Concerted Electronic and Nuclear Fluxes During Coherent Tunnelling

Article pubs.acs.org/JPCA

Concerted Electronic and Nuclear Fluxes During Coherent Tunnelling in Asymmetric Double-Well Potentials Timm Bredtmann,*,†,‡ Jörn Manz,*,†,§,∥ and Jian-Ming Zhao†,§ †

State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, §Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China ‡ Max-Born-Institut, 12489 Berlin, Germany ∥ Freie Universität Berlin, Institut für Chemie und Biochemie, 14195 Berlin, Germany ABSTRACT: The quantum theory of concerted electronic and nuclear fluxes (CENFs) during coherent periodic tunnelling from reactants (R) to products (P) and back to R in molecules with asymmetric double-well potentials is developed. The results are deduced from the solution of the time-dependent Schrödinger equation as a coherent superposition of two eigenstates; here, these are the two states of the lowest tunnelling doublet. This allows the periodic time evolutions of the resulting electronic and nuclear probability densities (EPDs and NPDs) as well as the CENFs to be expressed in terms of simple sinusodial functions. These analytical results reveal various phenomena during coherent tunnelling in asymmetric double-well potentials, e.g., all EPDs and NPDs as well as all CENFs are synchronous. Distortion of the symmetric reference to a system with an asymmetric double-well potential breaks the spatial symmetry of the EPDs and NPDs, but, surprisingly, the symmetry of the CENFs is conserved. Exemplary application to the Cope rearrangement of semibullvalene shows that tunnelling of the ideal symmetric system can be suppressed by asymmetries induced by rather small external electric fields. The amplitude for the half tunnelling, half nontunnelling border is as low as 0.218 × 10−8 V/cm. At the same time, the delocalized eigenstates of the symmetric reference, which can be regarded as Schrödinger’s cat-type states representing R and P with equal probabilities, get localized at one or the other minima of the asymmetric double-well potential, representing either R or P.

1. INTRODUCTION The purpose of this article is to extend the quantum theory for nuclear fluxes during coherent tunnelling in asymmetric doublewell potentials1 to corresponding concerted electronic and nuclear fluxes (CENFs). The derivations profit from results for CENFs during coherent tunnelling in symmetric double-well potentials2 (see also ref 3); those serve as a symmetric reference. The symmetric reference allowed for the discovery of rich phenomena;2 here, we shall investigate how they are affected by symmetry breaking. For example, the symmetric reference yields synchronicity of the CENFs during tunnelling isomerizations from the reactant (R) to the product (P), symmetry of the shapes of the CENFs in the domains of R and P, and maximum values of the CENFs half way between R and P, at the potential barrier. Will these properties survive in asymmetric double-well potentials? The present quantum dynamical derivation allows the first model application to CENFs during tunnelling isomerization in an external electric field. As an example, we consider a pericyclic reaction, specificly the Cope rearrangement4 of semibullvalene (SBV). The reaction is illustrated in Figure 1, in terms of the Lewis structures for R and P, with cyclic labels for carbon nuclei C1, C2, ...., C8 along an increasing angle ϕ. Both © XXXX American Chemical Society

R and P have Cs symmetry; accordingly, our model assumes conservation of Cs symmetry during the reaction. The importance of tunnelling reactions in organic chemistry has been demonstrated recently.5−9 The example, SBV, and several derivatives have already served as touchstones for various investigations of pericyclic reactivity: synthesis,10−14 spectroscopy,15−19 kinetics,15,20,21 electronic structure,12,13,17,18,22−31 thermochromicity,12−14,18,32 ab initio molecular dynamics,33 quantum dynamics of laser control,32,34−36 and imaging of chemically active valence electrons during a pericyclic reaction.37 The thermochromicity of SBV, for example, is supported by the rather low potential barrier between R and P and the rather large energy gap between the potential energy surfaces (PES) of the electronic ground and first excited states.18,26,27,31,34 Hence, the low barrier supports tunnelling in the electronic ground state. Tunnelling is also supported by the rather low value of the effective mass associated with the Special Issue: Ronnie Kosloff Festschrift Received: November 18, 2015 Revised: January 18, 2016

A

DOI: 10.1021/acs.jpca.5b11295 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

electric field generates an asymmetric effective double-well potential, with the well for R being higher than that for P. The present electric field thus breaks the symmetry of the effective double-well potential of SBV, even though it conserves Cs symmetry. The consequences for the CENFs will be derived below.

2. QUANTUM THEORY OF CENFS IN MOLECULAR SYSTEMS WITH ASYMMETRIC DOUBLE-WELL POTENTIALS For reference, the derivation of the quantum theory of CENFs in molecular systems with asymmetric double-well potentials starts by a summary of important results for symmetric (subscript s) systems2 (see also ref 1). In nonrelativistic approximation, the Hamiltonian Hs accounts for the kinetic energies of all of the electrons and nuclei and for their Coulomb interactions. Its eigenfunctions ϕk and eigenenergies e k are obtained as solutions of the time-independent Schrödinger equation (TISE) Hsϕk = ekϕk Figure 1. Cope rearrangement of semibullvalene (SBV) from reactant (R, left) to product (P, right). Carbon nuclei C1, C2, ..., C8 are labeled in a cyclic manner, with increasing values along the angle ϕ, starting from C1 at ϕ = 0. R and P are oriented with the C1−C5 carbon− carbon bridge parallel to the X axis, the Z axis perpendicular to the paper plane, and Y perpendicular to X and Z. The X−Z plane is the mirror plane in Cs symmetry. The nuclear center of mass (NCM) is at the origin. Also shown is the one-dimensional (1D) cut of the symmetric double-well potential energy surface V along the coordinate Q. Its two potential minima correspond to the classical structures of R and P; the coordinate Q leads straight from R to P. The difference (−ρn,R + ρn,P) of the reduced nuclear (subscript n) densities of P and R along Q is illustrated by the red curve (adapted from ref 2).

