1
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Suchan J, Liang F, Durden AS, Levine BG. Prediction challenge: First principles simulation of the ultrafast electron diffraction spectrum of cyclobutanone. J Chem Phys 2024; 160:134310. [PMID: 38573851 DOI: 10.1063/5.0198333] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2024] [Accepted: 03/18/2024] [Indexed: 04/06/2024] Open
Abstract
Computer simulation has long been an essential partner of ultrafast experiments, allowing the assignment of microscopic mechanistic detail to low-dimensional spectroscopic data. However, the ability of theory to make a priori predictions of ultrafast experimental results is relatively untested. Herein, as a part of a community challenge, we attempt to predict the signal of an upcoming ultrafast photochemical experiment using state-of-the-art theory in the context of preexisting experimental data. Specifically, we employ ab initio Ehrenfest with collapse to a block mixed quantum-classical simulations to describe the real-time evolution of the electrons and nuclei of cyclobutanone following excitation to the 3s Rydberg state. The gas-phase ultrafast electron diffraction (GUED) signal is simulated for direct comparison to an upcoming experiment at the Stanford Linear Accelerator Laboratory. Following initial ring-opening, dissociation via two distinct channels is observed: the C3 dissociation channel, producing cyclopropane and CO, and the C2 channel, producing CH2CO and C2H4. Direct calculations of the GUED signal indicate how the ring-opened intermediate, the C2 products, and the C3 products can be discriminated in the GUED signal. We also report an a priori analysis of anticipated errors in our predictions: without knowledge of the experimental result, which features of the spectrum do we feel confident we have predicted correctly, and which might we have wrong?
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Affiliation(s)
- Jiří Suchan
- Institute of Advanced Computational Science, Stony Brook University, Stony Brook, New York 11794, USA
| | - Fangchun Liang
- Institute of Advanced Computational Science, Stony Brook University, Stony Brook, New York 11794, USA
- Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, USA
| | - Andrew S Durden
- Institute of Advanced Computational Science, Stony Brook University, Stony Brook, New York 11794, USA
- Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, USA
| | - Benjamin G Levine
- Institute of Advanced Computational Science, Stony Brook University, Stony Brook, New York 11794, USA
- Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, USA
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2
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Pieroni C, Sangiogo Gil E, Ibele LM, Persico M, Granucci G, Agostini F. Investigating the Photodynamics of trans-Azobenzene with Coupled Trajectories. J Chem Theory Comput 2024; 20:580-596. [PMID: 38177105 DOI: 10.1021/acs.jctc.3c00978] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2024]
Abstract
In this work, we present the first implementation of coupled-trajectory Tully surface hopping (CT-TSH) suitable for applications to molecular systems. We combine CT-TSH with the semiempirical floating occupation molecular orbital-configuration interaction electronic structure method to investigate the photoisomerization dynamics of trans-azobenzene. Our study shows that CT-TSH can capture correctly decoherence effects in this system, yielding consistent electronic and nuclear dynamics in agreement with (standard) decoherence-corrected TSH. Specifically, CT-TSH is derived from the exact factorization and the electronic coefficients' evolution is directly influenced by the coupling of trajectories, resulting in the improvement of internal consistency if compared to standard TSH.
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Affiliation(s)
- Carlotta Pieroni
- CNRS, Institut de Chimie Physique UMR8000, Université Paris-Saclay, 91405 Orsay, France
- Dipartimento di Chimica e Chimica Industriale, Università di Pisa, via G. Moruzzi 13, 56124 Pisa, Italy
| | - Eduarda Sangiogo Gil
- CNRS, Institut de Chimie Physique UMR8000, Université Paris-Saclay, 91405 Orsay, France
| | - Lea M Ibele
- CNRS, Institut de Chimie Physique UMR8000, Université Paris-Saclay, 91405 Orsay, France
| | - Maurizio Persico
- Dipartimento di Chimica e Chimica Industriale, Università di Pisa, via G. Moruzzi 13, 56124 Pisa, Italy
| | - Giovanni Granucci
- Dipartimento di Chimica e Chimica Industriale, Università di Pisa, via G. Moruzzi 13, 56124 Pisa, Italy
| | - Federica Agostini
- CNRS, Institut de Chimie Physique UMR8000, Université Paris-Saclay, 91405 Orsay, France
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3
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Dines A, Ellis M, Blumberger J. Stabilized coupled trajectory mixed quantum-classical algorithm with improved energy conservation: CTMQC-EDI. J Chem Phys 2023; 159:234118. [PMID: 38117021 DOI: 10.1063/5.0183589] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2023] [Accepted: 11/26/2023] [Indexed: 12/21/2023] Open
Abstract
Coupled trajectory mixed quantum-classical (CTMQC) dynamics is a rigorous approach to trajectory-based non-adiabatic dynamics, which has recently seen an improvement to energy conservation via the introduction of the CTMQC-E algorithm. Despite this, the method's two key quantities distinguishing it from Ehrenfest dynamics, the modified Born-Oppenheimer momentum and the quantum momentum, require regularization procedures in certain circumstances. Such procedures in the latter can cause instabilities, leading to undesirable effects, such as energy drift and spurious population transfer, which is expected to become increasingly prevalent when the system gets larger as such events would happen more frequently. We propose a further modification to CTMQC-E, which includes a redefinition of the quantum momentum, CTMQC-EDI (double intercept), such that it has no formal divergences. We then show for Tully models I-III and the double arch model that the algorithm has greatly improved total energy conservation and negligible spurious population transfer at all times, in particular in regions of strong non-adiabatic coupling. CTMQC-EDI, therefore, shows promise as a numerically robust non-adiabatic dynamics technique that accounts for decoherence from first principles and that is scalable to large molecular systems and materials.
