1
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Huang DM, Green AT, Martens CC. A first principles derivation of energy-conserving momentum jumps in surface hopping simulations. J Chem Phys 2023; 159:214108. [PMID: 38047505 DOI: 10.1063/5.0178534] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2023] [Accepted: 11/06/2023] [Indexed: 12/05/2023] Open
Abstract
The fewest switches surface hopping (FSSH) method proposed by Tully in 1990 [Tully, J. Chem. Phys. 93, 1061 (1990)]-along with its many later variations-forms the basis for most practical simulations of molecular dynamics with electronic transitions in realistic systems. Despite its popularity, a rigorous formal derivation of the algorithm has yet to be achieved. In this paper, we derive the energy-conserving momentum jumps employed by FSSH from the perspective of quantum trajectory surface hopping (QTSH) [Martens, J. Phys. Chem. A 123, 1110 (2019)]. In the limit of localized nonadiabatic transitions, simple mathematical and physical arguments allow the FSSH algorithm to be derived from first principles. For general processes, the quantum forces characterizing the QTSH method provide accurate results for nonadiabatic dynamics with rigorous energy conservation, at the ensemble level, within the consistency of the underlying stochastic surface hopping without resorting to the artificial momentum rescaling of FSSH.
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Affiliation(s)
| | - Austin T Green
- University of California, Irvine, California 92697-2025, USA
| | - Craig C Martens
- University of California, Irvine, California 92697-2025, USA
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2
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Wu Y, Subotnik JE. A quantum-classical Liouville formalism in a preconditioned basis and its connection with phase-space surface hopping. J Chem Phys 2023; 158:024115. [PMID: 36641414 DOI: 10.1063/5.0124835] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/14/2023] Open
Abstract
We revisit a recent proposal to model nonadiabatic problems with a complex-valued Hamiltonian through a phase-space surface hopping (PSSH) algorithm employing a pseudo-diabatic basis. Here, we show that such a pseudo-diabatic PSSH (PD-PSSH) ansatz is consistent with a quantum-classical Liouville equation (QCLE) that can be derived following a preconditioning process, and we demonstrate that a proper PD-PSSH algorithm is able to capture some geometric magnetic effects (whereas the standard fewest switches surface hopping approach cannot capture such effects). We also find that a preconditioned QCLE can outperform the standard QCLE in certain cases, highlighting the fact that there is no unique QCLE. Finally, we also point out that one can construct a mean-field Ehrenfest algorithm using a phase-space representation similar to what is done for PSSH. These findings would appear extremely helpful as far as understanding and simulating nonadiabatic dynamics with complex-valued Hamiltonians and/or spin degeneracy.
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Affiliation(s)
- Yanze Wu
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Joseph E Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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3
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Solowan HP, Malý P, Brixner T. Direct comparison of molecular-beam versus liquid-phase pump-probe and two-dimensional spectroscopy on the example of azulene. J Chem Phys 2022; 157:044201. [DOI: 10.1063/5.0088365] [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
Although azulene's anomalous fluorescence originating from S2 rather than from S1 is the textbook example for the violation of Kasha's rule, the understanding of the underlying processes is still a subject of investigation. Here, we use action-based coherent two-dimensional electronic spectroscopy (2DES) to measure a single Liouville-space response pathway from S0 via S1 to the S2 state of azulene. We directly compare this sequential excitation in liquid phase detecting S2 fluorescence and in a molecular beam detecting photoionized cations, using the S2 anomalous emission to our advantage. We complement the 2DES study by pump-probe measurements of S1 excitation dynamics, including vibrational relaxation and passage through a conical intersection. The direct comparison of liquid and gas phase allows us to assess the effect of the solvent and the interplay of intra- and inter-molecular energy relaxation.
