Classical Scattering of an Atom from a Diatomic Rigid RotorR. J. Cross Jr. and D. R. Herschbach Citation: The Journal of Chemical Physics 43, 3530 (1965); doi: 10.1063/1.1696512 View online: http://dx.doi.org/10.1063/1.1696512 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/43/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Wave packet study of gas phase atom–rigid rotor scattering J. Chem. Phys. 89, 2958 (1988); 10.1063/1.455001 Virial theorem for inelastic molecular collisions. Atom–rigid rotor scattering J. Chem. Phys. 73, 3823 (1980); 10.1063/1.440613 Direct partition function of the rigid diatomic rotor J. Chem. Phys. 63, 5216 (1975); 10.1063/1.431305 Sum Rule for Interference Scattering from a Rigid Rotor J. Chem. Phys. 52, 5967 (1970); 10.1063/1.1672887 The Scattering of Atoms from Diatomic Molecules J. Chem. Phys. 20, 249 (1952); 10.1063/1.1700387

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 43, NUMBER 10 15 NOVEMBER 1965

Classical Scattering of an Atom from a Diatomic Rigid Rotor

R. J. CROSS, JR.,* AND D. R. HERSCHBACH

Department of Chemistry, Harvard University, Cambridge, M assachusetis

(Received 15 July 1965)

Collisions between an atom and a rotating rigid linear molecule were examined by computer integration of the classical equations of motion. An anisotropic potential consisting of a Lennard-Jones (6, 12) function multiplied by [1 +aP2 (COSI') ] was used in most of the calculations (1' is the angle between the atom and the axis of the molecule). The deviation in the scattering angle caused by the anisotropy of the potential energy was found to be small and approximately proportional to a, the asymmetry parameter. The devia­tions were large enough, however, to distort severely the differential cross section in the rainbow scattering region. The angular-momentum transfer was found to be appreciable (in the range ±1O fi) over a wide region of impact parameters extending well outside the repulsive core. For moderately heavy molecules a good approximation to the scattering angle and the angular-momentum transfer was obtained from a "flywheel" model which assumes that the molecule rotates independently of the collision and calculates the effect of the resulting time-dependent perturbation on the trajectory of the atom. Similarly, for light molecules it was found that the angular-momentum transfer could be approximated by evaluating the time-dependent perturbation of the molecular rotation caused by the atom moving in a trajectory calcu­lated from the spherically symmetric part of the potentia!.

MOST chemical experiments involve nonspherical molecules, but the influence of rotational energy

change and the anisotropy of the forces remains largely an open question in the interpretation of molecular beam scattering, transport properties, col­lision broadening and shifts of spectral lines, and ultra­sonic dispersion. A comprehensive review of the theo­retical approaches which have been made and their limitations has been given recently by Takayanagi.1.2 Although several quantum-mechanical treatments of the scattering of a particle by a rigid rotor have been developed,l-n few quantitative applications have been made. Most of the available calculations have had to invoke drastic idealizations concerning the trajectory of the interacting particles or the weakness of the anisotropic forces. However, for the simpler problem

* N.S.F. Predoctoral Fellow, 1962-1965. Present address: Department of Chemistry, University of Florida, Gainesville, Florida.

1 K. Takayanagi, Progr. Theoret. Phys. (Kyoto) Supp!. 25, 1 (1963).

2 K. Takayanagi, in Advances in Atomic and Molecular Physics, edited by D. R. Bates and I. Esterman (Academic Press Inc., New York, 1965).

8 A. M. Arthurs, A. Dalgarno, and R. J. W. Henry, Proc. Roy. Soc. (London) A256, 540 (1960); Atomic Collision Pro­cesses, edited by M. R. C. McDowell (John Wiley & Sons, Inc., New York, 1964), p. 914.

4 R. B. Bernstein, A. Dalgarno, H. Massey, and I. C. Percival, Proc. Roy. Soc. (London) A274,427 (1963).

6 C. S. Roberts, Phys. Rev. 131, 203, 209 (1963). 8 P. W. Anderson, Phys. Rev. 76,647 (1949). 7 J. Fiutak and J. Van Kranendonk, Can. J. Phys. 40, 1085

(1962) ; 41, 21, 433 (1963). 8 K. H. Kramer and R. B. Bernstein, J. Chern. Phys. 40, 200

(1964). 9 H. G. Bennewitz, K. H. Kramer, W. Paul, and J. P. Toennies,

Z. Physik. 177, 84 (1964). 10 M. Child, Mol. Phys. 8,517 (1964).

of elastic atom-atom scattering a complete set of working approximations is now available.12 This has been developed mainly from semiclassical theorY,l3 which greatly simplifies the calculations and also puts the results in a form that can be more readily visualized. As a step toward an analogous formulation of atom­diatomic-molecule scattering, we have carried out a computer study of the exact classical problem. The results provide a test of approximate models and bring out some simple features of the collision dynamics.

GENERAL THEORY

The scattering of two particles subject to a-spherically symmetric potential can be specified completely (in both quantum and classical mechanics) by three dy­namical parameters: the asymptotic relative kinetic energy, E; the impact parameter, b; and the reduced mass, m. In classical mechanics the angle of deflection (angle between the initial and final relative-velocity vectors) is given by

XO=1r-21O:>[1-~- V(r)]-i ~dr rm r2 E r2'

(1)

where rm=b[l- V(rm)/E]-i is the distance of closest approach in the collision. If the potential has the form V(r) = Ev(r/u-) , the scattering angle xo becomes a function of just two dimensionless variables, the re­duced energy, K= E/E, and the reduced impact param­eter, {3= b/IJ. The scattering of an atom from a diatomic molecule is subject to an anisotropic potential, but this result still holds if the relative orientation of the atom

11 K. Lawley and J. Ross, J. Chern. Phys. 43, 2930

12 See, for example, R. B. Bernstein, Science 144, 141 (1964). 18 K. W. Ford and J. A. Wheeler, Ann. Phys. (N.Y.) 7, 259

(1965). (1959). 3530

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SCATTERING FROM A DIATOMIC RIGID ROTOR 3531

and the molecule is held fixed or if the potential is averaged over all orientations. If orientational effects and torques are to be included, however, several ad­ditional dynamical parameters must be specified; in classical mechanics these are: the magnitude J .. and orientation angles OJ, cf>J of the:rotational angular,mo­mentum of the molecule; the phase angle w describing the initial orientation of the molecular axis in the plane perpendicular to the J vector; and the moment of inertia I of the molecule. Notation and definitions are collected in Table I. In this study the dependence of the scattering angle X on these various parameters and on the anisotropy of the potential has been examined in terms of the deviation, AX=x-xo, from the result obtained for scattering (at the same values of K and (3) from the spherically averaged part of the potential.

