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).
REFERENCES
1.
A. J.
Leggett
, S.
Chakravarty
, A. T.
Dorsey
, M. P. A.
Fisher
, A.
Garg
, and W.
Zwerger
, “Dynamics of the dissipative two-state system
,” Rev. Mod. Phys.
59
, 1
–85
(1987
).2.
M.
Bixon
and J.
Jortner
, “From isolated molecules to biomolecules
,” Adv. Chem. Phys.
106
, 35
–203
(1999
).3.
R. A.
Marcus
, “On the theory of oxidation-reduction reactions involving electron transfer. I
,” J. Chem. Phys.
24
, 966
–978
(1956
).4.
A.
Nitzan
, Chemical Dynamics in Condensed Phases
(Oxford University Press
, 2006
).5.
L. D.
Zusman
, “Outer-sphere electron transfer in polar solvents
,” Chem. Phys.
49
, 295
–304
(1980
).6.
J. T.
Hynes
, “Outer-sphere electron transfer reactions and frequency-dependent friction
,” J. Phys. Chem.
90
, 3701
–3706
(1986
).7.
J. E.
Straub
and B. J.
Berne
, “A statistical theory for the effect of nonadiabatic transitions on activated processes
,” J. Chem. Phys.
87
, 6111
–6116
(1987
).8.
E. S.
Medvedev
and A. A.
Stuchebrukhov
, “Inelastic tunneling in long-distance biological electron transfer reactions
,” J. Chem. Phys.
107
, 3821
–3831
(1997
).9.
S.
Jang
and M. D.
Newton
, “Theory of torsional non-condon electron transfer: A generalized spin-boson Hamiltonian and its nonadiabatic limit solution
,” J. Chem. Phys.
122
, 024501
(2005
).10.
A. O.
Lykhin
, D. S.
Kaliakin
, G. E.
dePolo
, A. A.
Kuzubov
, and S. A.
Varganov
, “Nonadiabatic transition state theory: Application to intersystem crossings in the active sites of metal-sulfur proteins
,” Int. J. Quantum Chem.
116
, 750
–761
(2016
).11.
T. J.
Penfold
, E.
Gindensperger
, C.
Daniel
, and C. M.
Marian
, “Spin-vibronic mechanism for intersystem crossing
,” Chem. Rev.
118
, 6975
–7025
(2018
).12.
C. M.
Marian
, “Spin-orbit coupling in molecules
,” Rev. Comput. Chem.
17
, 99
–204
(2001
).13.
C. M.
Marian
, “Spin–orbit coupling and intersystem crossing in molecules
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
2
, 187
–203
(2012
).14.
M. V.
Berry
and J. M.
Robbins
, “Chaotic classical and half-classical adiabatic reactions: Geometric magnetism and deterministic friction
,” Proc. R. Soc. London, Ser. A
442
, 659
–672
(1993
).15.
J.
Subotnik
, G.
Miao
, N.
Bellonzi
, H.-H.
Teh
, and W.
Dou
, “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.
151
, 074113
(2019
).16.
J.-T.
Lü
, M.
Brandbyge
, and P.
Hedegård
, “Blowing the fuse: Berry’s phase and runaway vibrations in molecular conductors
,” Nano Lett.
10
, 1657
–1663
(2010
).17.
Y.
Wu
and J. E.
Subotnik
, “Electronic spin separation induced by nuclear motion near conical intersections
,” Nat. Commun.
12
, 700
(2021
).18.
Y.
Georgievskii
and A. A.
Stuchebrukhov
, “Concerted electron and proton transfer: Transition from nonadiabatic to adiabatic proton tunneling
,” J. Chem. Phys.
113
, 10438
–10450
(2000
).19.
R. G.
Littlejohn
and W. G.
Flynn
, “Geometric phases in the asymptotic theory of coupled wave equations
,” Phys. Rev. A
44
, 5239
(1991
).20.
S.
Weigert
and R. G.
Littlejohn
, “Diagonalization of multicomponent wave equations with a Born-Oppenheimer example
,” Phys. Rev. A
47
, 3506
(1993
).21.
P. F.
Barbara
, T. J.
Meyer
, and M. A.
Ratner
, “Contemporary issues in electron transfer research
,” J. Phys. Chem.
100
, 13148
–13168
(1996
).22.
R. D.
Coalson
, D. G.
Evans
, and A.
Nitzan
, “A nonequilibrium golden rule formula for electronic state populations in nonadiabatically coupled systems
,” J. Chem. Phys.
101
, 436
–448
(1994
).23.
R.
Naaman
and D. H.
Waldeck
, “Spintronics and chirality: Spin selectivity in electron transport through chiral molecules
,” Annu. Rev. Phys. Chem.
66
, 263
–281
(2015
).24.
R.
Naaman
, Y.
Paltiel
, and D. H.
Waldeck
, “Chiral molecules and the electron spin
,” Nat. Rev. Chem.
3
, 250
–260
(2019
).25.
B.
Göhler
, V.
Hamelbeck
, T. Z.
Markus
, M.
Kettner
, G. F.
Hanne
, Z.
Vager
, R.
Naaman
, and H.
