REVIEW

Collision-induced spectra and current astronomical research1

Martin Abel and Lothar Frommhold

Abstract: Collision-induced spectra are the spectra of complexes of two or more atoms or molecules in a “fly-by” collisionalencounter. Collision-induced absorption (CIA) has been observed inmany dense gases and gasmixtures, inmost cases at infraredfrequencies in the form of quasi continua, and also in liquids and solids. CIA spectra of several binary complexes have beencomputed usingmodern quantum chemicalmethods, combinedwithmolecular scattering theory, which couples the collisionalcomplex to the radiation field as usual in other spectroscopic work. Binary collisional systems, such as H2 interacting withanother H2molecule, or with a helium or hydrogen atom, are first candidates for such computational work, owing to their smallnumber of electrons and the astrophysical interest in such systems. The computed CIA spectra are found to be in close agreementwith existing laboratory measurements of such spectra. Laboratory measurements exist at a limited selection of temper-atures around 300 K and lower, but theory currently also provides CIA data for temperatures up to 9000 K and for higherfrequencies (well into the visible), on a dense grid of temperatures and frequencies. For such calculations, detailed potentialenergy surfaces (PES) of the supermolecular complexes, along with the induced dipole surfaces (IDS), are needed so that therotovibrational matrix elements of PES and IDS may be computed for the molecules involved, which may be highlyrotovibrationally excited. Modern astronomical research needs opacity tables for analyses of the atmospheres of “cool” objects,such as cool white dwarfs, solar and extrasolar planets and their bigmoons, coolmain sequence stars, and “first” stars, which arebriefly described in a concluding section.

PACS Nos.: 78.20.Ci, 95.55.Qf, 34.90+q, 97.10.Ex, 96.15.Hy.

Résumé : Les spectres induits par collision sont les spectres de systèmes de deux ou plus d’atomes ou de molécules dans unequasi-rencontre de proximité. Nous avons observé l’absorption induite par collision (CIA) dans plusieurs gaz et mélanges gazeuxdenses, ainsi que dans des liquides et des solides, surtout dans l’infrarouge quasi-continu. Nous avons calculé les spectres CIA deplusieurs systèmes binaires en utilisant les méthodes de chimie quantique moderne en combinaison avec la théorie descollisions moléculaires qui couple le complexe collisionnel avec le champ de radiation, comme il est usuel en spectroscopie. Àcause de leur petit nombre d’électrons et de leur intérêt en astrophysique, les collisions de systèmes binaires H2–H2, H2–He etH2–H sont les premiers candidats pour ce type de calcul. Nous trouvons que nos spectres CIA calculés sont en bon accord avec lesrésultats expérimentaux existants pour ces systèmes. Les mesures expérimentales existent pour des températures autour de300 K et moins, mais les calculs CIA peuvent être étendus jusqu’a 9000 K et pour de plus hautes fréquences (jusqu’au visible), surdes échelles denses de température et de fréquence. De tels calculs requièrent une connaissance détaillée des surfaces d’énergiepotentielle (PES) et des surfaces de dipôle induit (IDS), afin de calculer les éléments de matrice rovibroniques PES et IDS pour lesmolécules impliquées qui peuvent être soumises a de fortes excitations rovibroniques. La recherche moderne en astrophysiquea besoin de tables d’opacité pour ces molécules, afin d’analyser les atmosphères d’objets froids, comme les naines blanches, lesplanètes solaires et les exoplanètes et leurs grosses lunes, les étoiles refroidies de la séquence principale et les premières étoiles,qui sont brièvement décrites dans la conclusion. [Traduit par la Rédaction]

1. IntroductionMolecules of common gases interact through the intermolecu-

lar van der Waals forces, which typically cause minute rearrange-ments of their normal charge distributions. As a consequence,nearly universally, weak interaction-induced electric dipole mo-ments arise duringmolecular collisions, whichmanifest asmicro-wave and infrared absorption or emission spectra of gases and gasmixtures, among other things. These interaction-induced spectraare widely referred to as collision-induced absorption or emission(CIA or CIE) spectra, if the interaction of the molecules exists onlyfor a short time, as in a fly-by encounter. If alternatively the inter-acting molecules are bound together by weak van der Waalsforces, the interaction-induced dipole moments give rise to thefamiliar spectra of van der Waals molecules [1–5]. Interaction-induced spectra typically consist of contributions from both col-lisional and bound molecular complexes of two or moremolecules. The latter may be recorded as molecular band spectra

when viewed with high resolution, with long absorption pathlengths and at low enough densities; the former appear typicallyas more or less structured quasi continua, especially at elevatedgas densities [6], even in gases and gas mixtures consisting ofmolecules that are infrared inactive (at low enough gas densities).

Collision-induced spectra were discovered in 1949 byWelsh andassociates in the near infrared, first in gaseous oxygen [7], butsoon thereafter it was shown that CIA is a universal feature atinfrared frequencies—not only of virtually all gases [8, 9], but alsoof many liquids and solids [10]: CIA is as universal and omnipres-ent as the van der Waals forces themselves [6, 9].

Interaction-induced dipoles are usually weaker than the perma-nent dipoles of common infrared active molecules. However, in-termolecular interactions involve two or more molecules andgenerate spectral intensities that are proportional to densitysquared, cubed, etc., so that at high enough gas densities thecollision-induced spectra are usually quite striking. In many cases

Received 4 December 2012. Accepted 31 January 2013.

M. Abel and L. Frommhold. Physics Department, University of Texas at Austin, TX 78712-1081, USA.

Corresponding author: Lothar Frommhold (e-mail: frommhold@utexas.edu).1This paper is part of a Special Issue on Mid- and far-infrared spectroscopy: Techniques and applications.

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Can. J. Phys. 91: 1–13 (2013) dx.doi.org/10.1139/cjp-2012-0532 Published at www.nrcresearchpress.com/cjp on 25 March 2013.

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a virial expansion of spectral intensities I(�, T, �) in powers of thegas density, �, is theoretically justifiable [11–13],

I(�, T, �) � I1(�, T)� � I2(�, T)�2 � I3(�, T)�

3 � … (1)

where � denotes angular frequency and T temperature. Theleading term to the right of the equal sign represents the “al-lowed” spectra where these exist; that term vanishes for theinfrared inactive gases. The second, third, etc., terms arecollision-induced contributions, which arise from molecularcomplexes of two, three, etc., molecules. Such complexes are“supermolecular” quantum systems with spectroscopic proper-ties of their own that may be quite different from those of theindividual, noninteracting molecules that make up the complex.Complexes of interacting molecules not only possess electric di-pole moments even when the individual molecules do not, theyalso respond to their own internal (van der Waals) forces andpossess new degrees of freedom relative the unperturbed (i.e.,well-separated) molecules, so that truly new optical features arise,even if collisional complexes exist only for a very short time (e.g.,the duration of a fly-by encounter; ≈10–13 s).

Very many examples of collision-induced infrared absorptionspectra have been reported in mixtures of monatomic gases, aswell as inmolecular gases andmixtures of gases [6, 9, 14]. But longbefore the discovery of rotovibrational CIA in 1949, unexpectedabsorption was observed in oxygen and air at high densities, atvisible and near ultraviolet frequencies [15–18]. The intensities ofthese absorption spectra varied as density squared in pureoxygen, and bilinearly for the contributions fromO2–X complexeswhen X is an atom or amolecule different fromO2. Such a densitydependence suggests that these spectra arise not from single mol-ecules, but from binary molecular complexes, just like the rotovi-brational CIA spectra. In the case of O2–X complexes the resultingspectra are nowunderstood to be interaction-induced spectra aris-ing from electronic transitions, involving the energetically low-lying a1�g and b1�g

� electronic states of O2 [19, 20]. Figure 1compares rotovibrational and electronic CIA by the examples ofFig. 1a, rotovibrational H2–X optical transitions and Fig. 1b, theO2–X electronic transitions. The potential energy curves of theelectronicmolecular states in the separated atom limit when bothatoms are in their lowest energetic states (the 1�g

� and 3�u� states

of H2 and the 3�g�, 1�g,

1�g�, and 3�u

�states of O2) are shown. (Notethat the O2–X curves are shown with a somewhat compressedenergy scale relative to the H2–X ones.) X stands for any arbitraryatom or molecule. It is important to remember that a collisionalpartner is essential for such optical transitions to occur; thesetransitions (arrows) are dipole forbidden in the noninteracting(well-separated) collisional partners. We note that in Fig. 1 onlythe vibrational levels of the nonrotating molecules are sketched.For example, the energy of the initial state of H2–X is given by thesum of the rotovibrational energy, E�j, of H2, the energy ET ofrelative motion of the collisional pair, and the internal energy EXof X; � and j are vibrational and rotational quantum numbers,respectively, of the H2 molecule; and a prime indicates the finalstate.

In the present paper we will be mainly concerned with thepurely rotovibrational–translational interaction-induced spectrain themicrowave and infrared regions; similarities with and com-mon features of the electronic spectra of oxygen and air will onlybe mentioned here in passing.

Soon after the discovery of the interaction-induced rotovibra-tional spectra [7], Herzberg pointed out direct evidence of thepresence of molecular hydrogen in Uranus' and Neptune's atmo-spheres, in the region of the “dipole forbidden” H2 fundamentalband [21], which is a striking feature of the CIA laboratory spectraof hydrogen, as well as in the spectra of the outer planets, owingto collision-induced absorption. Astronomers have by now discov-ered several other instances of the significance of CIA and CIE intheir research efforts, which we will briefly describe in Sect. 4below. Moreover, important applications of CIA also exist in theatmospheric and Earth sciences. We mention that various mono-graphs [6, 22, 23], conference proceedings [24, 25–30], bibliogra-phies [31–34], a database [14], and even a statistical mechanicstextbook [35] exist that deal more or less directly with variousfundamental aspects and applications of CIA–CIE.

2. Interaction-induced dipole moments

2.1. Van der Waals forces and binary induced dipolesConsider two dissimilar atoms, such as a helium and an argon

atom, approaching each other in a collisional encounter, as inFig. 2a. At initially large separations, no significant intermolecu-lar forces and induced dipole exist. However, as the separation

Fig. 1. Comparison of collision-induced (a) rotovibrational transitions of H2–X and (b) electronic transitions of O2–X complexes. Shown are thepotential energy curves of the molecules as functions of the internuclear separation r and the electronic molecular states, which areconnected to the lowest energy level in the separated atoms limit (r ¡ ∞). The X is any arbitrary atom or molecule with the internal energyEX. Vibrational levels of the (nonrotating) molecules are indicated.

r

H(1s) + H(1s)

3Σu+

1Σg+

Ev j + ET + EX

Ev´ j + E´T + E´X

photon energy

r

Σ

Σ

∆

Σ

3

1

1

3

+

+

–

u

g

g

g

O( P) – O( P)3 3

a b

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decreases, as in Fig. 2b, attractive dispersion forces arise as thecollisional partners enter the well region of the intermolecularpotential. Attractive forces arise from a slight enhancement of theelectronic charge in the space between the atoms [36]. The biggeratom (Ar) is likely to move slightly more negative charge to thespace between the atoms than the smaller one (He) so that anelectric dipole (arrow, Fig. 2b) will point to the bigger atom. As theatoms approach even more closely, as in Fig. 2c, exchange andoverlap forces dominate in the repulsive “wall” region of the in-termolecular potential so that the charge density in the spacebetween the atoms is reduced: the resulting dipole moment nowpoints in the opposite direction. (We note that the induced dipolewould be zero in a collision of identical atoms, owing to the in-version symmetry of a pair of like atoms, which is inconsistentwith an electric dipole moment.) Exchange and dispersion forcesgenerate similar dipoles if in the preceding example one or bothof the atoms were replaced by molecules.

Polarization of a collisional partner in a molecular multipolefield, Fig. 2d, is another, usually very important process that gen-erates induced dipole moments. All diatomic molecules possessrelatively strong electric quadrupole moments, along with hexa-decapole and higher-order moments that will induce significantinduced dipole moments in molecular encounters.

Bigger molecules may have (strong!) internal dipole momentsthat may add up to zero in noninteracting (and nonrotating, non-vibrating)molecules; CH4 (Fig. 3), CO2, andmany others. However,if the molecular frames are not very rigid, collisional interactionsmay distort the molecular symmetry enough that momentarilysizeable nonzero dipoles do exist for the duration of the interac-tion. These dipoles must be considered collision-induced dipolemoments, if collisional frame distortions occur.

Other mechanisms have also long been known to contributemore or less to collision-induced dipole moments (e.g., nonuni-form permanent multipolar fields of neighboring molecules,their field gradients, and nonlinear polarization effects [37]).

2.2. Ternary induced dipolesSeveral observations of ternary contributions to the CIA spectra

are known, based on a virial expansion, (1), of measured spectralintensities with increasing gas densities. Very little is known aboutthe spectral profiles of the ternary contributions, but the measure-ment of spectral moments (the integrals of spectral intensity over

frequency) have in a few cases been reliably measured. For com-plexes of three interacting hydrogen molecules, H2–H2–H2 [38, 39],semiempirical models of the induced ternary dipole componentshave been developed [40]. The results obtained may also be used forestimates of similar features of other diatomic molecular gases andgas mixtures (e.g., hydrogen and helium). Ternary induced dipolemoments consist of a pairwise additive component, arising from, insome cases, well-known binary induced dipoles, and some (usuallyrather poorly known, or totally unknown) “irreducible” compo-nents. For hydrogen gas, it was found that at low temperatures(<100 K) the ternary induced dipoles appear to arise mainly frompairwise additive processes. However, with increasing temperature,molecular collisions occur at closer range and spectral intensitiesfrom ternary complexes apparently increase much more rapidlywith temperature than the pairwise additive contributions; Fig. 4suggests a rapidly increasing irreducible contribution. Such a rapidincrease with temperature suggests that overlap (as opposed topurely long-range) processes are mainly responsible for the irreduc-ible contributions.

