Rotational Energy Transfer in Molecular Transitions in Parahydrogen Collisions : BY W. D. DAVISON Dept. of Theoretical Chemistry, University Chemical Laboratory, Lensfield Road, Cambridge Received 13th December, 1961 The theory of rotational transitions in the collision of two diatomic molecules is outlined, and the effect of quantum statistics discussed. A general expression is obtained for the matrix elements of the angular terms appearing in a non-spherical potential function. The full distorted wave method is employed to calculate the cross-sections for the 0,O +2,0 and 2,0 +4,0 rotational transitions in parahydrogen, the anisotropy of a " exp-six " potential being varied to secure agreement with ultrasonic dispersion measurements. ~~~~ ~ 1. INTRODUCTION In the collisions of light diatomic molecules in the gas phase, the inefficient transfer of rotational energy gives rise to the well-known phenomena of the rotational absorption and dispersion of ultrasonic waves.1 The rotational relaxation effects behind shock fronts2 and in thermal conductivity measurements 3 are also of current interest.The theoretical treatment of such problems has received much attention recently ; in particular, the cross-sections for transitions between the first few rotational levels in parahydrogen have been calculated with reasonable success by Beckerle,4 Brout 5 and Takayanagi,6 the latter author providing the most com- prehensive discussion. The chief source of uncertainty in such work is our limited knowledge of the details of the intermolecular potential function ; on the other hand, the many practical difficulties which arise in a detailed treatment have led previous workers to introduce several simplifying approximations.A more accurate analysis is here described. n 0 FIG. 1 .-Collision of two diatomic molecules. 2. COLLISION THEORY For the experimental situations in which we are interested, it is an excellent approximation to regard a diatomic molecule as a rigid rotator. The theory of 7172 ROTATIONAL ENERGY TRANSFER the scattering of an atom or ion by such a molecule has been examined in detail by Arthurs and Dalgarno,' and we first outline the straightforward extension of their treatment to the inelastic collision of two rigid rotators. Although we shall not fully exploit this more general formulation in the present work, it is required for other calculations ; for example, the estimation of the cross-section for simul- taneous transitions in the collision of polar molecules.For the moment we suppose that the two molecules are distinguishable (cf. 5 3 below). Atomic units are used throughout. The configuration of the system is described with respect to a set of space-fixed axes as indicated in fig. 1 ; GI, G 2 are the centres of mass of the two molecules, and we shall use the abbreviation f l = (61,41), etc. We make the following definitions : of the ith molecule I rotational Hamiltonian reduced mass moment of inertia rotational angular momentum quantum numbers reduced mass of the whole system, intermolecular potential, Laplacian for the relative motion of the two molecules, relative translational energy before the collision, quantum numbers for the resultant rotational angular momentum of the two molecules (j12 = j1+ j2), orbital (or relative) angular momentum, total angular momentum quantum numbers (J = j12+1).It is convenient to denote the frequently recurring groups of indices jl j2 j12Z and j2jlj121 by the single symbols n and ii respectively. If the molecules are initially in rotational states j1, ml and j2, m 2 , any wave function Y( j1,j2 I R,?& describing the complete system must satisfy the wave equation where and is the total energy of the system, the motion of its centre of mass having been separ- ated out in the usual way. We expand Y(j1, j2) in terms of the basic set of functions, CV~M(~~~P,,~J = C C ( j121mM-m I JW x + ~ I Z +it (4) m = - j l z r n l s - j l (jljzmlm - ml1 jlzni) q M -m&> Yjlml(il> Yj2m-ml(iz)y in which the rigorously good quantum numbers Jand M appear explicitly ; (abcrp I cy) is the Wigner vector-cou]i.ling coefficient, and the Y's are normalized spherical har- monics.As is usual in the stationary-state formulation of collision theory which we have adopted here, our ultimate aim is to obtain the solution of (1) which, for large values of R, represents an incoming plane wave carrying the initial rotational states, and outgoing spherical waves associated with various possible final states. First. however. it is convenient to consider the less general situation in which J, M ,W. D. DAVISON 73 j12 and I are sharply defined before the collision. Noting that J and M will remain unchanged throughout, the wave function for the system may then be written Y,"'(R,f,,f,) = CR- 'u$(R>Yi?