Isotope and potential energy surface effects in vibrational bonding

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J . Phys. Chem. 1984, 88, 2158-2164

2758

Isotope and Potential Energy Surface Effects in Vibrational Bonding D. C. Clary*+ Department of Chemistry, University of Manchester Institute of Science and Technology, Manchester M60 1 QD, England

and J. N. L. Connor*$ A. A. Noyes Laboratory of Chemical Physics,# California Institute of Technology, Pasadena, California 911 25 (Received: August 31, 1983)

Variational calculations of vibrational energy levels have been carried out in order to study vibrational bonding for the 1x1 and BrXBr molecules with X = Mu, H, and D. Extended London-Eyring-Polanyi-Sat0 (LEPS) and three-center diatomics-in-molecules(DIM) potential energy surfaces have been used for 1x1. For BrXBr, a three-center DIM potential surface was employed. The LEPS surface is purely repulsive, whereas the DIM potentials possess two van der Waals minima. The variational calculations use the exact Hamiltonian of Watson and provide rigorous upper bounds to the energy levels. Calculations have been performed both for the collinear configurations of the atoms as well as in three dimensions, with the total angular momentum quantum number equal to zero. The bond energies of YXY, where Y = Br and I, increase in the order D < H < Mu. This inverse isotope effect is opposite to that for conventional chemical bond energies. The bond energies for 1x1 also show a sensitive dependence on the nature of the potential surface. The presence of van der Waals wells as well as the lower barrier of the DIM surface increases the number of vibrationally bound states for 1x1 compared to the case of the LEPS surface. The accurate variational eigenenergies have also been approximately partitioned into normal-mode symmetric, bending, and asymmetric contributions. For all nine systems the symmetric frequency is less than the bending frequency, unlike the case for conventional triatomic molecules. The isotopic dependence of the normal-mode frequencies is consistent with the predictions of the valence force field approximation. The calculations are relevant to the experimental detection of vibrational bonds by conventional spectroscopic techniques or by photodetachment spectroscopy.

I. Introduction There has been considerable interest recently in the theory of vibrational bonding for heavy-light-heavy atom (HLH) triatomic molecules.'-* A vibrational bond is a new type of chemical bond with the novel feature that it can exist on a minimum-free potential energy surface.'-8 The stability of a vibrational bond comes from the fact that the zero point energy of HLH may be lower than the zero point energy of the isolated HL molecule, even for potential energy surfaces which have a barrier between the reactant The existence ( H LH) and product (HL + H) of a vibrational bond is thus a dynamical effect, unlike a conventional chemical bond, which arises from the overall attractive forces between the atoms. We have recently reported2 the first accurate three-dimensional (3D) calculation of the vibrationally bound energy levels of any molecule. We considered IHI using the extended London-Eyring-Polanyi-Sato (LEPS) potential surface A of ref 9. This LEPS surface is minimum free with a barrier of 4.62 kJ mol-' in the symmetric collinear I-H-I configuration? Our calculations2used a variational method which exactly includes Coriolis and vibrational angular momentum terms in the molecular Hamiltonian and provides a rigorous upper bound to the energy levels for a given potential surface. The aim of the present paper is to extend the work of ref 2 in two directions. First, we examine how vibrational bonding is affected by isotopic substitution of the light atom L. Second, we explore how a change in the potential energy surface affects the nature of a vibrational bond. We have carried out variational calculations of vibrational energy levels for the molecules 1x1and BrXBr, where X = Mu, H, and D. The muonium (Mu) atom can be considered to be a light isotope of hydrogen of mass 0.1 14 u with a lifetime of 2.2 X lo4 s. The low mass of Mu implies the existence of very large isotope effects for the IMuI and BrMuBr molecules. The gasphase chemistry of muonium has been reviewed by and by Fleming.'*

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'Present address: Department of Theoretical Chemistry, Lensfield Road, University of Cambridge, Cambridge CB2 IEW, England. *Permanent address: Department of Chemistry, University of Manchester, Manchester M13 9PL, England. 8 Contribution No. 6886.

0022-3654/84/2088-2758$01.50/0

For collinear 1x1and BrXBr, diatomics-in-molecules (DIM) potential surfaces have been reported recently by Last.13 These surfaces have been extended into 3D by Last and Baer.14 The DIM potentials contain wells in the reactant and product valleys. In order to assess the effect that these wells have on vibrational bonding, we have also performed variational calculations for the energy levels of the collinear and 3D 1x1and BrXBr systems on the DIM potentials. Any realistic potentials for these molecules will also contain wells in the reactant and product valleys because of van der Waals forces. In section I1 we describe the LEPS and DIM potential surfaces and outline the variational method used for the calculation of the vibrational energy levels. We present our results in section I11 for 1x1and BrXBr. This is the first time that an accurate 3D study of isotope effects for Mu, H, and D in vibrational bonding has been made. We present results for collinear 1x1and BrXBr as well as in 3D. For the collinear configuration of the atoms, only two coordinates corresponding to the symmetric and asymmetric stretching vibrational modes need be considered, whereas in 3D there is also the doubly degenerate bending mode. We discuss our results in section IV. In particular, we ap(1) J. Manz, R. Meyer, E. Pollak, and J. Romelt, Chem. Phys. Lett., 93, 184 (1982). (2) D. C. Clary and J. N. L. Connor, Chem. Phys. Lett., 94, 81 (1983). (3) E. Pollak, Chem. Phys. Lett., 94, 85 (1983). (4) J. Manz, R. Meyer, and J. Romelt, Chem. Phys. Lett. 96,607 (1983). ( 5 ) E. Pollak in "Proceedings of the 15th Jerusalem Symposium on Quantum Chemistry and Biochemistry: Intramolecular Dynamics", March 29-April 1, 1982, J. Jortner and B. Pullman, Eds., Reidel, Dordrecht, 1982, p 1; J. Chem. Phys., 78, 1228 (1983). (6) J. Romelt, Chem. Phys., 79, 197 (1983). (7) D. K. Bondi, J. N. L. Connor, J. Manz, and J. Romelt, Mol. Phys., 50, 467 (1983). (8) 0. Atabek and R. Lefebvre, Chem. Phys. Lett., 98, 559 (1983). (9) J. Manz and J. Romelt, Chem. Phys. Le??.,81, 179 (1981). (10) D. C. Walker, J . Phys. Chem., 85, 3960 (1981). (1 1) D. C. Walker, "Muon and Muonium Chemistry", Cambridge University Press, Cambridge, 1983. (12) D. G. Fleming in "Physics of Electronic and Atomic Collisions, Invited Papers of the XIIth International Conference on the Physics of Electronic and Atomic Collisions, Gatlinburg, TN, July 15-21, 1981", S. Datz, Ed., NorthHolland Publishing Co., Amsterdam, 1982, p 297. (13) I. Last, Chem. Phys., 69, 193 (1982). (14) I. Last and M. Baer, J. Chem. Phys., 80, 3246 (1984).

