Quantum entanglement and nonlocal proton transfer dynamics in ...

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.