Quantum Simulations for Isotope Effects of IR + UV Laser Pulses on Symmetry and Selective Hydrogen...

Z. Phys. Chem. 222 (2008) 1311–1331 . DOI 10.1524.zpch.2008.5390© by Oldenbourg Wissenschaftsverlag, München

Towards Toroidal Hydrogen Bonds

By Ingo Barth1, Jörn Manz1,*, Guillermo Pérez-Hernández2, andPeter Sebald3

1 Institut für Chemie und Biochemie, Freie Universität Berlin, Takustr. 3, 14195 Berlin,Germany

2 Institut für Physikalische Chemie, Friedrich Schiller Universität Jena, Lessingstr. 10, 07743Jena, Germany

3 Madenburgstr. 14, 76865 Insheim, Germany

Dedicated to Prof. Dr. Hans-Heinrich Limbach on the occasion of his 65th

birthday

(Received March 20, 2008; accepted May 8, 2008)

Hydrogen Bond . Linear Molecule FHF–. Circularly Polarized Laser Pulse .

Nuclear Dynamics . Pseudorotation . Vibrational and Pseudorotational States

We present hydrogen bonds with toroidal densities of the protons, called toroidal hydrogenbonds, in systems with cylindrical symmetry e.g. triatomic molecules AHB or ions AHB+ orAHB–. These may be prepared by excitation of the degenerate bending vibrations and relatedpseudorotation such that the hydrogen atom (or the corresponding proton) is no longer locatedon the symmetry axis between atoms A and B, but – classically speaking – it rotates aroundthat axis. Quantum mechanically, the toroidal hydrogen bond is represented by an excitednuclear eigenstate with nuclear wavefunction and corresponding nuclear density which havetoroidal shapes around the central nodal line which conincides with the AB symmetry axis. Theproperties of these bonds are analyzed, including the pseudorotational angular momentum.Toroidal hydrogen bonds may be excited by means of circularly polarized infrared (IR) laserpulses. The results are demonstrated exemplarily for the oriented model system FHF–, by meansof combined quantum chemistry calculations of the potential energy and dipole surfaces(adapted from L. González, G. Pérez-Hernández, J. González-Vázquez, (2008), submitted),calculations of the vibrational and pseudorotational states in the frame of Watson’s isomorphicHamiltonian for linear molecules (J. K. G. Watson, Mol. Phys. 19 (1970) 465), and quantumdynamics simulations of the laser driven nuclear dynamics, analogous to recent applications toCdH2 (I. Barth, J. Manz, P. Sebald, Chem. Phys. 346 (2008) 89).

1. Introduction

The double goal of this paper is to predict toroidal hydrogen bonds in systemswith cylindrical symmetry, e.g. triatomic molecules AHB or anions AHB– or

* Corresponding author. E-mail: [email protected]

1312 I. Barth et al.

cations AHB+, and their preparations by means of laser pulses. Traditional hydro-gen bonds have the hydrogen atom (or the corresponding proton) centered onthe symmetry axis [1]. In contrast, for toroidal hydrogen bonds, the probabilitydensity of finding the hydrogen atom on the symmetry axis is zero. Instead,classically speaking, the hydrogen atom rotates around that axis, with uni-direc-tional (right or left) sense of rotation. The corresponding quantum mechanicalnuclear wavefunction has the shape of a torus (or possibly a set of tori) aroundthat symmetry axis, in accord with the cylindrical symmetry. We shall showthat it represents a nuclear eigenstate with specific excitations of the degeneratevibrational bending vibrations describing uni-directional pseudorotation. Relatedsuperposition states representing combined bendings and pseudorotation havebeen predicted recently for the metastable molecule 114CdH2 [2]. We shall ana-lyze various properties of the toroidal hydrogen bonds, e.g. they are associatedwith pseudorotational angular momentum. Moreover, we anticipate that they rep-resent nuclear ring currents, comparable to the properties of toroidal electronicring currents around the axis of molecules with cylindrical symmetry [3], seealso Refs. [4], [5]. This analogy of excited electronic ring currents, and nuclearring currents of toroidal hydrogen bonds, shall allow us also to predict a methodfor the preparation of the toroidal hydrogen bond. Specifically, electronic ringcurrents in pre-oriented or pre-aligned molecules may be excited by circularlypolarized laser pulses in the ultraviolet (UV) or visible (vis) frequency domain[3–5]. Likewise, we shall present a quantum model simulation for the preparationof the toroidal hydrogen bond in a pre-oriented or a pre-aligned system in theelectronic ground state, by means of a circularly polarized infrared (IR) laserpulse. This approach is adapted from Ref. [2] which presents the rather similarexcitation of a bending and pseudorotational coherent (non-eigen-)state of114CdH2, by means of a circularly polarized IR laser pulse, analogous to thepreparation of electronic excited hybride (non-eigen-)states representing electroncirculation [6]. For comparison, laser excitations of nuclear pseudorotations inexcited electronic states have been demonstrated previously, e.g. for the triatomicmolecule Na3(B) [7–10], see also the discussion in Ref. [4]. The laser pulsesused in Refs. [7–10] are linearly polarized, however, causing interfering pseudor-otations with opposite left and right directions, different form the present goalof preparing toroidal hydrogen bonds which represent unidirectional pseudorota-tions of the hydrogen around the system’s symmetry axis.

The new phenomena shall be demonstrated, exemplarily, for the model sys-tem FHF–. This system has been suggested to us by H.-H. Limbach, as exampleof one of the strongest hydrogen bonds in symmetric systems [11]. Our choiceof FHF– is also motivated by the fact that it has already served as successfulprototype system for the prediction of other laser driven phenomena, e.g. for theprediction of spatial separation of photofragments F and HF following photode-tachment by means of combined IR+UV laser pulses [12–14]. Moreover, thespecial electronic and vibrational properties of the strong hydrogen bond of FHF–

provide a continuous challenge to quantum chemistry, see e.g. Refs. [15–22].

Towards Toroidal Hydrogen Bonds … 1313

Accordingly, our subsequent applications will profit from the quantum ab initioresults for the potential energy and dipole surfaces, specifically those which havebeen calculated in Ref. [22]. The present derivation of the toroidal hydrogenbond in FHF– then calls for an extended (at least) 4D model which takes intoaccount all vibrations including both degenerate bending vibrations and the re-lated pseudorotation. For this purpose, we shall adapt an approximate 4D modelwhich has been derived recently for the metastable molecule 114CdH2 [2], basedon Watson’s rigorous 6D quantum theory for linear triatomic molecules [23].

