Collision-Induced Absorption by H2 Pairs: From Hundreds to

Jan 5, 2011 - Each spectrum consists of very many optical transitions: hundreds of thousands at temperatures of interest here when all significant ...
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Collision-Induced Absorption by H2 Pairs: From Hundreds to Thousands of Kelvin Martin Abel,† Lothar Frommhold,*,† Xiaoping Li,‡ and Katharine L. C. Hunt‡ † ‡

Physics Department, University of Texas, Austin Texas 78712, United States Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, United States ABSTRACT: An interaction-induced dipole surface (IDS) and a potential energy surface (PES) of collisionally interacting molecular hydrogen pairs H2–H2 was recently obtained using quantum chemical methods (Li, X.; et al. Computational Methods in Science and Engineering, ICCMSE. AIP Conf. Proc. 2009, ; see also Li, X.; et al. Int. J. Spectrosc. 2010, ID 371201). The data account for substantial rotovibrational excitations of the H2 molecules, as encountered at temperatures of thousands of kelvin (e.g., in the atmospheres of “cool” stars). In this work we use these results to compute the binary collision-induced absorption (CIA) spectra of dense hydrogen gas in the infrared at temperatures up to several thousand kelvin. The principal interest of the work is in the spectra at such higher temperatures, but we also compare our computations with existing laboratory measurements of CIA spectra of dense hydrogen gas and find agreement.

’ INTRODUCTION Collision-induced absorption (CIA) of infrared radiation by dense gases was discovered in 1949 by Welsh and associates.1,2 CIA is effective whenever molecular gases and gas mixtures are dense enough.3,4 Even gases such as hydrogen, which are infrared inactive, absorb infrared radiation, if densities are high enough. Collisional interactions of two or more molecules induce transient electric dipole moments by multipolar induction, as well as by exchange and dispersion forces, i.e., by the same mechanisms that generate the intermolecular forces. These induced dipoles are modulated at the molecular rotovibrational frequencies of the colliding molecules and emit/absorb radiation. CIA intensities increase as gas density squared if collisions are binary. Collisions of three or more molecules may also contribute, which manifest themselves as higher-order terms in a virial expansion of CIA intensities.4,5 CIA spectra are typically quasi-continuous but consist of many very broad, overlapping “lines” at the rotovibrational transition frequencies of the individual molecules (that may be dipole forbidden in the isolated molecule), and at sums and differences of such frequencies, due to simultaneous transitions of the colliding molecules.3,4 Individual lines are very diffuse, because of the short duration (roughly 10-13 s) of molecular fly by collisions. The significance of CIA for astrophysics was immediately recognized. Herzberg pointed out the direct evidence of H2 molecules and even He atoms in the atmospheres of outer planets,3,6,7 based on knowledge obtained from laboratory CIA spectra.8-10 More recently various “cool” stellar objects were studied, such as cool white dwarfs,11-13 brown dwarfs,14 M dwarfs, and other cool main sequence stars,15 and also “hot” extrasolar planets.16 For detailed analyses of the emission of such objects, accurate knowledge of the CIA spectra of hydrogen and its mixtures with other gases (e.g., helium) over a broad range of r 2011 American Chemical Society

frequencies and temperatures is necessary. Laboratory measurements of CIA in hydrogen have provided valuable information, but especially at higher temperatures (>300 K) suitable measurements do not exist and one must rely on the fundamental theory for quantitative data for the analyses of astronomical observations.

’ SPECTRAL PROFILES: CALCULATIONS Optical Transitions (Spectral “Lines”). The CIA spectrum consists of the superposition of a great many spectral “lines” of the supermolecular complex H 2 -H 2 , which correspond to rotovibrational transitions of one molecule, or simultaneous transitions of both of the collisionally interacting molecules,

fv10 j10 v20 j20 g r fv1 j1 v2 j2 g

ð1Þ

Here ν1 and ν2 are the vibrational quantum numbers and the j1 and j2 the rotational quantum numbers of molecules 1 and 2 in the initial state of the collisional complex; a prime indicates the final state. The collisional van der Waals complex is viewed as a dissociated, transient supermolecule interacting with a single photon. Each line is very diffuse (Δω ≈ 1013 s-1), owing to the short lifetime of the supermolecule. Individual collision-induced lines typically cannot be resolved spectroscopically, except perhaps at low temperatures. The collision-induced absorption spectra appear as coarsely structured continua, as will be shown below. Nevertheless, conceptually it makes sense to talk about induced rotovibrational lines of a “supermolecule” (i.e., the Special Issue: J. Peter Toennies Festschrift Received: October 1, 2010 Revised: December 1, 2010 Published: January 05, 2011 6805

