We present numerical results for the dipole induced by interactions between a hydrogen molecule and a hydrogen atom, obtained from finite-field calculations in an aug-cc-pV5Z basis at the unrestricted coupled-cluster level including all single and double excitations in the exponential operator applied to a restricted Hartree–Fock reference state, with the triple excitations treated perturbatively, i.e., UCCSD(T) level. The Cartesian components of the dipole have been computed for nine different bond lengths r of H2 ranging from 0.942 a.u. to 2.801 a.u., for 16 different separations R between the centers of mass of H2 and H between 3.0 a.u. and 10.0 a.u., and for 19 angles θ between the H2 bond vector r and the vector R from the H2 center of mass to the nucleus of the H atom, ranging from 0° to 90° in intervals of 5°. We have expanded the interaction-induced dipole as a series in the spherical harmonics of the orientation angles of the H2 bond axis and of the intermolecular vector, with coefficients DλL(r, R). For the geometrical configurations that we have studied in this work, the most important coefficients DλL(r, R) in the series expansion are D01(r, R), D21(r, R), D23(r, R), D43(r, R), and D45(r, R). We show that the ab initio results for D23(r, R) and D45(r, R) converge to the classical induction forms at large R. The convergence of D45(r, R) to the hexadecapolar induction form is demonstrated for the first time. Close agreement between the long-range ab initio values of D01(r0 = 1.449 a.u., R) and the known analytical values due to van der Waals dispersion and back induction is also demonstrated for the first time. At shorter range, D01(r, R) characterizes isotropic overlap and exchange effects, as well as dispersion. The coefficients D21(r, R) and D43(r, R) represent anisotropic overlap effects. Our results for the DλL(r, R) coefficients are useful for calculations of the line shapes for collision-induced absorption and collision-induced emission in the infrared and far-infrared by gas mixtures containing both H2 molecules and H atoms.

When a hydrogen molecule and a hydrogen atom collide, their interactions distort the charge distributions of both H2 and H, producing a transient dipole during the collision.1–5 We have investigated the dependence of the interaction-induced dipole of the H2–H system on the bond length r of H2, the separation R between the centers of mass of H2 and H, and the angle θ between the H2 bond axis r and the vector R connecting the center of mass of H2 to the nucleus of the H atom.

The interaction energy of a hydrogen molecule with a hydrogen atom has been evaluated in multiple ab initio calculations of high accuracy,6–15 with nonadiabatic corrections included;16–18 however, we have found only two previous ab initio calculations of the dipole moment of H2–H. In 1973, Patch obtained the H2–H dipole at a full configuration-interaction (CI) level but in a minimal basis of three 1s functions.19 Patch determined the dipole for six relative orientations of H2–H and four separations between the H atom and the H2 center of mass, ranging from 1.0 a.u. to 4.0. a.u.19 All of these calculations were carried out with an H2 bond length r of 1.401 446 a.u.19 In 2003, Gustafsson, Frommhold, and Meyer (GFM) carried out a substantially more extensive study using a larger basis, variable bond lengths, and a wider range of H2 to H separations.20 

Our work is larger in scale than either previous study. We have calculated the interaction-induced dipole for 19 different angles θ from 0° to 90° in intervals of 5° vs four angles (0°, 30°, 60°, and 90°) used by GFM.20 Symmetry arguments make it possible to determine the dipole over the full range of angles out to 360°, based on the results from 0° to 90°. Our calculations cover nine different bond lengths of H2 from 0.942 a.u. to 2.801 a.u., while the GFM study covered five bond lengths from 1.111 a.u. to 1.787 a.u.20 We have also carried out calculations for a total of 16 different separations between H2 and H, from 3.0 a.u. to 10.0 a.u. vs 11 separations in the GFM work. Our calculations cover a total of 2736 geometrical configurations of H2–H. We have determined the dipole by finite-field methods with Molpro;21 the dipoles reported in the current work have been obtained from more than 43 000 ab initio calculations all together.

