J . Phys. Chem. 1989, 93,7699-7702
7699
Spectral Grid Study of Ro-Vibrational Coupling in H,-Metal Scattering Bret Jackson Department of Chemistry, University of Massachusetts, Amherst, Massachusetts 01003 (Received: April 3, 1989)
A theoretical technique is presented that allows one to quantum mechanically describe the rotational and vibrational behavior of diatomic molecules that experience nonnegligible ro-vibrational coupling. Such situations arise when molecules have a strong electronic interaction with a metal, leading to variations in the bond length, or dissociative adsorption. The approach uses the spectral-grid method, in which the molecular wave function is represented by its values on a fixed grid of points in space. This wave function is evolved in time, scattered from the surface, and used to compute all properties of interest. The ro-vibrational part of the wave function is treated exactly, avoiding the usual rigid-rotor approximation. The rotationally HD, and D2from a metal surface is examined. It is found that small variations in the molecular inelastic diffraction of H2, bond length near the surface can lead to significant variations in the rotationally inelastic scattering probabilities. This is important since many metals interact chemically with H2,and their scattering probabilities are often used to extract information about the gas-surface potential. Applications of the method to other problems are discussed.
I. Introduction Time-dependent quantum mechanical scattering theories have proven to be extremely useful in the study of gas-surface interactions.’-* In these methods, the wave function representing the gas molecule is written in an explicitly time-dependent fashion. It is then evolved via the time-dependent Schrijdinger equation and scattered from or reacted with the surface. All properties of interest can be computed from the wave function at any time. The center of mass translational degrees of freedom of the molecule are usually represented by a wave packet. This can be done in several ways, the most powerful of which is to use the In this approach, the wave function spectral-grid is represented by its values a t a fixed set of points on a threedimensional grid. The grid usually covers one surface unit cell and extends several angstroms from the surface into the asymptotic region. This formulation is exact for localized wave functions if the grid is sufficiently dense. A fast Fourier transform (FFT) algorithm is used to efficiently propagate in time the amplitude at each grid point. More information on the method can be found in section I1 and the literature.1”q+’2 These methods have been successfully used to study such processes as rotationally inelastic selective ads~rption,~ and the dissociative adsorption of diatomics on metal^.^ Time-dependent methods are very intuitive, as well as being computationally efficient. For example, an exact treatment of the rotationally inelastic diffraction of N2 from a corrugated surface was recently published that included 13 923 channels.2 Such a calculation would be intractable via any other method. For the most part, the FFT/grid method is used to treat molecular translation only. Internal molecular motion can be included (1) (a) Mowrey, R. C.; Kouri, D. J. Chem. Phys. Lett. 1985,119,285. (b) Mowrey, R.C.; Kouri, D. J. J . Chem. Phys. 1986,84,6466. (2) Mowrey, R.C.; Bowen, H. F.; Kouri, D. J. J. Chem. Phys. 1987,86, 2441. (3) Mowrey, R. C.; Kouri, D. J. J . Chem. Phys. 1987,86, 6140. (4) (a) Chiang, C.