The paper is organized as follows: Section 2 summarizes the model andtechniques for the quantum simulations of the bending vibrations and pseudorota-tional states of the pre-aligned FHF– by means of a circularly polarized laserpulse similar to the model and techniques for CdH2, adapted from Ref. [2] forself-consistency of the presentation. Section 3 summarizes the 4D model for thevibrational and pseudorotational eigenstates including the wavefunctions, eigen-energies and dipole matrix elements, starting from Watson’s isomorphic Hamil-tonian [23], again adapted from Ref. [2]. Section 4 has the results and discus-sions. The conclusions are in Section 5.

2. A simple model for excitation of the bending andpseudorotational states of the pre-aligned FHF– by acircularly polarized IR laser pulse

For the first goal of this paper, i.e. exemplary prediction of a toroidal hydrogenbond in FHF– and the analysis of its properties, it is helpful to anticipate alreadythe second goal, i.e. preparation of the toroidal hydrogen bond by means of acircularly polarized IR laser pulse. For this purpose, we assume that the centerof mass of the anion is located at the origin of the laboratory-fixed coordinatesX, Y, Z. These coordinates, and also the instant of the time t = 0, are specifiedby the circularly polarized laser pulse, as illustrated schematically in Fig. 1.

First, the direction of the propagation defines the coordinate Z. Second, theanion “sees” the pulse passing by, with increasing and then with decreasingintensity, and with electric field of the circularly right (or left) polarized laserpulse rotating in the plane perpendicular to the Z-direction, with right (or left)sense of rotation. The instant of maximum intensity defines the time t = 0. Third,the direction of the electric field at t = 0 defines the coordinate X. Finally, thecoordinate Y is perpendicular to the others such that X, Y, Z form a right-handedset of laboratory-fixed coordinates. Note that we employ capital letters for thesecoordinates. In contrast, small letters will be used in Section 3, to denote mole-cule-fixed coordinates, e.g. x, y, z. In general, the coordinates x, y, z are relatedto X, Y, Z in terms of Euler angles [24].

Specifically, the laser pulse, as “seen” by the anion, is defined in terms ofthe right (+) or left (–) circularly polarized vector potential [3,25]


Fig. 1. Preparation of the toroidal hydrogen bond of the pre-aligned FHF– by a circularlypolarized IR laser pulse (schematic). The laser pulse propagates along the Z-axis, parallel tothe symmetry axis of the pre-aligned FHF– anion. The cartesian displacements Q1,b, Q2,b, Q3,bfor the excited bend and pseudorotations along 41, 42, 43 of atoms 1(=F1), 2(=F2), 3(=H) areindicated by straight and curvilinear arrows, respectively. The laser parameters are ε0 = 81.9MVm–1 (Imax = 1.78 GWcm–2), ћω = 1303.8 hc cm–1 (duration of a laser cycle Δt = 25.6 fs),τ = 4.0 ps (tp = 33.8 ps, Γ = 3.75 hc cm–1). The arrows for the electric field indicate thesequence of the dominant laser cycles as they would be “seen” by the molecule when the pulsepasses by.

(1)

The time-dependent right (+) or left (–) circularly polarized electric field is thendefined as

(2)

with electric field amplitude ε0, carrier frequency ω and shape function

(3)

In the present application, we employ n = 20 such that s(t) approaches a Gaussianfunction very well [25]. The total pulse duration is tp and the initial time is setto – tp.2. The phase is equivalent to a rotation of the coordinates X, Y aroundthe Z-axis, by the angle η. This is irrelevant for the present application. Hence forconvenience, we set η to zero. Obviously, the electric field acquires its maximumintensity I(t) = cε0 - E± (t) -

2 at time t = 0, when it is oriented along the X-axis.Also note that the electric field (Eq. (2)) satisfies the condition of the far-fieldapproximation of Maxwell’s equations automatically [3,25–27],

(4)


The subsequent application employs a laser pulse with many cycles. The time-dependent intensity is then approximately given by

(5)

with peak value Imax ≈ cε0ε02 at time t = 0 and full width at half maximum

(fwhm) τ ≈ 0.118 tp. This effective duration of the laser pulse (Eqs. (1)–(3)) isrelated to the spectral width Γ by

(6)Moreover, we assume that the model FHF– has been pre-aligned along the

Z-coordinate, e.g. using the methods of Refs. [28–34], see also the reviews[35,36]. Hence the nuclei of the atoms F1, F2, H are distinguishable with respectto the direction of the laser pulse, and they are labeled as i = 1, 2, 3 with massesMi and equilibrium positions Re1 = (0, 0, –Re), Re2 = (0, 0, +Re), Re3 = (0, 0, 0),respectively. Specifically, M1 = M2 = 18.998403 amu, M3 = MH = 1.007825 amu,and Re = 1.1393 Å = 2.1530 a0 is the equilibrium distance between the H and Fatoms, implying the value Be = 0.3418 hc cm–1 for the rotational constant, cf.Section 4. Cartesian displacements from these equilibrium positions of the atomsi = 1, 2, 3 along α = X, Y, Z are described in terms of nine coordinates Qi,α, with

conjugate momenta Pi,α = –iћ∂

∂Qi,αand kinetic energy

(7)

For the following application, we assume that the anion keeps its initial pre-alignment during the laser pulse. Alternatively, the laser pulse can be fired duringone of the sequel rotational revivals, which occur periodically, with period ofrotational revivals, τrev = 1.(2cBe) [37]. The duration of the pre-alignment isactually short compared to τrev. Hence we require that the duration of the laserpulse should be much shorter than τrev, see the choice of the laser parameters inSection 4. Using the constraints for alignment, i.e. no rotations around the X- orY-axes, and separating the center of mass motion, the 9D expression (7) may bereduced to the 4D model with kinetic energy [2]

(8)

Here, Ps, Pb,X, Pb,Y and Pa denote the momenta conjugate to the coordinates Qs

for the symmetric stretch, Qb,X and Qb,Y for the bends along X and Y, and Qa forthe antisymmetric stretch, respectively, with reduced masses ms = MF.2 andma = mb = 2MHMF.(2MF + MH). The equivalence of the bends along X and Ysuggests to introduce polar coordinates Qb and φ such that

(9)


(10)with corresponding expression for the kinetic energy

(11)

where

(12)

denotes the angular momentum operator for the pseudorotation. The kinetic en-ergy operator (8) is thus rewritten as

(13)Hence the 4D model Hamiltonion of the anion is

(14)where V = V(Qs, Qb, Qa) is the potential energy surface of the model FHF– inthe electronic ground state X1Σg

