The infrared spectrum of the O⋯H⋯O fragment of H[sub 5]O[sub 2][sup +]: Ab initio classical...

JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 1 1 JANUARY 2001

The infrared spectrum of the O ¯H¯O fragment of H 5O2¿: Ab initio classical

molecular dynamics and quantum 4D model calculationsMikhail V. VenerHumboldt-Universita¨t zu Berlin, Institut fu¨r Chemie, Arbeitsgruppe Quantenchemie, Ja¨gerstrasse 10-11,10117, Berlin, Germany

Oliver KuhnInstitut fur Chemie, Physikalische und Theoretische Chemie, Freie Universita¨t Berlin, Takustr. 3,14195 Berlin, Germany

Joachim Sauera)

Humboldt-Universita¨t zu Berlin, Institut fu¨r Chemie, Arbeitsgruppe Quantenchemie, Ja¨gerstrasse 10-11,10117, Berlin, Germany

~Received 22 June 2000; accepted 16 October 2000!

The gas phase IR spectrum of the OH¯O fragment of H5O21 and its deuterated analogue are

calculated usingab initio classical molecular dynamics based on a MP2 potential energy surface.The assignment of the bands is made in terms of the quantum four-dimensional model calculationsof anharmonic frequencies and intensities. Comparing low and high kinetic temperature simulationsthe importance of anharmonicities of the potential energy surface for understanding the vibrationalband structure is highlighted. It is shown that any reasonable simulation of IR spectra of systemswith very strong hydrogen bonds has to account for the dipole moment function beyond the linearapproximation. ©2001 American Institute of Physics.@DOI: 10.1063/1.1330748#

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I. INTRODUCTION

The protonated water dimer, H5O21, is a prototype for

strong hydrogen-bonded systems whose structure, vitional spectrum and dynamics have extensively been stutheoretically.1–8 In contrast, little is known experimentally9

and the whole vibrational spectrum of H5O21 has not yet been

recorded in thegas phase. In particular, the vibrational fre-quencies of the O H¯O fragment are still not known. Onthe theoretical side the anharmonicity1,8 of the potential en-ergy surface~PES!, the strong mode coupling,8 and the mul-tidimensional nature of the proton transfer4 render the evalu-ation of the vibrational frequencies of the OH¯Ofragment difficult.

There are two possible approaches for overcominglimitations set by the harmonic approximation. The first ois based on molecular dynamics simulations. At presentquantum molecular dynamics can be applied to at mostatomic or tetra-atomic systems, while separable mean-fiapproximations are of limited applicability to strongcoupled modes.10 Classical molecular dynamics simulationon ab initio PES ~AIMD ! provide a full-dimensional treatment of the problem.2–4,6 Its essential restriction is the classical treatment of the nuclei. In addition, one often facdifficulties with the assignment of vibrational bands at hikinetic temperatures. The second approach consists incalculation of the quantum anharmonic frequencies omultidimensional PES.1 Here, theoretical studies can be pusued along several directions: The diffusion quantum MoCarlo method11,12 provides a full-dimensional treatment o

a!Author to whom correspondence should be addressed. [email protected]

2400021-9606/2001/114(1)/240/10/$18.00

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the problem, but only for the ground vibrational state. Tvibrational self-consisted field approximation is well-suitfor the calculation of excited vibrational levels of highfrequency modes.13–15

An alternative to a full-dimensional treatment is peforming accurate calculations of vibrational eigenstawithin a judiciously selected reduced subspace of the tPES.16–19 By focusing on a low-dimensional subspace~2–4vibrational degrees of freedom!, it is possible to describe, foinstance, tunneling processes16,20,21 or to assign vibrationalbands22,23 in H-bonded systems. For the reduced-subsptreatment to be physically meaningful, vibrations inside tsubspace must be weakly coupled to those lying outsidethat subspace, i.e., the so-calledsubstrate modes. The sub-strate modes can, however, be accounted for in harmapproximation using the idea of a reaction surfaHamiltonian17,20 as recently done in anab initio study of anintramolecular proton transfer reaction.24,25

In a previous study8 quantum anharmonic frequenciesthe O H¯O fragment of H5O2

1 were calculated using athree-dimensional~3D! PES obtained by second ordeMøller–Plesset perturbation theory~MP2!. The PES in-cluded one bending and two stretching coordinates ofO¯H¯O fragment. The main result of this study was tprediction of a large anharmonic shift of the OH¯O bend-ing vibration from the high-frequency to the low-frequenside of the asymmetric stretch band. The question arisethis effect would persist in a full-dimensional treatmentthe vibrational problem, i.e., if it is also seen in the IR spetrum obtained from MD trajectories on thesamePES. At thispoint it should be noted that the OH¯O fragment ofH5O2

1 is slightly nonlinear. Thus a comparative study ofil:

© 2001 American Institute of Physics

P license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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241J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 Infrared spectrum of O¯H¯ in H5O21

IR spectrum requires at least a four-dimensional treatmThe four degrees of freedom for the PES should includetwo stretching and two bending coordinates.

The aim of this study is to calculate the anharmonicspectrum of the O H¯O fragment of the isolated H5O2

1 ionas well as its deuterated counterpart H4O2D

1 using AIMDsimulations. The assignment of the vibrational bands is min terms of the quantum 4D calculations of anharmonic fquencies and intensities. The model adopted for solving4D problem is adiabatic separation of vibrational variabwhich leads to a 232 quantum problem. The essential fetures of the study are the use of the same MP2 PESconsideration of the dipole moment surface~DMS! beyondthe linear approximation.

