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Werther M, Choudhury SL, Großmann F. Coherent state based solutions of the time-dependent Schrödinger equation: hierarchy of approximations to the variational principle. INT REV PHYS CHEM 2020. [DOI: 10.1080/0144235x.2020.1823168] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Affiliation(s)
- Michael Werther
- Max-Planck-Institut für Physik Komplexer Systeme, Dresden, Germany
- Institut für Theoretische Physik, Technische Universität Dresden, Dresden, Germany
| | | | - Frank Großmann
- Institut für Theoretische Physik, Technische Universität Dresden, Dresden, Germany
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Stolyarov EV, White AJ, Mozyrsky D. Mixed quantum-classical approach to model non-adiabatic electron-nuclear dynamics: Detailed balance and improved surface hopping method. J Chem Phys 2020; 153:074116. [PMID: 32828087 DOI: 10.1063/5.0014284] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We develop a density matrix formalism to describe coupled electron-nuclear dynamics. To this end, we introduce an effective Hamiltonian formalism that describes electronic transitions and small (quantum) nuclear fluctuations along a classical trajectory of the nuclei. Using this Hamiltonian, we derive equations of motion for the electronic occupation numbers and for the nuclear coordinates and momenta. We show that, in the limit, when the number of nuclear degrees of freedom coupled to a given electronic transition is sufficiently high (i.e., the strong decoherence limit), the equations of motion for the electronic occupation numbers become Markovian. Furthermore, the transition rates in these (rate) equations are asymmetric with respect to the lower-to-higher energy transitions and vice versa. In thermal equilibrium, such asymmetry corresponds to the detailed balance condition. We also study the equations for the electronic occupations in the non-Markovian regime and develop a surface hopping algorithm based on our formalism. To treat the decoherence effects, we introduce additional "virtual" nuclear wave packets whose interference with the "real" (physical) wave packets leads to the reduction in coupling between the electronic states (i.e., decoherence) as well as to the phase shifts that improve the accuracy of the numerical approach. Remarkably, the same phase shifts lead to the detailed balance condition in the strong decoherence limit.
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Affiliation(s)
- E V Stolyarov
- Institute of Physics of the National Academy of Sciences of Ukraine, pr. Nauky 46, 03028 Kyiv, Ukraine
| | - A J White
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - D Mozyrsky
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Lévêque C, Madsen LB. Excitation spectra of systems of indistinguishable particles by the autocorrelation function technique: Circumventing the exponential scaling for bosons. J Chem Phys 2019; 150:194105. [DOI: 10.1063/1.5095991] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Camille Lévêque
- Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark
| | - Lars Bojer Madsen
- Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark
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Static Coherent States Method: One- and Two-Electron Laser-Induced Systems with Classical Nuclear Dynamics. APPLIED SCIENCES-BASEL 2018. [DOI: 10.3390/app8081252] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
In this report, we introduce the static coherent states (SCS) method for investigating quantum electron dynamics in a one- or two-electron laser-induced system. The SCS method solves the time-dependent Schrödinger equation (TDSE) both in imaginary and real times on the basis of a static grid of coherent states (CSs). Moreover, we consider classical dynamics for the nuclei by solving their Newtonian equations of motion. By implementing classical nuclear dynamics, we compute the electronic-state potential energy curves of H2+ in the absence and presence of an ultra-short intense laser field. We used this method to investigate charge migration in H2+. In particular, we found that the charge migration time increased exponentially with inter-nuclear distance. We also observed substantial charge localization for sufficiently long molecular bonds.
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Eidi M, Vafaee M, Rooein M. Complementary version of fermion coupled coherent states method and gram-schmidt algorithm: Theory and applications for electronic states of H2 and H2. J Comput Chem 2018; 39:679-684. [PMID: 29250810 DOI: 10.1002/jcc.25133] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2017] [Revised: 11/10/2017] [Accepted: 11/27/2017] [Indexed: 12/22/2022]
Abstract
In our previous report, we introduced a new version of Fermion coupled coherent states method (FCCS) which was especially suited for simulating the first symmetric spatial electronic state of two-electron systems. In this manuscript, we report a complementary version for FCCS method to simulate both of the first symmetric and antisymmetric spatial electronic states of two-electron systems. Moreover, the Gram-Schmidt orthogonalization process is employed to reach the excited states of the system. We apply this FCCS method and the original coupled coherent state method to simulate the energy of different electronic states of H2 and H2+, respectively. The results for the energy of computed electronic states of H2 and H2+ show a pretty good consistency with the exact values. © 2017 Wiley Periodicals, Inc.
