Quantum entanglement and nonlocal proton transfer dynamics in ...

Chemical Physics Letters 408 (2005) 302–306www.elsevier.com/locate/cplettQuantum entanglement and nonlocal proton transfer dynamicsin dimers of formic acid and analoguesFrançois FillauxLADIR-CNRS, UMR 7075 Universit´e Pierre et Marie Curie, 2 rue Henry Dunant, 94320 Thiais, FranceReceived 9 March 2005; in final form 19 April 2005Available online 13 May 2005AbstractWhether interconversion of hydrogen bonded centrosymmetric dimers of formic acid occurs through stepwise single-proton orpairwise double-proton transfer is a subject of controversy arising from a misleading treatment of proton dynamics. Quantumentanglement suggests a totally different mechanism. Dimers are in a superposition of indistinguishable tautomers for which interconversionis irrelevant. Disentanglement opens a tunneling channel for single-proton transfer across a nonlocal symmetrical doubleminimum potential. The nonlocal dynamics of a disentangled dimer is that of half-protons located at four sites with D 2h symmetry.Ó 2005 Elsevier B.V. All rights reserved.1. IntroductionProton transfer along hydrogen bonds is of fundamentalimportance to many physical, chemical and biochemicalprocesses. There is a general agreement thatthe dynamics is that of a light particle in a heavy frameworkexperiencing an effective potential along a reactioncoordinate and coupled to the motions of heavy atoms[1].For isolated centrosymmetric dimers, tautomers Iand II (Fig. 1) are identical and a symmetric double wellpotential is anticipated for simultaneous transfer of thetwo protons. A band splitting of 0.003 cm 1 , recentlyobserved in high resolution infrared spectra of dimersof formic acid (HCOOH) 2 in supersonic jet expansions,has been attributed to synchronous tunneling in theground state [2]. The measured moments of inertia accordwith a C 2h equilibrium geometry and the bandassignment scheme is consistent with numerical simulations[3–11]. The Ôminimum energy pathÕ correspondsto simultaneous transfer of the two protons moving likeE-mail address: fillaux@glvt-cnrs.fr.URL: http://ulysse.glvt-cnrs.fr/ladir/pagefillaux.htma perfectly rigid entity. In the D 2h transition state eachproton sits at the center of a hydrogen bond. The calculatedpotential barrier is in the range from 2500 to5000 cm 1 .This model requires an exceedingly strong couplingable to synchronize proton transfer across the barrier.By contrast, the infrared-Raman (u–g) band-splittingof 40 cm 1 for the OH stretching mode in the3000 cm 1 frequency range [12,13] is a representativeof the proton–proton coupling term, far too weak to excludea stepwise mechanism [14,15]. The reported synchronoustunneling [2] could be, if it were confirmed, abreakthrough in our understanding of proton transferdynamics across hydrogen bonds, proving the existenceof a so far unknown dramatic coupling betweenprotons.Alternatively, the interconversion scheme depicted inFig. 1 can be seriously questioned. First, it does not correspondto a minimum energy reaction path: energy-freeinterconversion is realized through rotations of thewhole dimer. Within the framework of quantummechanics, tautomers are indistinguishable and shouldbe thought of as a coherent linear superpositionfor which interconversion is pointless. Second, models0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.cplett.2005.04.069

F. Fillaux / Chemical Physics Letters 408 (2005) 302–306 303Fig. 1. Interconversion of a dimer of formic acid (HCOOH) 2according to [3]. Tautomers I and II have C 2h symmetry at equilibrium.The transition state has D 2h symmetry.within the framework of the Born-Oppenheimer approximationare also questionable for the electronic wavefunction is calculated in a field of nuclei represented asclassical point charges. As opposed to this approximation,proton tunneling requires a quantum mechanicaltreatment of nuclei [16].The purpose of this Letter is to rationalize protontransfer dynamics in terms of quantum entanglement–disentanglement, a concept at the heart of the profounddifference between classical and quantal physics.2. Quantum entanglement in hydrogen bondedcentrosymmetric dimersThe premises of the theoretical framework for quantumentanglement in centrosymmetric dimers are thefollowing [17–19].