Three types of ternary dipoles are considered in the semiempiri-cal model mentioned earlier: the exchange quadrupole-induceddipole (Fig. 5a) and the exchange dipole-induced dipole (Fig. 5b). Athirdmodel, which was first studied by van Kranendonk [12, 41] asa possible prototype of ternary dipoles, the quadrupole-inducedternary dipole (Fig. 5c), was shown to be insignificant for densehydrogen gas [40]. The quantitative model of the ternary induceddipole moments of three hydrogen molecules [40] was shown tobe quite successful in reproducing three independent spectro-scopic measurements of ternary spectra, namely: (i) the thirdvirial coefficient, (1), of themeasured CIA spectra of hydrogen [42];(ii) an observed triple transition Q1 + Q1 + Q1 of three interacting H2molecules [43]; and (iii) so-called intercollisional dips [44, 45], us-ing a model similar to exchange quadrupole-induced dipole intheir molecular dynamics calculations.

3. Calculation of CIA–CIE spectraFor various astrophysical applications it is desirable to generate

extensive tables of collision-induced opacities as a function of fre-quency and temperature ofmixtures of hydrogen and helium gases.Specifically, this means that the CIA contributions of the collisionalpairs H2–H2, H2–He, and — for “hot” environments, where some H2may be dissociated — H2–H and He–H need to be known accuratelyover a frequency range from the microwave region of the spectrumto the visible, and for temperatures of the coolest planets, say from20 K or so on, up to many thousands of Kelvin.

Calculations of the binary CIA spectra based on first principlesrequire as input the induced dipole surface (IDS) and the potentialenergy surface (PES) of the collisional binary systems mentionedeearlier. Modern quantum chemical methods are successful inproviding such data [46–51], certainly for the smaller molecularcomplexes (e.g., with but four electrons, such as H2–H2 andH2–He). Once the IDS and PES are available, amolecular scattering

Fig. 2. Dipoles induced by intermolecular forces. (a) At largeseparation of the collisional partners, induced dipoles are negligible.(b) As particles approach to distances where significant attractionexists, a dispersion force-induced dipole arises; (c) at even closerseparations repulsive exchange forces become significant and anexchange force-induced dipole of opposite polarity results. (d) Anexample of a multipole-induced dipole. The quadrupolar fieldsurrounding a symmetric diatomic molecule polarizes a collisionalpartner.

+–

–+

a d

b

c

+–

+–

Fig. 3. Permanent internal dipoles, when combined with collisionalframe distortions, may generate collision-induced dipoles.

CH4

–

+

+

+

+

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formalism, which couples the molecular complex to the photonfield is employed to obtain the spectra [6].

Our present interest is mainly in hydrogen–helium gas mix-tures. It is therefore useful to recall briefly some basic data of theH2 molecule, as shown in Fig. 6. The bond distance of the rotovi-brational groundstate (� = j = 0) is r = 1.449a0 (where a0 is the Bohrradius). Frequencies up to ≈20 000 cm–1 had to be considered,which correspond to excitations of the molecule up to the � = 6vibrational level. Electronic CIA is of no concern at such frequen-cies for hydrogen–helium mixtures.

3.1. IDSsVan derWaals forces are typically much weaker than the chem-

ical forces that bind commonmolecules together. For that reason,quantum chemical calculations of intermolecular forces requirespecial treatment. But for some time quantum chemical methodshave been known, which permit accurate calculations of van derWaals forces and interaction-induced electric dipole surfaces ofcomplexes of two (small) molecules [46–64], even if molecules arehighly rotovibrationally excited. Efforts to obtain IDSs and PESsfor complexes involving bigger molecules (N2, etc.) are underwayelsewhere in several research institutions.

New potential energy and IDSs of H2–H2 complexes have beencomputed [50, 61, 62] using finite-field coupled-cluster methods inMOLPRO 2000 [65], with a correlation-consistent, optimized aug-cc-pV5Z (spdf) basis set. Coupled-cluster wave functions have beencalculated at the CCSD(T) level, with single- and double-excitationoperators in the exponential applied to the reference state; tripleexcitations are treated perturbatively. Large basis sets are used(e.g., 248 functions [49, 51, 59, 66]). Points on the IDS have beencomputed for 15 intermolecular separations, R, from 3.4a0 to 10a0,at 28 bond length combinations (r1, r2) each, with r1 and r2 drawnfrom the set of eight separations (0.942a0, 1.111a0, 1.280a0, 1.449a0,1.787a0, 2.125a0, 2.463a0, and 2.801a0), for 17 geometries of the twoH2 molecules, for each set of r1, r2, R values.

Similarly, for the H2–He complex, the PES and IDS have beencomputed for 15 separations, R, from 2.2a0 to 10a0, for the preced-

ing eight different H2 bond lengths, r, and for each value of R andr, at 19 different angles between the H2 molecular axis and thevector from the center of mass of H2 to the He nucleus (0 to �/2, inincrements of �/36).

If in quantum chemical calculations (small) electric fields areintroduced, cartesian dipole components, x, y, z, may be ob-tained and converted to spherical tensor form, according to [6],

0 � z and ±1 � x ± iy

�2(2)

where i � ��1 is the imaginary unit. Spherical dipole tensorcoefficients A�1�2 L(r1,r2,R) are then computed at all fixed r1, r2, Rcombinations, for the 17 angular orientations, from a system oflinear equations,

� � ��1�2 L

A�1�2 L(r1, r2, R)Y�1�2 L1� (�1,�2,�) (3)

for H2–H2; similarly for H2–He [6]. The vector-coupling functionsare given by [67]

Y�1�2 L1� (�1,�2,�) �

(4�)3/2

�3�

MM1M2M

C(�1�2 ;M1M2M )

× C( L1;M M�)Y�1M1(�1)Y�2M2

(�2)YLM(�) (4)

where �1 and �2 are the orientation angles of the two molecules,� denotes the orientation angles of the intermolecular vector R,the Yℓmℓ

(�) are spherical harmonics, and the C are Clebsch–Gordan coefficients. Thirty-two A coefficients were computed forH2–H2 complexes, but only amuch smaller number were found tobe significant. For example, for H2–H2 the A coefficients with thelabels �1�2 L = 0001, 0221, 0223, 2021, 2023, 2211, 2233, 0443, 0445,4043, and 4045 affect the resulting spectra significantly [59]. Themost significant A coefficients are the quadrupole-induced 0223and 2023 components, but in certain frequency bands other com-ponents contribute significantly to the opacity. Othermore or lesssignificant dipole components are induced by polarization in theH2 hexadecapole field of the collisional partner (�1�2 L = 0445 and4045). All dipole components are affected by, or are entirely dueto, electron overlap and exchange of the interacting pair at nearrange, and also to a lesser extent by dispersion forces.