0 1984 American Chemical Society

Vibrational Bonding for 1x1 and BrXBr proximately partition our exact vibrational energies into normal-mode symmetric, asymmetric, and bending contributions. We also discuss the effect of isotopic substitution and potential surface variation on the nature of a vibrational bond. It is hoped that the trends revealed by our calculations will assist in the experimental detection of vibrational bonding. Our conclusions are in section V. Previous research on the nature of the vibrational bond in IHI includes exact quantum collinear',2 and 3D2calculations, classical collinear r e s ~ l t s ,as ~ well as vibrationally adiabatic collinear models'v6 that use plane polar (Delves' or hyperspherical) coordinate~;~~' see also ref 15-18 for additional studies on the dynamics of I H I using Delves' coordinates. There have also been approximate 3D adiabatic calculation^^^^ for vibrational bonding in IHI. Very recently, Manz et al.19920have carried out exact and approximate quantum calculations for collinear IHI and ID1 on the LEPS and (a slightly different) DIM potential surface as well as approximate adiabatic calculations for 3D IHI and ID1 on the LEPS surface. Their computations, which do not include an exact 3D treatment, complement those reported in this paper.

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11. Computations In this section, we compare the two potential surfaces used in our computations and outline the variational method for the calculation of the vibrational energy levels. Potential Energy Surfaces. Because accurate ab initio potential surfaces are not available for the 1x1and BrXBr molecules, we necessarily have to use semiempirical parametrized forms. For the 1x1 system, we used the same surface as in our earlier calculations* for IHI. This surface is of a standard extended LEPS form with Sato parameter S = 0.190 and asymptotic parameters given in Table I of ref 9 (surface A). The LEPS surface is minimum free with a symmetric collinear barrier of Eb = 4.62 kJ mol-'. A contour plot of the collinear surface is shown in Figure 1 of ref 9. Diatomics-in-molecules surfaces for the collinear 1x1 and BrXBr systems have been reported recently by L a d 3 and extended to 3D by Last and Baer.I4 Their DIM formalism includes a three-center term containing adjustable parameters which are used to fit the collinear potential barriers for the H + F2, H + C12, and F HF reactions. The equations for the DIM surfaces are much more complicated than those for the LEPS surface. In our calculations we used a Fortran code for the DIM surfaces that was kindly supplied to us by Dr. I. Last. In contrast, Manz et a1.19~20 used the equations for the collinear DIM surface given in ref 13. Unfortunately, eq 10 of ref 13 contains misprints2I with the result that the surface used by Manz et al.19,20is slightly different from the true DIM surface. Like the LEPS surface, the DIM potential for 1x1has a symmetric collinear barrier, and the classical barrier height is Eb = 1.26 kJ mol-'. In addition, the DIM surface has wells in the reactant and product valleys of depth E, = -3.79 kJ mol-l for the collinear configuration. Since all realistic potential surfaces will also possess wells in the reactant and product valleys because of van der Waals interactions, it is clearly important to investigate the effect of these wells on the vibrational bonding. The DIM potential for the BrXBr system is similar to that for 1x1with well

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(15) V. K. Babamov and R. A. Marcus, J . Chem. Phys., 74, 1790 (1981); V. K. Babamov, V. Lopez, and R. A. Marcus, ibid., 78, 5621 (1983). (16) J. A. Kaye and A. Kuppermann, Chem. Phys. Lett., 77, 573 (1981). (1 7) C. Hiller, J. Manz, W. H. Miller, and J. Rbmelt, J . Chem. Phys., 78, 3850 (1983). (18) V. Aquilanti, S. Cavalli, and A. Lagan& Chem. Phys. Lett., 83, 179 (1982). (19) J. Manz, R. Meyer, and H. H. R. Schor, J . Chem. Phys., 80, 1562 (1984). (20) J. Manz, R. Meyer, E. Pollak, J. Romelt, and H. H . R. Schor, Chem. Phys., 83, 333 (1984). (21) I. Last, private communication, 1983. In the numerator of eq 10 of ref 13 the term 4f4R4/ 15 should read Z4R4/15 and the term f2R should read {R. In the denominator, the term 4E4R,4/15 should read E4R,4/15. The expression for the DIM potential used by Manz et al. in ref 19 and 20 is the same as eq 10 of ref 13 except that the term FZRin the numerator was replaced by ZR (J. Manz, private communication, 1983).

The Journal of Physical Chemistry, Vol. 88, No. 13, 1984 2759 TABLE I: Comparison of Some Properties of the LEPS and DIM Potential Energy Surfaces for YXY (X = Mu, H,D and Y = Br. I) LEPSDIMDIMquantity (1x1) (1x1) (BrXBr) well depth, D,(XY)/kJ mol-' -308.400 -308.234 -378.072 position of barrier, rb(XY)/nm 0.1794 0.1818 0.1625 collinear barrier height, 4.62" 1.26" 3.35b &/kJ mol-' collinear well depth, -3.79a -3.356 E,/kJ mol-' zero point energy, 39.559 40.47 45.53 E,(MuY)/kJ mol-' zero point energy, 13.654 13.95 15.69 Eo(HY)/kJ mol-' zero point energy, 9.729 9.93 1 1.20 Eo(DY)/kJ mol-' ~