The paper is organized as follows: In Sec. II we intrduce the setup chosen for the AIMD simulations at two dferent kinetic temperatures. The reduced system for calcuing quantum vibrational states is specified in Sec. III togetwith the model for determination of the IR transition proabilities. Numerical results are given in Sec. IV and dcussed in Sec. V. The paper is summarized in Sec. VI.

II. CLASSICAL MOLECULAR DYNAMICS

AIMD simulations were performed using theTURBO-

MOLE package.26 The forces were calculated analyticalwithin the MP2 method27 using TZP and DZP basis sets othe oxygen and hydrogen atoms, respectively@T~O!DZP ba-sis set#. The Gaussian primitives were taken from Ref. 2Their number and contraction scheme is~O/H!@511111,411#/@31#. The smallestp exponent on oxygen is0.175 which makes the inclusion of additional diffuse bafunctions less important. The exponents of the polarizatfunctions are 1.2~d on O! and 0.8~p on H!. Sets of fivecomponents were used for thed-polarization functions. Thisis the least demanding basis set which gives a reliablescription of the physical properties of protonated water coplexes.

The simulations were performed at two temperaturesK and 360 K, defined as the time average of the kineenergy@see Eq.~4! in Ref. 4#. The initial conditions werechosen such as to provide access to those regions of thethat are responsible for the OH¯O vibrational motions.At 1 K we started with three different geometries of the iothe equilibrium and two slightly perturbed structures. Forlatter the displacement in a random direction of the hydroatoms from their equilibrium positions was 0.006 and 0.0a.u.; the displacement for the oxygen atoms was one ordemagnitude smaller. The initial momenta were distributrandomly according to the desired kinetic temperature.the largest displacement the initial momenta were set eto zero. At 360 K we started with equilibrium and four peturbed structures. For the latter the random displacemenhydrogen atoms from their equilibrium positions were 0.00.006, 0.009, and 0.012 a.u. Again, for the oxygen atomsdisplacements were one order of magnitude smaller thanH. Initial momenta were distributed randomly in correspodence to the desired kinetic temperature. The IR spectra wcalculated as the Fourier transform of the classical dip


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autocorrelation function29 obtained from theab initio dipolemoment calculated at every point of the MD trajectory. TMD time step was 0.5 fs and trajectories have been progated for about 4.5 ps resulting in a 8 cm21 resolution for theIR spectrum.

The calculated frequencies were virtually independenthe time step~from 0.5 fs to 1.5 fs!, the trajectory length~from 2 ps to 9 ps!, and the initial structure. The respectivintensities showed only a slight dependence on the timeand the trajectory length, but a strong dependence oninitial structure. This requires to calculate the spectrumaveraging over a number of trajectories~see, e.g., Ref. 6!. At1 K we obtained the IR spectrum after averaging ovetrajectories; at 360 K we included 14 trajectories to achieconvergence of the IR intensities.

The simulations were performed on a Pentium-basedThe wall clock time for one time step is around 1 min 40~20% SCF, 75% forces!. This allows the completion of 9000time steps in 10 days of CPU time.

III. FOUR-DIMENSIONAL MODEL HAMILTONIAN

A. Coordinates

The present state of the art of numerical methods permthe computation of~ro!vibrational spectra for systems including up to six fully coupled vibrational degreesfreedom.30,31 Hence, approximations cannot be avoidedany treatment of the motions of the bridging proton in H5O2

1.The vibration-rotational Hamiltonian of the considered sytem can be written as a sum of a vibrational part and a pinvolving vibration-rotational interactions.32 We will notconsider the rotational degrees of freedom33 and conse-quently neglect vibration-rotational interactions. We furthassume that the vibrational HamiltonianHvib can be decom-posed as

Hvib5Hsystem1Hsubstrate, ~1!

where Hsystem includes the vibrational coordinates of thO¯H¯O fragment, thesystem, andHsubstratestands for allother vibrational degrees of freedom which are not stroncoupled to the system coordinates and can be approximatreated as harmonic.24,34 This assumption isa posteriori jus-tified by our calculations, see Sec. IV. The system includfour degrees of freedom, i.e., the symmetric and asymmestretching and the two bending vibrations of the O¯H¯Ofragment. The four coordinates are defined as follows~seeFig. 1!: The Cartesian coordinate system, (X,Y,Z), is placedsuch that the vector connecting the oxygen atoms poalong theZ-axis. The origin is at the midpoint between thtwo oxygen atoms. The position of the bridging proton cthen be characterized by its three Cartesian coordinatesx, y,and z within the system (X,Y,Z). The fourth coordinate isthe distanceR between the oxygen atoms; its equilibriuvalue isReq52.4 Å. Note that the proton’s equilibrium position is at (0,yeq,0) with yeq being dependent onR. ~For R5Req we haveyeq50.08 Å.)

The vibrational Hamiltonian of the system can thenwritten as


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242 J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 Vener, Kuhn, and Sauer

Hsystem52\2

2m S ]2

]x2 1]2

]z2 1]2

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]R2

1U~x,y,z,R!, ~2!

where we introduced the reduced masses,

M5MOMO

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for the motion of the heavy particles, and

m5MH~MO1MO!