(1)

k = 0, 1, 2, .... For the symmetric reference, the wave functions ϕk have alternating parities, e.g., + and − for the wave functions ϕ0 and ϕ1 of the lowest doublet, respectively. In the following applications, we focus on the properties of this doublet. Its tunnelling splitting is Δe = e1 − e0

(2)

and the mean energy is

e ̅ = (e1 + e0)/2

(3)

The eigenstates (1) allow the wave functions to be defined, which represent the reactants and products77

coordinate Q for the main direction of the nuclear flux from R to P, μ = 0.208(MC + MH).2 (Compare this with the earlier estimate, μ ≈ (MC + MH)/4.16) The results for the CENFs for this reaction without an external field2 stimulated the present search for corresponding phenomena in the environment of an external electric field (see also ref 38). In a broader perspective,2 this article should add new discoveries to the quantum theory of intramolecular fluxes, including electronic fluxes,39−53 nuclear fluxes,1,54−61 and CENFs2,3,62−73 (see also refs 74−76). Before starting, it is helpful to explain two different meanings of the term symmetry in the context of this article. The first is the symmetry of the molecular point group of R and P, with corresponding symmetry elements. For the present application to SBV, this is Cs symmetry with its Cs mirror plane. When SBV is put into the external electric field, we choose its direction parallel to the C1−C5 carbon bridge of SBV, i.e., it is in the Cs mirror plane. This way the electric field conserves Cs symmetry. Subsequently, this will be important for the determination of the angular (ϕ) directions of the electronic fluxes during tunneling because they depend on the Cs symmetry plane.3,43 Indeed, conservation of Cs symmetry implies the same angular (ϕ) directions of the electronic fluxes in SBV for both scenarios, without and with electric field. The second meaning of symmetry concerns the double-well potential that supports R and P. Without electric field, it is symmetric. Accordingly, R and P have the same energies. When SBV is put into the present electric field, the energies of R and P increase and decrease, respectively, due to dipole coupling. Accordingly, the

ψR = (ϕ0 + ϕ1)/ 2

(4)

ψP = ( −ϕ0 + ϕ1)/ 2

(5)

together with their densities

ρR = |ψR |2

(6)

ρP = |ψP|2

(7)

Below, we shall consider the scenario where the system is prepared initially in the reactant state (4); this implies a twostate model. This preparation may be achieved by means of laser control78−85 including coherent control80 or different versions of optimal control;82−84 for a recent application to two-state models, see ref 85. The time evolution of the symmetric reference system is then obtained as the solution of the time-dependent Schrödinger equation (TDSE)86 ih

∂ ψ (t ) = HsψRs(t ) ∂t Rs

(8)

with the initial condition ψRs(t = 0) = ψR. The analytical solution is ψRs(t ) =

1 (ϕ e−ie0t / h + ϕ1 e−ie1t / h) 2 0

(9)

The corresponding density ρRs(t) = |ψRs(t)| can be written as1,2 2

ρRs (t ) = ρR + (ρP − ρR ) ·sin 2(πt /τ ) B

(10)


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The Journal of Physical Chemistry A Apparently, the symmetric system tunnels periodically from R at t = 0, τ, 2τ, ... to P at t = τ/2, 3τ/2, 5τ/2, etc. The tunnelling period τ is related to the tunnelling splitting Δe (2) by77

̇ (ϕ ,t ) + ρRs,ep

ρ̇Rs,n (Q ,t ) +

The wave functions and densities in eqs 1−10 depend on the coordinates of all electrons and nuclei. These total densities may be reduced to electronic (subscript e) and nuclear (subscript n) densities by integrating over the complementary nuclear and electronic densities, respectively. Further reduction of the electronic densities of all electrons to the familiar threedimensional (3D) one-electron density is achieved by integrating the all-electron density over the coordinates of all electrons but one. In favorable cases such as in the present application to SBV,2,3 the one-electron density may be separated into contributions from core and valence electrons. For the application to the pericyclic Cope rearrangement of SBV, the valence electron density may be separated further into contributions of pericyclic (subscript p) and other valence electrons.2,3 The pericyclic electrons occupy orbitals that contribute to the change of the Lewis structures from R to P (Figure 1), e.g., π and σ orbitals associated with carbon nuclei C2, C3, C4, C6, C7, and C8.3 In contrast, the other valence orbitals do not contribute, e.g., those that describe CH bonds. Finally, the 3D electron densities can be reduced to 1D electron densities along selected coordinates by integrating over the two complementary coordinates. These reductions conserve the structure of the fundamental solution (10) for the total densities. For example, the time evolution of the 1D pericyclic electron density along the cylindrical angle ϕ (Figure 1) is2

d j (Q ,t ) = 0 dQ Rs,n

(16)

2

for the nuclei. In eq 15, jRs,ep(ϕ,t) denotes the reduced (1D) angular flux density of the pericyclic electrons.2 Likewise, in eq 16, jRs,n(Q,t) is the reduced (1D) nuclear flux density along Q. The physical units of the 1D flux densities are the same as the units of fluxes, namely, 1/time; for this reason, the 1D reduced flux densities are also called fluxes.2 Inserting eqs 12 and 13 into eqs 15 and 16 yields jRs,ep (ϕ ,t ) = jRs,ep (ϕ0 ,t ) −

ϕ

∫ϕ

̇ (ϕ′,t ) dϕ′ρRs,ep

0

= jRs,ep (ϕ0 ,t ) −

∫ϕ

ϕ

dϕ′[ρP,ep (ϕ′) − ρR,ep (ϕ′)]

0

·(π /τ )sin(2πt /τ ) ≡ jRs,ep (ϕ0 ,t ) −

∫ϕ

ϕ

dϕ′Δρep (ϕ′)

0

·(π /τ )sin(2πt /τ ) (17)

and jRs,n (Q ,t ) = jRs,n (Q 0 ,t ) −

ρRs,ep (ϕ ,t ) = ρR,ep (ϕ) + [ρP,ep (ϕ) − ρR,ep (ϕ)]·sin 2(πt /τ )

= jRs,n (Q 0 ,t ) −

(12)

Likewise, the nuclear density for all nuclei can be reduced to the density for nuclear motions along a specific nuclear coordinate by integrating over all other nuclear coordinates. In the present application, we focus on the coordinate Q that leads from the potential minimum for R straight to P. The profile V(Q) of the symmetric double-well potential energy surface along Q is illustrated in Figure 1 (adapted from ref 2). The values of Q at the potential minima are QR = −QP = −1.06 Å. Again, the reduction does not change the general structure of the time-dependent total density of all particles (10), i.e., the nuclear density for motions along Q evolves as2

∫Q

Q

̇ (Q ′,t ) dQ ′ρRs,n 0

∫Q

Q

dQ ′[ρP,n (Q ′) 0

− ρR,n (Q ′)]·(π /τ )sin(2πt /τ ) ≡ jRs ,n (Q 0 ,t ) −

∫Q

Q

dQ ′Δρn (Q ′) 0

·(π /τ )sin(2πt /τ ) (18)

where Δρep (ϕ) = ρP,ep (ϕ) − ρR,ep (ϕ)