~~

~

aFrom ref 19. bFrom ref 13. depths of E, = -3.35 kJ mol-' and a symmetric barrier of Eb = 3.35 kJ mol-' for the collinear configuration. Contour plots of the collinear 1x1and BrXBr DIM potential surfaces are shown in Figure 3 of ref 13. In Table I, we compare some properties of the LEPS and DIM surfaces for YXY, where Y = Br and I. Notice that the well depth De of the isolated XI molecule is slightly different for the two surfaces. Note also that we usually measure energies from the three-atom dissociation limit or with respect to the minimum of the XY well. We also report in Table I the zero point energies Eo(XY) for the diatomic molecules XY on the three surfaces. They have been calculated for the DIM potentials by numerical solution of the one-dimensional radial Schrodinger equation by the Numerov algorithm as described by Cooley.22 For the LEPS potential, the analytic eigenvalue expression for a Morse oscillator was used. We checked the accuracy of the numerically determined DIM eigenvalues by applying the same Numerov-Cooley method to a Morse oscillator and obtained agreement for the energy levels to at least five significant figures. The magnitudes of the E,(XY) are important because the very existence of a vibrational bond requires that the zero point energy of the YXY triatomic molecule be less than Eo(XY). The magnitude of E,(XY) therefore provides an upper bound to the vibrational bond energy of YXY. As expected, the E,(XY) values in Table I follow the order Mu > H > D for a given Y, with ratios close to the reduced mass ratios of (PHY/PMuY)"~: :(hHY/MDY)'/'

(mH/mMu)'/2:1:(mH/mD)1/2 = 2.97:1:0.707 In our calculations we have used the following masses: mMu = 0.114 u, mH = 1.008 u, mD = 2.014 u, mBr= 79.91 u, and mI = 126.9 u.

Variational Calculations. We calculated the energy levels of the YXY systems by a modified version of the variational technique of Whitehead and Handy.23 We have computed the vibrational energy levels for both collinear and 3D YXY, but with the restriction in the 3D case that the YXY molecule has total angular momentum J = 0. We used the exact 3D Hamiltonian derived by Watson,24which is expressed in terms of normal coordinates. For all three potential surfaces the minimum reaction path occurs for the collinear configuration, and the Watson Hamiltonian for linear reference geometries24is therefore the appropriate one to use. Note that since we use an exact Hamiltonian, all Coriolis interaction terms are present in the 3D Hamiltonian (in the collinear calculations, the Coriolis terms are absent). (22) J . W. Cooley, Math. Comput., 15, 363 (1961). (23) R. J. Whitehead and N. C. Handy, J . Mol. Spectrosc., 55, 356 (1975). (24) J. K. G. Watson, Mol. Phys., 19, 465 (1970).

2760 The Journal of Physical Chemistry, Vol. 88, No. 13, 1984

TABLE 11: Exponents t , , t2, and t3 for the Basis Set 1" system tl/ao-2 m;' t2/ao-2m;' t3/aoW2m;' LEPS IMuI 0.1333 (-3) 0.3764 (-2) 0.8259 (-2) IHI 0.1904 (-3)' 0.9113 (-3)' 0.2285 (-2)b ID1 0.2285 (-3) 0.5355 (-3) 0.1343 (-2)

IMuI IHI ID1

0.1333 (-3) 0.1904 (-3) 0.2285 (-3)

DIM 0.3764 (-2) 0.9094 (-3) 0.5355 (-3)

0.8259 (-2) 0.2281 (-2) 0.1343 (-2)

BrMuBr BrHBr BrDBr

0.2857 (-3) 0.3047 (-3) 0.3047 (-3)

DIM 0.4862 (-2) 0.1455 (-2) 0.9179 (-3)

0.8652 (-2) 0.2053 (-2) 0.2302 (-2)

"The exponents have been optimized in 3D calculations with a basis set of size N = 7. The number in parentheses is the power of ten by which the entry must be multiplied. From ref 2.

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The first step in applying the Whitehead-Handy method is to relate the Cartesian displacement coordinates of the three atoms to normal coordinates. For symmetric YXY systems, the coefficients of the required transformation involve only the masses of X and Y and can be found in ref 25 for example. There are four normal coordinates, Q1, Q2a,Q2b,and Q3, for YXY, where Q, is the symmetric stretch mode, Qza and QZbare degenerate bending modes, and Q3 is the asymmetric stretch mode. It is also convenient to write Qza = Q2 cos 6 and QZb= Qz sin 6, where 6 is the vibrational angular momentum variable. The second step in the variational procedure is to specify the basis set. In the 3D case for J = 0, we use a basis set made up of elements of the form

where Hn,(Ql)is a Hermite polynomial of degree n,,Fn:(Q2) is is a normalizing a Laguerre polynomial of degree n2, and Cnlnz2 constant. In the collinear calculations the bending coordinate Q2 is absent and the basis set is

In the 3D calculations, basis functions having only even values of n2 and n3 contribute to the J = 0 solutions, while in the collinear calculations only even values of n3 are required for states of gerade symmetry. Thus, we did not attempt to calculate energy levels for states such as (01'0) or (OOol) with ungerade symmetry. Calculation of these levels would have required use of the Watson Hamiltonian with J I 1, and rotational eigenfunctions would be needed in the basis set expansion. The size of the 3D basis set is conveniently defined by a number N such that all possible n, combinations are included in the basis set where

n, + n2 + n3 + 1 I N

(3)

In a similar way, the size of the collinear basis set is defined by

nl

+ n3 + 1 5 M

(4)

The next step in the calculation is to construct the matrix elements of the secular equation. That requires the evaluation of integrals for the basis sets 1 and 2 over the potential energy surface. Gauss-Laguerre quadrature was used for the integrals involving Q2,and Gauss-Hermite quadrature was employed for the Q1 and Q3 coordinates. The final step is to optimize the exponents t l , t2, and t3 in the basis set (see Table 11). We determined near-optimum exponents by systematic variation of ti, t2, and t 3 in a 3D calculation with N = 7 for 1x1(LEPS) and BrXBr (DIM). The same exponents t l and t3 were then used in the collinear calculations. For IHI (25) D. C. Clary, J. Chem. Phys., 75, 209 (1981).