MH12MO!MO ~4!

for the proton vibration. In definition~2! we assumed that thkinematic coupling between the OO vibration and the pro-ton motion is negligible. The potential coupling, however,explicitly contained in the PESU(x,y,z,R). The couplingsbetween the system and substrate coordinates are impltaken into account sinceU(x,y,z,R) will be calculated foroptimized ~relaxed! values ofall substrate degrees of freedom. This so-called adiabatic approximation correspondthe zeroth-order level of the reaction surface model whhas been shown to be suitable for the description of stproperties of hydrogen-bonded complexes.17,20

B. Potential energy and dipole moment surface

The MP2 method with the T~O!DZP basis set was useto generate the 4D PESU(x,y,z,R) using theGAUSSIAN94

series of programs.35 The frozen core technique was employed, i.e., the computations were performed with the tlowest lying doubly occupied MOs excluded from the corlation space. The dipole moment vectorm(x,y,z,R)5(mX,mY,mZ) was calculated from Hartree–Fock and MPwave functions.

After choosing specific values of the four system codinates on a suitable grid~see below!, the structure of the ionwas optimized under the constraint of fixed mutual oriention of the two water molecules. It corresponds to the glominimum structure which is theC2 conformer shown in Fig.1.1,2,5 This assumption was made in order to exclude rerangements caused by an internal rotation of the wmolecules.2 Such motion will add fine structure to the pa

FIG. 1. Equilibrium geometry of H5O21 (C2 conformer! as obtained from

MP2 energy minimization. For the 4D model Hamiltonian we use the Odistance (R) as well as the proton’s position with respect to the shoCartesian coordinate system. The coordinate origin is atR/2 on the O Oline andy is theC2 axis ~see text!.


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ticular vibrational band, and indeed plays a role in the htemperature MD spectra~see below!, but does not change itfundamental frequency.

The width of the grid along the bending coordinatesxandy was 1.5 Å and the number of grid points within theintervals was 11. The width of the grid inz direction was0.35 Å, including 7 grid points.~The potential along thezcoordinate is symmetric, so that in total 13 points were avable.! Along R a range of 0.35 Å was covered with 6 gripoints. Hence a total of about 3200 structures spanningenergy range of 10 000 cm21 were computed to construct th4D PESU(x,y,z,R).

For all grid points also the vector components of tdipole moment were obtained36 and approximated by a Taylor expansion in the proton coordinates at the given valueR as follows:

mA~x,y,z,R!5(i j l

m i j lA ~R!xiyjzl . ~5!

Here, the superscriptA stands for theX,Y, or Z projection ofthe calculatedm at a given grid point. For the subsequediscussion it is important to note that due to the specchoice of the coordinate system, the three projections ofDMS give the values of the dipole moment along the procoordinatesx, y, and z for a particularR. The coefficientsm i j l (R) in Eq. ~5! are given by

m i j lA ~R!5

1

i ! j ! l !

] i 1 j 1 lmA~x,y,z,R!

]xi]yj]zl . ~6!

They can be calculated by standard numerical techniqfrom the dipole moment vector on the coordinate grid.37 Be-low we will discuss two different orders in the expansion fthe MP2 DMS@Eq. ~5!# starting from the following expression which includes up to fourth order terms~cf. Refs. 38,39!:

mX~x,y,z,Req!50.21x10.70z17.0xz2032xy26.84xz2

26.73xyz18.73xyz2296.7xz3,

mY~x,y,z,Req!521.7220.19y10.66y210.55x211.53z2

28.55y2z224.96z2x222.0z4, ~7!

mZ~x,y,z,Req!58.75z19.52xz28.25xyz10.27x27.5xz2

113.9xyz227.64z32101.0xz3116.3yz3.

Here all coefficients are given in units of D Å2n wheren isthe degree of the polynomial. In Eq.~7! the projections of thedipole vector on three proton coordinates at the equilibriO¯O distance (Req52.4 Å) has been given for brevity. Oucalculations show that theR dependence of all coefficient~except the linear term of themZ component! is strong. Thesecond approximation to the MP2 DMS is derived from E~7! by keeping only the linear and the largest second orterms, i.e.,]2mX/]x]z, ]2mY/]2z and]2mZ/]x]z. It shouldbe noted, that the use of the HF wave functions for the cculation of the DMS resulted in a difference of about 2.5%the equilibrium value and around 15% for the nonequilrium configurations in comparison with the MP2 values.


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The O H¯O fragment of H5O21 is slightly nonlinear.

In the considered coordinate systemY corresponds to theC2

axis ~see Fig. 1!. This implies that the two bending coordnatesX andY are essentially different. For this reason themX

projection contains a linear term along theZ coordinate aswell as a nonzero bilinear term, see Eqs.~7!. ThemY projec-tion does contain neither a linear nor a bilinear term alothe Z direction. Thus themY projection is decoupled fromtwo others in the low order approximation.

