Claude Leforestier et al- Determination of a flexible (12D) water dimer potential via direct...

8/3/2019 Claude Leforestier et al- Determination of a flexible (12D) water dimer potential via direct inversion of spectrosco

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Determination of a flexible 12D water dimer potential via direct inversionof spectroscopic data

Claude Leforestiera) and Fabien GattiLSDSMS (UMR 5636), CC 014, Universite Montpellier II, 34095 Montpellier Cedex 05, France

Raymond S. Fellers and Richard J. SaykallyDepartment of Chemistry, University of California, Berkeley, California 94720-1460

Received 10 July 2002; accepted 27 August 2002We report the determination of two dimer water potential energy surfaces via direct inversion ofspectroscopic data. The first surface, rigid, employs the MCY functional form originally fitted byClementi and co-workers from ab initio calculations, modified by adjunction of a fifth, uncharged,site to improve the dispersion component. The vibration-rotation-tunneling energy levels werecomputed by means of the pseudospectral split Hamiltonian method that we developed previously.The fitted surface shows considerable improvement as compared to the original one: transitionsamong the ground-state manifold are in error by at most 0.2 cm1, and excited state band originsup to 150 cm1) are reproduced to within 0.5 to 3 cm1. For the second surface, flexible, we usedthe same modified MCY functional form, considered now to depend on the 12 internal degrees offreedom, and augmented by the monomer potential energy terms. The water dimer is described inits full dimensionality by collision-type coordinates in order to access the whole configurationsampled by this floppy system. Internal motions of the monomers stretches and bends areexplicitly considered by invoking an adiabatic separation between the slow intermonomeric andfast intramonomeric modes. This (66)d adiabatic formulation allows us to recast thecalculations into an equivalent six-dimensional dynamics problem ( pseudorigid monomers on aneffective potential energy surface. The resulting, fitted, fully flexible dimer potential leads to a muchbetter agreement with experiment than does the rigid version, as examplified by the standarddeviation on all observed frequencies being reduced by a factor of 3. It is shown that monomerflexibility is essential in order to reproduce the experimental transitions. 2002 American Instituteof Physics. DOI: 10.1063/1.1514977

I. INTRODUCTION

Literally hundreds of potential energy surfaces PES ex-ist for use in computer simulations describing the behavior ofliquid and solid water, and such activities, along with abinitio molecular dynamics simulations, constitute a major ef-fort in contemporary science.1 The use of effective PES inthis context usually engenders several explicit approxima-tions, viz. 1 classical dynamics, 2 pairwise additive PES,and 3 frozen rigid monomers. Calculations with ab initiomolecular dynamics MD methods transcend the latter two,but both approaches are also subject to the limited accuracyof the ab initio methods used to describe the intermolecularforces e.g., density functional theory fails to account fordispersion interactions.2 These approximations have beenexamined in numerous treatments,1 but never in a truly de-finitive manner e.g., either the PES or the dynamical methodwas approximate.

Recently, several3 7 PES designed to accurately describethe water dimer have been developed by fitting or tuningab initio PES to the extensive data sets produced by terahertzvibration-rotation-tunneling VRT spectroscopy.8,9 Thesecan be used as the basis for developing more accurate PES

for condensed water, and they can provide more definitivetests of the approximations described above. The spectro-

scopic PES published thus far labeled VRT ASP-W I, II,and III and SAPT-5St are based on a rigid monomer de-scription and were determined via the associated 6D frozenmonomer dynamics calculation. In this paper we describe thedetermination of two new experimental water dimer PESrigid VRTMCY-5r and flexible VRTMCY-5f based onthe ab initio MCY PES of Clementi and co-workers,10 andwe explicitly examine the effects of including water mono-mer flexibility. The development of an accurate water dimerpotential is needed to compute the IR shifts of the mono-mers, and in the study of processes such as vibrational pre-dissociation, or dimerization in the gas phase. Once such an

accurate dimer potential is available, one also can address therole of flexibility in the prediction of observed tunneling fre-quencies in the dimer, and its possible consequences forlarger clusters.

The flexible calculations presented here can also serve asa benchmark for the nonrigid quantum Monte Carlomethod,11,12 or approximate formulations such as the vibra-tional self-consistent field1315 or the initial value representa-tion16,17 approaches.

The outline of this paper is as follows. In Sec. II, we firstdescribe the modification made to the original MCY PES,aElectronic mail: [email protected]

JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 19 15 NOVEMBER 2002

87100021-9606/2002/117(19)/8710/13/$19.00 2002 American Institute of Physics

Downloaded 02 Dec 2002 to 128.32.220.140. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp


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and present the rigid dimer potential VRTMCY-5r obtainedfrom fitting to (H2O)2 experimental transitions. In Sec. III,we introduce the exact quantum formulation describing theflexible water dimer, and discuss the adiabatic approximationinvoked to perform the calculations. This is then used in Sec.IV to derive a flexible dimer potential VRTMCY-5f fromthe (H2O)2 experimental data. In this section, we fully ad-dress the role of flexibility when comparing to experimental

VRT transitions. Finally, Sec. V concludes and considerssome directions for future work.

II. FITTING A RIGID DIMER POTENTIAL

In this section, we first recall the original rigid monomer6D formalism that we previously elaborated,18,19 and applyit to the MCY potential energy surface of Clementi andco-workers.10 A slight modification to this potential is thenintroduced. This modified potential is then fitted to experi-mental results and shown to lead to a much better agreement.

A. Rigid monomer formalism

This formalism has already been described in two previ-ous papers,18,19 the reader being referred to these for detailedinformation. We briefly sketch it below in order to show howit can be extended to flexible monomers, as in the next sec-tion.

