J . Phys. Chem. 1988, 92, 587-593
587
Vibrational and Rotational Wave Functions for the Triatomic van der Waals Molecules HeCI,, NeCI,, and ArCI, B. P. Reid,+ K. C. Janda,* Chemistry Department, University of Pittsburgh, Pittsburgh, Pennsylvania 1.5260
and N. Halberstadt Laboratoire de Photophysique MolPculaire, UniversitP de Paris Sud, 91 405 Orsay, France (Received: May 22, 1987; In Final Form: August 3, 1987)
Using realistic model intermolecular potentials for HeCI2, NeC12, and ArCI2, we have calculated vibrational wave functions for the modes associated with the van der Waals degrees of freedom. Pictorial representations of the vibrational wave functions show the excitations of the “stretching” and “bending” motions of the weak van der Waals bond. The spacings of the energy levels corresponding to the van der Waals stretches are used to verify the accuracy of models which assume a one-dimensional, Morse-like behavior for this coordinate. The calculated energy levels for different total angular momentum states with J I 3 are compared with rigid-rotor models, with and without centrifugal distortion effects. These calculations form a basis for understanding the structure and spectroscopy of these T-shaped van der Waals molecules.
Introduction Experimental studies have provided detailed information on the structures and vibrational predissociations of the rare gas-halogen van der Waals molecules.’V2 Electronic excitation spectra show rotational structure which is well described by rigid-rotor models, and hence yield effective molecular geometries. Studies of the dissociation of these molecules from metastable states with initial excitation in the diatomic halogen vibration show that the dissociation lifetimes depend strongly upon the vibrational excitation and vary widely for different molecules. Detailed theoretical treatmepts of the dissociation require accurate potential energy surfaces. The purpose of this paper is to explore the connection between experimentally observable rotational and van der Waals vibrational energy levels and the potential energy surface. We have constructed realistic, model intermolecular potential energy surfaces for HeCl,, NeCl,, and ArC1, which account for the available data. Using these potentials, we have calculated vibrational and rotational wave functions. The vibrational wave functions show the two modes associated with the van der Waals degrees of freedom. One is a “stretching” motion corresponding to a change in the distance between the rare gas atom and the center of mass of the halogen molecule. The other is a bend or librational motion which corresponds to the molecule deviating from its equilibrium T shape. The energy level spacings and wave functions corresponding to the stretching and bending modes of the van der Waals coordinates show the extent to which these two motions are separable. The spacing of the energy levels for the van der Waals stretch can be used to verify the accuracy of models which assume a one-dimensional, Morse-like behavior for this ~ o o r d i n a t e . ~The . ~ calculation of energy levels for different total angular momentum states allows comparison with rigid-rotor models. We compare the calculated rotational energy levels with a simple rigid-rotor model and with a model including centrifugal distortion effects. An important aspect of these loosely bound molecules is the relationship between parameters derived from experimental measurements and those derived from the potential energy function. A particular example of apparent inconsistency between different experimental results exists for the NeC12 system. Tshaped rigid rotor fits to excitation spectra of the NeC12 molecule yield a Ne to C12 center of mass distance of 3.565 f 0.015 8, in the ground electronic state.5 On the other hand, rotationally inelastic scattering data analyzed by using the infinite-order sudden approximation yield an intermolecular potential with the minimum Present address: Chemistry Department, Marietta College, Marietta, OH 45750.
0022-3654/88/2092-0587$01.50/0
in the T-shape configuration at 4.05 f 0.15 A.6 In this work we find that the rotational energy levels calculated for a potential based on the scattering results are fit by a rigid-rotor model with an effective bond length of 4.06 A, which suggests that the spectroscopic results should provide a good approximation to the potential minimum. This paper is divided into several sections. We begin by describing the model potential energy functions used, and then we briefly detail the method used to calculate the wave functions. The results which follow include pictorial representations of the vibrational wave functions, diagrams of the vibrational energy levels, and comparisons with rigid-rotor models. An overall summary concludes the paper.