-M.; Jackson, B. J . Chem. Phys. 1987,87, 5497. (b) Jackson, B.; Metiu, H. 1. J. Chem. Phys. 1987, 86, 1026. ( 5 ) (a) Jackson, B. J . Chem. Phys. 1988, 88, 1383. (b) Jackson, B. J . Chem. Phys. 1988,89,2473. (c) Jackson, B. J. Chem. Phys. 1989,90, 140. (6) (a) Kosloff, R.; Cerjan, C. J . Chem. Phys. 1984,81, 3722. (b) Yinnon, A. T.; Kosloff, R.;Gerber, R. B. Surf. Sci. 1984, 148, 148. (7) (a) Jackson, B.; Metiu, H. I. J. Chem. Phys. 1986, 84, 3535. (b) Jackson, B.; Metiu, H. I. J. Chem. Phys. 1986, 85, 4129. (8) (a) Drolshagen, G.;Heller, E. J . Chem. Phys. 1983, 79, 2072. (b) Drolshagen, G.; Heller, E. Surf. Sci. 1984, 139, 260. (9) (a) Feit, M. D.; Fleck, J. A., Jr.; Steiger, A. J. Comput. Phys. 1982, 47,412. (b) Feit, M. D.; Fleck, J. A., Jr. J . Chem. Phys. 1983, 78, 301. (c) Feit, M. D.; Fleck, J. A., Jr. J . Chem. Phys. 1984, 80, 2578. (IO) (a) Kosloff, R.;Kosloff, D. J . Chem. Phys. 1983, 79, 1823. (b) Kosloff, D.; Kosloff, R. J . Comput. Phys. 1983, 52, 35. ( I 1 ) Tal-Ezer, H.; Kosloff, R . J. Chem. Phys. 1984, 81, 3967. (12) (a) Kosloff, R. J . Phys. Chem. 1088, 92, 2087. (b) Bisseling, R.; Kosloff, R. J . Comput. Phys. 1985, 59, 136. (c) Sun, Y.; Mowrey, R. C.; Kouri, D. J. J. Chem. Phys. 1987, 87, 339.
0022-365418912093-7699$01 SO10
via a close coupling approach, with use of the usual expansions in rotational and vibrational eigenstates. One exception is a study of the dissociative adsorption of H2on Ni, where the molecule’s vibrational coordinate was represented on a grid, along with the translational motion perpendicular to the s ~ r f a c e . ~ Over the past few years, we have been interested in the dynamics of sticking and the dissociative adsorption of diatomics on metal surfaces. Whenever such trapping occurs, the molecule can experience severely hindered rotational motion. More importantly, molecules on or near the surface can experience a chemical interaction with the metal. That is, a redistribution of electron density can lead to a weakening of the molecular bond and an increase in the bond length. In cases such as these, the usual rigid-rotor approximation becomes invalid. One cannot ignore the vibrational coordinate, and it is important to properly couple rotational and vibrational motion. For these systems, the usual expansion in spherical harmonics is not the best approach. Also, an expansion in vibrational eigenstates is inadequate when the bond distortion is large or when dissociation can occur. In this paper, we explore the possibility of representing rovibrational motion on a three-dimensional Cartesian grid, using the FFT method. In Cartesian space, the ro-vibrational wave function is essentially a spherical shell and can be localized within a cube that is a few angstroms on each side. Such a representation makes no assumptions about ro-vibrational coupling or the functional forms of the eigenstates and is exact. Grid representations have been explored in the spherical coordinate system (r,0,+),but as yet, no fast Fourier algorithms have been developed for the 0 variable.12 In this paper, we wish to outline these ideas and discuss both their potential and their limitations. As an example, in sections I1 and 111, we present a study of the rotationally inelastic scattering of H2, HD, and D2from a metal surface. We allow for a weak chemical interaction with the surface and examine how this effects the rotational-transition probabilities. This study also applies to molecules scattered from metals where channels for dissociative adsorption exist but where the dissociative sticking probability is less than 1 at the given beam energy. The stability of the method for this problem, the computer requirements, propagation techniques, grid sizes, etc., are discussed. In section IV, further applications of the method are examined, particularly with regard to dissociative adsorption. Ways of improving the theory and of treating heavier molecules are proposed. 11. Method In this section, we develop a theory for the rotationally inelastic scattering of H2and its isotopes from a metal surface. This is not meant to be an exact or exhaustive study by any means. We wish, first of all, to illustrate some basic ideas behind the Cartesian grid method. Secondly, we wish to make an important point about studies of rotatioally inelastic scattering, which are often used to 0 1989 American Chemical Society
