J. Phys. Chem. 1987, 91, 2513-2518
2513
Structure and Vlbratlonal Spectroscopy of the Water Dimer Using Quantum Simulation D. F. Coked and R. 0. Watts* Research School of Physical Sciences, The Australian National University, Canberra, Australia (Received: June 17, 1986)
A method is described for calculating the vibrational frequencies of a molecular cluster using a combination of quantum simulation and local-mode variational theory. The method is applied to the water dimer and results are compared with normal-mode theory, ”frozen field” local-mode theory, and with experiment. It is shown that the new approach is more rapidly convergent than the other techniques. Also, providing a suitable potential surface is available, the method is readily applicable to other molecular clusters.
1. Introduction The most widely used theory of molecular vibrations is undoubtedly the method of normal modes.’ In this approach, the potential energy of the system under consideration is expanded to second order in displacements from the lowest energy structure, defining a set of coupled harmonic oscillators. The set of uncoupled oscillators corresponding to this approximate Hamiltonian defines the normal modes of the original system and the corresponding eigenvalues give the approximate vibrational frequencies. Corrections to the uncoupled oscillator potential surface arise from anharmonic terms in the full interaction as well as from centrifugal and Coriolis couplings.2 The eigenfrequencies can be corrected for these terms by using quantum variational theory. A difficulty with normal-mode theory arises from the fact that for many systems the anharmonic part of the interaction is very large. In such cases, the variational calculation is slowly convergent. In single molecule systems, the classic example of a strongly anharmonic interaction is the 0-H oscillator. As an example, Whitehead and Handy3 have examined the convergence of variational calculations of the water molecule vibrational frequencies using normal-mode theory. They find that if the first 20 eigenstates are to be obtained accurately, a variational calculation of at least 13th order, involving around 650 basis states, is required. Such slow convergence indicates that some alternative approach is needed, and Siebrand4 has suggested that a local-mode approach would be better. This suggestion was followed up by several group^,^ and it has been shown that local-mode models give greatly enhanced convergence. In the case of the H 2 0 molecule, Reimers and Watts6 reported a detailed study of the vibrational spectrum in terms of local modes. Their approach has some resemblance to a normal-mode analysis, in that a variational approach is used. However, rather than using harmonic oscillators as the reference state, they used Morse oscillators. Also, the normal-mode coordinates were replaced by nonrectilinear variables representing the radial and tangential components of the stretching and bending modes. Reimers and Watts6 showed that, in the case of the H 2 0monomer, a full local-mode calculation gave converged results for the first 56 eigenvalues using a basis set of 120 Morse oscillator states. Clearly, this approach is much more efficient than the normal-mode calculations referred to above.3 The question of calculating vibrational eigenvalues for van der Waals clusters is even more complicated. In such systems, there will be anharmonic high-frequency vibrators, such as the X-H modes, and highly anharmonic low-frequency vibrations associated with the intermolecular modes. Furthermore, the fact that systems such as water,7 benzene,* a m m ~ n i a and , ~ hydrogen fluoridelo clusters show strong frequency shifts on dimerization indicates that there are significant interactions between these disparate modes. Consequently, it is to be expected that simple approaches such as a normal-mode analysis may be unsatisfactory for van der Waals clusters. Reimers and Watts,’’ who have examined this matter in some detail for water clusters, show that this ex-
’
Permanent address: Department of Chemistry, Columbia University, New York, N Y .