Zacharias
, “Spin selectivity in electron transmission through self-assembled monolayers of double-stranded DNA
,” Science
331
, 894
–897
(2011
).26.
M. S.
Zöllner
, A.
Saghatchi
, V.
Mujica
, and C.
Herrmann
, “Influence of electronic structure modeling and junction structure on first-principles chiral induced spin selectivity
,” J. Chem. Theory Comput.
16
, 7357
–7371
(2020
).27.
E.
Medina
, L. A.
González-Arraga
, D.
Finkelstein-Shapiro
, B.
Berche
, and V.
Mujica
, “Continuum model for chiral induced spin selectivity in helical molecules
,” J. Chem. Phys.
142
, 194308
(2015
).28.
J.
Gersten
, K.
Kaasbjerg
, and A.
Nitzan
, “Induced spin filtering in electron transmission through chiral molecular layers adsorbed on metals with strong spin-orbit coupling
,” J. Chem. Phys.
139
, 114111
(2013
).29.
Y.
Wu
, G.
Miao
, and J. E.
Subotnik
, “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
124
, 7355
–7372
(2020
).30.
J.
Fransson
, “Vibrational origin of exchange splitting and chiral-induced spin selectivity
,” Phys. Rev. B
102
, 235416
(2020
).31.
G.-F.
Du
, H.-H.
Fu
, and R.
Wu
, “Vibration-enhanced spin-selective transport of electrons in the DNA double helix
,” Phys. Rev. B
102
, 035431
(2020
).32.
L.
Zhang
, Y.
Hao
, W.
Qin
, S.
Xie
, and F.
Qu
, “Chiral-induced spin selectivity: A polaron transport model
,” Phys. Rev. B
102
, 214303
(2020
).33.
T. P.
Fay
, “Chirality-induced spin coherence in electron transfer reactions
,” J. Phys. Chem. Lett.
12
, 1407
–1412
(2021
).34.
T. P.
Fay
and D. T.
Limmer
, “Origin of chirality induced spin selectivity in photoinduced electron transfer
,” Nano Lett.
21
, 6696
–6702
(2021
).35.
36.
H.-H.
Teh
, W.
Dou
, and J. E.
Subotnik
, “Spin polarization through a molecular junction based on nuclear Berry curvature effects
,” arXiv:2111.12815.37.
A. F.
Izmaylov
, D.
Mendive–Tapia
, M. J.
Bearpark
, M. A.
Robb
, J. C.
Tully
, and M. J.
Frisch
, “Nonequilibrium Fermi golden rule for electronic transitions through conical intersections
,” J. Chem. Phys.
135
, 234106
(2011
).38.
From Eq. (3.14), the nonequilibrium shift of the bath enters only in the dephasing function s(t). Thus, regardless of how far from equilibrium the bath is, the electron transfer rate is bounded byTherefore, for this particular model, whenever an equilibrium Fermi Golden Rule dynamics is valid for a thermalized bath, a nonequilibrium FGR calculation should also be valid – even for large initial displacements from equilibrium.
39.
T. K.
Das
, F.
Tassinari
, R.
Naaman
, and J.
Fransson
, “The temperature-dependent chiral-induced spin selectivity effect: Experiments and theory
,” J. Phys. Chem. C
, 126
(6
), 3257
–3264
(2022
).40.
H.
Koppel
, in Conical Intersections: Electronic Structure, Dynamics and Spectroscopy
, edited by W.
Domcke
, D. R.
Yarkony
, and H.
Koppel
(World Scientific Publishing Co.
, New Jersey
, 2004
), pp. 175
–204
.41.
D. R.
Yarkony
, in Conical Intersections: Electronic Structure, Dynamics and Spectroscopy
, edited by W.
Domcke
, D. R.
Yarkony
, and H.
Koppel
(World Scientific Publishing Co.
, New Jersey
, 2004
), pp. 41
–128
.42.
D. G.
Truhlar
and C. A.
Mead
, “Relative likelihood of encountering conical intersections and avoided intersections on the potential energy surfaces of polyatomic molecules
,” Phys. Rev. A
68
, 032501
(2003
).43.
D.
Truhlar
and C.
Mead
, “Comment on ‘Optical conversion of conical intersection to avoided crossing’ by Y. Arasaki and K. Takatsuka, Phys. Chem. Chem. Phys. 2010, 12, 1239
,” Phys. Chem. Chem. Phys.
13
, 4754
(2011
);
[PubMed]
Y.
Arasaki
and K.
Takatsuka
, “Reply to the “‘Comment on ‘Optical conversion of conical intersection to avoided crossing’” by D. G. Truhlar and C. A. Mead, Phys. Chem. Chem. Phys. 2011, 13
,” Phys. Chem. Chem. Phys.
13
, 4756
(2011
).44.
B.
Yang
, L.
Gagliardi
, and D. G.
Truhlar
, “Transition states of spin-forbidden reactions
,” Phys. Chem. Chem. Phys.
20
, 4129
–4136
(2018
).45.
R.
Ianconescu
and E.