(i) Bridging protons (deuterons) are indistinguishableat the time scale of vibrational dynamics.(ii) The time periodic dynamics is amenable to normalcoordinates.(iii) The bridging protons (deuterons) are fermions(bosons) amenable to the symmetry postulate ofquantum mechanics. This is a consequence of theionic character of the hydrogen bond (for exampleHCOO H + ). As opposed to the CH(D) covalentbond, it is allowed to permute bridging atomsindependently of electrons.(iv) The two protons (deuterons) are degenerate. Theoverlap of the vibrational wave functions is rigorouslynegligible and so is the exchange energy [20].Spin–spin interaction is also negligible (10 4 Hz).The vibrational Hamiltonian for a tautomer (say I) isexpanded asH I ¼ H IHðDÞ þ H IM þ H IHðDÞM ;ð1Þwhere H IHðDÞ represents the two protons (deuterons),H IM the molecular frame and H IHðDÞM a coupling betweenthe two subsystems. We temporally ignore theground state degeneracy arising from the superpositionof I and II, and concentrate on small amplitude oscillationsaround the mean positions. For the sake of simplicity,rotational and rovibrational dynamics are notincluded. Therefore, (1) is not intended to provide a detailedinterpretation of high resolution spectra. The purposeis limited to proton transfer dynamics for anisotropic probability distribution of dimer orientationsat T =0K.According to (i), the Hamiltonian is invariant uponpermutation of protons (deuterons): H IHðDÞM 0. Proton(deuteron) dynamics are totally decoupled fromthe molecular frame.According to (ii), dynamics are treated within theharmonic approximation. The cartesian axes x, stretchingm OH(D), y, in-plane bending d OH(D), and z,out-of-plane bending c OH(D), are closely parallel tothe inertia axes. Then, the Hamiltonian for two coupledharmonic oscillators in three dimensions, labelled 1 and2, respectively, located at distinguishable sites,fx 0 0 ; y0 0 ; z0 0 g and f x0 0 ; y0 0 ; z0 0g, can be written as[21]H IHðDÞ ¼ X H Ia ; a ¼ x; y; z;aH Ia ¼ 12m P 2 1a þ P 2 1h2a þ2 mx2 0aða 1 a I0 Þ 2 þ ða 2 þ a I0 Þ 2þ2k a ða 1 a 2 Þ 2i . ð2ÞP 1a and P 2a are kinetic momenta. Coordinates a 1 anda 2 are projections onto the a direction of positions, definedwith respect to the center of symmetry. The frequencyof uncoupled oscillators at equilibriumpositions ±a 0 is m 0a = x 0a /(2p). The coupling potentialproportional to k a depends quadratically on the distancebetween particles. The equilibrium positions ofthe coupled oscillators are shifted at ±a 0 0 ¼a 0 =ð1 þ 4k a Þ. Normal coordinates and conjugatedmomenta corresponding to symmetrical and antisymmetricaldisplacements,a s ¼ p 1 ffiffi ða 1 a 2 Þ; P sa ¼ p 1 ffiffiffi ðP 1a P 2a Þ;22a a ¼ p 1 ffiffi ða 1 þ a 2 Þ; P aa ¼ p1 ffiffi ðP 1a þ P 2a Þ;22ð3Þsplit H Ia into two orthogonal harmonicpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffioscillators atfrequencies m aa = m 0a and m sa ¼ m 0a 1 þ 4k a , respectively,each with an effective mass m:H Ia ¼P 2 aa2m þ 1 2 mx2 0a a2 aþP 2 sa2m þ 1 2 mx2 0a ð1 þ 4k pffiffi 2aÞ a s 2 a0I0þ mx 0a a 2 4k aI0. ð4Þ1 þ 4k a

304 F. Fillaux / Chemical Physics Letters 408 (2005) 302–306This Hamiltonian is a diagonal representation of thedynamics but information on particle positions is lost.In the ground state, the Gaussian wave functionsW a 0 ða aÞ and W s 0 ða pffiffiffis 2 a0p I0Þ represent nonlocal dynamics:a a =0ora s ¼ffiffi2 a0I0do not correspond to any physicallocation of particles. In addition, wave functions cannotbe factored into wave functions for each particle. It is nolonger possible to distinguish particles 1 and 2, and todecide which particle is at site a 0 0 or a0 0. Particles arefully entangled into EPR-like states [22]. Then, accordingto (iii), we have to distinguish whether the coupledoscillators are bosons (deuterons) or fermions (protons).For deuterons, the state vectors can be written as [21]:jI0 þi D¼jH I0þ ip1 ffiffiX1 X 1S 12 ði; jÞ½j1 : iij2 : jiŠ;6 i¼ 1 jPijI0 i D¼jH I0 ip1 ffiffiX1 X 1A 12 ði; jÞ½j1 : iij2 : jiŠ;3 i¼ 1 j>ið5Þwhere |1:iæ and |2:jæ are spin state vectors for particles 1and 2, respectively, with i,j = 1,0,1. The symmetricand antisymmetric projectors are, respectively,p1ffiffi S 12 ði; jÞ ¼ 2½1 þ P 12 Š; i 6¼ j;1; i ¼ j;andð6Þp1ffiffi A 12 ði; jÞ ¼ 2½1 P 12 Š; i 6¼ j;0; i ¼ j.The permutation operator P 12 is such as:P 12 ½j1 : iij2 : jiŠ ¼ j2 : iij1 : ji.The spatial wave function is"H I0 ¼ p1 ffiffi2 Y aY aW a 0 ða aÞW s 0 ða spffiffiffi2 a0I0 ÞW a 0 ða aÞW s 0 ða ps þffiffi2 a0I0#. Þ ð7ÞFor protons, i,j = ±1/2 and the antisymmetrizedstates are:jI0 þi H¼jH I0þ i X1=2i¼ 1=2X 1=2j>iA 12 ði; jÞ½j1 : iij2 : jiŠ;jI0 i H¼jH I0 ip 1 ffiffiX1=2 X 1=2S 12 ði; jÞ½j1 : iij2 : jiŠ.3i¼ 1=2 jPið8ÞAccording to (iv), entanglement occurs at no energycost, apart from the tiny spin–spin interaction. At theenergy scale of vibrations, spin states are degenerate.Dynamics of centrosymmetric pairs are intrinsicallydecoherence-free with respect to the molecular frame.Although there are many other sources of decoherence,such as collisions, photons, etc., [23], re-entanglement occursspontaneously (no energy cost) as long as the twoparticles are in the same state (not necessarily the groundstate) and the symmetry is preserved. It should be noticedthat the symmetry related spin correlation (8) is totallydifferent in nature from the spin states of ortho and paraH 2 molecules, which are different chemical entities.3. Tautomerism and quantum entanglementLet sites at f x 0 0 ; y0 0 ; z0 0 g and fx0 0 ; y0 0 ; z0 0g, correspondto II. Eqs. (1)–(8) apply formally upon substitutionof I by II. The bridging pair is a composed bosonand a dimer can be represented as a linear coherentsuperposition analogous to SchrödingerÕCat in a superpositionof dead-alive states:j0si ¼p 1 ffiffiffi ½jI0sisjII0siŠ.ð9Þ2Here, s = Ô±Õ. This superposition has C 2h symmetrybut it is totally different in nature from the mixture depictedin Fig. 1: there is no reaction path for interconversionin terms of classical particles moving alongtrajectories. The ground state is totally degenerate andthere is no transfer dynamics. This conclusion makessense for any centrosymmetric dimer, for it is impossibleto prepare, even in thought experiments, one tautomerexclusive of the other.A necessary condition for interconversion to occur isdisentanglement. This is realized through energy and/ormomentum transfer. Then, protons k = 1 and k = 2 becomedistinguishable and the disentangled wave functionsfor tautomers are:U Ik0 ¼ Y a kU IIk0 ¼ Y a kW a 0 ða kaÞW s 0 ða ksW a 0 ða kaÞW s 0 ða kspffiffi2 a0I0 Þ;pffiffi2 a0II0 Þ.ð10ÞDisentanglement destroys all consequences of the symmetrypostulate while the nonlocal nature of the wavefunctions is preserved. Each proton (deuteron) is a representativeof the whole dynamics. In counterintuitivewords, the dynamics of each particle is that of two ÔhalfparticlesÕ,one half at each site of a particular tautomer.Normal coordinates a ka and a ks represent anti-phaseand in-phase displacements of these half-particles. Eachtautomer is composed of two such two-half-protonssuperimposed at the same sites. The disentangled wavefunctions for the superposition of tautomers are then:XkU 0þ ¼ p1 ffiffi2U 0 ¼ p1 ffiffiX 2 kðU Ik0 þ U IIk0 Þ;ðU Ik0 U IIk0 Þ.ð11Þ

F. Fillaux / Chemical Physics Letters 408 (2005) 302–306 305Disentanglement removes the ground state degeneracy:the splitting (E 0 –E 0+ = hm 0t ) is determined by theoverlap of U Ik0 and U IIk0 , along the m OH coordinate.Since there is no energy cost, the zero-point energiesof entangled, |0 + æ, and disentangled, |U 0+ æ, states areequal. For each state (11), the two protons are equallydistributed over the four sites. Coherent superpositionof |U 0+ æ and |U 0 æ gives rise to oscillations, at frequencym 0t , between tautomers through coherent transfer of twohalf-particles, as sketched in Fig. 2. The overall balanceis the transfer of a nonlocal single bare proton over theinter site distance of 0.7 Å, across a nonlocal