(&,,i2).( 5 ) n' In order that (5) should represent the scattering of an incoming spherical wave, the radial functions are required to have the asymptotic form ui!'(R) - exp [ - i ( kI2R-- ; ) ] - ( z y S J ( n ; n ' ) exp R-+W SJ(n;n') is an element of the complex scattering matrix, the knowledge of which is sufficient for a complete description of the collision. Substitution of (5) in (1) yields the set of coupled equations where A (n'J I Y I n"J> = Yi?'VY,J.fndRdf,dt2 SSS and the wave numbers before and after a jl,jz+ji, j i transition, k12 and ki2 = k(j1,j2;ji, j ; ) , are given by We note that the matrix elements (8), as well as being diagonal in J and M, are also independent of M, since Vis invariant under rotation of the co-ordinate axes.To investigate the scattering of a pZane wave carrying the initial states and in- cident along the axis @ = 0, we combine the spherical waves (5) to give Q(j1,j2) = exp (ik,,R cos @)Y)12! Yj;;: (10) (jI2ZmO I JM)( j , j2mlm2 I j12m)( j;2Zrnz'M - m' I JM)( j ; j;m;mi I ji2m') x where TJ(n;nl) = Sn,nt - SJ(n;n!).74 ROTATIONAL ENERGY TRANSFER The total cross-section for the j l , j z + j i , jb transition is then given in terms of the scattering amplitudes by c C, J’ I 4(j1j2mIm2; j; iim;mk I*R) 1 2 ; ~ ~ (15) m l m ml,m2 where we have averaged over ml,m2, and summed over m;,rnL. (This procedure leads to an important simplification in subsequent expressions for the cross-section, but is, of course, only valid for an assembly of molecules under conditions such that there is a uniform distribution of molecules amongst the magnetic substates associ- ated with a given angular momentum.) After considerable analysis, in the course of which the summations over the magnetic quantum numbers are eliminated by an extension of the work of Blatt and Biedenharn,g (1 5) is finally reduced to The method used for the approximate evaluation of (16) depends on the specific problem under investigation. For parahydrogen, the change in relative translational energy during a rotational transition is large, while the departure from spherical symmetry is small, so that the non-diagonal matrix elements are much smaller than the diagonal elements ; the distorted wave approximation is then appropriate.We take as a first estimate to the solution of (7), u$ = ann,u$, (17) where o$ is the solution of the elastic scattering equation, which has the asymptotic form The phase shifts ~2 provide a measure of the distortion of the spherical waves by the potential. Inserting (17) in (7), we obtain an improved solution 9 with the asymptotic form, where Obtaining the scattering matrix by comparison with (6) and substituting in (16), the distorted wave approximation to the total inelastic cross-section is found to beW. D. DAVISON 75 The distorted wave approximation is essentially a first-order perturbation pro- cedure, based on a zero-order situation in which all non-diagonal matrix elements of the potential are neglected, so that only elastic scattering under the diagonal matrix elements is possible.The partial wave expansion of the familiar Born approxim- ation is obtained if in addition we neglect the diagonal matrix elements-in the classical limit this corresponds to replacing the actual curved orbit by a straight line path (the method of impact parameters). In most cases, however, the Born approximation leads to a gross overestimation of the cross-section, since the dis- tortion is much too great to be neglected in this way. This is because the inelastic scattering is controlled principally by the strong repulsive interactions at small separations, so that, of the partial waves which contribute significantly to the in- elastic cross-section, a