The description of nuclear motions in complexes com-prising two polyatomic fragments A and B was first studiedby Brocks et al.20 In the case where these fragments can beconsidered as rigid, these authors gave an explicit expressionfor the quantum Hamiltonian see Eqs. 24 and 30 in theirpaper, and also Ref. 21 for many examples, which reads as

Hrigid2

2AB

1

R

2

R2 RHrot

AHrot

BVR,A,B

1

2ABR2 J

2j22j.J, 1

where R is the distance between the centers-of-mass of thetwo monomers A and B , and AB their reduced mass; Hrot

X

and jX are, respectively, the rigidrotational Hamiltonian andtotal angular momentum of monomer X see Eq. 30 of Ref.20; jjAjB is the coupled internal rotational angular mo-mentum; JjL the total angular momentum L is the rela-tive angular momentum between the monomer centers-of-mass; and X(X,X,X) represents the Euler angles de-fining the orientation of monomer X in the dimer body-fixedBF frame.

1. Spectral representation

The kinetic energy operator displays a simple expressionin the overall basis set

Bn jA ,kA ,A jB ,kB ,B

J,K,M ABK, 2

where n is an appropriate basis for the interfragment dis-tance R; jX ,kX ,X is a Wigner basis set describing therotation of monomer X; and J,K,M is the Wigner basisset associated with the overall rotation of the complex. The

basis B can be projected onto the different irreducible repre-sentations irrep of the G16 molecular symmetry groupgoverning the energy levels22

B

n ;,

where each ; is a symmetry-adapted linear combinationof the angular functions jA ,kA ,A jB ,kB ,B

J,K,M.

2. Grid representation

The most compact representation for the potential energyV(R,A,B) is the six-dimensional grid gq

As

B

A

BRp, where

AB. This grid is restricted

to points where the potential energy is lower than somethreshold Vmax , and only non-symmetry-equivalent pointsare stored, which typically corresponds to a dimension ofaround 106.

3. Iterative calculation of energy levels

As very large bases have to be considered in order toconverge energy levels of interest for the water dimer, weresort to an iterative method based on the Lanczosalgorithm.23 It consists of repetitive actions of the Hamil-tonian operator on a seed vector u0

n1u n1 Hnu nnu n1, 3

in order to construct the requisite Krylov space Ku n.In this space, H is tridiagonal and symmetric

nu nHun,

n1un1Hun. 4

Diagonalization ofH in the K space first produces converged

energy levels which are located in the sparse part of thespectrum, i.e., its low energy region. The different symme-tries can be independently studied by defining a seed vec-tor u0 belonging to a given symmetry-adapted subspacen ;. As the Hamiltonian operator belongs to thetotally symmetrical irrep A1

of G16 , the above Lanczosscheme will confine the resulting Krylov space to the sameirrep as u0.

The action ofH on a vector un is performed by meansof the pseudospectral split Hamiltonian PSSH method asbriefly described here. The effect of the kinetic energy op-erator can be calculated in the spectral representation, whereit is analytical. To apply the potential term V, one switchesto the grid representation where it is diagonal, i.e., theamplitude of the wave function at each pointg ,q

A ,sB ,

A ,B ,Rp is simply multiplied by the value of

the potential at that point. After multiplication, the wavefunction is then back-transformed to the spectral representa-tion. Such a pseudospectral scheme is equivalent to a mul-tiple quadrature of Gaussian accuracy if the grid points co-incide with the zeros of analytical functions.24

B. The rigid MCY potential

Using the above scheme, energy levels have been com-puted up to 150 cm1 of internal excitation for the MCY

8711J. Chem. Phys., Vol. 117, No. 19, 15 November 2002 12D calculations on the water dimer



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dimer potential.10 Convergence of transition frequencies be-tween nondegenerate A and B symmetry levels to within0.01 cm1 was achieved by employing a Wigner basis setspecified by j max10 on each monomer. For this particularPES, degenerate E levels require a somewhat larger basis(j max11). Overall basis set sizes ranged from 73 000 forJ0 to 220000 for J1 calculations. The correspondingsymmetry-reduced grid independent of total J) spans ca.

106

points. Due to the use of a Lanczos iterative scheme, thelowest four J0 energy levels of each symmetry convergein about 15 min of CPU time on a Pentium IV Xeon PC. Thistime increases to 60 min for J1 calculations.

A comparison between experimental8,25,26 bold figuresand calculated figures in parentheses transition frequenciesappears in Fig. 1. Diagram a focuses on the splittings as-sociated with the J0,1 ground states, while b displays theband origins of low energy intermolecular vibrations. Asnoted before,18,19 this surface leads to interchange splittings(A iB i

transitions that are much too small, and acceptorsplittings such as the a0 and a1 values that are too large byalmost a factor of 2.

C. Modifying the rigid MCY potential

The original MCY pair potential10 corresponds to thesum of the Coulomb interaction in the form of point charges(q on the hydrogens and 2q located on the water C2axis near the oxygen, and a series of exponentials to modelthe short range exchange repulsion (AOO , AHH , and AOHterms and dispersion (AOH term at long range

VMCY charged sites

moleculesq iqj

Ri j

AOOeOOROO

AHH eHHRHHAOH eOHROH

AOH eOH ROH. 5

In the above equation, summations run on pairs of atomsbelonging to different water molecules.

This original MCY potential was modified by the ad-junction of a fifth sitea floating virtual uncharged site VSalong the C2 axis of each monomer as shown in Fig. 2. Thisvirtual site is then used in place of the oxygen atom in the

dispersion term (AOH eOH

RVSH) of the potential.

D. Fitting the modified MCY rigid potential:VRTMCY-5r

This modified potential depends on 11 parameters corre-sponding to the 9 coefficients q2,AOO ,OO ,AHH ,HH ,AOH ,OH ,AOH ,OH appearing in Eq. 5 and the two dis-tances dVC and dVS specifying the positions on the C2 axis ofthe charge2q and the virtual site, respectively. It should benoted that setting dVS to zero gives the original MCY poten-tial.

