1
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Chen S, Wu R, Fu HH. Persistent Chirality-Induced Spin-Selectivity Effect in Circular Helix Molecules. NANO LETTERS 2024; 24:6210-6217. [PMID: 38709107 DOI: 10.1021/acs.nanolett.4c00383] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/07/2024]
Abstract
The spin-orbit coupling (SOC), the dynamics of the nonequilibrium transport process, and the breaking of time-reversal and space-inversion symmetries have been regarded as key factors for the emergence of chirality-induced spin selectivity (CISS) and chirality-dependent spin currents in helix molecules. In this work, we demonstrated the generation of persistent CISS currents in various circular single-stranded DNAs and 310-helix proteins for the first time, regardless of whether an external magnetic flux is applied or not. This new CISS effect presents only in equilibrium transport processes, distinct from the traditional CISS observed in nonequilibrium transport processes and linear helix molecules; we term it as the PCISS effect. Notably, PCISS manifests irrespective of whether the SOC is chirality-driven or stems from heavy-metal substrates, making it an efficient way to generate chirality-locked pure spin currents. Our research establishes a novel paradigm for examining the underlying physics of the CISS effect.
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Affiliation(s)
- Song Chen
- School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China
| | - Ruqian Wu
- Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, United States
| | - Hua-Hua Fu
- School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China
- Institute for Quantum Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
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2
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Tao Z, Qiu T, Subotnik JE. Symmetric Post-Transition State Bifurcation Reactions with Berry Pseudomagnetic Fields. J Phys Chem Lett 2023; 14:770-778. [PMID: 36652556 DOI: 10.1021/acs.jpclett.2c02668] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
We investigate how the Berry force (i.e., the pseudomagnetic force operating on nuclei as induced by electronic degeneracy and spin-orbit coupling (SOC)) might modify a post-transition state bifurcation (PTSB) reaction path and affect product selectivity for situations when multiple products share the same transition state. To estimate the magnitude of this effect, Langevin dynamics are performed on a model system with a valley-ridge inflection (VRI) point in the presence of a magnetic field (that mimics the Berry curvature). We also develop an analytic model for such selectivity that depends on key parameters such as the surface topology, the magnitude of the Berry force, and the nuclear friction. Within this dynamical model, static electronic structure calculations (at the level of generalized Hartree-Fock with spin-orbit coupling (GHF+SOC) theory) suggest that electronic spin induced Berry force effects may indeed lead to noticeable changes in methoxy radical isomerization.
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Affiliation(s)
- Zhen Tao
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States
| | - Tian Qiu
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States
| | - Joseph E Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States
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3
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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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4
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Chandran SS, Wu Y, Subotnik JE. Effect of Duschinskii Rotations on Spin-Dependent Electron Transfer Dynamics. J Phys Chem A 2022; 126:9535-9552. [DOI: 10.1021/acs.jpca.2c06149] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Affiliation(s)
- Suraj S. Chandran
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States
| | - Yanze Wu
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States
| | - Joseph E. Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States
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5
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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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6
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Bian X, Qiu T, Chen J, Subotnik JE. On the meaning of Berry force for unrestricted systems treated with mean-field electronic structure. J Chem Phys 2022; 156:234107. [PMID: 35732536 DOI: 10.1063/5.0093092] [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
We show that the Berry force as computed by an approximate, mean-field electronic structure can be meaningful if properly interpreted. In particular, for a model Hamiltonian representing a molecular system with an even number of electrons interacting via a two-body (Hubbard) interaction and a spin-orbit coupling, we show that a meaningful nonzero Berry force emerges whenever there is spin unrestriction-even though the Hamiltonian is real-valued and formally the on-diagonal single-surface Berry force must be zero. Moreover, if properly applied, this mean-field Berry force yields roughly the correct asymptotic motion for scattering through an avoided crossing. That being said, within the context of a ground-state calculation, several nuances do arise as far interpreting the Berry force correctly, and as a practical matter, the Berry force diverges near the Coulson-Fischer point (which can lead to numerical instabilities). We do not address magnetic fields here.
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Affiliation(s)
- Xuezhi Bian
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Tian Qiu
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Junhan Chen
- 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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7
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Chandran SS, Wu Y, Teh HH, Waldeck DH, Subotnik JE. Electron transfer and spin-orbit coupling: Can nuclear motion lead to spin selective rates? J Chem Phys 2022; 156:174113. [PMID: 35525658 DOI: 10.1063/5.0086554] [Citation(s) in RCA: 13] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate a spin-boson inspired model of electron transfer, where the diabatic coupling is given by a position-dependent phase, eiWx. We consider both equilibrium and nonequilibrium initial conditions. We show that, for this model, all equilibrium results are completely invariant to the sign of W (to infinite order). However, the nonequilibrium results do depend on the sign of W, suggesting that photo-induced electron transfer dynamics with spin-orbit coupling can exhibit electronic spin polarization (at least for some time).