Figure 7 shows as an example the A0223(r1, r2, R) spherical in-duced dipole component of H2–H2 complexes at a separation ofR = 4a0. This component represents primarily the contribution aris-ing from the quadrupolar electric field surrounding molecule 2,which polarizes molecule 1 via the isotropic part (the trace) of thepolarizability tensor of molecule 1. An analogous A2023(r1, r2, R)component exists that describes the polarization of molecule 2 inthe quadrupolar field surrounding molecule 1. These two compo-nents are the strongest dipole components of the whole set. Atsmall separations, R ≈ 4a0, the absolute values of even the stron-gest A components are small, amounting to only ≈0.1ea0 at nearrange (where e designates the electronic charge). We note rapidand substantial variation of the dipole values with variation of thebond distances r1 and r2. With increasing separation, R, the |A|values fall off to zero as 1/R4. Figure 7 shows an equivalent (i.e.,quadrupole-induced) dipole surface of H2–He at R = 3a0.

3.2. Molecular scattering in the presence of photonsThe molecular scattering Hamiltonian is given by [68, 69]

H � H1(r1) � H2(r2) ��2

2m�R

2 � V(r1, r2,R) � �(r1, r2,R)·X � Hrad

(5)

Fig. 4. Comparison of measurement (squares) and calculation(heavy line) of the ternary integrated intensity (the so-called spectralmoment) in hydrogen gas as function of temperature. The dashedcurve represents the computation of the well-known pairwiseadditive ternary contributions — i.e., without the less well-knownirreducible components — which with increasing temperature fallmore and more short of the measurement rapidly [40].

10 100 1000 Temperature (K)

–4

–2

0

2

γ(3

) 0

(10

–10

cm–1

am

agat

–3)

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where H1 and H2 are the rotovibrational Hamiltonians of mole-cules 1 and 2, respectively; the third term to the right is the kineticenergy operator of relative motion of the collisional pair;m is thereduced mass; V(r1,r2,R) is the PES; is the interaction-inducedelectric dipole; –� · X couples the radiation field, X, to the colli-sional pair; and the last term is the Hamiltonian of the radiationfield. We are interested in weak radiation fields (i.e., small num-bers of photons), thus, the standard perturbation treatment issufficient, so an ordinary state-to-state molecular scattering prob-lem remains to be solved for initial and final states for the com-putation of optical transition probabilities.

If an isotropic interaction potential is assumed, the Hamiltonian [5]diagonalizes so that very simple radial Schrödinger equations of rel-ative motion of the partial waves, ℓ, have to be solved numerically,

���2

2m

d2

dR2� V000

c (R) ��2ℓ(ℓ � 1)

2mR2 ��c(R) � E�c(R) (6)

The superscript c = {�1j1�2j2} stands for the initial or final rotovibra-tional states of molecules 1 and 2; � are vibrational quantum num-bers; and j are rotational ones. This is the so-called isotropic potentialapproximation, if weakly anisotropic systems are considered [6].

For anisotropic potentials the Hamiltonian, (5), diagonalizesblockwise so that a close-coupled system of equations similar to(6) must be solved numerically [69–76].

3.2.1. PES and intermolecular potentialsHighly refined models exist of many intermolecular interaction

potentials [77, 78]. However, becausewe are concernedwith CIA–CIEspectra at elevated temperatures and higher frequencies than inmany previous works, we must account for the fact that initial andfinal states of the molecules involved are rotovibrationally highlyexcited: most existing refined models of intermolecular potentialsare not sufficient for our purposes. It is clear that the CIA spectra ofhydrogen and hydrogen–heliummixtures are strikingly structured,

especially at low temperatures (≤1000 K), if rotovibrational excita-tions of the H2 molecules take place in the photon absorption pro-cess [53, 79–82]. Use of quantum chemical calculations of the PES(Sect. 3.1) is necessary for applications at elevated temperatures andhighoptical frequencies (up to the visible). The isotropic PESs used inthis work were computed therefore at the same intermolecular sep-arations, R, and the 28 combinations of the eight bond distances r1and r2 noted in Sect. 3.1, so that the right rotovibrational matrixelementsmay be computed for each initial and final scattering state(i.e., before and after a photon is absorbed by the collisional pair).

In the case of the isotropic potential approximation the inter-molecular potential for a given initial scattering state c = �1, j1, �2,j2 is given by the rotovibrational matrix element of the isotropicpart of the PES

V000c (R) � ��1 j1�2 j2|V000(r1, r2, R)|�1 j1�2 j2� (7)

similar for the final scattering state c� � |��1 j�1��2 j�2�. A similar ma-trix element must be computed for the final state intermolecularpotential V000

c� (R) of H2–He complexes [62].Because the anisotropy of the H2 molecule is rather weak, iso-

tropic PESs may be used for much of this work. However, we notethat the anisotropy of the PESs of H2–H2 and H2–He are availablewhen needed; ignoring the anisotropy will very much simplifyand shorten the computational task at hand. Previous studies [69]at low temperatures (77 K) have shown that the effects of theanisotropy of the intermolecular potential on the CIA spectra israther weak in the regions of strong absorption. Figure 8 showsexamples of the isotropic PESs obtained at fixed separations, R.Again, we see rather strong dependences of the potential on thebond distances, much as this was seen earlier for the IDSs, Fig. 7.

For the (rotovibrational) ground state �1 = 0, j1 = 0, �2 = 0, j2 = 0 ofthe twomolecules of the H2–H2 collisional complex we obtain thepotential shown in Fig. 9 (solid line). The comparison with ahighly refined calculation shows close consistency.

Figure 9 also shows the intermolecular isotropic potential forthe rotovibrational ground state of H2–He (� = j = 0). For compar-ison, an existing, improved potential model [84] is also shown(dotted). The dashed curve is the intermolecular potential if one ofthe H2 molecules is in the � = 5, j = 0 rotovibrational state and theother in the rotovibrational groundstate (� = 0, j = 0).

3.2.2. Free versus bound molecular pairsInmost gases, bound dimers, so-called van derWaalsmolecules [1,

2], exist. Infrared spectra of van derWaalsmolecules arewell known[85], including those of the (H2)2 dimer [6, 85–89]. These spectra aredue to the very samemechanisms— the vanderWaals forces— thatgenerate the CIA spectra: collisional pairs as well as pairs bound bythe van der Waals forces interact with electromagnetic radiation inthe same way, through the interaction-IDS considered here. For ob-servation, long absorption paths and the lowest gas densities that

Fig. 5. Ternary irreducible dipoles. (a) Two atoms or molecules — particles 1 and 2 — collide; in the resulting exchange force-inducedquadrupole field a third particle, 3, is polarized (exchange quadrupole-induced dipole model). (b) Dissimilar particles 1 and 2 collide; in theresulting field of the exchange force-induced dipole a third particle is polarized (exchange dipole-induced dipole model). (c) In the permanentquadrupolar field of a diatomic molecule, 1, a collisional particle, 2, is polarized. The field of the resulting quadrupolar field-induced dipolepolarizes particle 3 (quadrupole-induced ternary dipole model), generating a third irreducible ternary dipole mechanism.