The six-dimensional energy level calculation was em-bedded in a standard nonlinear least-squares fitting routineLevenbergMarquardt algorithm27 in order to minimize thechi-square normalized

21

Nt

t

Transitions

EtobsEt

calc

t

2

,

where Etobs represents a transition between two rovibra-

tional states, and t corresponds to the associated uncer-tainty. In fact, the experimental uncertainty is negligible(1 MHz8105 cm1) as compared to the agreementEt

obsEt

calc which can be expected from the fit. In orderto handle transitions which differ largely in magnitudes, t

FIG. 1. Comparison between experimental Refs. 8, 25, 26 bold andcalculated ( H2O)2 transitions from the fitted VRTMCY-5r dimer potentialoriginal MCY results; figures given in parentheses. a Splittings in the J0,1 ground vibrational states; b band origins of the lowest excited states.

8712 J. Chem. Phys., Vol. 117, No. 19, 15 November 2002 Leforestier et al.



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has been set to 101Etobs for all observed transitions. For

the fit presented below, only transitions between nondegen-erate (A/B) levels were considered. The reason is that Elevels require significantly larger angular basis sets, and thus

would render the calculation much more costly. The experi-mental transitions nJK nJK actually used inthe fit are listed in Table I. In the above notation, n stands forthe n th vibrational level starting from n1) of symmetry associated with a total angular momentum projection numberK. Some of the transitions appearing in this table are actuallysymmetry forbidden, but were fit from experimental data andcorrespond to band origins. We also incorporated in the fit a

well depth D e of 4.95 kcal/mol as obtained in the largest abinitio calculations performed2831 at fixed equilibrium mono-mer geometry.

The matrix of derivatives is determined via theHellmanFeynman theorem

E

p k

V

p k.

As the potential expression is particularly simple in this case,analytical derivatives V/p k were used for the first nineparameters, while the last two were numerically computedby finite differences. The pseudospectral scheme previouslydescribed for handling the potential term can be straightfor-wardly applied here by expressing the derivatives on thesame six-dimensional grid.

Figure 1 presents a comparison between observed boldfigures and calculated transitions, obtained at the end of thefitting procedure. The transitions obtained from the originalMCY potential without the virtual site are recalled in pa-rentheses in this figure. It can be seen that all transitions

energies have been greatly improved, especially the donoracceptor splittings. The resulting standard deviation rms1/Nt Et

obsEt

calc2 takes the value 3.62 (cm1),down from 65.6 for the original MCY potential. Table IIgives the new set of parameters for the fitted potentialVRTMCY-5r, the main changes in these parameters con-cerning the repulsion and dispersion linear terms. The equi-librium geometry and the rotational constants reported in thistable were averaged over the ground A1

wave function

f0(A1)

fR,A,B0(A1)

, 6

the above integral being efficiently computed by the same

pseudospectral scheme as already used for the potential term.We used the naming conventions given in Fig. 3 to describethis equilibrium geometry.

III. FLEXIBLE MONOMER CALCULATIONS

In this section, we show how one can make use of anadiabatic decoupling between the intra- and intermolecularmodes to make the calculations feasible. This allows us torecast the flexible monomer formulation into that for therigid case, provided one uses for the intermolecular modesan adiabatic potential generated by the intramolecular coor-

dinates.A. Flexible monomer formalism

In order to describe the VRT dynamics of the waterdimer in its full dimensionality, six intramolecular coordi-nates must be added to the intermolecular coordinatesR,A,B considered in the previous section. The formercan be chosen as the Jacobi coordinates qXRX,rX ,Xdescribing the deformation of monomer X (A ,B) in itsown body-fixed frame, as shown in Fig. 4. The BF zX axis isdefined along the RX Jacobi vector, and yX is in the molecu-lar plane. The quantum exact Hamiltonian operator isthen20,32

FIG. 2. Modification of the MCY dimer potential by adjunction of a virtualuncharged site VS along the C2 axis of the water molecules.

TABLE I. Experimental (H2O)2 spectroscopic transitions Refs. 8, 25, 26used in the fit. nJ K means the n th vibrational level of symmetry associated to a total angular momentum projection number K.

Transition Frequency (cm1)

A2

100A2

110 1.057B2110B2

100 0.245A1100A1

110 1.163A2111B2

111a 0.541A1110B1

111a 13.666A1

100B1

111a 15.535A1

100B1

100a 0.752A2100B2

100a 0.652B2100B2

111a 0.290A1110A1

211b 86.62B1110B1

211b 88.48A1

100A1

211b 88.89B1

100B1

211b 87.03

A1

100B1

200b 109.78B1100A1

200b 106.08A2100B2

200b 98.04B2100A2

200b 97.38A2100A2

200b 114.03B2

100B2

200b 103.97A2

100B2

300b 143.71B2100A2

300b 141.18A2100A2

100b 54.94B2100B2

100b 51.74

Well depth De 1730.7

aCalculated from observed transitions.bBand origins obtained by fitting all the observed FIR data to the appropriateenergy level expressions.




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H2

2AB

1

R

2

R 2RTVR

ATVR

B

V qA,qB,R,A,B1

2ABR2 J

2j22j.J,

7

where TVRX represents the vibration-rotation kinetic energy

operator KEO of monomer X

TVRX

2

2

1

RX

2

RX2 RX

2

2m

1

rX

2

rX2 rX

lX2

2mrX2

1

2RX2 j

X2 lX22jX.lX. 8

In the above expressions, 2mH .mO/(2mHmO) and mmH /2 are the reduced masses associated with the R and rvariables, respectively, and l represents the angular momen-

TABLE II. Changes in the MCY potential parameters from original to fitted rigid rigid to fitted flexibledefinition, and resulting equilibrium geometry.