Model and Hamiltonian

Whittaker14 gives a formulation of the general clas­sical three-body problem which involves the minimum

r, e, .p

p, II, ¢

w

x

'Y

V(r, 'Y)

E, (1, a

m, p., I

b,(3

E,K

L,L*

TABLE I. Notation and coordinates."

Spherical polar coordinates locating Atom A with respect to the center of mass of the molecule BC

Spherical polar coordinates locating B with respect to C, p=constant for the rigid-rotor model

Spherical polar coordinates of the rotational angular momentum of the BC molecule

Phase angle giving orientation of the molecule in the plane perpendicular to the J vector

Scattering angle (between the initial and final relative-velocity vectors) ; xo refers to the spheri­cally averaged potential and AX=X-Xo

Angle between a line from A to the center of mass of BC and the internuclear axis of BC; 'Ym refers to the midpoint of the collision

Potential energy; Vo (r) refers to the spherically averaged potential

Potential parameters indicating well depth, radius, and anisotropy

Reduced mass of A+BC, reduced mass of B+C, and moment of inertia 1= p.p2 of the BC molecule

Impact parameter and reduced value, (3=b/u

Initial relative translational kinetic energy and reduced value, K =E/E

Reduced angular velocity parameter, n= (J /I) (m.r/E)!

a The X axis is along the initial relative velocity vector and the Z axis along the initial orbital angulat-momentum vector.

14 E. A. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge University Press, New York, 1960), pp. 339-357.

z

Zl

x

FIG. 1. Coordinate systems used in the rigorous solution of the equations of motion.

number of differential equations. This can be simplified further for the case considered here, the collision of an atom A with a rigid diatomic molecule Be. It is con­venient to make use of three Cartesian coordinate systems, all with origin at the center of mass of the BC molecule. In the XYZ system, used for the defini­tions of Table I, the Z axis is directed along the initial orbital angular-momentum vector, L, and the X axis along the initial relative-velocity vector. Figure 1 shows the other two systems. The xyz system is obtained from XYZ by an Euler angle transformation which puts the z axis along the total angular-momentum vector, L+J, and the x axis perpendicular to the plane defined by the Land J vectors. In a given col­lision the orientation of these axes remains fixed in space (although the origin translates along with the BC molecule). The x'y'z' system is a rotating system, defined such that, at any instant, the z' axis is perpen­dicular to the plane containing the three atoms (the X'y' plane) and the x' axis is parallel to the intersection of this plane with the xy plane. The relative motion of the three atoms (excluding over-all translation) in­volves six dynamical variables: ql and q2 are the x' and y' components, respectively, of the distance be­tween Band C; qa and q4 are the x' and y' components of the distance between A and the center of mass of BC; q6 is the angle between the x and x' axes; and q6 is the angle between the z and Zl axes.

The Hamiltonian in this formulation14 is

Here the p's are the momenta conjugate to the q's, P is the total angular momentum, and other quantities are defined in Table I. Since the variable q6 does not appear in the Hamiltonian, P.= constant; as shown by Whittaker,14 this constant is P, the total angular mo­mentum. Also, P6= 0 and

P COSq6= P'J!ll- Plq2+ P4qa- Paq4.

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3532 R. J. CROSS, JR., AND D. R. HERSCHBACH

The first three terms in Eq. (2) describe the motion in the instantaneous plane of the three bodies, whereas the terms which contain q6 describe the out-of-plane contribution (i.e., the rotation of the plane of the three bodies). Equation (2) is the general result and could be used to study vibrational excitation or chemical reaction. For our application to scattering from a rigid rotor, there is the special constraint that the distance p between Band C must be held fixed. Furthermore, the potential depends only on the radial distance r and angle 'Y between A and BC. We therefore introduce the following transformation:

p= (qI2+q22)i=constant (3)

r= (qs2+q42)1

'Y=tan-1(q2/ql) -tan-1(q4/qa)

1/;= tan-1 (q2/ql) ,

where I/; is the angle between the axis of BC and the x' axis. The Hamiltonian then becomes

H= Pr2+ P·l + (P'Y+P~)\ VCr, 'Y) 2m 2mr2 2p.p2

[sin2( 'Y-Y;) sin21/;] (P2_ p~2) + +--2' 2 . 2p.p2 2mr sm 'Y

(4)

The six derivatives of Eq. (4) with respect to the coordinates (r, 'Y, 1/;) and momenta (Pr, P 'Y, P~), and the derivative with respect to P (to give q6) form the equations of motion. This formulation thus reduces the problem to integration of seven simultaneous dif­ferential equations as compared to 12 for a treatment in Cartesian coordinates. Furthermore, in the coordi­nates used here the potential takes a simple form. The integration of the equations of motion is carried out with an IBM 7094 computer program which combines a fourth-order Runge-Kutta and a variable-step Adams-Moulton procedurel6 ; for the range of variables explored, the calculation of the deflection angle X (within ""0.1°) typically requires three seconds.

Choice of Potential

The form of the anisotropic potential can be reason­ably estimated1,12,16 for systems without strong chemical interactions. As in the spherically symmetric case, the potential has a long-range attractive branch and a short-range repulsive branch. For atom-molecule scat­tering the leading terms in the attractive branch take the form

16 SHARE Library routine D2RWINT. 16 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular

Theory of Gases and Liquids (John Wiley & Sons, Inc., New York, 1954), pp. 916-1104.

