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Reppert M, Reppert D. Equivalence of quantum and classical third order response for weakly anharmonic coupled oscillators. J Chem Phys 2023; 158:114114. [PMID: 36948800 DOI: 10.1063/5.0135260] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/25/2023] Open
Abstract
Two-dimensional (2D) infrared (IR) spectra are commonly interpreted using a quantum diagrammatic expansion that describes the changes to the density matrix of quantum systems in response to light-matter interactions. Although classical response functions (based on Newtonian dynamics) have shown promise in computational 2D IR modeling studies, a simple diagrammatic description has so far been lacking. Recently, we introduced a diagrammatic representation for the 2D IR response functions of a single, weakly anharmonic oscillator and showed that the classical and quantum 2D IR response functions for this system are identical. Here, we extend this result to systems with an arbitrary number of bilinearly coupled, weakly anharmonic oscillators. As in the single-oscillator case, quantum and classical response functions are found to be identical in the weakly anharmonic limit or, in experimental terms, when the anharmonicity is small relative to the optical linewidth. The final form of the weakly anharmonic response function is surprisingly simple and offers potential computational advantages for application to large, multi-oscillator systems.
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Affiliation(s)
- Mike Reppert
- Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, USA
| | - Deborah Reppert
- Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, USA
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2
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Pollak E, Cao J. $\hbar ^{2}$ Expansion of the transmission probability through a barrier. J Chem Phys 2022; 157:074109. [DOI: 10.1063/5.0106649] [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
Ninety years ago, Wigner derived the leading order expansionterm in $\hbar ^{2}$ for the tunneling rate through a symmetric barrier. Hisderivation included two contributions, one came from the parabolic barrier,but a second term involved the fourth order derivative of the potential atthe barrier top. He left us with a challenge which is answered in thispaper, to derive the same but for an asymmetric barrier. A crucial elementof the derivation is obtaining the $\hbar^{2}$ expansion term for theprojection operator which appears in the flux-side expression for the rate. It is also reassuring that an analytical calculation of semiclassical transition state theory (SCTST) reproduces the anharmonic corrections to the leading order of $\hbar^2$. The efficacy of the resulting expression is demonstrated for an Eckartbarrier, leading to the conclusion that especially when considering heavy atom tunneling, one should use the expansion derived in this paper, ratherthan the parabolic barrier approximation. The rate expression derived here reveals how the classical TST limit is approached as a function of $\hbar$ and thus provides critical insights to understand the validity of popular approximate theories, such as the classical Wigner, centroid molecular dynamics and ring polymer molecular dynamics methods.
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Affiliation(s)
- Eli Pollak
- Department of Chemical Physics, Weizmann Institute of Science Faculty of Chemistry, Israel
| | - Jianshu Cao
- Department of Chemistry, MIT, United States of America
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Abstract
Multidimensional optical spectra are measured from the response of a material system to a sequence of laser pulses and have the capacity to elucidate specific molecular interactions and dynamics whose influences are absent or obscured in a conventional linear absorption spectrum. Interpretation of complex spectra is supported by theoretical modeling of the spectroscopic observable, requiring implementation of quantum dynamics for coupled electrons and nuclei. Performing numerically correct quantum dynamics in this context may pose computational challenges, particularly in the condensed phase. Semiclassical methods based on calculating classical trajectories offer a practical alternative. Here I review the recent application of some semiclassical, trajectory-based methods to nonlinear molecular vibrational and electronic spectra. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
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Affiliation(s)
- Roger F. Loring
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York, USA
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Polley K, Loring RF. Two-dimensional vibronic spectra from classical trajectories. J Chem Phys 2019; 150:164114. [DOI: 10.1063/1.5093911] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023] Open
Affiliation(s)
- Kritanjan Polley
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA
| | - Roger F. Loring
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA
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Reppert M, Brumer P. Classical coherent two-dimensional vibrational spectroscopy. J Chem Phys 2018; 148:064101. [DOI: 10.1063/1.5017985] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Mike Reppert
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
