A Nonadiabatic Electron-Wavepacket Dynamics Study

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On the Elementary Chemical Mechanisms of Unidirectional Proton Transfers: A Nonadiabatic Electron-Wavepacket Dynamics Study Kentaro Yamamoto, and Kazuo Takatsuka J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 12 Apr 2019 Downloaded from http://pubs.acs.org on April 12, 2019

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The Journal of Physical Chemistry

On the Elementary Chemical Mechanisms of Unidirectional Proton Transfers: A Nonadiabatic Electron-Wavepacket Dynamics Study Kentaro Yamamoto∗ and Kazuo Takatsuka† Fukui Institute for Fundamental Chemistry, Kyoto University, Sakyou-ku, Kyoto 606-8103, Japan March 20, 2019

Abstract We propose a set of chemical reaction mechanisms of unidirectional proton transfers, which may possibly work as an elementary process in chemical and biological systems. Being theoretically derived based on our series of studies on charge separation dynamics in water splitting by Mn-oxides, the present mechanisms have been constructed after careful explore over the accumulated biological studies on cytochrome c oxidase (CcO) and bacteriorhodopsin. In particular we have focused on the biochemical findings in the literature that unidirectional transfers of approximately two protons are driven by one electron passage through the reaction center (binulear center) in CcO, while no such dissipative electron transfer is believed to be demanded in the proton transport in bacteriorhodopsin. The proposed basic mechanisms of unidirectional proton transfers are further reduced to two elementary dynamical processes, namely, what we call the coupled proton and electron-wavepacket transfer (CPEWT) and the inverse CPEWT. To show that the proposed mechanisms can indeed be materialized in a molecular level, we construct model systems with possible molecules that are rather familiar in biological chemistry, for which we perform the ab initio calculations of full-dimensional nonadiabatic electron wavepacket dynamics coupled with all nuclear motions including proton transfers.

∗ †

email: [email protected] email: [email protected]

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1 Introduction Unidirectional (one-way) proton transfers frequently identified in biological systems are among the most exciting challenges not only in biology but for chemists, since there can exist unknown “chemical mechanisms” hidden behind. By chemical mechanism we mean a one-way dynamics not based on the physical mechanisms such as diffusion and irreversibility due to ratchet machinery and so on, but making use of a series of chemical reactions. We here theoretically propose, not aiming at simulation of specific experimental results, a set of rather general chemical reaction models of unidirectional proton transfer in terms of nonadiabatic electron wavepacket dynamics. The proposed mechanisms have been attained through our nonadiabatic electron wavepacket studies on the elementary dynamical mechanism of charge separation in water splitting by Mn-oxides in photo-excited states. 1–4 In these dynamics, the cooperative transfers of protons with quantum electron wavepacket transfer play the critical roles. In particular, one of the major findings there is a mechanism of unidirectional electron transfer, which is pumped by the “reciprocal motion” of the relevant protons in the electronically ground state. 5 This study further suggested that exchanging the roles of electrons and protons in the similar context may naturally result in a dynamics of directional proton transfer. Besides such theoretical and computational studies as above, we have been inspired also by the accumulated experimental studies on the mechanism of the proton pumping in cytochrome c oxidase (CcO, also known as Complex IV), which is graphically summarized in Figure 1. CcO embedded in a mitochondrial membrane oxidizes cytochrome c, reduces molecular oxygen with the help of protons, and pumps protons across the membrane. 6–9 The experimental evidences suggest that the proton transfer there seems to be associated with electron transfers in a very characteristic manner: Each time when “one electron” passes through a channel as cytochrome c → CuA → heme a → BNC → O2 ,

(1)

where BNC stands for binuclear center consisting of heme a3 and CuB , up to “two protons” are transported after all; one from the negative side (N-side) of the membrane 9 (so-called chemical proton) is sent to reduce O2 , finally consumed to yield H2 O. The other proton (so-called pumped proton) is also supplied from the N-side and is released to the positive side (P-side), working to enhance the proton concentration gradient, which is eventually utilized in the production of the adenosine triphosphate (ATP). According to the experimental studies, 9 the amount of protons sent to the P-side and that to reduce O2 is roughly equal. This is quite interesting 2


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Figure 1: Schematic representation of the pathways of protons, electrons, molecular oxygen, and water molecules in cytochrome c oxidase (CcO). BNC stands for the binuclear center. and casts questions; Does one electron drive two channels? If there are two channels of proton pumping, are they different from each other? Do they work in a cooperative and/or synchronous manner as suggested by the ratio two protons to one electron? In this context we remind of another totally different manner of unidirectional proton transfer in bacteriorhodopsin, in which no external electrons and their dissipation are indicated to be required in order to pump the protons. Although the present study cannot resolve all these questions, the models proposed below may facilitate a partial understanding on what might be happening behind. There have been many experimental studies on CcO and also theoretical modeling studies and molecular simulations. See Ref. 10 and references therein. The mechanism of the so-called proton-coupled electron transfer (PCET) 10–15 in CcO has been understood mainly based on the Marcus picture of his theory of electron transfer. 16 However, dynamical coupling of protons and electrons along the real time is out of the scope in this theory. As suggested above, the proton pumping in CcO should not be recognized as a single electrostatic process, but some other dynamical coupling mechanisms of protons and electrons are likely involved. Assuming that this is the case, the chemical mechanism needs to be addressed by taking account of the dynamical coupling, which has not been attempted before in the context of biological proton pumping, to the best of our knowledge. Our studies consistently aim at extracting and establishing chemical principles or laws from those complicated phenomena such as water splitting in artificial and/or biological systems,

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in which electron-wavepacket dynamics is supposed to be involved. 1–4 To this goal, we have been applying the theory of nonadiabatic electron-wavepacket dynamics in the mixed quantum and classical representation. More precisely, the path-branching representation 17–19 and/or the semiclassical Ehrenfest theory 20–24 in full electronic and nuclear dimensions. Among the vast class of phenomena collectively called PCET, we in the present study rest on the notions of “coupled proton electron-wavepacket transfer (CPEWT) and inverse coupled proton electronwavepacket transfer (iCPEWT)”, because we track the real-time electron-wavepacket dynamics along with all the nuclear motions including proton transfers. This paper is structured as follows: We first describe the chemical models of unidirectional proton transfers in Section 2 in a rather general way. The notions of CPEWT and iCPEWT adopted in this section and throughout the paper are briefly reviewed in Appendix. Section 3 follows to materialize the elemental chemical processes using the practical molecules that appear frequently in biological chemistry. Full dimensional nonadiabatic electron wavepacket dynamics for these molecular systems constructed in order to show that the one-way proton transfer is indeed possible. The paper concludes in Section 4 with some remarks.

2 Dynamical mechanisms proposed for unidirectional proton transfer Inspired by the experimental studies for CcO, which suggests that at least two functions are involved in the unidirectional proton transfer, we here propose two theoretical models that partly account for it. In the first one, electrons supplied from a source flowing to a sink induces unidirectional proton transfer and referred here to as one electron and one proton (1E1P) mechanism. In the other, proton transfer is driven by reciprocal electronic motions without dissipative (external) electron flow, which is referred to as zero electron and one proton (0E1P) mechanism. Chemically these two mechanisms can work independently, or they also can serve in a cooperative manner. To emphasize that the mechanism does not depend on a specific property of certain molecules or proteins, we first present the mechanisms in a rather abstract manner. Examples of molecular materialization for concrete demonstrations of the mechanisms are shown in the next section.