pectation is upheld. A comparison of normal-mode and local-mode calculations, including corrections for anharmonicities and intermolecular couplings, showed that the normal method is inadequate for such clusters. The local-mode calculations on the water clusters were based on a “frozen field” approximation, in which the reference state is obtained by neglecting both intramolecular and intermolecular couplings.”-’* Reimers and Watts showed that, although this approximation is a considerable improvement over normal-mode theory, both types of interaction make significant contributions to intramolecular vibrations in the cluster. However, their results for the intermolecular couplings were based on a harmonic approximation, and hence there is considerable interest in pursuing the problem further. In this paper we do so by using quantum simulation to generate a reference state that includes such correlation effects directly in the basis f ~ n c t i o n s . ’ ~ J ~ Several related forms of quantum simulation have been reported in recent years.I5 That used here is based on an isomorphism between the time-dependent Schrodinger equation and a diffusion equation with source and sink terms. Although suggested many years ago by Metropolis and Ulam,16 the modern algorithm was reported quite recently, by Grimm and Storer” and Anderson.’*
(1) Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; Dover: New York, 1955. (2) Watson, J. K. G. Mol. Phys. 1968, 15, 479. Louck, J. D. J . Mol. Spectrosc. 1976, 61, 107. (3) Whitehead, R. J.; Handy, N. C. J . Mol. Spectrosc. 1975, 55, 356. 1976, 59, 459. (4) Siebrand, W. J . Chem. Phys. 1967, 46, 330. (5) Elert, M. L.; Stannard, P. R.; Gelbart, W. M. J . Chem. Phys. 1977, 67, 5395. Jaffe, C.; Brumer, P. J . Chem. Phys. 1980, 73, 5646. Heller, E. J.; Gelbart, W. M. J . Chem. Phys. 1980, 73, 626. Moller, H. S.; Mortenson, 0. S. Chem. Phys. Lett. 1979, 66, 539. Watson, I. A.; Henry, B. R.; Ross, I. G. Spectrochim. Acta 1981, A37, 857. (6) Reimers, J. R.; Watts, R. 0. Mol. Phys. 1984, 52, 357. (7) Vernon, M. F.; Krajnovich, D. J.; Kwok, H. S.; Lisy, J. M.; Shen, Y. R.; Lee, Y. T. J . Chem. Phys. 1982,77,47. Page, R. H.; Frey, J. G.; Shen, Y . R.; Lee, Y. T. Chem. Phys. Lett. 1984, 106, 373. (8) Johnson, R. D.; Burdenski, S.; Hoffbauer, M. A,; Giese, C. F.; Gentry, W. R. J . Chem. Phys. 1986,84, 2624. Vernon, M. F.; Lisy, J. M.; Kwock, H. S.; Krajnovich, D. J.; Tramer, A.; Shen, Y. R.; Lee, Y . T. J . Chem. Phys. 1981, 75, 3327. (9) Fraser, G. T.; Nelson, D. D., Jr.; Charo, A,; Klemperer, W. J. Chem. Phys. 1985, 82, 2535. (IO) Pine, A. S.; Lafferty, W. J. J . Chem. Phys. 1983, 78, 2154. Lisy, M. J.; Tramer, A,; Vernon, M. F.; Lee, Y. T. J . Chem. Phys. 1981, 75, 4733. Mitchell, D. W.; Dykstra, C. E.; Lisy, J. M. J . Chem. Phys. 1984,81, 5998. (11) Reimers, J . R.; Watts, R. 0. Chem. Phys. 1984, 85, 83. (12) Reimers, J. R.; Watts, R. 0.Chem. Phys. Lett. 1983,94,222. Chem. Phys. 1984, 91, 201. (13) Coker, D. F.; Miller, R. E.; Watts, R. 0. J . Chem. Phys. 1985,82, 3554. (14) Coker, D. F. Ph.D. Thesis, The Australian National University, 1985. (15) Schmidt, K. E.; Kalos, M. H. In Applications of Monte Carlo Methods i n Stntistical Physics, Binder, K., Ed.; Springer: Berlin, 1984. Ceperley, D. M.; Alder, B. J. Phys. Reu. Lett. 1980, 45, 566. Physic0 1981, 108B, 875. J . Chem. Phys. 1984.81, 5933. (16) Metropolis, N.; Ulam, S. J . A m . Stat. Assoc. 1949, 44, 335. (17) Grimm, R. C.; Storer, R. G. J . Comput. Phys. 1971, 7, 134.