Clary and Connor TABLE III: Collinear Bound-State Energy Levels for 1x1 (X = Mu, H, D) on the LEPS Potential Energy Surface" energv/kJ mol-' ~ 1 ~ M 3 = 10 M = 12 M = 17 M = 19 X = Mu 00 -286.522 -286.535 -286.546 -286.546 10 -285.049 -285.228 -285.255 -285.257 20 -283.149 -283.836 -283.978 -283.987 30 -280.787 -282.039 -282.701 -282.732 40 -278.889 -279.643 -281.413 -281.478 50 -275.421 -277.291 -280.130 -280.197 60 -268.872 -274.812 -278.431 -278.938 70 -258.525 -269.382 -275.988 -277.443 80 -242.288 -260.912 -272.912 -275.166 90 -212.004 -248.462 -270.465 -272.1 26 100 -1 95.661 -229.886 -266.492 -269.038

00 10 20 30

-299.168 -297.816 -296.521 -295.144

X=H -299.173 -297.837 -296.550 -295.296

-299.174' -297.845' -296.604b -295.450'

-299.174 -297.846 -296.608 -295.499

00 10 20 30

-300.660 -299.337 -298.090 -296.724

X-D -300.665 -299.360 -298 149 -296.948

-300.667 -299.380 -298.287 -297.352

-300.667 -299.381 -298.299 -297.440

I

" M defines the size of the basis set in the variational calculations, see definition (eq 2). The results for X = H are from ref 2. The solid line indicates the energy of XI(u=O): E,,,(u=O) = -268.841 kJ mol-'; EHI(u=O)= -294.746 kJ mol-'; EDl(u= 0) = -298.671 kJ mol-]. The quantities u1 and u3 are the usual quantum numbers used to characterize the symmetric and asymmetric stretching vibrational modes of a collinear triatomic molecule. ' M = 15.

on the DIM potential, we again optimized the exponents tl, t2, and t 3 and found they were very close to those for the LEPS potential. For IMuI and ID1 on the DIM surface we therefore simply used the values of the exponents already determined for the LEPS potential. The calculations using normal coordinates also require that we specify a "reference geometry" for symmetric YXY. We used the XY distance for the position of the saddle point on the three surfaces; these quantities are given in Table I. The reference geometry can be varied if desired to obtain the lowest variational energy.26 In test calculations we found the saddle point geometry gave close to optimum energies. 111. Results The energy levels obtained by the variational method are given in Tables 111-VIII. In particular, Tables 111 and IV display our results for collinear and 3D 1x1on the LEPS surface, Tables V and VI are for 1x1on the DIM surface, while the results for BrXBr are given in Tables VI1 and VIII. It is important to remember that since we are using a variational method together with exact kinetic energy operators, the results in Tables 111-VI11 are rigorous upper bounds to the collinear and 3D ( J = 0) vibrational energies for the potential energy surfaces used. The eigenvalues in Tables 111-VI11 are denoted by E ( v p 3 ) and E ( U ~ U for ~ ~the U ~collinear ) and 3D cases, respectively, where ul, u ~and ~ u3 , are the usual quantum numbers used to characterize the symmetric, bending, and asymmetric vibrational normal modes of a triatomic molecule. For higher vibrational states, the assignments are only approximate because of the severe failure of the normal-mode approximation. As explained in section 11, our choice of basis set means we have not performed calculations for states such as (OOol) or (01'0). The vibrational energies in the tables are shown for increasing values of the parameters M or N so that we can assess the convergence of the calculated eigenvalues. ( 2 6 ) T. C. Thompson and D. G. Truhlar, J. Chem. Phys., 77,3031 (1982).

Vibrational Bonding for 1x1and BrXBr

The Journal of Physical Chemistry, Vol. 88, No. 13, 1984 2761

TABLE I V Three-Dimensional Energy Levels for 1x1 (X = Mu, H, D) with J = 0 on the LEPS Potential Energy Surface" energy/kJ mol-' N = 10 N = 11 N = 12 u1u2xu3 X = Mu 0000 1000 20'0 30'0 40'0 50'0 60'0 70'0

-275.463 -274.088 -272.836 -27 1.425 -269.022 -266.466 -263.497 -260.854

00'0 10'0 20'0 30'0

-295.220 -293.982 -292.752 -291.561

-215.515 -214.288 -272.909 -27 1.664 -270.103 -267.525 -265.011 -261.890

-275.527 -214.317 -213.099 -21 1.758 -270.508 -268.726 -266.095 -263.496

-

X=H -295.225 -294.025 -292.841 -291.6 18

X=D 00'0 10'0 20'0 30'0

-291.836 -296.614 -295.410 -294.1 16

-291.843 -296.653 -295.485 -294.232

-295.227 -294.038 -292.926 -291.752

TABLE V: Collinear Bound-State Energy Levels for 1x1 (X = Mu, H, D) on the DIM Potential Enerev Surface" energy/kJ mol-l ulu3 M = 12 M = 16 M = 19 X = Mu 00 10 20 30 40 50 60 70 80 90 100 110 120

-292.21 1 -290.636 -289.01 8 -287.200 -284.669 -28 1.056 -278.161 -272.208 -261.016 -244.080 -218.868 -210.772 -202.617

00 10 20 30 40 50 60 70 80

-304.108 -302.619 -301.180 -299.702 -298.21 3 -296.1 18 -292.954 -289.369 -28 5.563

00 10 20 30 40 50 60

-305.427 -304.110 -302.784 -30 1.327 -299.718 -291.574 -294.782

-292.222 -290.611 -289.145 -287.605 -285.993 -284.297 -282.305 -279.23 1 -274.798 -271.471 -264.141 -252.221 -235.619

-292.223 -290.681 -289.161 -287.659 -286.164 -284.646 -283.020 -28 1.284 -279.302 -276.070 -27 1.168 -267.580 -260.652

X=H

-297.847 -296.681 -295.571 -294.353

a N defines the size of the basis set in the variational calculations; see definition (eq 1). The results for X = H are from ref 2. The solid = -268.841 kJ line indicates the energy of XI(u=O): EM:,~(v=O) mol-'; EHI(u=O)= -294.146 kJ mol-'; EDl(u=O)= -298.611 kJ mol-'. The quantities u I , uzA,and uj are the usual quantum numbers used to characterize respectively the symmetric, bending, and asymmetric vibrational modes of a triatomic molecule.

