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Littlejohn R, Rawlinson J, Subotnik J. Diagonalizing the Born-Oppenheimer Hamiltonian via Moyal perturbation theory, nonadiabatic corrections, and translational degrees of freedom. J Chem Phys 2024; 160:114103. [PMID: 38501907 DOI: 10.1063/5.0192465] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2023] [Accepted: 02/19/2024] [Indexed: 03/20/2024] Open
Abstract
This article describes a method for calculating higher order or nonadiabatic corrections in Born-Oppenheimer theory and its interaction with the translational degrees of freedom. The method uses the Wigner-Weyl correspondence to map nuclear operators into functions on the classical phase space and the Moyal star product to represent operator multiplication on those functions. These are explained in the body of the paper. The result is a power series in κ2, where κ = (m/M)1/4 is the usual Born-Oppenheimer parameter. The lowest order term is the usual Born-Oppenheimer approximation, while higher order terms are nonadiabatic corrections. These are needed in calculations of electronic currents, momenta, and densities. The separation of nuclear and electronic degrees of freedom takes place in the context of the exact symmetries (for an isolated molecule) of translations and rotations, and these, especially translations, are explicitly incorporated into our discussion. This article presents an independent derivation of the Moyal expansion in molecular Born-Oppenheimer theory. We show how electronic currents and momenta can be calculated within the framework of Moyal perturbation theory; we derive the transformation laws of the electronic Hamiltonian, the electronic eigenstates, and the derivative couplings under translations; we discuss in detail the rectilinear motion of the molecular center of mass in the Born-Oppenheimer representation; and we show how the elimination of the translational components of the derivative couplings leads to a unitary transformation that has the effect of exactly separating the translational degrees of freedom.
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Affiliation(s)
- Robert Littlejohn
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Jonathan Rawlinson
- School of Mathematics, University of Manchester, Manchester, United Kingdom
| | - Joseph Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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Schürger P, Renziehausen K, Schaupp T, Barth I, Engel V. Time-Dependent Expectation Values from Integral Equations for Quantum Flux and Probability Densities. J Phys Chem A 2022; 126:8964-8975. [DOI: 10.1021/acs.jpca.2c05995] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
Affiliation(s)
- P. Schürger
- Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Emil-Fischer-Strasse 42, 97074 Würzburg, Germany
| | - K. Renziehausen
- Friedrich Schiller University Jena, Helmholtzweg 4, 07743 Jena, Germany
- Theory Department, Max-Planck-Institute of Microstructure Physics, Weinberg 2, 06120 Halle (Saale), Germany
| | - T. Schaupp
- Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Emil-Fischer-Strasse 42, 97074 Würzburg, Germany
| | - I. Barth
- Theory Department, Max-Planck-Institute of Microstructure Physics, Weinberg 2, 06120 Halle (Saale), Germany
| | - V. Engel
- Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Emil-Fischer-Strasse 42, 97074 Würzburg, Germany
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Schaupp T, Engel V. Quantum flux densities for electronic-nuclear motion: exact versus Born-Oppenheimer dynamics. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2022; 380:20200385. [PMID: 35341310 DOI: 10.1098/rsta.2020.0385] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2021] [Accepted: 05/19/2021] [Indexed: 06/14/2023]
Abstract
We study the coupled electronic-nuclear dynamics in a model system to compare numerically exact calculations of electronic and nuclear flux densities with those obtained from the Born-Oppenheimer (BO) approximation. Within the adiabatic expansion of the total wave function, we identify the terms which contribute to the flux densities. It is found that only off-diagonal elements that involve the interaction between different electronic states contribute to the electronic flux whereas in the nuclear case the major contribution belongs to the BO electronic state. New flux densities are introduced where in both, the electronic and the nuclear case, the main contribution is contained in the component corresponding to the BO state. As a consequence, they can be determined within the BO approximation, and an excellent agreement with the exact results is found. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.