We have converted the interaction-induced Cartesian dipoles to a spherical-tensor form. Then, we have fit the results to a series in the spherical harmonics Yλμr) of the orientation angles Ωr of the H2 bond axis r and the spherical harmonics YLmR) of the orientation angles of the intermolecular vector R.22–25 The coefficients DλL(r, R) in this series depend only on λ, L, and the magnitudes of the bond length r and the H2–H separation R. The contributions to the dipole from various polarization mechanisms are separated out in the coefficients DλL(r, R).22–25 From the angle dependence of the interaction-induced dipole, we have determined the DλL coefficients D01, D21, D23, D43, D45, D65, D67, D87, and D89. The coefficient D01(r, R) gives the contribution to the H2–H dipole that is isotropic in the orientation of the H2 molecule. At long range, D01(r, R) gives the dominant term in the van der Waals dispersion dipole.26–34 At short range, D01(r, R) characterizes exchange and overlap effects, as well as dispersion. At long range, the coefficient D23(r, R) is determined by quadrupolar induction,22–25 while at short range, D23(r, R) is affected by anisotropic induction, overlap damping, and exchange effects. The coefficient D45(r, R) plays the analogous role for hexadecapolar induction.23–25 The leading long-range terms in the coefficients D21(r, R) and D43(r, R) vary as R−7 in the H2–H separation,23 but at short range, the relative importance of D21(r, R) and D43(r, R) increases, especially for the larger bond lengths r. These coefficients reflect anisotropic overlap and exchange effects on the total dipole moment.

The spherical harmonic series is needed to determine the line shapes for absorption and emission in the infrared and far infrared, due to the transient dipole that exists during collisions of H2 molecules with H atoms.1–5,19,20,35,36 Collision-induced absorption,1–5,37–74 emission,75–79 light scattering,80–92 and nonlinear Rayleigh and Raman scattering processes93 have been investigated experimentally for H2 interacting with helium atoms,41–48 with other inert gas atoms,44,45,49–54 with H2 molecules,46–48,55–74 or with other species, including CO,94–98 CO2,99 CH4,100–104 N2,96,104–106 NH3,107 and O2.54 The interaction effects on collision-induced spectra involving the isotopic variants HD and D2 have been studied experimentally,108–125 as well as interaction effects on the spectra of bound H2 dimers126–130 and of bound complexes of H2 with other molecules.131–138 Simultaneous vibrational transitions in all three molecules of an H2–H2–H2 cluster have been observed experimentally;140 these must be a consequence of irreducible three-body interactions.141–144 Interaction-induced transitions have also been studied in solid hydrogen,145–154 which shows rotational state changes with ΔJ = 4 (Ref. 153) and ΔJ = 6 (Ref. 154).

Rich and McKellar155 compiled an early bibliography of research on collision-induced absorption, starting with the first observation of the phenomenon in O2 gas by Crawford, Welsh, and Locke,37 followed shortly by the first observation of the rotovibrational infrared spectrum of H2,38 and continuing with the first detections of the collision-induced vibrational overtone in H2 gas39 and of the pure rotational absorption spectrum of H2.40 The bibliography was updated by Hunt and Poll in 1986.156 An overview of recent work in the field was provided by Hartmann and co-workers in 2018.5 Borysow and Frommhold compiled a bibliography of work on collision-induced light scattering through 1989.157 The literature on collision-induced absorption or emission, collision-induced light scattering, and collision-induced hyper-Rayleigh scattering or hyper-Raman scattering is quite extensive, even if limited to spectroscopic processes involving H2 or one of its isotopic variants.

The interaction-induced dipoles,158–170 interaction-induced polarizabilities,166–168,170–174 and interaction-induced hyperpolarizabilities166,167,170,171,175–177 that give rise to the collision-induced spectra for these species have been calculated with high accuracy ab initio, starting with work by Meyer, Frommhold, Borysow, and Birnbaum.158–165 For collision-induced absorption by H2–He and H2–H2, excellent agreement has been attained between experimental spectra and spectra calculated from ab initio results for the interaction-induced dipoles (see Refs. 158–160, 162–165, 168, and 178–182). A high level of agreement has also been found between the experimental and calculated collision-induced spectra of other molecules,183,184 including collision-induced vibronic transitions in O2–O2 and O2–N2.185,186 In addition, spectra have been successfully calculated with intermolecular potentials derived from transport coefficients.187–190 

Theory and experiment have converged in determining the scattering cross sections of H2 molecules and H atoms, based on calculations of the potential energy surface and quantum scattering theory,191 but theoretical work remains the sole source of information on the H2–H dipole to date. The H2–H complex is of interest as the smallest open-shell system where classical induction contributes to the dipole, in addition to exchange, overlap, and van der Waals dispersion effects. Comparisons of the H2–H dipole with the dipole of the small closed-shell system H2–He169 are included in this work.