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Affiliation(s)
- Aaron Dines
- Department of Physics and Astronomy and Thomas Young Centre, University College London, Gower Street, London WC1E 6BT, United Kingdom
| | - Matthew Ellis
- Department of Physics and Astronomy and Thomas Young Centre, University College London, Gower Street, London WC1E 6BT, United Kingdom
| | - Jochen Blumberger
- Department of Physics and Astronomy and Thomas Young Centre, University College London, Gower Street, London WC1E 6BT, United Kingdom
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4
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Arribas EV, Ibele LM, Lauvergnat D, Maitra NT, Agostini F. Significance of Energy Conservation in Coupled-Trajectory Approaches to Nonadiabatic Dynamics. J Chem Theory Comput 2023; 19:7787-7800. [PMID: 37853509 DOI: 10.1021/acs.jctc.3c00845] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2023]
Abstract
Through approximating electron-nuclear correlation terms in the exact factorization approach, trajectory-based methods have been derived and successfully applied to the dynamics of a variety of light-induced molecular processes, capturing quantum (de)coherence effects rigorously. These terms account for the coupling among the trajectories, recovering the nonlocal nature of quantum nuclear dynamics that is completely overlooked in traditional independent-trajectory algorithms. Nevertheless, some of the approximations introduced in the derivation of some of these methods do not conserve the total energy. We analyze energy conservation in the coupled-trajectory mixed quantum-classical (CTMQC) algorithm and explore the performance of a modified algorithm, CTMQC-E, where some of the terms are redefined to restore energy conservation. A set of molecular models is used as a test, namely, 2-cis-penta-2,4-dienimium cation, bis(methylene) adamantyl radical cation, butatriene cation, uracil radical cation, and neutral pyrazine.
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Affiliation(s)
| | - Lea M Ibele
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405 Orsay, France
| | - David Lauvergnat
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405 Orsay, France
| | - Neepa T Maitra
- Department of Physics, Rutgers University, Newark, New Jersey 07102, United States
| | - Federica Agostini
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405 Orsay, France
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5
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Villaseco Arribas E, Vindel-Zandbergen P, Roy S, Maitra NT. Different flavors of exact-factorization-based mixed quantum-classical methods for multistate dynamics. Phys Chem Chem Phys 2023; 25:26380-26395. [PMID: 37750820 DOI: 10.1039/d3cp03464j] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 09/27/2023]
Abstract
The exact factorization approach has led to the development of new mixed quantum-classical methods for simulating coupled electron-ion dynamics. We compare their performance for dynamics when more than two electronic states are occupied at a given time, and analyze: (1) the use of coupled versus auxiliary trajectories in evaluating the electron-nuclear correlation terms, (2) the approximation of using these terms within surface-hopping and Ehrenfest frameworks, and (3) the relevance of the exact conditions of zero population transfer away from nonadiabatic coupling regions and total energy conservation. Dynamics through the three-state conical intersection in the uracil radical cation as well as polaritonic models in one dimension are studied.
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Affiliation(s)
| | - Patricia Vindel-Zandbergen
- Department of Physics, Rutgers University, Newark 07102, New Jersey, USA.
- Department of Chemistry, New York University, New York, New York 10003, USA
| | - Saswata Roy
- Department of Physics, Rutgers University, Newark 07102, New Jersey, USA.
| | - Neepa T Maitra
- Department of Physics, Rutgers University, Newark 07102, New Jersey, USA.
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6
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Gardner J, Habershon S, Maurer RJ. Assessing Mixed Quantum-Classical Molecular Dynamics Methods for Nonadiabatic Dynamics of Molecules on Metal Surfaces. THE JOURNAL OF PHYSICAL CHEMISTRY. C, NANOMATERIALS AND INTERFACES 2023; 127:15257-15270. [PMID: 37583439 PMCID: PMC10424245 DOI: 10.1021/acs.jpcc.3c03591] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/27/2023] [Revised: 06/30/2023] [Indexed: 08/17/2023]
Abstract
Mixed quantum-classical (MQC) methods for simulating the dynamics of molecules at metal surfaces have the potential to accurately and efficiently provide mechanistic insight into reactive processes. Here, we introduce simple two-dimensional models for the scattering of diatomic molecules at metal surfaces based on recently published electronic structure data. We apply several MQC methods to investigate their ability to capture how nonadiabatic effects influence molecule-metal energy transfer during the scattering process. Specifically, we compare molecular dynamics with electronic friction, Ehrenfest dynamics, independent electron surface hopping, and the broadened classical master equation approach. In the case of independent electron surface hopping, we implement a simple decoherence correction approach and assess its impact on vibrationally inelastic scattering. Our results show that simple, low-dimensional models can be used to qualitatively capture experimentally observed vibrational energy transfer and provide insight into the relative performance of different MQC schemes. We observe that all approaches predict similar kinetic energy dependence but return different vibrational energy distributions. Finally, by varying the molecule-metal coupling, we can assess the coupling regime in which some MQC methods become unsuitable.