0022-365418712091-2513$01.50/0 0 1987 American Chemical Society
2514
The Journal of Physical Chemistry, Vol. 91, No. 10, 1987
We use the method to generate a numerically exact many-body ground-state wave function for the water dimer. This function is then projected into the local modes in the dimer, and the projections fitted to Morse oscillator wave functions which serve to define the reference state. Subsequent variational calculations give the required vibrational frequencies. In this paper, after outlining the potential surface used, we describe the quantum simulation algorithm and the method used to calculate the vibrational frequencies of the cluster. This description is followed by an analysis of the ground-state properties of the water dimer, including atomic pair distribution functions and the potential and kinetic energies. Finally, we discuss the vibrational frequencies and compare them with local-mode and normal-mode values. The result of the comparison is the suggestion that quantum simulation is an excellent method on which to base calculations of vibration frequencies in van der Waals clusters.
2. Potential Energy Surface The calculations were made using a pair potential for the water dimer that includes both intermolecular and intramolecular terms. For the intramolecular interactions, the potential takes the form
Coker and Watts
with
f ( R ) = 1 - 3.8845R2.326exp(-1.7921R)
(7a)
and R* = 0.948347R. The empirically determined parameters in V , have the values Am = 3.2049 X lo6 kcal mol-’ and am = 4.9702 8,-‘;the dispersion interactions are dampled by using the “universal” function of Douketis et with the dispersion coefficients taking the values recommended by Margofiash et al.?l C6= 625.45 kcal mol-’ 8,: C8= 3390 kcal mol-’ A,’ and Clo= 21200 kcal mol-’ In addition to the atom-atom terms there are Coulombic attractions between each H atom on one molecule and a negative charge located on the bisector of the HOH angle on the second molecule VHN(R) = --2QZ R and a repulsive interaction between these negative charges
where the local modes are defined as6
sl = R1 cos
(T )
sz = R 2 cos
(I)
8
- 80
8 - 60
(9)
- Ro
The position vector of the negative charge on a given molecule, in terms of a coordinate system centered on the oxygen atom, is
- Ro
d
R -- 2R0 cos (B0/2) (Rl + Rz)
The functions M ( s ) are Morse oscillator potentials
M,(s) = D,(1 - e-aJi)2
(3)
and (R,, R2, 6) are the two 0-H bond lengths and the included angle, respectively. Ro = 0.9572 8,is the equilibrium bond length and Bo = 104.52’ is the equilibrium bond angle. Reimers and Watts6 gave a detailed account of the local-mode theory used to a,)in the case when f12= 0. determine the parameters (Di, Essentially, they used a large basis set vibrational calculation to fit 56 vibrational levels for HzO, DzO, and HDO. Coker et a1.’3314 repeated the analysis, including the coupling parameterf,, in their potential. The result of their fitting produced the values D, = D2 = 131.25025 kcal mol-’, D, = 98.270 kcal mol-], al = a2 = 2.14125 A-I, a3 = 0.70600, andfiz = -15.1533 kcal mol-’ We have used these parameters in the present work. So far as the intermolecular potential is concerned, we use the semiempirical RWK2 model of Reimers et al.I9 This surface contains the correct monomer dipole and quadrupole moments and has good values for the dispersion coefficients. The potential includes both atom-atom interactions and Coulomb terms acting between point charges. It takes the form Q2
V H H W = AHH~ x P ( - ~ H H R+) R with Am = 631.92 kcal mol-’, kcal mol-’ 8,
aHH
(4)
= 3.2806 8,-‘,and Q‘ = 119.53
VOH(R) = A o H ( ~ ~ P ( - ~ o H-( R d ) - 1)’ - &-I ( 5 ) with AoH = 2.0736 kcal mol-’, 1.63781 8,
aOH
= 7.3615
where R I and R2 are the vector positions of the two H atoms, Ro is the equilibrium bond length given above, Bo is the equilibrium bond angle, and d = 0.26 8,. Earlier papers describing both the intramolecular and intermolecular potentials have contained small typographical errors in the defining equations6J’*’zJ9and consequently we report the full surface here. The model used to calculate the interaction between two distorted water molecules is obtained from the above equations, using the assumption that the various parameters are independent of geometry. Previous papers reporting properties calculated by this model suggest that this assumption is valid.”-I4 3. Ground-State Quantum Simulation The quantum simulation method used to generate the ground-state wave function is a variant of that described by Grimm and Storer” and Anderson.’* On introducing the transformation 7 = it/ h , the time-dependent Schriidinger equation becomes
where the “diffusion coefficients” Di = h2/2mi. The simulation begins by creating a large number of replicas of the system of interest, in our case the water dimer. At every time step the simulation algorithm consists of a diffusive part and a “birth/ death” process. In the diffusive part, every particle in a given cluster is given a Gaussian random displacement with variance 2DiA7. Next, the total energy of the new configuration, V,, is calculated by summing over both intramolecular and intermolecular potentials. If VT is less than some reference value, V,, copies of the replica under consideration are created with probability
A-l, and R, =
(18) Anderson, J. B. J . Chem. Phys. 1975, 63, 1499. 1976, 65, 4121. In?. J . Quantum Chem. 1979, 15, 109. (19) Reimers, J. R.; Watts, R. 0.;Klein, M. L. Chem. Phys. 1982, 64, 95.