7700 The Journal of Physical Chemistry, Vol. 93, No. 22, 1989 extract information about the molecule surface potential. A rigid-rotor close-coupled calculation is usually used to relate the observed scattering probability to some model p0tentia1.I~ We will demonstrate how these probabilities can vary significantly when gas-surface chemistry, leading to small changes in the bond length, occurs. We will attempt to uncover trends in the transition probabilities that suggest when this is happening. Our desire is that this study will eventually lead to better experimentally derived gas-surface potentials. It is worth noting that such studies, combined with low energy beam scattering from chemically active surfaces should be capable of providing information on the bamers that exist to dissociation. We begin with the usual Hamiltonian for the problem H=--
ti2
2ml
Vq2 -
h2
G v r ; + Vs(r1) + Vs(r2) + VI(b1 - r 2 0 (1)
where m l and m2 are the nuclear masses of the diatomic. V,(ri) is the interaction of nucleus i (with coordinate r, = x,, y,, z,) with a (100) surface and is written
The variable 0 is set equal to either 1 or 0, depending upon whether or not we choose to include attractive forces. The internuclear potential is also of the Morse form VI(r) = ~ ( e - 2 d ~ r-0 2e-d~r0') ') I (3) where r = Ir, - r2). We then make the usual transformation into a center of mass (COM) coordinate R and a relative coordinate r, both of which we express in Cartesian variables. The resulting total potential VT can be written 12
VT = Vs(ri) + VArd + Vdr) = Cfi(R) gj(r) + Vdlrl) J=l
(4)
where4 and gj are easily derived. In this initial study, we will treat the COM translational motion classically, although this can be avoided with a bit of extra work. However, earlier semiclassical studies7ahave shown that this approximation can still lead to good results for rotationally inelastic scattering, and our approach is adequate for what we wish to accomplish here. Our equations of motion for the COM coordinates are thus written
where R, and P,are the COM position and momentum vectors at time t and M is the total molecular mass. The above brackets denote an average over the ro-vibrational wave function $(r), at time t . The resulting Hamiltonian describing the evolution of $ is then
where is the reduced mass. The time dependence in H presents no problems since we are using a time-dependent method. We introduce the chemical interaction into the problem by allowing the equilibrium bond length r,,' to vary with molecular position and orientation. We have performed three sets of calculations. The first corresponds to setting r,,' = ro, where ro is the equilibrium bond length in the gas phase. In the second calculation, we write ro' = ro + be-(Zi-zo)/u (7)
Jackson where Z , is the COM distance above the surface. In the third calculation, we use ro' = To + &+5-Zo)/~ sin2 8 (8) where 0 is the polar orientation angle of the molecule. Thus, sin2 0 = (x2 + y 2 ) / r 2 ,where r = (x,y,z). This form reflects the observation that molecule-metal chemistry is favored when the molecular axis is parallel to the surface, allowing for the introduction of electron density from the metal into the antibonding orbital of the m01ecule.l~ The wave function $(r,t) is represented on a cubic grid that is 3.0 A and 32 grid points across on each side. A second-order difference propagation technique, which has performed well in a variety of other situation^,^^^^ was found to be unstable for this problem, except for extremely small integration time steps. We thus use the following scheme due to Feit and Fleck,g which is both stable and efficient. We wish to numerically integrate $(r,O) for N time steps At, to a final