Stockholm, Sweden.
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Sato K, Chuman H, Ten-no S. Comparative Study on Solvation Free Energy Expressions in Reference Interaction Site Model Integral Equation Theory. J Phys Chem B 2005; 109:17290-5. [PMID: 16853207 DOI: 10.1021/jp053259i] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The performance of the recently proposed partial wave (PW) free energy functional is compared with those of two previous expressions, Gaussian fluctuation (GF) and hypernetted chain (HNC), within the reference interaction site model framework. The applications to the calculations of ambient and supercritical water, solvation free energies of organic molecules, and partition coefficients clearly show that the PW and GF free energy expressions provide more reliable results than the HNC functional, indicating rather associative situations of geometry in ordinary liquids of medium-sized molecules.
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Affiliation(s)
- Kazuto Sato
- Graduate School of Information Science, Nagoya University, Chikusa-ku Nagoya, 464-8601 Japan
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