For an anisotropic van der Waals interaction (dis­persion or London forces) the force constant C may be estimated from the Slater-Kirkwood formula17 and the asymmetry parameter from the anisotropy of the polarizability,18

(6)

where all and a,L are the polarizability components parallel and perpendicular to the axis of the molecule. The maximum value given by Eq. (6) is unity and a"-'i for typical molecules. If the molecule has a permanent dipole moment, there is a further contri­bution to VA(r, 'Y) of the same form from the dipole­induced-dipole interaction.16 This has C' = J.l.m2aa and a' = 1, where P.m is the dipole moment of the molecule and aa the polarizability of the atom. Thus both the van der Waals interaction (Constants C, a) and the dipole-induced-dipole interaction (Constants C', 1) may be included in Eq. (5) by C~C+C' and a~ (aC+C')/(C+C').

The form adopted for the short-range repulsive poten­tial must be somewhat arbitrary, as the information available at present from either theory or experiment is quite meager. A convenient form for calculation is

By symmetry, the coefficient b can be nonzero only for a heteronuclear molecule. Recently, Roberts6 carried out a variational calculation of the repulsive part of the He-H2 potential and found the angular dependence was very well approximated by a P2 (cos'Y) term with c=0.375.

In most of our calculations we used for simplicity the same angular function for both the attractive and repulsive branches and a Lennard-Jones (6, 12) radial potential; thus

Also, we used a= 0.5 except where indicated otherwise. For one set of calculations, another attractive term of the form

(9)

was added to Eq. (8). This represents the quadrupole­induced-dipole interaction which is important in the interpretation of small-angle inelastic scattering9 of polar molecules and rotational line broadening.19 Ac­cording to the usual multipole expansion treatment,16 the coefficient is given by q=!QmJ.l.maa/EU7, where Qm is the quadrupole moment of the molecule and as before J.l.maa is the induced dipole moment of the atom; for typical values ofthe parameters, ~0.1 (Q= 5 X 10-26

17 J. Slater and J. Kirkwood, Phys. Rev. 37, 682 (1931). See also, K. S. Pitzer, Advan. Chern. Phys. 2, 59 (1959).

18 F. London, J. Phys. Chern. 46, 305 (1942). 19 P. W. Anderson, Phys. Rev. 80,511 (1950).

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SCATTERING FROM A DIATOMIC RIGID ROTOR 3533

esu, p.=5 D, a=2 As, e=O.4 kcal/mole, 0-=3 A). In another set of calculations the angular dependence of the repulsive branch of Eq. (8) was replaced by the more general form indicated in Eq. (7).

Results

In the main series of calculations, which used Eq. (8) with a=O.S, the deflection X and deviation ax=x-xo were tabulated as the molecular angular momentum J, moment of inertia I, and Orientation Angles OJ, c/>J, and w were each varied separately at fixed values of the reduced energy K and reduced impact parameter fJ. Figure 2 illustrates the general character of the results for scattering of a fairly "heavy system" such as Ar+ Br2. Despite the marked anisotropy of the po­tential, the deflection X is determined mainly by the

-\BOO~O ----,;l0.;;-5 -----:-:1.0,-----:'c1.5;-----:f:2.0t-----::'2 .. S

~

FIG. 2. The curves give the scattering angle for the spherically averaged potential, Xo, as a function of the reduced impact para­meter, {3, for two reduced energies. The bars indicate roughly the deviations caused by changing the position and angular momen­tum of the molecule for a=O.5, m=30 g mole-i , and 1=270 g mole-1 12. Glory scattering occurs where X passes through zero; rainbow scattering where X reaches a finite minimum; and orbiting at the singularity in the K=OA curve.

radial interaction and thus depends primarily on the K and fJ parameters. The curves in Fig. 2 give the scattering angle xo for the spherically averaged po­tential (equivalent to the a=O case). The bars indicate a rough estimate, based on many sample calculations, of the full range of the deviations in X obtained by varying the initial orientation angles of the molecule and angular momentum over all possible values (for the a=O.S case). For heavy systems, the deviation ax is limited to a rather small range which changes only slowly with K and fJ; this range is not enlarged by in­creasing the angular momentum beyond a certain range, since at high J the effect of the anisotropy is averaged out during the collision (see Fig. 7). Ex­ceptional behavior appears in the orbiting region (e.g., near the singularity in the K=0.4 curve of Fig. 2); here the deviations are expected to become larger, as the radial motion of the atom and molecule becomes

AXrT-r"rT-r'-rT-r"rT-r"rT-r"rT-rT'~

IS

-s , I

I

o

/' , .-.<tJII>----­, ..

2.0 2.5

FIG. 3. The deviation ax=x-xo as a function of {3 for two orientations. The "standard" conditions used here and in several subsequent figures are K=3, a=0.5, J=O, m=30 g mole-i ,

1=270 g mole-i 12; w=oo (--) corresponds to the molecule initially along the Z axis and w=95° (---) close to the Yaxis (see YZ case of Fig. 6). The sharp change in the curves occurs near the rainbow angle.

very slow and they rotate through large angles while close together. The range of ax shrinks as K is in­creased, since the anisotropy then has less time to act. The range vanishes for fJ-tO, corresponding to head-on collisions with Xo-t 1800

• It vanishes also for fJ-too, corresponding to grazing collisions with xo-tO. In this limit, ax/xo becomes roughly constant if the molecule is not rotating rapidly. This evidently occurs because the anisotropic term in the potential has the same de­pendence on radial distance as the spherically sym­metric term. If the molecule is rotating rapidly, ax/xo-tO at large impact parameters, since the aniso­tropic term is averaged out by rotation during the collision.