large proportion have associated phase shifts of the order of several radians.A notable exception is provided by the pressure broadening of microwave spectral lines ; this can be explained in terms of the rotational transitions induced by long-range interactions, the diagonal matrix elements of which are often zero. 3. THE COLLISION OF IDENTICAL MOLECULES When the molecules are of the same species, a complete discussion should take account of the modifications in the theory which are required by quantum statistics. For Bo s e-Eins t ein stat istics (e. g . , parahydro gen and other homo nuclear molecules) , any wave function describing the complete system must be unchanged by the trans- formation (R,rl,r2)-+( - R,rz,rl), corresponding to interchange of the two molec- ules; in the Fermi-Dirac case, such a function is changed in sign only.For sim- plicity we shall only consider the collision of homonuclear molecules with zero resultant nuclear spin; the more general situation has been discussed by Kerner.10 To facilitate comparison with the " unsymmetrized " theory of the previous section, it is convenient to retain the expansion (5) of Yi". We write ~ ' = j ; + j ; - - j ; ~ + l ' . Then, since the symmetry requirement introduces the restriction u,JI = (- 1y'u;lr: (24) except when j ; = j)2 and I' - j ; , is odd, in which case u$ is zero. We proceed with the analysis as before, bearing in mind that the order of j ; , j.; in the scattering matrix has now no significance (corresponding to the fact that it is no longer possible to " label " the molecules).We use a result of Gioumousis and Curtiss 11 to obtain the final expression for the " symmetrized " cross-section. In the distorted wave approximation this takes the form * where * Tildes are used to denote the " symmetrized " form of quantities previously defined in 5 2.76 ROTATIONAL ENERGY TRANSFER * ad? has the asymptotic form (19), as before, but is now the solution of (- l)”(G’J 1 V 1 dJ>} u:!’ = 0, (27) 1 where The cross-section (25) refeis to a collision in which two molecules, initially in states jI,j2, are finally in states j ; , j i , either the transition jl+ji,jz+ji or the transition jl+j;,j2+j having occurred. In many cases the exchange terms in (27) are small, so that ~ ~ ~ ~ c I I $ and (const.)x (- 1>.’(2J+ l)fl;!’&!! (30) Provided that a sufficient number of partial waves contribute significantly to the cross-section, the last term of (30) is negligible ; then J 1J’ j 1 2 , j i i and the effect of quantum statistics vanishes in the classical limit, as expected.Simultaneous transitions in both molecules have not been considered in the present calculations, so it is convenient to express the results in terms of the cross- section, Y ry Q(jl ,j2+ i A) = , j 2 ) U , j 2 Q f h ,h-4 , . j 2 ~ Q ( . i 1 J 2 -+j; ,L>. 4. MATRIX ELEMENTS OF ANGULAR-DEPENDENT POTENTIALS The potential between two rigid diatomic molecules may be expanded in terms of Legendre polynomials : ~ ( ~ ; 1 9 ; 2 ) = C urst(R)pr(cos X I 2 P s ( c o s )P~(cosx~>, (32) r,s,t where ~ 1 2 is the angle between rl and r2.At present, it is not possible to give more than rather rough estimates of the first few radial factors v,,t(R), and it is recognized that quite small variations in these functions can lead to marked changes in the calculated cross-sections. It is to be hoped, however, that following the recent progress in the ab initio calculation of molecular properties, it may soon be possible to carry out accurate investigations of the angular dependence of the potential between some simple systems. We therefore seek a general expression for the matrix elements of the angular factors in (32). The most compact formulation is in terms of the tensor algebraW. D. DAVISON 77 of Fano and Racah; 12 reference should be made to this book for a discussion of the notation employed.Expanding in spherical harmonic tensors, we have : s + t CAI Is1 Ctl CAI = c u [C(E)xC(Z)] I= I s-t I (34) where C(Z) denotes the spherical harmonic tensor associated with k, (abc)o is a contraction for the coefficient (abOO 1 cO), and CAI Crl Crl Is1 [tl [A1 u = (C( 1) C(2))[C( 1) x C(2)] . Then (n'J I Pr(cos xl2)Ps(cos xl)P,(cos x2) 1 n J ) = xi"+'-' (- 1)J+"+j12 (st40 x il