(19)

and

2

ρRs,n (Q ,t ) = ρR,n (Q ) + [ρP,n (Q ) − ρR,n (Q )]·sin (πt /τ )

Δρn (Q ) = ρP,n (Q ) − ρR,n (Q )

(13)

(20)

are the corresponding differences of the reduced electronic and nuclear densities of the products and the reactants, respectively. The lower integration limits ϕ0 = 0 and Q0 = −1.5 Å are chosen such that the reduced flux densities either vanish for symmetry reasons2,3 (jRs,ep(ϕ0 = 0,t) = 0) or are negligible (jRs,n(Q0 = −1.5 Å, t) ≈ 0.) More specifically, the Cs symmetry of SBV implies symmetry of the fluxes, jRs,ep(ϕ,t) = −jRs,ep(−ϕ,t) as well as jRs,n(Q,t) = jRs,n(−Q,t). Otherwise, the particles would flow systematically from one side of the Cs mirror plane to the other, thus breaking Cs symmetry, or the electrons would generate a ring current, which is impossible in the electronic ground state.43 As a consequence, the fluxes (17 and 18) are products of spatial times temporal factors. All temporal factors are the same, i.e., they are sinusoidal, with period τ and amplitude π/τ. Evidently, this implies synchronicity of the CENFs, with the same sinusoidal time evolutions. Again, this is a consequence of

Obviously, the time evolutions of the electronic density (12) and the nuclear one (13) are synchronous. This is an important analytical result of the two-state model (9).2 Otherwise, any significant contributions of more excited states in eq 9 may destroy the synchronicity (see also refs 2, 38, and 57). The TDSE (8) implies the continuity equation2,86 ρ̇(t ) + div j(t ) = 0

(15)

for the pericyclic electrons and

(11)

τ·Δe = h

d j (ϕ ,t ) = 0 dϕ Rs,ep

(14)

where div j(t) denotes the divergence of the flux density in full dimensionality (full-D). In analogy to the reductions of the fullD total density to 1D reduced electronic and nuclear densities, one can integrate eq 14 over all complementary nuclear and electronic degrees of freedom. Finally, one arrives at the corresponding reduced 1D continuity equations2 (see also ref 87) C


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The Journal of Physical Chemistry A E1 = e ̅ + f ·Δe/2 with scaling factor

the two-state model. Otherwise, any significant contributions of more excited states in eq 9 may destroy the synchronicity, see also refs 2, 38, and 57. The spatial factors are the (negative) integrals of the differences of the reduced (1D) densities of P and R. Applications to the concerted electronic and nuclear fluxes during the Cope rearrangement of SBV by tunnelling will be demonstrated in Section 3. The results for the symmetric reference are adapted from ref 2 (see also ref 1). Let us now proceed from the symmetric reference to CENFs in asymmetric double-well potentials. Symmetry is broken by adding an antisymmetric interaction W to the Hamiltonian of the symmetric reference Has = Hs + W

f=

∑ Cklϕk k

⎛ ⎞ ⎛ ψ0 ⎞ c s ⎜ ϕ0 ⎟ ⎜ ⎟= −s c ⎜ ϕ ⎟ ⎝ ψ1 ⎠ ⎝ 1⎠

(

(21)

α=

(24)

f → 2|v| /Δe

for

2|v| ≫ Δe

(36)

α = 0, π /8(=22.5°), 0.624(=35.8°) f = 1,

2,

10

(37)

These will be referred to as examples of zero, weak, and moderate interactions, respectively. The asymmetry of the system increases with increasing values of the parameters |2v/ Δe|, |α|, or f. Let us now determine the time evolutions of the wave function, the density and the flux of the asymmetric system, again starting from R. For this purpose, we apply steps analogous to those for the symmetric reference. Thus, first we solve the TDSE

(26)

(27)

ih

(28)

∂ ψ (t ) = HasψR (t ) ∂t R

(38)

subject to the initial condition

ψR (t = 0) = ψR

(39)

in TSA. (Note that the subscript s for the symmetric reference no longer appears in the asymmetric case.) For the solution, it is helpful to express the reactant and product states in terms of the eigenfunctions of the asymmetric system1

(29)

⎛ ψR ⎞ ⎛C −S ⎞⎛ ψ0 ⎞ ⎟⎜ ⎟ ⎜ ⎟=⎜ ⎝ ψP ⎠ ⎝ S C ⎠⎝ ψ1 ⎠

The validity of the TSA is essential for the subsequent application. Accordingly, marginal differences in the shapes of the wave functions in the different potential wells are negligible. For the present application to SBV in the electric field, the validity of the TSA will be demonstrated in Section 3. The eigenenergies El can be expressed as analytical solutions of eq 29 E0 = e ̅ − f ·Δe/2

and

2v /Δe = 0, −1, −3 (25)

Also, the couplings between levels 0 and 1 should be dominant. In this case, eqs 24−27 can be approximated well by the twostate approximation (TSA) for the lowest tunnelling doublet of the asymmetric system ⎛ C 0l ⎞ ⎛ e 0 v ⎞ ⎛ C 0l ⎞ ⎜ ⎟⎜⎜ ⎟⎟ = El ⎜⎜ ⎟⎟ ⎝ v e1⎠⎝ C1l ⎠ ⎝ C1l ⎠

(35)

In the applications below, the constraint (28) implies that |α| is always sufficiently below the upper limit (36). Exemplarily, we shall investigate three cases of 2v/Δe with corresponding values of α and f

As in ref 1, we consider scenarios with absolute values of the interaction energy |v| (27) that are not all too strong, e.g., not much larger than 10 times the tunnelling splitting. Moreover, this should be much less than the energy gap between e1 and the next excited level, e2 0 < |v| ≤ 10Δe = e 2 − e1

1 arctan( −2v /Δe) 2

|α| → π /4 (23)

but v = ⟨ϕ0|W |ϕ1⟩ ≠ 0

(34)

Note that negative values of v yield positive angles α. In the limit v → 0, we recover the case of the symmetric reference (α → 0; f → 1; ψ0 → ϕ0; ψ1 → ϕ1). In contrast, for very strong interactions, the solution (35) approaches the limit

(22)

where δjk denotes Kronecker’s symbol (δjk = 1 for j = k; otherwise, δjk = 0). The antisymmetry of W implies that ⟨ϕk |W |ϕk ⟩ = 0

)

where c = cos α, s = sin α, and

with matrix elements Has, jk = ejδjk + ⟨ϕj|W |ϕk ⟩

(33)

Furthermore, the TSA (29) yields analytical expressions for the wave functions (23)

