Theory of collision-induced translation-rotation spectra: H 2 -He

An adiabatic quantal theory of spectral line shapes in collision-induced absorption and emission is presented which incorporates the induced translation-rotation and translation-vibration spectra. The generalization to account for the anisotropy of the scattering potential is given. Calculations are carried out of the collision-induced absorption spectra of He in collisions with H2 with ab initio electric dipole functions and realistic potentials. The anisotropy of the interaction potential is small and is not included in the calculations. The predicted spectra are in satisfactory agreement with experimental data though some deviations occur which may be significant. The rotational line shapes have exponential wings and are not Lorentzian. The connection between the quantal and classical theories is written out explicitly for the isotropic overlap induction.


I. INTRODUCTION
In a collision between two dissimilar atomic systems, overlap forces induce an electric dipole which gives rise to a translational spectrum in the far infrared (FIR).If one of the colliding pair is a nonpolar molecule, a dipole is produced by the electric fields of its multipole moments which polarize the partner.In addition to the multipole component of the induced dipole, there is an overlap com- ponent (usually weaker) and both give rise to translation- rotation absorption bands forbidden in the individual mol- ecule.The spectra arise from free-free transitions, cou- pled with molecular transitions, in which the energy and momentum of the absorbed photon are transferred to the collisional atom-molecule pair.' Because induced dipoles exist only for times v. of the order of the duration of the collision, the spectra show a substantial bandwidth -õ rders of magnitude greater than ordinary Doppler or pressure broadening.Emission and absorption due to the induced dipoles in collisions of neutral particles are much weaker than the familiar bremsstrahlung spectra of elec- trons, and collision-induced emission has only recently been observed in the laboratory.
Apart from the general interest in collisional interac- tions which can be studied by collision-induced absorption (CIA) in novel ways, CIA is of astrophysical significance in regions of relatively high pressure and low temperature where ionization is weak.Trafton has pointed out that the opacity of the atmospheres of the outer planets is largely due to H2-H2 and Hz-He collisions.The substan- tial enhancement of induced absorption due to H2-He col- lisions ' offers an interesting possibility for determining the Hz to He ratio in dense, cold regions of space.
Collision-induced emission is significant in the atmospheres of cool stars.
Whereas the FIR collision-induced absorption spectra of pure H2 are well known from measurements at various temperatures and over extended frequency regions, ' '   the spectra due to H2-He collisions are less certain.Because of the significance of the H2-He absorption spectra, these are computed here from first principles to provide new information concerning the shape of these spectra and their variation with temperature.
Trafton ' has given an adiabatic description of the translational absorption spectrum of a gas of pure H2 and a gas mixture of H2 and He.By comparing line shapes calculated with the assumption of an isotropic Lennard- Jones interaction potential with experimental spectra, he obtained empirical parameters defining a simple analytical representation of the induced electric dipole.Wright and Dalgarno' calculated the induced electric dipole for H2- He from the wave functions of Gordon and Secrest' and resolved it into three symmetry components.They calcu- lated the adiabatic line shape of the translation-rotation spectra and obtained satisfactory agreement with measure- ment by scaling the theoretical dipole moment by a factor of 1.35.Recent calculations' '" show that the dipole mo- ment of Wright and Dalgarno was not in error by such a large factor.Instead, the discrepancy may be attributed to the choice of the interaction potential.Collision-induced absorption spectra are sensitive to the region of the in- teraction potential when it passes through zero.Improved H2-He interaction potentials show that the zero of the potential adopted by Wright and Dalgarno' is located at too large a distance.
Based on improved H2-He interaction potentials and accurate computations of the induced dipole mo- ment, ' we undertake here a quantal calculation of the in- duced translation-rotation spectrum for comparison with the measurement.
We present the theory for an anisotro- pic potential.The anisotropy of the Hz-He interaction is small, however, and the numerical calculations are based upon the isotropic component of the interaction.The spectra have three incoherent parts, one due to the isotro- pic component of the induced dipole, the others to aniso- tropic components.The computed spectra are tested by sum rules.' Although collision-induced vibrationrotation bands are not considered, the formalism can be generalized readily to include vibrational transitions./J, , (rr, RR ) = g A L (ir, R ) Yit (r,R ), Li, (7) The molecular Boltzmann factor P, =P,PJ.designates a normalized population probability of a vibration-rotation state, where -1 PJ =gje ' ggj (2j'+1)e -PE ., -PE.