Information on the energy of H2 interacting with an H atom is used to model processes in galactic gas clouds, stars, and planets with atmospheres that contain both hydrogen molecules and hydrogen atoms.192 Collision-induced absorption by H2–H2 and H2–He pairs193–196 and the absorption spectra of dimers197–199 are known to have astrophysical significance. For example, very old, very cool white dwarf stars emit less radiation in the infrared than predicted, based on the Planck radiation law and the temperatures of the stellar cores.194–196 The reduced intensity of emitted IR radiation is attributed to collision-induced absorption by H2–H2 and H2–He in the stellar atmospheres.194–196,200–205 Results for the H2–H2 and H2–He spectra have been included in the HITRAN database maintained by the Harvard-Smithsonian Center for Astrophysics.206 Effects of the interactions between H2 molecules and H atoms have been detected in the spectra of DA white dwarf stars,207,208 with outer shells of pure hydrogen. For these stars, a previously unexplained intensity of radiation in the ultraviolet has been traced to pressure-broadening of the Lyman alpha bands of H atoms, due to collisions with H2.207,208 The H2–H interactions alter both the transition dipole and the transition energy between the ground and excited states of the H atom. The current work focuses on a different property, the collision-induced dipole of H2–H in the ground electronic state. Collision-induced absorption by H2–H would occur in the same spectral region as absorption by H2–H2 or H2–He.

While the transition from atomic to molecular hydrogen is rather sharp under equilibrium conditions of astrophysical relevance,209 under nonequilibrium conditions, H2 and H may be present together in appreciable quantities.210–212 For example, star formation is driven by processes involving molecular hydrogen in cool galactic gas clouds;213 yet recent observations suggest that “atomic hydrogen has been dominating the cold-gas mass budget of star forming galaxies for at least the past three billion years.”214 Large gas reservoirs of atomic hydrogen have been detected in galaxies at red-shifts between 0.01 and 0.05 (Ref. 215) and between 0.17 and 0.25 (Ref. 216). Atomic hydrogen fractions are correlated with galactic dynamics, including recent mergers217 and disk-specific values of the angular momentum.218 

Molecular and atomic hydrogen are both found in the atmospheres of Jupiter,219 Saturn, and Saturn’s rings.220 Atomic hydrogen coronas have been detected around Ganymede,221,222 Callisto,223 Europa,224 Io,225 and Titan.226 Atomic hydrogen has also been detected at distances of ∼250 km from the surface of the Earth, closer than previously anticipated.227 Extrasolar “hot Jupiters” such as HD 209458b,228–231 HD 17156b,232 and HD 189733b233 show high concentrations of H atoms along with H2; for example, the concentration of H atoms in the atmosphere of HD 209458b is reported to be three orders of magnitude higher than in Jupiter’s atmosphere.228–231 The loss of H atoms from these exoplanets into space is a primary atmospheric escape mechanism234,235 leading to mass loss by the planets. “Warm Neptunes”236,237 have also been observed; GJ 436b exhibits a “giant comet-like cloud of hydrogen” escaping from the planet, and GJ 3470b shows detectable Rayleigh scattering that suggests a hydrogen/helium composition of the atmosphere. Collectively, these observations suggest that our results for the interaction-induced dipole of H2–H may find applications in astrophysical models.

In Sec. II of this paper, we describe our computational method and provide results for the Cartesian components of the dipole moment. Full results for the set of geometrical configurations in this work are included in the supplementary material deposited online. In Sec. III, we provide and analyze the results for the spherical-tensor coefficients DλL(r, R), again with full results in the supplementary material. Also in Sec. III, we check for convergence of D23, D45, and D01 to their known long-range forms. In both Secs. II and III, we compare our results with the earlier GFM calculations of the H2–H dipole. Section IV contains a brief summary, comparisons with the collision-induced dipole of H2–He, and conclusions.

We used an aug-cc-pV5Z basis238,239 and Molpro 200621 for our first set of production runs. We generated a wave function for the ground doublet state at the restricted Hartree–Fock (RHF) level and then constructed the unrestricted coupled-cluster wave function from that reference function240 by including all single and double excitations in the exponential operator applied to the reference wave function, plus triple excitations treated perturbatively, defining the RHF/UCCSD(T) level.241–243 From the RHF/UCCSD(T) energies, we obtained the dipole by finite-field methods, as opposed to direct calculation of the expectation value of the dipole moment.