symmetricaldouble well potential.According to quantum laws, any measurement ofproton dynamics automatically destroys quantumentanglement. Therefore, only the disentangled D 2hsymmetry is observable with spectroscopy techniques.However, moments of inertia are identical for the C 2hand D 2h structures depicted in Fig. 2, as opposed tothose in Fig. 1. Therefore, measured moments of inertiaare not conclusive with respect to the actual symmetry.A key issue is to determine the effective double-wellpotential. The sketch in Fig. 2 is so dramatically at variancewith that under consideration in theoretical works(Fig. 1) that previously calculated potentials are irrelevant.In addition, the present author is not aware ofany quantum chemistry method able to model nonlocaldynamics.The potential shape can be determined from experimentaldata. The double minimum potential splits them OH mode in the 3000 cm 1 range into two componentscorresponding to states |U 1+ æ and |U 1 æ, analogousto Eq. (11). Early analysis of the microwave rotationspectra of various carboxylic acid opened-dimers leadto potential barriers of 5000–6000 cm 1 and tunnelsplitting hm 0t 0.5–5 cm 1 for the ground state orhm 1t 50–500 cm 1 for the upper state [24]. For centrosymmetricdimers, the search for tunnel splitting hasbeen hampered by the prevailing opinion that the transferof a single proton yields an unrealistic complex composedof a de-protonated carboxylic group bound to adi-protonated one. Nevertheless, Robertson and Lawrence[25] have shown that infrared spectra of the formicacid dimer are consistent with a symmetrical doubleminimum potential with hm 0t 1.5 cm 1 , hm 1t 140 cm 1 , and H 6400 cm 1 . However, owing tothe manifold of band splitting mechanisms, the assignmentscheme of the OH stretching bands is controversial.To the best of the present authorÕs knowledge,observation of hm 0t has never been reported and moreinformation is necessary to corroborate this assignmentscheme.Quite similar effective potentials have been obtainedfor single-proton transfer in crystals composed ofhydrogen bonded centrosymmetric dimers. For potassiumhydrogencarbonate(KHCO 3 ) hm 0t =17cm 1 ,hm 01 = 130 cm 1 , and H = 4850 cm 1 [26,27]. For benzoicacid hm 0t =6cm 1 , hm 01 = 100 cm 1 , andH = 5000 cm 1 [28]. These values corroborate theassignment scheme of Robertson and Lawrence. Tothe least, a tunnel splitting of 0.003 cm 1 for the formicacid dimer [2] is quite out of range. Hopefully, thepresent Letter will be an incentive to seek tunnel splittingin the 1–10 cm 1 region.4. ConclusionNonlocality and quantum entanglement could be ofcentral importance to proton transfer dynamics, butthese concepts have been overlooked in previous works.With hindsight, this is surprising since wave functionsfor normal coordinates are nonlocal representation ofdynamics, logically amenable to entanglement. Thesymmetry related entanglement is extremely robust:environment or measurement induced decoherence iscontinuously counterbalanced by spontaneous re-entanglementat no energy cost. This leads to noncommonsensicalconclusions independent of the systemenergetics.Indistinguishability and quantum entanglement imposeperfect correlation of the two particles (either protonsor deuterons), irrespective of coupling terms. Anentangled dimer is a coherent superposition of tautomers(SchrödingerÕCat) and interconversion is meaningless.Disentanglement opens a tunneling channel forthe transfer of nonlocal entities with an effective massof 1 a.m.u. (two half-protons) across a nonlocal doubleminimum potential. This is a nonstatistical manifestationof quantum nonlocality.Fig. 2. Dynamics of indistinguishable nonseparable dimers. Thesystem oscillates between a superposition of C 2h entangled tautomersI and II in the ground state (a), and D 2h disentangled dimers (excitedstate at hm 0t ) with a half-proton at each site (b).References[1] S. Scheiner, Hydrogen Bonding: A Theoretical Perspective,Oxford University Press, Oxford, 1997.[2] F. Madeja, M. Havenith, J. Chem. Phys. 117 (2002) 7162.

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