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2.1 Coupled proton electron-wavepacket transfer (CPEWT) and inverse CPEWT (iCPEWT) as elementary dynamical processes As suggested in Figure 1, the present work falls into the category of the studies about PCET. PCET is used as a term covering any type of phenomena in which proton and electron transfers somehow couple to each other. 10–15 Numerous PCET reactions have been reported in literature, 11,12 and still actively investigated in a variety of context. 25–32 In particular, biological systems are known to utilize PCET in many important reactions such as photosynthesis and respiration. 33 In the standard theoretical studies and molecular dynamics (MD) simulations of PCET, the relevant energetics and kinetics seem to be of the central concern. 10,13–15 Our particular concern is with electron dynamics that nonadiabatically couples with proton transfer, both in electronically ground and excited states. Ultrafast redistribution of the electronic wavefunctions over and across molecules triggered by proton transfer and/or laser pulses are to be tracked with full dimensional time-dependent quantum chemistry along the all-particle nuclear paths. The system dependent mechanisms of charge separation and recombination can thus be directly discerned. We hence refer to these dynamics as CPEWT. To be more precise, we distinguish two classes of CPEWT; One is forward CPEWT (or simply CPEWT), in which electrons and protons to be transferred from a single molecular species to their individual destinations (see Figure 18a in Appendix). Thus a charge separation is materialized. The other one is inverse CPEWT (iCPEWT, Figure 18b), in which electronic wavepackets and protons sitting in different species get together in a single molecular system.

2.2 Unidirectional proton transfer with dissipative electron transfer from a source to a sink; 1E1P mechanism We first theoretically consider the mechanism of one-proton unidirectional transfer associated with one-electron transfer (referred to as the 1E1P mechanism) by taking account of the key features of the reactions in the vicinity of BNC of CcO (see Figure 1). We focus on the function of BNC that changes its acidity by redox reactions. For example, the acidity of propionates of heme a3 is influenced by the change of the oxidation state of heme a3 . 7,9 The O2 molecule that is attached between heme a3 and CuB is expected to have the same function, because it is reduced with the help of protons. 7,9 Thus it is suggested that the central molecule of our model system, denoted as Θ, through which both protons and electrons are supposed to pass, 5



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should be a weakly basic molecule and turns to be more basic when it is reduced. Now we schematically define the main body (referred to as the core subsystem in what follows) of the 1E1P mechanism as panel (a) of Figure 2, where X = (1e− buffer (1)), Y = (1e− buffer (2)), AH = (weakly acidic molecule), and BH = (weakly basic molecule). The asymmetry of acidity between AH and BH is crucial for the unidirectionality of proton transfer, as described below. “N” seen at the top left and bottom right corners in each panel of Figure 2 denotes the interface for a link to environmental electron transfer network, while “▽” at the top right and bottom left corners in each panel of Figure 2 indicates a link to a proton transfer network (i.e., hydrogenbond network). The pathway of protons is to be made along “▽ · · · AH· · · Θ· · · BH· · · ▽”, while that of electrons is assigned to the part “N X Θ Y N”. Note that Θ is involved both in the proton and electron transports. X and Y are implicitly designated to correspond to heme a and O2 of CcO, respectively. Thus the molecular orbital (MO) energy of X involved in the electron transfers should be higher than that of Y, because spontaneous electron transfers from heme a to O2 occur in CcO. X and Y are assumed to be attached individually on the environmental backbones like large proteins, which undergo large fluctuation and low frequency motions, by which X and Y are assumed to be brought close to or pulled apart from Θ with somewhat long while, although this model study does not specify those protein moieties. In a short time when X or Y come close to Θ, which should be about 100 fs long, the quantum mechanical coherent overlap between the electronic wavefunctions (simply termed as “coherence” in what follows) between them is realized for proton transfer or electron transfer to be driven as described below. Then, X or Y are carried and stay away from Θ, the duration of which should be expected to be long in biological systems, and the coherence is concomitantly broken and thereby neither proton nor electron transfer is assumed to take place in this time interval. The dynamical processes involved in the 1E1P mechanism are itemized below from (a) to (h), each of which is schematically represented in the correspondingly itemized panel of Figure 2: (a) X + e− −→ X− : An electron is prepared for injection, coming up to X before its transfer to Θ. X takes an electron form the environmental electron source, whose interface is expressed as a blue triangle on top left in Figure 2. This process corresponds to the reduction of heme a by CuA of CcO. (b) Θ + X− + AH −→ ΘH + X + A− : The resulting X− is brought close to Θ to induce coherent dynamics of an iCPEWT converging to Θ, that is, the X→Θ electron transfer coupling

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Figure 2: Schematic representation of the 1E1P mechanism. The core subsystem consists of A, B, Θ, X, and Y (see text for the definitions). The triangles “N” in blue indicate interfaces to the environmental electron transfer networks, whereas the red triangles “▽” link to the environmental proton transfer networks (i.e., hydrogen-bond network). “)” and “(” indicate that X and Y are sufficiently far from Θ for electronic interactions, respectively. to the A→Θ proton transfer. Such a proton transfer is induced since Θ turns to be strongly basic when it is reduced. AH precedes BH as a proton donor, because AH is supposed to be acidic. This demonstrates why the asymmetry of acidity between AH and BH is crucial. (c) (No chemical reactions): X is taken away from ΘH, and thereby the coherence between them decays to inhibit the reverse reaction due to the above iCPEWT, leaving only ΘH behind. (d) A− + H+ → AH: While ΘH is sufficiently far from both X and Y, a proton is provided to A− from the environmental proton-supplying moiety (the red triangle on the bottom left), because A− alone is not stable enough (recall that AH is defined as a weakly acidic molecule). (e) ΘH + Y + BH −→ Θ + Y− + HB+ H: Y is next set close to ΘH, and the coherent CPEWT diverging from Θ takes place, where the electron transfer of Θ→Y and the proton transfer of Θ→BH mutually couple. Notice that the iCPEWT (as in process (b)) is not induced this time, because the state of hydrogen-bond network has been already changed in process (d). Here BH precedes AH as a proton acceptor, because BH is supposed to be basic. This effect is again due to the asymmetric acidity between AH and BH. (f) (no chemical reactions): Y− is then supposed to be taken away from Θ and the reverse reaction is inhibited. Thus Θ remains as is for somewhat a long while.

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(g) HB+ H −→ BH + H+ : While Θ is sufficiently far from both X and Y− , HB+ H gives proton to the environmental proton-accepting moiety, (the red triangle on the top right), since HB+ H is not stable (recall that BH is defined as a weakly basic molecule). The proton thus transferred leaves away from Θ. (h) Y− −→ Y + e− : Y− then approaches the environmental electron sink to dissipate its captured electron. This process closes one cycle of the passage of an electron and the unidirectional transfer of one proton. By returning to the process (a), the cycles can resume. The key to this mechanism lies in the differences between processes (b) and (e), which correspond to iCPEWT and CPEWT, respectively. If A− generated by the iCPEWT (process (b)) does not take a proton as shown in the process (d), the proton of ΘH would just go back to A− (i.e., reverse reaction as for the proton) when Y approaches to ΘH as in the process (e). Accordingly, unidirectional proton transfer occurs because the situation of the hydrogen-bond network changes while the quantum coherence is switched off. In short, the main processes necessary to realize the one-way proton transfer are the coherent dynamics of iCPEWT and CPEWT, and the destruction of the coherent overlap to prevent the reverse reactions.