For open-shell systems, spin-unrestricted calculations typically yield more accurate energies than restricted calculations (where the orbitals for α and β spins are identical), especially when bonds are stretched or broken, but the wave functions in the spin-unrestricted case are not eigenfunctions of S2. They may be contaminated by other spin states (see Ref. 244). Spin contamination of the doublet state by the quartet does not pose a problem in our calculations of the collision-induced dipole. Energies obtained from RHF/UCCSD(T) calculations with a spin-unprojected wave function are identical to the energies obtained from a spin-projected wave function, as shown by Rittby and Bartlett245 and by Scuseria.246 The wave functions are spin-contaminated, but the energies are correct for the spin state of interest. Therefore, our finite-field results for the dipole should be unaffected by spin-contamination. Schlegel247 had shown that in a coupled-cluster calculation that starts from an unrestricted Hartree–Fock (UHF) reference state, the contamination of the wave function for one spin value by the next higher spin value S does not affect the energies. Since the H2–H system has a doublet ground state, and no spin states of H2–H with S > 3/2 are possible, a spin-unrestricted UHF reference state could have been used in the calculations, without causing spin contamination. In practice, we used the RHF/UCCSD(T) method implemented in Molpro.21 The Molpro code uses the perturbative triples defined by Watts, Gauss, and Bartlett242 although triples corrections of the type defined by Deegan and Knowles243 are also generated automatically.

Our first calculations were carried out with the default convergence criteria: 10−6 for the energy, 10−12 for the two-electron integrals, 10−12 for numerical zero, 10−12 for neglect of small two-electron integrals, and 10−4 for the coefficients in the UCCSD(T) wave function.248 The calculations were run for H2 to H separations of 3.4–4.0 a.u. in steps of 0.1 a.u., at 4.2 a.u. and 4.5 a.u., and then for 5.0 a.u.–10.0 a.u. in steps of 1.0 a.u. We set the bond lengths for the H2 molecule to 0.942, 1.111, 1.280, 1.449, 1.787, 2.125, 2.463, or 2.801 a.u. to facilitate the comparison with earlier calculations.20 The averaged bond length in the ground rovibrational state of H2 is 1.449 a.u.

We calculated the energies for six different values of a uniform field applied in the z direction (along R) and in the direction orthogonal to z in the plane containing H2–H. The field strengths were obtained from a reference electric-field value f by taking f, −f, 21/2 f, −21/2 f, 31/2 f, and −31/2 f.249 In the initial study, we set f = 0.002 a.u. The combinations ±f, ±21/2 f, and ±31/2 f make it possible to eliminate terms of even order in f from the calculated dipole. With results for six field strengths, we removed the odd-order hyperpolarization terms of orders f3 and f5, so the leading-order hyperpolarization effect that remains is O (f7). The RHF/UCCSD(T) results were obtained first for zero field and then for each of the fields listed above in sequence, using the converged orbitals from the previous calculation as the starting point for the next. Generally, this approach works quite well, but this sequence of calculations does not ensure precise equality between the energies calculated in fields f and −f perpendicular to R, for the linear and T-shaped configurations. In the first set of calculations, we obtained small, but nonzero dipole components perpendicular to R for T-shaped configurations when the bond length and the H2–H separations were both large.

We therefore undertook a second set of calculations with the aug-cc-pV5Z basis,238,239 using Molpro 2012 and tighter convergence criteria: 10−12 for the energy, 10−12 for the two-electron integrals (the default value), 10−14 for numerical zero, 10−5 for the coefficients in the UCCSD(T) wave function, and 10−10 for the density matrix. Results of these calculations are denoted by A5Z. With the tighter convergence thresholds, the symmetry requirements were met to at least six decimal places in the calculated dipoles. These calculations were run for nine H2 bond lengths (adding r = 1.618 a.u. to our previous set), eight H2–H separations R (from 3.0 to 10.0 a.u. in steps of 1.0 a.u.), and 19 angles θ. The base field strength f for these calculations was 0.002 a.u., as above. As a check on the results, calculations were run at r = 1.449 a.u. with the larger basis sets aug-cc-pV6Z and d-aug-cc-pV5Z.238,239 As an additional check, calculations were run for two geometrical configurations using Molpro 2015, the aug-cc-pV6Z basis, the tighter convergence criteria, and base field strengths f of 0.001 a.u. and 0.01 a.u.

The aug-cc-pV5Z basis has 8s, 4p, 3d, 2f, and 1g functions contracted to 5s, 4p, 3d, 2f, and 1g functions on each H center, for a total of 165 contracted functions for the H2–H system. In addition to accounting for the interaction-induced polarization along the vector R from the center of mass of H2 to the nucleus of the H atom, this basis should be flexible in representing the polarization perpendicular to R, which is nonzero when θ is different from 0° or 90°.

Our full results from the first and second sets of calculations are listed in Tables S1–S9 in the supplementary material. Results from calculations with the tighter convergence criteria are indicated by a superscript . Tables S1–S9 are organized in order by bond length, from Table S1 for r = 0.942 a.u. through Table S9 for r = 2.801 a.u. The results in each table are grouped by angle; then, for each angle, results are listed for a total of 16 separations R between the centers of mass of H2 and H. The vector from the center of mass of H2 to the nucleus of the H atom points vertically up along z. For the calculations in these tables, the H2–H complex lies in the yz plane, and the positive y axis points to the right, as shown in Fig. 1.