time t = N A f . That is $(r,t) = (e-(z/h)Haf)N$(r,O) (9) One can show that e-(1/hW&$(r,T) = , - ( , / z h ) K A , , - ( i / h ) v ~ - ( i / 2 h ) K A f ~ r , ~ ) + O[(At)3] (10) where the errors result from the fact that K and V, the kinetic and potential energy parts of H,,do not commute. After N applications of this short time propagator, the global error is negligible if At is small enough. Note that at all but the first and last integration steps the kinetic energy propagators can be combined and applied for a full time step At. The effect of the V part of the propagator on the amplitude at each grid point rJ is evaluated by a simple multiplication: e-(,/h) vA,$( rJ,7) = e-(,/h)vCrJu$( TI,7) (11) The kinetic part can be evaluated by first computing $(p,t), the Fourier transform of +(r,r). The effect is then also a simple multiplication at each momentum space grid point pJ: e - ( t / h ) K A f $ ( p 7) = e-(r/2#h)~?At$@~,~) (12) J'
An inverse Fourier transform then returns the $ to r space, where the potential part of the propagator is again applied and so on. Due to the high efficiency of the FFT algorithm, the full evolution requires only a few hours of computer time on a Celerity Model 1260D minieomputer. More details can be found in the literat~re.~,~ At t = 0, the molecule is several angstroms above the surface and allJ;(Ro) = 0. We start with the molecule in the ground rotational and vibrational states and thus write 1 $(r,O) = ;+o(r) 9 ( 4 4 ) (13) where
= (4a)-'I2and +o is the Morse oscillator ground state
a0(r) = [I'(2d-l)]-1/2a1/2 e~p[-de-~(~'o)] [2de-a(Fr~)]d-1/2 (14) where d = ( 2 p D ) 1 / 2 / a hand , I' is the gamma function. $(r,O) is evolved for approximately 1000 time steps, where At = 0.25 X s, until the wave function is again several angstroms above the surface. It is then projected on various free-space rotational-vibrational eigenstates to compute the inelastic scattering probabilities. For surfaces with corrugation, we run several trajectories to average over all possible surface impact sites. We thus choose several initial points &,Yo)within the surface unit cell. 111. Results
For H2 and its isotopes, V, is the same for both H and D. zi0 is arbitrary and we set zl0 = z20 = 0. We choose the same a,,
(13) (a) Whaiey, K. B.; Yu, C.; Hogg, C. S.; Light, J. C.; Sibener, S. J. f. Chem. Phys. 1985, 83, 4235. (b) Yu, C.; Whaley, K. B.; H o g , C. S.; Sibener, S. J. J. Chem. Phys. 1985,83,4217. (c) Liebsch, A,; Hams, J. SurJ
Sci. 1983, 130, L349.
(14) (a) Nerrskov, J. K.; Houm0ller, A.; Johansson, P. K.; Lundqvist, B. I. Phys. Rev. Lett. 1981, 46, 257. (b) Minot, C.; Sevin, A.; Leforestier, C.; Salem, L. f. Phys. Chem. 1988, 92, 904.
Spectral Grid Study of Ro-Vibrational Coupling in H2-Metal TABLE I: Revibrational Scattering Probabilities, Pa,,,,, for H2 Scattered from a Flat Surface, at Normal Incidence, with a Beam Enerev of 100 meV' model 2 model 3 H2 model 1 ~ooo Po20 ~040*
PlOO*
P120*
PMOS rmax/A
0.846 0.153 0.908(-3) 0.643(-7) 0.592(-4) 0.382(-5) 0.741
0.745 0.245 0.949(-2) 0.136(-4) OMS(-4) 0.408(-4) 0.821
0.772 0.225 0.324(-2) 0.879(-5) 0.735(-4) 0.103(-4) 0.795
" P,,,,,, corresponds to the nth vibrational state and the rotational state with quantum numbers I and m . The notation a(-N) stands for o X and the asterisks denote forbidden transitions.
TABLE 11: Ro-Vibrational Scattering Probabilities, P,,,, for HLY HD ~ooo POI0 Po20 p040* POSO* ~060*
PlOO* PlIO* p120* p130* Pl40*
rmax/A
model 1 0.707 0.181 0.370(-1) 0.468(-I) 0.231 (-1) 0.449(-2) 0.266(-3) 0.296(-6) 0.625(-5) 0.992(-5) 0.485(-4) 0.653(-4) 0.741
model 2 0.697 0.143 0.399(-1) 0.651(-1) 0.316(-1) 0.165(-1) 0.514(-2) 0.63 I(-4) 0.286(-4) 0.23 I(-4) 0.104(-3) 0.115(-3) 0.822
model 3 0.711 0.152 0.371(-1) 0.555(-1) 0.329(-1) 0.104(-1) 0.1 16(-2) 0.354(-4) 0.809(-5) 0.228(-4) 0.754(-4) 0.122(-3) 0.796
"Same conditions as Table I.