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Affiliation(s)
- James Gardner
- Department
of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Scott Habershon
- Department
of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Reinhard J. Maurer
- Department
of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
- Department
of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
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7
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Wu Y, Bian X, Rawlinson JI, Littlejohn RG, Subotnik JE. A phase-space semiclassical approach for modeling nonadiabatic nuclear dynamics with electronic spin. J Chem Phys 2022; 157:011101. [DOI: 10.1063/5.0093345] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Chemical relaxation phenomena, including photochemistry and electron transfer processes, form a vigorous area of research in which nonadiabatic dynamics plays a fundamental role. However, for electronic systems with spin degrees of freedom, there are few if any applicable and practical quasiclassical methods. Here, we show that for nonadiabatic dynamics with two electronic states and a complex-valued Hamiltonian that does not obey time-reversal symmetry (as relevant to many coupled nuclear-electronic-spin systems), the optimal semiclassical approach is to generalize Tully’s surface hopping dynamics from coordinate space to phase space. In order to generate the relevant phase-space adiabatic surfaces, one isolates a proper set of diabats, applies a phase gauge transformation, and then diagonalizes the total Hamiltonian (which is now parameterized by both R and P). The resulting algorithm is simple and valid in both the adiabatic and nonadiabatic limits, incorporating all Berry curvature effects. Most importantly, the resulting algorithm allows for the study of semiclassical nonadiabatic dynamics in the presence of spin–orbit coupling and/or external magnetic fields. One expects many simulations to follow as far as modeling cutting-edge experiments with entangled nuclear, electronic, and spin degrees of freedom, e.g., experiments displaying chiral-induced spin selectivity.
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Affiliation(s)
- Yanze Wu
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Xuezhi Bian
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Jonathan I. Rawlinson
- School of Mathematics, The University of Manchester, Oxford Rd., Manchester M13 9PL, United Kingdom
| | - Robert G. Littlejohn
- Department of Physics, University of California, 366 Physics North MC 7300, Berkeley, California 94720-7300, USA
| | - Joseph E. Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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8
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Villaseco Arribas E, Agostini F, Maitra NT. Exact Factorization Adventures: A Promising Approach for Non-Bound States. Molecules 2022; 27:molecules27134002. [PMID: 35807246 PMCID: PMC9267945 DOI: 10.3390/molecules27134002] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2022] [Revised: 06/07/2022] [Accepted: 06/09/2022] [Indexed: 11/29/2022] Open
Abstract
Modeling the dynamics of non-bound states in molecules requires an accurate description of how electronic motion affects nuclear motion and vice-versa. The exact factorization (XF) approach offers a unique perspective, in that it provides potentials that act on the nuclear subsystem or electronic subsystem, which contain the effects of the coupling to the other subsystem in an exact way. We briefly review the various applications of the XF idea in different realms, and how features of these potentials aid in the interpretation of two different laser-driven dissociation mechanisms. We present a detailed study of the different ways the coupling terms in recently-developed XF-based mixed quantum-classical approximations are evaluated, where either truly coupled trajectories, or auxiliary trajectories that mimic the coupling are used, and discuss their effect in both a surface-hopping framework as well as the rigorously-derived coupled-trajectory mixed quantum-classical approach.
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Affiliation(s)
| | - Federica Agostini
- Institut de Chimie Physique UMR8000, Université Paris-Saclay, CNRS, 91405 Orsay, France;
| | - Neepa T. Maitra
- Department of Physics, Rutgers University, Newark, NJ 07102, USA;
- Correspondence:
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9
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Talotta F, Lauvergnat D, Agostini F. Describing the photo-isomerization of a retinal chromophore model with coupled and quantum trajectories. J Chem Phys 2022; 156:184104. [DOI: 10.1063/5.0089415] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The exact factorization of the electron-nuclear wavefunction is applied to the study of the photo- isomerization of a retinal chromophore model. We describe such an ultrafast nonadiabatic process by analyzing the time-dependent potentials of the theory and by mimicking nuclear dynamics with quantum and coupled trajectories. The time-dependent vector and scalar potentials are the signature of the exact factorization, as they guide nuclear dynamics by encoding the complete electronic dynamics and including excited-state effects. Analysis of the potentials is, thus, essential - when possible - to predict the time-dependent behavior of the system of interest. In this work, we employ the exact time-dependent potentials, available for the numerically-exactly solvable model used here, to propagate quantum nuclear trajectories representing the isomerization reaction of the retinal chromophore. The quantum trajectories are the best possible trajectory-based description of the reaction when using the exact-factorization formalism, and thus allow us to assess the performance of the coupled-trajectory, fully approximate, schemes derived from the exact-factorization equations.