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Affiliation(s)
| | - Pavel Malý
- Institute of Physical and Theoretical Chemistry, Julius-Maximilians-Universität Würzburg, Germany
| | - Tobias Brixner
- Institut fuer Physikalische und Theoretische Chemie, Julius-Maximilians-Universität Würzburg, Germany
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4
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Subotnik J, Miao G, Bellonzi N, Teh HH, Dou W. A demonstration of consistency between the quantum classical Liouville equation and Berry’s phase and curvature for the case of complex Hamiltonians. J Chem Phys 2019; 151:074113. [DOI: 10.1063/1.5116210] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Joseph Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323, USA
| | - Gaohan Miao
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323, USA
| | - Nicole Bellonzi
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323, USA
| | - Hung-Hsuan Teh
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323, USA
| | - Wenjie Dou
- Department of Chemistry, University of California, Berkeley, California 94720, USA
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5
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Duan HG, Qi DL, Sun ZR, Miller RD, Thorwart M. Signature of the geometric phase in the wave packet dynamics on hypersurfaces. Chem Phys 2018. [DOI: 10.1016/j.chemphys.2018.09.016] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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6
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Qi DL, Duan HG, Sun ZR, Miller RJD, Thorwart M. Tracking an electronic wave packet in the vicinity of a conical intersection. J Chem Phys 2017; 147:074101. [DOI: 10.1063/1.4989462] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Da-Long Qi
- State Key Laboratory of Precision Spectroscopy, School of Physics and Material Science, East China Normal University, 3663 North Zhongshan Road, 200062 Shanghai, China
- Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Hong-Guang Duan
- Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
- I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
- The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Zhen-Rong Sun
- State Key Laboratory of Precision Spectroscopy, School of Physics and Material Science, East China Normal University, 3663 North Zhongshan Road, 200062 Shanghai, China
| | - R. J. Dwayne Miller
- Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
- The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
- The Departments of Chemistry and Physics, University of Toronto, 80 St. George Street, Toronto, Ontario M5S 3H6, Canada
| | - Michael Thorwart
- I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
- The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
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7
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Ryabinkin IG, Joubert-Doriol L, Izmaylov AF. Geometric Phase Effects in Nonadiabatic Dynamics near Conical Intersections. Acc Chem Res 2017; 50:1785-1793. [PMID: 28665584 DOI: 10.1021/acs.accounts.7b00220] [Citation(s) in RCA: 78] [Impact Index Per Article: 11.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
Abstract
Dynamical consideration that goes beyond the common Born-Oppenheimer approximation (BOA) becomes necessary when energy differences between electronic potential energy surfaces become small or vanish. One of the typical scenarios of the BOA breakdown in molecules beyond diatomics is a conical intersection (CI) of electronic potential energy surfaces. CIs provide an efficient mechanism for radiationless electronic transitions: acting as "funnels" for the nuclear wave function, they enable rapid conversion of the excessive electronic energy into the nuclear motion. In addition, CIs introduce nontrivial geometric phases (GPs) for both electronic and nuclear wave functions. These phases manifest themselves in change of the wave function signs if one considers an evolution of the system around the CI. This sign change is independent of the shape of the encircling contour and thus has a topological character. How these extra phases affect nonadiabatic dynamics is the main question that is addressed in this Account. We start by considering the simplest model providing the CI topology: two-dimensional two-state linear vibronic coupling model. Selecting this model instead of a real molecule has the advantage that various dynamical regimes can be easily modeled in the model by varying parameters, whereas any fixed molecule provides the system specific behavior that may not be very illustrative. After demonstrating when GP effects are important and how they modify the dynamics for two sets of initial conditions (starting from the ground and excited electronic states), we give examples of molecular systems where the described GP effects are crucial for adequate description of nonadiabatic dynamics. Interestingly, although the GP has a topological character, the extent to which accounting for GPs affect nuclear dynamics profoundly depends on topography of potential energy surfaces. Understanding an extent of changes introduced by the GP in chemical dynamics poses a problem of capturing GP effects by approximate methods of simulating nonadiabatic dynamics that can go beyond simple models. We assess the performance of both fully quantum (wave packet dynamics) and quantum-classical (surface-hopping, Ehrenfest, and quantum-classical Liouville equation) approaches in various cases where GP effects are important. It has been identified that the key to success in approximate methods is a method organization that prevents the quantum nuclear kinetic energy operator to act directly on adiabatic electronic wave functions.
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Affiliation(s)
- Ilya G. Ryabinkin
- Department
of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
- Chemical
Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
| | - Loïc Joubert-Doriol
- Department
of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
- Chemical
Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
| | - Artur F. Izmaylov
- Department
of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
- Chemical
Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
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8
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Kananenka AA, Hsieh CY, Cao J, Geva E. Accurate Long-Time Mixed Quantum-Classical Liouville Dynamics via the Transfer Tensor Method. J Phys Chem Lett 2016; 7:4809-4814. [PMID: 27934045 DOI: 10.1021/acs.jpclett.6b02389] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
In this Letter, we combine the recently introduced transfer tensor method with the mixed quantum-classical Liouville method. The resulting protocol provides an accurate, general, flexible and robust new route for simulating the reduced dynamics of the quantum subsystem for arbitrarily long times, starting with computationally feasible short-time mixed quantum-classical Liouville dynamical maps. The accuracy and feasibility of the methodology are demonstrated on a spin-boson benchmark model.