Potential Energy Function The potential energy function describing the interaction between a rare gas atom and a chlorine molecule can be defined as a function of three internal variables: R,the length of the vector from the center of mass of the chlorine to the rare gas; r, the length of the vector _between the two chlorine atoms; and 7,the angle between the R and r‘vectors. We are interested in the van der Waals motions and, as discussed below, we neglect the r dependence of the rare gas-chlorine interaction. To model the potential as a function of only R and y,we use a parameter expansion form.’ This type of potential is constructed by choosing a parametrized , expanding the parameters radial potential form, V ( R ; ( A ) )and as functions of y. ( A ) is the set of parameters of the radial potential. The expansion of these parameters is described in eq 1, in which the parameter Ai is expanded in the Legendre polyA , = Cai,J’x(cos A
Y)
(1)
nomials PA(c0sy) using the expansion coefficients ai,. Only even Legendre polynomials appear in the expansion of a homonuclear diatomic molecule such as T 1 2 . With this type of potential, only (1) Janda, K. C. Adv. Chem. Phys. 1985, 60, 201. (2) Cline, J. I.; Evard, D. D.; Reid, B. P.; Sivakumar, N.; Thommen, F.; Janda, K. C. Structure and Dynamics of Weakly Bound Complexes, Weber, A., Ed.;Reidel: Dordrecht, 1987; pp 533-551. (3) Kenny, J. E.; Johnson, K. E.; Sharfin, W.; Levy, D. H. J . Chem. Phys. 1980, 7 2 , 1109. (4) Drobits, J. C.; Skene, J. M.; Lester, M. I. J . Chem. Phys. 1986, 84, 2896. Drobits, J. C.; Lester, M. I. J . Chem. Phys. 1987, 86, 1662. (5) Evard, D. D.; Thommen, F.; Janda, K. C. J . Chem. Phys. 1986, 84, 3630. (6) Reid, B. P. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1986. (7) Pack, R. T Chem. Phys. Letr. 1978, 55, 197.
0 1988 American Chemical Society
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TABLE I: Potential Parameters parameter
HeC12 30 -1 5 4.26 0.8 -0.16 7.0 133000 18900 1090000 3 15000
4
NeC12
ArC12
59 -29 4.52 0.8 -0.17 7.0 277000 43800 2090000 1050000 4
162 -90 4.46 0.8 -0.16 7.0 923000 140000 9050000 3620000 4
a few terms in the angular expansions are needed to produce very anisotropic potentials. The radial potential chosen is a Morse-van der Waals form. A Morse function with a reduced shape parameter, p, describes the short- and medium-range interactions: vM(R) = te-bx[e-@x- 21 (2) t is the depth of the well at the minimum R, and X = (R R,)/R,. At long range the potential assumes the van der Waals expansion form: c6
c8
Vw(R) = -- - R6 R8 The complete form is given by if R IRi V(R) = VM
= Vw
+ [VM - Vw] exp(-6[(Ri - R ) / R , ] ~ )
(3)
if R IRi (4)
where Ri = R , (1 + In 2 / p ) is the inflection point of the Morse function, V,. For R larger than Ri, the function gradually switches from a Morse function to the van der Waals form. The parameter 6 controls how fast this change occurs. This isotropic form provides a realistic well and repulsive wall and a proper long-range de'pendence using relatively few parameters. Values of the parameters for the He-, Ne-, and Ar-C12 systems were chosen by considering information from several sources. Scattering experiments include total differential cross sections for HeC12 and NeC128and inelastic time-of-flight data for NeC12.6 These data provide information about the potential wells, the lower part of the repulsive walls, and the potential anisotropy. Electronic excitation spectra provide effective structural parameters for each of these van der Waals molecule^^-^*^^ and product state distributions of vibrational predissociations provide dissociation energy estimates for NeC12 and ArC12.'OJ1 The well depths, eo, for NeC12 and ArC12were chosen so that the calculated dissociation energies equalled those found spectroscopically. In the case of NeC12 this also closely matched that found by scattering experiments. The scattering results were used to estimate eo for HeC12 and t2 for NeC12. The t2 values for HeC12 and ArC12 were selected so that the anisotropy ratios, e2/tO, were similar to that for NeC12. The Rm,O values for HeC12 and NeC12were taken from the scattering fits, while the value for ArC12 was taken as the value obtained by spectroscopic studies using a rigid-rotor model. The values for all three systems were taken as those deof Rm,2and termined by the inelastic NeC1, scattering results. Also for all three systems, the value for @ was chosen as 7.0, a reasonable value for weakly bound systems. The van der Waals coefficients, given in the direct Legendre expansion form, were calculated by using simple formulas8with input coming from the polarizabilities and ionization,potentials appropriate for the ground states of C12 and the rare ga~es.'~-'~ The value for 6 was selected so that the change (8) O'Loughlin, M. J. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1986. (9) Cline, J. I.; Evard, D. D.; Thommen, F.; Janda, K. C. J. Chem. Phys. 1986,84, 1165. (10) Evard, D. D., Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1987. (1 1) Cline, J. I.; Sivakumar, N.; Evard, D. D.; Janda, K. C. J. Chem. Phys. 1987,86, 1636. (12) Miller, T. E.; Bederson, B. Adu. A t . Mol. Phys. 1977, 13, 1.