+– + –1 2

3

–+1 2

3

1

2

3

+–

+–

a b c

Fig. 6. The intramolecular potential of the H2 molecule. Shown arethe vibrational levels of the (nonrotating) molecule.

a0 1 2 3 4 5

1

0

–1

–2

–3

–4

V(R

)(1

000

0 cm

–1)

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still permit a recording of the rotovibrational bands of the van derWaals molecules without excessive pressure broadening of the linesare chosen. In infrared inactive molecular gases the rotovibrationalband spectra of van derWaalsmolecules appear superimposed withthe quasi-continuous CIA spectra near themolecular rotovibrationalline frequencies of the monomers.

In previous work we have computed the spectra of the bound-to-bound, bound-to-free, free-to-bound, and free-to-free transitions ofbinary van derWaals complexes, such as H2–H2 andmany others [6].At low temperatures (roughly 100 K or less) the dimer structures arequite striking if high enough resolution is employed, whereas at themuch higher temperatures of our present interest these bounddimer structures are not significant and are broadened to the pointof being barely discernible; they amount tomuch less than 1% of thetotal intensity, so the bound dimer structures are ignored here.

3.3. Optical transitionsWhen two hydrogen molecules collide and a photon is ab-

sorbed, energy is conserved

E�1 j1� E�2 j2

� Er � �� � E��1 j�1� E��2 j�2

� Er� (8)

where E�ijirepresents the rotovibrational energy of the hydrogen

molecule, i, in the rotovibrational state, �iji, with i = 1 and 2; Er isthe energy of relative motion; �� is the photon energy; and aprime indicates the final state. Optical transitions give rise to aspectral “line”, which in the CIA case is extremely broad, owing tothe short duration (≈10–13 s) of a fly-by collision. The center fre-quency of such a line is given by

��0 � (E��1 j�1� E�1 j1) � (E��2 j�2

� E�2 j2) (9)

that is, the sums or differences of the well-known rotovibra-tional frequencies of H2. One of the two terms in parentheses iszero, unless both �1 > 0 and �2 > 0 for a given A coefficient. Inthat case both terms may be nonzero, and one term may actu-ally be negative, when a single photon is absorbed; these cor-respond to simultaneous transitions in both molecules. Wenote that energy changing simultaneous rotovibrational tran-sitions are not very significant in the overall picture, but at lowtemperatures and low densities striking dimer features due tosimultaneous transitions have been observed. The most strik-ing features of the spectra, after summation over all such tran-sitions, are the rotovibrational bands of the unperturbed H2

Fig. 7. (a) The spherical tensor component A0223 of the induced dipole of H2–H2 complexes, at the intermolecular separation of R = 4a0, whichrepresents primarily the quadrupole-induced dipole. The H2 bond lengths r1 and r2 are given in atomic units (a0), and A0223 is plotted in10−3 atomic units; the circle marks the point, where both molecules are in the rotovibrational ground state. (b) The analogous quadrupole-induced spherical dipole component A23(r, R) of H2–He complexes at R = 3a0, is reproduced; the dot marks the point where the molecule is inthe rotovibrational ground state.

11.5

22.5

31

1.5

2

2.5

30

50

100

A02

23

r1

r2

1 1.5 2 2.5 3Bond length, r (a0)

0

50

100

150

–A23

(r,R

)

a b

Fig. 8. (a) The isotropic part, V000, of the PES of H2 pairs at the separation of R = 4a0 is shown; V000 is here expressed in atomic units of energy.The H2 bond lengths r1 and r2 are given in atomic units of length; the square marks the point, where both molecules are in the rotovibrationalgroundstate. (b) The analogous isotropic part of the PES of H2–He complexes is shown as function of the H2 bond length at the separation of 3a0; thedot marks the point, where the molecule is in the rotovibrational groundstate.

1

1.5

2

2.5

3 11.5

22.5

3

00.10.20.30.4

V000

r1

r2

1 1.5 2 2.5 3Bond length, r (

0.02

0.03

0.04V

00(r

,R)

(

au)

)a0

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molecules. (Note that these optical transitions are dipole-forbidden in unperturbed, noninteracting H2 molecules.) Hun-dreds of thousands of optical transitions must be consideredfor two colliding H2 molecules; for H2–He collisions tens ofthousands of optical transitions shape the resulting CIA spec-tra, especially at higher temperatures and at near-infrared andvisible frequencies.

The profiles of the thermally averaged individual lines (whichare very diffuse!) are computed according to [6]

VGSc�c(�� , T) � �0

3��ℓℓ�

(2ℓ � 1)C(ℓLℓ�;000)2�0

∞

exp�Er

kT× |�ℓEr|BS

cc� × (R)|ℓ�E�r�|2dEr (10)

where � is the frequency shift relative to the line center (9); thesubscript S is short for the spherical tensor labels �1�2 L; Er is theenergy of the translational motion of the initial state; a primeindicates the final state; ��� Er

′ � Er; ℓ is the partial wave angularmomentum quantum number; and

�0 � �2��2

mkT

is the thermal de Broglie wavelength [6].In (10), the dipole operator B is the rotovibrational matrix ele-

ment of the spherical dipole coefficients, A [6]

B�1�2 L�1 j1�2 j2��1 j1′��2 j�2(R) � ��1j1�2j2�A�1�2 L(r1, r2, R)���1j1

′��2 j�2� (11)

Note that only the radial parts of the H2 rotovibrational statefunctions enter the right-hand side of (11); the angular parts definethe selection rules

�j1 � ���1,��1 � 2,Ê, �1 �j2 � ���2,��2 � 2,Ê, �2

(12)

The Clebsh–Gordon coefficients appearing in (15) result from suit-able summations of the angular parts.

As an example, Fig. 10 shows the normalized profile of thestrong quadrupole-induced (�1�2 L = 0223) optical transition�1j1�2j2 = 0101 ¡ 0103 of molecular hydrogen pairs at three tem-peratures. The enormous widths of CIA “lines” — roughly fiveorders of magnitude larger than the widths of many of the morefamiliar lines — is because of the short duration of fly-by colli-sions, which decreases with increasing temperature.