+, within the Born-Oppenheimer approximation.The restriction of the model Hamiltonian (Eq. (14)) to the electronic ground stateis made because we assume that the circularly polarized IR laser pulse will notinduce any electronic excitations. Note that the cylindrical symmetry impliesthat V does not depend on φ. The corresponding vibrational-pseudorotationaleigenfunctions Ψvl(Qs, Qb, Qa, φ) = ) Qs, Qb, Qa, φ - Ψvl * and eigenenergies Evlare solutions of the time-independent Schrödinger equation

(15)Here l = 0, ±1, ±2, ±3, … and v denote the pseudorotational and vibrationalquantum numbers, respectively. We anticipate that v can be assigned to a set of“good” quantum numbers v = (vs, vb, va) for the symmetric stretch vs = 0, 1, 2,3, …, the (radial) bend vb = 0, 1, 2, 3, …, and for the antisymmetric strech va =0, 1, 2, 3, …, respectively, at least for low energies Evl, see Sections 3 and 4.Note that [Hvib,pr, Lpr] = 0, hence

(16)with wavefunctions Фvl (Qs, Qb, Qa) and

(17)

for the vibrations and for the pseudorotation, respectively. Accordingly, positiveand negative quantum numbers l denote right and left pseudorotations, respec-tively.

The pre-aligned molecule and its interaction with the laser pulse is describedby the time-dependent Hamiltonian

(18)


with semiclassical coupling of the dipole operator M and the electric fieldE± (t), cf. Eq. (2). In the present scenario, with the laser pulse propagating alongZ,

(19)i.e. the Z-component of the dipole operator Mz is irrelevant. Moreover, for sym-metry reasons,

(20)

(21)where Mt = Mt(Qs, Qb, Qa) denotes the component of the dipole operatorperpendicular to the Z-axis. Note that Mt depends, in principle, on the vibra-tional coordinates Qs, Qb, Qa but not on φ.

The laser driven vibrational and pseudorotational dynamics is described bythe time-dependent Schrödinger equation,

(22)

assuming that initially, at time t = –tp.2, the pre-aligned model FHF– is in thevibrational and pseudorotational ground state, v, l = (vs = 0, vb

l = 00, va = 0).Inserting the ansatz

(23)

and operating with ) Ψvl - yields the equivalent set of equations for the time-dependent coefficients,

(24)

with initial condition

(25)The matrix elements Hvl,v’l’ (t) are evaluated as in Ref. [2],

(26)

implying the selection rule l / l ± 1 for pseudorotational transitions, by analogywith corresponding expressions for laser induced electron circulations and elec-tric ring currents [3–6,25]. We anticipate the equivalent propensity rule, i.e. right(+) and left (–) circularly polarized laser pulses will induce dominant transitionsl / l + 1 and l / l – 1 for absorption, or l / l – 1 and l / l +1 for stimulatedemission, respectively.

Equations (24) and (25) are then solved by means of the Runge-Kutta methodof fourth order (Simpson rule) [38], beyond the rotating wave approximation(RWA), with a test of the convergence of the results with respect to the number


of bending and pseudorotational states which are included in the expansion(Eq. (23)). In practice, vb ≤ 4 is sufficient for the present application. The result-ing coefficients yield populations

(27)of the bending and pseudorotational states - Ψvl *, and the corresponding meanvalues

(28)

(29)

3. Eigenstates and dipole matrix elements

The double goal of this paper, i.e. prediction of the toroidal hydrogen bond aswell as its preparation by means of a circularly polarized laser pulse, with exem-plary demonstration for the aligned model FHF–, now call for the calculationsof the vibrational and pseudorotational eigenstates including the wavefunctionsΨvl and the eigenenergies Evl as well as the dipole matrix elements Mt,vl,v’l’ =) Φvl - Mt - Φv’l’ *, in the frame of the simple 4D model of Section 2. In thisSection, these quantities will be derived, starting from (i) quantum chemistry abinitio results for the potential energy and dipole surfaces which are adapted from[22], and (ii) using Watson’s isomorphic Hamiltonian for linear molecules [23],together with approximations which have been tested previously, for the 114CdH2molecule [2]. Both tasks are carried out in terms of molecule-fixed coordinates.Hence, they invoke another step i.e. (iii) the definition of molecule-fixed coordi-nates and their relation to the laboratory-fixed ones, cf. Section 2. The order ofsolving the tasks is (i), (iii), (ii).

(i) Suffice it here to say that the ab initio calculations are carried out at theCoupled Cluster including Single, Double and non-iteratively Triple excitations(CCDS(T)) level of theory, using a Dunning type of basis-set aug-cc-pVTZ [39].They start with the search for the minimum of the potential energy surface,which is determined with D∞h symmetry and equilibrium bond length re = rHF =1.1393 Å, corresponding to the rotational constant Be = 0.3418 hc cm–1, cf.Section 2.

(iii) Next, we introduce a right-handed set of molecule-fixed axis a, b, zusing the equilibrium geometry, with the position of the central H atom as itsorigin, the symmetry axis as z-axis, an arbitrary direction perpendicular to z asaxis a, and the third one b perpendicular to a and z. Later on, the condition ofpre-alignment allows us to identify the molecule-fixed and laboratory-fixed axesz and Z. Likewise, a specific choice of molecule-fixed coordinates a, b will becalled x, y, to be identified with the laboratory-fixed coordinates X, Y, respec-


tively. Accordingly, the equilibrium positions of atoms i = 1, 2, 3 are re1 =(0, 0, –re), re2 = (0, 0, +re), re3 = (0, 0, 0), and cartesian displacements of atomsi = 1, 2, 3 from these positions along α = x, y, z are described in terms of nine

coordinates qi,α with conjugate momenta pi,α= –iћ∂

∂qi,α.

Expansion of the potential energy surface V to second order in terms of thecartesian displacements qi,α, and diagonalization of the associated Hessian matrixyields the ubiquitious three translational modes for the center of mass, two modesfor librations or small amplitude rotations, and most importantly, four normalmodes for the vibrations of the linear triatomic molecule, see e.g. Ref. [40]. Thevibrational normal modes are along mass-weighted normal mode coordinates qs,qba, qbb, qa for the symmetric stretch, two degenerate bends and the antisymmet-ric stretch, respectively. The masses associated with these normal modes are ms,mb and ma = mb, and the harmonic frequencies are ωs, ωb (degenerate) andωa, respectively. The notation qba and qbb indicates two degenerate bends inperpendicular planes denoted a-z or b-z. Specific examples are the x-z and y-zplanes with corresponding notations qbx and qby for the normal bending coordi-nates. The degenerate bending normal modes can also be re-expressed in termsof polar coordinates i.e.