| | - Paul Brumer
- Chemical Physics Theory Group, Department of Chemistry, and Center for Quantum Information and Quantum Control, University of Toronto, Toronto, Ontario M5S 3H6, Canada
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Loring RF. Mean-trajectory approximation for electronic and vibrational-electronic nonlinear spectroscopy. J Chem Phys 2017; 146:144106. [DOI: 10.1063/1.4979621] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Roger F. Loring
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA
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Affiliation(s)
- Timothy J. H. Hele
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY, USA
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Hele TJH, Ananth N. Deriving the exact nonadiabatic quantum propagator in the mapping variable representation. Faraday Discuss 2016; 195:269-289. [DOI: 10.1039/c6fd00106h] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023]
Abstract
We derive an exact quantum propagator for nonadiabatic dynamics in multi-state systems using the mapping variable representation, where classical-like Cartesian variables are used to represent both continuous nuclear degrees of freedom and discrete electronic states. The resulting Liouvillian is a Moyal series that, when suitably approximated, can allow for the use of classical dynamics to efficiently model large systems. We demonstrate that different truncations of the exact Liouvillian lead to existing approximate semiclassical and mixed quantum–classical methods and we derive an associated error term for each method. Furthermore, by combining the imaginary-time path-integral representation of the Boltzmann operator with the exact Liouvillian, we obtain an analytic expression for thermal quantum real-time correlation functions. These results provide a rigorous theoretical foundation for the development of accurate and efficient classical-like dynamics to compute observables such as electron transfer reaction rates in complex quantized systems.
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Affiliation(s)
| | - Nandini Ananth
- Department of Chemistry and Chemical Biology
- Cornell University
- Ithaca
- USA
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Xu S, Du J, Wang C, Wang Z, Cui Y. Bonding and Spectral Character of Cr Doped Cdte, Cdse and Cds Nano-Clusters. JOURNAL OF CHEMICAL RESEARCH 2015. [DOI: 10.3184/174751915x14472431980683] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
The geometrical structures, bonding characteristics, Raman spectra, and molecular orbitals of hexagonal and tetrahedral Cr-doped CdX molecules have been investigated using density functional theory calculations at the B3LYP/LANL2DZ level. The structures of Cr-doped CdX molecules had obvious distortions compared with pure CdX clusters, especially CdCr2Se3 and CdCr2S3 clusters. The Mulliken charges of Cr atoms were negative and were lower than those of Cd atoms in the corresponding structures. Raman spectra showed that there was the vibration of the Cr–Cr bond in Cr-doped CdX molecules. Wiberg bond indices (WBI) values show that there were Cr–Cr bonds in some clusters. WBI values have also revealed that bonds between Cr and Te, Se, S were stronger than those between Cd and X atoms. Finally, molecular orbital observations indicate that the transitions in doped structures were from d orbital to d orbital, which is different from undoped structures.
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Affiliation(s)
- Shuhong Xu
- Advanced Photonics Center, School of Electronic Science and Engineering, Southeast University, Nanjing 210096, P.R. China
| | - Jinhua Du
- Advanced Photonics Center, School of Electronic Science and Engineering, Southeast University, Nanjing 210096, P.R. China
| | - Chunlei Wang
- Advanced Photonics Center, School of Electronic Science and Engineering, Southeast University, Nanjing 210096, P.R. China
| | - Zhuyuan Wang
- Advanced Photonics Center, School of Electronic Science and Engineering, Southeast University, Nanjing 210096, P.R. China
| | - Yiping Cui
- Advanced Photonics Center, School of Electronic Science and Engineering, Southeast University, Nanjing 210096, P.R. China
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Alemi M, Loring RF. Two-Dimensional Vibrational Spectroscopy of a Dissipative System with the Optimized Mean-Trajectory Approximation. J Phys Chem B 2015; 119:8950-9. [PMID: 25275943 PMCID: PMC4383732 DOI: 10.1021/jp5076884] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2014] [Revised: 09/10/2014] [Indexed: 11/30/2022]
Abstract
The optimized mean-trajectory (OMT) approximation is a semiclassical method for computing vibrational response functions from action-quantized classical trajectories connected by discrete transitions representing radiation-matter interactions. Here we apply this method to an anharmonic chromophore coupled to a harmonic bath. A forward-backward trajectory implementation of the OMT method is described that addresses the numerical challenges of applying the OMT to large systems with disparate frequency scales. The OMT is shown to well reproduce line shapes and waiting time dynamics in the pure dephasing limit of weak coupling to an off-resonant bath. The OMT is also shown to describe a case where energy transfer is the predominant source of line broadening.
