#### Abstract

Collision-induced absorption by hydrogen and helium in the stellar atmospheres of cool white dwarfs causes the emission spectra to differ significantly from the expected blackbody spectra of the cores. For detailed modeling of radiative processes at temperatures up to 7000 K, the existing induced dipole and potential energy surfaces of high quality must be supplemented by calculations with the bonds stretched or compressed far from the equilibrium length. In this work, we describe new dipole and energy surfaces, based on more than 20 000 *ab initio* calculations for . Our results agree well with previous *ab initio* work (where those data exist); the calculated rototranslational absorption spectrum at 297.5 K matches experiment similarly well. We further report the calculated absorption spectra of for frequencies from the far infrared to 20 2009;000 , at temperatures of 600 K, 1000 K, and 2000 K, for which there are no experimental data.

#### 1. Introduction

It is well known that dense gases of infrared inactive molecules such as absorb infrared radiation. Absorption continua range from the microwave and far infrared regions of the spectrum to the near infrared and possibly into the visible. Collisionally interacting pairs of hydrogen molecules possess transient electric dipole moments, which are responsible for the observed absorption continua [1, 2]. Planetary scientists understood early on the significance of collision-induced absorption (CIA) for the modeling of the atmospheres of the outer planets [3, 4]. More recently, it was shown that the emission spectrum of cool white dwarf stars differs significantly from the expected blackbody spectrum of their cores: CIA in the dense helium and hydrogen atmospheres suppresses (filters) the infrared emissions strongly [5–10]. Detailed modelling of the atmospheres of cool stars with proper accounting for the collision-induced opacities is desirable, but it has been hampered heretofore by the highly incomplete or nonexisting theoretical and experimental data on such opacities at temperatures of many thousands of kelvin.

Quantum chemical calculations of the induced dipole surfaces of , He and other complexes have been very successful [11–14]. Based on such data, molecular scattering calculations accounting for the interactions of the molecular complexes with photons have been under-taken which accurately reproduced the existing laboratory measurements at low temperatures (T 300 K or so) [2]. At higher temperatures, virtually no suitable laboratory measurements of such opacities exist, but reliable data are needed. We therefore decided to extend such quantum chemical calculations of the induced dipole (ID) and potential energy surfaces (PES) of complexes to highly rotovibrationally excited molecules, as encountered at high temperatures (up to 7 000 K) and photon energies up to ~2.5 eV.

#### 2. *Ab Initio *Calculations of the Induced Dipole and Potential Energy Surfaces

At the temperatures characteristic of cool white-dwarf atmospheres, the CIA spectra depend on transition dipole matrix elements with vibrational quantum numbers up to . To evaluate these matrix elements, we have determined the induced dipoles and interaction energies of pairs of hydrogen molecules with bond lengths ranging from 0.942 a.u. to 2.801 a.u. (1 a.u. m). For comparison, the vibrationally averaged internuclear separation in is 1.449 a.u., in the ground vibrational state. We have used MOLPRO 2000 [15] to calculate the PES for and to calculate the pair ID by finite-field methods, at coupled-cluster single and double excitation level, with triple excitations treated perturbatively . In this work, we have employed MOLPRO’s aug-cc-pV5Z(spdf) basis, consisting of (9s 5p 4d 3f) primitive Gaussians contracted to [6s 5p 4d 3f]; this gives 124 contracted basis functions for each of the molecules. The basis gives accurate energies and properties [16]; yet it is sufficiently compact to permit calculations on pairs with 28 different combinations of bond lengths, at 7 different intermolecular separations, in 17 different relative orientations (the orientations listed in Table 1), and at a minimum of 6 different applied field strengths for each geometrical configuration.

In the calculations, the centers of mass of the two molecules are separated along theaxis by distances ranging from 4.0 to 10.0 a.u. The vector joins molecule 2 to molecule 1. The molecular orientations are characterized by the angles (, , ), where is the angle between the axis and the symmetry axis of molecule 1, is the angle between the axis and the symmetry axis of molecule 2, and is the dihedral angle between two planes, one defined by the axis and the symmetry axis of molecule 1 and the other defined by the axis and the symmetry axis of molecule 2.