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Affiliation(s)
| | - David Lauvergnat
- Institut de Chimie Physique, UMR 8000, CNRS Délégation Ile-de-France Sud, France
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10
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Ha JK, Min SK. Independent Trajectory Mixed Quantum-Classical Approaches Based on the Exact Factorization. J Chem Phys 2022; 156:174109. [DOI: 10.1063/5.0084493] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Mixed quantum-classical dynamics based on the exact factorization exploits the "derived" electron-nuclear correlation (ENC) term aiming for the description of quantum coherences. The ENC contains interactions between the phase of electronic states and nuclear quantum momenta which depend on the spatial shape of the nuclear density.The original surface hopping based on the exact factorization (SHXF) [\textit{J. Phys. Chem. Lett.} \textbf{2018}, \textit{9}, 1097] exploits frozen Gaussian functions to construct the nuclear density in the ENC term while the phase of electronic states is approximated as a fictitious nuclear momentum change.However, in reality, the width of nuclear wave packets varies in time depending on the shape of potential energy surfaces.In this work, we present a modified SHXF approach and a newly-developed Ehrenfest dynamics based on the exact factorization (EhXF) with time-dependent Gaussian functions and phases by enforcing total energy conservation.We perform numerical tests for various one-dimensional two-state model Hamiltonians.Overall, the time-dependent width of Gaussian functions and the energy conserving phase show a reliable decoherence compared to the original frozen Gaussian-based SHXF and the exact quantum mechanical calculation.Especially, the energy conserving phase is crucial for EhXF to reproduce the correct quantum dynamics.
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Affiliation(s)
- Jong-Kwon Ha
- Chemistry, Ulsan National Institute of Science and Technology, Korea, Republic of (South Korea)
| | - Seung Kyu Min
- Ulsan National Institute of Science and Technology, Korea, Republic of (South Korea)
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11
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Vindel-Zandbergen P, Matsika S, Maitra NT. Exact-Factorization-Based Surface Hopping for Multistate Dynamics. J Phys Chem Lett 2022; 13:1785-1790. [PMID: 35170972 DOI: 10.1021/acs.jpclett.1c04132] [Citation(s) in RCA: 13] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
A surface-hopping algorithm recently derived from the exact factorization approach, SHXF [Ha et al. J. Phys. Chem. Lett. 2018, 9, 1097], introduces an additional term in the electronic equation of surface hopping that couples electronic states through the quantum momentum. This term not only provides a first-principles description of decoherence, but here we show it is crucial to accurately capture nonadiabatic dynamics when more than two states are occupied at any given time. Using a vibronic coupling model of the uracil cation, we show that the lack of this term in traditional surface-hopping methods, including those with decoherence corrections, leads to failure to predict the dynamics through a three-state intersection, while SHXF performs similarly to the multiconfiguration time-dependent Hartree quantum dynamics benchmark.
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Affiliation(s)
| | - Spiridoula Matsika
- Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122, United States
| | - Neepa T Maitra
- Department of Physics, Rutgers University, Newark, New Jersey 07102, United States
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12
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Esch MP, Levine BG. An accurate, non-empirical method for incorporating decoherence into Ehrenfest dynamics. J Chem Phys 2021; 155:214101. [PMID: 34879667 DOI: 10.1063/5.0070686] [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
In mixed quantum-classical nonadiabatic molecular dynamics methods, the anchoring of the electronic wave function to a single nuclear geometry results in both quantitative and qualitative errors in the dynamics. In the context of both Ehrenfest and trajectory surface hopping methods, methods for incorporating decoherence are widely used to eliminate these errors. However, the accuracy of these methods often depends strongly on the parameterization of the decoherence time and/or other related quantities. Here, we present a refinement of the recently introduced collapse to a block (TAB) scheme for incorporating decoherence into Ehrenfest dynamics. The proposed approach incorporates an approximation to the history of the population dynamics and treats the coherence decay as Gaussian, rather than exponential. This method uses parameters that can be obtained from first principles, rather than empirical fitting. Application to one-dimensional models indicates excellent agreement with numerically exact simulations. We also introduce a second refinement to the TAB method: a robust linear least-squares algorithm for determining collapse probabilities.
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Affiliation(s)
- Michael P Esch
- Department of Chemistry and Institute for Advanced Computational Science, Stony Brook University, Stony Brook, New York 11794, USA
| | - Benjamin G Levine
- Department of Chemistry and Institute for Advanced Computational Science, Stony Brook University, Stony Brook, New York 11794, USA
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13
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Vindel-Zandbergen P, Ibele LM, Ha JK, Min SK, Curchod BFE, Maitra NT. Study of the Decoherence Correction Derived from the Exact Factorization Approach for Nonadiabatic Dynamics. J Chem Theory Comput 2021; 17:3852-3862. [PMID: 34138553 PMCID: PMC8280698 DOI: 10.1021/acs.jctc.1c00346] [Citation(s) in RCA: 22] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/03/2023]
Abstract
![]()
We present a detailed
study of the decoherence correction to surface
hopping that was recently derived from the exact factorization approach.
Ab initio multiple spawning calculations that use the same initial
conditions and the same electronic structure method are used as a
reference for three molecules: ethylene, the methaniminium cation,
and fulvene, for which nonadiabatic dynamics follows a photoexcitation.
A comparison with the Granucci–Persico energy-based decoherence
correction and the augmented fewest-switches surface-hopping scheme
shows that the three decoherence-corrected methods operate on individual
trajectories in a qualitatively different way, but the results averaged
over trajectories are similar for these systems.