(30)

(31)in the general case or,

(32)

(33)for the specific example. We shall call motions along qb and δ or 4 bendingmotions and vibrational (bending) angular motions or, by analogy with the nota-tion of Section 2, pseudorotations, respectively. It is also convenient to introducecorresponding dimensionless normal coordinates q–s = sqs, q–b = bqb, q–a = aqa

where k = √ωk

ћfor k = s, b, a, respectively.

The potential energy surface may be separated into the harmonic (h) plusanharmonic (ah) parts,

(34)where

(35)

Note that, for symmetry reasons, V, Vh and Vah depend exclusively on coordinatesqs, qb and qa. Likewise, the molecule-fixed components of the dipole functions


depend just on qs, qb and qa. In particular, the “vertical” components may beexpressed as

(36)

(37)or, for the special choice of coordinates qba = qbx, qbb = qby,

(38)

(39)In so called double harmonic (h) approximation (DHA), one uses Eq. (35)

for the PES together with the linear approximation for the dipole,

(40)

(ii) The calculation of the molecular eigenstates and dipole matrix elementsis based on the ab initio results and on Watson’s isomorphic Hamiltonian Hiso

for a linear molecule or ion [23], together with a set of approximations whichhave been introduced in Ref. [2]:

(Aprx1) Using the complete Hamiltonian Hiso. Watson’s isomorphic Hamilto-nian implies that this corresponds to nothing but the Born-Oppenheimer approxi-mation. Here, the pseudorotational quantum number l must be equal to the rota-tional quantum number K. It follows that in full treatment of the rovibrationalproblem, vibrational eigenstates with l ≠ 0 can only occur if the molecule isrotationally excited (J ≥ K). In a good approximation, the rotationless vibrationalenergies evl (= Gv

l in spectroscopic notation) for the states of FHF– with vb = lcan then be calculated from the energetically lowest rovibrational termenergiesEv,J, as follows:

(41)

(Aprx2) Neglecting the J-dependent rotational and Coriolis coupling termsin Hiso. Note that Aprx2 implies the decoupling of the vibrational and pseudorota-tional degrees of freedom from the rotational ones. In accord with the modelscenario of pre-aligned FHF–, see Section 2, this allows us to reduce Watson’sisomorphic Hamiltonian from 6D to 4D, implying frozen rotations similar to the6D / 4D reduction of Ref. [2]. At the same time, we identify the remaininglaboratory-fixed and molecular-fixed coordinates, as anticipated in item (iii), thusX = x, Y = y, Z = z for the special choice where x = a, y = b. As a consequence,we also set Qs =

qs

√ms

, Qb =qb

√mb

, Qa =qa

√ma

with factors1

√mk

for compensation of

the mass-weighting of the normal modes qk, compared to the motions along non-mass-weighted Jacobi coordinates Qk, with associated mass mk. Likewise, we setφ = 4 for the pseudorotational angles.


(Aprx3) Considering only the momenta conjugate to qi and δ in Hiso, i.e.neglecting the rotational and Coriolis coupling terms as well as the term describ-ing Watson’s vibrational angular momenta [23].

(HA) i.e. the harmonic (h) approximation, thus using Aprx3 with harmonicpotential Vh. Â This simple model yields the eigenenergies

(42)

which are independent of the pseudorotational quantum number l. Thus, for vb≠ 0 the vibrational state is l + 1-fold degenerate. The related harmonic wavefunc-tions are written as a product of 1D and 2D harmonic oscillator wavefunctionsfor the symmetric(s) and antisymmetric(a) stretches and for the bend(b), respec-tively,

(43)

with quantum numbers v = (vs, vb, va) for the vibrations and l = –vb, –vb + 2, …,vb – 2, vb for the vibrational (bending) angular momentum or pseudorotationalong δ, and with Hermite polynomials Hn (x) and associated Laguerre polynomi-als Ln

k (x), with the normalizations Nvs, Nva and Nvb of the 1D and 2D harmonicoscillator wavefunctions [2], see also Ref. [41].

By extrapolation of the excellent experience gained in Ref. [2], we adaptApprx2 for the calculations of the vibrational eigenstates, in particular for theprediction of the toroidal hydrogen bond. After separation of the rotational wave-functions YJM, the 4D anharmonic vibrational wavefunctions are expanded interms of the harmonic ones,

(44)

and the eigenenergies and expansion coefficients are obtained by diagonalizingthe Hamiltonian matrix. Here M denotes the size of the vibrational basis (43).For comparison, for Aprx1, Watson’s isomorphic Hamiltonian is diagonalizedusing the vibrational and rotational basis - ψvlYJM *. The wavefunctions- ψvl * are labeled by quantum numers v, l for the maximum absolute value ofthe coefficients, - cvl - = maxv’l’ - cvl, v’l’ -. These quantum numbers may be consid-ered as “good” (but not rigorous) if the coefficient - cvl - is rather large, say- cvl -

2 > ½. This is indeed the case for all the relevant approximate eigenstates- ψvl * where v, l = (vs = 0, vb

l, va = 0) with vb = 0, 1, …, 4 and l = –vb, –vb +2, …, vb – 2, vb, which are considered in the subsequent application for thetoroidal hydrogen bond, i.e. for the first task. Note, however, that the criterion


is strongly violated for more excited states, in particularly for states with vb ≠- l - (vb > 4).

For the second task, i.e. the design of a circularly polarized laser pulse forpreparation of the toroidal hydrogen bond, we shall employ the same anharmoniceigenenergies which have been calculated according to Apprx2. In addition, weneed the relevant matrix elements for the perpendicular dipole component. Inprinciple, these could be evaluated at the same level of Aprx2, thus

(45)

For convenience, we shall employ, however, the much simpler so calleddouble harmonic approximation DHA, which consists of the HA approximation(Eqs. (42)–(43)) together with the linear approximation for the dipole function(Eq. (40)). The resulting dipole matrix elements in DHA approximation (v, l =(vs, vb

l, va) and v’, l’ = (v’s, v’bl’, v’a)) are

(46)

see Ref. [2]. The use of the DHA is supported by good experience for the quan-tum dynamics simulations, as demonstrated for 114CdH2 in Ref. [2]. Moreover,the DHA yields simple selection rules

(47)

(48)

(49)For the anharmonic FHF–, these harmonic selection rules serve as propensityrules. Accordingly, we neglect transitions from vibrational states v, l = (vs = 0,vb

l, va = 0) to other states v’, l’ = (v’s, v’bl’, v’a) where l’ = l ± 1 and v’s + v’a >

0. In this approximation, the ansatz (23) and the propagation of the set of equa-tions (24) for the time-dependent coefficients Cvl (t) is, therefore, restricted toquantum numbers v, l = (vs = 0, vb

l, va = 0).As a summary, the simple 4D model of pre-aligned FHF– allows to evaluate

the eigenstates, in particluar the toroidal hydrogen bond, by means of Aprx2 andusing molecule-fixed coordinates, for the first task. For the second task, i.e. thequantum model simulation of the preparation of the toroidal hydrogen bond bythe circularly polarized laser pulse, we have identified the molecule-fixed coordi-nates of the non-rotating anion with the laboratory-fixed ones. Likewise, thecorresponding vibrational wavefunctions are also set equal to each other, i.e.