C. Transition frequencies and IR intensities

It is well known that the quantum-mechanical assigment of vibrationally excited states of systems with threemore coupled, anharmonic vibrations is nontrivial, often abiguous, and sometimes impossible.40 Therefore, the 4D vi-brational problem was solved using a Born–Oppenheimtype separation between motions of light and heavy nu~adiabatic separation of vibrational variables!.41,42The appli-cability of the adiabatic separation of vibrational coordinahas been convincingly demonstrated in a seriespapers.21,33,43–50A general condition of the adiabatic separtion of variables of two subsystems is51

S V

n DBn,n8!1, ~8!

where V and n are the characteristic frequencies of t‘‘slow’’ and ‘‘fast’’ subsystems, respectively, andBn,n8 arethe matrix elements of the nonadiabaticity operator withspect to the states of the ‘‘fast’’ subsystem. For vibratioproblems, in particular for hydrogen-bonded systems,44 Bn,n8frequently turns out to be much smaller than unity andthis reason the conditionV/n!1 need not necessarily bsatisfied. For instance, according to Ref. 47 the adiabapproximation gives good results forV/n50.8. Generallyspeaking, it could be applicable even forV/n>1 if Bn,n8!1.49

Specifically, we will use the ‘‘adiabatic stretchmethod52 which assumes an adiabatic separation of thebending~x andy! from the two stretching~z andR! motions.Hence the total vibrational wave function can be written

Fnmvk~x,y,z,R!5xnmvk~z,R!wnm~x,y;z,R!. ~9!

Using this ansatz together with the Hamiltonian~2! the wavefunction wnm(x,y;z,R) for the bending vibrations are giveby the solutions of the stationary Schro¨dinger equation,

F2\2

2m S ]2

]x2 1]2

]y2D1U~x,y,z,R!Gwnm~x,y,z,R!

5«nm~z,R!wnm~x,y;z,R!, ~10!

for fixed stretching coordinatesz andR. We note in passingthat the cuts ofU(x,y,z,R) along the bending and stretchinproton coordinates behave rather different; a detailed dission of this point is given in Ref. 8. As a consequence of E~10! we have a parametric dependence of the adiabatic suwnm& and adiabatic energies«nm(z,R) on z andR. The en-ergies enter as the adiabatic PES into the Schro¨dinger equa-tion for the stretching coordinates which reads


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5Enmvkxnmvk~z,R!. ~11!

Equations~10! and ~11! have been solved numericallusing the Fourier grid method53 employing a 32332 repre-sentation of the different PES. In order to get well-localizwave functions it was necessary to increase the numbegrid points as well as the width of the grid inz andR direc-tion for the adiabatic potentials«nm(z,R). For 2.26 Å,R,2.61 Å, «nm(z,R) was approximated by polynomials up tfourth power. These polynomials were used for computthe energies foruzu ranging from 0.35 Å to 0.84 Å. The dependence of the polynomials coefficients on R was callated as well and used to obtain the potential in the regi1.77 Å,R,2.26 Å and 2.61 Å,R,3.38 Å. For the lowestvibrationally excited states we expect that the error intduced by this procedure is rather small. Note that the ponomial inter- and extrapolation of multidimensionalab initioPES has been extensively used for anharmonic frequecalculations~see, e.g., Refs. 18, 22, 23, 54–59!.

The frequencies for the deuterated ion containingO¯D¯O fragment were obtained by substituting the maMH by the deuterium massMD in Eq. ~4!.

Having the adiabatic wave function, Eq.~9! at hand, thetransition probabilityPnmvk from the ground stateu0000& to agiven excited stateunmvk& state can be written as

Pnmvk5(A

u(lE dzdRx0000~z,R!zl

3Fl ,mnA ~z,R!xnmvk~z,R!u2, ~12!

with xnmvk(z,R) being the wave function of the stretchinvibrations@Eq. ~11!# and

Fl ,mnA ~z,R!5(

i jm i j l

A ~R!E dxdyw00~x,y;z,R!

3xiyjwnm~xy;z,R!. ~13!

Herewnm(x,y;z,R) is the wave function of the bending vibrations obtained from Eq.~10!.

IV. NUMERICAL RESULTS

A. Ab initio MD simulations

In the following we present results of our classical dnamics calculations of the IR spectra as outlined in Secfor T51 K andT5360 K. The calculated spectrum of H5O2

1

at low kinetic temperature (T51 K) agrees well with the oneobtained within the harmonic approximation as can be sfrom Table I. This enables us to assign the IR ‘‘bandcorresponding to the fundamental vibrational transitionsthe O H¯O by comparing the IR spectrum~autocorrela-tion function of the total dipole moment! with spectra result-ing from projections of the dipole moment onX andZ coor-dinates as shown in Fig. 2. Note that the vibration alongZ axis is equivalent to the asymmetric stretch (nas) and thevibrations along theX(nx) andY(ny) axis correspond to thetwo bending modes.


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The vibrations of the O H¯O fragment between 100and 2000 cm21 show three important peculiarities~Fig. 2!:~i! The intensity of the asymmetric stretching vibrationabout twice as large as those of the two bending vibratio~ii ! The mY dipole moment component (ny) is basically de-coupled frommX andmZ. ~iii ! There appears to be a stroncoupling between themX andmZ projections; about one-halof the intensity of the bending vibration along theX axis (nx)is caused by the asymmetric stretching vibration along thZaxis (nas).

These findings are readily explained by the polynomexpressions of the dipole moment function in Eq.~7!. At 1 Kthe proton is displaced from its equilibrium position by abo

TABLE I. Comparison of the IR active harmonic vibrational frequencwith those obtained using the MD simulations of H5O2

1 and H4O2D1 at low

kinetic temperature~1 K!. The units are cm21. Relative intensities are givenin parentheses@intensity of the~OHO!as stretch equals to 1, vibrations having intensities less than 0.03 are not reported#.