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Affiliation(s)
- Suraj S Chandran
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Yanze Wu
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Hung-Hsuan Teh
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - David H Waldeck
- Department of Chemistry, University of Pittsburgh, Pennsylvania 15260, USA
| | - Joseph E Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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8
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Wu Y, Subotnik JE. Semiclassical description of nuclear dynamics moving through complex-valued single avoided crossings of two electronic states. J Chem Phys 2021; 154:234101. [PMID: 34241259 DOI: 10.1063/5.0054014] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The standard fewest-switches surface hopping (FSSH) approach fails to model nonadiabatic dynamics when the electronic Hamiltonian is complex-valued and there are multiple nuclear dimensions; FSSH does not include geometric magnetic effects and does not have access to a gauge independent direction for momentum rescaling. In this paper, for the case of a Hamiltonian with two electronic states, we propose an extension of Tully's FSSH algorithm, which includes geometric magnetic forces and, through diabatization, establishes a well-defined rescaling direction. When combined with a decoherence correction, our new algorithm shows satisfying results for a model set of two-dimensional single avoided crossings.
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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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9
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Bian X, Wu Y, Teh HH, Zhou Z, Chen HT, Subotnik JE. Modeling nonadiabatic dynamics with degenerate electronic states, intersystem crossing, and spin separation: A key goal for chemical physics. J Chem Phys 2021; 154:110901. [DOI: 10.1063/5.0039371] [Citation(s) in RCA: 19] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022] Open
Affiliation(s)
- Xuezhi Bian
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Yanze Wu
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Hung-Hsuan Teh
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Zeyu Zhou
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Hsing-Ta Chen
- 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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10
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Abstract
When nonadiabatic dynamics are described on the basis of trajectories, severe trajectory branching occurs when the nuclear wave packets on some potential energy surfaces are reflected while those on the remaining surfaces are not. As a result, the traditional Ehrenfest mean field (EMF) approximation breaks down. In this study, two versions of the branching corrected mean field (BCMF) method are proposed. Namely, when trajectory branching is identified, BCMF stochastically selects either the reflected or the nonreflected group to build the new mean field trajectory or splits the mean field trajectory into two new trajectories with the corresponding weights. As benchmarked in six standard model systems and an extensive model base with two hundred diverse scattering models, BCMF significantly improves the accuracy while retaining the high efficiency of the traditional EMF. In fact, BCMF closely reproduces the exact quantum dynamics in all investigated systems, thus highlighting the essential role of branching correction in nonadiabatic dynamics simulations of general systems.
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Affiliation(s)
- Jiabo Xu
- Center for Chemistry of Novel & High-Performance Materials, and Department of Chemistry, Zhejiang University, Hangzhou 310027, China
| | - Linjun Wang
- Center for Chemistry of Novel & High-Performance Materials, and Department of Chemistry, Zhejiang University, Hangzhou 310027, China
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11
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Miao G, Bian X, Zhou Z, Subotnik J. A "backtracking" correction for the fewest switches surface hopping algorithm. J Chem Phys 2020; 153:111101. [PMID: 32962370 DOI: 10.1063/5.0022436] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We propose a "backtracking" mechanism within Tully's fewest switches surface hopping (FSSH) algorithm, whereby whenever one detects consecutive (double) hops during a short period of time, one simply rewinds the dynamics backward in time. In doing so, one reduces the number of hopping events and comes closer to a truly fewest switches surface hopping approach with independent trajectories. With this algorithmic change, we demonstrate that surface hopping can be reasonably accurate for nuclear dynamics in a multidimensional configuration space with a complex-valued (i.e., not real-valued) electronic Hamiltonian; without this adjustment, surface hopping often fails. The added computational cost is marginal. Future research will be needed to assess whether or not this backtracking correction can improve the accuracy of a typical FSSH calculation with a real-valued electronic Hamiltonian (that ignores spin).
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Affiliation(s)
- Gaohan Miao
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Xuezhi Bian
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Zeyu Zhou
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Joseph Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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12
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Wu Y, Miao G, Subotnik JE. Chemical Reaction Rates for Systems with Spin-Orbit Coupling and an Odd Number of Electrons: Does Berry's Phase Lead to Meaningful Spin-Dependent Nuclear Dynamics for a Two State Crossing? J Phys Chem A 2020; 124:7355-7372. [PMID: 32869999 DOI: 10.1021/acs.jpca.0c04562] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023]
Abstract
Within the context of a simple avoided crossing, we investigate the effect of a complex-valued diabatic coupling in determining spin-dependent rate constants and scattering states. We find that, if the molecular geometry is not linear and the Berry force is not zero, one can find significant spin polarization of the products. This study emphasizes that, when analyzing nonadiabatic reactions with spin orbit coupling (and a complex-valued Hamiltonian), one must consider how Berry force affects nuclear motion-at least in the context of gas phase reactions. Work is currently ongoing as far as extrapolating these conclusions to the condensed phase, where interesting spin selection has been observed in recent years.