3.4. Computation of CIA spectra from first principlesCIA spectra are composed of a great number of very diffuse,

strongly overlapping lines, the bands of the rotovibrational tran-sitions of the (unperturbed) molecules of the collisional complex,and bands at sums and differences of these when simultaneoustransitions in both molecules take place

��1 j�1��2 j�2 ¢ �1j1�2j2 (13)

The CIA spectrum of a diatomic gas, such as hydrogen, is given by[6, 60]

�(�, T) �2�2Na

2

3�c�2��1 � exp�

��

kT �Vg(�, T) (14)

The function Vg(�, T) appearing to the right of the equals sign isgiven by the sum over all optical transitions, (10),

Vg(�, T) � �S

�cc�

(2j1 � 1)P1(T)C(j1�1 j�1;000)2(2j2 � 1)P2(T)

× C(j2�2 j�2;000)2VGScc�(� � �0

cc�, T) (15)

Fig. 9. (a) The solid line shows the intermolecular potential of H2–H2 complexes for the rotovibrational ground state (�1 = j1 = �2 = j2 = 0), (7).This result is compared with a refined model [83] (°). The dashed curve shows the result for the case of �1 = 5, j1 = �2 = j2 = 0. (b) Similar resultsare shown for the H2–He complex: the solid line shows the intermolecular potential of H2–He for the rotovibrational ground state (� = j = 0).A refined potential is also shown (dotted) [84]. The dashed curve is for the case of � = 5, j = 0.

2 3 4 5 6 7 8 9 10R

– 0.1

0

0.1

1

10

100

V(R

)

(10

–3

au)

2 3 4 5 6 7 8 9 10R

– 0.1

0

0.1

1

10

100

V(R

)

(10

–3

au)

a b ( )a0( )a0

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The P1(T) and P2(T) are the thermal population probabilities ofthe initial rotovibrational states of molecules 1 and 2. The C(...)are the Clebsch–Gordan coefficients, which define the selectionrules, (12).

Expressions similar to (8)–(15) have been given elsewhere fordissimilar collision partners, such as H2–He [62, 90].

3.4.1. H2–H2 CIA spectraFigure 11 shows the calculated CIA spectrum for H2 pairs for

temperatures from 300 to 3000 K on a highly compressed inten-sity scale. At lower temperatures several bands of relatively highintensity are observed at frequency bands that roughly coincidewith the H2 rotovibrational bands (which are of course dipoleforbidden in the noninteracting molecule). Between these strongintensity bands very deep minima exist at lower temperatures,which do, however, tend to fill in with increasing temperature.We note that work is underway to extend the spectra in Fig. 11 tohigher temperatures and frequencies.

Early ab initio computations of CIA spectra of H2–H2 complexeswere based on Meyer's advanced IDS and PES [6]. That work wasused mainly by planetary scientists and was limited at the time tolow temperatures (≈300 K and below) and frequencies correspond-ing to the pure rotational band [48], the fundamental band [80],and the first overtone band [82] of the H2 molecule. In the mean-time, new astrophysical research to be briefly outlined hereincalled for data at much higher temperatures and increasinglyhigher frequencies.

CIA spectra of H2–H2 complexes at temperatures up to thou-sands of kelvin, at frequencies corresponding to the second H2overtone band, were first computed by Borysow [91]. To that end,Meyer's IDS was supplemented down to smaller intermolecularseparations (R = 2.5a0) of the collisional pair, and data points atseveral larger H2 bond distances (r up to 2.150a0) were simplyadded to Meyer's IDS to obtain an extended IDS. That extendedIDS was widely criticized for employing inconsistent basis sets.Laboratory measurements of the H2 CIA spectra in the second H2overtone band were not reproduced very well from theory withthe extended IDS [61, 92]. Semiempirical dipole operators werethen constructed and used for the extended opacity tables result-ing from these efforts, in preference to the theoretical dipole data[93, 94]. Moreover, the interactionpotentialmodel used for theopacitycomputations was but loosely based on the theoretical PES.

New IDS and PES data were recently obtained for H2–H2 com-plexes by Hunt and associates over a substantially extended rangeof bond distances, r1 and r2, and collisional separations, R [59, 61].Further expansion of the H2–H2 opacity tables has been possible.The current opacity tables [14] are based on these efforts; thepresent work is an ongoing substantial extension of the HITRANopacity tables. We note that there is full consistency of Hunt'snew, enlarged IDS and PES with Meyer's earlier data where directcomparisons of the two quantum chemical results for H2–H2 com-plexes can be made.

Some laboratory measurements of the binary CIA spectra ofhydrogen exist for room temperature and below in the rotational,fundamental, and first and second overtone bands of the H2 mol-ecule, which are seen to be in close agreement with the spectracomputed from first principles [61]. The far wings of these spectrafall off rapidly to very low intensities; see the deep minima be-tween these rotovibrational bands in Fig. 11. Some unexplainedinconsistencies of theory and laboratory measurement were ob-served at the time in the far “blue” wing of the pure H2 rotationalband. It was not clear at the time whether the measurementmight have been in error. It is well known that such measure-ments of small absorption are quite difficult, and interference byundetected three-body contributions cannot be excluded. It isequally likely that theory was insufficient: accurate calculationsof far wings of spectra require higher-order spherical dipole compo-nents than were available at the time [73]. Recent work based on themore elaborate IDS yielded satisfactory agreement of calculation andmeasurement [76]. Inotherwords, it isnowclear that themeasurementwas not at fault; theory at the time was somewhat insufficient in thebluewing of the rototranslational spectrum.

Spectral signatures of the van der Waals molecule, (H2)2, of hy-drogenhavebeen investigated in somedetail. The features of the vander Waals molecule could be resolved using long absorption paths(154 m) and relatively low pressures (0.37 to 0.74 atm at 77 K) [88].Figure 12a shows the recording of the dimer signatures in the farinfrared, where the rototranslational CIA spectrum appears. Super-imposed with the CIA spectrum, in the immediate vicinity of the(forbidden) H2 S0(j) lines, with j = 0 and 1 [88]. An early ab initiocalculation [87] is shown in Fig. 12b. The most striking part of thedimer signature, the two spikes (the thin dotted trace in Fig. 12a),which occur over a relatively small band of frequencies to the rightand left of the H2 S0(j) transition frequencies. The spikes arise fromoptical transitions from a bound dimer state (H2)2 to a predissociat-

Fig. 10. Spectral profiles of the quadrupole-induced opticaltransition �1j1�2j2 � 0101 ¡ ��1 j�1��2 j�2 � 0103 of collisional H2–H2

complexes at three temperatures, which is one of the strongertransitions of the far infrared region of the spectrum.

–1000 0 1000 2000Frequency shift (cm–1)

0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

inte

nsity

λ1 λ2 ΛL = 0223

7000 K1500 K

300 K

Fig. 11. Calculated binary CIA coefficients, �, normalized by gasdensity squared (�2), of hydrogen gas at temperatures of 300, 600,1000, 2000, and 3000 K, in the infrared region of the spectrum.