(20) Douketis, C.; Scoles, G.; Marchette, S.; Zen, M.; Thakkar, A. J. J. Chem. Phys. 1982, 76, 3057. Ahlrichs, R.; Penco, R.; Scoles, G. Chem. Phys. 1977, 19, 119. Hepburn, J.; Scoles, G.; Penco, R. Chem. Phys. Lett. 1975, 36. - - , 45-~I .
(21) Margofiash, D. J.; Proctor, T. R.; Zeiss, G. D.; Meath, W. J. Mol. Phys. 1978, 35, 747.
The Journal of Physical Chemistry, Vol. 91, No. 10, 1987 2515
Spectroscopy of the Water Dimer exp((VR - V,)AT) - 1 ; if V, is greater than VR, the replica is destroyed with probability 1 - exp((VR - VT)AT).The reference potential, VR, is adjusted during the simulation to keep the total population of replicas approximately constant. Once equilibrium is established, the average value of VR over the simulation is the estimator of the ground-state energy. Details of the algorithm have been given by Anderson.I8 In the case of the present simulation, one or two details must be given. First, the time step has to be chosen sufficiently small to ensure that the factorization into uncoupled diffusive and birth/death steps is an accurate approximation. In the case of water clusters, where the intramolecular potential varies rapidly by comparison with the intermolecular potential, this constraint means that accurate intermolecular properties are only obtained after rather long calculations. A similar problem arises in a classical molecular dynamics simulationz2of water. As well as the disparity between the steepness of the two interactions, and as a consequence of the atomic masses of the two atoms, there is a factor of 16 difference between the 0-atom and H-atom diffusion coefficients. One result of this disparity is that the intermolecular part of the wave function equilibrates more slowly than the intramolecular part. Again, the solution is to use long calculations. Quantum random walks were performed for the water dimer using ensembles stabilized near 200 replicas and near 400 replicas, and calculations were completed with AT corresponding to 0.004 and 0.002 fs. Values of all the properties calculated with these parameters were found to agree to within the statistical fluctuations of the averages. To test the accuracy of the simulations further, we computed the ground-state energy of the water monomer, using 100OOO time steps. The result Eo = 4615 f 15 cm-l, is in excellent agreement with the value of 4623 cm-' obtained from the converged variational calculation of Coker et al.l39I4 All the results reported in this paper were obtained with an ensemble of 400 replicas, a time step of 0.004 fs, and a calculation extending over at least 100000 time steps. As well as calculating the ground-state wave function and energy, we estimated the kinetic and potential components of the energy and atomatom distribution functions. In order to do these calculations, it was necessary to generate the ground-state probability density function, $o*(R)$o(R). In the case that J/o(R) is real, as in the present study, quantities of the form ( A ) = I $ ? ( R ) A(R) dR
(12)
where A is not a differential operator, can be evaluated from a sequence of configurations, as follows. Once a stable ensemble of replicas has been obtained, the positions of the particles define the wave function, rLo(R). Under these circumstances, systems replicate most rapidly in the regions where $o(R) is large. However, during the remainder of its lifetime every replica then proceeds to generate a subdistribution that is also a sample from &(R). It follows that if a quantity A(R) is calculated for every replica at time to, and then weighted by the number of descendants the replica produces, the resulting average is an estimate of (A). K a 1 0 s ~has ~ described a similar method for generating the $02 distribution using the Green's function Monte Carlo method. We have used this approach to calculate atom-atom distribution functions. 4. Intramolecular Vibrations in the Water Dimer