-6 -4 -2 0
2
4
Chlorine Axis
(A)
6
Figure 1. (a) A three-dimensional plot of the NeC12 potential as a function of R and y. (b) The corresponding plot of equipotential contours. Along the dashed contour the potential is zero. The contours at large values of R (thin lines) represent -0.1, -0.3, -0.5, -0.8, and -0.95 times ~ ( 7 = ? ~ / = 2 )73.5 cm-I, while those at smaller values of R (thick lines) represent 5 and 10 times t(y=7r/2). The dots at -1 and 1 A along the chlorine axis represent the locations of the chlorine nuclei.
from the Morse to van der Waals function is smooth but not excessively slow. The parameters used are listed for the He-, Ne-, and Ar-C12 systems in Table I. Higher order parameters were not estimated and were set to zero. Although much of the experimental work involving rare gas-halogen complexes has involved both ground and excited electronic states, here we consider only the ground state, where the neglect of the dependence of Yupon r should be most appropriate. A three-dimensional plot and an equipotential contour plot for the NeC12 system are shown in Figure 1. Contour plots for the other systems appear very similar especially when the contours are plotted at the same fractions of the potential minimum, e(7=7r/2). These potentials contain much of the experimental information known about these systems and should be reasonable representations of the true interactions. Although some parameters, such as Rm(7=7r/2)for NeC12, may be modified as further information is obtained, the qualitative results which follow should not be significantly altered. Calculation of Wave Functions and Energies This section outlines the calculation of the vibrational wave functions and energies. The method is similar to the one used previously in modeling the NeC12 predissociation.16 We are concerned with the vibrational levels of the van der Waals modes and hence the stretching of the C12 molecule has been neglected by fixing r at ro = 1.991 A, the effective bond length of the ground vibrational level of the ground electronic state (X) of C12? This represents a separation of the fast chlorine stretching motion (a,= 560 cm-', T = 0.06 ps) from the slower (13) Hirschfelder, J. 0.; Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. (14) Weast, R. C. Ed. Handbook of Chemistry and Physics; 59th ed.; CRC Press: West Palm Beach, FL, 1987. (15) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure: ZV. Constants of Diatomic Molecules; Van Nostrand Reinhold: New York, 1979. (16) Halberstadt, N.; Beswick, J. A.; Janda, K: C. J. Chem. Phys. 1987, 87, 3966. Note that the angles used in this reference, (f#q&,C#),O), correspond to the angles (b,d,x,~) used in the present work. (17) Coxon, J. A. J. Mol. Spectrosc. 1980, 82, 264.
The Journal of Physical Chemistry, Vol. 92, No. 3, 1988 589
Wave Functions for HeC12, NeC12, and ArC12 van der Waals motions (we 5 40 em-', 7 2 0.8 ps), and is equivalent to assuming the diatomic to be a rigid rotor. In this case, we can solve the time-independent Schrodinger equation, %$ = E$, with the Hamiltonian in body-fixed coordinates:'*
yy =-- h2a2
12
i2 ++ V(R97) 2mdR2 + % 2mR2
(5)
In this equation p is the reduced mass occhlorine, m is the reduced mass of the rare gas-chlorine system, j is the rotational acgular momentum which is associated with the rotation of 3, and 1 is the orbital angular momentum associated with_the+rota_tionof R. The total angular momentum is defined as J = j + 1. The overall solution for the system, in this approximation, is Q(r,R,8,$,~,7) = cp(r)(l/R)$(R,8,$,~,7), where q(r) is the gcound-state wave function for C12,8 and $ are the polar angles of R in a space-fixed reference frame, and x and-? are the polar angles of 3 in the molecule-fixed frame with R as the polar axis. This Schrodinger equation was solved by using the secular equation method1*which solves eq 5 by diagonalization of a matrix representation of the Hamiltonian. The set of basis functions used for the matrix representation was constructed by forming products of radial harmonic oscillator functions with the angular functions given byI9 qFw,e,7,x) =
[ 2J+
I"'
"-3Yjn(Y,X)
+
8 4 1 + hl) P(-1 )JG-,($,8,0) Yj-n(r,x)l ( 6 )
Figure 2. (a) Ground-state, J = 0, HeClz wave function, and (b) the probability density. The.energy of this state is -15.25 cm-I. The halfcircles along the chlorine axis locate the chlorine nuclei.