Figures 3-S illustrate the systematic variation of .6x for scattering from a nonrotating molecule (J = 0) with

60'

300

-30'

-6011,1

" I I , ,

I ,

~-

-OO~0~~-L~Q~5LJ-L~l~OJ-~-L~t5-L~~~a~0~~-LJ2~· {3

FIG. 4. ax as a function of {3 for two orientations at K ... 0.4, with other conditions the same as in Fig. 3. The singularities occur in the orbiting region.

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3534 R. J. CROSS, JR., AND D. R. HERSCHBACH

�4'r---,--,----,---.----,.---,

4'

AX 2'

0'

-2'

-6'

-s'

-10' 0~'--,3~O'.--,6~O·'-~90~'-~~-~.-~ISif'

FIG. 5. LlX as a function of the phase angle, w, for collisions with the molecule initially in the X Y, XZ, or Y Z planes (see Fig. 6). Other conditions are the same as in Fig. 3 except that 1=540 g mole l 12 and /3=1.07 (corresponding to xo=23°). -- indi­cates X Y plane, - - - indicates YZ plane, and - - indicates XZplane.

the various initial orientations as indicated in Fig. 6. Thus the calculations of Figs. 3 and 4 refer to "broad­side" collisions, in which the atom approaches along the X axis and the molecule is initially in the YZ plane, with its axis either parallel to the initial orbital angular momentum L (along the Z axis, w=O° case) or approximately perpendicular to it (close to the Y axis, w=95° case). The variation of AX with impact parameter is seen to be small and fairly uniform except near the rainbow and orbiting regions, where AX dis­plays undulations or extrema which are quite different for the two orientations. For the calculations of Fig. 5,

v~

X

d

/

l

/ I I

I

/

Yl PLANE

XY PLANE

Y

Xl PLANE

FIG. 6. Diagram showing the "standard" orientations of the molecule used in the calculations of Fig. 5 and elsewhere.

the collision energy and impact parameter are fixed, and the whole spectrum of initial orientations in the XV, XZ, and YZ planes is scanned by varying the phase angle w (defined with the conventions indicated in Fig. 6). Again, the variation AX is seen to be smooth and regular but very sensitive to the orientation of the molecule. The curves overlap at values of w correspond­ing to alignment along one of the three axes. The X axis corresponds to w=O° on the XYplane and w=90° on the XZ plane, the Y axis corresponds to w= 90° on the XY and YZ planes, and the Z axis to w=O° on the XZ:and YZ planes. Since the potential has even parity, the curves have a period of 180°, and since the scattering must be the same if the coordinates are re­flected through the XY plane, the curves for orienta­tions in the XZ and YZ planes are symmetric about 90°.

.... . ................. .

-10'

o· 30' 60' 90" 120' 150" 180' w'

FIG. 7. LlX as a function of w for various values of the rotational angular momentum, J, for collisions with the molecule initially in the YZ plane (see Fig. 6) and rotating about the X axis. Other conditions are the same as in Fig. 3 and /3=1.17 (corresponding to xo=OO). - J =0 (!2=0),-- - J=80 (!2=20.7), ••• J= 150 (!2=38.8).

Figure 7 illustrates the dependence of Ax on the rotational angular momentum. The molecule again takes various initial orientations in the Y Z plane; the curve for j = 0 is analogous to that for the Y Z case in Fig. 5. In the presence of rotation the initial phase angle w becomes irrelevant, since rotation of course introduces a purely kinetic shift in the phase angle which is unrelated to the action of the potential. To eliminate this, Ax is plotted against w*, the value which the phase angle would have at the midpoint of the collision (distance of closest radial approach) if there were no interaction. For the moment of inertia used in these calculations (I = 270 g mole-1 Az), the maximum in the thermal distribution over rotational states of the molecule occurs at about j = 40fi for T= 300°K. Except for light molecules at extreme tem­peratures, the average duration of a collision (typically ,.....,10-13 sec) is shorter than the average rotational period (typically ,.....,10-12 sec at j,.....,40fi and inversely pro­portional to j); thus the effect of rotational averaging

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SCATTERING FROM A DIATOMIC RIGID ROTOR 3535

does not become apparent until J becomes quite large, and ~x~ rather slowly as J increases.

The variation of ~x with the asymmetry parameter is found to be approximately linear for a<2, as il­lustrated in Fig. 8. The linearity holds for a wide range of initial conditions, although the proportionality con­stant varies markedly with the collision energy, impact parameter, orientation, and angular momentum. For values of a> 2, the potential of Eq. (8) becomes physically unrealistic since the angular factor, 1+ aP2( cos,),), is then negative for certain orientations and therefore gives long-range repulsion and short­range attraction.

The results obtained in calculations using the po­tentials of Eqs. (7) and (9) are qualitatively the same except that some of the symmetry present in Figs. 5 and 7 is destroyed because these potentials contain terms of odd parity. As expected, the quadrupole­induced-dipole term in Eq. (9) has maximum effect near the region of glory scattering, where xo= 0 and the contributions of the Lennard-Jones terms cancel out. Also, the contribution of this term is approxi­mately linear in the parameter q (within 10% up to q<0.5).

Rainbow and Glory Scattering

Once the deflection function X is evaluated in terms of the collisional parameters, classical calculations of the angular distribution and other properties of the scattering may be carried out by making use of Monte Carlo procedures to average over distributions in the various parameters. Detailed calculations have not been done yet, but some qualitative features may be pointed out. The effect of the anisotropic forces and rotational perturbations is of particular interest in the region of the rainbow and glory scattering (see caption of Fig. 2). In atom-atom scattering these regions give rise to characteristic features which yield

6.x

10

2 A.,mmltry Parameter, a

3

FIG. 8. fox as a function of the asymmetry parameter, a, for two orientations with the molecule initially in the YZ plane (see Fig. 6). Other conditions are the same as in Fig. 3 and fJ = 1.17 (corresponding to xo=OO). 0, w=40o; £:,., w=10°.