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Affiliation(s)
- Thomas Schaupp
- Universität Würzburg, Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Strasse 42, Würzburg 97074, Germany
| | - Volker Engel
- Universität Würzburg, Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Strasse 42, Würzburg 97074, Germany
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Hanasaki K, Takatsuka K. On the molecular electronic flux: Role of nonadiabaticity and violation of conservation. J Chem Phys 2021; 154:164112. [PMID: 33940814 DOI: 10.1063/5.0049821] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Analysis of electron flux within and in between molecules is crucial in the study of real-time dynamics of molecular electron wavepacket evolution such as those in attosecond laser chemistry and ultrafast chemical reaction dynamics. We here address two mutually correlated issues on the conservation law of molecular electronic flux, which serves as a key consistency condition for electron dynamics. The first one is about a close relation between "weak" nonadiabaticity and the electron dynamics in low-energy chemical reactions. We show that the electronic flux in adiabatic reactions can be consistently reproduced by taking account of nonadiabaticity. Such nonadiabaticity is usually weak in the sense that it does not have a major effect on nuclear dynamics, whereas it plays an important role in electronic dynamics. Our discussion is based on a nonadiabatic extension of the electronic wavefunction similar in idea to the complete adiabatic formalism developed by Nafie [J. Chem. Phys. 79, 4950 (1983)], which has also recently been reformulated by Patchkovskii [J. Chem. Phys. 137, 084109 (2012)]. We give straightforward proof of the theoretical assertion presented by Nafie using a time-dependent mixed quantum-classical framework and a standard perturbation expansion. Explicitly taking account of the flux conservation, we show that the nonadiabatically induced flux realizes the adiabatic time evolution of the electronic density. In other words, the divergence of the nonadiabatic flux equals the time derivative of the electronic density along an adiabatic time evolution of the target molecule. The second issue is about the accurate computationability of the flux. The calculation of flux needs an accurate representation of the (relative) quantum phase, in addition to the amplitude factor, of a total wavefunction and demands special attention for practical calculations. This paper is the first one to approach this issue directly and show how the difficulties arise explicitly. In doing so, we reveal that a number of widely accepted truncation techniques for static property calculations are potential sources of numerical flux non-conservation. We also theoretically propose alternative strategies to realize better flux conservation.
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Affiliation(s)
- Kota Hanasaki
- Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 606-8103, Japan
| | - Kazuo Takatsuka
- Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 606-8103, Japan
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Takatsuka K. Electron Dynamics in Molecular Elementary Processes and Chemical Reactions. BULLETIN OF THE CHEMICAL SOCIETY OF JAPAN 2021. [DOI: 10.1246/bcsj.20200388] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/28/2022]
Affiliation(s)
- Kazuo Takatsuka
- Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 606-8103, Japan
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Schaupp T, Renziehausen K, Barth I, Engel V. Time-dependent momentum expectation values from different quantum probability and flux densities. J Chem Phys 2021; 154:064307. [PMID: 33588545 DOI: 10.1063/5.0039466] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Based on the Ehrenfest theorem, the time-dependent expectation value of a momentum operator can be evaluated equivalently in two ways. The integrals appearing in the expressions are taken over two different functions. In one case, the integrand is the quantum mechanical flux density j̲, and in the other, a different quantity j̲̃ appears, which also has the units of a flux density. The quantum flux density j̲ is related to the probability density ρ via the continuity equation, and j̲̃ may as well be used to define a density ρ̃ that fulfills a continuity equation. Employing a model for the coupled dynamics of an electron and a proton, we document the properties of the densities and flux densities. It is shown that although the mean momentum derived from the two quantities is identical, the various functions exhibit a very different coordinate and time-dependence. In particular, it is found that the flux density j̲̃ directly monitors temporal changes in the probability density, and the density ρ̃ carries information about wave packet dispersion occurring in different spatial directions.
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Affiliation(s)
- Thomas Schaupp
- Institut für Physikalische und Theoretische Chemie, Universität Würzberg, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
| | - Klaus Renziehausen
- Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle (Saale), Germany
| | - Ingo Barth
- Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle (Saale), Germany
| | - Volker Engel
- Institut für Physikalische und Theoretische Chemie, Universität Würzberg, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
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Schaupp T, Engel V. On the calculation of time-dependent electron momenta within the Born-Oppenheimer approximation. J Chem Phys 2019; 150:164110. [DOI: 10.1063/1.5092562] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Thomas Schaupp
- Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Emil-Fischer-Str. 42, Würzburg 97074, Germany
| | - Volker Engel
- Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Emil-Fischer-Str. 42, Würzburg 97074, Germany
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Matsuzaki R, Takatsuka K. Electronic and nuclear fluxes induced by quantum interference in the adiabatic and nonadiabatic dynamics in the Born-Huang representation. J Chem Phys 2019; 150:014103. [DOI: 10.1063/1.5066571] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Rei Matsuzaki
- Fukui Institute for Fundamental Chemistry, Kyoto University, Sakyo-ku, Kyoto 606-8103, Japan
| | - Kazuo Takatsuka
- Fukui Institute for Fundamental Chemistry, Kyoto University, Sakyo-ku, Kyoto 606-8103, Japan
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Matsuzaki R, Takatsuka K. Electronic and nuclear flux analysis on nonadiabatic electron transfer reaction: A view from single-configuration adiabatic born-huang representation. J Comput Chem 2018; 40:148-163. [DOI: 10.1002/jcc.25557] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2018] [Revised: 07/16/2018] [Accepted: 07/24/2018] [Indexed: 12/13/2022]
Affiliation(s)
- Rei Matsuzaki
- Fukui Institute for Fundamental Chemistry; Kyoto University; Sakyou-ku Kyoto Japan
| | - Kazuo Takatsuka
- Fukui Institute for Fundamental Chemistry; Kyoto University; Sakyou-ku Kyoto Japan
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