2.3 Unidirectional proton transfer with interior reciprocal motion of electrons; 0E1P mechanism As mentioned above, in CcO, two protons are pumped up from one side of the hydrogen-bond network associated with one-electron transfer. Reality however is that two-proton transfer seems to be driven by only one-electron transfer in CcO, 9 which may be referred to as “1E2P” process. The 1E1P mechanism is obviously not sufficient to explain the “1E2P” process, and additional mechanism that gives additional “0E1P” process seems to be missing. Thus the following 0E1P mechanism has been derived to complement this aspect. We also recall that the unidirectional proton transfer in bacteriorhodopsin is believed to proceed without net electron transport passing across the reaction center, 34 which also suggests the existence of the process(es) like 0E1P. We therefore consider another mechanism of unidirectional proton transfer which does not require the passage of electrons through the core subsystem. The core subsystem of the 0E1P mechanism with Θ and nearby molecules in terms of electron and proton transfer networks is defined as in panel (a) of Figure 3. As immediately noticed, this core subsystem is simpler than that of the 1E1P counterpart in that the electron 8


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Figure 3: Schematic representation of the 0E1P mechanism. The core subsystem consists of A, B, Θ, and X (see the text for definitions). The red triangles indicate the interfaces to the environmental proton transfer networks. There is no interface with respect to electron source or sink. “)” and “(” indicate that X and Y are sufficiently far from Θ for electronic interactions, respectively. acceptor Y is not present. It is shown though that unidirectional proton transfer is made possible in this 0E1P model under the similar spirit of the 1E1P machinery. All the processes involved in the 0E1P mechanism are abstractly represented in Figure 3. The alphabet on each item below corresponds to that of Figure 3, in which we symbolically depict the motions of electron, protons, and X happening in each elementary process: (a) Θ + X− + AH −→ ΘH + X + A− : X− is brought close to Θ, and the coherent interaction of iCPEWT converging to Θ begins; electron transfer of X→Θ and proton transfer of A→Θ. This proton transfer is induced because Θ turns to be strongly basic when it is reduced. AH precedes BH as a proton donor, because AH is supposed to be an acidic molecule. (b) (No chemical reactions): X leaves from ΘH and the reverse reaction of the iCPEWT is inhibited. Consequently, proton is left bound to Θ, for a sufficiently long time. (c) A− + H+ → AH: While ΘH is far from X, A− takes proton from the environmental proton-supplying moiety (the red triangle on the bottom left), since A− alone is not stable (recall that AH has been chosen to be a weakly acidic molecule). (d) ΘH + X + BH −→ Θ + X− + HB+ H: After some long while X is brought back to ΘH, inducing the coherence of CPEWT diverging from Θ, that is, electron transfer of Θ→X coupled

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with proton transfer of Θ→BH. The coherence of iCPEWT (process (a)) is not induced this time, because the state of hydrogen-bond network has been changed (process (c)). Here BH precedes AH as a proton acceptor, because BH is supposed to be basic. (e) (no chemical reactions): X− leaves from Θ, which destructs the coherence in CPEWT. As in the process (d), proton stays away from Θ, and thereby Θ can live for somewhat a long while. (f) HB+ H −→ BH + H+ : While Θ is sufficiently far from X− , HB+ H donates a proton to the environmental proton-accepting moiety, (the red triangle on the top right), because HB+ H is not stable (recall that BH is chosen so as to be weakly basic). This concludes the one circuit of unidirectional proton transfer from A to B. As noticed immediately, the processes (a) through (f) of the 0E1P mechanism are similar to the processes of (b) through (g) of the 1E1P mechanism. Obvious differences are: First, the 0E1P mechanism cannot use the free energy possibly provided by the electron translocation through Θ, since the electrons involved in this mechanism undergo just reciprocal motion in between X and Θ. The energy to push the protons should be provided from somewhere else, which we do not specify within the present model construction. Second, the molecular orbital (MO) of X needs to be appropriate both for donating and accepting electrons. Incidentally, the 0E1P mechanism is closely related to the mechanism of unidirectional electron transfer that we have proposed before, 5 with which proton reciprocal motions are associated.

3 Molecular materialization of the abstract models We next show that the rather abstract chemical models of 1E1P and 0E1P can be indeed reproduced within actual molecular levels. We further perform the full-dimensional nonadiabatic electron-wavepacket dynamics to see how the mechanisms actually work. Elementary chemical reactions in solvents and/or protein moiety are generally triggered by a contact (collision) between reactant molecules and the coherent overlap of their electronic wavefunctions, and then proceed according to the principle of quantum mechanics. These processes are more or less affected by their environments such as solvation energy, random phase of the wavefunctions, stochastic dissipation of energy, and so on. However, we do not equipped with a computational methodology that makes it possible to track the fate of reactions

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throughout such a long time as the time-scale of large amplitude and slow frequency modes of proteins. (Incidentally, we recall, just for reference, that the typical time scales of the successive sequence of electron and hole transfers in PSII dynamics are very long, extending from ns to ms. 15 ) Therefore we concentrate here in this particular study only on quantum wavepacket dynamics of electrons (not only transferring electrons but all the other involved ones) along with the classical-like nuclear path dynamics including proton transfers. By this method we track the real time dynamics for a relatively short while up to about 100 fs for a single molecular encounter and try to extract the quantum mechanical mechanism and associated dynamical features of the reactions. Therefore thermodynamical quantities like the chemical potential with which the environments provide external electrons to the reaction center (see Figure 1) are totally disregarded. If the product molecules remain to contact each other for a long time, the reversed reactions can take place. This is particularly the case for reaction in high viscous solvents and a molecular cage. On the other hand, if the product molecules are separated swiftly so that their quantum interference due to the coherent overlap of their electronic wavefunctions is switched off, such reverse reactions will be suppressed. The spatial separation between the product molecules are often realized by the intervention of solvent molecules or conformational change of large proteins to which one of the product molecules clings. Although we cannot simulate such receding processes of the product molecules at all, we assume implicitly throughout this paper that such a process of destruction of the coherent overlap of electronic wavefunctions is critically vital for the reverse reaction to be prohibited or at least reduced, and thereby for a unidirectional proton transfer to be effectively achieved.