FIG. 1.

Geometrical configuration of the H2–H complex in the yz plane. The H2 bond length is r, the separation between the centers of mass of H2 and H is R, and the angle between the z axis along R and the H2 bond axis is θ.

FIG. 1.

Geometrical configuration of the H2–H complex in the yz plane. The H2 bond length is r, the separation between the centers of mass of H2 and H is R, and the angle between the z axis along R and the H2 bond axis is θ.

Close modal

When the distance r between the nuclei in H2 is greater than or equal to 2.125 a.u. and R and θ are both small, the correct nuclear coordinates are still given by the nominal label H2–H, but the system should be identified as H3, H–H2 (pairing different H nuclei into H2), or H–H–H. The dipoles μy and μz are tabulated in these cases, but to indicate that they do not refer to H2–H, the results are printed in red in Tables S1–S9. In a small number of cases, no value is listed because the calculations did not converge.

To exemplify the results, values obtained from the first and second sets of calculations for a bond length r of 1.449 a.u. (the average bond length in the ground rovibrational state) are listed in Table I for R from 4.0 a.u. to 10.0 a.u. and for angles θ from 0° to 90° in steps of 15°. This table contains 84 nonzero values of μy and μz from the two main sets of calculations. The corresponding results agree to 10−6 a.u. in 51 of the 84 cases, differ by ±1 × 10−6 a.u. in 18 additional cases, and differ by more than ±1 × 10−6 a.u. but agree to 10−5 a.u. in the remaining 15 cases. In two cases of the last set, the difference between the two calculations is just ±2 × 10−6 a.u. The largest differences are found for the dipole when R = 9.0 a.u. or 10.0 a.u., where the results from the calculations with the tighter convergence criteria (identified by the superscript in the column headers) are preferable.

TABLE I.

Cartesian dipole components of H2–H for H2 bond length r = 1.449 a.u.; R denotes the separation between the centers of mass of H2 and H along the z axis, and θ is the angle between the H2 bond axis r and the z axis pointing along R. Results labeled have been obtained with an aug-cc-pV5Z basis set using Molpro 2012 and convergence criteria tighter than the default criteria (see text).