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collisional molecular complex). We note that the state of relative motion, |k, læ, of the collisional complex generally also undergoes a transition in the process; here k is the magnitude of the wave vector, which is related to the translational kinetic energy of relative motion, Et = p2k2/2m, and l is the angular momentum quantum number, but thermal averaging will hide these dependences in the end. We will call ωctr the center frequency of an optical transition, which is given by pωctr ¼ ðEv10 j10 - Ev1 j1 Þ þ ðEv20 j20 - Ev2 j2 Þ

ð2Þ

The Eνj are rototranslational energies of molecules 1 and 2. The approximate width (in energy units) of the “red” wing of the highly asymmetric line profile is roughly κT, where κ is the Boltzmann constant; the “blue” wing may be of a similar width, but it could also be broader.4 Besides the collisional (free) molecular complexes, bound complexes (so-called dimers or van der Waals molecules) exist and undergo similar transitions, including free-to-bound and bound-to-free transitions. Dimer signatures are well-known at low temperatures.4,17-20 In many common gases, the mean lifetime of a van der Waals molecule is of the order of the time between collisions or often much longer, so if bound-to-bound supermolecular transitions are discernible, they typically appear as fairly sharp lines or bands. Since our present focus is on elevated temperatures and hydrogen gas, where the opacity contributions involving van der Waals molecules are known to be rather insignificant and to occur in narrow frequency bands only,4 we may here exclude optical transitions involving bound H2 pairs from further consideration. Molecular Scattering Process. The spectral line intensities are obtained from a state-to-state molecular scattering calculation where the collisional pair is coupled to the radiation field through the induced dipole moments.21-25 The scattering Hamiltonian is of the form p2 H ¼ H 1 ðr1 Þ þ H 2 ðr2 Þ - rR 2 þ V ðr1 ;r2 ;RÞ 2m - μðr1 ;r2 ;RÞ 3 X þ H

rad

ð3Þ

The H1 and H2 are rotovibrational Hamiltonians of molecules 1 and 2; r1 and r2 are the molecular bond vectors; the third term is the kinetic energy operator of relative motion of the pair; m is the reduced mass; R is the center-to-center separation vector of the interacting molecules; V (r1,r2,R) is the potential energy surface (PES); μ(r1,r2,R) = {μx, μy, μz} is the interaction-induced electric dipole moment, also called the “induced dipole surface” (IDS); -μ 3 X couples the radiation field X (ω) to the collisional pair; and the last term is the Hamiltonian of the radiation field. We approximate the intermolecular potential with an isotropic form, V ðr1 ;r2 ;RÞ  V 000 ðr1 ;r2 ;RÞ

and

pffiffiffi μ ( 1 ¼ -ðμx ( ιμy Þ= 2

λ1 λ2 ΛL

The vector coupling functions are given by26 Yλ1n1 λ2 ΛL ðΩ1 ;Ω2 ;ΩÞ ¼

ð4πÞ3=2 X pffiffiffi Cðλ1 λ2 Λ;M1 M2 MΛ Þ 3 MM1 M2 MΛ

CðΛL1;MΛ MnÞ YλM1 1 ðΩ1 Þ YλM2 2 ðΩ2 Þ YLM ðΩÞ

ð7Þ

where Ω1 and Ω2 are the orientation angles of molecules 1 and 2; Ω denotes the orientation angles of the intermolecular are spherical harmonics; and the C are vector; the Ym l Clebsch-Gordan coefficients. Twenty-six A coefficients were actually computed,27,28 but only 15 were found to be significant here: λ1λ2ΛL = 0001, 0221, 0223, 2021, 2023, 2211, 2233, 0443, 0445, 4043, 4045, 2465, 2467, 4265, 4267. We are interested in weak radiation fields (i.e., single-photon absorption) so that the standard perturbation treatment of the -μ 3 X term is sufficient. After applying the usual separation of variables procedures to the Schr€odinger equation of the above Hamiltonian, eq 3, the numerical work begins with the integration of the radial Schr€odinger equations of initial and final state for a given optical transition, eq 1 H S ψðRÞ ¼ EψðRÞ