3.1 Computational Methods Prior to the molecular realization of the proposed mechanisms, we here present a nonadiabatic electron wavepacket methodology we have taken. We again stress that the aim of this study is to propose possible molecular machinery enabling unidirectional proton transfers, and therefore the core subsystems to be constructed are as minimal as possible, and large peripheral protein moiety is not considered here. As for solvent effects, the method of quantum mechanics/molecular mechanics (QM/MM) 35 is among the most practical approaches to take into account. Again we disregard the environmental effects here, in spite of the former formulation of nonadiabatic dynamics theory in condensed phases. 36 11



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3.1.1 Theory of nonadiabatic electron-wavepacket dynamics The semiclassical Ehrenfest theory (SET) 20–24 is employed for the numerical dynamics calculations. The SET can be derived as an approximation to the path-branching representation theory, 17,18,37,38 in which electron dynamics is represented by quantum wavepackets to be evolved in time along nuclear branching paths. Herein we briefly outline the theory, and extensive reviews are available elsewhere. 17,18 The electronic wavefunction Ψelec (r, t; R(t)) is expanded in a series of time-independent wavefunctions {ΦI (r; R)} as Φelec (r, t; R(t)) =

∑

CI (t)ΦI (r; R)

I

(2) R=R(t)

in which CI (t) is the Ith time-dependent coefficient to be evaluated at each time step. r and R denote the collective representation of all electronic and nuclear coordinates, respectively. R(t) represents a nuclear path on which the electronic wavefunctions are to be propagated at time t. Any types of orthonormal many-electron basis functions including Slater determinants, configuration state functions (CSFs), and adiabatic states may be employed as {ΦI (r, R)}. The coupled equations of motion for the electron-wavepacket are expressed as [ ] 2 ∑ ∑ ∑ dCI ~ (el) k k k∗ HIJ − i~ i~ = R˙ k XIJ − (YIJ + YJI ) CJ dt 4 J k k in which the matrix elements are evaluated as ⟩ ⟨ ⟩ ⟨ ∂ ˆ (el) (el) k ΦJ , HIJ = ΦI H ΦJ , XIJ = ΦI ∂Rk

and

k YIJ

⟨ 2 ⟩ ∂ = ΦI 2 ΦJ ∂Rk

(3)

(4)

ˆ (el) denotes the electronic Hamiltonian. The bra-ket notation here is defined as The operator H k an integral over the electronic coordinates. The terms including YIJ in eq 3 are the nontrivial

corrections to the conventional SET regarding electronic motions, 17,18 but they are usually neglected in practical computations because it is multiplied by the small quantity ~2 . k The nuclear paths are driven by the force matrix FIJ expressed as

k FIJ

] (el) ) k ∑( ∑ [ ∂X l ∂HIJ ∂XIJ (el) (el) k IJ k ˙ =− − XIK HKJ − HIK XKJ + i~ − Rl ∂Rk ∂R ∂Rl k K l

(5)

k The off-diagonal elements of FIJ induce path-branching at every single time step. Infinite

number of path-branchings are necessary to obtain the exact solution of eq 5. Various practical methods have been proposed to avoid such computational difficulties. 17,18,38 The SET is also 12


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derived as a drastically simple approximation, which employs wavepacket-averaged force ⟨Fk ⟩ expressed as ⟨Fk ⟩ =

∑

k CI∗ FIJ CJ = −

IJ

∑

( ) ∑ ∂HIJ (el) (el) k k CJ − CI∗ XIK HKJ − HIK XKJ CJ ∂Rk IJ (el)

CI∗

IJ

(6)

If the basis set {ΦI (r; R)} was complete, eq 6 can be rewritten in the form of Hellmann– Feynman force

⟩ ∂H (el) ˆ ⟨Fk ⟩ = − Ψelec (r, t; R(t)) Ψelec (r, t; R(t)) ∂Rk ⟨

(7)

3.1.2 Useful theoretical tools for the analysis of nonadiabatic electron dynamics Although the dynamics of nuclei in SET is easily monitored, that of electrons is somewhat technical. To extract and monitor the electrons involved in the chemical reactions, we have used two quantities, namely, unpaired electron density D(r) and electron flux j(r, t). The unpaired electron density D(r) is defined as 39 ∫ D(r) = 2ρ(r, r) − dr′ ρ(r, r′ )ρ(r′ , r)

(8)

along a path, in which ρ(r, r′ ) is the first order spin-less density matrix in the coordinate representation. The number of unpaired electrons ND is estimated as: ∫ ND = drD(r)

(9)

This quantity has been proved to be very useful in the previous study, 5 because D(r) has a one-to-one correspondence to the radical location, and it is useful in the present study too. The electron flux j(r, t) (also known as Schiff probability current density of electron 40 ) satisfies the continuity equation as: ∂ρ(r, t) + ∇ · j(r, t) = 0 ∂t

(10)

In quantum mechanics, j(r, t) is defined as j(r, t) =

~ [ψ ∗ ∇ψ − ψ∇ψ ∗ ] 2ime

(11)

in which me and ψ are the mass and wave function of electrons. For many-electron systems it is redefined as

~ ′ ′ j(r, t) = [∇r ρ(r , r) − ∇r′ ρ(r , r)] 2ime r′ →r 13


(12)


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where ∇r and ∇r′ are the nabla with respect to r and r′ , respectively. In eq 12 the coordinates r′ are to be replaced with r after the derivative is done. As seen from eq 11, only complex-valued wave functions can give non-zero flux. Indeed, stationary-state electronic wavefunctions like ˆ (el) in quantum chemistry are commost of the eigenfunctions of the electronic Hamiltonian H puted to be real-valued, thereby electron flux given by them are identically zero everywhere. Many applications of electron flux in chemical dynamics have been reported. 41–46 Incidentally, the present work treats only the electronic flux at each nuclear configuration. Manz and his group have been studying the total electronic and nuclear flux. 44,47,48 One of the latest comprehensive descriptions of the electronic and nuclear fluxes to analyze nonadiabatic dynamics in the Born-Huang representation has been given by Matsuzaki and Takatsuka. 49,50

3.2 Illustrative model systems for 1E1P mechanism Since the 1E1P and 0E1P mechanisms are similar to each other, we first concentrate on the 1E1P mechanism, and then address the 0E1P mechanism later, minimizing repeated descriptions. In the molecular materializations, we concentrate on the processes in which the dynamical coupling of protons and electrons is involved, namely, the processes (b), (c), (e), (f) of the 1E1P mechanism (see Figure 2). Recall that the processes (b) and (e) are iCPEWT and CPEWT, whereas the processes (c) and (f) switches off the coherent overlap of electronic wavefunction in each of iCPEWT and CPEWT, respectively. All the dynamical processes other than (b), (c), (e), and (f) such as electron and proton supply from the outside to the core subsystem through their individual channels are not considered explicitly in this paper. We simply assume that there are appropriate flow channels for external electrons and protons as schematically drawn in Figure 1. Yet, our preliminary studies on the relevant energetics about the reaction cycles with respect to the core subsystem support that the constructed models are sufficiently feasible energetically (see Section S1 of Supporting Information). The initial structure of the molecular model system for the 1E1P mechanism is schematically shown in Figure 4. In case where we consider the iCPEWT (relevant to processes (b) and (c)), Y is intentionally disregarded under the assumption that Y is sufficiently far from the pair of X-Θ. Similarly, X is neglected when the CPEWT (relevant to processes (e) and (f)) is considered. We represent the 1E1P mechanism (see Figure 2 for the abstract expressions) with our molecular model system in Figure 5. A molecule whose acidity changes when it is involved in redox reactions is to be chosen as 14


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Figure 4: Schematic representation of the molecular model for the 1E1P mechanism in the initial configuration to realize the cycle of 1E1P mechanism of Figure 2a.

Figure 5: Schematic molecular representation of the chemical scheme for the 1E1P mechanism. Each panel has a one-to-one correspondence to one panel of Figure 2.