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Affiliation(s)
- Alexei A Kananenka
- Department of Chemistry, University of Michigan , Ann Arbor, Michigan 48109, United States
| | - Chang-Yu Hsieh
- Department of Chemistry, Massachusetts Institute of Technology , Cambridge, Massachusetts 02319, United States
| | - Jianshu Cao
- Department of Chemistry, Massachusetts Institute of Technology , Cambridge, Massachusetts 02319, United States
| | - Eitan Geva
- Department of Chemistry, University of Michigan , Ann Arbor, Michigan 48109, United States
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9
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Gherib R, Ye L, Ryabinkin IG, Izmaylov AF. On the inclusion of the diagonal Born-Oppenheimer correction in surface hopping methods. J Chem Phys 2016; 144:154103. [DOI: 10.1063/1.4945817] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Rami Gherib
- Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
| | - Liyuan Ye
- Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
| | - Ilya G. Ryabinkin
- Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
| | - Artur F. Izmaylov
- Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
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10
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Krčmář J, Gelin MF, Domcke W. Simulation of femtosecond two-dimensional electronic spectra of conical intersections. J Chem Phys 2015; 143:074308. [PMID: 26298135 DOI: 10.1063/1.4928685] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We have simulated femtosecond two-dimensional (2D) electronic spectra for an excited-state conical intersection using the wave-function version of the equation-of-motion phase-matching approach. We show that 2D spectra at fixed values of the waiting time provide information on the structure of the vibronic eigenstates of the conical intersection, while the evolution of the spectra with the waiting time reveals predominantly ground-state wave-packet dynamics. The results show that 2D spectra of conical intersection systems differ significantly from those obtained for chromophores with well separated excited-state potential-energy surfaces. The spectral signatures which can be attributed to conical intersections are discussed.
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Affiliation(s)
- Jindřich Krčmář
- Department of Chemistry, Technische Universität München, D-85747 Garching, Germany
| | - Maxim F Gelin
- Department of Chemistry, Technische Universität München, D-85747 Garching, Germany
| | - Wolfgang Domcke
- Department of Chemistry, Technische Universität München, D-85747 Garching, Germany
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11
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Kapral R. Quantum dynamics in open quantum-classical systems. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:073201. [PMID: 25634784 DOI: 10.1088/0953-8984/27/7/073201] [Citation(s) in RCA: 49] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Often quantum systems are not isolated and interactions with their environments must be taken into account. In such open quantum systems these environmental interactions can lead to decoherence and dissipation, which have a marked influence on the properties of the quantum system. In many instances the environment is well-approximated by classical mechanics, so that one is led to consider the dynamics of open quantum-classical systems. Since a full quantum dynamical description of large many-body systems is not currently feasible, mixed quantum-classical methods can provide accurate and computationally tractable ways to follow the dynamics of both the system and its environment. This review focuses on quantum-classical Liouville dynamics, one of several quantum-classical descriptions, and discusses the problems that arise when one attempts to combine quantum and classical mechanics, coherence and decoherence in quantum-classical systems, nonadiabatic dynamics, surface-hopping and mean-field theories and their relation to quantum-classical Liouville dynamics, as well as methods for simulating the dynamics.
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Affiliation(s)
- Raymond Kapral
- Department of Chemistry, Chemical Physics Theory Group, University of Toronto, Toronto, ON, M5S 3H6 Canada
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12
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Ryabinkin IG, Hsieh CY, Kapral R, Izmaylov AF. Analysis of geometric phase effects in the quantum-classical Liouville formalism. J Chem Phys 2014; 140:084104. [PMID: 24588145 DOI: 10.1063/1.4866366] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We analyze two approaches to the quantum-classical Liouville (QCL) formalism that differ in the order of two operations: Wigner transformation and projection onto adiabatic electronic states. The analysis is carried out on a two-dimensional linear vibronic model where geometric phase (GP) effects arising from a conical intersection profoundly affect nuclear dynamics. We find that the Wigner-then-Adiabatic (WA) QCL approach captures GP effects, whereas the Adiabatic-then-Wigner (AW) QCL approach does not. Moreover, the Wigner transform in AW-QCL leads to an ill-defined Fourier transform of double-valued functions. The double-valued character of these functions stems from the nontrivial GP of adiabatic electronic states in the presence of a conical intersection. In contrast, WA-QCL avoids this issue by starting with the Wigner transform of single-valued quantities of the full problem. As a consequence, GP effects in WA-QCL can be associated with a dynamical term in the corresponding equation of motion. Since the WA-QCL approach uses solely the adiabatic potentials and non-adiabatic derivative couplings as an input, our results indicate that WA-QCL can capture GP effects in two-state crossing problems using first-principles electronic structure calculations without prior diabatization or introduction of explicit phase factors.