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Affiliation(s)
- Mallory Alemi
- Department
of Chemistry and
Chemical Biology, Baker Laboratory, Cornell
University, Ithaca, New York 14853, United
States
| | - Roger F. Loring
- Department
of Chemistry and
Chemical Biology, Baker Laboratory, Cornell
University, Ithaca, New York 14853, United
States
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Alemi M, Loring RF. Vibrational coherence and energy transfer in two-dimensional spectra with the optimized mean-trajectory approximation. J Chem Phys 2015; 142:212417. [PMID: 26049437 DOI: 10.1063/1.4916644] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
Abstract
The optimized mean-trajectory (OMT) approximation is a semiclassical method for computing vibrational response functions from action-quantized classical trajectories connected by discrete transitions that represent radiation-matter interactions. Here, we extend the OMT to include additional vibrational coherence and energy transfer processes. This generalized approximation is applied to a pair of anharmonic chromophores coupled to a bath. The resulting 2D spectra are shown to reflect coherence transfer between normal modes.
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Affiliation(s)
- Mallory Alemi
- Baker Laboratory, Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
| | - Roger F Loring
- Baker Laboratory, Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
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Grossmann F. Quantum effects in intermediate-temperature dipole-dipole correlation-functions in the presence of an environment. J Chem Phys 2014; 141:144305. [DOI: 10.1063/1.4896835] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- F. Grossmann
- Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
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Gerace M, Loring RF. Two-dimensional spectroscopy of coupled vibrations with the optimized mean-trajectory approximation. J Phys Chem B 2013; 117:15452-61. [PMID: 23924378 PMCID: PMC3865215 DOI: 10.1021/jp405225g] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The optimized mean-trajectory (OMT) approximation is a semiclassical representation of the nonlinear vibrational response function used to compute multidimensional infrared spectra. In this method, response functions are calculated from a sequence of classical trajectories linked by discontinuities representing the effects of radiation-matter interactions, thus providing an approximation to quantum dynamics using classical inputs. This approach was previously formulated and assessed numerically for a single anharmonic degree of freedom. Our previous work is generalized here in two respects. First, the derivation of the OMT is extended to any number of coupled anharmonic vibrations by determining semiclassical approximations for pairs of double-sided Feynman diagrams. Second, an efficient numerical procedure is developed for calculating two-dimensional infrared spectra of coupled anharmonic vibrations in the OMT approximation. The OMT approximation is shown to reproduce the fundamental features of the quantum response function including both coherence and population dynamics.
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Affiliation(s)
- Mallory Gerace
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York, 14853, USA
| | - Roger F. Loring
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York, 14853, USA
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Gruenbaum SM, Loring RF. Semiclassical quantization in Liouville space for vibrational dynamics. J Phys Chem B 2011; 115:5148-56. [PMID: 21375227 DOI: 10.1021/jp104872r] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Semiclassical approximations to quantum mechanics can include quantum coherence effects in dynamical calculations based on classical mechanics. The Herman-Kluk (HK) semiclassical propagator has been demonstrated to reproduce quantum effects in nonlinear vibrational response functions of anharmonic oscillators but does not provide a practical numerical route to calculations for multiple degrees of freedom. In an HK calculation of a response function, quantum coherence effects enter through interference between pairs of classical trajectories. We have previously elucidated the mechanism by which the HK approximation reproduces quantum effects in response functions in the regime of quasiperiodic dynamics. We have applied this understanding to significantly simplify the semiclassical calculation of response functions in this dynamical regime. The phase space difference between trajectories is treated perturbatively in anharmonicity, allowing integration over these differences to be performed analytically and leaving integration over mean trajectories to be performed numerically. This mean-trajectory (MT) approximation has been applied to linear and nonlinear vibrational response functions for isolated and coupled anharmonic motions. Here, we derive an MT approximation for the Liouville space time evolution operator or superoperator that propagates the density operator. This analysis provides a form of the MT approximation that is readily applicable to other dynamical quantities besides response functions and clarifies the connection between semiclassical quantization of propagators for the wave function and for the density operator.
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Affiliation(s)
- Scott M Gruenbaum
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA
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