Rμyμyμzμz
θ = 0° 
4.0 −0.011 906 −0.011 907 
5.0 0.002 905 0.002 904 
6.0 0.003 829 0.003 829 
7.0 0.002 683 0.002 683 
8.0 0.001 689 0.001 690 
9.0 0.001 064 0.001 069 
10.0 0.000 693 0.000 696 
θ = 15° 
4.0 −0.004 294 −0.004 294 −0.014 923 −0.014 923 
5.0 −0.002 243 −0.002 243 0.001 540 0.001 539 
6.0 −0.001 234 −0.001 234 0.003 205 0.003 205 
7.0 −0.000 702 −0.000 702 0.002 368 0.002 369 
8.0 −0.000 416 −0.000 416 0.001 513 0.001 515 
9.0 −0.000 260 −0.000 260 0.000 957 0.000 963 
10.0 −0.000 170 −0.000 171 0.000 625 0.000 614 
θ = 30° 
4.0 −0.007 211 −0.007 211 −0.022 959 −0.022 960 
5.0 −0.003 811 −0.003 811 −0.002 064 −0.002 065 
6.0 −0.002 101 −0.002 101 0.001 544 0.001 544 
7.0 −0.001 194 −0.001 194 0.001 524 0.001 524 
8.0 −0.000 708 −0.000 708 0.001 038 0.001 038 
9.0 −0.000 442 −0.000 442 0.000 669 0.000 662 
10.0 −0.000 290 −0.000 291 0.000 438 0.000 428 
θ = 45° 
4.0 −0.008 003 −0.008 003 −0.033 411 −0.033 412 
5.0 −0.004 298 −0.004 298 −0.006 694 −0.006 694 
6.0 −0.002 381 −0.002 381 −0.000 614 −0.000 614 
7.0 −0.001 355 −0.001 355 0.000 409 0.000 409 
8.0 −0.000 805 −0.000 805 0.000 404 0.000 404 
9.0 −0.000 503 −0.000 503 0.000 280 0.000 273 
10.0 −0.000 330 −0.000 331 0.000 186 0.000 176 
θ = 60° 
4.0 −0.006 677 −0.006 677 −0.043 229 −0.043 229 
5.0 −0.003 638 −0.003 638 −0.011 002 −0.011 002 
6.0 −0.002 025 −0.002 025 −0.002 654 −0.002 654 
7.0 −0.001 156 −0.001 156 −0.000 662 −0.000 662 
8.0 −0.000 689 −0.000 688 −0.000 210 −0.000 210 
9.0 −0.000 430 −0.000 430 −0.000 101 −0.000 108 
10.0 −0.000 282 −0.000 283 −0.000 063 −0.000 072 
θ = 75° 
4.0 −0.003 755 −0.003 754 −0.050 009 −0.050 007 
5.0 −0.002 065 −0.002 065 −0.013 964 −0.013 964 
6.0 −0.001 154 −0.001 153 −0.004 075 −0.004 075 
7.0 −0.000 661 −0.000 661 −0.001 419 −0.001 419 
8.0 −0.000 394 −0.000 394 −0.000 647 −0.000 647 
9.0 −0.000 247 −0.000 247 −0.000 374 −0.000 381 
10.0 −0.000 162 −0.000 163 −0.000 243 −0.000 252 
θ = 90° 
4.0 −0.052 405 −0.052 404 
5.0 −0.015 011 −0.015 010 
6.0 −0.004 580 −0.004 580 
7.0 −0.001 690 −0.001 690 
8.0 −0.000 805 −0.000 805 
9.0 −0.000 473 −0.000 480 
10.0 −0.000 308 −0.000 317 
Rμyμyμzμz
θ = 0° 
4.0 −0.011 906 −0.011 907 
5.0 0.002 905 0.002 904 
6.0 0.003 829 0.003 829 
7.0 0.002 683 0.002 683 
8.0 0.001 689 0.001 690 
9.0 0.001 064 0.001 069 
10.0 0.000 693 0.000 696 
θ = 15° 
4.0 −0.004 294 −0.004 294 −0.014 923 −0.014 923 
5.0 −0.002 243 −0.002 243 0.001 540 0.001 539 
6.0 −0.001 234 −0.001 234 0.003 205 0.003 205 
7.0 −0.000 702 −0.000 702 0.002 368 0.002 369 
8.0 −0.000 416 −0.000 416 0.001 513 0.001 515 
9.0 −0.000 260 −0.000 260 0.000 957 0.000 963 
10.0 −0.000 170 −0.000 171 0.000 625 0.000 614 
θ = 30° 
4.0 −0.007 211 −0.007 211 −0.022 959 −0.022 960 
5.0 −0.003 811 −0.003 811 −0.002 064 −0.002 065 
6.0 −0.002 101 −0.002 101 0.001 544 0.001 544 
7.0 −0.001 194 −0.001 194 0.001 524 0.001 524 
8.0 −0.000 708 −0.000 708 0.001 038 0.001 038 
9.0 −0.000 442 −0.000 442 0.000 669 0.000 662 
10.0 −0.000 290 −0.000 291 0.000 438 0.000 428 
θ = 45° 
4.0 −0.008 003 −0.008 003 −0.033 411 −0.033 412 
5.0 −0.004 298 −0.004 298 −0.006 694 −0.006 694 
6.0 −0.002 381 −0.002 381 −0.000 614 −0.000 614 
7.0 −0.001 355 −0.001 355 0.000 409 0.000 409 
8.0 −0.000 805 −0.000 805 0.000 404 0.000 404 
9.0 −0.000 503 −0.000 503 0.000 280 0.000 273 
10.0 −0.000 330 −0.000 331 0.000 186 0.000 176 
θ = 60° 
4.0 −0.006 677 −0.006 677 −0.043 229 −0.043 229 
5.0 −0.003 638 −0.003 638 −0.011 002 −0.011 002 
6.0 −0.002 025 −0.002 025 −0.002 654 −0.002 654 
7.0 −0.001 156 −0.001 156 −0.000 662 −0.000 662 
8.0 −0.000 689 −0.000 688 −0.000 210 −0.000 210 
9.0 −0.000 430 −0.000 430 −0.000 101 −0.000 108 
10.0 −0.000 282 −0.000 283 −0.000 063 −0.000 072 
θ = 75° 
4.0 −0.003 755 −0.003 754 −0.050 009 −0.050 007 
5.0 −0.002 065 −0.002 065 −0.013 964 −0.013 964 
6.0 −0.001 154 −0.001 153 −0.004 075 −0.004 075 
7.0 −0.000 661 −0.000 661 −0.001 419 −0.001 419 
8.0 −0.000 394 −0.000 394 −0.000 647 −0.000 647 
9.0 −0.000 247 −0.000 247 −0.000 374 −0.000 381 
10.0 −0.000 162 −0.000 163 −0.000 243 −0.000 252 
θ = 90° 
4.0 −0.052 405 −0.052 404 
5.0 −0.015 011 −0.015 010 
6.0 −0.004 580 −0.004 580 
7.0 −0.001 690 −0.001 690 
8.0 −0.000 805 −0.000 805 
9.0 −0.000 473 −0.000 480 
10.0 −0.000 308 −0.000 317 