TABLE 111: Ro-Vibrational Scattering Probabilities, Pa,,,,,for D2" D2
pooo Po20 Po40
PMO* PlGQ* p120*
PIN*
rmx/A
model 1 0.750 0.228 0.219(-1) 0.662(-4) 0.103(-6) 0.327(-4) 0.335(-4) 0.741
model 2 0.679 0.239 0.801(-1) 0.220(-2) 0.432(-4) 0.816(-4) 0.171(-3) 0.821
model 3 0.669 0.283 0.475(-1) 0.375(-3) 0.176(-4) 0.687(-4) 0.102(-3) 0.795
'Same conditions as Table I.
D,,c,, and cy as were used in some previous exact and semiclassical studies of this p r ~ b l e m . ~Thus, ~ , ~ ~D1 = D2 = 20 eV, al = a2 = 1.833 15 A-', c, = 4.0 A, and cy = 2.0 A. We also use the values for VI that reproduce the gas-phase spectrosco ic data for H2: D = 4.745 eV, a = 1.943 ,&-I, and ro = 0.741 . All trajectories are for normal incidence and a beam kinetic energy of 0.1 eV. We will consider both flat (& = p2 = 0) and corrugated surfaces. As in the earlier studies, the attractive part of the potential has been dropped (6 = 0). The flat surface results for H2, HD, and D2 are presented in Tables 1-111, respectively. P,Im is the probability of scattering into the nth vibrational state and the rotational state described by YIm(6,4), where n = 1 = m = 0 initially. The notation a(-N) represents a X Model I corresponds to r,,' = ro, and models 2 and 3 correspond to eq 7 and 8, respective&. We choose 6 = 0.1 A, v = 0.5 A, and Zo= 1.6 A. As the molecule approaches the surface, its average bond length ro increases to a maximum value of r,,, at the turning point. The molecular wave function a t this part of the trajectory is nonspherical (flattened in the z direction), and rotational motion is hindered. The maximum variation of r,' is between 0.054 and 0.081 A for the systems studied. The foruse similarare in agreement with Other calculations that potentials and beam energies.15q16 The
w
(15) (a) Lill, J. V.;Kouri, D. J. Chem. Phys. Let?. 1984, 112, 249. (b) Proctor, T. R.;Kouri, D. J.; Gerber, R. B . J. Chem. Phys. 1984, 80, 3845. (c) Whaley, K. B.; Light, J. C. J . Chem. Phys. 1984, 81, 3334.
The Journal of Physical Chemistry, Vol, 93, No. 22, 1989 7701 TABLE IV: Ro-Vibrational Scattering Probabilities, P,,,,, for ,a Corrugated Surface"
H2 ~ooo Po20
Po21 Po22 Pw*
model 1 0.846 0.152 0.210(-3) 0.283(-3) 0.899(-3)
'Same conditions as Table I.
model 2 0.744 0.244 0.320(-3) 0.453(-3) 0.940(-2)
model 3 0.771 0.224 0.304(-3) 0.451(-3) 0.323(-2)
= & = 0.05.