J
For hydrogen, gj --1 if j is even and g& --3 if j is odd.At low temperatures, where vibration can be ignored, P, o --1 to a good approximation.The vibration-rotation matrix element of p in ( 2) is given by p"=(s ~/J, (rr, RR ) s'), where the vector r = rr describes the orientation and separation of the hydrogen nuclei and R = RR is the vec- tor joining the centers of mass of the colliding atom and molecule.
If the induced dipole p, has components p"and p~p er- pendicular to R, and p, parallel to R, the spherical com- ponents of the induced dipole can be expanded in the formi6''7 where po -p, and p+& --+-(p"+ip")/v 2. , The interaction between H2 and He consists mainly of the spherically symmetric component of the potential Vo(R), and in our numerical calculations we ignored the orientationdependent terms.By contrast, the dipole moment has strong orientation-dependent components.
To take the orientation dependence of the dipole moment into ac- count, we introduced in Eq. ( 7) the vector-coupled func- tion Yir.These are eigenfunctions of the total angular momentum J and projection M, composed of the spherical harmonics YJ (r) and Yi (R ), according to J The absorption spectrum at temperature T and angular frequency to = 2vrcv generated by collision-induced dipole moments can be written as ' 4m a(co) = n, nba(1e ~) Vg(co), (1) where n, and nb are number densities of H2 and He, /3=1/kT, Vis the volume, a(ro) is expressed in cm ', and   g(co) is the spectral density.The product Vg(ro) is a func- tion of the temperature T and depends upon specific molecular properties.The spectral density is defined in terms of the matrix elements (t ~p"~t ') of the electric dipole moment p by the formula Yjt (r,R)= g C(j,/, J;m~, mt)YJ (r)Yt, (R), (8)   Nl Nlh where M=mj+mi and C(j&,j j2, ' are Clebsch- Gordan coefficients. A more general scattering theory of collision-induced absorption and emission is sketched in Appendix A. We calculate the matrix elements of (7) using the wave function of the collisional complex VJtk(rr, RR)=u"(v) ukt(kR)Y~~(r, R-), R where we have assumed separability of vibrational, rota- tional, and translational motion.We have, furthermore, suppressed for convenience of notation the j dependence of the vibrational wave function u"(r).Using the Wigner- Eckart theorem we may write for the angularly dependent part of the dipole matrix element (j,/, J,M ~YiL(r, R) ~j ', /', J',M') =C(J, 1, J', M, v)(j, /, J ~~YiL(r, R) ~~j', /', J') .
g(o))= gP, gP, [ (t   ~p"~t ') )'5(co"+io«ro), (2) ss t, t' where the subscript s= -Iu, j,mj I denotes the vibration- rotation states of the molecule, and t = IE, or k, /, mt, J,MI is an analogous vector designating the translational state of the collisional complex consisting of the inolecule H2 and atom He.Here / »d mt define the orbital angular momentum and its projection, and J and M are the coupled internal rotational and orbital angular momentum quantum numbers.A prime indicates a final state.Also, we have m"= (E, E, )/irt andco« (E; E, )/k w-hich a-re molecular and translational transition frequencies, respectively, E, is the energy of rel- ative motion, and P, is the normalized translational Boltzmann factor at the temperature T. It may be written where A, o is the de Broglie wavelength.This is given in terms of the reduced mass m of the collisional complex and temperature T by (j,/, J ~~/t ~j j', /', J') may be obtained for each com- ponent I. , A, of (7) by multiplying (10) by C(J, 1, J', M, v)   and summing over M, M', v to give j' /' J' &j,/, J I I I'iL where I I is a 9-j symbol identical to Fano's X(j ', /, J', j,/, J;iL,L",1).