As discussed above, the quantum simulation algorithm generates on ensemble of systems whose coordinates represent a sample distributed according to the ground-state wave function. This ensemble can be used to estimate the vibrational frequencies within a cluster providing certain assumptions are made. The idea behind these assumptions is to determine an approximate analytic form for the ground-state wave function that can be used to define a
basis set for subsequent variational calculations of higher eigenstates. The ground-state wave function of the cluster is approximated as a product of an intermolecular part and a set of single oscillator local-mode eigenfunctions &(si)
We assume that the single oscillator states are Morse functions24
where the normalizing constant N = I'(2K)/(2K - 1)2 and x = 2K exp(-a(s - so)). The parameters K , a, and so are determined from the quantum simulation results. The numerical many-body wave function is projected onto each of the local-mode coordinates in the cluster, by accumulating histograms of their values during the course of the simulation. A least-squares method is then used to determine the parameters in the analytic single oscillator wave function. Once these parameters are found, the corresponding Morse potential well depth can be calculated from the relationship K = AD'I2 where A = 21/2/ahG'/2, with G the diagonal component of the G matrix1L6corresponding to the local-mode s. The effective single oscillator states obtained in this way define a reference state for the subsequent variational calculation. Note that, assuming the fit to the numerical projections is accurate, this reference state represents the best independent oscillator model for the groundstate vibrations. By writing the ground-state wave function as a product of uncorrelated functions we have neglected any direct coupling between the various local modes and between the intramolecular modes and the intermolecular modes. However, by construction the set of effective oscillators defined in this way gives the best possible separable wave function for the ground-state dimer and allows for such couplings in the mean-field sense. Reimers and Watts"J2 found that for the potential surface used here coupling between intermolecular and intramolecular modes is quite weak, although that between OH vibrations on the same molecule is strong. However, when we defined the reference state using the mean-field method, the explicit coupling between the two O H oscillators is averaged out. Consequently, when making the variational calculations, we included thef12 coupling, and kinetic energy couplings due to off-diagonal terms in the G matrix, but neglected other terms. The result of this assumption is that the variational matrix becomes block diagonal, each block representing a single water molecule. When a water molecule combines to form a cluster, its rotational degrees of freedom are restricted by the presence of neighboring molecules. The influence of these hindered rotations is also implicitly included in the mean-field reference state defined above. Consequently, the centrifugal and Coriolis coupling terms in the Hamiltonian are not included in the variational calculation. Having made these simplifying assumptions, we made the variational calculations using a basis set of 120 states (vl, v2, v3), where v 1 + vz + v3 I 5. Previous results for the water monomer show that a basis set of such a size is converged for this pr0blem.6.'~ 5. Results
As stated in section 3, the ground-state energy of the system being simulated is obtained as the average value of VR needed to stabilize the replica population.lS In the case of the water dimer, this method gives Eo = 28.92 f 0.1 kcal/mol when the energy of the minimum energy geometry dimer is taken as zero. The ground-state energy can also be found by making use of the quantum virial theorem, which states that
where pis the kinetic energy operator. By calculating the virial, (22) Stillinger, F.H.; Rahman, A. J . Chem. Phys. 1972, 57, 1281. 1971, 55, 3336. 1974, 60,1545. 1974, 61, 4913. (23) Kalos, M.H.J . Comput. Phys. 1967, 2, 196. Phys. Rev. 1970, A2, 250.
(24) Morse, P.M.Phys. Rev. 1929, 34, 57. ter Haar, D.Phys. Rev. 1946, 70. 222.
2516 The Journal of Physical Chemistry, Vol. 91, No. 10, 1987
Coker and Watts
Potential enerav
-
n
a
20.