ð8Þ

where E is the total energy of the initial or final state. The radial scattering Hamiltonian for the initial state is given by   p2 d p2 lðl þ 1Þ 2 d c R þ V000 ðRÞ HS ¼ þ dR 2mR 2 2mR 2 dR þ Eν1 j1 þ Eν2 j2 þ pω

ð9Þ

A similar expression may be given for the radial Hamiltonian Hs0 of the final state. Radial scattering wave functions ψck,l(R) are obtained using Cooley's method29,30 of integration. Radial wave functions are energy-density normalized,4,31 Z¥ c ψc ð10Þ k00 ;l ðRÞψk;l ðRÞdR ¼ δðEt 00 - Et Þ 0

where δ(x) is Dirac's δ function. We use initial and final state labels, c = ν1j1ν2j2 and c0 = ν0 1j01ν02j02; k and k0 are the wave vectors of the initial and final states; k00 in eq 10 is the wave vector of an arbitrary energy; and pω is the energy of the incident photon. The intermolecular potential is given by the radial rotovibrational matrix element of the PES, c ðRÞ ¼ Æν1 j1 ν2 j2 jV 000 ðr1 ;r2 ;RÞjν1 j1 ν2 j2 æ V000

ð4Þ

ð11Þ 0

which makes the extensive molecular scattering computations manageable. The Cartesian dipole components μx, μy, and μz, obtained by the quantum chemical calculations are converted to spherical dipole components μ0 and μ(1, according to μ0 ¼ μz

The dipole spherical-tensor coefficients Aλ1λ2ΛL are computed by fitting the spherical dipole components to the expansion X Aλ1 λ2 ΛL ðr1 ;r2 ;RÞ Yλ1n1 λ2 ΛL ðΩ1 ;Ω2 ;ΩÞ ð6Þ μn ¼

Once the radial scattering wave functions ψckl(R) and ψck0 l0 (R) are computed, the profiles of individual spectral lines may be obtained according to4 X 0 VGsc c ðω;TÞ ¼ λ0 3 p ð2l þ 1ÞCðlLl0 ;000Þ2 Z 

ð5Þ

ll ¥ 0

6806

c0 k0 l0

0

e - Et =kT jÆψ ðRÞjBccs ðRÞjψckl ðRÞæj2 dEt

ð12Þ

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The label s stands for one of the set of dipole expansion parameters λ1λ2ΛL and ωh is 2π times the frequency shift relative to the line center ωctr, given by eq 2. The partial wave angular momentum quantum number is called l and λ0 is the thermal de Broglie wavelength.4 The factor V of G to the left of the equation is typical of all second virial coefficients;32 it is the volume. The dipole operator B(R) is given by4 0

Bsc c ðRÞ ¼ Æν0 1 j0 1 ν0 2 j0 2 jAλ1 λ2 ΛL ðr1 ;r2 ;RÞjν1 j1 ν2 j2 æ

ð13Þ

where the c0 and c superscripts of B designate the transition eq 1. Note that only the radial parts of the H2 rotovibrational state functions enter the right-hand side of eq 13; the angular parts defining the selection rules can be obtained in closed form and are included in eq 15 via the Clebsch-Gordan coefficients C. The functions As are the dipole spherical-tensor coefficients defined by eq 6 The resulting absorption spectrum is the superposition of the profiles of all lines (c0 c) and dipole components (s), Rðω;TÞ ¼

2π2 Na 2 2 F ωð1 - e - pω=kT ÞVgðω;TÞ 3pc

ð14Þ

Na = 2.67  1018 cm-3 is Loschmidt's number, and4 XX Vgðω;TÞ ¼ ð2j1 þ 1ÞP1 ðTÞ Cðj1 λ1 j0 1 ;000Þ2 s