Vibration

H5O21 H4O2D

1

Harm. MD Harm. MD

H–O–H wag 318~0.11! 324 ~0.02! 305 ~0.15! ¯

H–O–H wag 450~0.06! 447 ~0.09! 430 ~0.05! 429 ~0.11!H–O–H rock ¯ ¯ 468 ~0.03! 467 ~0.04!H–O–H rock 499 ~0.03! 501 ~0.07! 495 ~0.08! 490 ~0.16!~OHO!as stretch 1058~1.0! 1056 ~1.0! 828 ~1.0! 835 ~1.0!(OHO)x bend 1435~0.12! 1434 ~0.45! 1129 ~0.06! 1134 ~0.23!(OHO)y bend 1523~0.05! 1527 ~0.09! 1206 ~0.05! 1210 ~0.06!OH2 asymm bend 1781~0.58! 1781 ~0.25! 1685 ~0.40! 1685 ~0.07!O–H stretch 3714~0.11! 3732 ~0.11! 3710 ~0.19! 3739 ~0.52!O–H stretch 3818~0.14! 3840 ~0.43! 3816 ~0.22! 3838 ~0.91!

FIG. 2. Results of AIMD simulations for H5O21 at T51 K. Dipole autocor-

relation function for the total dipole momentm ~top!, and the dipole projec-tions mZ ~middle!, andmX ~bottom!.


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0.01 Å on average. Therefore, only the linear terms in Eq.~7!have to be considered. First, we notice that the leading tis about an order of magnitude larger in themZ projectionthan in the other two components. This explains in the donant contribution of the asymmetric stretching vibrationthe IR spectrum of the O H¯O fragment. Second, to lowest ordermY is decoupled from two others projections, i.ethe bending vibrationny contributes only to the dipole projection onto theY axis. Third, the large coupling between thasymmetric stretch and the bending vibrationnx can betraced to the fact that in linear ordermX is essentially pro-portional toz, i.e., the IR intensity of thenx mode is partlydue to motion of O H¯O asymmetric stretching type. Wconclude that at low kinetic temperature the fluctuationsthe O–H bond distances and theO–H–O angle within theO¯H¯O fragment are small enough to render the effectmechanical and electric anharmonicity negligible. Thespectrum is correctly described within the harmonic appromation.

The spectrum of H5O21 simulated at high kinetic tem

perature (T5360 K! is shown in the upper panel of Fig. 3.resembles the spectrum obtained in Ref. 2 by MD simutions on a PES calculated using density functional theory.this temperature the proton coordinates are subject to lafluctuations ~on average, the proton is displaced fromequilibrium position by about 0.2 Å!, i.e., a much widerrange of the PES is explored by the trajectories. This caumechanical and electric anharmonicities to come into pwhich makes the assignment of the simulated spectrum inregion between 900 and 2200 cm21 extremely difficult.60 Inaddition the internal rotation of the lateral water molecusignificantly contributes to the IR bands. The separate csideration of the dipole moment projections does not provany hint concerning a possible assignment, in contrast tolow temperature case discussed above.

Nevertheless, a comparison of the simulated spectrlow and high temperature~Figs. 2 and 3! enables us to drawthe following conclusions:~i! The vibrational spectrum o

FIG. 3. Results of AIMD simulations for the IR spectrum of H5O21 at T

5360 K and power spectrum, obtained by a Fourier transform of the csical velocity autocorrelation function.


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the O H¯O fragment shows a pronounced temperaturependence.~ii ! The transitions observed at low temperaturethe region around 1500 cm21 (nx ,ny) virtually disappear.Instead, new bands appear in the region between 9001000 cm21. ~iii ! The most intense band attributed to tasymmetric stretching modenas shifts by about 100 cm21 toblue upon increasing the kinetic temperature.~iv! The bandaround 1800 cm21 (OH2 asymmetric bending at low temperature, cf. Table I! only broadens upon increasing the temperature. The development of a consistent picture leadin~i!–~iv! is postponed until the next section where we repthe results of the quantum 4D model calculations.

The H/D substitution of the bridging proton can be epected to result in a strong red-shift of the asymmestretching and bending vibrations of the OD¯O as com-pared with the O H¯O fragment. Table I and Fig. 4~lowerpanel! show results for low kinetic temperature. Low-~,600cm21! and high-frequency~.3700 cm21! vibrations are in-sensitive to the H/D substitution. This implies, that thevibrations of the lateral water molecules are only weacoupled to the bridging proton’s motion. The only exceptiis the asymmetric bending of the lateral water molecules1780 cm21. It shifts by about 100 cm21 to red after H/Dsubstitution. As in the normal case, however, the spectcan be described rather well within the harmonic approximtion ~see Table I!.

At high kinetic temperature (T5360 K) the spectrum ofH4O2D

1 is markedly different from the low temperature rsult as can be seen by comparing the two panels in FigThe observations~i!–~iv! made above for H5O2

1 apply to thepresent situation as well. The only significant differencetween the spectra of the two species at 360 K is the bshape of the asymmetric bending vibration of the lateral wter molecules around 1700 cm21 ~compare Figs. 3 and 4!. Aninterpretation of this phenomenon will be given in the nesection.

The results of this section also show that the vibratiocoordinates of the O H¯O fragment are not appreciablcoupled to the substrate vibrational coordinates of the H5O2

1

FIG. 4. Results of AIMD simulations for the IR spectrum of H4O2D1 at

T51 K andT5360 K.