Pollak
, “Photoinduced cooling of polyatomic molecules in an electronically excited state in the presence of Dushinskii rotations
,” J. Phys. Chem. A
108
, 7778
–7784
(2004
).46.
R.
Torres-Cavanillas
, G.
Escorcia-Ariza
, I.
Brotons-Alcázar
, R.
Sanchis-Gual
, P. C.
Mondal
, L. E.
Rosaleny
, S.
Giménez-Santamarina
, M.
Sessolo
, M.
Galbiati
, S.
Tatay
, A.
Gaita-Ariño
, A.
Forment-Aliaga
, and S.
Cardona-Serra
, “Reinforced room-temperature spin filtering in chiral paramagnetic metallopeptides
,” J. Am. Chem. Soc.
142
, 17572
–17580
(2020
).47.
M.
Kettner
, V. V.
Maslyuk
, D.
Nurenberg
, J.
Seibel
, R.
Gutierrez
, G.
Cuniberti
, K.-H.
Ernst
, and H.
Zacharias
, “Chirality-dependent electron spin filtering by molecular monolayers of helicenes
,” J. Phys. Chem. Lett.
9
, 2025
–2030
(2018
).48.
R.
Naaman
and D. H.
Waldeck
, “Chiral-induced spin selectivity effect
,” J. Phys. Chem. Lett.
3
, 2178
–2187
(2012
).49.
S.
Mai
, P.
Marquetand
, and L.
González
, “A general method to describe intersystem crossing dynamics in trajectory surface hopping
,” Int. J. Quantum Chem.
115
, 1215
–1231
(2015
).50.
M. R.
Hoffmann
and G. C.
Schatz
, “Theoretical studies of intersystem crossing effects in the O+H2 reaction
,” J. Chem. Phys.
113
, 9456
–9465
(2000
).51.
X.
Bian
, Y.
Wu
, H.-H.
Teh
, Z.
Zhou
, H.-T.
Chen
, and J. E.
Subotnik
, “Modeling nonadiabatic dynamics with degenerate electronic states, intersystem crossing, and spin separation: A key goal for chemical physics
,” J. Chem. Phys.
154
, 110901
(2021
).52.
H.
Guo
and D. R.
Yarkony
, “Accurate nonadiabatic dynamics
,” Phys. Chem. Chem. Phys.
18
, 26335
–26352
(2016
).53.
C.
Xie
, J.
Ma
, X.
Zhu
, D. R.
Yarkony
, D.
Xie
, and H.
Guo
, “Nonadiabatic tunneling in photodissociation of phenol
,” J. Am. Chem. Soc.
138
, 7828
–7831
(2016
).54.
Interestingly, even for a two-state purely real-valued Hamiltonian with a conical intersection, the notion of complex-valued wavefunctions was suggested long ago by Mead and Truhlar.55 The basic premise was that, as you circle a conical intersection, the adiabatic states can be chosen in a well-defined manner by not implementing parallel transport and instead choosing electronic states with a complex-valued phase. This approach has been used by Gherib et al. recently to effectively “isolate” the effect of geometric phase.56 Nevertheless, we must emphasize that, for a two-state real-valued Hamiltonian, there is no Berry force: the Berry force is independent of the phase (or so-called “gauge”) of the electronic adiabats. The Mead–Truhlar mapping in Ref. 55 allows for computational ease without introducing any new physics.
55.
C. A.
Mead
and D. G.
Truhlar
, “On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei
,” J. Chem. Phys.
70
, 2284
–2296
(1979
).56.
R.
Gherib
, I. G.
Ryabinkin
, and A. F.
Izmaylov
, “Why do mixed quantum-classical methods describe short-time dynamics through conical intersections so well? Analysis of geometric phase effects
,” J. Chem. Theory Comput.
11
, 1375
–1382
(2015
).57.
I. G.
Ryabinkin
, L.
Joubert-Doriol
, and A. F.
Izmaylov
, “Geometric phase effects in nonadiabatic dynamics near conical intersections
,” Acc. Chem. Res.
50
, 1785
–1793
(2017
).58.
D. A.
Fedorov
and B. G.
Levine
, “A discontinuous basis enables numerically exact solution of the Schrödinger equation around conical intersections in the adiabatic representation
,” J. Chem. Phys.
150
, 054102
(2019
).59.
G. A.
Meek
and B. G.
Levine
, “Wave function continuity and the diagonal Born-Oppenheimer correction at conical intersections
,” J. Chem. Phys.
144
, 184109
(2016
).60.
D. R.
Yarkony
, in Conical Intersections: Electronic Structure, Dynamics and Spectroscopy
, edited by W.
Domcke
, D. R.
Yarkony
, and H.
Koppel
(World Scientific Publishing Co.
, New Jersey
, 2004
), pp. 129
–174
.61.
D. R.
Yarkony
, “Diabolical conical intersections
,” Rev. Mod. Phys.
68
, 985
–1013
(1996
).62.
Y.
Wu
, X.
Bian
, J.
Rawlinson
, R. G.
Littlejohn
, and J. E.
Subotnik
, “A phase-space semiclassical approach for modeling nonadiabatic nuclear dynamics with electronic spin
,” arXiv:2202.13973 (2022
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