• ;;. c c

! c

~

180·r---.,----....,-----,---,--__ .... ---,

30

r:! =O--~2---4~--~6--~8~--~IO~~

K

FIG. 9. The curve gives the rainbow angle, Xr, for a spherically symmetric potential as a function of reduced energy. The bars indicate roughly the deviations caused by varying the initial orientation of the molecule (with other conditions the same as in Fig. 3).

direct information about the potential. Near the rain­bow angle the angular distribution shows a maximum13•20

and since the position of the rainbow angle depends primarily on the reduced energy K = Ej E, the depth of the potential well can be determined from a measure­ment of the variation of Xr with energy.I2•21 The glory scattering produces interference with the small-angle scattering from the long-range attractive branch of the deflection function; this gives rise to small undulations in the velocity dependence of the total cross section which provide a measure of EIT, the "area" of the potential well.13 •22

Since ~x is small, in regions where the angular distri­bution is changing slowly the blurring effect of the anisotropy of the potential may be estimated from

I(x)"'Io(xo)+1o'(xo) (~x )+!1o"(xo) (~X2), (10)

where Io(xo) is the angular distribution calculated for the spherically symmetric part of the potential, and the correction terms are averaged over the ap­propriate range of angular momentum and orientations. Since ~x is distributed about zero in many cases, <~X) may be small, and the (~X2) term is often the dominant correction. In regions where the angular distribution changes rapidly the derivatives 10' (xo) and Io"(xo) may become very large and thereby amplify the blurring; thus, the rainbow and glory features are expected to be drastically affected unless the anisotropy is weak or effectively averaged out by rapid rotation. A rough estimate of the blurring of the rainbow maximum in the angular distribution may be obtained by calculating the range of rainbow angles for various orientations. Figure 9 shows the range of

20 E. A. Mason, J. Chern. Phys. 26, 667 (1957). 21 D. Beck, J. Chern. Phys. 37, 2884 (1962). 22 R. B. Bernstein, J. Chern. Phys. 37, 1880 (1962).

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3536 R. J. CROSS, JR., AND D. R. HERSCHBACH

x. (indicated by the bars) as a function of K for J=O and a=0.5 (the solid curve gives the result for a spherically symmetric potential). The maximum x. occurs in collisions in which the atom and molecule are collinear at the midpoint; the minimum x. in collisions with the molecule perpendicular to the plane of the collision (the XY plane of Fig. 6). At each energy, the range in x. is appreciably larger than the full width of the rainbow maxima observed for typical atom-atom systems.12.~I.~ Thus in atom­m.olecule scattering it is expected that the anisotropy wIll usually smear out the rainbow pattern. The available experiments have not yet established whether this happens. The atom-molecule systems for which rainbow maxima have been observed are all examples for which the effective anisotropy is expected to be weak; those for which the rainbows are severely dis­torted or absent are all examples for which damping by chemical reaction may occur.21 ,28,24

. The. impact parameters, (3.(K) and (3g(K), which gIve nse to rainbow and glory scattering are found to vary only slightly with the orientation of the molecule. For glory scattering, the results may be closely approxi­~ated by fl{3~-flxg/ag, where ag= (aXO/a{3)E, flXg IS calculated with {3 and K such that xo=O, and fl{3 is the increment in impact parameter required to mak~ Xg=O. Thus here again the effect of the anisotropy is a small, essentially first-order perturbation. Since the small-angle scattering requires a quantum treatment a classical calculation of the effect of the anisotrop; on the total cross section is not possible. Rotationally inelastic scattering is important in the glory region, and this is expected to diminish the undulations in the total cross section since the inelastic scattering cannot interfere coherently with the small-angle scattering from the long-range attractive branch of the deflection function to produce the undulations.

APPROXIMATE SOLUTIONS

From the viewpoint of computational convenience and physical insight, approximate solutions are more useful than the rigorous numerical calculation; also if an approximation is designed to be correct for ~ limiting case, it provides a welcome check on the com­puter program. Two approximations have been ex­amined: a "fixed orientation" approximation which reduces the calculation to superposition of scattering from a series of spherically symmetric potentials and a "flywheel" model which simplifies the equations of motion by means of classical perturbation theory.

liE. F. Greene, A. L. Moursund, and J. Ross, Advan. Chern. Phys.9, (1965).

"R. W. Roberts, Ph.D. thesis, Brown University Providence Rhode Island, 1959; M. Ackermann, E. F. Greene,'A. L. Mour~ sund, and J. Ross, Symp. Combust., 9th Cornell University Ithaca, N.Y. 1962,669 (1963). '

Fixed-Orientation Approximation

Mason and Monchick26 26 have developed a tractable treatment of transport properties of polar gases based on the fixed-orientation approximation. They point out that for most of the relevant range of {3 and K the largest contribution to the deflection angle comes from a small region near r=rm , the distance of closest ap­proach in the collision, since here the integrand of Eq. (1) becomes very large. Ordinarily the molecule do.es n~t have time to rotate much while traversing thIS regIOn. Thus the deflection in a given collision may be approximated by a calculation which proceeds as if the molecule remained fixed throughout the collision in the orientation appropriate to the distance of closest approach. For a fixed value of the orientation angle 'Y the potential becomes spherically symmetric, and the problem reduces to evaluating Eq. (1) for various values of 'Y and averaging the results over a suitable distribution of orientations .

This procedure is readily tested by comparison with the rigorous computer calculations since the fixed-. . ' onentatIOn approximation requires that at a given energy and impact parameter the scattering angle should be a function only of 'Ym, the orientation angle ~t ~he midpoint of the collision. As illustrated in.Fig. 10, It IS found that the variation of flx with 'Ym obtained from the computer results is usually quite different than that given by the fixed-orientation approximation. The calculations in Fig. 10 were done for various

.. ,

t:.x 0

Orientation An,le, Ym FIG. 10 .. Test of the fixed orientation approximation: .!lx is

pl?tted ~ga.mst 'Ym ,for ~he exact computer solution and the fixed onentatJon approximation. The dashed curve gives the out-of­plane an~le . (,:,~lue ~f !11'-e at the end of the collision). The mo~cule IS IDltJaIly ill a pl~n~ .45° to ~~e XYZ axes (OJ=45°, cf>{-45) and ,8=1.0; other illitial conditions are the same as in Fig .. 3. -.-, real trajectory; ••• - - -. ", fixed orientation ap­proXimatIon; ---, out-of-plane angle.