The wave function pki(kR ) in ( 9) is the solution of the radial Schrodinger equation as R~ao ( 14) where 5i is the scattering phase shift.It is convenient to use the energy normalization where iri k =2mE, and Vo(R) is the spherical average of the interaction potential.The equation is solved subject to the boundary conditions where i, f signify the initial and final state, respectively, Ql Qd (R)=& IA ( R) I ( 17) is the vibrational matrix element, which we specialize now to the case in which v =v'=0.Note that the square of the sums in ( 16) is the same as a multiple sum over L,A, and L', A, '.From (2) and ( 16) we get g(co) =Pi g (2j+ 1)P.X g f dE,P, g(2J'+1) l, l' 2 I &i I/.If& I', M, M', v QkI kR Qk~I R dR = E&~-E& so that the sums over energy in (2) may be replaced by the integrals f f dE dE'.
The sum of squares of the matrix elements of dipole components (7) can be written where we have already integrated over E;.The factor A stems from converting the 5 function of frequency in (2) to one of energy.The 5 function imposes the condition ~=(E, , -E, )+(EJ.EJ ) on the r-emaining integration in (18).Since the radial wave functions are independent of J and J', the matrix elements in Eq. ( 16) can be written as a product and summed over J and J' according to X(2J'+l)&i / J Energy conservation is imposed by the 5 function in (2) which defines an upper state k' under the integral for any fixed k, j, j, and ca.The radial wave functions uik(kR )/R and matrix elements must be obtained by numerical integration.
-J J+J For fixed L,A,, the functions gI i(co) are the same and are given by ( 21) but shifted by the rotational frequencies co~~( which will be negative if j &j).From GLz(R), the absorption coefficient a(co) is computed according to a(co) = g ar z (co),  Each aL z(co) features a translational component and a number of rotational lines just as GI z.The frequency-dependent factor [1exp( pirico)]accounts for stimulated emission.
The computations to be presented below are based on numerical solutions of Eq. ( 22) for selected isotropic potentials.
From the solutions, the radial matrix elements are computed and substituted into the translational spectral density ( 22) to yield rotation-translation spectra ( 21) and ( 24).Moment relations ' (sum rules) are used to test the computed results.Since translational and rotational profiles are given in terms of the same function gI z(co), it is sufficient to consider the unshifted spectral function for each given I. , A, .
Their zeroth and first spectral moments Go and 6& are given by ' These are translational quantities which are not affected by the molecular state or transition.The function g(R) is usually taken as the low-density limit of the classical radi- al distribution function, g(R) =exp[ -PVo(R)], with wave-mechanical corrections ' to the order of h .This is sufficiently accurate for Hz-He at T )190 K.
Collision-induced vibrational bands can also be obtained by evaluating the vibrational matrix elements (17).The molecular frequencies co" then must include vibrational transitions and the molecular Boltzmann factor must in- clude vibrational states.
Although the theory is developed for the atom-diatom pair, it also describes the essential features of the diatom- diatom (Hz-Hz) collision-induced spectra.The atom is formally replaced by the spherical average of the second diatom which is assumed to be without internal structure.
While simultaneous transitions in both collisional partners cannot be described in this formulation, they are known from experiment to be extremely weak and can be ignored in a first approximation.An accurate description of the Hz-Hz translation-rotation spectra based on (20) may be obtained as suggested by preliminary computations.' Specialization of the results obtained here to the in- teraction of dissimilar atoms is straightforward.
In such cases, the induced dipole (7) consists solely of the term I.=1,A, =O, and rotational excitation is not possible.The resulting translational spectrum is given by ( 22) and ( 24 Sample computations at negative frequencies were seen to be consistent with (27).For each computed line shape, zeroth and first moments are obtained by integration of the spectral intensity and compared with the sum rules (25) and ( 26), and are typically found to be in agreement to within -2%.Higher precision has been obtained in trial runs at the expense of increased computer time.
At room temperature, about thirty partial waves contri- bute to the H2-He spectrum.Computations of the 60000 matrix elements needed for a line shape at 300 K, based on three induced parts L,A, =1,0, 3,2, and 1,2, require about 2000 sec of CPU (central processor unit) time on a Control Data Corporation Cyber-750/150 computer.Line Adiabatic line shapes ( 22) can be obtained if the isotro- pic part of the interaction potential is known, along with the induced components AL, z(R).We use a computer code which is an extension of an existing program developed originally for collision-induced light scattering3 and ab- sorption.