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Affiliation(s)
- Ilya G Ryabinkin
- Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
| | - Chang-Yu Hsieh
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
| | - Raymond Kapral
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
| | - Artur F Izmaylov
- Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
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13
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Subotnik JE, Ouyang W, Landry BR. Can we derive Tully's surface-hopping algorithm from the semiclassical quantum Liouville equation? Almost, but only with decoherence. J Chem Phys 2014; 139:214107. [PMID: 24320364 DOI: 10.1063/1.4829856] [Citation(s) in RCA: 133] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this article, we demonstrate that Tully's fewest-switches surface hopping (FSSH) algorithm approximately obeys the mixed quantum-classical Liouville equation (QCLE), provided that several conditions are satisfied--some major conditions, and some minor. The major conditions are: (1) nuclei must be moving quickly with large momenta; (2) there cannot be explicit recoherences or interference effects between nuclear wave packets; (3) force-based decoherence must be added to the FSSH algorithm, and the trajectories can no longer rigorously be independent (though approximations for independent trajectories are possible). We furthermore expect that FSSH (with decoherence) will be most robust when nonadiabatic transitions in an adiabatic basis are dictated primarily by derivative couplings that are presumably localized to crossing regions, rather than by small but pervasive off-diagonal force matrix elements. In the end, our results emphasize the strengths of and possibilities for the FSSH algorithm when decoherence is included, while also demonstrating the limitations of the FSSH algorithm and its inherent inability to follow the QCLE exactly.
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Affiliation(s)
- Joseph E Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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14
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15
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Joubert-Doriol L, Ryabinkin IG, Izmaylov AF. Geometric phase effects in low-energy dynamics near conical intersections: A study of the multidimensional linear vibronic coupling model. J Chem Phys 2013; 139:234103. [DOI: 10.1063/1.4844095] [Citation(s) in RCA: 39] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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16
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Kelly A, Markland TE. Efficient and accurate surface hopping for long time nonadiabatic quantum dynamics. J Chem Phys 2013; 139:014104. [DOI: 10.1063/1.4812355] [Citation(s) in RCA: 79] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
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17
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Yonehara T, Takatsuka K. Path-Branching Representation for Nonadiabatic Electron Dynamics in Conical Intersection. J Phys Chem A 2013; 117:8599-608. [DOI: 10.1021/jp402655q] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Takehiro Yonehara
- Department of Basic
Sciences, Graduate School of Arts and Sciences, The University of Tokyo, Komaba, 153-8902 Tokyo, Japan
| | - Kazuo Takatsuka
- Department of Basic
Sciences, Graduate School of Arts and Sciences, The University of Tokyo, Komaba, 153-8902 Tokyo, Japan
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18
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Hsieh CY, Kapral R. Analysis of the forward-backward trajectory solution for the mixed quantum-classical Liouville equation. J Chem Phys 2013; 138:134110. [DOI: 10.1063/1.4798221] [Citation(s) in RCA: 61] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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19
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Jang S. Nonadiabatic quantum Liouville and master equations in the adiabatic basis. J Chem Phys 2012; 137:22A536. [DOI: 10.1063/1.4748142] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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20
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Malhado JP, Hynes JT. Photoisomerization for a model protonated Schiff base in solution: Sloped/peaked conical intersection perspective. J Chem Phys 2012; 137:22A543. [DOI: 10.1063/1.4754505] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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21