c0 c

 ð2j2 þ 1ÞP2 ðTÞ Cðj2 λ2 j0 2 ;000Þ2 0

 VGcs c ðω- ωc0 c ;TÞ

ð15Þ

The Pi(T) are population probabilities of the initial rotovibrational states of molecules i =1, 2. The Clebsch-Gordan coefficients imply the selection rules, | j0i - ji| e λi, for i = 1, 2. Quantum Chemical Calculations. For the actual computations of spectral profiles the potential energy surface V 000(r1,r2, R) must be known, along with the dipole tensor coefficients Aλ1λ2ΛL(r1,r2,R); eq 11 and eq 13 describe the quantities that actually enter the calculations. The dipole tensor coefficients are obtained from the induced dipole surface μ(r1,r2,R). The PE and ID surfaces of H2-H2 complexes have been computed using finite-field coupled-cluster methods in MOLPRO 2000,33 with an aug-cc-pV5Z (spdf) basis set, which is a correlation-consistent, optimized set. Coupled-cluster wave functions are calculated at the CCSD(T) level, that is with single- and doubleexcitation operators in the exponential applied to the reference state; triple excitations are treated perturbatively. The basis set includes a total of 248 functions. At present, about 40 000 individual ab initio calculations have been run.27,28 Points on the induced dipole surface of H2-H2 have been computed for intermolecular separations R from 4 to 10 au in steps of 1.0 au, for 28 bond length combinations (r1, r2), with r1 and r2 drawn from the set {0.942, 1.111, 1.280, 1.449, 1.787, 2.125, 2.463, 2.801 au}, for 17 dipolar relative orientations of the two H2 molecules, for each of the r1, r2, and R values. Previous calculations of the H2-H2 induced dipole surface exist.31,34,35 Agreement with the present results is excellent where a comparison is possible.28 The present work evaluates a greater set of dipole coefficients Aλ1λ2ΛL(r1,r2,R) over a greater range of bond distances r1, r2, as necessary for work at elevated temperatures and high frequencies (e.g., near the higher H2 overtones).

Figure 1. Calculated collision-induced absorption spectrum of H2 pairs in the infrared at room temperature (solid line). Also shown is an earlier calculation36 (dotted).

’ RESULTS Figure 1 shows the calculated absorption spectrum of H2 pairs at room temperature, from the microwave region to the nearinfrared. The absorption coefficient R(ω,T) is normalized by gas density squared as usual. As expected, individual lines cannot be resolved, but the spectrum exhibits striking coarse structures which, from left to right, correspond roughly to the H2 pure rotational band, the H2 fundamental band, and the first and second H2 overtone bands. Preliminary results at even higher frequencies suggest the presence of a few similar higher H2 overtone bands (not shown). All of these bands are of course dipole-forbidden in an unperturbed (noninteracting) H2 molecule. Other bands, which arise from simultaneous transitions, are less striking but are included in the figure. The normalized absorption coefficient varies over a range of nearly six orders of magnitude. Between the four bands just mentioned are deep minima. Also shown in Figure 1 is an earlier calculation of the CIA spectrum at room temperature,36 which will be compared with present calculations below. Comparison with Laboratory Measurements. Laboratory measurements of CIA spectra of hydrogen gas exist at selected temperatures from roughly 20 to 300 K, for various frequency bands ranging from the microwave region to the near-infrared region of the spectrum. Previous theoretical calculations based on first principles are also known, which were shown elsewhere to agree closely with existing measurements.4 For our present purpose, we compare our computations to laboratory measurements at the highest available temperatures (≈300 K). (As temperatures are lowered, the opacity contributions involving the van der Waals molecules show up more and more strikingly. Since we do not include these in our present calculations, comparisons of low-temperature measurement and theory are not directly relevant to this work.) Figure 2 shows the rototranslational CIA spectrum in the farinfrared region. The agreement of measurement and theory is excellent. In this frequency band, dipoles are induced almost exclusively by polarization of a molecule in the electric quadrupole field of the collisional partner. The other induction mechanisms, hexadecapolar induction, dispersion, field nonuniformity, and exchange forces, barely affect the spectral profiles, except in the wings. 6807

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Figure 2. Collision-induced absorption by pairs of H2 molecules in the rotational band of H2 in hydrogen gas at room temperature. Calculation: solid trace. Measurements: dots37 and circles.38

Figure 3. Collision-induced absorption by pairs of H2 molecules in the fundamental band of H2 in hydrogen gas at room temperature. Calculation: solid (smooth) trace. Measurements: dots,39,40 circles,41 squares,42 red trace (slightly noisy).43