The spectral functions VgL z(co) are obtained at 15 different frequencies (i.e. , 0, 3, 8, 20, 40, 70, 100, 140,   200, 280, 400, 560, 800, 1000, and 1200 cm ').A third- order spline interpolation of the logarithm of gL&(co) is employed, followed by exponentiation, to compute the spectral function at positive frequencies.Logarithmic in- terpolation proved to be more accurate than direct inter- polation.A straightforward exponential extrapolation is used at the high frequencies, which, however, has little ef- fect on the zeroth and first moments, or on the appear- ance of the spectra.To compute g(co) at negative frequen- cies, we use detailed balance shapes are computed from the recorded matrix elements for a variety of temperatures in seconds.

A. Induced Dipole
The induced dipole components Al ~have been comput- ed from first principles.' ' ' We base our computations on the recent work of Wormer and van Dijk' which   shows that for L =A. +1 the induced dipoles are of the form in a.u. and (29) p)p = 0.61 1 5ap The leading dispersion term is C7 ---61.8a.u.'  The other nonvanishing coefficients of the induced di- pole are those with I.=k -1 which are purely short range.
Detailed discussions of the vari- ous He-Hz potentials can be found elsewhere.' ' For our computations of the absorption spectra we chose the empirical' and ab initio potentials as representative examples of the H2-He interaction.
IV. RESULTS
The rotational lines are due to the quadrupole induction (3-2), and a weaker quadrupolar overlap contribution from the 1-2 component.
At the high temperatures, the two potentials lead to spectra which are in reasonable agreement.At the lowest temperature (77 K), differences of 15% occur in the translational part, but the rotational-line intensities are ap- parently not affected by the small differences of the potentials.The rotational-line intensities are dominated by the long-range quadrupolar induction term Cz/R .Line spectra based on this induction mechanism effectively average potentials over a relatively large range of distance which often reduces the effects of differences between po- tential models.The isotropic part, in contrast, has a much shorter range owing to the small p&p and has an almost negligible dispersion part.The translational profile, which is basically due to this induction mechanism, mag- nifies the differences of the potentials over a very small range of separations near their collision diameters, which, however, in the present case differ by only 1.3%.
Theoretical potentials tend to overestimate the magni- tude of o and, indeed, the ab initio value of cr is 5.745ao, in a.u., with oo --5.742ao, they are not significant.
Other components (L=5,A, =4 and L=5,A, =6) have been determined.' However, the associated spectral in- tensities amount to only a few hundredths of one percent of the total intensity, owing to the smallness of high-order overlap dipoles' and the higher multipole moments, and we ignore them.and transport data ' have been proposed.That by Gengenbach and Hahn, ' which is based on molecular beam scattering over a range of energies up to 2.9 eV is probably the most reliable in the vicinity of the collision diameter which is so important for CIA intensi- ties.Shafer and Gordon adopted it to determine an given by the sum of the three components shown as light curves marked L-A, =1-0, 1-2, and 3-2.The computation is based on the ab initio induced dipole (28)(29)(30)(31)(32) and the empirical potential of Gengenbach and Hahn (Ref. 18).The dashed curve is, simi- larly, the total computed intensity, but based instead on an ab initio potential of Meyer et al. (Ref. 20).The dots are mea- surements by Birnbaum (Ref.6).The arrows indicate the rotational-line positions of hydrogen.FKJ. 2. Absorption spectrum at 195 K due to H2-He col- lisions.Details are as for Fig. 1.