The TISE (22) may then be rewritten in matrix form HasCl = ElCl

(32)

ΔE = E1 − E0 = f ·Δe

The wave functions ψl may be expanded in terms of the eigenfunctions ϕk for the symmetric reference ψl =

1 + (2v /Δe)2

Accordingly, the tunnelling splitting ΔE of the asymmetric system is f times larger than the symmetric reference

Subsequently, we shall use the coupling between the molecular dipole and the external field as a special application, but the present general derivation is valid for arbitrary antisymmetric interactions W. The eigenfunctions ψl and eigenenergies El of the asymmetric system are obtained as solutions of the TISE

Hasψl = Elψl

(31)

(40)

where C = cos(α − π/4) and S = sin(α − π/4). As a consequence, the analytical solution of eqs 38 and 39 is ψR (t ) = C e−iE0t / hψ0 − S e−iE1t / hψ1

(41)

The corresponding density ρR(t) = |ψR(t)|2 can be written as

(30) D


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The Journal of Physical Chemistry A ρR (t ) = (1 − 1/f 2 )ρR + (1/f 2 ) {ρR + [ρP − ρR + (2v /Δe) ·2ψR ψP]· sin 2(πt /T )} ≡ ρR,ntun + ρR,tun (t )

(42)

0

for

(43)

0

[ρP,ep (ϕ′) − ρR,ep (ϕ′)]·(π /T )sin(2πt /T )}

2v ≫ Δe

ρR (t ) → ρR,tun (t ) = [ρR + (ρP − ρR ) ·sin 2(πt /τ )]

0

·(π /T )sin(2πt /T )} (51)

and

(44)

0

= jR,n (Q 0′ ,t ) − (1/f 2 ) Q

∫Q ′ dQ ′[ρP,n (Q ′) − ρR,n (Q ′)]·

{

0

(π /T )sin(2πt /T )}

0

·(π /T )sin(2πt /T )} for

(52)

Apparently, the solutions (51 and 52) are closely analogous to the symmetric reference (17 and 18). They have the same spatial factors. The temporal factors have the same form, except that the tunnelling time τ of the symmetric reference is replaced by the shorter T in the system with an asymmetric double-well potential. This indicates the synchronicity of the CENFs. Again, this is a consequence of the TSA. Otherwise, any significant contributions of more excited states in eq 41 may destroy the synchronicity, analogous to the discussion of the equations 9, 17, and 18. The factor 1/f 2 accounts for the dying out of the tunnelling in asymmetric systems, compared to the symmetric reference. In the limit of zero asymmetry (v → 0, f → 1), the solutions (51 and 52) for the asymmetric system approach those (17 and 18) of the symmetric reference. In the present application to SBV in the electric field parallel to the C1−C5 bond, Cs symmetry is conserved. This implies the same boundary conditions and, therefore, also the same directions of the fluxes during tunneling of SBV for both scenarios, without or with an electric field. Specifically, the locations of zero fluxes are the same, (ϕ0′ = ϕ0 = 0) and jR,ep(0,t) = 0. Likewise, one can set Q′0 = Q0 for the location of zero nuclear fluxes, jR,n(Q0,t) = 0 (cf. eqs 51 and 52).

(46)

analogous to eq 11. Comparison with eq 33 yields T = τ /f

(47)

i.e., the period T for tunnelling in asymmetric systems decreases by the factor f (compared to τ for the symmetric reference). The tunnelling term in eq 42 contains the contribution +(2v/ Δe)·2ψRψP. This can safely be neglected because the wave functions representing R and P have almost zero overlap. This allows us to employ the zero-overlap approximation ρR (t ) ≈ (1 − 1/f 2 )ρR + (1/f 2 )[ρR + (ρP − ρR ) · sin 2(T )]

(48)

In order to determine the CENFs, we employ again the continuity eq 14, reduced from full-D to 1D.2 The results are d j (ϕ ,t ) = 0 dϕ R,ep

3. APPLICATIONS TO CENFS DURING COPE REARRANGEMENT OF SEMIBULLVALENE BY TUNNELLING IN ELECTRIC FIELDS The general theory of CENFS during tunnelling of systems with asymmetric double-well potentials (Section 2) will now be applied to CENFs during a pericyclic reaction, exemplarily, the Cope rearrangement of SBV,2,3,10−38 by tunnelling in external electric fields. The presentation will follow the order of Section 2, i.e., for reference, we shall first show the results for the symmetric system (SBV without external field), followed by those for the asymmetric system (SBV in weak and moderate electric fields.) Here, we focus on effects of asymmetric double-

(49)

for the pericyclic electrons and ρ̇R,n (Q ,t ) +

d j (Q ,t ) = 0 dQ R,n

Q

∫Q ′ dQ ′Δρn (Q ′)

≡ jR,n (Q 0′ ,t ) − (1/f 2 ){

For asymmetric systems, τ is replaced by the tunnelling period T (cf. eq 42). It is related to the tunnelling splitting ΔE by1

T ·ΔE = h

Q

̇ (Q ′,t )] ∫Q ′ dQ ′ρR,n

jR,n (Q ,t ) = jR,n (Q 0′ ,t ) − (1/f 2 )[

(45)

ν→0

ϕ

∫ϕ′ dϕ′Δρep(ϕ′)

≡ jR,ep (ϕ0′,t ) − (1/f 2 ){

which means that the initial density is frozen: tunnelling in very asymmetric systems dies out. The other term ρR,tun(t) is called the tunnelling term.1 It consists of the expression in wavy brackets {} in eq 42 times the factor 1/f 2. Hence, the tunnelling term decreases with increasing strength of the antisymmetric interaction. For the symmetric reference (v = 0, f = 1), we recover the analytical solution for tunnelling in the symmetric double-well potential2

̇ (ϕ ,t ) + ρR,ep

ϕ

∫ϕ′ dϕ′

= jR,ep (ϕ0′,t ) − (1/f 2 ){

Compare this with ref 1. Apparently, it consists of a timeindependent term ρR,ntun and a time-dependent term ρR,tun(t). The term ρR,ntun is called the nontunnelling term.1 It consists of the initial density ρR times a factor (1 − 1/f 2) that increases with asymmetry from 0 for the symmetric reference (v = 0, f = 1) to 1 in the case of very strong antisymmetric interaction (2v ≫ Δe, f → |2v/Δe| → ∞). In this limit ρR (t ) → ρR,ntun = ρR