It may be shown that r + s r + t [a1 IS1 CAI - W(upA/tsr)[C(l) x C(2)] . (38) Substituting in (37) and evaluating the reduced matrix elements, we deduce that {n'J 1 P,(cos xl2)Ps(cos xl)P,(cos x2) I n J ) = (- 1)J+j12+ji+ji+r~ a,BJ (rsa>o(rtP>o(st4o( j i j al0(ji j2P) o(l' Wo x [(2j1 + 1)(2j; + 1)(2j2 + 1)(2j; + 1)(2jI2 + 1)(2ji2 + 1)(2Z+ 1)(21'+ I)]* x - w(uPL/tsr)%ji 2 j 1 ~ W J ) X C ~ L ~ l a ~ j i j2~/ji2.il (39) For any given problem, the matrix elements (including, if necessary, those of multipole interactions) can be straightforwardly derived as particular cases of (39) ; when a zero appears in any coefficient, the indices can always be permuted so as to make use of such relations as (aOb)o = 8ab, - W(abc/deO) = (- l)"+b+c6a,6bd[(2a + 1)(2b + 1)]-+, X(abc/def/ghO) = (- l)b+c+d+g6cf6gh[(2f + 1)(2g + l)]-*i%(abc/edg).(40) For the present calculations we iequire only (n'J 1 Ps(cos xl) I n J ) = (- 1)J+i2+s6j,ii(2s+ l)-'( j; jls)o(lfZs)o x [(2j1 + 1)(2j; + 1)(2j12 + 1)(2ji2 + 1)(2Z+ 1)(2Z'+ I)]+ x - W( j i 2 j 2s1 u J>WC j i z j sij j i j 2). (41) The Racah W-coefficients have been tabulated for limited ranges of the indices only, but can be computed using finite series expressions.78 ROTATIONAL ENERGY TRANSFER 5.ROTATIONAL TRANSITIONS I N PARAHYDROGEN The collisional excitation of the first vibrational level in hydrogen has been studied by Salkoff and Bauer,l3 these authors being the first to employ the distorted wave method without approximation in such a problem. As a simple application of the foregoing theory, we have carried out an analogous investigation of the rotational problem, confining ourselves to the 0,0+2,0 and 2,0+4,0 transitions in para- hydrogen. The numerical work was performed on the EDSAC 2 computer in the University Mathematical Laboratory, Cambridge. (i) H2-H2 POTENTIAL We have used two simple representations of the potential between two hydrogen molecules.The first has the Morse form V(R,t1,i2) = A exp [ - 2a(R - R,)] - 2A exp [ - cc(R - R,)] + PA exp c - 24R - Ro)IP2(cos Xl) + mcos X2)J (42) where A = 1.1 x 10-4, a = 0.935, Ro = 6.4 and B = 0.075. This potential has already been used by Takayanagi ; 6 the repulsive terms provide a reasonable fit to the theoretical calculations of Evett and Margenau 14 when higher-order terms in the expansion (32) are neglected. In Evett and Margenau’s work, however, the approximation of certain of the rnulticentre integrals reduces the predicted anisotropy of the potential, while in the subsequent calculations of Mason and Hirschfelder 15 there are other approximations which lead to an abnormally high angular dependence -corresponding to PeO-4 in (42).The “ shape ” of the hydrogen molecule is certainly not far from spherical, and we might reasonably expect B to lie in the range 0.1 to 0.2. In the second potential used in our calculations, we have to some extent allowed for this uncertainty in the short-range potential terms, at the same time taking more careful account of the long-range attractive forces. We use the “ exp-six ” function V(R,t1,+2) = A exp [ - 2a(R - R,)] - BR-6 + (PA exp [ - 2a(R - R,)] - DR+) x where A, a and Ro have the same values as before, B = 11.0, D = 0.8 and p is re- garded as a parameter to be varied. The long-range R-6 terms are based on the calculations of Britton and Bean.16 The quadrupole-quadrupole interaction and other higher order terms have been omitted, their matrix elements being very small or zero for the rotational transitions considered in this paper.These terms are, of course, important in simultaneous transitions in the two molecules, but con- sideration of these transitions must be deferred until the higher-order terms in the short-range potential are known with some accuracy. [PZ(COS Xl) +P,(cos X2)IY (43) (ii) NUMERICAL INTEGRATION The Adams method 17 was employed for the numerical integration of the radial eqn. (27). The potential was taken as effectively infinite for R<2-0, so that the integration of each equation was begun at R = 2.0 with the initial value of the solu- tion zero and with a suitably small initial gradient, and continued outwards to R = 15.0, beyond which point the potential is quite negligible.The integration interval was given its smallest value