B. Potential
compared to the empirical value' of 5.671ao.A collision diameter too large by l%%uo produces an intensity deficiency of about 10% in the case of short-range induction.The longer-ranged inductions usually show a lesser sensitivity to o.Our preference is for the spectra based on the empirical potential.' Zeroth and first spectral moments may be calculated from the profiles using the left-hand sides of Eqs. ( 25) and ( 26).The results for 77, 195, and 297 K are given in the columns labeled g(co) in Table I.The moments can also be obtained directly from the interaction potential and induced-dipole component, without computing the spectral function, by using the right-hand sides of ( 25) and ( 26).These results are given for T=195 and 297 K in columns marked g in Table I.The former are exact wave-mechanical values, while the latter are based on a classical expression of the pair distribution function with wave-mechanical corrections to the order -h .Compar- isons are carried out for several contributions of the dipole moment and potential functions.For the isotropic com- ponents, the lowest-order quantal corrections amount to -10% at 297 K, -15% at 195 K, and 30 -40% at 77 K.

B. Moments
Whereas at the two highest temperatures the corrections reduce the error to an acceptable limit of 2%, at 77 K a much greater error occurs and the present sum rules with lowest-order quantum corrections are less useful. (Sum- rule moments are, therefore, not given in Table I for the lowest temperature.) For the quadrupole-induced part, the lowest-order quantum corrections are smaller and often negative.
Apparently, the long-range nature of the quadrupole-induced dipole is associated with substantial positive corrections at small distances, and negative corrections at large distances and these corrections tend to cancel.The small anisotropic component with L =1,A, =2 shows the same short-range behavior as the isotropic part, with quantal corrections that are large and positive.
At 297 K the moments based on the spectral function and sum rules agree within 1 -3%, the sum rules giving the greater values.The agreement is within the uncertainty of the line-shape computation and confirms the correctness of the computed spectra Qu. antal corrections are significant for the H2-He system even at 300 K.

C. Comparison with experiment
The measurements of a(co) are shown as dots in Figs.
1 -3.They lie consistently above theory by as much as 20% except for the strong Sp(1) line at 77 K, where the measured absorption is less than the computed value.The H2-He spectra are obtained by subtracting two comparable measurements, one in the mixture and the other in pure hydrogen.Consequently, the difference spectra for H2-He collisions is of lower accuracy than for neat systems.However, since the most prominent difference of the H2- He spectrum from the H2-H2 spectrum is the translational component generated by the isotropic part of the induced dipole, a feature which is absent in pure hydrogen, errors should be relatively small in the measurement of the translational spectrum of the mixture.The rotational lines, on the other hand, are in order of magnitude more intense in pure hydrogen at comparable densities, so that the measurements of the rotational lines in the mixture are of lesser accuracy even when optimal mixture ratios are employed.
With the help of a model line-shape function and a least-mean-squares fitting procedure it is possible to decompose the measured spectra into approximate isotropic and anisotropic components.
The empirical isotropic line shape can be compared with the computed spectral function g~p(co).The model line shape is more intense than the calculated line shape particularly at lower tem- peratures and falls off faster than the theoretical profiles, suggesting a slightly longer range p&o for the induced iso- tropic dipole than that given in Eq. (29).
We were led to a similar conclusion from an empirical determination of the range parameter p=0.337A, a value which is nearly unaffected by the uncertainties of the in- teraction potentials.
However, the falloff of the model line shape is even steeper than this empirical range would suggest.Thus the ab initio range p&0 is probably too small.The comparison of moments computed from the quan- tum line shape and from sum rules at the higher tempera- TABLE I. Computed spectral moments of Hz-He absorption spectra.compared for various induced models, AI ~, and isotropic in- teraction potentials, labeled by the equation (in parentheses) or reference number.g(m) signifies the moments Go and G~obtained by integrating spectral density profiles, and g indicates moments obtained from the sum rules, Eqs. ( 25) and ( 26). (Eq.) Ref. tures indicates a precision of the line-shape calculations of better than 2%.This precision is nearly uniform over the frequency range considered, and only in the far wings of the spectral function g(co), where the intensities have fal- len off to less than 10 of the peak value, do the numeri- cal inaccuracies increase.
This does not significantly affect the precision of the computed absorption spectra shown in Figs. 1 -3.