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Θ, which corresponds to BNC of CcO. Incidentally, BNC has several such sites. For example, propionates of heme a3 and O2 bound to the space in between heme a3 and CuB . 7,9 Here we adopt ΘH = 4-methylphenol (a functional group of tyrosine (Tyr)) as a simple model system to be involved in the 1e− /1H+ redox reaction. Tyr is one of the proteinogenic amino acids and is known to change its acidity by redox reactions. 15 Therefore 4-methylphenol must fit as ΘH; 4-methylphenoxyl (CH3 –C6 H4 –O• ) is a strong acid (weak base) while 4-methylphenol (CH3 –C6 H4 –OH) is a weak acid (relatively strong base). As AH (to be weak acid) and BH (to be weak base), we simply choose AH = (acetic acid) (a functional group of glutamic acid (Glu) and aspartic acid (Asp)) and BH = imidazole (a functional group of histidine (His)). Neither of them is involved in redox reactions as easily as Tyr, and this is indeed the case in our present numerical calculations, as will be explicitly shown later. Such a configuration “Asp· · · PRAa3 · · · His” has been suggested to be a part of hydrogen-bond network for the proton pumping in a certain type of CcO by means of MD simulations with a fixed redox state, 51 in which PRAa3 stands for propionate A of the active site heme a3 . Molecules that are involved in 1e− redox reactions need to be employed for the electron buffers X and Y, which correspond to heme a and O2 of CcO, respectively. However, it is not technically easy to track the time evolution of spin states in both cases. Instead of such a direct simulation, we try to find a model system that is qualitatively equivalent to them but technically feasible. A desirable model should have an MO that can serve to donate or accept electrons in response to the change of the state of Θ. As for CcO, it would be one of the d orbitals of metal center of heme a3 , the conjugated π orbitals of heme a3 , and the π ∗ orbitals of O2 . And yet any other MOs involved in the electron transfer within BNC of CcO should work as such. Taking these requirements into account, we have surveyed various models and found a concise model that is conceived to be appropriate for our analysis, which consists of X = protonated flavin moiety (FH+ ) and Y = protonated 1,4-benzoquinone (QH+ ). FH+ and QH+ are partial systems of flavin adenine dinucleotide and ubiquinone of biological cofactors, which are originally involved in 2e− /1H+ and 2e− /2H+ redox reactions, respectively. In the present paper, we restrict them to be 1e− redox agents by choosing the molecular configurations appropriately to suppress proton transfers. The reason why X = FH+ is chosen instead of X = QH+ is that the energy of the lowest unoccupied MO (LUMO) of FH+ that can accept an electron is slightly higher than that of QH+ , as shown in Figure 6.

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Figure 6: Spatial distributions of LUMOs and their orbital energies of (a) X = FH+ (protonated flavin moiety) and (b) Y = QH+ (protonated 1,4-benzoquinone).

3.3 Full-dimensional nonadiabatic dynamics of 1E1P mechanism 3.3.1 Setup for numerical calculations We carry out the nonadiabatic electron-wavepacket dynamics as well as energetics for the systems chosen as above. The matrix elements necessary for such computations are implemented in GAMESS quantum chemistry package. 52,53 The electric charge of the system is either neutral or positive depending on the target process. For the electron-wavepacket dynamics, the adiabatic states obtained with the CISD/RHF/6-31G level of calculation are used as the basis set of the electron-wavepacket {ΦI (r, R)}, where CISD stands for configuration interactions of single and double excitation represented in terms of the restricted Hartree–Fock (RHF) molecular orbitals. A special care should be exercised here, since the character of the adiabatic states can change drastically before and after the passage of a nonadiabatic region like conical intersection. By tracking the electronic characters faithfully, with making use of the overlap integrals of the adiabatic wavefunctions before and after the nonadiabatic transition, the ordering of the suffix I in {ΦI (r, R)} must be appropriately made anew. In this study, CISD active orbitals are set to be all the orbitals occupied by valence electrons and the LUMO. More precisely, in the case of either iCPEWT of both 1E1P and 0E1P mechanisms (X− = FA• ) or CPEWT of the 0E1P mechanism (X = FA+ ), HOMO−84 to LUMO is chosen to be the CISD active space, with HOMO being the highest occupied MO. The number of CSFs amounts to 3,741. In the case of CPEWT of the 1E1P mechanism (Y = QH+ ) is involved, HOMO−65 to LUMO 17



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is chosen to be the CISD active space. The number of CSFs is then 2,278. These settings are similar to our previous study on the unidirectional electron transfer. 5 The atomic basis set and the CISD active space are carefully chosen to be a good compromise between the quality and computational cost (see Section S2 of Supporting Information for the details of the basis set). Obviously it is computationally too demanding to sample SET paths many enough to make statistical ensembles to discuss the iCPEWT and CPEWT reactions. Besides, for many paths to be sampled, these reactions do not always occur even when X or Y are brought close to Θ. Therefore in stead of accumulating statistical ensembles, we would rather pay a special attention to the qualitative features of the relevant chemical reactions. To do so, we first locate the nonadiabatic regions by means of static analysis, and then ab initio MD trajectories are run backward to the reactant direction on the ground state potential surface. Details for the static analysis and the backward ab initio MD are described below. The static analyses in an extracted two dimensional space for each iCPEWT and CPEWT are conducted to locate the nonadiabatic regions. We define one coordinate, denoted as RDH , which is the distance between the proton and the donor atom along the linear interpolation of the reactant and the product geometries of either iCPEWT (O atom nearest to the proton) and CPEWT (phenolic O atom). RDH is supposed to serves as a reaction coordinate. The other coordinates chosen to characterize the relevant reactions are RXΘ (RYΘ ), which are a distance between Θ and X (Y). The geometries to be monitored in the iCPEWT dynamics are as follows. • AH· · · Θ· · · HB + X+ (reactant) • A− · · · HΘ· · · HB + X (product) Similarly, those for the CPEWT are: • AH· · · ΘH· · · HB + Y (reactant) • AH· · · Θ· · · HB+ H + Y− (product) X = FH+ or Y = QH+ and the other part consisting of A = acetate, BH = imidazole, Θ = 4-methylphenoxyl with protons are independently optimized in energy before the dynamics calculations, since Θ and X (and Y as well) are slightly attractive with each other. And then they are set to appropriate places, thus giving rise to RXΘ or RYΘ . The initial geometrical configurations of molecules are set so as the planes of π conjugated systems of Θ to be parallel 18