0000100050Frequency (cm–1)

10–9

10–8

10–7

10–6

10–5

H2 – H23000 K2000 K1000 K600 K300 K

α / ρ

2(c

m–1

am

agat

–2 )

H2

rota

tiona

l ban

d

H2

fund

amen

tal b

and

H2

1. o

vert

one

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ing state of the dimer in the continuum. Perhaps less striking, butstill discernible are other bound-to-free and free-to-bound contribu-tions to the dimer spectrum, the differences between the free-to-free (marked f–f) contributions (dashed curve) with the dottedone. The thick line represents the convolution of the CIA + dimerspectrum with an instrumental profile of 4.3 cm−1 resolution andit agrees closely with the Voyager IRIS spectrum of Jupiter [87].That study [87] was based on the isotropic potential approxima-tion. Later theoretical work [95] accounted for the anisotropy ofthe intermolecular potential in detail.

3.4.2. H2–He CIA spectraThe early detailed calculations of the CIA spectra of H2–He colli-

sional complexes were based on Meyer's IDS quantum chemical cal-culations for the H2–He complex [47, 53, 96]. Recently newcalculations of the IDS and PES of H2–He complexes were pub-lished that consider a greater set of spherical dipole compo-nents [50], which permitted calculations of CIA spectra over anextended range of temperatures (up to 9000 K) and frequencies(up to frequencies in the visible) [62], see Fig. 13. The agreementof laboratory measurements (at room temperature and below)with theory is again excellent.

We note that an earlier exploratory study attempted coarseestimates of collision-induced spectra to be expected in very denseand hot sonoluminescence environments [97, 98]. TheH2–He com-plex was chosen as a suitable prototype of a collisional complex,but a very small basis set was used in the quantum chemicalcalculations so that the resulting spectra were expected to showcertain general tendencies of what one might expect to see. How-ever, those quantum chemical calculations were never intendedto provide opacity tables of H2–He complexes for astrophysicaluse as the data were deemed too qualitative and inaccurate forthat purpose. Nevertheless, opacity tables based on this IDS weremade available and used for analyses of zero-metallicity stellaratmospheres [99]. At low temperatures somewhat reasonable con-sistency was observed of these opacities with the accurate data[62], but with increasing temperatures the former CIA calculated

spectra [99] were too high by factors up to three [62] so that theseolder data should no longer be used for any serious work.

3.4.3. Hydrogen isotope spectraCIA spectra of the isotopic systems, the D2–He and T2–He com-

plexes, are computed from the same IDS and PES obtained forH2–He [63, 64, 100], see Fig. 14. The resulting spectra look muchlike those of H2–He complexes, Fig. 13, except that for the isotopicsystems the spacings of the vibrational bands get narrower withincreasing isotope mass, as expected. For deuterium–helium gasmixtures some laboratory measurements attributed to D2–Hecomplexes exist at room temperature and below, which are in

Fig. 12. The spectroscopic signature of the hydrogen van der Waals molecule (H2)2 in the close vicinity of the H2 S0(j) lines in the far infrared;(a) the measurement with j = 0 and 1 [88]; (b) a calculation from first principles, in the isotropic potential approximation, with j = 0 [87]. Thedashed curve represents the free-to-free transitions, where both the initial and final states of the optical transition are collisional complexes — i.e.,the “true” CIA contribution. The dotted line represents the superposition of the free-to-free part with the optical bound-to-free and free-to-bound contributions, where either the initial or the final state is a van der Waals molecule. (Bound-to-bound transitions do not occur in thisapproximation.) The thick line represents the convolution of the dotted line with an instrumental profile of 4.3 cm−1 halfwidth and is in closeagreement with the observation.

S0 (0)120 K

f – f

Total

340 360 380Frequency (cm )–1

2.5

3

3.5

αρ

(10

cm

a

mag

at

)–6

–1–2

2/

a

b

Fig. 13. The binary CIA spectrum of H2–He complexes at varioustemperatures.

0 5000 10000 15000 20000Frequency (cm–1)

10–13

10–11

10–9

10–7

10–59000 K6000 K4000 K2000 K1000 K600 K300 K

α / ρ

1 ρ2

(cm

–1 a

mag

at –

2)

H2

rota

tiona

l ban

d

H2

fund

amen

tal b

and

H2

1. o

vert

one

band

H2

2. o

vert

one

H2

3. o

v.

H2

4. o

.

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satisfactory agreement with calculations based on the fundamen-tal theory [64].

We note that intracollisional interference effects that occur withthe HD–He complex [70, 71] were studied both experimentally [101,102] and theoretically. TheHDmoleculepossesses aweakpermanent(nonadiabatic) dipole moment whose interaction with the collision-induced dipolemoment gives rise to interesting spectroscopic signa-tures [101]. The permanent dipole of HD and induced dipole ofHD–He complexes are of comparable strength.

3.4.4. CIA spectra of other systemsOpacity tables of CIA spectra of many other systems of interest

in the Earth and planetary sciences have recently been collectedin a new section of the HITRAN database [14]. For the astrophysicalapplicationsmentioned herein, the CIA spectra of He–H andH2–Hare of some interest. CIA spectra of many other molecular com-plexes have been collected in the HITRAN database. Specifically,selected data of rotovibrational CIA spectra of themolecular com-plexes N2–N2, N2–H2, N2–CH4, H2–H2, H2–He, H2–CH4, H2–H, He–H,O2–O2, O2–N2, O2–CO2, CO2–CO2, CH4–CH4, and CH4–Ar, are tabu-lated; newspectra of thekindwill be addedas theybecomeavailable.

4. Significance of CIA–CIE in astrophysics

4.1. PlanetsThe significance of CIA for astrophysical research was recog-

nized immediately after its discovery in 1949. Herzberg pointedout direct evidence of H2 molecules in the atmospheres of theouter planets [21, 103], based on knowledge obtained from labora-tory CIA spectra [22, 104, 105]. It is important to know the compo-sition of the atmospheres well, but there are very few ways tomeasure the presence of molecular hydrogen spectroscopically insuch a cold environment — CIA offers itself naturally for thepurpose. Moreover, the importance of CIA in the atmospheres ofthe inner planets (including Earth) [22, 29], and also of Saturn's bigmoon Titan with its atmosphere of predominantly nitrogen [106],has long been recognized.

In recent years the discovery of extrasolar planets has stimu-lated similar research, making use of the existing CIA data [107–110]. The atmospheres of the known giant exoplanets somewhatresemble Jupiter's atmosphere, but most orbit much closer totheir center stars so that many are much hotter, about 1000 K ormore. (For comparison, the temperature of Jupiter's upper atmo-sphere amounts to a mere ≈125 K.).