21 E. A. Mason, J. T. Vanderslice, and J. M. Yos Phys Fluids 2, 688 (1959). ' .

28 L. Monchick and E. A. Mason, J. Chern. Phys. 35 1676 (1961); 36,2746 (1962). '

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SCATTERING FROM A DIATOMIC RIGID ROTOR 3S37

orientations of the molecule in a plane at 4So to the X, Y, and Z axes. Each value of -Ym corresponds to two orientations of the molecule in space, since the atom may be going toward or away from the molecule at the particular relative orientation specified by -Ym. Each of these two orientations gives a different AX in the rigorous solution, but the same AX in the fixed­orientation approximation. The rigorous solution de­pends on the orientation of the molecule in three­dimensional space and not merely on the relative orientation of the atom and the molecule. Sample calculations for other impact parameters and orienta­tions of the molecules showed no better agreement with the fixed-orientation approximation. Also, since the fixed-orientation approximation makes use of a spherically symmetric potential, it requires the col­lision to take place in a plane perpendicular to the initial orbital angular momentum. However, as shown in Fig. 10, in the rigorous calculation the out-of-plane scattering angle is found to be comparable to AX, so that a good part of the total deflection due to anisotropy arises from the out-of-plane scattering.

The reason the fixed-orientation approximation proves inadequate can be seen by examining the equations of motion. If the potential is divided into a spherically symmetric term and an anisotropic term, Vo(r)+ AV (r, -y), the fixed-orientation approximation amounts to including the anisotropic contribution to the radial force, proportional to aAV I ar, but neglecting the angular torque, proportional to aAV la-y. The radial and angular derivatives are comparable in magnitude, and the aAvla-y term gives rise to out-of-plane scatter­ing if the molecule is out of the plane. An approximation that goes beyond the spherically symmetric approxi­mation must include these torque effects.

Flywheel Approximation

The fact that atom-diatomic-molecule scattering is very similar to atom-atom scattering suggests that a perturbation approximation might be used. The un­perturbed problem is the scattering from the spherically averaged potential,

Vo(r) =! (" VCr, -y) sin-yd-y, 210

(11)

and the perturbation is the anisotropic term of the potential, AV=V(r, -y)-Vo(r). This procedure saves computer time since AX is obtained directly as a small difference which need not be calculated very accurately to get a good result for the total scattering angle. The anisotropic term perturbs slightly the trajectories of both the atom and the molecule. The perturbation of the molecule produces a smaller second-order perturbation of the trajectory of the atom and so on. The flywheel approximation consists of ignoring this second-order effect and assuming that the rotation of

the molecule proceeds independently of the collision. This allows a simple formulation in which the aniso­tropic term imposes a time-dependent perturbation on the trajectory of the atom.

For the flywheel model it is convenient to set up the Hamiltonian in spherical coordinates as

(12)

with

and

where 8, I/> specify the rotation of the molecule and r, 8, <I> locate the atom with respect to the molecule (see Table I). The angle 'Y is given by

cos-y=cos(} cos8+sin8 sin8 cos(I/>-<I». (14)

Since the rotation of the molecule is regarded as un­perturbed, the "molecule" and "atom" parts of Eq. (12) are separately constant. In terms of the initial orientation of the molecule and its angular momentum, the flywheel approximation gives Hm= J2121 and

wet) =wo+Jt/I (lSa)

cos(}= -cosw sin8J (lSb)

tan (I/>-I/>J) =tanw/cos8J. (lSc)

The equations of motion for the atom as obtained from Ha are given by

r= prim; 8= Pelmr2; 4>= pif!/mr2 sin28 (16a)

p,= [pif!2/(mr3 sin28) J+ (pe2/ mr3) - (av lar) (16b)

pe= [cos8pif!2/(mr2 sin38) J- (av la8);

(16c)

The sole link to the rotation of the molecule occurs via Eq. (14). In zeroth order, the anisotropic part of the potential is omitted and since the atom then under­goes central-force motion in the plane perpendicular to the initial orbital angular momentum, 8 0= l7l', poo= 0, Pif!o= L; thus Ha reduces to

and Eqs. (16) are simplified accordingly. The equations for the perturbation corrections, obtained by simply subtracting these zeroth-order terms from Eqs. (16),

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3538 R. J. CROSS, JR., AND D. R. HERSCHBACH

are given by

Lli'= I1pr/m,

(l8a)

_aV(r,'Y)+aVo(ro). ( ) ar aro' 18b

I1p~= -al1v /a<I>;

I1Pe=[COS8p~2/(mr2 sin38)J-aI1V/a8; (18c)

where 111'=1'-1'0, etc. Integration of these equations was carried out with another IBM 7094 computer program, again based on a standard procedure,16

The results obtained from the flywheel approxima­tion are found to agree closely with the rigorous solu­tion over a surprisingly wide range of initial conditions. For example, for the conditions of Fig. 2 and reduced energy K=3, the values of I1x (flywheel) agreed to within 5% of the maximum I1x (exact) over the whole range of impact parameters; for K = 0.4 the agreement was within 10%. This degree of fractional agreement in the orbiting region is remarkable, in view of the large size of the deviations there. The flywheel approximation is designed, of course, to give the proper limit not only when the anisotropy is feeble or the rotation is rapid but also whenever the reduced mass of the molecule is much larger than the reduced mass of the atom and the molecule (p,»m). Since the convergence of the perturbation calculation depends on many variables, it is difficult to specify the region in which the flywheel approximation is useful; however, in practice this often can be judged from the size of the perturbation correction for any given case.