Calculations were performed first for two molecules with bond lengths of a.u., the ground-state, vibrationally averaged internuclear separation. The interaction energies were evaluated in the absence of an applied field; then the pair dipoles were obtained from finite-field calculations, grouped into three sets of 40. Within each of the sets, the fields were confined to the , , or planes, and the two components of the applied field were selected randomly, in the range from 0.001 a.u. to 0.01 a.u., for a total of 120 calculations. For each fixed set of the bond lengths, orientation angles, and intermolecular separation, the total energies were fit (by least squares) to a quartic polynomial in the applied field :

where the Einstein convention of summation over repeated Greek subscripts is followed. The coefficients of the linear terms were selected from each fit, to obtain the Cartesian components of the induced dipole moments , , and . In Table 1, our results for the components of the pair dipole are given for pairs with a.u.

In earlier work on the polarizabilities for [16], we conducted several tests of this fitting procedure: we compared results from quartic fits with 120 different field strengths, quartic fits with 200 different field strengths, and quintic and sixth-order fits with 200 field strengths (at one set of orientation angles and an intermolecular distance of 2.5 a.u., where the differences between the calculations were expected to be magnified); we found excellent agreement among the results from all of the fits. We also compared the results from the random-field calculations with the values obtained analytically, based on calculations with 6 or 8 selected values of the field strengths, for fixed orientation angles and the full range of intermolecular separations. The field values were grouped into the sets , , , , , , , , , , , , and , , , , , , , , with 0.001, 0.002, 0.003, and 0.004 a.u. At the shortest intermolecular distance ( a.u.), the results for = 0.001 a.u. – 0.003 a.u. were affected by numerical imprecision in the hyperpolarization contributions; at larger , they agreed well with the random-field results. Agreement between the random-field results and the results obtained with = 0.004 a.u. was excellent for all values. On this basis, we have used random-field fits in the work with a.u., but we have used analytic fits with 6 different field values for the computations with or a.u. In [16], we also compared the results obtained via analytic differentiation at the self-consistent field (SCF) level using Gaussian 98 versus the results from our SCF calculations, for the full range of intermolecular separations and three different relative orientations, again with excellent agreement. Basis set superposition error (BSSE) has been shown to be negligible [16], as tested by function counterpoise (“ghost-orbital”) methods. BSSE occurs when the pair basis provides a better representation of than the single-molecule basis provides for an isolated molecule. In these calculations, BSSE has been suppressed by the large size of the single-molecule basis.

The interaction mechanisms that determine the induced dipole include classical multipole polarization, van der Waals dispersion, and short-range exchange, overlap, and orbital distortion. At long range, the leading term in the collision-induced dipole comes from quadrupolar induction, which varies as in the separation between the molecular centers [2]. The next long-range polarization term is of order ; it results both from hexadecapolar induction and from the effects of the nonuniformity of the local field gradient (due to the quadrupole moment of the collision partner). The magnitude of the latter term depends on the dipole-octopole polarizability tensor E. At order , back-induction [17, 18] and dispersion [17–21] affect the pair dipole. Back-induction is a static reaction field effect: the field from the permanent quadrupole of molecule 1 polarizes molecule 2, which sets up a reaction field that polarizes molecule 1 (and similarly, with molecules 1 and 2 interchanged). The van der Waals dispersion dipole results from *dynamic* reaction-field effects, combined with the effects of an applied, static field [21], via two physical mechanisms. Spontaneous, quantum mechanical fluctuations in the charge density of molecule 1 produce a fluctuating field that acts on molecule 2; then molecule 2 is hyperpolarized by the concerted action of the field from 1 and the applied field . This sets up a field-dependent dynamic reaction field at molecule 1, giving a term in the van der Waals energy that is linear in the applied field . The correlations of the fluctuations in the charge density of molecule 1 are altered by the static field acting on 1; molecule 2 responds linearly to field-induced changes in the fluctuations of the charge density of 1, again giving a term in the van der Waals energy that is linear in the applied field . The precise functional forms of the short-range exchange, overlap, and orbital-distortion effects on the dipole are not known; however, these contributions are expected to drop off (roughly) exponentially with increasing [2].