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Affiliation(s)
| | - Lea M Ibele
- Department of Chemistry, Durham University, South Road, Durham DH1 3LE, U.K
| | - Jong-Kwon Ha
- Department of Chemistry, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Ulsan 44919, Republic of Korea
| | - Seung Kyu Min
- Department of Chemistry, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Ulsan 44919, Republic of Korea
| | - Basile F E Curchod
- Department of Chemistry, Durham University, South Road, Durham DH1 3LE, U.K
| | - Neepa T Maitra
- Department of Physics, Rutgers University, Newark, New Jersey 07102, United States
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14
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Lee IS, Ha JK, Han D, Kim TI, Moon SW, Min SK. PyUNIxMD: A Python-based excited state molecular dynamics package. J Comput Chem 2021; 42:1755-1766. [PMID: 34197646 PMCID: PMC8362049 DOI: 10.1002/jcc.26711] [Citation(s) in RCA: 19] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/07/2021] [Revised: 06/10/2021] [Accepted: 06/16/2021] [Indexed: 01/17/2023]
Abstract
Theoretical/computational description of excited state molecular dynamics is nowadays a crucial tool for understanding light-matter interactions in many materials. Here we present an open-source Python-based nonadiabatic molecular dynamics program package, namely PyUNIxMD, to deal with mixed quantum-classical dynamics for correlated electron-nuclear propagation. The PyUNIxMD provides many interfaces for quantum chemical calculation methods with commercial and noncommercial ab initio and semiempirical quantum chemistry programs. In addition, the PyUNIxMD offers many nonadiabatic molecular dynamics algorithms such as fewest-switch surface hopping and its derivatives as well as decoherence-induced surface hopping based on the exact factorization (DISH-XF) and coupled-trajectory mixed quantum-classical dynamics (CTMQC) for general purposes. Detailed structures and flows of PyUNIxMD are explained for the further implementations by developers. We perform a nonadiabatic molecular dynamics simulation for a molecular motor system as a simple demonstration.
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Affiliation(s)
- In Seong Lee
- Department of Chemistry, Ulsan National Institute of Science and Technology (UNIST), Ulsan, South Korea
| | - Jong-Kwon Ha
- Department of Chemistry, Ulsan National Institute of Science and Technology (UNIST), Ulsan, South Korea
| | - Daeho Han
- Department of Chemistry, Ulsan National Institute of Science and Technology (UNIST), Ulsan, South Korea
| | - Tae In Kim
- Department of Chemistry, Ulsan National Institute of Science and Technology (UNIST), Ulsan, South Korea
| | - Sung Wook Moon
- Department of Chemistry, Ulsan National Institute of Science and Technology (UNIST), Ulsan, South Korea
| | - Seung Kyu Min
- Department of Chemistry, Ulsan National Institute of Science and Technology (UNIST), Ulsan, South Korea
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15
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Schirò M, Eich FG, Agostini F. Quantum-classical nonadiabatic dynamics of Floquet driven systems. J Chem Phys 2021; 154:114101. [PMID: 33752379 DOI: 10.1063/5.0043790] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We develop a trajectory-based approach for excited-state molecular dynamics simulations of systems subject to an external periodic drive. We combine the exact-factorization formalism, allowing us to treat electron-nuclear systems in nonadiabatic regimes, with the Floquet formalism for time-periodic processes. The theory is developed starting with the molecular time-dependent Schrödinger equation with the inclusion of an external periodic drive that couples to the system dipole moment. With the support of the Floquet formalism, quantum dynamics is approximated by combining classical-like, trajectory-based, nuclear evolution with electronic dynamics represented in the Floquet basis. The resulting algorithm, which is an extension of the coupled-trajectory mixed quantum-classical scheme for periodically driven systems, is applied to a model study, exactly solvable, with different field intensities.
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Affiliation(s)
- Marco Schirò
- JEIP, USR 3573 CNRS, Collège de France, PSL Research University, 11 Place Marcelin Berthelot, 75321 Paris Cedex 05, France
| | - Florian G Eich
- HQS Quantum Simulations GmbH, Haid-und-Neu-Straße 7, D-76131 Karlsruhe, Germany
| | - Federica Agostini
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405 Orsay, France
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16
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Pieroni C, Marsili E, Lauvergnat D, Agostini F. Relaxation dynamics through a conical intersection: Quantum and quantum-classical studies. J Chem Phys 2021; 154:034104. [PMID: 33499611 DOI: 10.1063/5.0036726] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023] Open
Abstract
We study the relaxation process through a conical intersection of a photo-excited retinal chromophore model. The analysis is based on a two-electronic-state two-dimensional Hamiltonian developed by Hahn and Stock [J. Phys. Chem. B 104 1146 (2000)] to reproduce, with a minimal model, the main features of the 11-cis to all-trans isomerization of the retinal of rhodopsin. In particular, we focus on the performance of various trajectory-based schemes to nonadiabatic dynamics, and we compare quantum-classical results to the numerically exact quantum vibronic wavepacket dynamics. The purpose of this work is to investigate, by analyzing electronic and nuclear observables, how the sampling of initial conditions for the trajectories affects the subsequent dynamics.