at small R (where the potential is varying rapidly), and was increased twice as the integration proceeded outwards; it was, however, kept sufficiently small over the whole range so that at higher impact energies theW. D. DAVISON 79 rapid oscillations in the solutions were accurately reproduced. The phase shifts and normalization constants for each solution were found by comparing the values of the solution at R1 = 14.5 and R2 = 15.0 with the values of the corresponding spherical Bessel functions at these points. Thus, if a solution with the asymptotic form, and The spherical Bessel functions required were generated without appreciable error by means of the standard recurrence relation.(iii) c A L c u LA TIONS In our first calculations, the cross-section &0,0-+2,0) was evaluated using the potential function (42). J is restricted by symmetry to even values only; the four radial equations for each value of J (corresponding to I’ = J+2,J,J-2 and I = J ) were integrated simultaneously. The required angular matrix elements were first computed and then combined with the appropriate radial factors at each step of the integration to form the diagonal and coupling matrix elements. On completion of the integration, the contributions to the cross-section from the current value of J were printed, together with details of the solutions. The calculations were then repeated for further values of J until the contributions to the cross-section became negligibly small.The computed cross-sections for several incident energies are included in table 1. TABLE 1 .-CROSS-SECTIONS FOR ROTATIONAL TRANSITIONS IN PARAHYDROGEN (a;). incident wave number, kl2 244 2-60 2.80 3.10 3.50 3.73 400 4-50 5.00 5-50 6.25 T = 197.1”K <Q,y> { T = 298.4”K e” (0,@+2,0; kl2) Morse exp-six (16 = 0-14) 0~0oO0 0*0000 0.0362 00551 0.0755 0.1335 0.1508 0.2964 0.27 13 0.583 1 04416 1.0146 0.8118 2.0416 - 3.5324 3.3x 10-2 6 . 8 ~ 10-2 2.2 x 10-1 - - - - - - - 0~0000 0 * 0 0 5 0 2 0.0374 0.1121 0.2331 0.5006 2.2 x 10-4 2 . 8 ~ 10-3 When we come to consider transitions involving states of higher angular momentum, we are faced with the practical difficulty of integrating a much larger number of radial equations for each value of J.The consequent increase in com- puter time is hardly worthwhile in view of the rather rough potential functions available. If, however, in the radial eqn. (27), we neglect angular distortion (re- placing the diagonal matrix elements by the spherical part of the potential), the80 ROTATIONAL ENERGY TRANSFER number of equations to be solved is greatly reduced. Provided that the departure from spherical symmetry is small, the error in the final cross-section introduced by this approximation is not serious; some sample calculations have indicated that in the present case it is at the most a few pzr cent. Following this simplified procedure, we have evaluated the cross-sections Q(0,0--+2,0) and 5(2,0-+4,0) using the potential function (43). It was found convenient to carry out the calculations in two stages.The distorted waves were first computed for many values of J, and transferred to permanent storage. In the second part of the programme, they were called down to the fast store as required, and integrated together with the appropriate angular matrix elements to form the integrals (26). This scheme was made sufficiently general (especially with regard to the evaluation of the Racah coefficients) that, given the relevant angular terms in the potential function, the cross-sections for other transitions could be computed with equal facility. Furthermore, the programme was so devised that the cross- section corresponding to any desired values of the potential parameters /? and D could be easily deduced from the final output.The variation of a and A exp (2aRo) Y J FIG. 2.