The dipole moment of the pair can be cast into a symmetry-adapted form, as a series in the spherical harmonics of the orientation angles of molecules 1 and 2 and the orientation angles of the intermolecular vector:

where the sum runs over all values of , , , , and ; , 0, or , corresponding to the dipole components,

In (2), and denote the orientation angles of molecules 1 and 2, that is, the orientation angles of the axes of the molecule-fixed frames, is the orientation angle of the vector (note that runs from molecule 2 to molecule 1, in this work), and the quantities and are Clebsch-Gordan coefficients. Equation (2) follows immediately from the fact that the collision-induced dipole of is a first-rank spherical tensor, which is obtained by coupling functions of , and . Therefore , , , and the magnitudes of , , and completely determine the dipole expansion coefficients

The dipole coefficients arising from various long-range polarization mechanisms are categorized in Table 2, through order . In this table, Θ denotes the molecular quadrupole moment; is the trace of the single-molecule polarizability; is the polarizability anisotropy, which is equal to in the molecular axis system, where is the symmetry axis; Φ is the hexadecapole moment; E is the dipole-octopole polarizability, which has a second-rank spherical tensor component and a fourth-rank component . The van der Waals dispersion dipole is given by an integral over imaginary frequencies, where the integrand is a product of the polarizability at imaginary frequency and the dipole-dipole-quadrupole hyperpolarizability B. The B tensor is a fourth-rank Cartesian tensor with spherical-tensor components of ranks 0, 2, and 4.

For distinct molecules 1 and 2, or for chemically identical molecules that have different bond lengths, all of the dipole coefficients listed in Table 2 are nonzero, although some of the coefficients may be quite small numerically. For chemically identical molecules, when , the coefficients , with , with , and vanish; the remainder are nonzero. The coefficients and can be obtained from the coefficients and via the relations

where interchanges the labels of molecules 1 and 2. For centrosymmetric molecules such as , the dipole coefficients vanish unless and are both even. Also, due to the Clebsch-Gordan coefficients in (2), nonvanishing contributions are found only if , , or . Coefficients with higher values of and than those listed are of higher order than at long-range, although they may represent significant short-range overlap effects.

From the dipole values in Table 1, we have obtained a set of coefficients by least-squares fit (at each value) to (2), for a.u. From the fit, we have been able to determine the coefficients _{2021}, _{0221}, _{2023}, _{0223}, _{2211}, _{2233}, _{4043}, _{0443}, _{4045}, _{0445}, _{4221}, _{2421}, _{4223}, _{2423}, _{4233}, _{2433}, _{4243}, _{2443}, _{4245}, _{2445}, _{4255}, _{2455}, _{4265}, _{2465}, _{4267}, and _{2467}. We have kept all of these coefficients, as well as _{0001} and _{2201}, in the calculations with unequal bond lengths for molecules 1 and 2. However, for a.u and a.u., the least squares fit shows that the first ten coefficients are numerically important, while the remaining coefficients are essentially negligible. At a.u., the remaining coefficients do not exceed a.u. in absolute value, and the values drop off rapidly with increasing . Table 3 gives our results for _{2021}, _{2023}, _{2211}, _{2233}, _{4043}, and _{4045}; the other numerically significant coefficients are given by the relations _{0221} = _{2021}, _{0223} = _{2023}, _{0443} = _{4043}, and _{0445 }= _{4045}.

In Table 3, the results are also compared with results from two earlier *ab initio* calculations of the dipole with a.u., reported by Meyer et al. [12], Meyer et al. [13], and Fu et al. [22]. (The signs in Table 3 follow from our choice of the positive direction of the intermolecular vector .) Meyer et al. [12, 13] used configuration-interaction wave functions including single, double, and triple excitations from a reference Slater determinant, in a (7s 1p) basis of Gaussian primitives on each H center, contracted to [3s 1p] and augmented by a (3s, 2p, 2d) basis at the center of the H–H bond, giving a total of 31 basis functions for [11]. They performed calculations for 18 relative orientations that provided 9 nonredundant Cartesian dipole components. Fu et al. [22] employed the same basis to generate the CCSD (T) wave functions, in calculations for in 13 relative orientations, selected so that in all cases. To find the dipoles, they used finite-field methods, with two fields that were equal in magnitude but opposite in sign. From Table 3, it is apparent that the results of Fu et al. (FZB) [22] agree well with the earlier results given by Meyer et al. (MBF) [13].