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Affiliation(s)
- Carlotta Pieroni
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405 Orsay, France
| | - Emanuele Marsili
- Department of Chemistry, Durham University, South Road, Durham DH1 3LE, United Kingdom
| | - David Lauvergnat
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405 Orsay, France
| | - Federica Agostini
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405 Orsay, France
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17
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Martinez P, Rosenzweig B, Hoffmann NM, Lacombe L, Maitra NT. Case studies of the time-dependent potential energy surface for dynamics in cavities. J Chem Phys 2021; 154:014102. [PMID: 33412864 PMCID: PMC7968936 DOI: 10.1063/5.0033386] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2020] [Accepted: 12/10/2020] [Indexed: 11/14/2022] Open
Abstract
The exact time-dependent potential energy surface driving the nuclear dynamics was recently shown to be a useful tool to understand and interpret the coupling of nuclei, electrons, and photons in cavity settings. Here, we provide a detailed analysis of its structure for exactly solvable systems that model two phenomena: cavity-induced suppression of proton-coupled electron-transfer and its dependence on the initial state, and cavity-induced electronic excitation. We demonstrate the inadequacy of simply using a weighted average of polaritonic surfaces to determine the dynamics. Such a weighted average misses a crucial term that redistributes energy between the nuclear and the polaritonic systems, and this term can in fact become a predominant term in determining the nuclear dynamics when several polaritonic surfaces are involved. Evolving an ensemble of classical trajectories on the exact potential energy surface reproduces the nuclear wavepacket quite accurately, while evolving on the weighted polaritonic surface fails after a short period of time. The implications and prospects for application of mixed quantum-classical methods based on this surface are discussed.
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Affiliation(s)
- Phillip Martinez
- Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
| | | | - Norah M. Hoffmann
- Department of Physics, Rutgers University, Newark, New Jersey 07102, USA
| | - Lionel Lacombe
- Department of Physics, Rutgers University, Newark, New Jersey 07102, USA
| | - Neepa T. Maitra
- Department of Physics, Rutgers University, Newark, New Jersey 07102, USA
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18
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Marsili E, Olivucci M, Lauvergnat D, Agostini F. Quantum and Quantum-Classical Studies of the Photoisomerization of a Retinal Chromophore Model. J Chem Theory Comput 2020; 16:6032-6048. [PMID: 32931266 DOI: 10.1021/acs.jctc.0c00679] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
We report an in-depth analysis of the photo-induced isomerization of the 2-cis-penta-2,4-dieniminium cation: a minimal model of the 11-cis retinal protonated Schiff base chromophore of the dim-light photoreceptor rhodopsin. Based on recently developed three-dimensional potentials parametrized on ab initio multi-state multi-configurational second-order perturbation theory data, we perform quantum-dynamical studies. In addition, simulations based on various quantum-classical methods, among which Tully surface hopping and the coupled-trajectory approach derived from the exact factorization, allow us to validate their performance against vibronic wavepacket propagation and, therefore, a purely quantum treatment. Quantum-dynamics results uncover qualitative differences with respect to the two-dimensional Hahn-Stock potentials, widely used as model potentials for the isomerization of the same chromophore, due to the increased dimensionality and three-mode correlation. Quantum-classical simulations show, instead, that three-dimensional model potentials are capable of capturing a number of features revealed by atomistic simulations and experimental observations. In particular, a recently reported vibrational phase relationship between double-bond torsion and hydrogen-out-of-plane modes critical for rhodopsin isomerization efficiency is correctly reproduced.
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Affiliation(s)
- Emanuele Marsili
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, Orsay 91405, France.,Department of Chemistry, Durham University, South Road, Durham DH1 3LE, United Kingdom
| | - Massimo Olivucci
- Department of Biotechnology, Chemistry and Pharmacy, Università degli Studi di Siena, Via A. Moro 2, I-53100 Siena, Italy.,Department of Chemistry, Bowling Green State University, Bowling Green, Ohio 43403, United States
| | - David Lauvergnat
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, Orsay 91405, France
| | - Federica Agostini
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, Orsay 91405, France
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19
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Talotta F, Agostini F, Ciccotti G. Quantum Trajectories for the Dynamics in the Exact Factorization Framework: A Proof-of-Principle Test. J Phys Chem A 2020; 124:6764-6777. [PMID: 32786992 DOI: 10.1021/acs.jpca.0c03969] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
In the framework of the exact factorization of the time-dependent electron-nuclear wave function, we investigate the possibility of solving the nuclear time-dependent Schrödinger equation based on trajectories. The nuclear equation is separated in a Hamilton-Jacobi equation for the phase of the wave function, and a continuity equation for its (squared) modulus. For illustrative adiabatic and nonadiabatic one-dimensional models, we implement a procedure to follow the evolution of the nuclear density along the characteristics of the Hamilton-Jacobi equation. Those characteristics are referred to as quantum trajectories, since they are generated via ordinary differential equations similar to Hamilton's equations, but including the so-called quantum potential, and they can be used to reconstruct exactly the quantum-mechanical nuclear wave function, provided infinite initial conditions are propagated in time.