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ion which may be approximately treated as being harmoThe decomposition of the vibrational Hamiltonian in Eq.~1!appears to be justified.

B. Quantum 4D model calculations

We will limit the discussion to the transitions from thground vibrational stateu0000& to some excited state havinthe transition frequency

nnmvk5~Enmvk2E0000!/h. ~14!

Here Enmvk is given by the solutions of Eq.~11!. Furtheronly those transitions are considered for whichn1m1v1k,3, i.e., we include the first overtones and complex trasitions corresponding to the simultaneous excitation of tdifferent vibrations. The relative intensities will be calclated according to

I nmvk}nnmvkPnmvk , ~15!

wherePnmvk is given by Eq.~12!. Note that due to the use othe adiabatic approximation the relative accuracy of callated IR intensities has an error of about 10%.33 Table II andFig. 5 show the calculated frequencies and intensities offerent transitions.

The anharmonic frequency of the fundamental O¯Ostretching vibration is 587 cm21. This value agrees fairlywell with the results of previous theoretical studies usingabinitio multidimensional PESs~Refs. 1, 8, 59! ~cf. Table III!.The frequency increases slightly upon H/D substitution,accord with the previous 3D study.8 This can be explained tobe a consequence of the decrease of the effective O¯O dis-tance upon deuteration. This frequency increase for the dterated species in complexes with extremely strohydrogen-bonds is known as Ubbehlode effect.61 It can beinterpreted if one takes into account the strong couplingtween the symmetric and asymmetric stretching vibrationsthe O H¯O fragment.62,63

The frequency of the fundamental asymmetric stretchvibration is 1158 cm21 in good agreement with the results oprevious theoretical studies~cf. Table III!. Note that stepping

TABLE II. Anharmonic vibrational frequencies of the OH¯O fragmentcomputed using the 4D PES and DMS function. Relative intensitiesgiven in parentheses@intensity of the~OHO!as stretch equals to 1#. Relativeintensities, obtained using the truncated DMS@cf. Eqs. ~7!#, are given initalics ~* , evaluated on theextrapolatedregion of the 4D PES, see text!.

nmvk Assignment H5O21 H4O2D

1

0001 O O stretch 587~0.08;0.07! 607 ~0.08;0.07!1000 (OHO)x bend 968~0.28;0.27! 682 ~0.20;0.21!0100 (OHO)y bend 1026~0.29;0.29! 717 ~0.21;0.21!0010 ~OHO!as stretch 1158~1.0; 1.0! 778 ~1.0; 1.0!0002 2O O stretch 1158~0.34;0.32! 1162 ~0.15;0.14!1001 (OHO)x1O¯O 1543~0.06;0.06! 1263 ~0.03;0.03!0101 (OHO)y1O¯O 1606~0.02;0.02! 1306 ~0.03;0.03!0011 ~OHO!as1O¯O 1671* ~0.59;0.57! 1312 ~0.42;0.40!2000 2(OHO)x 1971 ~0.45;0.44! 1376 ~0.49;0.48!1010 (OHO)x1~OHO!as 2001 ~0.19;0.16! 1388 ~0.09;0.08!1100 (OHO)x1(OHO)y 2003 ~0.48;0.47! 1401 ~0.51;0.50!0200 2(OHO)y 2083 ~0.54;0.54! 1446 ~0.33;0.33!0110 (OHO)y1~OHO!as 2107 ~0.19;0.17! 1457 ~0.10;0.09!


id

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from a 3D to a 4D model results in a red-shift of the consered frequency by about 50 cm21. Obviously, anharmoniccorrection and mode coupling work in opposite directionthe asymmetric stretching vibration.1,8 Increasing the numbeof system coordinates gives a stronger coupling, but doeschange the anharmonic correction.

The anharmonic frequencies of the fundamental bendvibrations are 968 and 1026 cm21, i.e., by about 500 cm21

lower than predicted by the harmonic approximation~cf.Table I!. As a result the bending frequencies are now belthat of the asymmetric stretching vibration. This was alreafound in the previous 3D treatment8 but not in a recentcorrelation-corrected vibrational self-consistent field stud59

~cf. Table III!. However, the disappearance of the banaround 1500 cm21 ~the region of the fundamental frequecies of theharmonicbending vibrations! and the absence oany intensive bands between 2500 and 3000 cm21 ~the re-gion of the first overtones and complex transition of thehar-monic bending vibrations! in the IR spectrum simulated btheab initio MD method atT5360 K supports the results othe present quantum 4D model study.

There is a large number of complex transitions and fiovertones above the anharmonic frequency of the asymric stretching vibration~see Table II and Fig. 5!. With respect

FIG. 5. Stick spectrum of quantum 4D model calculations for H5O21 ~upper

panel! and the deuterated species~lower panel!. The harmonic frequencyand intensity of the asymmetric vibrations of the lateral water moleculesTable I, is plotted as a solid bar.

TABLE III. Comparison of the calculated anharmonic vibrational frequecies of the O H¯O fragment computed in the present study with tavailable theoretical data: 2D, discrete variable Representation methothe MP2/pVTZ PES~Ref. 1!; 3D, adiabatic separation of the vibrationvariables on the MP2/T~O!DZP PES ~Ref. 8!; CC-VSCF, correlation-corrected vibrational SCF method on the MP2/TZP PES~Ref. 59!. Thecorresponding harmonic frequencies are given in parentheses; all freqcies in cm21.