Significant features of the results for the Cartesian components of the collision-induced dipole are described next. Details on the dependence of the dipole components on r, R, and θ are presented in the supplementary material. The y component of the dipole is negative for all angles θ from 5° to 85°, when 3.0 a.u. ≤ R ≤ 10.0 a.u. and r ≤ 1.787 a.u., but in some cases, μy is positive at short range for larger bond lengths. The absolute value |μy| of the dipole in the y direction tends to decrease with increasing H2–H separation R at constant r and θ; this is true for all r ≤ 1.618 a.u. and any angle θ. From θ = 0° to θ = 90° at constant R and r, the absolute value of μy increases monotonically with increasing θ to a maximum between θ = 30° and θ = 45° and then decreases monotonically with further increases in θ. Figure 2 shows |μy| as a function of θ and R when r = 2.125 a.u. For R ≥ 6.0 a.u., the absolute value of μy increases monotonically with bond length r over the full range from 0.942 a.u. to 2.801 a.u. at constant θ.

FIG. 2.

Absolute value of μy for H2–H multiplied by 103 (in a.u.), for R between 3.4 a.u. and 10.0 a.u. and θ between 0° and 90°, at bond length r = 2.125 a.u.

FIG. 2.

Absolute value of μy for H2–H multiplied by 103 (in a.u.), for R between 3.4 a.u. and 10.0 a.u. and θ between 0° and 90°, at bond length r = 2.125 a.u.

Close modal

The dipole component μy shows the expected behavior for a quadrupole-induced dipole at long range. Exceptions with positive values of μy have been found at short range, for large bond lengths r and moderate to large angles θ. The exceptions at short range are consistent with substantial positive contributions to μy from exchange and overlap. These contributions become increasingly important as r increases and as the H2 molecule rotates toward the y axis (while 0° < θ < 90°), causing the H nucleus in H2 that has a positive z coordinate to move further out in the +y direction.

The dipole component in the z direction (along R) typically changes sign from negative to positive as R increases, for θ < 50° and r ≤ 2.125 a.u. Exceptions are found for several of the values of μz listed in red in Tables S1–S9. Figure 3 shows μz as a function of θ and R, at r = 1.449 a.u. When θ ≥ 60°, |μz| decreases monotonically with increasing R, for r from 0.942 to 2.125 a.u. When θ ≤ 55°, |μz| typically decreases with increasing R, then increases, and finally decreases again, a pattern that holds in a little over 90% of the cases. The sign changes in μz in this case lead to a complex pattern of variations with r at fixed R and θ.

FIG. 3.

Dipole component μz for H2–H multiplied by 103 (in a.u.) for R from 3.4 to 10.0 a.u. and θ decreasing from 90° to 0°, at r = 1.449 a.u. Each point on this surface has a potential energy ΔE above the minimum for H2–H that satisfies ΔE ≤ kBT at T = 2600 K.

FIG. 3.

Dipole component μz for H2–H multiplied by 103 (in a.u.) for R from 3.4 to 10.0 a.u. and θ decreasing from 90° to 0°, at r = 1.449 a.u. Each point on this surface has a potential energy ΔE above the minimum for H2–H that satisfies ΔE ≤ kBT at T = 2600 K.

Close modal

If μz were determined by quadrupolar induction effects alone, then we would find μz > 0 when θ < θm, and μz < 0 when θ > θm. Here, θm denotes the quadrupole “magic angle,” where the quadrupole field in the z direction vanishes [3 cos2m) − 1 = 0, θm ≈ 54.7356°]. This pattern of signs is generally followed at long range. For example, Fig. 4 shows μy and μz as functions of θ at R = 10.0 a.u. and r = 1.449 a.u. The results from θ = 0° to θ = 90° have been calculated directly ab initio, and results for larger θ have been deduced by symmetry. The oscillatory pattern characteristic of quadrupolar induction is evident in this figure.

FIG. 4.

Dipole components μy and μz for H2–H vs θ for R = 10.0 a.u. and r = 1.449 a.u.

FIG. 4.

Dipole components μy and μz for H2–H vs θ for R = 10.0 a.u. and r = 1.449 a.u.