ϕ

̇ (ϕ′,t )] ∫ϕ′ dϕ′ρR,ep

jR,ep (ϕ ,t ) = jR,ep (ϕ0′,t ) − (1/f 2 )[

(50)

for the nuclei, analogous to eqs 15 and 16. The resulting pericyclic electronic and nuclear fluxes in the asymmetric system are E


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Figure 2. Results for the Cope rearrangement of SBV by tunnelling in zero (left column, ε = 0.0 V/cm), weak (middle column, ε = −0.218 × 10−8 V/cm), and moderate (right column, ε = −0.654 × 10−8 V/cm) electric fields. The corresponding values of the semiclassical dipole (d) couplings times 2, 2ε·d(QR) of the reactant (R), are equal to 0, −Δe, and −3Δe, respectively, where −Δe = 2.13 × 10−18 eV is the tunnelling splitting of the lowest tunnelling doublet, corresponding to the long tunnelling time τ = 1940 s in zero field.7 The values of the corresponding factors f = 1 + (2ε·d(Q R )/Δe)2 are equal to 1, 2 , 10 , respectively. The first row (panels a−c) shows the dipole couplings −ε·d(Q) versus the nuclear coordinate Q which leads from the reactant (QR = −1.06 Å) straight to the product (QP = +1.06 Å) (red curves). The black curves show magnifications of the resulting potential curves V(Q) − ε·d(Q), near the potential minima (compare with Figure 1). The second row (panels d−f) shows an increase in the tunnelling splittings from Δe to ΔE = f·Δe. The energy scale is the same as that in panels a−c, but the zero energy has been shifted to the center of the tunnelling splittings. The third row (panels g−i) shows the time evolution of the nuclear tunnelling dynamics illustrated by contour plots of the nuclear probability densities. Efficient tunnelling in the ideal zero field limit is suppressed by the increasing absolute values of field strengths. The last row (panels j−l) shows contour plots of the nuclear flux densities. Increasing field strengths reduce the tunnelling periods and the maximum flux amplitudes by factors 1/f and 1/f 2, respectively.

rotate and that its nuclear center of mass is fixed at the origin. Effects of translations have been discussed in refs 97 and 98. Some important results for the symmetric reference are adapted from ref 2. In particular, Figure 1 shows the symmetric double-well potential V(Q), which is obtained as a 1D cut of the full-D potential energy surface along Q. The potential minima at Q = QR = −1.06 Å and at QP = −QR correspond to the classical configurations of R and P, respectively. (We note that the value 1.06 Å is evaluated for the tunnelling mass m = 0.208(mC + mH), as derived in ref 2; this implies a marginal correction of the value 1.04 Å deduced from equation 130 of ref 2). For convenience, the values of the potential minima are set equal to zero, V(QR) = V(QP) = 0. Magnifications of V(Q) at

well potentials on the reduced CENFs. The main directions of these fluxes have already been determined in ref 2. Accordingly, pericyclic electrons flow preferably along the angular coordinate ϕ, whereas the direct flux of the nuclei from R to P is along the coordinate Q (=Q1 in ref 2) (cf. Figure 1). For comparison, ref 2 has the results for the flux of all valence electrons; this is dominated by the flux of pericyclic electrons along the angle ϕ. The model SBV is adapted from ref 2 (see also ref 3). Accordingly, we assume molecular preorientation as illustrated in Figure 1, e.g., by the methods of refs 88−96. The C5−C1 carbon−carbon bridge is thus parallel to the x axis, and the xz plane serves as the mirror plane for Cs symmetry during tunnelling. Furthermore, we assume that the molecule does not F


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Figure 3. Time evolutions of the angular (ϕ; cf. Figure 1) densities (top row) and flux densities (bottom row) of the pericyclic electrons during the Cope rearrangement of SBV by tunnelling in zero (left column), weak (middle column), and moderate (right column) electric fields. The notations are as in Figure 2. Comparison with the nuclear fluxes and flux densities (Figure 2g−l) documents the synchronicity of the concerted electronic and nuclear fluxes (CENFs). The vertical arrows indicate the angular positions of carbon nuclei C1, C2, ..., C8.

The difference of the reduced nuclear densities ρnP(Q) − ρnR(Q) of P and R is illustrated in Figure 1. The resulting time evolution of the reduced nuclear density ρRs,n(Q,t) is shown in Figure 2g by means of a contour plot (cf. eq 13). Periodic tunnelling from R at t = 0 to P at t = τ/2 and back to R at t = τ is obvious from this figure. Apparently, the nuclear density is always centered at R and/or P; in contrast, it is entirely negligible at the barrier (Q = 0). Figure 2g also exhibits the symmetry relations

the bottoms of the two potential wells are shown in Figure 2a; on this scale, the two branches of the potential wells to the left and right of the potential minima coincide within graphical resolution. For quantum mechanical simulations, the related domains of the left (Q < 0) and right (Q > 0) potential wells are attributed to R and P, respectively. The local maximum of V(Q) at Q = 0 is the barrier for the nuclear flux from R to P along Q. The method of quantum chemistry for calculations of V(Q) has been elucidated in refs 2 and 31. Here, we use the same method to evaluate the x component d(Q) of the molecular dipole. Molecular symmetry implies that d(Q) is antisymmetric d (Q ) = − d ( − Q )

ρRs,n (Q ,t ) = ρRs,n (Q ,τ − t ) = ρRs,n ( −Q ,τ /2 + t ) = ρRs,n ( −Q ,τ /2 − t )

(53)

(54)

which are consequences of the periodic time evolution (13) and Cs symmetry. The corresponding angular pericyclic electron density ρRs,ep(ϕ,t) is illustrated in the top left panel of Figure 3 (cf. eq 12). To facilitate the analysis, the angular positions of carbon nuclei C1, C2, ..., C8 are also indicated in this panel by vertical arrows. The domains from C4 to C6 and from C8 to C2 (cyclic) may be attributed to the “old” bond C4−C6 and to the “new” bond C2−C8, which are broken or made during tunnelling, respectively. The other domains from C2 to C3, etc. belong to bonds C2C3, etc. This assignment allows the main trends in pericyclic electron transfers during the Cope rearrangement of SBV by tunnelling from R to P to be deduced. In the time interval from 0 to τ/2, electrons in bond C4−C6 disappear, indicating that this bond is broken. In the same time interval, electrons appear in bond C2−C8, indicating bond formation. At the same time, the double bonds C2C3 and C7C8 shift to the neighboring bonds C3−C4 and C6− C7, respectively. All processes are reversed during the back