Reduction of Parameters

In the flywheel approximation the perturbation of the collision trajectory by the molecular rotation is governed solely by the relative position of the atom and the molecule. Thus the deviation I1x will depend on the angular velocity of the molecule rather than the angular momentum. The moment of inertia of the molecule is important only for light molecules where the molecular motion is severely distorted by the col­lision and the flywheel approximation breaks down. To obtain a suitable reduced angular velocity we may examine the ratio, X, of the period of the molecular rotation to the duration of the collision which is given by

X=~(;b)f(K, ~)=j(m:S(~Sf(K, ~), where f is a function of K and {3. Possible alternative definitions of the duration of the collision involve

changes in the function f. This suggests that a reduced angular velocity, n, may be used where

(19)

Exploratory calculations have shown that as long as n, K, and {3 are held constant m, u, E, I, and J can be varied over a considerable range without appreciably changing the scattering angle, as required by the fly­wheel approximation. It is not correct, however, to assume that I1x depends only on x. Figures 2, 3, and 7 show that I1x depends much more strongly on K and {3 than on J. Thus if K and {3 are changed keeping Iv/2bJ constant, I1x is changed much more than if K and (3 are held fixed while Iv/2bJ is varied.

Since I1x is linear in the asymmetry parameter a, a reduced deviation function I1x' can be defined such that

x= Xo(K, (3) +al1x' (K, (3, n, OJ, CPJ, w). (20)

Table II shows a test of this reduction scheme. Several exact calculations are compared for which the reduced parameters are the same although the absolute values of various parameters are varied over a considerable range. The flywheel model and Eq. (20) give the same result in all cases and the exact solution deviates only slightly from this. Since the reduction scheme has the same region of validity as the flywheel approximation, it will fail for light molecules or strongly anisotropic forces.

ANGULAR-MOMENTUM TRANSFER

The final angular momentum of the molecule, J', is also readily obtained from the computer program. Of course, a classical calculation is necessarily too lenient, as it does not restrict the increment I1J = I J' - J I to be an integer multiple of n and it allows all three components of the angular-momentum vectors to be determined. However, sample calculations were carried out to examine the qualitative functional de­pendence of I1J and to look for possible "pseudo­quantization" or "selection rule" effects. It was found that for a=0.5 and a heavy-molecule case (e.g., as specified by the parameters given in the caption of Fig. 2) the classical angular-momentum transfer is in the range I1J",,±O to lOn for a wide range of J (0 to lOOn), K (0.4 to 5), and {3 (0 to 2). This includes impact parameters which correspond to distances of closest approach well beyond the repulsive core of the potential. The range G-Wn is roughly that observed by Steinfeld and Klemperer27 in their recent studies of rotational momentum transfer in 12 perturbed by various gases. For both heavy and light molecules, the range in I1J is found to be approximately linear in the asymmetry parameter but the scatter is somewhat

27 J. I. Steinfeld and W. Klemperer, J. Chern. Phys. 42, 3475 (1965) •

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SCATTERING FROM A DIATOMIC RIGID ROTOR 3539

TABLE II. Test of reduction of parameters.

AX Ax E iT b m I J L w=30° w=70° (kcal) (mole-I) (1\.) (1\.) (g mole-I) (g mole-I 1\.2) h h

6.95 5.93 0.431 1.293 3.0 3.519 30 270 20 100 7.71 6.33 1.724 5.172 3.0 3.519 120 270 20 400 6.71 6.26 0.431 1.293 3.0 3.519 30 1080 80 100 6.95 5.93 1.724 5.172 3.0 3.519 30 270 40 200 6.40 6.27 0.108 0.324 3.0 3.519 7.5 270 20 25 6.99 5.93 0.431 1.293 1.5 1. 760 30 270 10 50 6.73- 6.28"

" Calculated from the flywheel approximation with reduced parameters: a=O.S, K=3.0, J9=1.173,fl=s.17,IJJ=4S, <PJ=O, xo=O.OSo (near the glory angle).

larger than that found for .lx (see Fig. 8). Also, as J is increased to very large values the angular-momentum transfer approaches zero because the asymmetry is averaged out during the collision. This is again in analogy to the behavior found for .lx and the rotational quenching sets in at about the same range of J (see Fig. 7). No significant tendency towards spatial polari­zation of .lJ with respect to the initial orbital angular­momentum vector or the initial plane of the collision could be discerned.

In the flywheel approximation total angular mo­mentum is not conserved, since the torque exerted on the atom by the freely rotating molecule is included but the corresponding torque on the molecule is omitted from the equations of motion. However, the change in the orbital angular momentum, .lL, can be evaluated from the trajectory of the atom and then .lJ can be obtained by use of the conservation law. This devious procedure works, as shown in Fig. 11 which provides a comparison with .lJ obtained from the exact computer solution. This treatment of the flywheel model (the "heavy mass" case in Fig. 11) closely approximates the exact result except when the reduced mass of the atom and the molecule is rather small (IL<20 g/mole).

A similar perturbation treatment may be constructed for the "low mass" case. Again, the zero-order problem is the free rotation of the molecule and the collision of the atom and molecule in the spherically averaged potential. In this case, however, the perturbation of the molecular rotation is examined, and the atom is assumed to collide in the unperturbed trajectory. The equations are very similar to (11)-(18) given above for the flywheel model; 8 is replaced by 0, and <P is replaced by cf>. As before, 00 and cf>o are calculated from the free rotation of the molecule. Using this method one can obtain .lJ but, of course, not .lx. This becomes the exact solution in the limit of an infinitesimal mo­ment of inertia, and would be expected to be a good approximation to the classical solution for small molecules. Figure 11 shows the variation of .lJ with the reduced mass of the molecule, IL, for a given orien­tation impact parameter, and energy. It is easily seen that ;he flywheel model is a good approximation for high f.L and the low-mass approximation is good for

small f.L. As f.L goes to zero, .lJ also goes to zero as might be expected.