Vibration 2D 3D Present~4D! CC-VSCF

O¯O stretch 572~619! 588 ~628! 587 ~628! 599 ~633!~OHO!as stretch 1275~983! 1226 ~1054! 1158 ~1054! 1209 ~871!(OHO)x bending ¯ ¯ 968 ~1435! 1442 ~1511!(OHO)y bending ¯ 1025 ~1524! 1026 ~1524! 1494 ~1552!


-

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s

tet-

to these transitions there appears to be a major differebetween the normal and deuterated species which willvery important for the discussion of the MD simulated spetra given below. In the case of H5O2

1 these transitions covethe region from about 1600–2100 cm21, i.e., they overlapwith the asymmetric bending vibration of the lateral wamolecules which is at 1781 cm21 in the harmonic approxi-mation~cf. Fig. 5!. Note that in our 4D model this transitiois not included. In the case of H4O2D

1 these transitions arelocated below 1500 cm21, i.e., they do not overlap with theasymmetric bending vibration of the lateral water molecul

Next, we consider the calculated intensities of the anhmonic frequencies. First we notice that the fundamenstretching vibration of the O O bridge does have a nonzerintensity in both species. This is an effect of the strong cpling between the two stretching vibrations of the OH¯Ofragment.64 As a result, the symmetric stretching mode aquires some IR oscillator strength from the asymmetric oThe relative intensities of the two fundamental bendingbrations of the O H¯O fragment are virtually equal toeach other. The H/D substitution basically decreases thetensities of the symmetric stretching and bending vibratioas can be seen in Table II.

A striking observation concerns the intensities of the fiovertones and complex transition of the bending vibratioThey turn out to be larger than those of the fundamentransitions~Table II!. In fact this effect was predicted manyears ago.65 It is caused by strong mechanical and electanharmonicities~see, e.g., Ref. 66!. Comparison of the tran-sition probabilities obtained with the MP2 and HF dipomoments shows that the changes in the relative intensitiethe first overtones and binary transitions are rather mi~less than 10%!, in accord with Ref. 39.

The intensities of the overtones and complex transitioare only weakly affected by the H/D substitution. Here, tonly exception is the first overtone of the symmetric streting vibration. Indeed the first overtone of the normal specat 1158 cm21 has a rather large intensity. This behavior mbe interpreted in terms of aFermi resonance between thiovertone and the fundamental OH¯O asymmetric stretch-ing vibration.~Fermi resonances have been widely discusin model studies of vibrational spectra of different hydrogebonded systems.67–69! Since in the deuterated species the fquency of the asymmetric stretch is much lower than thathe first overtone of the symmetric stretch the intensity oflatter drops which supports our interpretation.

Besides providing insight into the properties of thO¯H¯O fragment itself, the quantum 4D model calcultion delivers the key for the understanding ofab initio MDspectra at 360 K shown in Figs. 3 and 4. The emergpicture can be summarized as follows: The IR spectrumthe isolated H5O2

1 in the region of the O H¯O fragment~600–2500 cm21! consists of two bands:~i! The most intenseand relatively wide band is located in the region between 9and 1300 cm21. It includes the two O H¯O bending vi-brations~on the left-hand side of the band! and the asymmet-ric stretching vibration plus the first overtone of the symmric stretching vibration~in the center of the band!; ~ii ! Thesecond band having about the same intensity occurs aro

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1700 cm21. It comprises the complex transitions corresponing to the excitation of two stretching vibrations of thO¯H¯O fragment~on the left-hand side of the band!, theasymmetric bending vibration of the lateral water molecu~in the center of the band! and the first overtones and complex transitions of the bending vibrations of the OH¯Ofragment~on the right-hand side of the band!.

Upon H/D substitution the red-shift of the OH¯Orelated transitions lead to a considerable narrowing ofhigher energetic band which basically consists of the asmetric bending vibration of the lateral water molecules. Nthat in the MD spectra there is in addition to the transitiodiscussed so far an appreciable contribution to the ovefine structure of the O H¯O related bands from the internal rotations of the lateral water molecules.2 For complete-ness we would like to mention that the IR spectrum offull H5O2

1 system shows two additional bands. In the lofrequency region below 500 cm21 we have contributionsfrom the rocking and wagging vibrations of the lateral wamolecules. For frequencies above 3500 cm21 the stretchingvibrations of the lateral water molecules can be found.

V. DISCUSSION

The predicted IR spectra do not only depend onmethod adopted for treatment of the nuclei’s dynamics,also on the quantum chemical method used for evaluatingPES. Harmonic frequencies of the OO stretching andO¯H¯O bending vibrations are quite insensitive to the bsis set and theoretical method employed.5 In contrast, theharmonic frequency of the asymmetric OH¯O stretchingmode exhibits a strong dependence on the applied level oabinitio theory. It is 1054 cm21 at the MP2/T~O!DZP level oftheory8 and drops to 736 cm21 for the B-CCD/TZ2P treat-ment ~coupled cluster method with double excitations athe triple-z plus double polarization basis set!.5 This effectmay be explained by difficulties in the approximation ofvery flat anharmonic potential along the asymmetric strecoordinate1,8,59 by a harmonic function. Allab initio calcu-lations give the following order of the frequencies of tfundamental vibrations of the OH¯O fragment:O¯O stretch, ~OHO!as stretch, ~ OHO! x bend, ~ OHO! y

bend. According to the results of the present stuanharmonic correction and strong mode–mocoupling change the order of these vibratioto O¯O stretch,~OHO!x bend,~OHO!y bend,~OHO!as

stretch. Also for another system with an extremely strohydrogen bond, FHF2, harmonic calculations predict thbending mode to be the highest frequency vibration, wherexperimentally, the asymmetric stretch is the highestquency mode.70,71