In the high-frequency (“blue”) wing of the rototranslational band, Figure 2, agreement of theory and measurement deteriorated slightly in a previous study.45 This previous slight inconsistency is now understood to be related to the increasing relative importance of the hexadecapole-induced dipole components which were previously not known so well but are now accurately evaluated.46 Planetary scientists suspecting unknown contributions at these frequencies in the emission of the outer planets, as recorded by the Voyager space probes,47 were particularly interested in the precise details of this wing. The present data27,28,45 reproduce the more recent, refined measurements of the blue wing (given by circles in Figure 2) quite well and permit a better prediction of the interband minima of the absorption coefficient seen in Figure 1. Figure 3 shows the collision-induced absorption spectrum in the H2 fundamental band. Agreement of theory and measurement is nearly perfect. We note that near the frequency of 4150 cm-1 the “intercollisional dip” appears in the measurements (difficult to see in Figure 3). It is a many-body feature involving dipole interference in successive molecular collisions48 that is not present in a binary scattering theory as employed here. The intercollisional dip is limited to a few small frequency bands and is ignored for our present purposes.

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Figure 4. Collision-induced absorption by pairs of H2 molecules in the first overtone band of H2 at room temperature in hydrogen gas. Calculation: solid trace. Measurements: open squares,2 dots,40 circles.44

Figure 5. Collision-induced absorption by pairs of H2 molecules in the second overtone band of H2 at room temperature in hydrogen gas. Calculation (solid line). Measurement: circles.49

A similar picture emerges for the collision-induced absorption spectra in the first and second overtone bands of H2, Figures 4 and 5: in the broad vicinity of the absorption peaks, agreement of theory and measurement is remarkably good. The principal dipole components responsible for the absorption at the H2 S lines discernible in the spectra are the quadrupole-induced dipole components, λ1λ2ΛL = 0223 and 2023. At frequencies away from peak absorption other, generally weaker dipole components are relatively more important, namely 2233, 0221, and 2021 and a few others. As a consequence, accurate predictions of collisioninduced absorption by H2 pairs require a greater computational effort, especially near the interband absorption minima. Summarizing this section, agreement of the fundamental theory and existing laboratory measurements is observed as Figures 2-5 demonstrate. Because of the agreement, reliable predictions of the opacity of hydrogen pairs at higher frequencies and temperatures appear to be quite feasible. Laboratory measurements of the interband minima seen in Figure 1 do not exist. Intermolecular Potential. It has long been known that for the calculations of CIA spectra reliable intermolecular potentials must be used.4 Advanced methods have been used to compute intermolecular forces accurately, especially the repulsive wall region and a number of dispersion coefficients. If the well region of an intermolecular potential needs to be known accurately, 6808

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Figure 6. Intermolecular potential, eq 11, for ν1 = j1 = ν2 = j2 = 0 (solid line) and ν1 = j1 = j2 = 0, ν2 = 5 (dashed line). Also shown is an “effective” potential50 (dotted) and an earlier calculation by Meyer31 (circles).

empirical information has often been added, such as the rotovibrational spectra of van der Waals dimers, if known.51 For our work with elevated temperatures, most H2 molecules are initially rotovibrationally excited. After the absorption of a photon of 2.5 eV energy, the final rotovibrational state of an H2 molecule of the collisional complex is even more highly excited, up to vibrational levels of ν = 7 or 8, eq 1. Under such conditions a potential energy surface V (r1,r2,R) must be known as a function of the H2 bond distances, r1 and r2, so that suitable rotovibrational averaging of initial and final scattering state potentials is possible, eq 11. For the recent most highly refined intermolecular potentials, the rotovibrational matrix elements eq 11 are not available, but vibrational dependences for the lowest vibrational states (ν = 1 and 2, but not for rotation) have been given previously.31,35 These vibrational corrections affect the calculations of the spectra significantly, but they are not sufficient for our present purposes. It was therefore necessary for this work to obtain a new ab initio PES27,28 that permits calculations of the ν and j dependences, eq 11. Figure 6 compares our intermolecular H2-H2 potential for ν1 = ν2 = j1 = j2 = 0, eq 11, with an advanced model by Meyer31 (also for the rotovibrational ground states). The agreement is a very close one for R g 4 bohr, the smallest separation presently considered.27 Smaller separations (R < 4) are presently obtained by an exponential extrapolation so that the small deviations from the advanced previous work31 are understandable (but not significant). We also show in Figure 6 as an example a calculation, eq 11 for a vibrationally excited H2 molecule, ν2 = 5, ν1 = j1 = j2 = 0 (dashed curve). Figure 7 compares CIA spectra at three temperatures, calculated with the Sch€afer-K€ ohler potential52,53 (dotted line) and with our present one (solid line). The latter accounts for the exact rotovibrational excitations of the two H2 molecules of both initial and final state of the two H2 molecules. The former model neglects all rotovibrational dependences of the intermolecular potentials of initial and final states. All other choices (the dipole functions, eq 13) are the same for both runs. The spectra are nearly indistinguishable at the temperatures and frequencies