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Affiliation(s)
- Francesco Talotta
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405, Orsay, France.,Université Paris-Saclay, CNRS, Institut des Sciences Moléculaires d'Orsay, 91405, Orsay, France
| | - Federica Agostini
- Université Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405, Orsay, France.,Université Paris-Saclay, CNRS, Institut des Sciences Moléculaires d'Orsay, 91405, Orsay, France
| | - Giovanni Ciccotti
- CNR, Institute for Applied Computing "Mauro Picone" (IAC), Via dei Taurini 19, 00185 Rome, Italy.,School of Physics, University College of Dublin UCD - Belfield, Dublin 4, Ireland.,Dipartimento di Fisica, Università di Roma La Sapienza, P. le A. Moro 5, 00185 Roma, Italy
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20
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Talotta F, Morisset S, Rougeau N, Lauvergnat D, Agostini F. Internal Conversion and Intersystem Crossing with the Exact Factorization. J Chem Theory Comput 2020; 16:4833-4848. [PMID: 32633509 DOI: 10.1021/acs.jctc.0c00493] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
We present a detailed derivation of the generalized coupled-trajectory mixed quantum-classical (G-CT-MQC) algorithm based on the exact-factorization equations. The ultimate goal is to propose an algorithm that can be employed for molecular dynamics simulations of nonradiative phenomena, as the spin-allowed internal conversions and the spin-forbidden intersystem crossings. Internal conversions are nonadiabatic processes driven by the kinetic coupling between electronic states, whereas intersystem crossings are mediated by the spin-orbit coupling. In this paper, we discuss computational issues related to the suitable representation for electronic dynamics and the different natures of kinetic and spin-orbit coupling. Numerical studies on model systems allow us to test the performance of the G-CT-MQC algorithm in different situations.
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Affiliation(s)
- Francesco Talotta
- Institut de Chimie Physique, UMR8000, CNRS, Université Paris-Saclay, 91405 Orsay, France.,Institut des Sciences Moléculaires d'Orsay, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | - Sabine Morisset
- Institut des Sciences Moléculaires d'Orsay, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | - Nathalie Rougeau
- Institut des Sciences Moléculaires d'Orsay, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | - David Lauvergnat
- Institut de Chimie Physique, UMR8000, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | - Federica Agostini
- Institut de Chimie Physique, UMR8000, CNRS, Université Paris-Saclay, 91405 Orsay, France
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21
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Talotta F, Morisset S, Rougeau N, Lauvergnat D, Agostini F. Spin-Orbit Interactions in Ultrafast Molecular Processes. PHYSICAL REVIEW LETTERS 2020; 124:033001. [PMID: 32031839 DOI: 10.1103/physrevlett.124.033001] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2019] [Indexed: 06/10/2023]
Abstract
We investigate spin-orbit interactions in ultrafast molecular processes employing the exact factorization of the electron-nuclear wave function. We revisit the original derivation by including spin-orbit coupling, and show how the dynamics driven by the time-dependent potential energy surface alleviates inconsistencies arising from different electronic representations. We propose a novel trajectory-based scheme to simulate spin-forbidden non-radiative processes, and we show its performance in the treatment of excited-state dynamics where spin-orbit effects couple different spin multiplets.
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Affiliation(s)
- Francesco Talotta
- Laboratoire de Chimie Physique, UMR 8000 CNRS/University Paris-Sud, University Paris-Saclay, 91405 Orsay, France
- Institut de Sciences Moléculaires d'Orsay, UMR 8214 CNRS/University Paris-Sud, University Paris-Saclay, 91405 Orsay, France
| | - Sabine Morisset
- Institut de Sciences Moléculaires d'Orsay, UMR 8214 CNRS/University Paris-Sud, University Paris-Saclay, 91405 Orsay, France
| | - Nathalie Rougeau
- Institut de Sciences Moléculaires d'Orsay, UMR 8214 CNRS/University Paris-Sud, University Paris-Saclay, 91405 Orsay, France
| | - David Lauvergnat
- Laboratoire de Chimie Physique, UMR 8000 CNRS/University Paris-Sud, University Paris-Saclay, 91405 Orsay, France
| | - Federica Agostini
- Laboratoire de Chimie Physique, UMR 8000 CNRS/University Paris-Sud, University Paris-Saclay, 91405 Orsay, France
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22
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Lian C, Zhang SJ, Hu SQ, Guan MX, Meng S. Ultrafast charge ordering by self-amplified exciton-phonon dynamics in TiSe 2. Nat Commun 2020; 11:43. [PMID: 31896745 PMCID: PMC6940384 DOI: 10.1038/s41467-019-13672-7] [Citation(s) in RCA: 34] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2019] [Accepted: 11/14/2019] [Indexed: 11/24/2022] Open
Abstract
The origin of charge density waves (CDWs) in TiSe\documentclass[12pt]{minimal}
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\begin{document}$${}_{2}$$\end{document}2 has long been debated, mainly due to the difficulties in identifying the timescales of the excitonic pairing and electron–phonon coupling (EPC). Without a time-resolved and microscopic mechanism, one has to assume simultaneous appearance of CDW and periodic lattice distortions (PLD). Here, we accomplish a complete separation of ultrafast exciton and PLD dynamics and unravel their interplay in our real-time time-dependent density functional theory simulations. We find that laser pulses knock off the exciton order and induce a homogeneous bonding–antibonding transition in the initial 20 fs, then the weakened electronic order triggers ionic movements antiparallel to the original PLD. The EPC comes into play after the initial 20 fs, and the two processes mutually amplify each other leading to a complete inversion of CDW ordering. The self-amplified dynamics reproduces the evolution of band structures in agreement with photoemission experiments. Hence we resolve the key processes in the initial dynamics of CDWs that help elucidate the underlying mechanism. The physical origins of charge density waves in 1T-TiSe2 and their response to ultrafast excitation have long been a topic of theoretical and experimental debate. Here the authors present an ab initio theory that successfully captures the observed dynamics of charge density wave formation.