We consider the collinear results for 1x1on the LEPS surface in Table I11 first. Our vibrational energies for X = H and D are evidently converged to fO.OO1 kJ mol-' for the lowest two states, and we find there are four and two vibrationally bound states for H and D, respectively. As we discussed earlier,2 the H results are in very close agreement with an independent calculation using a different numerical technique by Manz et al.' More recently, Manz et aLzohave obtained for X = D exact values of -300.6668 and -299.3815 kJ mol-' for E(O0) and E(10), respectively, which are in excellent agreement with our variational values. For X = Mu, our two lowest eigenvalues are evidently converged to f 0.002 kJ mol-', but for the higher eigenvalues, the convergence is much worse. Nevertheless, our calculations show that the IMuI molecule possesses ut leasf 11 bound states on the LEPS surface. The 3D results in Table IV for 1x1on the LEPS surface show that inclusion of the bending degree of freedom decreases the number of bound states for each isotope. In fact, for X = D we could not obtain a bound state in 3D. This same conclusion has also been obtained from an approximate 3D adiabatic calculation?0 For X = Mu, we find at least five bound states in 3D. Again, the convergence of our results for the higher eigenvalues is much worse than for the lower ones. Next we consider the results for 1x1on the DIM surface in Tables V and VI. For the collinear case our results are well converged for low values of the quantum number u l , but the convergence rapidly decreases as u1 increases. Our results show that there are at least 11, 8, and 6 bound states for Mu, H, and D, respectively. For u1 = 0, 1, and 2 our eigenvalues for X = H agree well with those obtained by an alternative exact method in Table 2 of ref 19. Manz et al.'9,20have also analyzed the number of bound states for IHI and ID1 on their DIM surface by the diagonally corrected vibrationally adiabatic hyperspherical (DIVAH) appro~imation.~,' This is a mathematically one-dimensional approximation using plane polar coordinates that exploits the separation of fast and slow motions in H L H systems. According to the DIVAH app r o x i m a t i ~ n 16 , ~ bound ~ ~ ~ ~states of the type E(u,O) exist for H and there are 14 for D. For the 3D 1 x 1 molecules in Table VI, we find there are at least 7, 5, and 4 bound states for Mu, H, and D, respectively.

-304.109 -302.632 -301.288 -300.082 -298 3 3 3 -291.291 -295.636 -293.892 -291.342

-304.109 -302.633 -301.297 -300.1 18 -299.030 -297.871 -296.478 -294.762 -292.935

X=D -305.431 -304.171 -303.132 -301.981 -300.552 -298.890 -291.143

-305.432 -304.189 -303.203 -302.209 -301.045 -299.664 -298.045

-

"The notation is the same as for Table 111. EMUI(u=O) = -267.76 kJ mol-'; EHI(v=O)= -294.28 kJ mol-'; EDI(v=O) = -298.30 kJ mol-'. TABLE VI: Three-Dimensional Energy Levels for 1x1 (X = Mu, H, D) with J = 0 on the DIM Potential Energy Surface" enerw/kJ mol-' ululxvl N = 10 N = 11 N = 12 X = Mu 00'0 10'0 20'0 30'0 40'0 50'0 60'0 70'0

-280.032 -218.423 -276.808 -275.160 -272.332 -269.3 18 -265.048 -261.634

00'0 10'0 20'0 30'0 40'0 50'0

-299.9 14 -298.528 -297.060 -295.609 -293.942 -291.738

00'0 10'0 20'0 30'0 40'0

-302.500 -301.239 -299.830 -298.374 -297.011

-280.055 -278.599 -276.972 -275.320 -273.500 -270.534 -267.307 -262.868

-280.065 -278.658 -277.160 -275.529 -273.856 -27 1.755 -268.807 -265.153

X=H -299.920 -298.602 -297.246 -295.721 -294.258 -292.3 73

-299.923 -298.622 -297.397 -295.966 -294.389 .___ -292.868

X=D -302.520 -301.368 -300.022 -298.5: 1 -297.188

-302.528 -301.449 -300.233 -298.127 -297.255

-

"The notation is the same as for Table IV. EMuI(u=O) = -267.16 kJ mol-I; EHI(u-0) = -294.28 kJ mol-'; ED,(v=O) = -298.30 kJ mol-'.

Clary and Connor

2762 The Journal of Physical Chemistry, Vol. 88, No. 13, 1984 TABLE VI1 Collinear Bound State Energy Levels for BrXBr (X = Mu, H, D) on the DIM Potential Energy Surface' energy/kJ mol-' M = 19 u1u3 M = 12 M = 16 X = Mu -359.006 00 -359.000 -359.005 -356.858 -356.865 10 -356.843 -354.766 20 -354.723 -354.739 -352.648 -352.704 30 -352.589 -350.582 -350.657 40 -350.058 -348.21 3 -348.6 19 50 -346.816 -346.576 60 -343.193 -345.120 -341.398 -344.125 70 -340.127 -340.895 80 -3 35.262 -337.929 -334.156 -336.993 90 -326.966 -333.1 50 100 -3 14.914 -327.208 -3 16.966 -3 29.446 110 -297.944 00 10

20 30 40 50 60 00 10 20 30 40 50

-371.673 -369.720 -367.847 -365.957 -363.632 -360.448 -356.593

X=H -37 1.687 -369.822 -368.230 -366.590 -364.637 -362.505 -360.274

-371.688 -369.827 -368.291 -366.909 -365.329 -363.418 -361.215

-313.169 -371.779 -370.209 -368.225 -365.885 -363.01 8

X=D -373.173 -371.865 -370.505 -368.726 -366.599 -364.204

-373.175 -371.934 -370.771 -369.276 -367.466 -365.343

"The notation is the same as for Table 111. EMuBr(u=O)= -332.54 kJ mol-'; EHBr(u=O) = -362.38 kJ mol-'; EDBr(u=O)= -366.87 kJ mol-'. TABLE VIII: Three-Dimensional Energy Levels for BrXBr (X = Mu, H, D) with J = 0 on the DIM Potential Energy Surface" enernv/kJ mol-' U~C','VI N = 10 N = 11 N = 12 X = Mu -345.221 -345.148 -345.208 00'0 -343.3 16 -343.039 -343.229 10'0 -341.3 18 -341.070 -341.125 20'0 -339.188 -339.270 30'0 -338.903 -336.897 -337.323 40'0 -335.799 -333.55 1 -334.851 50'0 -332.134 -331.351 -328.500 -330.026 60'0 00'0