For the largest coefficients, _{2023} and _{0223}, our results are in excellent agreement with both of the earlier calculations: The percent differences between our results and those of Meyer et al. [13] are largest at a.u. (1.52%) and a.u. (0.99%); the remaining differences in these two coefficients average to 0.48%. We have obtained results at a.u., which were not given previously. The differences between our values for _{2233} and those of Meyer et al. [13] are typically ~3% (smaller at a.u.). Differences in the values of _{2021} and _{0221} are ~5% or less at short range ( a.u.), where these coefficients have their largest values. At longer range, the absolute discrepancies are smaller, although the differences are larger on a relative basis. The principal differences in the dipole coefficients are attributable to the inclusion of _{4043} and _{0443} in our work; this affects the values of _{4045}, _{0445}, and _{2211} (to a lesser extent).

In Table 3, the *ab initio *values of the coefficients are also compared with values based on the quadrupole-induced dipole model (QID) and the long range model (LR), which is complete through order . The LR calculations include hexadecapolar induction, back-induction, and van der Waals dispersion effects, in addition to quadrupolar induction. The QID and LR calculations are based on the value of the quadrupole computed by Poll and Wolniewicz [23], the value of Θ interpolated to a.u. given by Visser et al. [24], the hexadecapole computed by Karl et al. [25], the polarizabilities and -tensor values given by Bishop and Pipin [26], and the dispersion dipoles computed from the polarizability and dipole-dipole-quadrupole polarizability at imaginary frequencies, also given by Bishop and Pipin [27].

The coefficient _{2023} depends primarily on the quadrupole-induced dipole: the difference between the QID approximation and our result is ~12% at a.u., ~10.6% at a.u., ~6.5% for a.u., and smaller at larger . The QID model gives remarkably good values for this coefficient, even when is quite small. Agreement with the full long-range model is somewhat better, with errors of ~5.25% at a.u. and only 1.88% at a.u. Quadrupole-induced dipole effects are also present in the coefficient _{2233}; this coefficient fits the QID and LR models quite well for a.u., but the percent errors in these approximations are larger than those in _{2023} for and 5.0 a.u. It should be noted that the back-induction and dispersion contributions have the same sign in _{2023} but opposite signs in _{2233}.

At long range the values of _{4045} and _{0445} depend on hexadecapolar induction, which varies as ; there are no other contributions through order . We find strong agreement between the values of these coefficients and the hexadecapole-induced dipole terms (which determine LR), for a.u.; short-range effects become significant when is reduced to 4.0 a.u. In contrast, _{2021}, _{0221}, _{2211}, _{4043}, and _{0443} seem to reflect the short-range exchange, overlap, and orbital distortion effects predominantly. For these coefficients, the leading long-range terms of back-induction and dispersion vary as ; and they contribute with opposite signs in each case, further reducing the net effect of the long-range polarization mechanisms, in these particular dipole coefficients.

As noted above, we have carried out calculations with 28 different combinations of bond lengths in molecules 1 and 2. *Ab initio* calculations have been completed for pairs with each bond length combination, in each of the 17 relative orientations, at each of 7 separations between the centers of mass, and for at least six values of the applied field in the , , or direction.