Parametersa.u.

OriginalMCY Ref. 10

VRTMCY-5r

VRTMCY-5f

q2 0.514 783 0.475 651 0.583 599AOO 1734.20 1998.13 2376.94OO 2.726 70 2.523 48 2.717 70

AHH 1.061 90 0.884 429 0.986 42

HH 1.460 97 1.522 05 1.567 04AOH 2.319 39 1.942 69 2.913 18OH 1.567 37 1.520 02 1.504 67

AOH 0.436 01 0.547 74 0.475 543OH 1.181 79 1.226 86 1.182 16dVC 0.505 78 0.462 31 0.517 424dVS 0 0.369 80 0.038 57

rms (cm1) 65.6 3.62 1.26

Equilibriumgeometry b

ROO () 2.87 2.91a 2.99 3.10a 3.04 3.10a

HbODOA

4.17 9.63 9.97

(zA

,OAOD

) 142.2 107.3 118.0

D 2.99 3.33a 1.75 2.98a 2.14 3.11a

De(cm1) 2059.7 1741.9 1734.5

D 0(cm1) 1351.2 1043.5 1231.6

Rot.constants(cm1)

A 7.938.21a 6.977.82a 7.237.93a

(BC)/2 0.2170.211a 0.2050.187a 0.1960.187a

BC 5.2(4)(2.6(2) )a 2.6(4)(1.7(2) )a 4.1(5)(1.7(2) )a

aThese quantities are averaged over the ground A1 wave function Eq. 6.

bAssuming gas phase equilibrium geometry for the monomers (dOH0.9579 , HOH104.50).

FIG. 3. Equilibrium geometry of the water dimer and naming conventionsused in the text: d means the donor molecule, and a the acceptor one; Hfand Hb correspond, respectively, to the free and bound hydrogens in thedonor molecule.

FIG. 4. Internal Jacobi coordinates used for describing the intramolecularvibrations on each monomer.




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tum associated with the r motion. jX, jjAjB, and JjL have the same meaning as in the rigid rotor formulationSec. II A.

An exact quantum calculation explicitly treating all 12internal degrees of freedom is not possible for the time be-ing, due to the computational demands. However, the aboveexpression Eq. 7 can serve as a starting point for an ap-proximate treatment as discussed below.

The central concept is that for the intramolecular modesof the water dimer, a single quantum of excitation energywould break the hydrogen bond. This implies that only theground vibrational state for these modes should be consid-ered in a first approximation as long as one is not interestedin vibrational predissociation of the dimer.

Such an adiabatic approximation was recently appliedby Klopper, Quack, and Suhm33,34 to a (42)d treatment ofthe HF dimer, in which the eigenvalues of the two HFstretches define effective potential surfaces for the intermo-lecular dynamics (4d). In this approach, different rotationalconstants Be were used for the monomer subunits dependingon the number of excitation quanta.

A similar adiabatic approximation, denoted (66)d,can be invoked here in order to separate out the internaldegrees of freedom into two groups:

i the fast intramolecular coordinates qA,qB;ii the slow intermolecular coordinates R,A,B,

denoted collectively Q, which will govern the dynam-ics on the adiabatic potential defined with respect tothe fast motion. Our approach will, however, differ inthe way we treat the rotational constants Bx , By , andBz associated with each monomer, retaining their ex-plicit dependence on the Euler angles A and B, aswill be shown later on.

In order to apply this adiabatic decoupling scheme, onefirst separates out the total Hamiltonian Eq. 8 into twoparts

HHintra QTinter , 9

where Hintra(Q) represents the intramolecular vibrationalHamiltonian at fixed intermolecular geometry Q

Hintra QTVATV

BV qA,qB;Q, 10

TVX

2

2

2

RX2

2

2m

2

rX2 12RX2

1

2mrX2

2

sin X

Xsin X

X, 11

that is, TVX corresponds to reducing TVR

X to the jX0 case,and Tinter is to be specified later on. The adiabatic decouplingthen consists of repetitively solving the six-dimensional vi-brational equation for the the intramolecular modes

Hintra QnAnB qA,qB;QEnAnB QnAnB q

A,qB;Q,12

in order to define the adiabatic potential EnAnB(Q), where nXstands for the three indices ns symmetric stretch, n b bend-ing, and na antisymmetric stretch labeling the vibrationalmodes of monomer X.

Such a six-dimensional calculation has to be performedat every point of the six-dimensional intermolecular Q grid(106 points. In order to make this efficient, we used thefollowing semiperturbative scheme at each geometry Q:

i We first optimize the intramolecular geometry ( qoptA ,qopt

B ;Q) by means of the Powell method,27 which does not requirethe energy derivatives;

ii the intramolecular potential is then expanded around the minimum

V qA,qB;QV qoptA ,qopt

B ;Q V0 Q

V qA,qoptB ;QV0 Q VA qA;Q

V qoptA ,qB;QV0 Q VB qB;Q

V qA,qB;QVA qA;QVB qB;QV0 Q VAB qA,qB;Q ; 13iii the vibrational states of monomer X are exactly computed for monomer YX at its minimum energy geometry qopt

Y bymeans of a sequential truncation-reduction scheme35,36

TVXVX qX;QnX q

X;QEnXX

QnXqX;Q; 14

iv the remaining term VAB (qA,qB;Q) can be taken into account by perturbation

EnAnB QV qopt

A ,qoptB ;QEnA

A QEnB

B QnA

A QnB

B QVAB nA

A QnB

B Q. 15

In fact, we will show in Sec. III C that this correction, which is very small but very demanding on computation, as it resultsfrom a six-dimensional integration, can be omitted with negligible effects on the transition energies.