The values d(QR) and d(QP) at R and P are +0.0924ea0 (= 0.394D) and −0.0924ea0, respectively. The value d(QR) is positive because the reactant SBV accumulates part of the pericyclic electrons in the bond C4−C6, at the expense of the broken bond between carbon nuclei C2 and C8; the opposite is true for P. It turns out that the x component of the dipole is nearly linear, d(Q) ≈ d′·Q with derivative d′ = −0.0924/1.06e = −0.0872e. By comparison, its z component is negligible compared to the x component. The y component vanishes, for symmetry reasons. The tunnelling splitting Δe between the levels e1 and e0 of the lowest tunnelling doublet is illustrated in Figure 2d. Note that panels a and d in Figure 2 have the same scaling but that the ordinate of Figure 2d has been shifted by the mean value e ̅ of the tunnelling splitting. The value Δe = 2.13 × 10−18 eV = 18.4 × 10−15 hc cm−1 = 0.0784 × 10−18Eh is small, in accord with the long tunnelling time τ = 1940 s7 (cf. eq 11). G


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flow forward from the double bond C2C3 to the neighboring bond C3−C4, resulting in double bond shifting. The equivalent electron flux backward achieves double bond shifting from C7C8 to C6−C7. Altogether, the directions of the angular flux changes six times from ϕ = 0 to 2π, resulting in pincer-type electron transfer (cf. refs 2 and 3). All directions of the flux are reversed during tunnelling from P back to R in the time interval from τ/2 to τ. Time integration of the flux from 0 to τ/2 yields the number of pericyclic electrons that have been transferred during tunnelling from R to P, also called the yield

reaction from P to R by tunnelling, from t = τ/2 to τ. The panel also displays the symmetries ρRs,ep (ϕ ,t ) = ρRs,ep (ϕ ,τ − t ) = ρRs,ep (2π − ϕ ,t ) = ρRs,ep (2π − ϕ ,τ − t )

(55)

which are again consequences of the periodic time evolution (12) and molecular symmetry. Finally, the reduced nuclear flux density, or flux jRs,n(Q,t), is illustrated in Figure 2j, again by a contour plot (cf. eq 18). According to the derivation in Section 2, it is the product of a spatial factor times a temporal factor. The spatial profile is obtained by integrating the (negative) difference of the reduced densities of P and R (shown in Figure 1). The temporal factor is π/τ·sin(2πt/τ). Accordingly, the nuclear flux is periodic, with period τ. It is positive (direction R → P) during the time interval 0 < t < τ/2 and negative (direction P → R) during the back reaction (τ/2 < t < τ). It vanishes at t = 0, τ/2, and τ. It has its maximum half way between R and P, at t = τ/4, Q = 0. On first glance, this appears to be paradoxical because there is hardly any nuclear density below the barrier. None the less, the result may be rationalized by saying that in the classical limit the system does not tunnel; instead, it prefers configurations R and P. However, since quantum mechanics dictates that it must tunnel,77 it avoids the barrier by flowing through it as fast as possible.59 The temporal periodicity and Cs symmetry imply the symmetry relations of the nuclear flux

YRs,ep(ϕ) =

dt jRs,ep (ϕ ,t ) = −

∫0

ϕ

dϕ′Δρep (ϕ′)

Equation 58 reveals the meaning of the spatial factor of the flux (17): it is the electron yield. Maximum and minimum values of the yield are obtained for angles ϕ close to positions of carbon nuclei C2, C2, C4, C6, C7, and C8 (compare with the vertical arrows in Figure 3). This provides an a posteriori confirmation of the observer planes for the electronic fluxes between neighboring bonds, as defined in ref 3 (see also ref 38). Accordingly, the maximum number of electrons that flow out of the “old” bond C4−C6 into the two neighboring bonds is equal to 2 × 0.23 = 0.46. The same number of electrons flow from two neighboring bonds into the “new” bond C2−C8. For comparison, the number of electrons that contribute to double bond shifting is much fewer, 2 × 0.09 = 0.18. The total number of transferred electrons is 4 × 0.23 + 2 × 0.09 = 1.1. The present numbers, 0.23 and 0.09, of pericyclic electrons that flow between neighboring bonds during bond breaking and forming and during double bond shifting by tunnelling agree with those for the reaction with energy above the barrier.3 The corresponding numbers for all valence electrons are 0.63 and 0.23, respectively.2 Let us now investigate the CENFs during the Cope rearrangement of SBV by tunnelling in asymmetric doublewell potentials. For this purpose, the previous symmetric model is extended to SBV in weak or moderate x-polarized fields ε. Note that this modification conserves Cs symmetry but breaks the + and − parities of the eigenstates of the symmetric reference; hence, the model is called asymmetric. In semiclassical dipole approximation, the antisymmetric interaction of the field ε and the molecular dipole d is W = −ε·d. The Hamiltonian of the asymmetric model SBV in electric field ε is thus Has = Hs − ε·d (cf. eq 21). In the frame of the TSA (Section 2), the matrix element of this interaction for the states of the lowest tunnelling doublet is

(56)

compare with eq 54. Otherwise, the nuclear fluxes would break Cs symmetry, analogous to the discussion of eq 18. The associated angular flux of the pericyclic electrons is illustrated in the lower left panel of Figure 3 (cf. eq 17). One readily notices the symmetry relations jRs,ep (ϕ ,t ) = −jRs,ep (ϕ ,τ − t ) = −jRs,ep (2π − ϕ ,t ) = jRs,ep (2π − ϕ ,τ − t ) = jRs,ep (π − ϕ ,t ) = −jRs,ep (π − ϕ ,τ − t )

τ /2

(58)

jRs,n (Q ,t ) = −jRs,n (Q ,τ − t ) = jRs,n ( −Q ,t ) = −jRs,n ( −Q ,τ − t )