Table 1. Vibrational energies – ZPEa (hc cm–1) for FHF–.

state Aprx1b Aprx2b Aprx3b HAc HAresd

ZPEa 2366.2 2366.2 2366.2 2354.2 2291.6(0, 11, 0) 1304.6 1304.6 1304.2 1367.2 1304.6(0, 22, 0) 2613.2 2613.2 2612.5 2734.4 2609.2(0, 33, 0) 3925.3 3925.3 3924.2 4101.6 3913.8(0, 44, 0) 5240.0 5238.5 5468.8 5218.4a Zero point energy.b Aprx1: full rovibrational approach, Aprx2: rotationless approach, Aprx3: approach neglect-

ing rotations and Watson’s vibrational angular momenta πx and πy [23]. For details, seeSection 3.

c Harmonic approximation.d Harmonic approximation using the resonance energy of the bend, Eq. (53).

(50)

where ψ(vs,vbl,va) (q–s, q–b, q–a, 4) is calculated variationally (c.f. Eq. (44)) on the level

of Aprx2 (with proper renormalization to account for the different dimensionsof the variables Qk and q–k). Accordingly, the vibrational energies

(51)are also identified with each other, whereas we set

(52)

for the dipole matrix elements.

4. Results and discussions

In this Section, we shall present the results, first for the vibrational and pseudoro-tational eigenstates with selected eigenfunctions and related eigenenergies asderived in Section 3 – this will include the first goal of this paper, i.e. theprediction of the toroidal hydrogen bond, together with a discussion of its proper-ties. Second, using these results as well as the DHA approximation for the dipolematrix elements as input for the set of time-dependent equations (24) which areequivalent to the time-dependent Schrödinger equation (22), we shall present theresults for our second goal, the quantum dynamics simulation of the preparationof the toroidal hydrogen bond.

Table 1 lists the zero point energies and the calculated vibrational energiesevl for the series v, l = (vs = 0, vb

l, va = 0) with vb = l = 0, 1, …, 4, for theapproximations Aprx1, Aprx2, Aprx3 and for the harmonic approximation (HA),Eq. (42), where ωs = 640.3 hc cm–1, ωb = 1367.2 hc cm–1, ωa = 1333.7 hc cm–1.The results for Aprx1 (calculated for vb = 0, 1, …, 3) serve as a reference forthe others. Accordingly, Aprx2 is perfect, confirming similar excellent experi-


Fig. 2. The initial vibrational and pseudorotational ground state - Ψ(0,00,0) * of the pre-alignedFHF– (panel a), and the excited states - Ψ(0,vb

l,0) * (vb = l = 1, 2, 3) with the toroidal hydrogenbonds (panels b–d), represented by nuclear equidensity contours for the atoms F1, F2 and H.States - Ψ(0,vb

l,0) * (vb = l = 1, 2, 3) represent unidirectional pseudorotations of the proton aroundthe symmetry axis, associated with pseudorotational angular momenta, l = 1, 2, 3, respectively,see Section 4.

ence which we have gained previously for 114CdH2 [2]. Hence we shall employthe energy values of Aprx2 for the subsequent quantum dynamics simulations.Table 1 shows that even Aprx3 and HA are very good or good approximations,respectively. In fact, the HA approximation can be much improved rather easily,simply by replacing the harmonic quantum (1367.2 hc cm–1) by the resonanceenergy

(53)

adapted from Aprx1 or Aprx2, see the entry HAres in Table 1. Again, this sup-ports the corresponding choice of the DHA approximation for the dipole matrixelements, for the subsequent quantum dynamics simulations, compare with Ref.

[2]. Specifically, we shall employ the DHA value∂μt

∂qb -

qb=0

= 0.0869e, implying

e.g. μt,(0,00,0),(0,11,0)h = 0.0869ea0, cf. Eq. (46). Note that we employ the DHA

dipole matrix elements but not the HA energies, because the anharmonicity ofthe Aprx2 energies (cf. Table 1) will be helpful for the second task, selectiveexcitation of the toroidal hydrogen bond, see below.

The vibrational and pseudorotational eigenfunctions are illustrated in Fig. 2,exemplarily for the ground state - Ψ(0,00,0) * and for the first excited states- Ψ(0,vb

l,0) * (vb = l = 1, 2, 3), as aligned along the Z-axis in the laboratory frame,by means of equidensity contour plots. Apparently, for the ground state, thecentral H atom (or the corresponding proton) carries out rather large amplitudemotions, preferably in a nearly spherical domain around the equilibrium position.In contrast, the F atoms (or their nuclei) move with small amplitudes in prolate


ellipsoidal domains along the Z-axis. The large versus small domains of motionsare a consequence of the much larger masses of the F atoms compared to hydro-gen, respectively. Moreover, the shape of - Ψ(0,00,0) * is dominated by the harmonic- Ψ(0,00,0)

h* (with coefficient 0.9586 in the expansion (44)), and the accidental near

degeneracy of the corresponding harmonic frequencies ωa and ωb for the anti-symmetric stretch and the bends imply almost identical amplitudes of the corre-sponding nearly harmonic motions of the hydrogen atom along the Z and perpen-dicular X, Y axes, respectively, hence the nearly spherical equidensity contourfor the H atom. In contrast, the motions of the F atoms along the Z axis areenabled by two dominant normal mode vibrations, the high frequency antisym-metric stretch and the low frequency symmetric stretch, whereas each of theperpendicular motions are due to a single normal mode i.e. the bending vibrationwith high frequency – hence the larger amplitudes of motions of the F atomsalong Z compared to X or Y, i.e. hence the prolate ellipsoidal shape.