results for models 2 and 3 clearly show that rotationally inelastic transition probabilities increase when ro becomes larger near the surface. This is not surprising, since the larger bond length increases the molecule's moment of inertia, decreasing the rotational energy spacings and thus the transition energies. For H2, the P020probability increases by almost 50% when r,' increases by only 0.05 A. For D2, PW increases by more than a factor of 2 for model 3 and almost a factor of 4 for model 2. The effect on the allowed transition probabilities for HD is noticeable but not as large as for H2 and D2. Models 2 and 3 show the same trends, except that the effects are larger for model 2 because r,, is larger for a given 6 (the average of sin2 6 is less than 1). We thus performed calculations using model 2 with 6 = 0.0675 A, such that rm = 0.795, the model 3 value. The resulting transition probabilities were similar to the model 3 values listed in the tables, showing small deviations with no clear pattern. Thus, the sin2 6 term serves primarily to lower r,, slightly (for a given 6), and changes in the rotational energy spacings have a larger effect on PnImthan molecular orientation terms in VI. One can conclude that small variations in the equilibrium bond length can lead to large changes in the computed rotationaltransition probabilities. If such variations occur and are ignored, the potential surface fit to the experimental data may contain significant errors. It is thus important to be able to predict when this might be happening. One clue is in the forbidden-transition probabilities (marked with asterisks in the tables), which change dramatically when r,' is modified. This is expected, since these transitions are very sensitive to the difference between the available energy and the transition energy, which changes with r,'. For H2, P, increases by a factor of 10, Po@ increases by a factor of 20 for HD, and Poaoincreases by a factor of 33 for D2 (all model 2). Since the chemical interaction with the surface directly modifies the molecule's vibrational amplitude, the probabilities corresponding to vibrational excitation show the strongest increase. For both models this increase can be over 2 orders of magnitude for the An = 1, A1 = 0 transition but is much smaller when AI # 0. These large forbidden-transition probabilities are thus indicative of bond length variation during scattering. It would not be easy to reproduce these large values with a model potential that assumed r,' to be constant. Unfortunately, the transition probabilities that show the largest change are the most difficult to detect. However, it should be possible to resolve some of the An = 0 forbidden transitions, which show large effects. In Table IV, we present results for H2 scattered from a corrugated surface, where PI = p2 = 0.05. For AI = 2, the Am # 0 transitions demonstrate the same behavior as the Am = 0 transitions. Thus, no new information is gained.
Iv. Further Applications Clearly, one could take a close-coupled approach to the above problem, although many ro-vibrational basis functions would need to be included. Our intent is to apply this method to problems where the usual basis set expansion is out of the question. Various models for dissociative adsorption are being developed from these For example, an approach is being examined where the "strong" variables (r, the bond length; Z , the COM distance above the surface; and 6, the polar orientation angle of the molecule) are fully coupled and represented on a three-dimensional grid. (16) (a) Richard, A. M.; DePristo, A. E. Surf.Sci. 1983, 134, 338. (b) DePristo, A. E. Surf.Sci. 1984, 137, 130.
1702
The Journal of Physical Chemistry, Vol. 93, No. 22, 1989
L
Jackson
............................. . . . . ......... .. .. . . . ......... .. . . . .*. .. .. . . . . 1.0 . . . . . . . . . . . . . ... .. .. .. .. . . . . . . . . . . . v,tA> . . . .
...F:.-: : /I
~~~~
0
-1.0
I
.
-1.0
.: . .
.
x (A,
The remaining Yweaknvariables are on a second grid or represented classically. Ways of including correlation between the two sets of variables are being examined. The primary problem of the Cartesian grid method, as applied to ro-vibrational problems, is one of wasted space. For I = 0, the wave function is localized about a spherical shell. For H2 and its isotopes, this shell is thick and fills the grid fairly evenly. For heavier molecules like N2, however, the shell is very narrow and has a longer radius. Thus, most of the grid volume is unused. In fact, in order to get a sufficient number of grid points into this region of large 1+12, we must increase the number of points along each axis to 64 or even 128. While this is still computationally tractable, it is certainly not desirable. A possible solution is to define a coordinate transformation that takes a regular rectangular array of points and moves them where they are needed, Le., where the wave function is largest. Consider NZ, where V ( r ) can be approximated by the values D = 9.759 eV, CY = 2.7 k-l, and ro = 1.098 A. In Figure 1, we show the z = 0 cross section of our Cartesian coordinate system. For reference, three circles are plotted that correspond to r = 0.9, 1.098, and 1.3 A. Most of the probability density, which peaks at the middle circle, is localized between the inner and outer circles. One usually defines the grid points to be located at x,,’ = xmin nAx, where Ax = (x xmi,)/N and n = 1, 2, .... N. We choose x, = xmin= 1.6 for this example and use N = 32 as before. The y’ coordinate is treated in an equivalent fashion. Using this rectangular grid in Figure 1 would result in a very low density of points between the circles and many wasted points. As an example, consider the following transformation (rf to r)
+
+ [0.23 + y ( r f - r l ) ] X
‘
[+ 1
0.35 cos ( 4 tan-l
0
1.0
X.CA>
Figure 1. The circles correspond to radii of 0.9, 1.098, and 1.3 A in this our ro-vibrationalcoordinate system. Most of the wave function is localized between the inner and outer circles. The plotted points (x,,,~,,,)correspond to the square grid of points ( x k , y,,,’) defined in the text and transformed according to eq 15.