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Figure 7. Comparison of CIA spectra calculated for three temperatures with different intermolecular potential models; solid line, present ab initio model with full accounting for the rotovibrational states involved; dotted line, from ref 52.

shown, an illustration of the near-equivalence of these intermolecular potential models under these conditions. We note, however, that (not surprisingly) at higher temperatures and frequencies substantial differences are seen (not shown here). High Temperature Spectra. For the calculations of CIA spectra of hydrogen at higher temperatures (>300 K) the numerical methods (Fortran codes) used in previous work4 had to be modified substantially. Each spectrum consists of very many optical transitions: hundreds of thousands at temperatures of interest here when all significant combinations of the eight quantum numbers ν1, j1, ν2, j2, ν01, j01, ν02, j02, are considered, eq 1. Use is made of symmetry relations, e.g., interchange of molecules 1 and 2, eq 13, Æv20 j20 v10 j10 jAλ2 λ1 ΛL ðr2 ;r1 ;RÞjν2 j2 ν1 j1 æ ¼ ð - 1Þ1þΛ Æv10 j10 v20 j20 jAλ1 λ2 ΛL ðr1 ;r2 ;RÞjν1 j1 ν2 j2 æ

ð16Þ

For each of the nonredundant optical transitions, the radial matrix elements of the scattering process in eq 12, 0

0

Æψck0 l0 ðRÞjBccs ðRÞjψckl ðRÞæ are then computed, where the radial wave functions ψ(R) for initial and final states are obtained with the appropriate interaction potentials, eq 11, with B functions according to eq 13. As final steps, an integration over free-state energies and a summation of these line profiles are done, eqs 12, 14 and 15. Figure 8 shows first results thus obtained (solid curves). At the lower temperatures, the H2 rotational band, fundamental band, and first overtone structures are quite striking, but with increasing temperatures the interband minima become less and less pronounced and the H2 bands merge more and more. Preliminary results (not shown) suggest that such studies can be extended to higher frequencies and temperatures, to the H2 second and higher overtone bands. However, as was briefly alluded to above, Figure 1, the existing tables of the A coefficients, eq 6, must be supplemented substantially at small values of the intermolecular separation (R ≈ 4 bohr and smaller) to obtain reliable high-frequency opacity data. With increasing frequencies >10 000 cm-1 accurate values of the A coefficients need to be known at smaller R values for an accurate description of the CIA 6809

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Figure 8. High temperature spectra. Comparison of present (solid lines) and previous data54 (dashes) at three temperatures.

spectra; efforts are underway to supplement the existing IDS and PES of hydrogen pairs.

’ COMPARISON WITH PREVIOUS WORK Early quantum computations of CIA spectra of H2 pairs at low temperatures (≈300 K and below) of the H2 rotational, fundamental, and first overtone bands were based on an a very accurate induced dipole surface31,34,35 and the Meyer potential,31 or the Sch€afer-K€ohler potential,52 which is based on Meyer's ab initio calculations (unpublished) with small empirical corrections. These calculations were shown to reproduce closely the existing laboratory measurements in the broad vicinity of the absorption peaks in the H2 rotational, fundamental, and first overtone bands.4 However, due to limitations of that IDS,31,34,35 the far wings of these bands (and more generally the deep absorption interband minima) were not rendered with the desired precision, but our present, expanded IDS reproduced the wing intensities closely.45,55 An independent attempt was made elsewhere to supplement Meyer's IDS over a greater range of intermolecular separations and bond distances, namely, down to R = 2.5 bohr, and up to r1, r2 = 2.150 bohr.56 These additional data were simply added to Meyer's existing data set to obtain an expanded dipole surface, suitable for describing stronger H2 vibrations. That composite IDS was criticized for employing inconsistent basis sets. The hope was that with the extended data set one would be in a position to calculate reliable collision-induced absorption spectra at higher frequencies and much higher temperatures. However, even with this extended input, accurate laboratory measurements of the H2 second overtone band could not be reproduced very well.49