-Partial cross-sections as a function of J. (a) Z(0,O +2,0 ; 2.60) X 10, (b) Z(0,O +2,0 ; 6*25), (c) Z(2,O +4,0 ; 4.50) X 102, ( d ) &2,0 +4,0 ; 5.50)X 10. Note that contributions to -&0,0-+2,0) come from even values of J only. is not so straightforward and was not attempted. For a potential of the form (42), the cross-section (neglecting angular distortion) is proportional to /?2 ; with the in- clusion of a long-range angular term in (43), the cross-section is considerably reduced for a given value of 2, particularly near the threshold, and is somewhat more sensitive to variation of p. The cross-sections corresponding to p = 0.14 lead to satisfactory agreement with experiment (see below), and are included in table 1.In fig. 2, the contributions to the cross-sections from individual values of J have been plotted against J for some typical energies. It will be seen that at the highest energies con- sidered there are significant contributions from values of J up to Jfi30.W. D . DAVISON 81 (iv) COMPARISON WITH EXPERIMENT In correlating our results with the dispersion measurements of Rhodes,ls we assume that the probability of a molecule making a transition is independent of the rotational state of its collision partner (in particular, we ignore the possibility of simultaneous transitions). This assumption is consistent with the simple form we have adopted for the potential function, and allows us to interpret the rotational dispersion in terms of a simple three-level mechanism.The deviations from perfect gas behaviour are sufficiently small to be neglected. incident wave number, k12 FIG. 3.-Averaging over the Maxwellian distribution at 197.1 O K . (4 M(kl2)Y (6) G(0,O +2,0 ; 4 2 ) x 10-1, (4 Z(2,04,0; k12), (4 M(k12) x e'C0,o +2,0 ; k12) x 10, (e) M(k12) x S(2,O +4,0 ; k12) x 103. It may then be shown that the probability per second that a molecule in the rotational state j will make a transition to the state j' is where p is the pressure, k the Boltzmann constant, and ( Q j y ) the cross-section for the j-j' transition averaged over the Maxwellian distribution of incident wave numbers ; < Q j j * ) = F ( k 1 2 ) i i ~ , o + j i o ;k, 2 ~ 1 2 ; Fig. 3 illustrates the formation of the integmnd in (48).82 ROTATIONAL ENERGY TRANSFER Solving the relaxation equations for a periodic disturbance of frequency 0/2n.we obtain the following expression for the effective rotational specific heat per mole at temperature T : E2, E4 are the energies of the j = 2,4 levels respectively, and ni is the number of molecules per mole in the j = 2 level at equilibrium. The ultrasonic dispersion curve may be calculated from the expression for the complex sound velocity V,. As expected, the Morse potential (which was considered chiefly for the sake of comparison with Takayanagi's less rigorous cal- culations) yields a value for ((302) which is clearly too low. If we choose p = 0.14 in the potential (43), the calculated dispersion curve is in satisfactory agreement with Rhodes' measurements at 197-1"K (see fig.4). At room temperature, however, the theoretical curve (with the same value of p) is too high at lower frequencies. I I 1 1 I I I f I 1 1 1 I I I I I I 10 100 4 2 n p (Mclatrn) FIG. 4.-Rotational dispersion in parahydrogen ; expt. points (Rhodes) : x , 197-1°K ; 0, 298.4"K. This can be partly attributed to the limitations of the potential (43), especially the assumption that the spherical and angular terms have the same exponential de- pendence on R. However, as pointed out by Takayanagi,lg the discrepancy probably arises for the most part from the neglect of simultaneous transitions, particularly the 2,2++4,0 transition, where the change in relative translational energy is com- paratively small. 6. DISCUSSION Takayanagi has made extensive use of the " modified wave number " approxim- ation, in which the centrifugal potential Z'(Z' + l)/R2 in the radial equations is replacedW.D. DAVISON 83 by its value at a distance R,, chosen to be of the order of the distance of closest approach in a typical collision. The problem is thereby reduced to that of a one- dimensional collision with effective wave number kLff given by if the s-wave (I’ = 0) radial equation can be solved in closed form, the direct numerical solution of the radial equations is avoided, and considerable simplification results. This procedure (with some refinements) has proved valuable in exploratory cal- culations, for vibrational as well as rotational transitions, but has two obvious drawbacks. First, it is useful only within the very restricted range of potentials for which an analytic solution of the s-wave equation is available (thus, it is of no help in calculations