A4 is the quantum number ?associated with the projection of the total angular momentum J along-the space-fixed 2 axis, R is associated with the projection of J along the body-fixed z axis, p is either +1 or -1, am is a Kronecker delta, the dM, are Wigner functions, and qnare spherical harmonics. In this equation, 8 is restricted to nonnegative values. In the diagonalization of thy matrix, the parameters characterizing the harmonic oscillator set, a fundamental frequency, w , and an equilibrium length, Re,were variationally optimized. In order to reduce the dimensionality of the matrix needed for accurate description of the wave functions and energies, the Hamiltonian was factorized as follows. The angular basis functions are eigenfunctions of the total angular momentum J so that diagonalization of the Hamiltonian may be performed separately for each value of J . The eigenvalues are independent of M and only one value of M need be considered for each J . Two symmetries allow further factorization of the Hamiltonian matrix. First, the Hamiltonian is invariant with respect to inversion of the coordinates through the origin. The functions in the basis set described by eq 6 are eigenfunctions of the inversion operator with eigenvalues, p , of +1 or -1. There will be no coupling between functions with different values of p, and separate diagonalizations are performed for each value. Second, the Hamiltonian is invariant with respect to interchange of the identical chlorine atoms. This symmetry is manifested in the angular basis by the even or odd parity of j . The Hamiltonian will not couple functions with j even to those having j odd, so that separate diagonalizations are performed for the sets of even and odd functions. Thus, overall, for each value of J , four diagonalizations need to be done and the wave functions for each symmetry will be labeled as follows: Je+ for p = +1, j even; J,+ for p = +1, j odd; J, for p = -1, j even; and J , for p = -1, j odd. For the J = 0 calculations, 25 radial functions and angular basis functions with the lowest 22 even or odd values of j were used. The lowest vibrational levels for J = 0 appear to be converged to within about 0.005 em-' for all three systems. Higher vibrational levels, and especially those near the dissociation limit, are of course less well converged. The vibrational levels of the van der Waals modes were analyzed in detail only for J = 0. The main aim of the calculations for J > 0 was to compare the energy (1 8) See,for example: Le Roy, R. J.; Carley, S. J. A h . Chem. Phys. 1980, 42, 354. (19) Launay, J. M. J . Phys. B 1976, 9, 1823.
--6 - 4 -2
0
3
Chlorine Axis
4
6
(A)
Figure 3. (a) A three-dimensional plot of the first vibrationally excited state, E = -11.17 cm-', of HeCl,; (b) a diagram with contours of equal wave function amplitude. The dashed lines correspond to regions of negative amplitude. In this and all other contour plots of wave functions, the contours are at h0.2, h0.4, h0.6,and A0.8 times the maximum amplitude, unless otherwise noted.
levels for the ground vibrational state to rigid-rotor models. Since functions with $2 up to J must be included, the numbers of radial functions and spherical harmonics used for higher values of J were slightly smaller than those for J = 0. The resulting rotational energy levels are also estimated to be converged to within 0.005 cm-'. Wave Functions and Energy Levels for J = 0 The vibrational wave functions for levels with the total angular momentum equal to zero are relatively simple since no basis functions with 8 > 0 are needed. These wave functions can be plotted as functions of the variables R and y, using either three-dimensional or contour plotting techniques. Such plots show the extent of delocalization of the wave functions as well as the modes of the vibrationally excited states. Figures. 2-4 illustrate the J = 0 wave functions for HeC12. Figure 2a shows a three-dimensional representation of the ground-state vibrational wave function, (1/R)$.It is interesting to note that the binding energy of this state, 15 em-', is only 40% of the well depth, 37.5 em-': the zero-point energy is 60%of the
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-6 -4 -2 0
2
4
-6 -4 -2 0
2
4 6
6
I
\
m-7
-6 -4 -2 0
2
Chlorine Axis
4
-6 -4 -2
6
2
0
Chlorine Axis
(A)
Figbe 4. Contour plots for the bound states of HeC12: (a) two (E = -8.25 cm-’), (b) three (E = -5.46 cm-’), and (c) four (E = -2.56 cm-’) quanta of energy in the bending mode.
6
4
(A)
Figure 6. The first excited stretching state for NeC12 shown three dimensionally in (a) and by a contour plot in (b). The energy of this state is -30.05 cm-’.
-5
-3
-1
3
1
Chlorine Axis
5
(A)
Figure 7. Contour plot displaying the two radial nodes associated with the second excited stretching state of NeC12 (E = -11.10 cm-’). In addition to the contour levels described in the caption of Figure 3, additional contours at fO.l times the maximum amplitude are shown.