4.2. Cool white dwarf starsStars that burn hydrogen are called main sequence stars —

these are by far the most common objects in the night sky. Oncethe hydrogen fuel is exhausted and temperatures begin to fall, theobject undergoes various transformations and a “white dwarfstar” is eventually born, the ember of the expired main sequencestar. Temperatures of a newborn white dwarf may be in the hun-dreds of thousands of kelvin, but unless further fusion processesoccur, such as 4He burning to 12C or 16O, that is, if the mass of thewhite dwarf is not greater than just a few solar masses, the whitedwarf must cool down — a process that takes many billions ofyears. It is interesting to note that the coolest white dwarfs dis-covered — thousands of cool white dwarfs are known and newones are being discovered — have temperatures as high as ≈4000 K.The coolest are the oldest, and their temperatures, if well known,maybe consideredameasureof the ageof theoldest objects, becausethe cooling process is conceptually simple and therefore relativelyeasy tomodel. Such a new estimate of the age of our oldest stars is amajor goal of current astronomy research.

Gravity at the surface of a white dwarf is about five orders ofmagnitude greater than the Earth's gravity so that the residual atmo-spheres, which aremixtures ofmolecular hydrogen (a left-over fromthe outer regions of the originalmain sequence star) andhelium, arevery dense. The emission spectra of “cool” white dwarfs do not lookat all like Planck blackbody spectra. Instead, nearly the whole infra-red (from wavelengths of ≈1 m up) is attenuated or missing alto-gether [111–115], Fig. 15, owing to CIA in the hydrogen–heliumatmospheres surrounding the cores [116–119]. More accurate temper-ature measurements must therefore be corrected for the effects ofthe atmospheric opacity, which is caused by CIA.

4.4. First starsAttempts to model the formation of the “first” star from the

pure hydrogen and helium gas clouds below ≈10 000 K show thatthe heat generated in the gravitational contraction phasemust besomehow radiatively released. This is no problem as long as tem-peratures are still high enough that free electrons exist. However,at the lower temperatures in neutral gases H2 molecules mustform from hydrogen atoms, a process that generates enormousamounts of heat that must somehow be radiated away in CIEprocesses; if CIEwere nonexistent,molecules formation could nottake place and temperatures could not fall much in the availabletime. CIE is indispensable for further cooling, which eventually

Fig. 14. The binary CIA spectrum of (a) D2–He and (b) T2–He complexes at various temperatures (from top to bottom: 9000, 2000, 600, and150 K).

0 5000 10000 15000 20000Frequency (cm–1)

10–13

10–11

10–9

10–7

10–5

α / ρ

1 ρ2

(cm

–1 a

mag

at –

2 )

0 5000 10000 15000 20000Frequency (cm–1)

10–13

10–11

10–9

10–7

10–5

α / ρ

1 ρ2

(cm

–1 a

mag

at –

2)

a b

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will create a protostellar environment so that “first stars” canactually be formed [120–128].

4.5. Cool main sequence stars and other cool starsThe atmospheres of low metallicity cool stars are composed

primarily of hydrogen and helium. CIA, for example by H2–H2 andH2–He collisional complexes, is often a principal opacity source insuch atmospheres. For example, CIA in the H2 fundamental band,which falls on top of an opacity window between H2O–CH4 orH2O–CO (depending on the temperature), plays an important rolein shaping brown dwarf spectra [108, 129, 130]. Higher gravitybrown dwarf stars often show even stronger CIA, owing to thedensity squared dependence of CIA intensities, when many otheropacity sources are linearly dependent on density. CIA is alsoimportant in low-metallicity brown dwarfs, because “low metal-licity” means reduced CNO (and other) elemental abundancescompared to H2 and He, and thus stronger CIA compared to H2O,CO, and CH4 absorption. CIA absorption of H2–X collisional com-plexes is thus an important diagnostic of high-gravity and low-metallicity brown dwarfs [131, 132]. All of this is also true of theM dwarfs, but to a lesser extent. M dwarf atmospheres are hotterso that some increased portion of the H2 molecules is in the dis-sociated state, which weakens CIA by H2–X complexes. The signif-icance of CIA for cool astronomical objects was long suspected orknown to some degree [133, 134].

4.6. Dark matterIt is well known that most of the matter in the universe is

“dark” (i.e., it interacts with the rest of the world gravitationally,but not electromagnetically). Darkmatter particle concentrationstend to be high in the central regions of groups and clusters ofgalaxies. A massive star, such as a cool white dwarf, in a region ofhigh dark matter concentration would be heated by bombard-ment of darkmatter particles, but a similar star farther away fromthose centers would be heated less and thus cool much moreslowly [135–137]. The comparison of the temperatures of cool

white dwarf stars near these centers with the temperatures ofsuch stars near the solar system should, therefore, show somedifference, owing to the difference of the heating by dark matterparticles. If this could be demonstrated, it would be new, directevidence of dark matter processes.

5. ConclusionNew ab initio calculations of the collision-induced spectra of

H2–H2 andH2–He complexes were obtained in the isotropic poten-tial approximation over a wide range of temperatures and fre-quencies. The work differs from all similar previous calculationsby the use of intermolecular potentials according to (7), whichemploys the appropriate rotovibrational matrix elements of thePES for every one of the hundreds of thousands of optical transi-tions that must be considered. We note that previous work choseinstead a single intermolecular potential for each H2 band (rota-tional, fundamental, overtones,…), regardless of the rotational (ordouble) transitions considered in that band. That previous proce-dure was acceptable at low temperatures for the lower vibrationalbands justmentioned, but in “hot” environments and for frequen-cies of the higher H2 overtone bands CIA intensities were previ-ously calculated that were too high by factors of up to three.Moreover, striking discontinuities existed in some previous datain the wings of these CIA bands, which are all avoided whenpotentials according to (7) are used. No such discontinuities of theCIA–CIE spectra exist at any frequency or temperature now, be-cause for every optical transition the correct (isotropic) interac-tion potential is employed in the molecular scatteringcalculations.

The effects the anisotropic terms of the interaction potentialhave on CIA spectra have been considered in some detail [69, 74,76, 138–144]. No discernible effects were observed in direct calcu-lations with and without accounting for the anisotropic terms ofthe interaction potential at frequencies where absorption isstrong, (i.e., near the peaks of absorption and in the H2 rotovibra-tional bands) certain differences were noticed in the far “blue”wings of these bands at low temperatures. In other words, theintensity fall-offs to the deep minima seen in the computed spec-tra, Figs. 11 and 13,may not be as reliable as the various absorptionpeaks, where the anisotropy matters less. This conclusion is alsosupported by the observation that the best (i.e., low density, trulybinary) measurements agree very closely with the fundamentaltheory in the broad vicinity of the peaks of the bands [6].

AcknowledgementsIt is a pleasure to acknowledge the collaboration with quantum

chemists K.L.C. Hunt and associates, without which this workwould not have been possible. Profound thanks are also due toastronomers Saumon, who suggested the work many years ago,and Winget and associates, for support, insight, and understand-ing. In the early phases of the work NSF support, grant no.0709106, is acknowledged. Support by the Center for ComplexQuantumSystems at the University of Texas is also acknowledged.

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