DISCUSSION

The fact that .lx is so small means that it will be difficult to determine the anisotropy of the potential from scattering data unless some form of rotational state selection or analysis is used9 or unless the potential is highly anisotropic. On the other hand, molecular scattering data taken without state selection can be analyzed with a fair degree of accuracy in terms of a spherically symmetric potential.

The limitations of use of classical mechanics in this problem should be emphasized. Classical mechanics fails for small-angle scattering because of an un­certainty principle limitation. The methods of this paper thus cannot be used for calculations of aniso­tropic effects in small-angle scattering or total cross sections. Also, if the molecule is very light, the quan­tized rotational-energy-Ievel spacings are comparable to the potential energy and this should properly be taken into account in the theory. The classical treatment, in

-; 12 1;

~ 10

10 20 15O 100 200 l500 IOO(

1'. Rtducod Mall .1 R.I.r (9lmaltl

FIG. 11. Angular-momentum transfer as a function of !he reduced mass of the molecule, p., for the exact computer solutlOn (-) the flywheel approximation (- - -, "high mass" case), and ~ "low mass" perturbation approximation (-.-.-). In these calculations, the angular velocity has been kept c~mstant a~ p. is varied by increasing the initial angular momentum In p~oporbon to p. (such that J=Nh for p.=N g mole-I). The other lrubal c~m­ditions are the same as in Fig. 3 except that J9 = 1.0 (corresponding to xo=37°).

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3540 R. J. CROSS, JR., AND D. R. HERSCHBACH

this case (d. Fig. 11), gives a small boJ, not an integral multiple of n. A quantum-mechanical analog of the low-mass approximation has recently been applied to K+HBr scattering. l1 Thus it is gratifying that the classical low-mass approximation agrees closely with the rigorous solution for such systems.

It should also be noted that the various quantum­mechanical perturbation theories differ fundamentally from the classical perturbation treatments examined here. The quantum perturbation theories such as the Born approximation or the distorted wave approxi­mation assume that the probability of rotational exci­tation is small compared to unity. Except for very small-angle scattering (where the potential is small) or hydride systems (where the rotational levels are widely spaced) the probability of rotational excitation is large, and therefore quantum perturbation theories are drastically restricted in application.1.2 The flywheel

model, on the other hand, assumes that the motion of the molecule is not severely affected by the collision, which is a less stringent requirement.

ACKNOWLEDGMENTS

This work was begun at the University of California at Berkeley under the auspices of the Inorganic Ma­terials Division of the Lawrence Radiation Laboratory, supported by the Division of Research, U. S. Atomic Energy Commission, and was continued at Harvard with the support of the National Science Foundation. Computer time was provided by the Harvard Com­puting Center under National Science Foundation grant GP-2723 and by the Massachusetts Institute of Technology Computation Center. The authors have enjoyed many discussions with Dr. R. G. Gordon and Dr. J. I. Steinfeld.

THE JOURNAL OF CHEMICAL PHYSICS VOLUME 43, NUMBER 10 15 NOVEMBER 1965

Molecular Orbitals of Diborane in Terms of a Gaussian Basis LOUIS BURNELLE* AND JOYCE J. KAUFMAN

Research Institutefor Advanced Studies (Martin Company), 1450 South Rolling Road, Baltimore, Maryland

(Received as Note 12 May 1965, Article received 21 July 1965)

The molecular orbitals of diborane have been calculated by the nonempirical SCF-LCAO method, taking all electrons of the molecule into account and calculating exactly all the integrals. The basis functions were Gaussians centered on the various atoms of the molecule. Two sets of Gaussians have been used, the largest one containing 54 functions, distributed as follows: 9'+3p on each boron, and 3' on each hydrogen. The ordering of the molecular orbitals on the energy scale agrees with that found by Newton, Boer, Palke, and Lipscomb who have used an almost optimized minimal STO basis. The calculated total molecular energy, -52.753 a.u., is 0.468 a.u. higher than the experimental value. A value of 12.9 eV is obtained for the ionization potential, which is to be compared with experimental results ranging from 11.9 to 12.1 eV. An electron-population analysis indicates the presence of very small gross atomic charges (qB= -0.088e; /]H(bridee) = -0.002e; qH(terminal) = 0.045e). The overlap populations are as follows: n (B, B) = 0.064; n (B, Hterm) =0.880; n (B, Hbridee) =0.345. These suggest a structure involving practically no bond between the borons, the bonding in the central part of the system being mainly concentrated in three-center banana bonds.

INTRODUCTION

THE electronic structure of diborane has already been the object of a certain number of theoretical

studies. Among the quantitative approaches which have been made within the frame of MO theory, it is worth mentioning the semiempirical treatment of Hoffman Lipscomb, l which rested on an extended Huckel-type method, and the nonempirical studies of Hamilton2

and Yamazaki.3 Our attention has been drawn by the referee to an article by Newton, Boer, Palke, and

* Present address: Department of Chemistry New York Uni­versity Washington Square College New York, New York 10003.

1 R. Hoffman and W. N. Lipscomb, J. Chern. Phys. 37, 2872 (1962) .

2 W. C. Hamilton, Proc. Roy. Soc. (London) Al35,395 (1956). 3 M. Yamazaki, J. Chern. Phys.17, 1401 (1957).

Lipscomb,4 which appeared after submitting our origi­nal Note, and which mentions results of an SCF-LCAO treatment of diborane which includes all the electrons of the molecule. Comparison with the results of Newton et al. are made as we proceed along the present paper.

Hamilton2 approximated the molecule to a four­center, four-electron system, by neglecting the terminal hydrogens. The two-electron BH bonds were considered as part of the core acting on the four valence electrons and were assimilated to boron'hybrids pointing toward the hydrogens and containing" one electron~each. How­ever, it appeared later that the excIusionof the elec­trons of the terminal BH bonds from the valence electrons was not justified. Indeed, Yamazaki,3 who

4 M. D. Newton, F. P. Boer, W. E. Palke, and W. N. Lipscomb, Proc. Nat!. Acad. Sci. 53, 1089 (1965).

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