Figure 5. NeC12 ground-state wave function showing more localization than the analogous HeC12 function and having the energy E -55.40 cm-’ . f
well depth. This is accompanied by an extensive delocalization of the He about the chlorine. The wave furiction has maximum intensity at y = ~ / 2 but , there is observable amplitude even at y = 0. A somewhat more physical description of the delocalization is given by Figure 2b which is a plot of the probability density, (1/R)fi2 sin y, for finding the system in a given (I?,?) configuration, independent of x and the other angular coordinates. This is the square of the wave function integrated over the x coordinate which introduces a factor of R sin y. Plots of the wave function itself will be used in further figures since the positive and negative features are clearly apparent in such plots. Figure 3a shows the first excited vibrational level of HeC12. The placement of the node through the line y = 7r/2 indicates that this excited level corresponds to a quantum of vibration in the bending mode. Nodes in the three-dimensional plots become more difficult to discern as more excitations are involved and contour plots of these wave functions bring out the nodal structure more clearly. Figure 3b shows a contour plot of the same wave function shown in Figure 3a, with the dotted lines showing the parts of the function with negative amplitude. Figure 4 shows wave functions with progressive excitations in the bending mode. These are the only bound levels for our HeC12 potential. Figure 5 shows the NeC1, ground-state wave function which is noticeably more localized than the corresponding HeC12 function of Figure 2a. The nodal patterns for NeC12 wave functions with bending excitations are similar to those of the HeC12 system. The seventh excited NeC1, function, depicted in the plots of Figure
-6 -4 -2
, *,
2
0
a
-
4
6
ll
Figure 8. Nine angular nodes are apparent in the excited bending state of NeC12 shown in (a) which has the energy -17.23 cm-’. (b) Shows a combination state with one angular node and tvio radial nodes. This state has the energy -6.23 cm-’.
6, shows a node in the radial direction which corresponds to an excitation of the van der Waals stretch. Further overtones of both the bend and the stretch occur, including combinations. Figure 7 shows two quanta in the stretch while Figure 8a shows 11 quanta in the bend and 8b shows two quanta in the stretch and one in the bend. Some more highly excited modes show interesting “mixtures” of the “pure” bending and stretching modes of similar energies. (Pictures of this type of mixed function are given below for the ArC12 system.) Figure 9 shows that the ground-state wave function for ArC12 is even more localized than that for NeC12. Figure 10 shows the
Wave Functions for HeC12, NeC12, and ArC1,
The Journal of Physical Chemistry, Vol. 92, No. 3, 1988 591
A
1
t
V
Figure 9. ArC12 ground state showing even more localization than the NeC12 ground state and having the energy -180.8 cm-'.
-4
-2
0
2
4
-4
-2 0
2
4
Chlorine Axis
(A)
Figure 11. Two highly mixed levels of ArClz (a and b). The states
correlate primarily to two "pure" states, one with no stretching excitations and 12 quanta in the bend, and another with two stretching quanta and two bending quanta. The energy of the state shown in (a) is -83.7 cm-' while that of the state shown in (b) is -82.8 cm-'.
-4 h\
-2 0
2
4
HeCI,
NeCI,
ArCI,
T
v/
..:,
.,
-4 -2 0 T
2
4
-
+2 0
- +I 1
=+lo - +o 2
+o
2
4
Chlorine Axis
(A)
-4
-2 0
Figure 10. The first three excited stretching states of ArC12: (a) E = -142.5 cm-'; (b) E = -106.5 cm-'; and (c) E = -73.6 cm-I. Contours of 10.1 times the maximum amplitudes are included.
first three excited stretching levels. Although successively higher energy states span more of the angular coordinate, the nodal patterns of these states are those of simple stretching excitations. While many of the high-energy states have nodal patterns that clearly define the number of stretch and bend quanta, there are some which are mixed by Fermi resonance. Such an example is shown in Figure 11. The two functions shown correspond primarily io a mixture of two puke states, one with no stretching and 12 bending quanta, and the other with two stretching and two bending quanta. Two more nearby states, one with one stretching and six bending quanta, the other with one stretching and eight bending quanta are also slightly mixed in. These assignments were determined from the energy levels of vibrational progressions and the observed nodal patterns. The states for all three systems are summarized in the energy level diagram of Figure 12. This figure shows the energy level spacings for the bending and stretching modes. To aid in identifying the nature of each level, dotted lines indicate states composed of o d d j functions (states with an odd number of bending quanta) while solid lines indicate states composed of even j functions. Several of the levels are marked with stretch and bend quantum numbers (in that order) to help identify progressions. In all there are 5 bound levels for HeC12, 26 for NeCl,, and 85 for ArC1,. Although the calculations are not coriverged for the
I
j
1
-
Figure 12. Diagram showing the J = 0 vibrational energy levels for all three systems, HeC12,NeC12,and A T C The ~ ~ even j states are solid while the odd states are dotted. The notation labeling selected states is the
number of quanta in the van der Waals stretching and bending vibrations, in that order. The minimum of each potential energy surface is also indicated and labeled with an asterisk.