As prominent result of the paper, - Ψ(0,11,0) *, - Ψ(0,22,0) *, - Ψ(0,33,0) * show largeand small toroidal shapes for the densities representing the motions of the H andF atoms, respectively, as illustrated in Fig. 2. The prominent tori for the protonsuggest the term “toroidal hydrogen bond” – this yields the first goal of thispaper. Again, the different large and small tori of the central hydrogen versusthe neighbouring F atoms are a consequence of the corresponding light versusheavier masses, respectively. Moreover, the toroidal shapes are due to thedominant contributions of the harmonic basis functions - Ψ(0,11,0)

h*,

- Ψ(0,22,0)h

*, - Ψ(0,33,0)h

* (with coefficients 0.9754, 0,9855, 0.9898 in the expansions(44)), which in turn have toroidal shape due to the property of the 2D harmonicoscillator wavefunction for the bend, implying the nodal line which coincideswith the symmetry axis, cf. Eq. (43). Apparently, the nuclear tori resemble elec-tronic tori for electrons in atomic (e.g. 2p+1) or molecular (e.g. π+1) orbitals, seethe discussions e.g. in Refs. [3,4,6,42]. This analogy is quite useful for the subse-quent analysis of the toroidal hydrogen bond.

The toroidal shapes for the nuclear densitites represented by - Ψ(0,11,0) *,- Ψ(0,22,0) * and - Ψ(0,33,0) * imply unusual properties of the corresponding toroidalhydrogen bond of the FHF– anion. First, the eigenstates have pseudorotationalangular momenta Lz = l = 1ћ, 2ћ and 3ћ, respectively, cf. Eqs. (12) and (17),corresponding to uni-(right,+)directional classical pseudorotations of the nucleiaround the symmetry axis. Second, the radii of the tori increase systematicallywith the angular momenta – this is in accord with classical expectations, seealso [41]. Moreover, we anticipate that the pseudorotations imply correspondingnuclear ring currents, which induce the magnetic fields. These will be investi-gated elsewhere, by analogy with electronic ring currents and magnetic fields,e.g. in excited Π± states of linear molecules, see the discussions in Refs. [3,42].

Last but not least, the toroidal hydrogen bond of the FHF– anion is a vibra-tional and pseudorotational excited eigenstate, i.e. it has a finite lifetime. In thegas, this corresponds to the radiative lifetime which is estimated, e.g. for the firstexcited state, as [42,43]


(54)

with transition frequency ω(0,11,0),(0,00,0) = 2πc · 1304.6 cm–1, and the square of theabsolute value of the dipole matrix element. In DHA approximation, this valueis

(55)

thus τ(0,11,0) ≈ 59 ms, see also the conclusions in Section 5. In general, the life-times of the excited pseudorotational states - Ψ(0,vb

l,0) * for vb = - l - ≥ 1 may beestimated, using the DHA approximation and Eqs. (54), (55) and (46), asτ(0,vb

l,0) = τ(0,11,0) / - l - (vb = - l - ≥ 1), i.e. the lifetime decreases with - l - as 1 . - l -.Next we do the second task, quantum model simulation of the preparation of

the toroidal hydrogen bond by means of the circularly polarized IR laser pulse,starting from the ground state - Ψ(0,00,0) *. For simplicity, we consider the targetstate - Ψ(0,11,0) *. For this purpose, we shall again exploit the analogy of the excitedbending and pseudorotating nuclear eigenfunction presenting nuclear ring cur-rents, and excited electronic eigenfunctions presenting electronic ring currents,i.e. the design of the laser pulse for populating - Ψ(0,11,0) * will be made analogousto the design of laser pulses for electronic ring currents. Specifically, we haveshown previously that the latter can be excited by means of circularly polarizedso called π laser pulses, with corresponding photon energies near resonant to thetransition frequency, and with electric field strengths ε0 and durations of thecorresponding cos2 laser pulses t~p close to the references which satisfy the condi-tion of π pulses [4]

(56)

where M =12

Mt,(0,11,0),(0,00,0), see Eq. (26). A necessary condition for the approxi-

mate validity of Eq. (56) is that the transition to the target state does not induceany other significant competing processes, e.g. two- or multiphoton transitionsto higher excited states, or transitions to any other states which have the samesymmetry as the target state, and which are near-degenerate to the target state,within the widths Γ of the laser pulse. For the present analogous task, this condi-tion implies that the widths Γ of the laser pulse should be smaller than theanharmonicity of the bending and pseudorotational levels,

(57)

because otherwise the laser pulse might climb the ladder of the pseudorotationaland bending states - Ψ(0,vb

l,0) * with vb = l by means of sequential IR multiphotontransitions, see the effect demonstrated in Ref. [2]. Obviously, condition (57)cannot be satisfied for ideal models with equidistant level spacings, such as theHA model – this is why we apply the energy levels for Aprx2, not HA, cf.Table 1. Condition (57) combined with Eq. (6) imply that, on one hand, thepulse duration must be sufficiently long,


(58)

On the other hand, however, τ must be much shorter than the time of rotationalrevivals,

(59)

In the present case, as a compromise we choose tp = 33.8 ps such that τ = 4.0ps and Γ = 3.75 hc cm–1, in order to satisfy both conditions (58) and (59).Smaller values of Γ would be favourable for condition (57), (58), but in conflictwith the constraint (59). Obviously, it is not easy to satisfy both competingconditions (58) and (59) such that one may anticipate that the goal of perfectpopulation of the target state - Ψ(0,11,0) * can only be reached approximately,whithin the simple form and parametrization of the laser pulse given by Eqs. (1)–(3). The other two parameters of the laser pulse, i.e. ε0 and ω, are determined byoptimization, starting from the reference values for the resonant photon energyћω(0,11,0),(0,00,0) = 1304.6 hc cm–1 (cf. Eq. (53)) and the field strength ε0 = 81.8MVm–1 which satisfies the condition of the π pulse, Eq. (56). In practice, thepulse with these reference parameters yields significant but not ideal populationsof the target state. The yields may be improved by systematic re-optimization,i.e. fine-tuning of the laser parameters. In the present case, we have determinedthe optimal laser parameters ћω = 1303.8 hc cm–1, ε0 = 81.9 MVm–1 corre-sponding to Imax = 1.78 GWcm–2, and tp = 33.8 ps, see above. The correspondinglaser intensity I(t) and the resulting populations of the pseudorotational and bend-ing states - Ψ(0,vb

l,0) * are shown in Fig. 3.One readily notices the continuous rise of the target population, at the ex-

pense of the initial one. Nevertheless, the transient populations of the state- Ψ(0,22,0) * indicates the onset of double (pseudorotational and bending, cf. Ref.[2]) ladder climbing which is, however, suppressed quite successfully, at the endof the pulse. The final populations at time t = tp.2 are P(0,vb

l,0)(tp / 2)= 0.011, 0.951, 0.038, 0.000 and 0.000 for vb = l = 0, 1, 2, 3, 4, close to the idealvaules 0, 1, 0, 0, 0, …, respectively. The populations of all other states arenegligible, on the grafical scale of Fig. 3. As a consequence, the mean values of) vb (t) * and ) l (t) * are also indistinguishable in Fig. 3. The non-monotonicbehaviour of ) vb (t) * ≈ ) l (t) * i.e. small overshooting before reaching the targetvalue (= 1) is due to the transient population of - Ψ(0,22,0) *. Snapshots of the laserdriven wavepacket dynamics - Ψ (t) * (cf. Eq. (23)) are illlustrated by means ofequidensity contour plots, in Fig. 4. One readily recognizes the nearly perfecttransition form the initial ground state - Ψ(0,00,0) * to the target toroidal state - Ψ(0,11,0) *, compare with Fig. 2.