∫0

(57)

which are again consequences of the periodicity and molecular symmetry (compare with eq 55). The Cs symmetry implies that the angular flux must vanish on the symmetry plane; hence, jRs,ep(ϕ0,t) = jRs,ep(ϕ1,t) = 0 at ϕ0 = 0 and ϕ1 = π. This fixes the boundary condition in eq 17. Analysis of Figure 3 reveals the mechanism of electron transfer during the Cope rearrangement of SBV by tunnelling from R to P during the time interval 0 < t < τ/2. Apparently, the electrons flow out of the bond C4−C6 and into the bond C2−C8; hence, the “old” bond C4−C6 is broken, whereas the “new” bond C2−C8 is formed. More specifically, equal parts of the pericyclic electrons that leave the “old” bond C4−C6 flow forward (jRs,ep > 0) to the neighboring bond C6−C7 and backward to bond C3−C4. At the same time, the same amount of pericyclic electrons flow from bonds C2 C3 and C7C8 (backward and forward, respectively) into the “new” bond C2−C8. Also at the same time, pericyclic electrons

v = −ε⟨ϕ0|d|ϕ1⟩

(59)

cf. eq 27. The ratio of v and the tunnelling splitting Δe or, equivalently, the factor f = 1 + (2v /Δe)2 determines the concerted tunnelling dynamics of the electrons and nuclei, in particular the CENFs. The resulting full-D densities and flux densities are reduced to those for nuclear motions along Q or to the motions of pericyclic electrons along ϕ, according to the general theory of Section 2. For the present application, the symmetric doublewell potential V(Q) is thus replaced by the asymmetric effective double-well potential Vas(Q ) = V (Q ) − ε ·d(Q ) H

(60) DOI: 10.1021/acs.jpca.5b11295 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A with dipole function d(Q) of the symmetric reference. Its antisymmetry (53) and the approximate linearity d(Q) ≈ d′·Q, together with the pretty good localization of the eigenfunctions and the corresponding densities at R and P (cf. Figure 1), allow the approximation1 v ≈ −ε ·d(Q P) = ε ·d(Q R )

intermediate case of half tunnelling, half nontunnelling to nontunnelling occurs at the present weak field (2v = Δe , f = 2 ). Hence, this specific weak field is also called the critical field. Analogous effects of the time evolutions of the densities of pericyclic electrons depending on zero, weak, and moderate electric fields are documented in the top row of Figure 3. Already for the critical field (middle panel) the electrons transferred out of the “old” bond C4−C6 and into the “new” bond C2−C8 are pretty much suppressed, in comparison to the zero field reference (left panel). For the moderate field (right panel), the signatures of bond shifting also disappear. The electronic structure of the molecule remains essentially (90%) in its initial R configuration; tunnelling is frozen. Finally, let us consider the effects from zero via weak to moderate electric fields on the CENFs during the Cope rearrangement of SBV by tunnelling. The resulting nuclear fluxes are documented in Figure 2j−l, respectively. Three effects are obvious immediately: (i) the spatiotemporal patterns are robust. In particular, the symmetry of the nuclear fluxes (56) is conserved. This is surprising in view of symmetry breaking of the related nuclear densities (48) (cf. Figure 2g−i). This effect can be explained by noting that the symmetry of the flux densities is determined by the symmetry of the time derivatives of the densities (cf. eqs 16 and 50 as well as 15 and 49). Those time derivatives are robust, except for the factor 1/ f 2 (cf. eq 48); hence, they conserve symmetry. Moreover, the maxima of the nuclear fluxes remain at the barrier (Q = 0). (ii) The tunnelling time τ decreases to T by the factor f (cf. eq 47). The same effect can also be deduced from close inspection of Figure 2g,h and eventually also Figure 2i for the nuclear densities, but it is much more obvious from Figure 2j−l for the fluxes. (iii) The magnitudes of the fluxes decrease by the factor 1/f 2. As a consequence, the resulting yields, i.e., the number of the nuclei that are transferred from R to P during a half period T/2 of tunnelling (analogous to eq 58), also decrease by the factor 1/f 2. (iv) Combination of effects (ii) and (iii) yields the reduction of the nuclear transfer rate by tunnelling (i.e., the number of nuclei that are transferred per unit time, not per tunnelling period T) by the factor 1/f 2/(1/f) = 1/f, proportional to the decrease of the tunnelling time. The effects of zero, weak, and moderate electric fields on the fluxes of pericyclic electrons are documented in the left, middle, and right panels of the bottom row of Figure 3. One readily recognizes effects analogous to those that have just been discovered for the nuclear fluxes, i.e., (i) conservation of spatiotemporal symmetries (57), (ii) decrease of the tunnelling period by the factor f (47), (iii) decrease of the magnitudes of the fluxes (51) and the corresponding yields (58) by the factor 1/f 2, and (iv) decrease of the electron transfer rate by tunnelling by the factor 1/f. (v) Moreover, comparison of the last rows of Figures 2 and 3 shows that all CENFs are synchronous. We close this section by considering some related experimental and practical aspects. Measuring CENFs is an important task. Encouraging steps toward this goal include the first determination of nuclear fluxes by means of pump−probe spectroscopy60 and the first observation of electron migration by high harmonic generation.104 An experiment to monitor effects (i)−(v) of the CENFs during the Cope rearrangement of SBV in zero, weak, and moderate external electric fields would face several challenges. Here, we focus only on two of them. First, one has to generate

(61)

Apparently, the electric field ε may be used as control parameter that determines the value of v. Equation 37 calls for the corresponding zero, weak, and moderate electric fields ε = 0.0, −0.218 × 10−8, and −0.654 × 10−8 V/cm (= 0.0, −0.424 × 10−18, and −0.272 × 10−18Eh/ea0 in atomic units) in order to satisfy the definitions of zero, weak, and moderate interactions, respectively. Explicitly, these electric fields satisfy the conditions 2ε ·d(Q R ) = 0, −Δe , −3Δe f=

1 + [2ε ·d(Q )/Δe]2 = 1,

2,

10 (62)

respectively (cf. eq 37). The small value of the tunnelling splitting Δe implies that the values of the so-called weak and moderate electric fields are also small. In fact, they are so small that the resulting asymmetric double-well potential (60) and the symmetric reference are indistinguishable at the graphical resolution of Figure 1. The differences between V(Q) and Vas(Q) for zero, weak, and moderate electric fields are obvious, however, from the enlarged regions shown in Figure 2a−c, respectively. The energetic increase of the potential well of R compared to that of P suggests a working hypothesis in which switching on the electric field should promote tunnelling from the energetically enriched R to P. The related increase of the tunnelling splittings (33) by the factor f (62), from zero via weak to moderate electric fields, is documented in Figure 2d−f, respectively. Let us now investigate the consequences of the weak and moderate electric fields on the concerted electronic and nuclear dynamics during the Cope rearrangement of SBV by tunnelling in asymmetric double potential wells, compared to the symmetric reference. A necessary prerequisite is to verify the validity of the TSA in eqs 23 and 29. As a test, we have calculated the coefficients of the expansion (23) beyond the TSA, using perturbation theory. It turns out that, as expected, the absolute values of the coefficients decrease rapidly with excitation energy. Even for the largest (so-called moderate) electric field strength considered in the applications, the largest coefficient beyond the TSA is as small as 2 × 10−18. The corresponding probability (= square of coefficient) is below 10−35. This confirms that the present system, SBV in weak to moderate fields, is described perfectly in the frame of the TSA. The resulting time evolutions of the nuclear densities for zero, weak, and moderate electric fields are documented in Figure 2g,h,i, respectively. Obviously, the spatial symmetry (55) of the symmetric reference (Figure 2g) is broken by the electric fields (Figure 2h,i). This is a consequence of the fact that the nontunnelling term of the density (42) (or its approximation (48)) increases by the factor 1 − 1/f 2 from 0 via 1/2 to 9/10 whereas the tunnelling terms decrease from 1 via 1/2 to 1/10 from zero via weak to moderate electric fields, respectively. In other words, tunnelling is frozen out from zero via the weak to moderate electric field. This falsifies the naive working hypothesis that energy supply to the reactants should promote tunnelling. The turnover from efficient tunnelling via the I