In the work of Meyer et al. on the absorption spectra of pairs in the fundamental band, results for the Cartesian components of the pair dipoles are listed for four nonredundant pairs of bond lengths, (, ), (, ), (, ), and (, ), with a.u., a.u., and a.u. [13]. Fu et al. [22] augmented this set by the addition of a larger bond length, a.u., and reported results for all ten nonredundant pairs of configurations with the bond lengths drawn from the set . In the current work, we have included , three bond lengths smaller than (1.280 a.u., 1.111 a.u., and 0.942 a.u.), and four bond lengths larger than (1.787 a.u., 2.125 a.u., 2.463 a.u., and 2.801 a.u.), in order to examine new portions of the dipole surface, particularly those that may become significant for photon absorption at higher temperatures. The specific nonredundant length combinations used in the calculations are = (2.801, 2.125), (2.801, 1.787), (2.801, 1.449), (2.801, 1.280), (2.801, 1.111), (2.801, 0.942), (2.463, 2.125), (2.463, 1.787), (2.463, 1.449), (2.463, 1.280), (2.463, 1.111), (2.463, 0.942), (2.125, 1.787), (2.125, 1.449), (2.125, 1.280), (2.125, 1.111), (2.125, 0.942), (1.787, 1.449), (1.787, 1.280), (1.787, 1.111), (1.787, 0.942), (1.449, 1.449), (1.449, 1.280), (1.449, 1.111), (1.449, 0.942), (1.280, 1.111), (1.280, 0.942), and (1.111, 0.942), with all bond lengths in a.u.

To illustrate the results for pairs with one or both bond lengths displaced from (the averaged internuclear separation in the ground vibrational state of ), in Table 4 we list our values for the dipole expansion coefficients when = 1.787 a.u. and = 1.449 a.u., and we compare with the values given earlier by Fu et al. [22]. In general, we find excellent agreement. The values of _{0001}, _{2021}, _{0221}, _{2023}, _{0223}, _{2233}, _{2243}, and _{2245} agree quite closely, particularly given the extension of the basis set and the corrections for hyperpolarization effects included in the current work. A few of the coefficients show larger differences, based on differences in the fitting procedures. In the current work, we have omitted the coefficients _{2221} and _{2223}, which were included by Fu et al.; this contributes to the difference in the fitted values of _{2211}. On the other hand, we have included _{4043} and _{0443}, which were omitted by Fu et al. [22]; this probably accounts for the difference in the values of _{4045} and _{0445} shown in Table 4. Our inclusion of _{4221}, _{2421}, _{4223}, _{2423}, _{4243}, _{2443}, _{4245}, _{2445}, _{4265}, _{2465}, _{4267}, and _{2467} in the fitting procedure also causes slight shifts in the values of the other coefficients.

No previous results are available for comparison when one or both of the molecules in the pair have bond lengths of 0.942 a.u., 1.280 a.u., 2.125 a.u, 2.463 a.u., or 2.801 a.u. In Table 5, we provide results for one such combination of bond lengths, with a.u. and a.u. The coefficients listed in the top line of each set (and the corresponding coefficients for other pairs of bond lengths) were used in generating the rototranslational and vibrational spectra. These were obtained from fits that included 26 dipole coefficients all together (with _{2211} and _{2233}, but not _{2221} and _{2223}); immediately below those results in each set, we list values obtained from fits with 27 dipole coefficients (including _{2221} and _{2223}, but not _{2211}). We find that the coefficients _{0001}, _{2021}, _{2023}, _{2243}, _{2245}, _{4043}, and _{4045} are numerically “robust;” these coefficients are little affected by the difference in the fitting procedure. The coefficients _{0221}, _{0223}, _{2233}, _{0443}, and _{0445} show greater sensitivity, although the agreement tends to improve as the separation between the molecular centers increases (particularly for _{0223} and _{2233}). The full results for the new potential energy surface and the pair dipoles, with individual bond lengths ranging from 0.942 a.u. to 2.801 a.u., will be reported and analyzed in a subsequent paper. However, here we note that the coefficients _{2023}, _{0223}, _{2233}, _{4045}, and _{0445} appear to be dominated by long-range induction mechanisms, specifically quadrupolar induction for _{2023}, _{0223}, and _{2233}, hexadecapolar induction for _{4045} and _{0445}, and -tensor induction for _{2245}. When the logarithms of the absolute values of these coefficients are plotted versus the logarithms of the separations *R* between the molecular centers of mass, over the range from 8.0 a.u. to 10.0 a.u., the slopes are for _{2023}, for _{0223}, and for _{2233}, all close to the quadrupolar-induction value of . Similarly, the slopes are for _{4045} and for _{0445}, close to the value of for hexadecapolar induction; and the slope is for _{2245}, close to the value of for -tensor induction [17].