Due to the strong mechanical and electric anharmonties the unambiguous assignment of the MD spectrum at hkinetic temperature is impossible even if projections ofdipole moment on the individual axis are considered. Hoever, the predictions of the quantum 4D model treatmenthe nuclear motions give valuable guidance. Indeed, thespectrum virtually does not show any intense bands ab2100 cm21. This implies that the first overtones and bina


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complex transitions of the bending vibrations are below tvalue. This means, the fundamental frequencies of two being vibrations must be locatedbelow1100 cm21.

To our knowledge the presentab initio study is the firstthat shows the significance of the specific form of the dipmoment surface in the case of very strong hydrogen bonAccording to our calculations, the nonlinear and nondiagoterms in the dipole moment surface cannot be neglected iassignment of the IR spectrum of the OH¯O fragment ofH5O2

1. This conclusion is also relevant in the following conections:~i! It is well known that electric anharmonicities arnot important for systems with medium-strength hydrogbonds ~see, e.g., Refs. 57, 58!; ~ii ! In recent sophisticatedMD studies of the dynamics of an excess protonwater72–74 attention was mainly focused on the proper aproximation of the potential energy surfaces, while for tdipole moment only linear terms were considered.

According to our calculations most of the intensitiesthe first overtones and binary complex transition of the being vibration derive from the low-order linear and nonlineterms of the DMS, see Eq.~7! and Table II. The importanceof the nondiagonal elements of the dipole momentcharged complexes with a strong symmetric hydrogen bwas already highlighted in Ref. 75. On the basis of expemental IR spectra in different solvents and the tentative trement it was found that about 85% of total absorption arifrom two nonequivalent bilinear electro-optical parameters75

The idea of the relatively intense overtones of the being vibrations is well appreciated in the literature76–79 whenit comes to aqualitative interpretation of IR spectra of systems with strong hydrogen bonds. The large relative intenties of the first overtones of the bending vibrations obtainin this study support this picture.

Finally, we would like to point out that in previouAIMD studies ~except Ref. 2! the density of states for thevibrational modes4,6 and so-called power spectra wecalculated.3 We show a power spectrum obtained from tclassical velocity autocorrelation function in the lower panof Fig. 3. In the O H¯O spectral region there appears tono direct relation between the IR spectrum and the powspectrum. We note, however, that in the high-frequencygion ~.3500 cm21! the density of states and the IR spectruare very similar since the relevant O–H stretching vibratioof the lateral water molecules have significant oscillastrength. Instead the power spectrum resembles the intedistribution of the Raman spectrum, computed in the hmonic approximation using the MP2/T~O!DZP level oftheory. The Raman spectrum has peaks in particular atcm21~4.0!, 1054 cm21~0.3!, 1436 cm21~0.6!, 1524cm21~5.7!, 1680 cm21~2.0!, and 1781 cm21~2.0! ~in paren-thesis Raman intensities in Å4/a.u.!.

VI. CONCLUSIONS

The IR spectra of H5O21 and its deuterated analogue a

determined by strong mechanical and electric anharmonties. A reasonable simulation of IR spectra of systems lthese with strong hydrogen bonds requires to account fordipole moment function beyond the linear approximatio


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We predict that in the region above 500 cm21 and below2500 cm21 the IR spectrum of H5O2

1 has two broad andintense bands. The low-frequency band~between 1000 cm21

and 1250 cm21! includes fundamental vibrations of thasymmetric OH stretching and of the two OH¯O bendingmotions. The high-frequency band~1750–1950 cm21! is dueto the asymmetric HOH bending vibrations of the two ouwater molecules and the first overtones and combinabands of the O H¯O bending vibrations. In the deuteratespecies H4O2D

1 the latter band is much narrower and icludes the asymmetric HOH bending of the two outer wamolecules only, while all other transitions give rise tobroad band extending from 600 to 1200 cm21. We do hopethat these predictions encourage experimentalists to obsthese bands characteristic of a prototype system for vstrong hydrogen bonds.

ACKNOWLEDGMENTS

This work has been supported by the ‘‘Max-PlanGesellschaft,’’ the ‘‘Fonds der Chemischen Industrie,’’ athe Deutsche Forschungsgemeinschaft~Sfb450!. One of theauthors ~M.V.V.! thanks the Alexander von HumboldStiftung for a fellowship, Professor N. D. Sokolov for usefdiscussions and Dr. S. Vyboishchikov, Dr. S. D. Elliot, aMarek Sierka for help with the numerical computations. TKonrad Zuse-Zentrum fu¨r Informationstechnologie in Berlinis acknowledged for providing computing time on theCRAY T3D.

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The infrared spectrum of the O⋯H⋯O fragment of H[sub 5]O[sub 2][sup +]: Ab initio classical...

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Transcript of The infrared spectrum of the O⋯H⋯O fragment of H[sub 5]O[sub 2][sup +]: Ab initio classical...