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To reproduce the laboratory measurements in the H2 second overtone band, semiempirical dipole components were constructed, on the basis of laboratory measurements at two low temperatures.49,57,58 The results of this approach are shown (by the dotted line) in the H2 second overtone band in Figure 1. The other parts of Figure 1, the H2 rotational band, the fundamental band and the first overtone band, were constructed with model line shapes (as opposed to exact quantum line shape calculations; see ref 4) obtained with Meyer's IDS and PES.31,34,35 In other words, in the earlier work the different band structures, in Figure 1, were obtained in different ways with different accuracies. Model line shapes were seen in the past to closely reproduce exact line shape calculations in the broad vicinty of the peaks of the lines, but less well in the wings.59 The close agreement of the present calculation and the composite model54 is therefore somewhat surprising. On the other hand, the discrepancies, especially in the dips between the H2 fundamental and the H2 first overtone band and between the H2 first and second overtone band can readily be understood in this way. We also note that collision-induced opacities at frequencies of H2 overtones higher than the second one were assumed to be insignificant, even at temperatures of thousands of kelvin.56 The present IDS27,28 includes more high-order dipole spherical-tensor coefficients; this helps to explain and correct these defects of the previously available induced dipole surfaces, as seen at the lower frequencies, ≈2000 cm-1.46 We note that the refs 45 and 46 consider the “blue wing” of the H2 rotational band near 2000 cm-1, but analogous situations exist in the wings of the other H2 bands and interband minima: the weaker induced dipole components with λ1, λ2 = 4 become more important in the wings, because of selection rules, which generate wing intensities.4 The present IDS is based on one consistent set of ab initio calculations and includes a larger set of intranuclear separations. With that IDS, the laboratory measurement in the H2 second overtone band is reproduced closely, Figure 5, except in the far blue wing where our presently incomplete IDS does not yet permit accurate results; efforts to supplement the IDS are underway. Furthermore, preliminary calculations suggest that third and fourth H2 overtones, even though weaker than the other absorption bands, should be clearly discernible. In the present work all line profiles of the optical transitions are computed separately, using exact quantum line shape calculations (as opposed to so-called model profiles4), the correct dipole matrix elements, eq 13, taking into account the rotovibrational excitations of the molecules, and the correct intermolecular potentials, eq 11, for the initial and final states. The solid curve in Figure 1 shows a superposition of all these contributing transitions. The whole spectrum is based on the single IDS and PES27,28 for all transitions. Summarizing the situation at the low temperatures, it may be said that the previous IDS31,34,35 reproduces the laboratory measurements as well as the present one27,28 in the broad vicinity of absorption peaks, Figure 2-5, except for the H2 second overtone band. However, the previous calculations were lacking in the interband minima, owing to an insufficient number of relatively weak spherical dipole components, which nevertheless become important far away from the absorption peaks. We note that the interband minima of the hydrogen CIA spectra have been of special interest for planetary scientists, as explained elsewhere.4 Previous work on CIA spectra focused on temperatures near 300 K and lower.36,55,60,61 For temperatures of 1000 K and 6810

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higher, on the other hand, two other calculations of CIA spectra were available in the past. One of them was an estimate62 to determine whether CIA is important at the higher temperatures. The other was a more detailed calculation,54 based on the extended IDS and an effective potential.50 This means that at 1000 K two inconsistent calculated CIA spectra existed. The lowtemperature spectra reproduced closely all laboratory results. The high temperature data were meant to be used strictly for the higher temperatures. The two data sets were meant to be used for either one task or the other, but inadvertently could not be joined smoothly at 1000 K. This is unphysical and unacceptable in applications that require a reliable description in the 1000 K temperature range. The present results do not suffer from such an incompatibility at any frequency or temperature. For previous high temperature calculations54 an effective isotropic potential50 was chosen, along with the above-mentioned composite IDS. The potential was thought to average the H2 rotovibrational dependences; it was obtained from shock wave measurements in gaseous and liquid hydrogen and deuterium, where temperatures up to 7000 K were achieved. Based on these empirical data, attempts were made to develop a potential model that fits the equation of state in situations of very high hydrogen gas pressures. The hope was that the effective pair potential thus obtained would account approximately for high densities and elevated temperatures of several thousand degrees, but it is not well suited for the calculation of CIA spectra based on state-tostate scattering events, as needed for the current work. Since for the molecular scattering calculations54 the potential50 is the same for initial and final states, the quantum profiles of each optical transition were computed for positive frequency shifts only. For negative frequency shifts intensities were then obtained by detailed balance, Gð- ωÞ ¼ expð- pω=kTÞGðωÞ