with the potential (43)).Secondly, the arbitrariness of the parameter R, introduces an added uncertainty into the final cross-sections, in con- trast to an accurate numerical solution, where the adoption of a given potential function leads to unambiguous values for the cross-sections. For the present problem, Takayanagi 6 has used the Morse potential (42) and obtained cross-sections proportional to x = p2Rz ; R, was chosen so as to provide the best fit to expe. I rment at lower temperatures.19 The final dispersion curves are very similar to those in fig. 4. This semi-empirical procedure gives us little information as to the “ best ” values of p and R, ; the former quantity is of interest in that it provides a measure of the actual anisotropy of the potential, the latter in that it might serve as a guide in further applications of the modified wave number approximation, particularly in cases where comparison with experiment is not yet possible.The present calculations were undertaken in the hope of securing closer agree- ment with experiment at room temperature. In fact, although there are differences in detail which are only to be expected, our final results are in overall agreement with those of Takayanagi, and it would appear that no further improvement is possible within the limitations imposed by the potentials (42) and (43). In retrospect, the chief value of our investigation would seem to lie in the fact that, having successfully overcome many of the computational difficulties associated with this kind of work (for example, the accurate generation and manipulation of a large number of distorted waves, and the evaluation of the matrix elements), we can be confident that the wave eqn.(1) has been solved with reasonable accuracy, and that defects in the potential function are therefore responsible for most of the remaining discrepancy between theory and ex@riment. It is clear that accurate calculations of this kind may soon provide an important link between the theoretical prediction of inter- molecular potentials and, for example, the direct measurement of individual cross- sections by molecular beam techniques. For rotational transitions in the collision of heavier molecules, the effect of higher-order coupling in the eqn.(7) cannot be neglected. For vibrational transi- tions, on the other hand, the distorted wave procedure is probably adequate in most cases. As an outgrowth of this work, programmes to treat such problems are at present being developed. I am grateful to Prof. H. C. Longuet-Higgins, F.R.S., for his help and encourage- ment throughout this work, and to Prof. A. Dalgarno and Dr. A. M. Arthurs for some valuable discussions. I also wish to express my thanks to the Director and Staff of the University Mathematical Laboratory, Cambridge, for the generous facilities afforded to me on EDSAC 2, and to Gonville and Caius College for the award of a Rhondda Studentship.84 ROTATIONAL ENERGY TRANSFER 1 Herzfeld and Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic Press, New 2 Andersen and Hornig, Mol. Physics, 1959, 2,49. 3 Srivastava and Barua, Proc. Physic. SOC., 1961, 77, 677. 4 Beckerle, J. Chem. Physics, 1953, 21, 2034. 5 Brout, J. Chem. Physics, 1954, 22, 934. 6 Takayanagi, Proc. Physic. SOC. A, 1957, 70, 348 ; Sci. Reports Saitama Univ. A, 1959, 3, 37 ; 7 Arthurs and Dalgarno, Proc. Roy. Sac. A, 1960, 256, 540. 8 Blatt and Biedenharn, Rev. Mod. Physics, 1952, 24, 258. 9 Mott and Massey, The Theory of Atomic CoZIisions (Clarendon Press, Oxford, 1949). 10 Kerner, Physic. Rev., 1953, 91, 1174. 11 Gioumousis and Curtiss, J. Chem. Physics, 1958, 29, 996. 12 Fano and Racah, Irreducible TensoriaZ Sets (Academic Press, New York, 1959) ; see also 13 Salkoff and Bauer, J. Chem. Physics, 1958, 29, 26. 14 Evett and Margenau, Physic. Rev., 1953, 90, 1021. 15 Mason and Hirschfelder, J. Chem. Physics, 1957, 26, 756. 16 Britton and Bean, Can. J. Physics, 1955, 33, 668. 17 see, for example, Booth, Numerical Methods (Butterworths, London, 1955). 18 Rhodes, Physic. Rev., 1946, 70, 932. 19 Takayanagi, J. Physic. Soc. Japan, 1959, 14, 1458. York, 1959). and further references therefrom. Edmonds, Angular Momentum in Quantum Mechanics (University Press, Princeton, 1957).