Fig. 3. Preparation of the target state - Ψ(0,11,0) * representing the toroidal hydrogen bond of themodel FHF– by means of the right circularly polarized IR laser pulse, cf. Fig. 1. Panels a andb show the laser intensity I (t) and the resulting population dynamics Pvl (t) for thedominant excited states with short-hand notation vb

l for the quantum numbers v, l =(vs = 0, v

bl = vb, va = 0), respectively. All other states have negligible populations. The events

marked by the vertical lines correspond to the snapshots of the laser driven wavefunction- Ψ (t) *, Eq. (23), as illustrated in Fig. 4. Panel c shows the mean excitations of the bend) vb (t) * and pseudorotation ) l (t) * – they coincide on the grafical scale.

5. Conclusions

This paper presents the toroidal hydrogen bonds (Fig. 2, panels b–d) and quan-tum dynamics simulations of the preparation by means of a well-designed circu-larly polarized IR laser pulse (Figs. 1, 3, 4), exemplarily for the pre-alignedanion FHF– with target state - Ψ(0,11,0) *. Unlike traditional hydrogen bonds AHBwhere the proton is located on the line between atoms A and B, protons intoroidal hydrogen bonds avoid this line. Instead, they carry out pseudorotationsaround it, with specific consequences for the properties, i.e. pseudorotationalangular momentum, nuclear ring current, and induced magnetic field, as repre-sented by the toroidal wavefunction. Toroidal hydrogen bonds exist in excited


Fig. 4. Preparation of the target state - Ψ(0,11,0) * representing the toroidal hydrogen bond of themodel FHF– by means of the right circularly polarized IR laser pulse, cf. Fig. 1. Snapshots ofthe time-dependent nuclear probability densities for atoms F1, F2 and H illustrate excitation ofthe bend and pseudorotation of the model FHF–. The times for the snapshots are marked inFig. 3.

nuclear eigenstates, with lifetimes in the ms time domain – more than longenough for spectroscopic analysis. The unidirectional right or left sense of thepseudorotation can be controlled by the corresponding choice of the right or leftcircular polarizations of the laser pulse, respectively. The results documented inFigs. 1–4 correspond to right circular polarization; left circular polarizationwould induce the opposite sense of pseudorotation.

It is easy to predict several extensions of the discoveries for the presentmodel system, aligned FHF– in the excited states - Ψ(0,vb

l,0) * (vb = l = 1, 2, 3).First, similar excited bending and pseudorotational states should exist in mosttriatomic or linear polyatomic anions, molecules or cations with hydrogen bondsAHA–, AHA, AHA+ or AHB–, AHB, AHB+ with D∞h or C∞v symmetries, respec-tively, irrespective of the charge. The common cylindrical symmetry supportsdoubly degenerate states with pseudorotational angular momentum quantumnumbers l = vb or l = –vb which are essential for this phenomenon. Similardegenerate states may also exist in hydrogen bonded molecules where the ABaxis has threefold or higher symmetry. More excited pseudorotational states offera rich world of toroidal hydrogen bonds, possibly with several tori depending onthe vibrational and pseudorotational quantum numbers. Superpositions of severalpseudorotational and vibrational eigenstates extend these scenarios to time-de-pendent coherent states, e.g. a 1:1 superposition of the states (shown in Fig. 2)would correspond to a hybride state representing time-dependent circulations ofthe hydrogen bond.

The construction and preparation of the toroidal hydrogen bonds has profitedfrom an analogy with corresponding electronic ring currents in excited eletronicstates in molecules with C∞v, D∞h, or other symmetries supporting doubly degen-erate states, cf. [3,4]. In spite of this qualitative analogy, the quantitative proper-ties of these effets, i.e. magnitudes of the ring currents or induced magnetic


fields are much larger for the electronic excited states than for the present nuclearexcited states, compare e.g. the results of Refs. [3,4,42] with the present ones.This is mainly a consequence of the small electronic mass compared to nuclearmasses. The design of laser pulses for preparing the selected toroidal hydrogenbonds may be considered as yet another extension of many previous pioneeringquantum dynamics studies demonstrating selective laser driven manipulations ofhydrogen bonds or, more generally, hydrogen in molecules, including also iso-tope effects, cf. Refs. [12–14,44–56], see also Ref. [57], for a review see [58] inRef. [1]. It is a challenge to verify these predictions experimentally.

Acknowledgement

We would like to dedicate this paper to Professor Hans-Heinrich Limbach, onthe occasion of his 65th birthday, with gratitude. We would also like to expresssincere thanks to Mr. A. Schild for efficient preparation of a movie demonstrat-ing the laser driven pseudorotations of FHF–, as documented by snapshots forthe wavepacket dynamics shown in Figs. 2 and 4, and Dr. J. González-Vázquezfor helpful discussions. Financial support by Deutsche Forschungsgemeinschaft(project GK 788, TP A1), by Fonds der chemischen Industrie and by Agenturfür Arbeit Berlin-Nord is also gratefully acknowledged.

References

1. Hydrogen-Transfer Reactions. J. T. Hynes, J. P. Klinman, H.-H. Limbach, R. L.Schowen (Eds.), Wiley-VCH, Weinheim (2006).

2. I. Barth, J. Manz, P. Sebald, Chem. Phys. 346 (2008) 89.3. I. Barth, J. Manz, L. Serrano-Andrés, Chem. Phys. 347 (2007) 263, doi:10.1016.

j.chemphys.2007.09.037.4. I. Barth, J. Manz, Y. Shigeta, K. Yagi, J. Am. Chem. Soc. 128 (2006) 7043.5. I. Barth, J. Manz, in: Femtochemistry VII: Fundamental Ultrafast Processes in

Chemistry, Physics, and Biology. A. W. Castleman (Jr.), M. L. Kimble (Eds.), Elsev-ier, Amsterdam (2006) p. 441.