higher energy levels, the diagram qualitatively illustrates how rapidly the density of states increases with vibrational excitation for these species. While the HeCl, potential well does not support any stretching levels in the van der Waals coordinate, both the NeC1, and ArCl, potentials support multiple excitations in the stretching mode. If we assume these stretching excitations to originate from an effective one-dimensional Morse potential, the energy levels can be used to estimate the corresponding Morse parameters. These Birge-Sponer estimates can then be compared with the actual potential parameters. This process simulates the experimental determination of well depths from observed vibrational progressions. For NeC1, the sixth excited state corresponds to almost pure stretching and lies 25.36 cm-' above the ground state. The fourteenth excited level is predominantly two quanta in the van der Waals stretch, as discussed above, and lies 18.94 cm-* above the sixth state. When these levels are assumed to originate from a one-dimensional Morse potential, the fundamental frequency, we, of the oscillator is found to be 31.78 cm-' and the anharmonicity constant wexe is 3.21 cm-'. The value for we is close to a value estimated directly from the potential, since along the y = a / 2 line, the Morse part of the potential function has parameters
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The Journal of Physical Chemistry, Vol. 92, No. 3, 1988
TABLE II: Asymmetric Top Symmetry Species (a) and Symmetry Correlation of Wave Functions with Asvmmetric TODFunctions (b)
-
TABLE III: Rieid-Rotor Constants (Enereies . - in cm-')" HeCI, NeCI, ArC1, RR CD RR CD RR CD 0.2433 0.2577 0.2433 0.2488 0.2433 0.2457 0.2792 0.2907 0.0658 0.0647 0.0419 0.0410 0.1300 0.1264 0.0518 0.0499 0.0358 0.0347 -7.2 x 10-7 -7.2 x 10-7 -7.2 x 10-7 -2.1 x 10-4 -4.8 X lo'? -6.8 X lo-? -3.3 x 10-2 -3.7 x 10-5 -2.5 x 10-4 0.00 0.00 0.00 3.99 3.91 4.06 4.09 3.97 4.02 1.99 1.97 1.99 1.93 1.99 1.98
asym top function K
D'
even even odd odd
+I -1 +1 -1 wave function sym
svm sDecies A
B, Bx B"
i
R
asym top sym species
even even odd
+1 -1 +1 -1
A
odd
Bz BY
Bx
which yield we = 30.8 cm-'. With the values calculated from the energy levels, the formula, De = 4 4 x e for the well depth of a Morse potential, yields De = 79 cm-'. This value can be compared to the minimum of the potential along y = n/2,73.5 cm-I. Given that this approximation ignores any effects of the angular coordinate, the one-dimensional Morse model appears to give a remarkably accurate estimation of the stretching potential. For ArC12, however, the same sort of analysis shows an interesting difference. When the first three ArC12stretching energy levels are used, the estimate for we, 41.0 cm-I, is the same as the value for the potential along y = a / 2 . But De = 318 cm-' for the Morse estimate, while the actual potential minimum along y = n / 2 is 207 cm-I. The lack of agreement can be attributed to the fact that the excited stretching states are not quite pure stretching states but are mixed with excited bending levels very close in energy. If the potential is slightly modified by changing to and t2 so that the overall minimum remains constant but the bending frequency is changed, the coincidence of the energy levels can be changed. For example, if we decrease e2 by 10 cm-I and eo by 5 cm-I, e(y=n/2) is unchanged. However the bending frequency is increased by 0.5 cm-' and the Morse estimate for De, using the first two stretching levels, becomes De = 260 cm-l, a significantly closer value. When t2 is increased by 10 cm-', Fermi resonance effects are so great that it is impossible to identify any state as the first excited stretch. A further increase of t2 by 10 cm-' gives an estimate of De = 160 cm-I. Unfortunately, these results force us to conclude that a Birge-Sponer estimate of the well depth may not always be accurate. It will probably not be obvious from experimental data whether a Fermi resonance is affecting the analysis unless an extensive progression is measured.
Energy Levels for Higher Angular Momentum States Energy levels up to J = 3 were calculated for all three systems. These energies correspond to overall rotations of the molecule and are compared with two models. One is a simple rigid-rotor model and the other is a rigid-rotor model with centrifugal distortion included. First, we correlate the symmetries of the calculated wave functions to the symmetries of the rigid-rotor wave functions. We then compare the simple rigid-rotor model and finally consider the centrifugal distortion effects. If the He-, Ne-, and Ar-C12 molecules were rigid, they would be asymmetric tops whose wave functions could be expressed as linear combinations of symmetric top wave functions.20 Usually a basis set of symmetrized symmetric top functions very similar in form to the angular basis set of eq 6 is used to describe the asymmetric top function, $&:
The c;ip coefficients determine the expansion; p'is either +1 or (20) For a discussion of asymmetric tops, see, for example: Gordy, W.; Cook, R. L. "Microwave Molecular Spectra"; In Techniques of Chemistry, 3rd ed.; Wiley: New York, 1984; Vol. XVIII.