1 2r’ r’+ E
:)I u:
-1.0
z = 0 cross section of
r = -irl
....,
.. .. .. .. .. .. .. .. .................... .... : ::: .. .. .. .. .. .. ......... . . . .. ......... . . . ............. .. . . .
1
1.0
0
... ...
(:) I(.-
r l ) / (15)
where r ’ = Ir‘l, r , = 0.9 A, t = 0.035 A, and y = 0.046 A-l. In Figure 1, we plot the points (x,y,) which correspond to the rectangular grid points (x,,’,~,,,’) defined above, using the transformation of eq 15. We now have a sufficiently high density and few wasted points. However, the FFT algorithm can only be applied to a rectangular array of points. We thus perform an inverse transform to the primed coordinate system shown in Figure 2. The points are now regularly spaced and the three circles of Figure 1 become deformed. The result is that both the potential VIand the wave function are “stretched out” to fill the grid, and few points are wasted. The idea, then, is to transform the initial
Figure 2. Same as Figure 1, except that the points and circles in the x-y plane have been transformed to the x’-y’coordinate system via eq 15.
system to a primed space, evolve it in time with use of the usual FFT rectangular grid techniques, and then back-transform the final state to the original space for examination. One of course needs to know the Hamiltonian in the primed space. A smooth and continuous transformation (like eq 15) can in general be concocted, and computing V ( f )on a grid is straightforward. Using standard metric techniques,” one can transform the Laplacian to the new coordinate system in the following way:
Thus, the Hamiltonian becomes more complex, and the Feit method9 cannot be used. It is still possible to use the second-order difference algorithm,I0 although seven F W s are required per step instead of two. The resulting increase in the number of FFT’s, however, is overshadowed by the advantage of lowering N by a factor of 2 or more in each dimension. The Chebychev method of propagation,” which is highly stable and efficient, can also be used with the transformed Hamiltonian and would probably be the best choice for this problem. We have yet to fully explore these ideas, but they may eventually prove useful, since many systems studied by the FFT method have difficulties with wasted space. In summary, we have shown that a Cartesian-grid/FFT representation of ro-vibrational motion works and may eventually find a useful application in studies of strong gas-surface interactions. The development of grid transformation and propagation techniques may make this approach more attractive in the future. We have also demonstrated the importance of considering chemical molecule-surface interactions when studying nonreactive rotationally inelastic scattering. Small variations in the bond length that result from modifications of the molecular electronic structure can lead to significant variations in the rotationally inelastic scattering probabilities, particularly for transitions that are energetically forbidden or nearly forbidden. Acknowledgment. This work was supported in part by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy, under Grant No. DE-FG02-87ER13744. I also thank the National Science Foundation for an equipment grant used to upgrade the departmental (Chemistry) computer facility on which this work was performed. Registry No. H2,1333-74-0; D2,7782-39-0; HD, 13983-20-5. (17) See, for example: Arfkin, G. Mathematical Methods for Physicists, 2nd ed.;Academic Press: New York, 1970; p 76.
