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Tao Z, Qiu T, Bhati M, Bian X, Duston T, Rawlinson J, Littlejohn RG, Subotnik JE. Practical phase-space electronic Hamiltonians for ab initio dynamics. J Chem Phys 2024; 160:124101. [PMID: 38526114 DOI: 10.1063/5.0192084] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2023] [Accepted: 02/14/2024] [Indexed: 03/26/2024] Open
Abstract
Modern electronic structure theory is built around the Born-Oppenheimer approximation and the construction of an electronic Hamiltonian Ĥel(X) that depends on the nuclear position X (and not the nuclear momentum P). In this article, using the well-known theory of electron translation (Γ') and rotational (Γ″) factors to couple electronic transitions to nuclear motion, we construct a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, ĤPS(X,P). While classical Born-Oppenheimer dynamics that run along the eigensurfaces of the operator Ĥel(X) can recover many nuclear properties correctly, we present some evidence that motion along the eigensurfaces of ĤPS(X,P) can better capture both nuclear and electronic properties (including the elusive electronic momentum studied by Nafie). Moreover, only the latter (as opposed to the former) conserves the total linear and angular momentum in general.
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Affiliation(s)
- Zhen Tao
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Tian Qiu
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Mansi Bhati
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Xuezhi Bian
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Titouan Duston
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Jonathan Rawlinson
- Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom
| | - Robert G Littlejohn
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Joseph E Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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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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Resta R. Adiabatic electronic flux in molecules and in condensed matter. J Chem Phys 2022; 156:204118. [DOI: 10.1063/5.0087883] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The theory of adiabatic electron transport in a correlated condensed-matter system is rooted in a seminal paper by Niu and Thouless [J. Phys A {\bf 17}, 2453 (1984)]; I adopt here an analogous logic in order to { retrieve the known expression for the adiabatic electronic flux in a molecular system [L. A. Nafie, J. Chem. Phys. {\bf 79}, 4950 (1983)]. Its derivation here is considerably simpler than those available in the current quantum-chemistry literature; it also explicitly identifies the adiabaticity parameter, in terms of which the adiabatic flux and the electron density are both exact to first order. It is shown that continuity equation is conserved to the same order. For the sake of completeness,} I also briefly outline the relevance of the macroscopic electronic flux to the physics of solids and liquids.
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Affiliation(s)
- Raffaele Resta
- Istituto Officina dei Materiali, CNR, Istituto Officina dei Materiali Consiglio Nazionale delle Ricerche, Italy
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Schaupp T, Engel V. Correlated three-dimensional electron-nuclear motion: Adiabatic dynamics vs passage of conical intersections. J Chem Phys 2022; 156:074302. [DOI: 10.1063/5.0082597] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Thomas Schaupp
- Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
| | - Volker Engel
- Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
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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. Born–Oppenheimer and non-Born–Oppenheimer contributions to time-dependent electron momenta. J Chem Phys 2020; 152:204310. [DOI: 10.1063/5.0004560] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Thomas Schaupp
- Universität Würzburg Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
| | - Volker Engel
- Universität Würzburg Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
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Schaupp T, Engel V. Electronic and nuclear flux dynamics at a conical intersection. J Chem Phys 2019; 151:084309. [DOI: 10.1063/1.5111922] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Thomas Schaupp
- Universität Würzburg Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
| | - Volker Engel
- Universität Würzburg Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, 97074 Würzburg, Germany
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