"RR, rotational constant 2,fixed; d, varied to fit exact levels. CD, incudes centrifugal distortion ( T coefficients) and vibration-rotation interaction.*O Effective bond lengths, in angstroms. TABLE I V HeC12 Rotational Energy Levels and Comparison with Rigid-Rotor Models (Energies in cm-I)
J 1
exact calcn jp energy eO-
O+
2
o+
e+ e0-
e+ 3
0-
e-
e+ O+ 0-
eO+
0.3979 0.3980 0.5575 1.0166 1.0448 1.4855 1.4878 1.6248 1.8981 1.9027 2.5899 2.6950 3.0304 3.0321 3.1609
rigid-rotor calcn energies level
1 (RR)
l,, 101
0.4093 0.3733 0.5225 1.0426 1.0353 1.3825 1.4903 1.5749 1.9489 1.9498 2.6102 2.5156 2.9013 3.1160 3.1699
11,
2,, 2,, 2,, 22, 2,, 303 3,, 3,, 312 321 3,, 330
2 (CD) 0.4162 0.3833 0.5483 1.0482 1.0440 1.4404 1.5386 1.6402 1.9418 1.9452 2.6695 2.6485 3.0028 3.2007 3 2660
-1; and K is the projection of on the body-fixed z axis. If the z-axis for the symmetric top representation is chosen to be along R, the symmetries of the rigid-rotor functions of eq 7 can be directly correlated to the non-rigid-rotor functions described in eq 6. This choice for the z axis is convenient, since it is one of the principal axes for the T-shaped molecules. The rigid-rotor wave functions belong to the D2group which includes four symmetry species, A, B,, B,, and By, which are defined by the behavior of the wave functions upon application of the symmetry operations, q,Cjs, and q.This results in the four species being defined by the parity of K and the value of p', as shown in Table IIa. In the limit of rigid T-shaped molecules, we can use the functions e$@from eq 6 as a basis to describe the asymmetric top functions if we set y = a/2. Using the properties of the spherical harmonics and the Wigner functions, we can write
For Y$a/2,0) to be nonzero, j and i2 must be both even or both odd. This correlates the parity of j with the parity of Q in the rigid-rotor limit. The functions of eq 8 are of the same form as the symmetric top basis functions used in eq 7 with Q in place of K and ~ ( - 1 )in~ place of p'. This allows us to immediately correlate the rigid-rotor symmetry species with the parity of j and the value of p , as shown in Table IIb. The simple rigid-rotor model includes only three parameters, and 8,rotational constants of a molecule. To compare the B,, 8,, with this model, we fixed the 8,rotational constant at the value for the free diatomic chlorine and then varied 8, so that the rigid-rotor energy levels most closely matched the "exact" calculations. 8, was determined by the planarity relation, By-' = 8,-' + 8;'. This procedure is particularly interesting because it is analogous to the procedure used to fit the rotational structure in the electronic excitation spectra of these m o l e ~ u l e s . ' ~The ~~'~
Wave Functions for HeCl,, NeCl,, and ArC1,
The Journal of Physical Chemistry, Vol. 92, No. 3, 1988 593
TABLE V NeC12 Rotational Energy Levels and Comparison with Rigid-Rotor Models (Energies in cm-')
J 1
exact calcn jp energy e0-
O+
2
e+ O+
oe-
3
e+ eoO+
e+ e-
oO+
0.1112 0.3047 0.3185 0.3328 0.5128 0.5541 1.1344 1.1351 0.6634 0.8246 0.9070 1.4649 1.4683 2.4665 2.4665
rigid-rotor calm energies level 1 (RR) 2 (CD) lnl
0.1176 0.2951 0.3091 0.3520 0.5163 0.5584 1.0908 1.0916 0.7017 0.8477 0.93 17 1.4437 1.4476 2.3666 2.3666
0.1145 0.2986 0.3 135 0.3427 0.5123 0.5569 1.1092 1.1101 0.6829 0.8324 0.9215 1.4498 1.4540 2.4092 2.4092
TABLE VI: AKl2Rotational Energy Levels and Comparison with Rigid-Rotor Models (Energies in cm-I)
exact calm energy
J
jp
1
eoO+
2
e+ O+
oe-
e+ 3
e0O+
e+ e-
oO+
0.0745 0.2837 0.2897 0.2233 0.4267 0.4446 1.0723 1.0724 0.4463 0.6410 0.6768 1.2953 1.2959 2.3565 2.3565
rigid-rotor calcn energies level
1 (RRI
2 (CDI
0.0777 0.2791 0.2852 0.2330 0.4284 0.4469 1.0509 1.051 1 0.4656 0.6522 0.6892 1.2841 1.2848 2.3064 2.3064
0.0757 0.2804 0.2867 0.2270 0.4255 0.4444 1.0585 1.0586 0.4537 0.6431 0.6807 1.2852 1.2859 2.3246 2.3246
procedure determines an effective distance, Reff,between the rare gas and the chlorine center of mass, since 9, = h2/2pRef?. The rigid-rotor constants obtained are shown in Table 111, and the energy levels are listed in Tables IV, V, and VI for the He, Ne, and ArC1, systems, respectively. Comparison of the HeClz exact levels and the rigid-rotor fit shows some interesting results. Individual levels differ by as much as 0.1 cm-I, and this lack of agreement is accentuated by the fact that not all of the symmetry-correlated energy levels are ordered the same way. For example, the lol level is the lowest rigid-rotor level for J = 1. It has B, symmetry, which according to Table I1 correlates with J,. However, the ,1 state is the second lowest level of the exact calculations. This implies that the symmetries of the exact energy levels, when ordered according to energy, correspond to a rigid