#### 3. About the Spectra

The absorption spectrum is a quasicontinuum, consisting of many thousand highly diffuse, unresolved “lines,” corresponding to rotovibrational transitions from an initial state , to a final state , of the binary collision complex. Under the conditions encountered in cool stellar atmospheres, vibrational quantum numbers from 0 to about 5 occur with significant population numbers, with rotational quantum numbers up to 20 or so, for molecules.

The isotropic potential approximation (IPA), which neglects the anisotropic terms of the intermolecular potential, is used for the calculation of the spectra [2]. Each “line” requires as input the matrix elements of the spherical dipole components [2]

and the isotropic component of the intermolecular potential for the initial (unprimed) state

the potential for the final state is given by a similar expression, where all rotovibrational quantum numbers are primed. The line shape calculations proceed with these expressions as described elsewhere [2]. In (5), (6), as above, designates the intermolecular separation and , the intramolecular separations. The indices are the expansion parameters of the spherical dipole components in (2).

Figure 1 shows the calculated absorption coefficient , normalized by the numerical density squared, at the temperature of 297.5 K, and frequencies from 0 to 3000 (the “rototranslational band”). Laboratory measurements [28] are shown for comparison (). Good agreement of theory and measurements is observed.

We note that similarly good agreement of theory and measurement was previously observed, based on an earlier *ab initio* ID surface and a refined intermolecular potential [2, 12]. In the present work, a more complete induced dipole surface has been obtained and used, although the extension has not significantly affected the rototranslational band, shown in Figure 1. Additionally, a new potential energy surface has been obtained and used in the current work. This new potential surface (as well as the new ID surface) accounts for highly rotovibrationally excited mole-cules; and the new surfaces will be essential for our high-temperature opacity calculations—but again, the extensions of the potential surface are of little consequence for the rototranslational band, Figure 1, near room temperature.

The new potential surface is believed to be accurate in the repulsive region of the interaction, but it is not as extensively modeled in the well region, and at long range (dispersion part). Nevertheless, the measurements of the absorption spectra are as closely reproduced by the new *ab initio* input, Figure 1, as they are by the earlier advanced models. Apparently, the collision-induced absorption spectra arise mainly through interactions in the repulsive part of the potential, which is certainly consistent with previous observations [2].

The new opacity calculations of the fundamental and overtone bands [29] show similar agreement with measurements. Figure 2 shows the calculated normalized absorption coefficients over a frequency band ranging from the microwave region of the spectrum to the visible. In these calculations, we have used the exact equilibrium populations for the initial states, which at 2000 K consist of , 1, and 2, with many different rotational states, including highly excited states. For the final states (after a photon of energy up to 2.5 eV has been absorbed), we have included much higher rotovibrational states of the molecules. We have accounted for all of these states rigorously, using the new intermolecular potential and induced dipole surfaces.

The coarse structures seen in the spectrum correspond roughly to the rototranslational band (peak near 600 ), the fundamental band of (peak near 4200 ), and the first through fourth overtone bands of (remaining peaks). Unfortunately, no measurements exist for these high-frequency data, but we feel that the results shown are of comparable reliability to the results in Figure 1.

Calculations of the type shown supplement previous estimates, especially at the highest frequencies [10, 30]. Presently, we are attempting calculations of opacities at still higher temperatures (up to 7000 K). Moreover, similar calculations are planned for −He and −H collisional complexes.

#### 4. Conclusion

We report opacity calculations of collisional complexes for temperatures of thousands of kelvin and a frequency range from the microwave to the visible regions of the electromagnetic spectrum. The calculations are based on new *ab initio* induced dipole and potential energy surfaces of rotovibrating molecules, and are intended to facilitate modeling the atmospheres of cool stars. Agreement with earlier theoretical work and laboratory measurements, where these exist, is excellent.

#### Acknowledgments

This work has been supported in part by the National Science Foundation Grants AST-0709106 and AST-0708496 and by the National Natural Science Foundation of China Grant NSFC-10804008.