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Let us now consider the definition of the Tinter term,which appears in Eq. 9 as the result of separating the totalHamiltonian into intra- and intermolecular contributions. Themain consequence of having defined a Q-dependent vibra-tional basis set Eq. 12 is to make the Bs coefficientsentering the rotational KEO depending explicitly on the Eu-ler angles A and B, as well as on the R separation. It isthus necessary to reformulate these operators in a consistent

way. Within an adiabatically constrained model for the KEO,Gatti et al.37 have shown that in such a case the correct ex-pression for the rotational KEO associated with molecule Xis given by

TRXjx

X ,jyX ,jz

X.Bx

X 0 0

0 ByX Byz

X

0 BzyX Bz

X . j

xX

jyX

jzX , 16

where Bx , By , ByzBzy and Bz are the rotational constantsassociated with the instantaneous internal geometry specifiedby the (r,R ,) Jacobi coordinates

Bx12 R

2mr21, 17

By1

2R2, 18

Bz1

2 sin2 cos2

R2

1

mr2 , 19

Byzcot

2R2. 20

The presence of the off-diagonal Byz term originates from

the loss of the C2v symmetry for the water molecule in thebound dimer. Equation 16 can be recast into the more com-pact expression

TRXjX t.B(X).jX. 21

In the following, we will neglect the effect of the ByzBzycontributions, as those terms are at least 2 orders of magni-tude smaller due to being very close to /2 in the opti-mized geometry qopt defined earlier. This approximation typi-cally changes the rotational energies by less than 10 4 inrelative values.

The final expression retained for the intermolecular

KEO, denoted Tinter , is

Tinter2

2AB

2

R 2jA t.B(A).jAjB t.B(B).jB

1

2ABR2 J

2j22j.J. 22

If we denote XM the basis set associated with the intermo-lecular coordinates, and N a shorthand notation for the nAnB(q

A,qB;Q) adiabatic states defined in Eq. 12, the ma-trix elements of the total Hamiltonian in the overall basis setwill read as

XMNHNXMXM NHintra NXM

XMNTinterNXM.

In a near-adiabatic approximation, we do not allow for cou-pling between different adiabatic N states, which simplifiesthe above expression as

XMNHNXM

ENMMXM NTinterNXM.

One can then invoke a FranckCondon-type approximation

NTinterNTinter qoptA ,qopt

B ,Q, 23

where the Tinter operator is evaluated for the optimized(qopt

A ,qoptB ) internal geometry corresponding to the instanta-

neous Q intermolecular configuration. This simplification isjustified because each molecule is in its ground vibrationalstate, for which the harmonic approximation is almost exact.The B matrices appearing in Eq. 22 will thus be evaluated

for the optimized internal geometry corresponding to the Qgeometry

NBX qA,qB;QNqAqBB

X qoptA ,qopt

B ;Q.

Computation of the TRATR

B terms is about one order of mag-nitude more expensive than in the rigid case because one hasto switch from the spectral to the grid representation severaltimes in order to compute them. As this transformation is themost time-consuming part of the algorithm, this also slowsdown the whole calculation by one order of magnitude. Wehave thus investigated the approximation of averaging the BX

matrices over the Euler angles

FIG. 5. Changes in the intermolecular potential due to flexibility: Vrlx isthe static decrease in energy upon relaxation of the monomer geometries;ZPE corresponds to the lowering of the zero-point energy of the mono-mers embedded in the dimer; E0 corresponds to the sum of these contri-

butions. These R-dependent quantities have been obtained from averagingover the Euler angles Eq. 32.




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BXR dA dB BXR,A,B e(V(R,

A,B)Vmin), 24

where is some constant (103 a.u.), while explicitly re-taining the R dependence. This approximation allows us torecast the flexible formulation into a rigid one Eq. 1, ex-cept for the rotational constants Bx

X , ByX , and Bz

X , whichdepend now on the separation R

HrigidnR,A,B EnnR,A,B. 25

The actual angular dependence

BXR,A,BBXR,A,BBXR, 26

is retrieved by first-order perturbation theory applied to theintermolecular eigenstates n(R,

A,B)

EnnjA t.BA.jAjB t.BB.jBn. 27

We will show in Sec. III C that this simplification essentiallygives the exact transition energies. We will also investigatefurther approximations for the definition of the rotationalconstants BX.

B. Definition of the flexible MCY potential

For definition of the flexible potential, we employ theexpansion

V qA,qB,R,A,B VH2O qA VH2O q

B

VMCY qA,qB,R,A,B , 28

where VH2O(q) is the JPT2 empirical potential determined byTennyson and collaborators,38 and VMCY is the modifiedMCY potential defined in Sec. II C. This form guaranteesthat the dimer potential will exhibit the correct asymptoticbehavior upon dissociation. In the above equation, the VMCY

potential is now evaluated for the instantaneous 12D geom-etry, and no longer for the water molecules in their equilib-rium geometry (qeq

A ,qeqB ) as in Sec. II. It should be noted here

that no new parameters have been introduced in the defini-tion of the flexible potential Eq. 28, that is, it depends onthe same 11 parameters see Sec. II D as in the rigid case.

As discussed in Sec. III A, flexibility will modify theintermolecular potential in two ways:

i First, at every Q geometry the dimer energy is low-

ered due to relaxation of the monomer geometriesVrlx QV qopt

A ,qoptB ,QV qeq

A ,qeqB ,Q; 29

ii the zero-point energies ZPE of the water moleculesembedded in the dimer differ from their isolated val-uesZPEQZPE Q2ZPEH2O . 30

These two quantities represent the change from a rigid to aflexible potential

E0 QVMCY qeqA ,qeq

B ,QVrlx QZPE Q.31

In Fig. 5 we present the dependence of these two quan-tities, Vrlx and ZPE, on the separation distance R, result-ing from averaging over the Euler angles

VR0VR,A,B 0AB, 32

where 0 is the ground (A1) intermolecular wave function

as determined from this potential.In order to fit such a flexible potential, the Hellman

Feynman procedure referred to in Sec. II D will require thederivatives of the adiabatic potential EnAnB(Q) with respectto the parameters p k. From the definition 15 of this po-tential, its derivatives will be given as

TABLE III. Comparison of different definitions of the rotational constants B see the text for a selected set of

transitions. BR, : explicit angular dependence of the B matrices Eq. 21; B(R)B: perturbative

calculation of the angular dependence Eqs. 2427; B(R): constants averaged over the angles Eq. 24;Bequ : constants from the averaged monomer equilibrium geometry in the dimer; B : asymptotical values

dissociated dimer of the rotational constants; B(R), VAB: effect of the intramolecular six-dimensionalVAB(qA,qB) term Eq. 15.