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Affiliation(s)
- Mohammadreza Eidi
- Department of Chemistry, Tarbiat Modares University, P.O.Box 14115-175, Tehran, Iran
| | - Mohsen Vafaee
- Department of Chemistry, Tarbiat Modares University, P.O.Box 14115-175, Tehran, Iran
| | - Mitra Rooein
- Department of Chemistry, Tarbiat Modares University, P.O.Box 14115-175, Tehran, Iran
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Eidi M, Vafaee M, Niknam AR, Morshedian N. A new version of fermion coupled coherent states method: Theory and applications in simulation of two-electron systems. Chem Phys Lett 2016. [DOI: 10.1016/j.cplett.2016.04.054] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Kondorskiy AD, Nanbu S. Electronically nonadiabatic wave packet propagation using frozen Gaussian scattering. J Chem Phys 2015; 143:114103. [DOI: 10.1063/1.4930923] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Alexey D. Kondorskiy
- P. N. Lebedev Physical Institute of the Russian Academy of Sciences, 53, Leninsky Prospekt, Moscow 119991, Russia
| | - Shinkoh Nanbu
- Department of Materials and Life Sciences, Faculty of Science and Technology, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102-8554, Japan
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Tatchen J, Pollak E, Tao G, Miller WH. Renormalization of the frozen Gaussian approximation to the quantum propagator. J Chem Phys 2011; 134:134104. [DOI: 10.1063/1.3573566] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Reed SK, González-Martínez ML, Rubayo-Soneira J, Shalashilin DV. Cartesian coupled coherent states simulations: NenBr2 dissociation as a test case. J Chem Phys 2011; 134:054110. [DOI: 10.1063/1.3532407] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Thompson AL, Punwong C, Martínez TJ. Optimization of width parameters for quantum dynamics with frozen Gaussian basis sets. Chem Phys 2010. [DOI: 10.1016/j.chemphys.2010.03.020] [Citation(s) in RCA: 47] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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Shalashilin DV, Child MS, Kirrander A. Mechanisms of double ionization in strong laser field from simulation with Coupled Coherent States: Beyond reduced dimensionality models. Chem Phys 2008. [DOI: 10.1016/j.chemphys.2007.11.006] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Shalashilin DV, Child MS. Basis set sampling in the method of coupled coherent states: Coherent state swarms, trains, and pancakes. J Chem Phys 2008; 128:054102. [DOI: 10.1063/1.2828509] [Citation(s) in RCA: 64] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Wang L, Pollak E. Frozen Gaussian Wavepacket Study of the Ground State of the He Atom. J Chem Theory Comput 2007; 3:344-9. [PMID: 26637022 DOI: 10.1021/ct600332v] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The Rayleigh-Ritz functional is used in conjunction with an approximate time evolution to improve ab initio estimates of ground-state energies. The improvement is due in part to the introduction of a novel variational "normalization function" for the approximate propagator. An additional variational parameter was introduced in the form of a constant shift energy of the Hamiltonian. The approximate propagator used was the frozen Gaussian propagator; however, the trajectories evolved on the coherent-state averaged Hamiltonian (Q representation). For Coulombic forces, this removes the singularity, easing the computation. An additional variational parameter was the width parameter used for the coherent states appearing in the frozen Gaussian propagator. Using an initial combination of nine Gaussian functions for He, with an initial energy of -2.5115 au, the variational method, with a very short time interval of integration, led to an improved energy of -2.81 ± 0.04 au.
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Affiliation(s)
- Ling Wang
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel
| | - Eli Pollak
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel
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Martinazzo R, Nest M, Saalfrank P, Tantardini GF. A local coherent-state approximation to system-bath quantum dynamics. J Chem Phys 2006; 125:194102. [PMID: 17129084 DOI: 10.1063/1.2362821] [Citation(s) in RCA: 50] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A novel quantum method to deal with typical system-bath dynamical problems is introduced. Subsystem discrete variable representation and bath coherent-state sets are used to write down a multiconfigurational expansion of the wave function of the whole system. With the help of the Dirac-Frenkel variational principle, simple equations of motion--a kind of Schrodinger-Langevin equation for the subsystem coupled to (pseudo) classical equations for the bath--are derived. True dissipative dynamics at all times is obtained by coupling the bath to a secondary, classical Ohmic bath, which is modeled by adding a friction coefficient in the derived pseudoclassical bath equations. The resulting equations are then solved for a number of model problems, ranging from tunneling to vibrational relaxation dynamics. Comparison of the results with those of exact, multiconfiguration time-dependent Hartree calculations in systems with up to 80 bath oscillators shows that the proposed method can be very accurate and might be of help in studying realistic problems with very large baths. To this end, its linear scaling behavior with respect to the number of bath degrees of freedom is shown in practice with model calculations using tens of thousands of bath oscillators.
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Affiliation(s)
- Rocco Martinazzo
- Department of Physical Chemistry and Electrochemistry, University of Milan, Via Golgi 19, 20133 Milan, Italy.
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Sherratt PA, Shalashillin DV, Child MS. Description of multidimensional tunnelling with the help of coupled coherent states guided by classical Hamiltonians with quantum corrections. Chem Phys 2006. [DOI: 10.1016/j.chemphys.2005.06.050] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Shalashilin DV, Child MS. A version of diffusion Monte Carlo method based on random grids of coherent states. II. Six-dimensional simulation of electronic states of H2. J Chem Phys 2005; 122:224109. [PMID: 15974653 DOI: 10.1063/1.1926269] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We report a new version of the diffusion Monte Carlo (DMC) method, based on coherent-state quantum mechanics. Randomly selected grids of coherent states in phase space are used to obtain numerical imaginary time solutions of the Schrodinger equation, with an iterative refinement technique to improve the quality of the Monte Carlo grid. Accurate results were obtained, for the appropriately symmetrized two lowest states of the hydrogen molecule, by Monte Carlo sampling and six-dimensional propagation in the full phase space.
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Affiliation(s)
- Dmitrii V Shalashilin
- Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom.
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