The QCCQpocp of the computed hne shapes ls controlled by the input functions.The calculated and measured line shapes of the translational spectra differ somewhat as Figs. 1 -3 illustrate.At the higher temperatures, the difference in the peak translational intensities of between 10% and 15% may exceed the experimental uncertainties and be significant.At 77 K the differences amount to about 25% and probably are outside the errors of the mea- surement.The observed differences are due presumably to uncertainties in the potential, to approximations such as the neglect of electron correlation in the calculation of the induction models, and to measurement error.The failure to account for the anisotropy of the potential is not likely to lead to an error which is significant compared with the experimental error.
The uncertainty of present potential models amounts to a 10% change for the isotropic and a smaller change for the anisotropic components.This is illustrated in the fig- UI'cs by thc so11d RIld dashed 11Ilcs, whcI'c computed spec- tra based on different potentials' ' are compared.Table I compares the computed spectra, 1 moments which show thc same tcndcnc1cs, 11ncs 2 Rnd 3 fol thc lsotrop1c part, and lines 7 and 8, and 9 and 10 for the anisotropic com- ponents.Earlier potential models like the HFD model give rise to substantial uncertainties, from 30% to 50%, if comparable induced dipoles are input as shown by lines 3 and 5 of Table I.The unsuitability of a Lennard-Jones po- tential' ' is seen by comparing lines 3 and 6 of Table I.
An estimate of the effect of uncertainties associated with the ub initio induced dipoles can be obtained in simi- lar ways.Bcrns et aI.Used a variatlonal approach to calcUlRtc induced-dipole components.SpcctI'Rl moments computed with thclr 1sotI'oplc 1ndUccd component dl ffcl" by almost 20% at high temperatures; compare lines 2 and 3 of Table I.The main anisotropic component is even morc Unccrtaln as thc comparison of 11ncs 10 Rnd 11 shows.%'c note that Berns et ah.do not obtain an aniso- tropic overlap contribution, the quadrupole induction con- sisting of the 8 term only.Recent preliminary results by Meyer apparently support the results of Wormer and van Dijk at the separations of interest.The agreement is at the 5% level which indicates an uncertainty of 10% for the computed spectra, because of the quadratic depen- dcncc oIl thc dlpolc stlcngth.

V. CONCLUSION
We have formulated a wave-mechanical theory of the line shape of collision-induced absorption in the adiabatic RppI'oxlm ation.Thc thcoI'y 1s Rppl1c able to thc translat1on-rotat1on spectra of coll1slonal systems cons1st™ ing of an atom and a diatomic molecule.It was used to compute CIA spectra of H2-He, for which accurate dipole functions and interaction potentials are available.The ro- tational line shapes, which are computed here for the first time on the basis of a rigorous theory, show exponential wings and are not Lorentzian as previously assumed.' While we have not considered vibrational CIA spectra, the formalism is applicable directly to that case.It can also be used to calculate the line shapes of pairs of collid- ing diatomic molecules.
For reliable predictions of CIA intensities, accurate values of the repulsive interaction for separations near the collision diameter must be used.The exceedingly strong dependence of CIA spectra, particularly the isotropic and other short-range parts, on the repulsive interaction suggests that it may be interesting to test the assumption re- garding the neglect of anisotropy.
), with the two nonvanishing coefficients (2l+1)[C(l, l, l +1;0,0)] =/+1, and (2l+1)[C(l, l, l -1;0,0)] =I.The resulting expressions are in agreement with previous work.' It can be shown similarly that collision-induced light scattering by pairs of atoms results in a spectral function (22), with the induced dipole operator replaced by diatom polarizability invariants, and I.=0 and 2. III.DETAILS OF THE LINE-SHAPE COMPUTATIONS gl.z. ( co) =e -~gl.z. (co) (27) 75 A and ejk =19.7 K (Ref.39) has been used for analyzing collision-induced absorption in He-H2.Recent theoretical work' ' ' " has resulted in much improved potentials with significantly larger collision diameters o. of =3.0 A. Empirical potentials based on nuclear magnetic resonance ' FIG.1.Absorption spectrum at 77.4 K due to H2-He col- lisions.The heavy line represents the total theoretical intensities FIG. 3. Absorption spectrum at 292.4 K due to H2-He col- lisions.Details are as for Fig. 1.
For the The helium polarizability a is given inRef.35andthehydrogen quadrupole moment 0 inRef.36.