Transition Exp. ( cm1) BR, B(R)B B(R) Bequ B B(R), VAB

A1

100B1

100 0.752 0.596 0.596 0.602 0.593 0.623 0.597A1100A1

110 1.163 0.971 0.972 0.978 0.969 0.999 0.972B2100B2

111 0.290 0.280 0.279 0.338 0.362 0.047 0.376A2111A2

111 0.541 0.460 0.460 0.465 0.458 0.495 0.461A1110A1

111 13.666 13.748 13.754 13.698 13.645 14.095 13.684A1

100A1

111 15.535 15.312 15.318 15.278 15.205 15.706 15.253B2

100B2

100 51.74 50.70 50.70 51.03 51.02 50.52 51.27A1100A1

211 88.89 88.96 88.97 89.22 88.98 88.91 89.34A2100B2

200 98.04 98.17 98.17 97.81 97.73 99.19 97.85A1100B1

200 109.78 106.49 106.50 106.14 105.95 108.19 106.14A2100A2

200 114.03 114.69 114.69 114.44 114.27 114.02 114.54A2100B2

300 143.71 143.11 143.11 143.29 141.64 142.95 143.43

D0 1231.7 1232.4 1232.4 1233.3 1227.8 1233.9

rms 1.23 1.23 1.31 1.32 0.89 1.31

2

2.05 2.04 2.15 2.29 2.89 2.22




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p kEnAnB

Q

p kV qopt

A ,qoptB ;Q

p kEnA

A Q

pkEnB

B Q, 33

where we have ignored the VAB coupling term shown aboveto be negligible. The derivatives of the EnX

X contributions are

obtained from a second HellmanFeynman scheme appliedto the intramolecular coordinates, that is

EnXX

p knX

VX

p knX. 34

Any approximation made in the evaluation of the energyderivatives E/pk will not change the value of the minimumeventually obtained at the completion of the LevenbergMarquardt scheme, but only modify the path followed toreach it.

C. Test calculations

In Table III we present the results of calculations testingthe different approximations made in the flexible formula-tion. These results correspond to tests conducted with thepotential energy surface obtained at the end of the fittingprocedure.

In this table

i B(R,) means the actual angular dependence of theB matrices Eq. 21;

ii B(R)B corresponds to retrieving the angular de-pendence effect from perturbation theory Eqs. 2427;

iii B(R) ignores this angular dependence, and only usesthe values averaged over the angles Eq. 24;

iv Bequ defines the constants from the averaged mono-mer equilibrium geometry in the dimer;

v B uses the asymptotical values dissociated dimerof the rotational constants; and

vi B(R), VAB , which should be compared to the B(R)results, tests the influence of the intramolecular six-dimensional VAB (qA,qB) term Eq. 13.

From this series of tests, it first appears that a perturbativetreatment of the angular dependent term B(R,) essen-

tially gives the exact transition values, the difference being atmost 0.006 cm1 for all the transitions considered in the fit.The next column of Table III shows that ignoring this Bterm constitutes a very good approximation, which repro-duces the exact values to better than 1%, except for theB2

100B2

111 . Using the values corresponding to thedimer equilibrium geometry (Bequ) basically reproduces theB(R) results, but one can note larger discrepancies for thehigher transitions. Interestingly, using the monomer equilib-rium geometry (B) leads to the smallest standard deviation,but to the largest 2 value. Apart, from the B case, the

2

values show that the results stay essentially similar as long asa reasonable definition of the rotational constants is used.

The last column (B (R), VAB

), which should be com-pared to the B(R) one, displays the results of explicitly con-sidering the VAB coupling term in the evaluation of the six-dimensional ZPE of the intramolecular modes Eq. 15. Itcan be seen that neglect of this term produces marginalchanges, being at most 0.04 cm1 for the ground transitions,and 0.2 cm1 for the band origins.

IV. FULL DIMENSIONALITY RESULTS

We show in this section that considering a flexible po-tential with the same number of parameters 11 as for therigid case allows us to achieve a much better fit of the VRT

FIG. 6. Same as Fig. 1 for the fitted VRTMCY-5f pair potential. Thenumbers in parentheses recall the values obtained from the fitted rigidVRTMCY-5r potential.




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transitions. We further assess the effect of flexibility by per-forming calculations on this fitted flexible potential but ri-gidified, that is, reduced to the rigid case 6D, and show thatthe results are essentially identical to those obtained from therigid SAPT5S dimer potential untuned of Szalewicz et al.4

This flexible potential is then used to compute the infraredspectral shifts, and we compare them to experimental values.Finally, we use the flexible potential, fitted exclusively withthe (H2O)2 transitions, to predict those of the (D2O)2 spe-cies, and compare against experimental results.

A. Fitted flexible potential VRTMCY-5f

Using the formalism given in Sec. III, we fit a flexiblepair potential VRTMCY-5f without considering the transi-tions involving degenerate E states. An attempt to includethose transitions in the fit yielded only marginal changes inthe results. E-state data are not independent of those alreadyfit, as they involve transitions between E states associated

with the A/B states explicitly considered in the analysis.Very small bifurcation splittings do depend explicitly on theE states, but these were not included in the present analysis.