rotor with 9, > 9, > 9,. The tendency for this behavior can be understood by considering the ground-state J = 0 wave function, which has a large amount of amplitude at large R. This would tend to decrease 9,. For NeC1, and ArCl, the rigid-rotor fits get progressively better. The symmetries line up properly for both of these systems, and the numerical deviations are generally smaller. To see if simple corrections significantly improve the rigid-rotor approximation, we considered centrifugal distortion to terms quartic in the angular momentum, and the vibration-rotation interaction corrections to the rotational constants, including harmonic, cubic, and Coriolis corrections.20 We performed a simple normal-mode analysis, estimated the harmonic force constants for the van der Waals coordinates, calculated centrifugal distortion and vibration-rotation interaction constants, and then calculated the rotor energy levels. The three normal modes are
proportional to the chlorine stretch, the van der Waals stretch, and the van der Waals bend. The harmonic force constants were taken directly from the second derivatives of the potential at equilibrium. The results for HeCl,, NeCl,, and ArCl, are also tabulated in Tables 111-VI. These results, which include no varying of parameters, are better than the fits to the simple rigid-rotor model. For HeCl, the improvement is small. For NeC1, and ArC1, the approximation including centrifugal distortion, based essentially on the assumption of small-amplitude vibrations, appears to predict, fairly accurately, the rotational energy levels of these van der Waals molecules. As mentioned in the Introduction, the simple rigid-rotor analysis has not provided a simple explanation of the discrepancy between the NeCl, potential obtained from scattering experiments and the effective bond length obtained spectroscopically. As seen in Table 111, the effective bond length obtained in our simulated rigid-rotor fits is essentially the same as the location of the potential minimum. This suggests that spectroscopically determined bond lengths for these van der Waals molecules do correspond closely to the potential minima.
Summary Using realistic potentials to describe the rare gas-chlorine interactions and treating the chlorine molecule as rigid, we have calculated vibrational and rotational wave functions ( J I3) for the HeCl,, NeCl,, and ArCl, van der Waals molecules. The HeCl, potential supports five vibrational states when the total angular momentum is zero. The ground vibrational state is significantly delocalized, and all the vibrational states correspond to excitations of the van der Waals bending vibration which has a fundamental frequency of about 3.5 cm-'. The rotational energy levels are not quantitatively fit by rigid-rotor models. The NeCl, molecule has at least 26 vibrational levels, with the ground-state wave function being more localized than that for HeC1,. The vibrational states include, in addition to excited bending states, two states with essentially pure van der Waals stretching excitation, as well as combinations of bends and stretches, and states in Fermi resonance. The van der Waals bend has a fundamental frequency of -7 cm-' while the stretch is around 32 cm-I. A one-dimensional Morse potential can model the van der Waals stretches and provides an accurate estimate for De. The rotational energy levels are reasonably well reproduced by rigid-rotor models. Nearly 100 bound vibrational levels were found for the ArC1, molecule with bends, stretches, combinations, and Fermi resonances similar to NeCl,. The ground vibrational state is more localized than for NeCl,, the bending motion has a frequency near 12 cm-', and the stretching frequency is 41 cm-'. Fermi resonance distorts the one-dimensional Morse model of the first three van der Waals stretches. The estimate of De using this model is over 50% larger than the true value. The rigid-rotor model even more accurately predicts the ArC1, rotational energy levels than it does the HeCl, and NeCl, levels. In conclusion, this work has shown the nature of the vibrations associated with the van der Waals coordinates as well as the rotational energy levels of three triatomic van der Waals molecules. These results can be used to guide and interpret further experimental and theoretical studies of these and other van der Waals systems. Acknowledgment. We thank Professor J. A. Beswick for extensive discussions on the problems addressed in this paper. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. The Fulbright Foundation and NATO provided funds that enabled our international collaboration. The work received further support from the National Science Foundation. Registry No. He, 7440-59-7; Ne, 7440-01-9; Ar, 7440-37-1; CI,, 7782-50-5.