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The Journal of Physical Chemistry A and control rather small linearly (x) polarized electric fields, near the critical weak field ε = −0.218 × 10−8 V/cm (= −0.424 × 10−18 Eh/ea0) for half tunnelling, half nontunnelling. This could be achieved by the setup illustrated schematically in Figure 4: A power source generates an input voltage on the

however, when the symmetric reference is distorted to a system with asymmetric double-well potentials. In spite of this symmetry breaking for EPDs and NPDs, spatial symmetry is conserved for CENFs. This is one of the most surprising results of this article. The analytical results for tunnelling in asymmetric doublewell potentials, which have been derived in this article, are valid during very long periods of time: the present applications are in the domains of kiloseconds. This is much beyond the capabilities of numerical techniques, which are taylored to rather short time domains of molecular reaction dynamics, typically pico-, femto-, or even attoseconds.99−103 Exemplary application to the Cope rearrangement of SBV shows that tunnelling of the ideal symmetric system can be suppressed by asymmetries induced by rather small external electric fields. In the present case, the amplitude of the critical field that yields the half tunnelling, half nontunnelling border is as low as 0.218 × 10−8 V/cm. Even slightly larger fields (say larger than 10−8 V/cm) suppress tunnelling completely. At the same time, the delocalized eigenstates of the symmetric reference, which can be regarded as Schrödinger’s cat-type states representing R and P with 50:50% probabilities, get localized at one or the other minima of the asymmetric doublewell potential, representing either R or P with 100% versus 0% probabilities. We conclude this article with a related speculation that goes much beyond the present application to SBV: under prebiotic conditions, similar weak (as well as stronger) electric fields may suppress tunnelling between two forms (R and S or D and L) of enantiomers, e.g., between D- and L-alanine. Instead, they should induce localizations of enantiomers in either one or the other form (e.g. L-alanine), thus destroying the ideal Schrödinger’s cat state, which represents 50:50% of one and the other forms. This type of de facto symmetry breaking may have been an important early step during evolution (see the extended, in-depth discussions in refs 105−108). By extrapolation, the results of ref 109 suggest an analogous breaking of parity by weak fields, combined with the generation of huge dipoles in highly excited Rydberg atoms or molecules.

Figure 4. Generation of the small weak (or critical) and moderate electric fields to investigate the effects of symmetry breaking during the Cope rearrangement of SBV by tunnelling, as documented in Figures 2 and 3. The output voltage of the power source (of the order of 0.001 V) is reduced by a series of slide resistors (R); the last slide resistor is used for fine-tuning the voltage, which is then applied to generate the target field strengths between two parallel grids at distance d.

order of Vin = 1 mV. A series of N slide resistors then reduces the voltage by factors m = Rout/Rin. For example, if m = 1/10 for a reduction of 1 order of magnitude, then a cascade of N = 5 resistors generates the output voltage Vout = Vin·mN = 10−8 V. Application of −Vout to two parallel grids perpendicular to x and spaced at d generates the x-polarized electric field ε = −Vout/d, for example, ε = −0.5 × 10−8 V/cm for d = 2 cm. Fine tuning of the ratio Rout/Rin in the last slide resistor generates, e.g., the critical weak target field strength, ε = −0.218 × 10−8 V/cm. Another challenge to this experiment would be the suppression of ubiquitious stray fields that may arise, e.g., from residual charges on the walls of the volume that contains the molecules or from ions in nonperfect vacuum. For example, the critical field strength ε = −0.218 × 10−8 V/cm = −0.424 × 10−18 Eh/ea0 could be compensated by the reverse field strength ε = e/ ( 4 π ε 0 R 2 ) o f a s i n g l e a n i o n a t d i st a n c e R = 1/(0.424 × 10−18) a0 = 1.54 × 109a0 = 8.12 cm the left of the molecule.

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AUTHOR INFORMATION

Corresponding Authors

*(T.B.) E-mail: [email protected]. *(J.M.) E-mail: [email protected]. Phone: +49(0) 3083853338.

to

Notes

4. CONCLUSIONS The quantum theory of CENFs during coherent periodic tunnelling from R to P and back to R in molecules with asymmetric double-well potentials has been developed. The results are deduced from the solution of the time-dependent Schrödinger equation (38) as a coherent superposition of two eigenstates (41); here, these are the two states of the lowest tunnelling doublet, but applications can be extended to analogous rather general superpositions of two eigenstates. This allows the periodic time evolutions of the resulting electronic and nuclear probability densities (EPDs and NPDs) as well as the flux densities (EFDs and NFDs) to be expressed in terms of simple sinusodial functions. These analytical results reveal various phenomena during coherent tunnelling, e.g., all EPDs and NPDs as well as all CENFs are synchronous, not only in symmetric2 but also in asymmetric double-well potentials. Spatial symmetry of the EPDs and NPDs is broken,

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to ChunMei Liu (Berlin), Annerose Polinske (Berlin), and Prof. Yonggang Yang (Taiyuan) for their great help in the preparation of the manuscript. Financial support by Deutsche Forschungsgemeinschaft (project Ma 515/27-1), the Natural Science Foundation of Shanxi, China (2014021004), the National Natural Science Foundation of China (NSFC, grant nos. 11004125 and 61378015), and the Program for Changjiang Scholars and Innovative Research Team (IRT13076) is also gratefully acknowledged. T.B. acknowledges partial financial support from U.S. Air Force Grant No. FA9550-12-1-0482. Finally, J.M. thanks Prof. P. Luger and Dr. D. Zobel (Berlin) for experimental searches for Schrödinger cat states of substituted semibullvalene back in 1992this paper J


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explains why they could not detect those states, due to the ubiquitous crystal fields

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