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to that of X (Y) (i.e., π-π stacking). Keeping the face-to-face configuration, X (Y) has a degree of freedom for the rotational orientation with respect to the axis normal to the plane of Θ, and we determined the initial relative angle so that the quantum phases of their MOs allow for the maximum electron transfer. Then a backward ab initio MD is started from the point at which the energy difference is minimum along the RDH of each of iCPEWT and CPEWT. The initial momentum is given to the RDH stretching in each case of iCPEWT and CPEWT. More precisely, momenta that are the same magnitude with the opposite direction are given to the H and D atoms with respect to RDH so that the kinetic energies are equal to 0.5 eV and 0.25 eV for iCPEWT and CPEWT, respectively. These backward trajectories are run for 5 fs. And then the momenta of them are inverted towards the product side, which are regarded as the initial conditions for the nonadiabatic electron-wavepacket dynamics. All the initial electron-wavepackets are commonly prepared in the adiabatic ground state. By varying RXΘ we will discuss the significance of the coherence in iCPEWT. As initial conditions, we take 3.3 Å, 4.0 Å, and 4.7 Å for RXΘ . The value RXΘ = 3.3 Å is around the point that gives the minimum energy along the RXΘ axis. This value also gives nearly the largest energy gap between the ground and the excited states, due to the maximum level repulsion, which indicates the large interaction between X and Θ between the donor and acceptor. RYΘ is used for CPEWT in the similar manner. 3.3.2 Dynamics of CPEWT and iCPEWT We show the results of the full-dimensional nonadiabatic electron dynamics along a SET path of RXΘ = 3.3 Å for iCPEWT and RYΘ for CPEWT, where the coherent interactions are strong (see Figures 5b and 5e). As shown below, these cases turns out to be almost adiabatic in that the dynamics appears to proceed mostly on a single potential surface. The geometrical pathways of proton and electron wavepackets in the cases of the longer RXΘ or RYΘ are more or less the same, but the efficiency of the proton transfer is significantly different from each other because of the presence of nonadiabatic transition (see Section 3.3.3). Both the iCPEWT and CPEWT take place as expected in Figures 5b and 5e, respectively. The unpaired electron density D(r) and the electron flux j(r, t) along the relevant nuclear paths demonstrate well how the dynamics proceed, selected snapshots of which are shown in Figure 7a for iCPEWT (process (b)) and 7b for CPEWT (process (e)). The computational results are summarized as follows. (i) At t = 4.0 fs after the first collision between Θ and X− (X

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Figure 7: Selected snapshots of the unpaired electron density D(r) superposed with the electron flux j(r, t) along the SET paths starting from RXΘ = RYΘ = 3.3 Å, respectively, in (a) the inverse CPEWT and (b) CPEWT in the 1E1P mechanism. The box of each panel highlights which proton is involved in the inverse CPEWT or CPEWT. Note that electron flux j(r, t) can oscillate with less than 1.0 fs of frequency, and the net electron transfer is represented by integrating them over space-time (see Refs. 49 and 50 for a general discussion about oscillatory behavior of electron flux).

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is assumed to have already received an electron as in the process (a)), an electron begins to jump from X− = FH• to Θ = 4-methylphenoxyl along with the proton transfer from AH = (acetic acid) to Θ = 4-methylphenoxyl, thus performing the iCPEWT converging to Θ in the ground state (process (b)). This process is displayed in Figure 7a. (ii) After X = FH+ leaves significantly far from ΘH = 4-methylphenol (process (c)), a proton is assumed to be supplied from the environmental proton source, because AH = (acetic acid) is a weak acid (process (d)). Note that the A− = acetate at the end of iCPEWT (t = 30 fs of Figure 7a) changes to the AH = (acetic acid) before the beginning of CPEWT (t = 0 fs of Figure 7b). (iii) At t = 6.0 fs after the second collision between Y = QH+ and ΘH = 4-methylphenol (processes (e), time is reset to t = 0 fs), an electron moves from ΘH = 4-methylphenol to Y = QH+ along with the proton transfer from ΘH = 4-methylphenol to BH = imidazole. It has been confirmed that the present CPEWT can proceed with a low energy profile (see Figure S3 panel (a) in Supporting Information, the energy difference between the process (e) and (f)). This completes the CPEWT diverging from Θ in the ground state (process (e)). Note that the proton involved in the iCPEWT is different form that of the CPEWT, because the A− = acetate is protonated by the environmental source as mentioned above. Thus unidirectional proton transfer can be indeed induced. These iCPEWT and CPEWT happen to occur mostly in the ground state, because the electron wavepacket is dominated by the electronic ground state (more than 99% in state population). These are directly seen in Figure 8. Plotted in Figure 8a is RHD along the path. Recall that RHD is defined as the distance between the donor atom and the proton relevant to either the iCPEWT or CPEWT process. We judge that the proton transfer actually has occurred if RHD is sufficiently longer than the typical OH bond length, say, 1.2 Å. Figure 8a and c show that the iCPEWT and CPEWT occurs, respectively, and besides no reverse reactions are numerically observed for at shortest 30 fs. In the very early stages of the iCPEWT and CPEWT, the energy gap between the ground and the first excited states is more than 2.0 eV (see Figure 8b and d for iCPEWT and CPEWT, respectively). This suggests that the electronic interaction between the donor and acceptor is sufficiently strong enough to suppress nonadiabatic transition. This means that the dynamics described above for RXΘ = RYΘ = 3.3 Å happens to be mostly represented with those methods such as single-surface MD simulations. However, this is not the case in longer RXΘ and RYΘ , as shown below.

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Figure 8: Time evolution of the selected properties along a SET path for RXΘ = RYΘ = 3.3 Å of the 1E1P mechanism. (a) RHD and (b) adiabatic potential energy levels Vn for the inverse CPEWT of the 1E1P mechanism, whereas (c) and (d) correspond to (a) and (b) for the CPEWT, respectively.

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Figure 9: Potential energy (the electronic energy) Vn and state populations |Cn |2 given along the SET paths of different RXΘ for the inverse CPEWT of the 1E1P mechanism. These graphs show that the nonadiabatic transition is more intensified as the coherence is lost. 3.3.3 The roles of nonadiabatic transitions It is natural to conceive that when a donor and an acceptor are sufficiently apart from each other, electron transfer should be hard to occur. Nevertheless we need to take account of the effects of nonadiabatic interactions even in those cases. As shown in Figure 9 for the iCPEWT and Figure 10 for the CPEWT, the more adiabatic natures are numerically observed for the longer RXΘ and RYΘ . The barriers also become higher when RXΘ or RYΘ do longer. If the barrier height alone is taken into account, it appears that higher the kinetic energy may be needed for the higher reaction probability. On the other hand, the nuclear velocity R˙ k contributes to the higher nonadiabatic transition amplitude due to the nonadiabatic coupling ∑ k element k R˙ k XIJ in eq 3. Therefore nonadiabatic electron dynamics is necessary, in particular to discern a possibility for the irreversibility of proton and electron to take place. Nevertheless we here do not track the SET paths with nonadiabatic transitions any further, because the path would run on a unphysical mean potential surface. 38 If path-branching methodologies are invoked, more natural dynamics of both electrons and protons can be tracked, in which some paths on the excited state undergo the reverse reaction while the other paths reach the product side. See Refs. 38 and 2 for the discussion on the path-branching. Once the reaction is over and the quantum coherent overlap is lost, the nonadiabatic transition turns out to serve to inhibit the reverse reactions. After the iCPEWT (process (b)), the resulting A− cannot pull the proton back from ΘH, because Θ− is maintained to be a strong base unless it is oxidized again. This is another critical role of nonadiabatic interactions. 23