1
m
0
x
10.
1
Kinetic energy
W
I
0.
2.5
5.0
7.5
10.0
Time ( f s ) Figure 1. The kinetic and potential energies of the water dimer as a
function of imaginary time. TABLE I: Calculated Ground-State Energies for the Water Dimer, in kea1 mol-'
method quantum simulation local mode, frozen field" normal mode" CI surfacez6 LCAO-SCF25
EO
intra
inter
28.92 28.91 29.14 29.23 30.38
26.10 26.00 26.22 27.03 28.13
2.82 2.91 2.92 2.20 2.25
and calculating ( V ) ,one finds the ground-state energy as ( f) + (V). Calculations of both terms constituting Eo require the use of the descendant-weighting algorithm described in section 3. Figure 1 shows the two quantities ( T ) and ( V ) as a function of the number of time steps used to generate descendants. Clearly, both quantities converge if the simulation is sufficiently long. The errors in this approach to the ground-state energy are larger than those inherent in averaging VreRfor the replica population distribution is squared. Despite, this problem, we find ( T ) = 13.5 f 0.4 kcal/mol and ( V) = 15.6 f 0.2 kcal/mol, giving the ground-state energy as Eo = 29.1 f 0.4 kcal/mol. This estimate agrees with that obtained from V,, to within numerical error, thus confirming the accuracy of the simulation. Table I compares several calculations of the ground-state energy of the water dimer. In addition to the present result, we give estimates based on normal-mode and local-mode calculations, by Reimers and Watts," results from the LCAO-SCF plus localmode calculation of Curtiss and P ~ p l eand , ~ a~ similar calculation The by SlaniaZ6based on the C I surface of Matsuoka et results of Reimers and Watts" are directly comparable with the present value, for their calculations were based on almost the same potential surface. Quantum simulation and the local-mode analysis give essentially the same result for Eo,although the distributions between intramolecular and intermolecular interactions differ by about 0.1 kcal mol-'. The a b initio result for Eo reported by SlaniaZ6is close to that generated by the semiempirical model although there are significant differences between the intramolecular and intermolecular components. The LCAO-SCF value for Eo is significantly larger than the other estimates. The ground-state energy was calculated with respect to the total energy of the equilibrium dimer. This quantity was found by minimizing the potential energy of the dimer as a function of all the atom coordinates and is -6.1 kcal mol-'. A detailed comparison of the dimer geometry with available experimental data has been given by Reimers and Watts.17 Returning to the quantum calculations, we noted earlier that the ground-state energy of the water monomer is 4615 cm-I or 13.2 kcal mol-'. Hence the zero ( 2 5 ) Curtis,, L. A,; Pople, J. A. J. Mol. Specrrosc. 1975, 55, 1. (26) Slania, Z. Adv. Mol. Relax. Interact. Process 1981, 19, 117. (27) Matsuoko, 0.;Clementi, E.; Yoshime, M. J . Chem. Phys. 1976.64,
1351.
2
,I
,I
I ,,
4
6
4
6
, II
2 r / A
Figure 2. Comparison between the intermolecularatom-atom distribution functions calculated by using quantum simulation and the cIassical Monte Carlo method with T = 10 K.