10'0 20'0 30'0 00'0 10'0

20'0 30'0

-367.022 -365.193 -363.409 -361.443

X=H -367.078 -365.322 -363.51 1 -361.680

-367.108 -365.458 -363.711 -361.823

-369.959 -368.375 -366.348 -363.839

X=D -370.017 -368.533 -366.681 -364.427

-370.066 -368.694 -366.922 -364.871

"The notation is the same as for Table IV. EMuBr(~=O) = -332.54 kJ mol-'; EHBr(u=O)= -362.38 kJ mol-'; EDBr(u=O) = -366.87 kJ mol-'. Again, the convergence of our results deteriorates as u1 increases. Comparing the results for the DIM surface in Tables V and VI with those for the LEPS surface in tables I11 and IV shows that, for a given isotope, the number of vibrationally bound states is greater on the DIM surface. Evidently, the wells and lower barrier height on the DIM surface enhance the formation of vibrational bonds.

TABLE I X Vibrational Bond Energies (AE,,) for Collinear (1D) and 3D ( J = 0) YXY AEOI LEPSDIMDIMkJ mol-' dimension (1x1) (1x1) (BrXBr) Mu 1D 17.71 24.46 26.47 3D 6.69 12.31 12.68 H 1D 4.43 9.83 9.31 3D 0.48 5.64 4.73 D 1D 2.00 7.13 6.31 not bound 4.23 3.20 3D

Finally, we consider the collinear and 3D results for BrXBr on the DIM surface that are presented in tables VI1 and VIII. The trends in these tables are similar to those already discussed for 1x1. Thus, in ID the number of vibrationally bound states follows the order Mu > H > D and there are fewer bound states in 3D than in the corresponding collinear case. Notice also that (as before) the convergence of our results deteriorates as u1 increases. Some of the vibrational energies in Tables 111-VI11 lie above the zero point energies of XY(u = 0). They arise because we are limited to a finite basis set in our calculations. For a larger basis set, some of these states would drop in energy and become true bound states. It is also likely that some of these states correspond to long-lived quasi-bound states which would be observed as resonances in scattering calculations. Resonance effects in the collinear heavy-light-heavy atom reaction C1+ XCl ClX C1 (X = Mu, H, D) have been analyzed in detail in ref 7 while Garrett et al.27have studied resonances in the collinear H FH H F H reaction. We can also use the results in Tables 111-VI11 to calculate the bond energy AE,, for the YXY molecule. The vibrational bond energy is defined as the difference in the zero point energies of XY and YXY; i.e.

-

-

+

+

+

AEo = Exu(v=O)- E(O0)

(collinear)

(5)

or AEo = EXy(u=O)

- E(00'0)

(3D) Our results for the bond energies are presented in Table IX. It is interesting to note that all three potential surfaces show the same trends. For example, the collinear bond energies increase in the order D < H < Mu. The zero point energy of the bending modes also increases in the order D < H < Mu, but this effect is not large enough to alter the order of bond energies for the 3D molecules which is D C H C Mu, as in the collinear case. In a conventional chemical bond between two atoms, increasing the mass of one of the atoms also increases the bond energy. However, for vibrational bonds, Table IX shows that we have the opposite result (Le., the inverse isotope effect): increasing the mass of the light atom decreases the bond energy.

IV. Normal-Mode Approximation In section 111, we presented and discussed our variational results for the eigenenergies E(u1~13)and E(u1~2'03). In this section, we describe how we can partition these vibrational energies into normal-mode symmetric, bending, and asymmetric contributions. This partitioning is necessarily approximate since a normal-mode description of vibrational energies is itself an approximation. The vibrational energy levels of a linear triatomic molecule in the normal-mode approximation are given by28 E(uIu3) = D

+ (ul + j / J h ~+, (u3 + j/2)h~3

(collinear) (7)

or E ( u , u ~ ' u ~ )D

+ (VI + j/2)fi.~1+ ( ~ +2 l ) h ~ +2 (u3 + )/2)hW3

(3D) (8)

(27) B. C. Garrett, D. G. Truhlar, R. S. Grev, G. C. Schatz, and R. B. Walker, J . Pbys. Cbern., 85, 3806 (1981). (28) G. Herzberg, "Infrared and Raman Spectra of Polyatomic Molecules", Van Nostrand, New York, 1945, p 80.

Vibrational Bonding for 1x1 and BrXBr

The Journal of Physical Chemistry, Vol. 88, No. 13, 1984 2763

TABLE X Normal-Mode Energies 1/2h~1, h w z , and 1 / 2 h ~for 3 YXY Calculated from Eq 9-13 with D = De4 '/2hwiD('/2hwiD)/kJ hwz/kJ '/2hw:D('/2hw~D)/kJ system mol-' mol-' mol-' LEPS IMuI 0.58 (0.64) 11.02 21.28 (21.21) IHI [OS91 (0.66) 3.95 [8.63] (8.56) ID1 [OS81 (0.64) [2.82] [7.15] (7.09)

DIM IMuI IHI ID1

0.70 (0.77) 0.65 (0.74) 0.54 (0.62)

12.16 4.19 2.90

15.31 (15.24) 3.47 (3.39) 2.26 (2.18)

0.95 (1.07) 0.83 (0.93) 0.69 (0.62)

13.79 4.58 3.1 1

18.11 (18.00) 5.56 (5.45) 4.21 (4.28)

"Values in square brackets use energies above the zero point energy of XY in part of the calculation. where D is the depth of the potential energy function. Using eq 7, we can estimate 1/2hwland ' / 2 h ~ from 3 the collinear vibrational energies in Tables 111, V, and VI1 by means of the relations I/ZhWp = f/Z[E(lO) - E(00)]