Figure 6 presents a comparison between observed andcalculated transitions obtained from the final fit. The transi-tions obtained from the previously fitted rigid potentialVRTMCY-5r are recalled in parentheses in this figure.Comparing with these latter values, it can be seen that inclu-sion of flexibility systematically improves the agreementwith the observed transitions it is reiterated that only thesame set of 11 parameters was used in this fit, the PJT2monomer potential being left unchanged during the fittingprocess, yielding a standard deviation of 1.23 cm1, down

from 3.62 cm

1 for the fitted rigid potential. The new param-eters are displayed in Table II, together with the equilibriumgeometry and rotational constants averaged over the groundA1 wave function; see Eq. 6. The virtual site introduced in

our modified MCY potential Sec. IIC is now very close tothe oxygen atom (dVS0.04 ). Although this value is verysmall, an attempt to fit the experimental data while fixing thissite on the oxygen atom significantly degraded the results,yielding a standard deviation of 2.5 cm1, and a 2 value of6.04. Table IV gives the 12D equilibrium geometry of therelaxed dimer. Flexibility increases the D e value by only65 cm1 0.18 kcal/mol, but results in an increase muchlarger in D0 (190 cm

1). The reason stems from the

change in the value of the intramolecular ZPE, which islowered in the dimer due to hydrogen bonding see Fig. 5.

Of particular interest in the ground-state splitting dia-gram Fig. 6a is the sum a0a1 of the (H2O)2 acceptortunneling splittings, which has been measured experimen-tally to be 13.92 cm1. This quantity was not used in thefitting process as it involves E states, and can be defined as

a0a1EE100 EE100 EE211

EE111 , 35

using our conventions see Table I. The value computedfrom the fitted surface is 13.87 cm1, in nearly perfect coin-cidence with the experiment. This result supports our preced-ing remark concerning the nonindependence of transitionsinvolving E states with respect to those involving A/B states.Rigid calculations performed on the fitted potentialVRTMCY-5r lead to a value of 14.08 cm1, also in closeagreement.

Figure 7 provides a global picture of all VRT levels of

FIG. 7. Global picture of all nondegenerate levels up to 200 cm1 of exci-tation. Dotted lines indicate the width of the donoracceptor interchangetunneling splitting.

TABLE IV. Equilibrium geometry of the relaxed dimer for the VRTMCY-5f dimer potential; the notations are defined in Fig. 3. Distances are in ,angles in degrees.

Relaxed geometry

ROO 3.034

HbODOA

9.67

(zA

,OAOD

) 117.7

ROA

H 0.965RO

DH

b0.968

ROD

Hf

0.959

HbODHf 102.8

HOAH 102.4

De(cm1) 1799.0




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(H2O)2 up to 200 cm

1 of excitation. Each fork represents apair of levels split by acceptor tunneling, the dotted linesrecalling the further splitting by donoracceptor interchangetunneling. For sake of clarity, only nondegenerate levels aredepicted in this figure.

B. Assessing the role of flexibility

The most important result deduced from this study con-cerns the much better agreement achieved when using a flex-ible potential as compared to a rigid one. As the functionalform with 11 free parameters retained in the two cases wasidentical, except for the presence of the monomer contribu-

tions VH2O(q), these latter terms must be responsible for theimprovement. They enter the definition of the adiabatic po-tential through the change in the ZPE that they contribute to,and the intramolecular geometry relaxation effect they gen-erate see Fig. 5.

In order to assess the role of these flexible monomerterms, we have run rigid 6D calculations using theVRTMCY-5f potential restricted to the monomer equilib-rium geometry, that is the rigid version of our fitted flexible

pair potentialVrigid

fR,A,B

VRTMCY5f qeqA ,qeq

B ,R,A,B.

The results, depicted in Fig. 8, show that the acceptor tun-neling splitting sum a0a1 increases to a value of20.3 cm1 in the rigid case. This behavior reflects the in-crease of both individual splittings a0 and a1

a0 : 11.6616.33,

a1 : 2.213.97.

In the same figure we compare with the results obtained from

the original untuned ab initio SAPT5S rigid pair potentialof Szalewicz et al.4 given in parentheses. The striking pointis that both sets of transitions coincide within a few hun-dredths of a wave number for all transitions involving theJ0 and J1 ground vibrational states Fig. 8a. Notice,however, that some of these transitions are in error by severalwave numbers with respect to the experimental values.

In order to rationalize the coincidence of the two sets ofresults, let us consider as a working hypothesis that our fittedflexible potential is essentially exact at least for the lowenergy range considered in this work. Similarly, considerthe SAPT5S potential as a nearly exact representation of thetrue pair potential when the monomers are fixed at their equi-

FIG. 8. Same as Fig. 1 for the rigidified VRTMCY-5f pair potential. Thenumbers in parentheses recall the values obtained from the rigid SAPT5SRef. 4 ab initio pair potential.

TABLE V. Band origins for the IR excitations: b bending, O HbD donoroxygenbound hydrogen, OHfD donor oxygenfree hydrogen, ss Aacceptor symmetric stretch, as A acceptor antisymmetric stretch. Values inparentheses correspond to frequencies computed at the equilibrium geom-etry of the dimer.

IR excitation

H2O

Expt. Calc.

bH2O 1594.7 1595.4ss H2O 3657.1 3659.3as H2O 3756.0 3758.8

(H2O)2

Expt. Calc.

bD 1615a 16301637O HbD 3530,

b 3601b 35733561O HfD 3730,

b 3735c 37273729bA 1601a 16241624

ss A 3600,b 3660c 35923581as A 3745b 37033683

aReference 40.bReference 39.cReference 41.




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librium geometry. This hypothesis is substantiated by thehigh level ofab initio calculations used for its definition. Ourrigidified flexible potential should then coincide with thislatter surface, as demonstrated by the above results.

