2974
J . Phys. Chem. 1985,89, 2974-2975
Weakly Bound, Strongly Anisotropic van der Waals Molecules Phillip R. Certain* and Nimrod Moiseyed Theoretical Chemistry Institute, Department of Chemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706 (Received: April 23, 1985)
Bound and resonance rotational states of a model potential for an atom-diatomic molecule are calculated for varying strengths of the anisotropic potential, ranging from a spherical potential to a strongly anisotropic potential. It is found that resonance states do not exist for the most anisotropic potential, and the complex rotates essentially as a linear, triatomic molecule. The relationship of the model to the system He-LiH is briefly discussed.
Introduction This paper contains a quantum mechanical description of the bound rotation-vibration states of a van det Waals complex formed between an atom and a polar diatomic molecule. Our results are for an idealized complex and are intended to provide a general description of polar complexes of this type. Subsequent work will employ accurate potential energy surfaces and include all molecular degrees of freedom in the dynamics. Our work was motivated by calculatiolis of the potential energy surface of the helium-lithium hydride system.] This complex is perhaps the simplest polar van der Waals complex for which definitive theoretical and experimental results can be obtained. Calculations reveal a ptential well for helium at the lithium end of the diatomic which is only some 120-cm-' deep. The potential is strongly anisotropic, however, and the potential at the hydrogen end is repulsive. Thus, the anisotropic potential is comparable in magnitude to the spherical potential, since it is sufficiently strong to cancel the binding at one end of the molecule. For a weakly anisotropic atom-diatom potential, rotational excitation of the diatomic weakens the van der Waals bond and, for sufficiently strong excitation, leads to dissociation.* The original intent of our work was to study rotational predissociation resonances in the He-LiH system. In the course of our study, however, we discovered that the potential is sufficiently anisotropic so that the complex rotates as a linear triatomic molecule, albiet with one weak bond. There are only a few bound rotation-vibration states and no resonance states. Thus, all dissociations are direct and rapid. The properties of He-LiH contrast sharply with van der Waals complexes in which. the anisotropy is weak compared to the binding energy. In the latter case, rotational predissociation resonances are easily found. We therefore constructed a model potential which allowed us to study the transition between weakly and strongly anisotropic potentials in order to understand the nature of the bound and resonance states of such systems. Theoretical Methods Formulation. We have chosen a space-fixed formulation in which the triatomic complex is viewed as an atom bound to a diatomic molecule whose angular momentum is referred to a space-fixed coordinate ~ y s t e m . ~This formulation is most appropriate for weakly anisotropic complexes, but by retaining a sufficiently large number of coupled angular momentum states, it can be used to study strongly anisotropic complexes as well. In the space-fixed formulation, the relevant quantum numbers are j , the rotational angular momentum quantum number of the diatomic; I , the quantum number for rotation of the atom about the center of mass of the diatomic; and J , the total angular momentum quantum number. In the spherical limit, all are good quantum numbers, while for anisotropic potentials, only J is. Model Potential. The potential energy function for a triatomic complex is denoted by V(R,r,y). Here R is the distance from the atom to the center of mass of the diatomic, r is the internuclear 'Permanent address: Department of Chemistry, Technion-Israel Institute of Technology, Haifa, Israel.
0022-3654/85/2089-2974$01 .SO10
TABLE I J 0 1 2
bound states," cm-'
resonances
-21.0986 (6 1.55) -1 1.4375 (6 = 1.50) -1.81389 (6 = 1.50)
none none none
6 is the nonlinear parameter in the harmonic oscillator basis. distance of the diatomic, and y is the angle between 1? and ?. For the present paper, we shall make the following simplifications: 1. The diatomic is in its ground vibrational state. Hence, V(R,r,y)is replaced by V(R,y),which is the average of the total potential for the ground vibrational state. 2. V(R,y) is represented by V(R,y) = VOW
+ ~ l ( ~ ) P I ( C O Y) S
(1)
3. The forms of Vo(R)and V l ( R )are Vo(R) = 46[(a/R)12 - (1 - X ) ( U / R ) ~ ] V](R) = -4Xt(a/R)6 cos y
(2)
(3)
where u = 3.9573 a. and e = 122.8337 cm-I. For X = 0, the potential V(R,y)is spherically symmetric. In this paper, we study the range 0 6 X 6 1. The value of X for which V(R,y)most closely matches the calculated surface for He-LiH is X = 1.9. This value is too large to be considered effectively with a space-fixed formulation, and therefore we treat it separately within a body-fixed f o r m ~ l a t i o n . ~ Computational Methods. With the potential introduced above, the wave equation in the space-fixed formulation reduces to a set of coupled equations
J'I(R) Xf~Gl,j'~',J)+;.,(R) =0 j'l'
where p = 2.6708 amu, B,,, = 4.1878 cm-' -is the rotational constant of LiH, and j1(j,lJ',l',J) is the Percival-Seaton coeffi~ i e n t .For ~ J = 0 states, we included the terms GJ) = (O,O), (l,l), and (2,2); for J = 1, ( j , l ) = (O,l), (l,O), (2,1), (1,2); and for J = 2, G , l ) = (0,2), (2,0), ( l , l ) , (2,2). Each was expanded in a common basis of the first 20 harmonic oscillator functions X,,[P(R- RO)], and the bound state energies were determined by diagonalizing the resulting martrix equation. Resonance states (1) D. M. Silver, J . Chem. Phys., 72, 6445 (1980). (2) See, for example, articles in D. G. Truhlar, "Resonances", American Chemical Society, Washington, DC, 1984, ACS Symp. Ser., No. 263. (3) R. T. Pack, J . Chem. Phys, 60, 633 (1974). (4) B. K. Holmer and P. R. Certain, J. Phys. Chem., accepted for publication. (5) I. C. Percival and M. J. Seaton, Proc. Cambridge Philos. Soc., 53, 654
(1957).
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 14, 1985 2975
Letters E2
I
=O
- i4 '
'
'
'
0.5 '
'
'
'
' IO
E (cm-') Figure 2. &trajectories for the J = 1 state arising from the (I = 0, j = 1) state in the isotropic limit. The arrows denote the estimate of the resonance energy width for X = 0.65 and X = 0.70.
the continuum. For a narrow range of A, they can be characterized as resonances when they are close enough to the dissociation threshold, but as the width grows, this becomes a less useful picture. For example, in Figure 2 , shows &trajectories for J = 1 resonances for two values of X just greater than the value at which the bound state which correlates with j = 1, 1 = 0 in the spherical limit crosses into the continuum. Clearly as X increases, the resonance position and width increase rapidly. In a picture based on weak anisotropic coupling, this resonance would be described as resulting from rotational excitation 0' = 1) of the diatomic. The value of X is so large, however, that this picture is no longer valid, since the molecule rotates essentially as a linear triatomic molecule. This behavior is undoubtedly characteristic of weakly bound, strongly anisotropic complexes. For X = 1, it is clear from Figure 1 that there are no resonance states possible for the model potential we have used. To the extent that the potential mimics the He-LiH system, no rotational predissociation resonances can be expected to occur, and, in fact, one may question whether even bound states exist for this system. Further electronic structure calculations are needed to refine the potential in order to address this question and to encourage experimental studies of this system. (6) N. Moiseyev and P. R. Certain, J . Phys. Chem., accepted for publication. See also S. I. Chiu, J . Chem. Phys., 72, 4772 (1980).
Acknowledgment. This research was supported by the National Science Foundation (CHE 8 1-07628) and the US.-Israel Binational Science Foundation