(9)

y2hwlD = E(O0) - f/2hwiD - D

(10)

In a similar way hw,, hw2,and hw3 for the 3D YXY molecules can be calculated from 1/2hW:D

= f/2[E(lOOO)- E(OOOO)]

(1 1)

ha2 = E(0000) - E(O0)

(12)

) / 2 h ~=$E(00'0) ~ - f / z h ~-: ha2 ~ -D

(13)

Superscripts 1D or 3D have been added to h w , and h w 3 because slightly different values for these quantities are expected, depending on whether accurate 1D E(vlO) or 3D E(c1O00)are used. If the normal-mode ap roximation were exact, then we would have hwiD hwiD and hw3D h i D . It should also be noticed that we need to specify D in order to calculate hwlD or hwiD. The quantity D defines the minimum of the normal-mode potential energy function V, In the present case, Vis not a static potential but rather corresponds to an effective potential. Since our calculations do not determine D,in the following we have simply set D = De, the well depth of the isolated XY molecule (see Table I). Notice this choice for D (or any other) merely defines a zero of energy from which hwiD or hwiD is measured. Further discussion on the choice of D is given below. Table X shows 1 / 2 h ~ hw2, 1 , and ' / 2 h ~ 3calculated from eq 9-13. The table shows that L / 2 h ~ := D L / 2 h ~ to ! Dwithin 15% and 1/2hw:D= '/2hoiDto within 4%, with maximum absolute differences of 0.12 and 0.1 1 kJ mol-', respectively. This demonstrates that the coupling of the bending mode with the symmetric and asymmetric stretch modes is weak for the ground-state energy levels of the present molecules. Next we consider the isotope effects in our results. Table X shows that ' j 2 h w I depends only weakly on whether X is Mu, H, or D, whereas h w 2 and ' / 2 h ~ 3both show a strong dependence on the choice of X. In order to understand these trends, we use the valence force field a p p r o ~ i m a t i o nfor ~ ~the potential energy E

P

Y = j/zkl(qi2 + 4z2) + '/2k6S2+ D

(14)

where q1 and q2 are the changes in the two X-Y distances from the reference geometry and S is the change in the bond angle from 180'. The quantities k l and k6 are the corresponding force (29) Reference 28, pp 168, 172

IMuI IHI ID1

[0.97] (0.97) [I1 (1) [0.98] (0.97)

IMuI IHI ID1

1.08 (1.04) 1 (1)

0.83 (0.84)

BrMuBr BrHBr BrDBr

1.15 (1.15) 1 (1) 0.83 (0.67)

X=Mu X=H X=D

1

2.79 1 [0.71]

[2.47] (2.48) D l (1) [0.83] (0.83)

DIM 2.90 1

4.41 (4.50) 1 (1) 0.65 (0.64)

3.01

3.26 (3.30) 1 (1) 0.76 (0.78)

0.69 DIM

DIM BrMuBr BrHBr BrDBr

TABLE XI: Ratio of Normal-Mode Energies for YXY with D = De Relative to X = H 4

1 1

1

0.68

Mass Ratios 2.97 1 0.7 1

2.97 1

0.71

4Valuesin square brackets use energies above the zero point energy of XY in part of the calculation. The mass ratios are those given by eq 18-20. constants. For this model the angular frequencies w I , w2, and w3 are given by29 wI2 = kl/my w22

(15)

= (2/mY)(1 + 2mY/mX)(k6/12)

(16)

= ( l / m y ) t l + 2my/mx)k,

(17)

w32

where 1 is the reference XY distance. Notice that w2 and w 3 depend on the masses of both X and Y , whereas w1 involves only the mass of Y. The physical reason for this is that the normal modes characterized by o2and w3 involve the motion of both X and Y, whereas for w l , the atom X is stationary in the symmetric stretch motion with only the end atoms Y moving.30 In the limit of a heavy-light-heavy molecule, eq 15-17 give rise to the following ratios for '12hwl, ha2,and ' / 2 h ~for3 X = Mu, H, and D. 1/2hW I (MU):1/2hW I (H) :j/ZhwI (D) = 1:1:1 (18) hw2(Mu):hw2(H):hw2(D)= (mH/mM,)'~2:1:(mH/mD)1~2 = 2.97:1:0.71 (19) '/zh~3 (MU):'/zh~ 3H): ( 1/'2 h w j ( D) = (mH/mM,)'/2:1:(mH/mD)'~2 = 2.97:1:0.71 (20)

In Table XI, we report ratios of the normal-mode energies 'l2hw1,hw2,and ' / 2 h ~ for 3 X = Mu, H, and D together with the corresponding mass ratios predicted by eq 18-20. It can be seen that ' / z h w l and hw2 follow the mass ratios quite closely, implying that the normal-mode description is a reasonable approximation for these two modes for the ground-state energy levels 3 larger deviations from of the molecules. However, 1 / 2 h ~shows the mass ratios. Part of the reason for the discrepancies exhibited by '12hw3 arises from our choice of D,namely D = De.The quantity D is the well depth of the normal-mode potential. This potential is not obtained from a Taylor expansion of the static LEPS or DIM potentials, rather it corresponds to an effective potential. Using the DIVAH approximation, Manz et al.4919have obtained D = -304.520 kJ mol-' for IHI on the LEPS potential surface. The values of D for X = Mu and D will be different from D for X = H in the DIVAH model, thereby emphasizing that Vis an effective potential, not a static Born-Oppenheimer potential (which would be mass independent). In general, one expects D to lie in the range De E , 5 D 5 De + E,,. We have also calculated ' / 2 h ~ 3for X = Mu, H, and D using D = De + E, and D = De

+

(30) Reference 28, p 66, Figure 25b.