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Affiliation(s)
- Chao Lian
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, P. R. China
| | - Sheng-Jie Zhang
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, P. R. China
| | - Shi-Qi Hu
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, P. R. China
| | - Meng-Xue Guan
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, P. R. China
| | - Sheng Meng
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, P. R. China. .,School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, P. R. China. .,Songshan Lake Materials Laboratory, Dongguan, Guangdong, 523808, P. R. China.
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23
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Ibele LM, Curchod BFE. A molecular perspective on Tully models for nonadiabatic dynamics. Phys Chem Chem Phys 2020; 22:15183-15196. [DOI: 10.1039/d0cp01353f] [Citation(s) in RCA: 38] [Impact Index Per Article: 9.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
Abstract
We present a series of standardized molecular tests for nonadiabatic dynamics, reminiscent of the one-dimensional Tully models proposed in 1990.
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Affiliation(s)
- Lea M. Ibele
- Department of Chemistry
- Durham University
- Durham DH1 3LE
- UK
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24
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Ibele LM, Nicolson A, Curchod BFE. Excited-state dynamics of molecules with classically driven trajectories and Gaussians. Mol Phys 2019. [DOI: 10.1080/00268976.2019.1665199] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
Affiliation(s)
- Lea M. Ibele
- Department of Chemistry, Durham University, Durham, UK
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25
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Lacombe L, Hoffmann NM, Maitra NT. Exact Potential Energy Surface for Molecules in Cavities. PHYSICAL REVIEW LETTERS 2019; 123:083201. [PMID: 31491208 DOI: 10.1103/physrevlett.123.083201] [Citation(s) in RCA: 21] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2019] [Indexed: 06/10/2023]
Abstract
We find and analyze the exact time-dependent potential energy surface driving the proton motion for a model of cavity-induced suppression of proton-coupled electron transfer. We show how, in contrast to the polaritonic surfaces, its features directly correlate to the proton dynamics and we discuss cavity modifications of its structure responsible for the suppression. The results highlight the interplay between nonadiabatic effects from coupling to photons and coupling to electrons and suggest caution is needed when applying traditional dynamics methods based on polaritonic surfaces.
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Affiliation(s)
- Lionel Lacombe
- Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
| | - Norah M Hoffmann
- Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science and Department of Physics, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Neepa T Maitra
- Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
- Physics Program and Chemistry Program, Graduate Center of the City University of New York, New York 10016, USA
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26
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Agostini F, Curchod BFE. Different flavors of nonadiabatic molecular dynamics. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2019. [DOI: 10.1002/wcms.1417] [Citation(s) in RCA: 42] [Impact Index Per Article: 8.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/03/2023]
Affiliation(s)
- Federica Agostini
- Laboratoire de Chimie Physique UMR 8000 CNRS/University Paris‐Sud Orsay France
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27
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Gossel GH, Lacombe L, Maitra NT. On the numerical solution of the exact factorization equations. J Chem Phys 2019; 150:154112. [PMID: 31005081 DOI: 10.1063/1.5090802] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The exact factorization (EF) approach to coupled electron-ion dynamics recasts the time-dependent molecular Schrödinger equation as two coupled equations, one for the nuclear wavefunction and one for the conditional electronic wavefunction. The potentials appearing in these equations have provided insight into non-adiabatic processes, and new practical non-adiabatic dynamics methods have been formulated starting from these equations. Here, we provide a first demonstration of a self-consistent solution of the exact equations, with a preliminary analysis of their stability and convergence properties. The equations have an unprecedented mathematical form, involving a Hamiltonian outside the class of Hermitian Hamiltonians usually encountered in time-propagation, and so the usual numerical methods for time-dependent Schrödinger fail when applied in a straightforward way to the EF equations. We find an approach that enables stable propagation long enough to witness non-adiabatic behavior in a model system before non-trivial instabilities take over. Implications for the development and analysis of EF-based methods are discussed.
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Affiliation(s)
- Graeme H Gossel
- Department of Physics and Astronomy, Hunter College and the City University of New York, 695 Park Avenue, New York, New York 10065, USA
| | - Lionel Lacombe
- Department of Physics and Astronomy, Hunter College and the City University of New York, 695 Park Avenue, New York, New York 10065, USA
| | - Neepa T Maitra
- Department of Physics and Astronomy, Hunter College and the City University of New York, 695 Park Avenue, New York, New York 10065, USA
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28
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Agostini F, Gross E, Curchod BF. Electron-nuclear entanglement in the time-dependent molecular wavefunction. COMPUT THEOR CHEM 2019. [DOI: 10.1016/j.comptc.2019.01.021] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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