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Affiliation(s)
- Yanze Wu
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States
| | - Gaohan Miao
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States
| | - Joseph E Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States
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13
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Chen HT, Zhou Z, Subotnik JE. On the proper derivation of the Floquet-based quantum classical Liouville equation and surface hopping describing a molecule or material subject to an external field. J Chem Phys 2020; 153:044116. [PMID: 32752688 DOI: 10.1063/5.0013873] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate different approaches to derive the proper Floquet-based quantum-classical Liouville equation (F-QCLE) for laser-driven electron-nuclear dynamics. The first approach projects the operator form of the standard QCLE onto the diabatic Floquet basis and then transforms to the adiabatic representation. The second approach directly projects the QCLE onto the Floquet adiabatic basis. Both approaches yield a form that is similar to the usual QCLE with two modifications: (1) The electronic degrees of freedom are expanded to infinite dimension and (2) the nuclear motion follows Floquet quasi-energy surfaces. However, the second approach includes an additional cross derivative force due to the dual dependence on time and nuclear motion of the Floquet adiabatic states. Our analysis and numerical tests indicate that this cross derivative force is a fictitious artifact, suggesting that one cannot safely exchange the order of Floquet state projection with adiabatic transformation. Our results are in accord with similar findings by Izmaylov et al., [J. Chem. Phys. 140, 084104 (2014)] who found that transforming to the adiabatic representation must always be the last operation applied, although now we have extended this result to a time-dependent Hamiltonian. This paper and the proper derivation of the F-QCLE should lay the basis for further improvements of Floquet surface hopping.
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Affiliation(s)
- Hsing-Ta Chen
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Zeyu Zhou
- 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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14
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Zhou Z, Jin Z, Qiu T, Rappe AM, Subotnik JE. A Robust and Unified Solution for Choosing the Phases of Adiabatic States as a Function of Geometry: Extending Parallel Transport Concepts to the Cases of Trivial and Near-Trivial Crossings. J Chem Theory Comput 2020; 16:835-846. [PMID: 31869225 DOI: 10.1021/acs.jctc.9b00952] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
We investigate a simple and robust scheme for choosing the phases of adiabatic electronic states smoothly (as a function of geometry) so as to maximize the performance of ab initio non-adiabatic dynamics methods. Our approach is based upon consideration of the overlap matrix (U) between basis functions at successive points in time and selecting the phases so as to minimize the matrix norm of log(U). In so doing, one can extend the concept of parallel transport to cases with sharp curve crossings. We demonstrate that this algorithm performs well under extreme situations where dozens of states cross each other either through trivial crossings (where there is zero effective diabatic coupling), or through non-trivial crossings (when there is a non-zero diabatic coupling), or through a combination of both. In all cases, we compute the time-derivative coupling matrix elements (or equivalently non-adiabatic derivative coupling matrix elements) that are as smooth as possible. Our results should be of interest to all who are interested in either non-adiabatic dynamics, or more generally, parallel transport in large systems.
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Affiliation(s)
- Zeyu Zhou
- Department of Chemistry , University of Pennsylvania , Philadelphia , Pennsylvania 19104 , United States
| | - Zuxin Jin
- Department of Chemistry , University of Pennsylvania , Philadelphia , Pennsylvania 19104 , United States
| | - Tian Qiu
- Department of Chemistry , University of Pennsylvania , Philadelphia , Pennsylvania 19104 , United States
| | - Andrew M Rappe
- Department of Chemistry , University of Pennsylvania , Philadelphia , Pennsylvania 19104 , United States
| | - Joseph Eli Subotnik
- Department of Chemistry , University of Pennsylvania , Philadelphia , Pennsylvania 19104 , United States
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15
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Bellonzi N, Alguire E, Fatehi S, Shao Y, Subotnik JE. TD-DFT spin-adiabats with analytic nonadiabatic derivative couplings. J Chem Phys 2020; 152:044112. [PMID: 32007078 PMCID: PMC7043850 DOI: 10.1063/1.5126440] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2019] [Accepted: 12/22/2019] [Indexed: 11/14/2022] Open
Abstract
We present an algorithm for efficient calculation of analytic nonadiabatic derivative couplings between spin-adiabatic, time-dependent density functional theory states within the Tamm-Dancoff approximation. Our derivation is based on the direct differentiation of the Kohn-Sham pseudowavefunction using the framework of Ou et al. Our implementation is limited to the case of a system with an even number of electrons in a closed shell ground state, and we validate our algorithm against finite difference at an S1/T2 crossing of benzaldehyde. Through the introduction of a magnetic field spin-coupling operator, we break time-reversal symmetry to generate complex valued nonadiabatic derivative couplings. Although the nonadiabatic derivative couplings are complex valued, we find that a phase rotation can generate an almost entirely real-valued derivative coupling vector for the case of benzaldehyde.
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Affiliation(s)
- Nicole Bellonzi
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | | | - Shervin Fatehi
- Department of Chemistry, The University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA
| | - Yihan Shao
- Department of Chemistry and Biochemistry, University of Oklahoma, Norman, Oklahoma 73019, USA
| | - Joseph E. Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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