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Figure 10: Potential energy Vn and state populations |Cn |2 given along the SET paths for different RYΘ in the CPEWT process of the 1E1P mechanism. The amount of nonadiabatic transition is seen to increase as the coherence is lost. 3.3.4 Critical role of the large fluctuation in RXΘ and RYΘ to prevent the backward proton-electron transfers Since it is virtually impossible to perform the entire calculations throughout the dynamics, we instead try to visualize the characteristic properties in the energy profile behind the unidirectional proton transfer. Herein we show the two-dimensional adiabatic potential surfaces for the ground and the first excited states in Figure 11, the character of the corresponding electronic states of which are depicted in terms of color according to the number of unpaired electrons ND . The potential surfaces colored red, for instance, are of ND ∼ 2, indicating a biradical character. Tracking their color continuously, one can identify the diabatic nature of those adiabatic potential surfaces. According to the description in Section 3.3.1, the distances (RHD ) between the donor atom and the proton relevant to either the iCPEWT or CPEWT and the distance between π-π stacking Θ = 4-methylphenoxyl and X = FH+ (RXΘ ) are employed to draw those two dimensional potential energy surfaces. The landscape of the two-dimensional potential surfaces clearly illustrates the importance of the contacting and receding of the pair of (Θ = 4-methylphenoxyl, X− = FH• ) for the iCPEWT (process (b)), and that of (ΘH = 4-methylphenol, Y = QH+ ) for the CPEWT (process (e)). Let us focus on the iCPEWT in a greater detail. The barrier height is lowered by the close contact of the electron donor (X− = FH• ) to the acceptor (Θ = 4-methylphenoxyl). This is represented by shorter RXΘ (see Figure 11a). The energy gap between the ground and the first excited state near the top of the barrier becomes larger to be more like avoided crossing 24


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Figure 11: Two-dimensional adiabatic potential energy surfaces, which are colored according to the number of unpaired electrons ND for the 1E1P mechanism; blue for ND = 0.2, white for ND = 1.0, and red for ND = 2.0.

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Figure 12: Schematic representation of the molecular model for the 0E1P mechanism, which represents the initial configuration of the catalytic cycle. See Figure 3a for the general scheme. than conical intersection as RXΘ becomes longer, and the chance of electron transfer becomes smaller. Similar results are obtained for the CPEWT. The barrier height is lowered as RYΘ becomes smaller as seen in Figure 11b.

3.4 Materialization of the 0E1P mechanism We next proceed to a molecular materialization of the 0E1P mechanism. The molecular model system is designated to be built so that as many components are shared with those of the 1E1P mechanism as possible. This is because “1E2P” process observed in the CcO system suggests that there are two elementary mechanisms of one-way proton transfers that may coupled with each other. In other words, 1E1P and 0E1P might share common components. The initial structure of the molecular model system for the 0E1P mechanism is schematically shown in Figure 12. This system consists of X− = FH• , Θ = 4-methylphenoxyl, AH = (acetic acid), and BH = imidazole. There are two differences between the initial condition of the 1E1P and 0E1P mechanisms (see Figure 4 for the comparison to that of the 1E1P mechanism). First, the system has already had an electron on the electron buffer X. In other words, X− = FH• is employed at the beginning instead of X = FH+ . Second, Y is not used in this system, since the role of electron acceptor is played by X itself. We represent the 0E1P mechanism with our molecular model system as shown in Figure 13. Herein we briefly discuss only the CPEWT (process (d) of Figure 13), because the molecular materialization for the iCPEWT (process (a) in Figure 13) is the same as that of 1E1P process (panel (b) in Figure 5). (Revisit, if necessary, Section 2.3 for the explanation of the iCPEWT in detail.) This CPEWT also proceeds with a low energy profile (see Figure S3 panel (b) in Supporting Information, the energy difference between the process (d) and (e)). The full-

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Figure 13: Schematic molecular representation of the series of chemical reactions for the 0E1P mechanism, making use of the molecular components of Figure 12. dimensional nonadiabatic electron dynamics is also performed by means of the SET with three values of RXΘ , namely, 3.3 Å, 4.0 Å, and 4.7 Å for CPEWT. RXΘ = 3.3 Å also corresponds to a strong coherent case (see Figure 13a). The initial conditions are prepared, prior to the full SET calculations, by the backward ab initio MD from the point with minimum energy gap between the ground state and the first excited state. The initial momentum is given to the backward trajectories in the RDH stretching mode in the similar manner to those adopted in the study of the 1E1P mechanism, but the kinetic energy here is chosen to be 0.75 eV for CPEWT. These trajectories are run for 5 fs. Then the momenta are reversed towards the product side. The results are similar to those of the 1E1P mechanism as shown below. A couple of selected snapshots of the unpaired electron density and the electron flux along a SET path of the 0E1P mechanism (Figure 14) indicates that the electron turns back to X = FH+ , which is supported by the appearance of the unpaired electrons to be Θ = 4-methylphenoxyl and X− = FH• . See Figure 7b for the comparison to those of the 1E1P mechanism, in which the asymptotic biradical state is formed on Θ = 4-methylphenoxyl and Y− = QH• . As for the electron dynamics, this reaction looks like a reverse reaction of the iCPEWT of Figure 7a. Being coupled to this electron transfer, the proton moves from Θ = 4-methylphenoxyl to BH = imidazole, whereas the iCPEWT of Figure 7a involves proton transfer from A− = acetate to Θ = 4-methylphenoxyl. Thus it is confirmed that the 0E1P mechanism in which unidirectional proton transfer is induced by electron reciprocal motions can work in a realistic molecular system. The time evolution of the distance (denoted RDH ) between the donor atom (phenolic O atom) and the proton involved in CPEWT (see Figure 15a) indicates that the reverse proton 27



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Figure 14: Selected snapshots representing the unpaired electron density D(r) and the electron flux j(r, t) along the SET path of CPEWT of RXΘ = 3.3 Å in the 0E1P mechanism.

Figure 15: Time evolution of the selected properties of along the SET path of RXΘ = RYΘ = 3.3 Å of the 1E1P mechanism. (a) RHD and (b) adiabatic potential energy Vn are shown for the CPEWT of 0E1P mechanism. transfer does not occur at least for 30.0 fs. The potential energy curves Vn and state populations |Cn |2 show that the present CPEWT of the 0E1P mechanism with RXΘ = 3.3 Å is regarded mostly as a reaction in the electronic ground state as shown in Figure 15b. Recall Figure 8c and d for the comparison to those of the 1E1P mechanism to find the similar graphs. As in the cases of the 1E1P mechanisms, the nonadiabatic transition amplitude increases when RXΘ is elongated as shown in Figure 16. Therefore the coherent overlap of the electronic wavefunctions is important for the efficient occurrence of the CPEWT reaction, here also in the 0E1P mechanism. The transition amplitude depends on several factors including the velocity of protons and relative positions of the electron donor and acceptor. The nonadiabatic transition amplitude of the 0E1P mechanism is larger than or equal to that of 1E1P mechanism for the 28


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Figure 16: Potential energy Vn and state populations |Cn |2 given along the SET paths of different RXΘ for the CPEWT of the 0E1P mechanism. The amount of nonadiabatic transition is seen to increase as the coherence is lost. case of RXΘ = RYΘ as far as the present model systems are concerned. The molecular structure of ΘH = 4-methylphenol is more similar to that of Y− = QH• than X− = FH• , and therefore the electronic interaction between Θ and Y may be stronger, because it is expected that the energy difference is smaller and the MO overlap is larger in this case. The dynamics results of the 0E1P mechanism described above is intuitively comprehended with the two-dimensional adiabatic potential surfaces shown in Figure 17. The discussion goes parallel to that for the CPEWT of 1E1P mechanism (see Figure 11) except for a rather minor difference; the nonadiabatic region of the 0E1P mechanism is shifted to the product side. As a result, the CPEWT reaction looks as if it is a sequential event, that is, electron transfer follows after proton transfer in contrast to the case of the 1E1P mechanism (see Figure 11b).