J. Phys. Chem. 1984,88, 2764-2771

2764

+

Ebinstead of D = De. The absolute values of ‘ / 2 h ~are 3 then found to differ considerably (for example, by a factor of 6 for ID1 (DIM)) although the trends in the isotopic ratios are still consistent with those expected from the mass ratios ( 2 0 ) . In principle, a better procedure would be to calculate D for each isotope by the DIVAH model. It is interesting to note from eq 15 and 17 that the ratio w ~ / w I in the valence force field approximation is independent of k l and is given by w ~ / w I = (1 + 2 m y / m x ) 1 / 2 (21)

= (2my/mx)’I2

(22)

for a heavy-light-heavy molecule. The results in Table X show trends consistent with eq 22. Again, part of the reason for the discrepancies is in the choice of D discussed above. Next, we consider some of the spectroscopic consequences of the results in Table X. First, we make the obvious point that all bound-to-bound transition energies must be smaller than the zero point energies E,(XY) (see Table I). However, bound-to-resonance state transitions can result in energies larger than Eo(XY). The width of these transitions will depend on the lifetime of the resonance. Table X also shows that hwl < hw2 whereas the usual rule is h w , > ha2. Another stable molecule which has h a l < hw2 is the F X F anion31s32with X = H , and D. As already analyzed above, h w , is weakly dependent on isotopic substitution, whereas hw2shows a pronounced isotopic dependence. It is also interesting to note that ‘/2Awl is smaller than the zero point energy of the Y2 molecule. Equations 7 and 8 can also be used to estimate the number of states with v2h# 0 and v3 # 0, although the harmonic approximation is expected to become very poor for highly excited vibrational states. Comparing I / 2 h ~ 1 ( Xand ) hw2(X) for the LEPS and DIM surfaces shows that 1/2hwl(LEPS)= 1 / 2 h w l (DIM) and hw2(LEPS) = hw2(DIM) for a given isotope, although the corresponding vibrational bond energies (Table IX) exhibit a pronounced sensitivity to the potential surface. V. Conclusions In this paper we have made a systematic study of isotope and potential surface effects on vibrational bonds for 1x1and BrXBr, (31) J. J. Rush, L. W. Schroeder, and A. J. Melveger, J. Chem. Phys., 56, 2793 (1972). (32) R. A. More OFerrall, “Proton Transfer Reactions”,E. Caldin and V. Gold, Eds., Chapman and Hall, London, 1975, Chapter 8; J. Emsley, Chem. SOC.Rev., 9, 91 (1980).

with X = Mu, H, and D. For 1x1, we used extended LEPS and three-center DIM potential surfaces, and for BrXBr a three-center DIM potential surface was employed. For each of the nine systems, variational calculations of the vibrational energy levels were carried out both for the collinear configuration and in 3D (J = 0). The variational method uses the exact Watson Hamiltonian and provides a rigorous upper bound to the vibrational eigenenergies. The bond energies of the YXY molecules increase in the order D < H < Mu, which is the opposite effect to that normally encountered. Comparison of the LEPS and DIM results for 1x1 shows that the presence of van der Waals wells and lower barrier height of the DIM surface increases the number of bound states. The bond energies also show a sensitive dependence on the nature of the potential surface. In addition, we approximately partitioned our exact vibrational energies into normal-mode contributions: l/zhwl, hw2, and 1 / 2 h ~ 3For . all nine systems, we found that h a l < hw2, unlike the case for normal molecules where h w , > haz. The isotopic , and ‘ l 2 h w ,is consistent with that dependence of 1 / 2 h ~ 1hw2, expected for the valence force field model. We hope the calculations reported in this paper will help in the spectroscopic detection of vibrationally bonded species. Our calculations should also be relevant to the spectroscopic properties of the triatomic anions YXY- which have been studied in noble gas matrices.33 Finally, we note that another possible route to the experimental study of the neutral molecules YXY is by photodetachment spectroscopy of the anion YXY-.34 Acknowledgment. We thank Dr. I. Last (Rehovot) for sending

us a copy of his Fortran code for the DIM potentials and Dr. J. Manz (Munich) for helpful correspondence and for preprints. J.N.L.C. thanks NATO for a Senior Scientist Award and Professor R. A. Marcus for his hospitality at the A. A. Noyes Laboratory of Chemical Physics, California Institute of Technology, where part of this research was carried out. Support of this research by the National Science Foundation is also gratefully acknowledged. The numerical calculations were performed on the CDC 7600 computer at the University of Manchester Regional Computer Centre. Registry No. IMu, 79104-08-8; IHI, 12694-71-2; IDI, 391 17-75-4; BrMu, 12587-64-3; BrHBr, 11071-85-5; BrDBr, 391 17-76-5. (33) L. Andrews, Annu. Reu. Phys. Chem., 30, 79 (1979). (34) J. L. Beauchamp, private communication, 1983; R. R. Coderman and W. C. Lineberger, Annu. Reu. Phys. Chem., 30, 347 (1979).

Spillover of Deuterium on Pt/Ti02. 1. Dependence on Temperature, Pressure, and Exposure D. D. Beck and J. M. White* Department of Chemistry, University of Texas, Austin, Texas 78712 (Received: February 21, 1984) The adsorption of D2on Pt/Ti02 powders has been studied by thermal desorption spectroscopy in an ultrahigh-vacuum system. In addition to chemisorption on Pt, two thermally activated states are observed and are attributed to spillover states on the oxide. Application of a surface diffusion model yields diffusion activation energies of 5.9 and 7.6 kcal/mol, respectively. One state is ascribed to spillover on a special oxide site, possibly TiO, associated with the Pt metal. The degree of spillover is dependent upon the condition of the catalyst surface with respect to a variety of factors including reduction and hydroxylation. Introduction The use of TiOz as a support in heterogeneous catalysts has been the object of much attention in recent years. Of particular interest are the photocatalytic properties of the Pt/TiO, (1) S. Sat0 and J. M. White, J . Phys. Chem., 85, 336 (1981).

0022-3654/84/2088-2764$01.50/0

and the strong metal-support interaction (SMSI) in reduced R/Ti02!’S In addition, a mmber of recent Papers have W g e s t d (2) A. V. Bulatov and M. L. Khidakel, Tzu. Akad. Nauk SSSR, Ser. Khim. 1902 (1976). (3) T. Kawai and T. Sakata, Nature (London), 282, 283 (1979).

0 1984 American Chemical Society