ð17Þ

of the frequency shift ω h (instead of the absolute frequency ω). This is technically correct, if the intermolecular potentials for initial and final states are the same. However, the initial and final rotovibrational states differ, especially at high frequencies, and the scattering potential really should reflect this fact.4,63 Quantum line shape calculations are necessary for both negative and positive frequency shifts, the more so the higher the frequencies to be considered are. We have compared our CIA spectra that were computed with existing (proven) potential models with spectra computed with the effective pair model.50,54 We find the resulting absorption of the latter too strong by up to a factor of 3, as seen in Figure 8 (compare dashed and solid lines). In other words, the principal difference of our high-temperature data with those of ref 54 is related to the use of different intermolecular potential models. The effective model50 features a much softer repulsive wall, which causes the higher intensities, see Figure 6. Moreover, the neglect of the rotovibrational states dependence is to a lesser extent also responsible for the high intensities. If realistic pair spectra are desired, the effective potential should not be used in such calculations. Intermolecular potential models that account for the rotovibrational states dependence are necessary for such calculations.4

’ CONCLUSION In the present work we computed the complete collisioninduced absorption spectrum, from near zero frequency to the

highest frequencies given, taking proper account of all rotovibrational initial and final states of the interacting molecules, for both IDS and PES. The PES used here permits very accurate reproduction of the measured spectra, comparable in accuracy to the spectra calculated with highly refined isotropic potentials. All rotovibrational dependences are treated properly in our present calculation, including hot bands, i.e., absorption resulting from molecular pairs when one or both molecules are rotovibrationally excited, as encountered under conditions of higher temperatures and at the higher frequency regions shown above. We have set up a program that generates the binary absorption coefficient R(ω,T), normalized by density squared, for a system of two interacting linear molecules at temperatures above ≈200 K. Prerequisites are tables of the isotropic potential energy surface (PES), V 000(r1,r2,R) and the induced dipole surface (IDS), or a sufficiently complete set of the dipole spherical-tensor coefficients A(r1,r2,R), for the collisional system under consideration. Since at elevated temperatures, and especially at high frequencies, many different rotovibrational states of the colliding molecules are involved, an attempt is made to account consistently for the rotovibrational state dependence not only of the induced dipole functions but also of the intermolecular potential. Previously, the dependence of the potential functions on the rotovibrational states was occasionally ignored, or else dealt with summarily by first-order corrections, something that may be acceptable at low temperatures and frequencies in the far-infrared. Previous calculations of H2-H2 absorption spectra at elevated temperatures were based on an induced dipole surface with a limited set of dipole tensor components and in some cases effective intermolecular potentials, which do not directly account for the dependences on the rotovibrational excitations of the interacting molecules. This accounts for the differences between the current work and certain previous results.49,57,58

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We thank Dr. D. Saumon for suggesting the work. The work was supported in part by the National Science Foundation, grants AST 0709106 (L.F.) and AST 0708496 (K.L.C.H.). Support by the Center for Complex Quantum Systems, University of Texas, is also acknowledged. ’ REFERENCES (1) Welsh, H. L.; Crawford, M. F.; Locke, J. L. Phys. Rev. 1949, 76, 580. (2) Welsh, H. L.; Crawford, M. F.; MacDonald, J. C. F.; Chisholm, D. A. Phys. Rev. 1951, 83, 1264. (3) Welsh, H. L. In MTP International Review of Science;Physical Chemistry, Series one; Vol. III, Spectroscopy; Buckingham, A. D., Ramsay, D. A., Eds.; Butterworths: London, 1972; Chapter 3, pp 33-71. (4) Frommhold, L. Collision-induced Absorption in Gases; Cambridge University Press: Cambridge, NY, 1993 and 2006. (5) Moraldi, M. Virial expansion of correlation functions for collision induced spectroscopies. In Spectral Line Shapes 6; Frommhold, L., Keto, J., Eds.; AIP Conference Proceedings 216; American Institute of Physics: New York, 1990; pp 438-452. 6811

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