point energy of the dimer relative to two isolated molecules is (28.9
- 2 X 13.2 - 6.1) = -3.6 kcal mol-'. That is, the destabilization
of the dimer due to zero point motion is 2.5 kcal mol-'. Figure 2 compares the ground-state intermolecular atom-atom distribution functions calculated from the quantum simulation with the corresponding classical Monte Carlo functions calculated at a temperature of 10 K. The quantum distributions, which were obtained by using the descentant-weighting algorithm, are very much broader than the classical functions. This is due to large amplitude zero point motions, which are particularly important in the H-atom distributions. The distribution functions are consistent with the lowest energy structure predicted by the pair potential, see Reimers and Watts." That is, one molecule acts as a "donor" on forming a "hydrogen bond" and the other acts as an "acceptor". As stated earlier, the random walk simulation was used to define a reference state for calculating the vibrational spectrum of the cluster. For every dimer in the ensemble, the shortest intermolecular 0-- -H distance was used to define both the hydrogen bond and the donor molecule. Separate histograms were accumulated for all six intramolecular local modes, and, in addition, a histogram of the 0-- -0separation was calculated. From symmetry arguments, it is expected that the local modes (sI.sz)on the acceptor molecule will be equivalent, so that the projected wave functions for these two modes should be identical. In practice, these two projections differed a little due to fluctuations. However, the lowest vibrational frequencies associated with the corresponding
The Journal of Physical Chemistry, Vol. 91, No. 10, 1987 2517
Spectroscopy of the Water Dimer TABLE 11: Calculated and Experimental Vibrational Frequencies, in cm-I, for the Water Dimer
mode 0-- -0 stretch acceptor bend donor bend acceptor bend overtone donor bend overtone donor stretch acceptor, symmetric acceptor, asymmetric stretch donor stretch -
1
this work 150 1590 1610 3130 3201 3535 3610 3714 3721
It
a.
II
Experiment
',,
1500
I
I
I' '.~i , \ :
I 1 I
',
i '1
d.
-
I .i!
:
I
111111
b.
Frozen field
1593, 16Ol3O 1611, 161930 317013 3215l' 3532,'' 3545' 36OO7J3 3714,' 3722l' 3730'*"
I
Simulation
expt 1 5OZ9
I
1 1750
3500
I
I
4000
Frequency ( w a v e n o s . )
Figure 3. Stick diagram comparing various methods for calculating the
water dimer bending and stretching frequencies with experiment. independent Morse oscillators were always within 25 cm-' of each other throughout the simulation. Consequently, the basis states for the (sI,s2)coordinates of the acceptor molecule were obtained from the combined histograms. In the case of the donor molecule, the "hydrogen bonded" and free coordinates, (sI,s2),differed by almost 200 cm-' and were averaged separately. There are eight equivalent conformations of the water dimer, corresponding to all the possible permutations of hydrogen atoms as donors in the hydrogen bond, and rotation of the acceptor molecule about its symmetry axis. In principle, the quantum simulation should allow for quantum tunneling between these conformers. We found no evidence for the hydrogen atom involved in the hydrogen bond changing its identity during the time scale of the simulation. Values of the vibrational frequencies associated with the intramolecular 0-H stretches, the bending modes and overtones, and the 0-0intermolecular mode are given in Table 11. Also shown is the available experimental data for these vibrations. It can be seen from this table that the agreement between theory and experiment is good, lying within the scatter of the measurements. In order to understand this agreement, and interpret the theory correctly, we now consider the calculations in more detail. Figure 3a shows the v = 0 1 transition energies for the basis states associated with the donor and acceptor molecules. The lowest frequency mode corresponds to the bending mode, s3,on the acceptor molecule and the next frequency is that of the s3mode on the donor molecule. There are three frequencies in the region 3250-4000 cm-l, the lowest of which is associated with the hydrogen bonded mode, sl,on the donor molecule. Next, there are two degenerate modes associated with the two modes (sI,s2)on the acceptor molecule. Finally, the highest frequency mode is that associated with the second 0-H stretch, s3,on the donor molecule. Figure 3b shows the frequencies obtained after completing the variational calculations described in the previous section. It can be seen that the hydrogen-bonded oscillator is shifted a few wavenumbers to the red, the free donor mode moves
-
to the blue, and the degenerate acceptor states are now separated into two states. The lower frequency mode corresponds to a symmetric combination of basis states and the other to an antisymmetric Combination? Both bending modes are moved to the red by several wavenumbers. The experimental data in the 0-H stretch region shown in the figure is taken from the paper of Coker et al.,I3 where vibrational frequencies from infrared predissociation measurements on water clusters are reported. As discussed by Coker et al., similar results for the OH stretching frequencies are also found in the papers of Vernon et ale7and Page et al.,' the two sets of data agreeing to within a few cm-I. Matrix isolation experiments in this region, using Ar and N2 hosts, also given OH stretching frequencies that are close (