4 Concluding remarks We have proposed a set of chemical reaction mechanisms of unidirectional proton transfers, each of which can serve as an elementary process in chemical and biological systems. These theoretical mechanisms have been theoretically derived based on our series of studies on charge separation dynamics in water splitting by Mn oxides 5 and by being logically inspired by the biological functions of CcO and bacteriorhodopsin. In the first mechanism we refer to as 1E1P mechanism, both protons and electrons are supplied from external sources (donors, reservoirs) and pass across the core subsystem, and

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Figure 17: Two-dimensional adiabatic potential energy surfaces, which is colored according to the number of unpaired electrons ND for the CPEWT of 0E1P mechanism. Color is assigned at each point as blue for ND = 0.2, white for ND = 1.0, and red for ND = 2.0. eventually both are dissipated to their individual sinks (acceptors). The basic chemical machineries that we think make unidirectional proton transfer are inverse coupled proton electronwavepacket transfer (iCPEWT), in which electrons and protons get together to the central molecule (Θ in Figure 2), and coupled proton electron-wavepacket transfer (CPEWT), in which both electrons and protons diverge from the central molecule to other molecules different from the source. These elementary dynamics take place quantum mechanically only for a short while, during which the electronic wavefunction of the central molecule retain its coherent overlap with the electronic wavefunctions of the peripheral molecular moiety that are responsible for iCPEWT or CPEWT. These quantum coherent processes should be swiftly finished after each event and thereby suppress their reverse dynamics. Such destruction of the coherent overlap will be realized by separation of the product molecules and/or possibly the intervention of solvents or protein molecules in between them. Hence, a consecutive occurrence of [injection of proton and electron into the core subsystem → iCPEWT → destruction of the coherent overlap → CPEWT → destruction of the coherent overlap] can materialize unidirectional proton transfer from one site to another. After one circuit of the directional proton and electron transfer, the core subsystem returns as it was before. In a little independent process, what we call the 0E1P mechanism, only protons are injected into the core subsystem. Unidirectional proton transfer is driven by a reciprocal motion of electrons in between the central molecule (Θ in Figure 3) and another molecule within the core subsystem. 30


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To show that the above abstract mechanisms can be practically (theoretically yet) materialized, we have constructed molecular models for both 1E1P and 0E1P processes in terms of molecules that are rather familiar in biochemistry. Since we are not able to carry out the real-time dynamics for a comprehensive system including solvents and/or proteins, we simply have extracted the relevant iCPEWT and CPEWT processes without solvents and carried out the full-dimensional nonadiabatic electron wavepacket dynamics. Other processes have been considered in the level of static energetics. Thus our calculations are practically about dynamics mimicked in vacuum. Nevertheless, the results show that all the processes constructed can proceed within a chemically favorable energy range. The better choice of modeling molecules and the possible solvent effects would get the energetics more favorable down to the level of far milder chemical conditions. Hence, the present model calculations should be conceived to support our proposed mechanism. Although not aiming at simulation of any actual biological unidirectional proton transfers, we have taken account of the biochemical studies in the literature, particularly for the stoichiometry with respect to protons and electrons passing through the binuclear center (BNC) of cytochrome c oxidase (CcO) and in bacteriorhodopsin as well. The experimentally observed “1E2P” process in CcO seems to suggest the presence of a set of cooperative elementary procedures of “1E1P” and “0E1P”. However, we cannot conclude within our models alone that these two processes cooperatively or synchronously work as a single set. The “0E1P” process biologically claimed in the bacteriorhodopsin certainly needs a mechanism in which external electrons are not demanded as in our 0E1P mechanism. It would be interesting to consider the 1E2P process in CcO and 0E1P for bacteriorhodopsin in the context of biological evolution from the view point of complexity, efficiency and so on of chemical reactions. The more comprehensive understanding of the biological proton pumping needs not only the studies of mechanism of (locally) unidirectional proton transfer such as those we have proposed in the present paper but also other investigations on those biological functions that serve as a “valve” to inhibit the reverse flow of protons. In bacteriorhodopsin, for instance, a molecule working as a valve is known to exist at the interface of the membrane. 34 In CcO, such functions have been also actively investigated. 7,9,54 Together with these insights, the present mechanism will set a theoretical foundation for the deeper understanding of the biological proton pumping.

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Acknowledgments The authors thank Professor Takefumi Yamashita of The University of Tokyo, the coauthor of Ref. 54, for useful discussions. This work was supported by JSPS KAKENHI (Grant Number JP15H05752). The computations were partly performed using Research Center for Computational Science, Okazaki, Japan.

Supporting Information The characters of the ground electronic states and energy profiles of all the states involved in the 1E1P and 0E1P mechanisms, along with their dependence on the atomic basis sets and CI active spaces.

A CPEWT and inverse CPEWT in unidirectional electron transfer Coupled proton electron-wavepacket transfer (CPEWT) and inverse coupled proton electronwavepacket transfer (iCPEWT) are schematically depicted in Figure 18a and Figure 18b, respectively in a general context. CPEWT was first discussed as a single event in the study on early-stage dynamics in coupled proton-electron transfer from π − π ∗ state of phenol to solvent ammonia clusters. 41 (See Ref. 55 for extensive references with respect to experimental studies on proton transfer in the ground and excited states of the related molecular systems.) In CPEWT the electrons undergo quantum mechanical transfer as electron wavepackets with broad spatial distribution, while proton dynamics are rather localized. Electron transfer can take place only when the electron accepting molecules have vacancy in state space (orbitals) with appropriate energy levels and the vacant state has sufficient quantum mechanical overlap with the wavefunction of the donating state (orbitals). Note that the iCPEWT of Figure 18b is not meant to be the reversal process of the forward CPEWT of Figure 18a. Direct reversal process of the forward CPEWT (Figure 18a) corresponds to a direct charge recombination without proton and/or electron pumping. The essential difference between CPEWT and iCPEWT is that the molecular species B (proton acceptor) and C (electron acceptor) in panel (a) are not necessary the same, respectively, as D (proton donor) and E (electron donor) in panel (b). 32


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Figure 18:


a) An abstract scheme of coupled proton and electron-wavepacket transfer

(CPEWT); electron wavepackets penetrates to molecular species C and protons are carried to B from a molecular species A, in which nonadiabatic interaction through conical intersections are usually involved. The B and C may be the same molecule. b) inverse CPEWT, electron wavepackets are provided from E and protons are from D, getting together in A. The timings of the electron and proton transfers depend on systems under study. Rather, they should be different for one-way proton transfer to be achieved. As an illustrative example, suppose a case where the iCPEWT of Figure 18b first takes place and then the CPEWT of Figure 18a follows. It should end up with a proton transfer from D to B passing through A.

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A Nonadiabatic Electron-Wavepacket Dynamics Study

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