It thus appears that the tuning of the SAPT5S potential,5

in order to reproduce the experimental VRT data, constitutesan artificial improvement of the surface: it compensates forthe missing key ingredient of the actual pair potential, viz.,flexibility of the monomers.

C. Infrared shifts

The flexible dimer potential allows us to compute the IRshifts, within the adiabatic approximation invoked earlier. At

each intermolecular geometry Q, one identifies the acceptor(a) and donor ( d) molecules from the shortest HO dis-tance between unbound atoms see Fig. 3; namely, the hy-drogen defines the donor and the oxygen the acceptor. Onecan then attribute the bound (OHb) and free (OHf)hydrogenoxygen frequencies of the donor molecule to thelowest and highest stretch frequencies, respectively. Simi-larly, the symmetric and antisymmetric stretches of the ac-

ceptor molecule are assigned according to increasing fre-quencies. This procedure allows us to define six excitedadiabatic potentials

En(X)

Q, Xa or d,

n1,0,0 ,0,1,0 or 0,0,1 ,

each one corresponding to a single excitation quantum in oneof the six intramolecular modes. Band origins for the IRtransitions are computed as the energy difference betweenthe lowest tunneling (A1

) level associated with such an ex-cited adiabatic potential En

(X) and the lowest tunneling (A1)

level of the lowest adiabatic potential (E0)

n(X)En

(X)A1

100 E0A1

100 .

The computed frequencies are compared to the experimentalvalues3941 in Table V.

The first three rows of Table V present a comparisonbetween experimental monomer transition frequencies andthose obtained from the method used to calculate the dimerintramolecular ZPE Sec. IIIA. It constitutes a check thatthe parameters used in the calculations provide a sufficientaccuracy. The following rows report the calculated band ori-gins, as well as their experimental values.3941 The values inparentheses correspond to frequencies calculated at the equi-

librium geometry, i.e., without averaging over the intramo-lecular wave function. On the whole, averaging tends to in-crease these values, except for the bending frequencies. Onecan note that the calculated transitions involving the donormolecule (d) are in better agreement with the experimentalvalues, particularly the oxygen-free hydrogen (OHf) fre-quency. For the acceptor transitions ( a), the calculated shiftsin energy appear exaggerated as compared to the experimen-tal values : measured acceptor transitions are almost identicalto those of the free monomer.

D. D2O

2results

In order to assess the accuracy of the fitted flexible sur-face, we used it to predict VRT transitions of the (D2O)2isotopomer. No change was made in the code except for themass of the hydrogen atom. The results are compared in Fig.9 to the experimental values of Braly et al.9 It can be seenfirst that the donoracceptor interchange splittings are quitewell described, even though they are reduced by almost afactor of 20 as compared to (H2O)2 . One does, however,observe a systematic error of 0.015 cm1 (40%) forthese values. The relative errors are much smaller (6%)concerning the acceptor tunneling splittings. Finally, errorsfor the excited vibrational states range from 1 to 7 cm1.

FIG. 9. Comparison between experimental Ref. 9 and calculated (D2O)2transitions using the VRTMCY-5f potential fitted on (H2O)2 transitions.




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V. DISCUSSION

In this work, we showed that the wealth of experimentalresults accumulated for the water dimer (H2O)2 allowed usto determine, or more precisely, to refine an existing poten-tial energy surface via direct inversion of spectroscopic data.The functional form initially retained in this study was therigid MCY potential of Clementi and co-workers, originallyfitted from ab initio calculations. This PES displays the Cou-

lomb interaction, short-range repulsion, and dispersionterms, but no polarization. A fit of a polarizable rigid MCYPES to the same set of experimental data is under progress.

The determination of a PES by fitting to experimentaldata is particularly efficient for the flexible case consideredhere. As the potential now depends on 12 degrees of free-dom, fitting from ab initio calculations would require sam-pling a tremendous number of geometries. Furthermore, oneknows a priori a zero-order description of this PES by meansof the one-body terms Eq. 28. Spectroscopic accuracy hasalready been achieved for these terms from previous fittingto experimental results. The fit of the flexible surface thusdeals only with the correction with respect to this zero-orderdescription.

Flexibility was included in our calculations by definingJacobi-type intramolecular coordinates, which minimize thecoupling to intermolecular ones. This 12-dimensional systemwas handled by means of a (66)d adiabatic separationbetween intra- and intermolecular coordinates. This formula-tion allowed us to recast the flexible calculation into a rigidone using the adiabatic energy as an effective potential. Ourcalculations have shown that flexibility plays a crucial role inthe description of the acceptor tunneling splitting: its value ismodified by the change in the adiabatic zero-point energywith respect to the intermolecular geometry. This was dem-

onstrated by doing test calculations using a rigidified versionof the fitted flexible dimer potential: it produced transitionsnearly identical to those originating from the high qualityrigid SAPT5S untuned potential of Szalewicz and co-workers.

The flexible dimer potential obtained in this studyreaches a near-spectroscopic accuracy for all transitions re-ported in the microwave and far-infrared regions. However,its predictions of intramolecular IR shifts are less satisfac-tory. This can be attributed to the very simple functionalform used: the expression we retained, identical to the rigidcase, has to account for the changes in intramolecular fre-quencies of the molecules imbedded in the dimer. A better

expression for the potential is under study, which will allowus to include the IR shifts in the experimental fitting data, aswell as the transitions measured for the (D2O)2 isotopomer.

ACKNOWLEDGMENTS

Dr. Linda Braly, Nir Goldman, and Serena Anderson aregratefully acknowledged for very helpful discussions. Thiswork was partially funded by a CNRS-NSF grant. The Ber-

keley effort is supported by Experimental Physical Chemis-try Program of the National Science Foundation.

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