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Mandal A, Hunt KLC. Adiabatic and nonadiabatic contributions to the energy of a system subject to a time-dependent perturbation: complete separation and physical interpretation. J Chem Phys 2012; 137:164109. [PMID: 23126697 DOI: 10.1063/1.4750045] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state ∣0> of an unperturbed Hamiltonian H(0), the wave function acquires excited-state components ∣k> with coefficients c(k)(t) exp(-iE(k)t/ℏ), where E(k) denotes the energy of the unperturbed state ∣k>. It is well known that each coefficient c(k)(t) separates into an adiabatic term a(k)(t) that reflects the adjustment of the ground state to the perturbation--without actual transitions--and a nonadiabatic term b(k)(t) that yields the probability amplitude for a transition to the excited state. In this work, we prove that the energy at any time t also separates completely into adiabatic and nonadiabatic components, after accounting for the secular and normalization terms that appear in the solution of the time-dependent Schrödinger equation via Dirac's method of variation of constants. This result is derived explicitly through third order in the perturbation. We prove that the cross-terms between the adiabatic and nonadiabatic parts of c(k)(t) vanish, when the energy at time t is determined as an expectation value. The adiabatic term in the energy is identical to the total energy obtained from static perturbation theory, for a system exposed to the instantaneous perturbation λH'(t). The nonadiabatic term is a sum over excited states ∣k> of the transition probability multiplied by the transition energy. By evaluating the probabilities of transition to the excited eigenstates ∣k'(t)> of the instantaneous Hamiltonian H(t), we provide a physically transparent explanation of the result for E(t). To lowest order in the perturbation parameter λ, the probability of finding the system in state ∣k'(t)> is given by λ(2) ∣b(k)(t)∣(2). At third order, the transition probability depends on a second-order transition coefficient, derived in this work. We indicate expected differences between the results for transition probabilities obtained from this work and from Fermi's golden rule.
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Affiliation(s)
- Anirban Mandal
- Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA
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22
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Minezawa N, Gordon MS. Optimizing conical intersections of solvated molecules: The combined spin-flip density functional theory/effective fragment potential method. J Chem Phys 2012; 137:034116. [DOI: 10.1063/1.4734314] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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23
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Kelly A, van Zon R, Schofield J, Kapral R. Mapping quantum-classical Liouville equation: Projectors and trajectories. J Chem Phys 2012; 136:084101. [DOI: 10.1063/1.3685420] [Citation(s) in RCA: 82] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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24
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Zgrablić G, Novello AM, Parmigiani F. Population Branching in the Conical Intersection of the Retinal Chromophore Revealed by Multipulse Ultrafast Optical Spectroscopy. J Am Chem Soc 2011; 134:955-61. [DOI: 10.1021/ja205763x] [Citation(s) in RCA: 35] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Goran Zgrablić
- T-ReX Laboratory, Sincrotrone Trieste, S.S. 14 km 163.5 in Area Science
Park, I-34012 Basovizza Trieste, Italy
| | - Anna Maria Novello
- T-ReX Laboratory, Sincrotrone Trieste, S.S. 14 km 163.5 in Area Science
Park, I-34012 Basovizza Trieste, Italy
- Department of Condensed Matter
Physics, University of Geneva, Rue du Général-
Dufour 24, 1204 Geneva, Switzerland
| | - Fulvio Parmigiani
- T-ReX Laboratory, Sincrotrone Trieste, S.S. 14 km 163.5 in Area Science
Park, I-34012 Basovizza Trieste, Italy
- Department of Physics, Università degli studi di Trieste, Piazzale
Europa 1, I-34127 Trieste, Italy
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25
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Yonehara T, Hanasaki K, Takatsuka K. Fundamental Approaches to Nonadiabaticity: Toward a Chemical Theory beyond the Born–Oppenheimer Paradigm. Chem Rev 2011; 112:499-542. [DOI: 10.1021/cr200096s] [Citation(s) in RCA: 154] [Impact Index Per Article: 11.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Takehiro Yonehara
- Department of Basic Sciences, Graduate School of Arts and Sciences, The University of Tokyo Komaba, 153-8902 Tokyo, Japan
| | - Kota Hanasaki
- Department of Basic Sciences, Graduate School of Arts and Sciences, The University of Tokyo Komaba, 153-8902 Tokyo, Japan
| | - Kazuo Takatsuka
- Department of Basic Sciences, Graduate School of Arts and Sciences, The University of Tokyo Komaba, 153-8902 Tokyo, Japan
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26
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Chapman CT, Liang W, Li X. Open-system electronic dynamics and thermalized electronic structure. J Chem Phys 2011; 134:024118. [DOI: 10.1063/1.3526027] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
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