6. I. Barth, J. Manz, Angew. Chem. 118 (2006) 3028; Angew. Chem. Int. Ed. 45 (2006)2962.

7. J. Gaus, K. Kobe, V. Bonačić-Koutecký, H. Kühling, J. Manz, B. Reischl, S. Rutz,E. Schreiber, L. Wöste, J. Phys. Chem. 97 (1993) 12509.

8. T. Baumert, R. Thalweiser, G. Gerber, Chem. Phys. Lett. 209 (1993) 29.9. J. Schön, H. Köppel, Chem. Phys. Lett. 231 (1994) 55.

10. J. Schön, H. Köppel, J. Chem. Phys. 103 (1995) 9292.11. I. G. Shenderovich, H.-H. Limbach, S. N. Smirnov, P. M. Tolstoy, G. S. Denisov,

N. S. Golubev, Phys. Chem. Chem. Phys. 4 (2002) 5488.12. N. Elghobashi, L. González, J. Manz, Z. Phys. Chem. 217 (2003) 1577.13. N. Elghobashi, J. Manz, Isr. J. Chem. 3 (2003) 293.14. N. Elghobashi, L. González, J. Manz, J. Chem. Phys. 120 (2004) 8002.15. J. Almlöf, Chem. Phys. Lett. 17 (1971) 49.16. S. A. Barton, W. R. Thorson, J. Chem. Phys. 71 (1979) 4263.17. V. C. Epa, W. R. Thorson, J. Chem. Phys. 93 (1990) 3773.18. J. E. Del Bene, M. J. T. Jordan, Spectrochim. Acta 55 (1999) 719.


19. K. Yamashita, K. Morokuma, C. Leforestier, J. Chem. Phys. 99 (1993) 8848.20. V. Spirko, M. Sindela, R. N. Shirsat, J. Leszczynski, Chem. Phys. Lett. 376 (2003)

595.21. N. Elghobashi, L. González, J. Chem. Phys. 124 (2006) 174308.22. G. Pérez-Hernández, J. González-Vázquez, L. González, (2008) submitted.23. J. K. G. Watson, Mol. Phys. 19 (1970) 465.24. R. N. Zare, Angular Momentum Understanding Spatial Aspects in Chemistry and

Physics. Wiley, New York (1988).25. I. Barth, L. González, C. Lasser, J. Manz, T. Rozgonyi, in: Coherent Control of

Molecules. B. Lasorne, G. A. Worth (Eds.), CCP6, Daresbury (2006) p. 18.26. J. Rauch, G. Mourou, Proc. Am. Math. Soc. 134 (2005) 851.27. S. Chelkowski, A. D. Bandrauk, Phys. Rev. A 71 (2005) 053815.28. B. Friedrich, D. Herschbach, Phys. Rev. Lett. 74 (1995) 4623.29. L. Cai, J. Marango, B. Friedrich, Phys. Rev. Lett. 86 (2001) 775.30. M. Leibscher, I. S. Averbukh, H. Rabitz, Phys. Rev. A 69 (2004) 013402.31. S. Zou, Q. Ren, G. G. Balint-Kurti, F. R. Manby, Phys. Rev. Lett. 96 (2006) 243003.32. K. F. Lee, D. M. Villeneuve, P. B. Corkum, A. Stolow, J. G. Underwood, Phys.

Rev. Lett. 97 (2006) 173001.33. S. Fleischer, I. Sh. Averbukh, Y. Prior, J. Mod. Opt. 54 (2007) 2641.34. I. Barth, L. Serrano-Andrés, T. Seideman, J. Chem. Phys. (2008) in press.35. H. Stapelfeldt, T. Seideman, Rev. Mod. Phys. 75 (2003) 543.36. T. Seideman, E. Hamilton, Ad. At. Mol. Opt. Phys. 52 (2006) 289.37. T. Seideman, Phys. Rev. Lett. 83 (1999) 4971.38. P. Deuflhard, F. Bornemann, Numerische Mathematik II: Gewöhnliche Differential-

gleichungen. de Gruyter, Berlin (2002).39. D. E. Woon, T. H. Dunning, Jr., J. Chem. Phys. 100 (1994) 2975.40. G. Herzberg, Molecular Spectra and Molecular Structure. Vol. II, Van Nostrand

Reinhold Company, New York (1945).41. W. H. Shaffer, Rev. Mod. Phys. 16 (1944) 245.42. I. Barth, J. Manz, Phys. Rev. A 75 (2007) 012510.43. A. Hinchliffe, R. W. Munn, Molecular Electromagnetism. John Wiley and Sons Ltd.,

Chichester (1985).44. J. E. Combariza, B. Just, J. Manz, G. K. Paramonov, J. Phys. Chem. 95 (1991)

10351.45. Y. Fujimura, L. González, K. Hoki, D. Kröner, J. Manz, Y. Ohtsuki, Angew. Chem.

Int. Ed. 39 (2000) 4586.46. N. Elghobashi, L. González, J. Manz, J. Chem. Phys. 120 (2004) 8002.47. N. Došlić, O. Kühn, J. Manz, K. Sundermann J. Phys. Chem. A 102 (1998) 9645.48. N. Došlić, K. Sundermann, L. González, O. Mo, J. Giraud-Girard, O. Kühn, Phys.

Chem. Chem. Phys. 1 (1999) 1249.49. N. Došlić, O. Kühn, Chem. Phys. 255 (2000) 247.50. S. Gräfe, C. Meier, V. Engel, J. Chem. Phys. 122 (2005) 184103.51. P. Marquetand, V. Engel, J. Chem. Phys. 127 (2007) 084115.52. M. Abe, Y. Ohtsuki, Y. Fujimura, Z. Lan, W. Domcke, J. Chem. Phys. 124 (2006)

224316.53. W. Jakubetz, B. L. Lan, Chem. Phys. 217 (1997) 375.54. I. Vrábel, W. Jakubetz, J. Chem. Phys. 118 (2003) 7366.55. C. A. Marx, W. Jakubetz, J. Chem. Phys. 125 (2006) 234103.56. A. Datta, C. A. Marx, C. Uiberacker, W. Jakubetz, Chem. Phys. 338 (2007) 237.57. N. E. Henriksen, Adv. Chem. Phys. 91 (1995) 433.58. O. Kühn, L. González, in: Hydrogen-Transfer Reactions. J. T. Hynes, J. P. Klinman,

H.-H. Limbach, R. L. Schowen (Eds.), Wiley-VCH, Weinheim (2006) p. 79.

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