Close modal

At short range (small R), the observed sign pattern of μz is somewhat different, for θ < θm and r from 0.942 to 1.787 a.u. We find μz < 0 in cases where the quadrupolar induction model would predict positive values of μz. In this range, there must be negative overlap and exchange contributions to μz that exceed the damped quadrupolar induction effects. But as R increases, μz converts to positive values for θ < θm, consistent with quadrupolar induction. The value of R where μz first becomes positive ranges from 3.9 a.u. to 9.0 a.u. for the bond lengths listed above.

For θ > θm, the quadrupole induction mechanism gives μz < 0. From the calculations, we have found μz < 0 for all θ ≥ 60° when the bond length r is between 0.942 and 1.787 a.u. inclusive, and μz < 0 for all θ ≥ 55° when r = 2.125, 2.463, or 2.801 a.u. (except when R = 3.0 a.u. and r = 2.801 a.u.). The overlap and exchange contributions to μz when θ ≥ 60° cannot be deduced on the basis of sign arguments alone, but the opposite signs of the short-range exchange and overlap effects for μy and μz in cases with θ < θm suggest that the H atom gains a partial negative charge at short range, while the region around the nucleus in H2 that is closer to the H atom gains a partial positive charge.

In Table II, we compare our results for the H2–H dipole with ab initio results obtained previously by GFM,20 with our results converted to the xz plane. The full comparison of the 288 values of μx or μz obtained in the calculations is provided in Tables S10–S14 of the supplementary material. The overall patterns of the dipole components are quite similar, but the several of the specific values differ. Comparing all 288 values from our work with the results of GFM,20 we find an average difference of 27.5% between the two calculations. However, nine of the μz values are distinct outliers in terms of the magnitudes of the differences. In each of these cases, the geometrical configuration is close to the point where μz changes sign as R increases, at fixed θ. A slight displacement in the location of the R value where μz crosses zero leads to large percent differences. Excluding these nine points from the comparison set reduces the average absolute value of the discrepancy between our results and the earlier results20 to 6.45%, a more realistic representation of the differences. The magnitude of the differences varies noticeably with the angle θ and the dipole component μx or μz. The closest agreement is found for μz with θ = 90°, where the percent difference is only 0.40%. In general, the differences in μz values are smaller than the differences in μx values. The averages of the absolute values of the percent differences in μz are 2.02% at 0° (excluding one outlier), 4.92% at 30° (excluding eight outliers), and 6.17% at 60°. The differences in the μx values are 8.91% at 30° and 10.2% at 60°.

TABLE II.

Comparison of Cartesian components of the dipole moment of H2–H in this work with the results of Gustafsson, Frommhold, and Meyer (GFM).20 We have reoriented the H2–H complex in our work to correspond to the orientation used by GFM. Results are listed from our calculations with the aug-cc-pV5Z basis set, Molpro 2012, and the tighter convergence criteria, except for R = 3.5 and 4.5 a.u.; those results were obtained with Molpro 2006 and the default convergence criteria. The H–H bond length is r = 1.449 a.u., the angle between the H2 bond axis r and the pair-fixed z axis (along R) is θ, and the distance between the centers of mass of H2 and H is R.

Rμxμx(GMF)μzμz(GMF)
θ = 0° 
3.0 −0.068 855 −0.069 411 
3.5 −0.034 992 −0.035 530 
4.0 −0.011 906 −0.012 267 
4.5 −0.001 161 −0.001 367 
5.0 0.002 905 0.002 792 
6.0 0.003 829 0.003 781 
7.0 0.002 683 0.002 662 
8.0 0.001 689 0.001 684 
9.0 0.001 064 0.001 061 
10.0 0.000 693 0.000 688 
θ = 30° 
3.0 −0.014 908 −0.013 912 −0.092 080 −0.092 580 
3.5 −0.010 327 −0.009 463 −0.050 876 −0.049 865 
4.0 −0.007 211 −0.006 546 −0.022 959 −0.020 867 
4.5 −0.005 195 −0.004 811 −0.008 620 −0.006 361 
5.0 −0.003 811 −0.003 491 −0.002 064 −0.000 860 
6.0 −0.002 101 −0.001 746 0.001 544 0.001 708 
7.0 −0.001 194 −0.001 065 0.001 524 0.001 506 
8.0 −0.000 708 −0.000 654 0.001 038 0.001 013 
9.0 −0.000 442 −0.000 414 0.000 669 0.000 649 
10.0 −0.000 290 −0.000 274 0.000 438 0.000 421 
θ = 60° 
3.0 −0.012 731 −0.011 852 −0.143 668 −0.142 866 
3.5 −0.009 222 −0.008 541 −0.081 857 −0.080 270 
4.0 −0.006 677 −0.006 297 −0.043 229 −0.040 993 
4.5 −0.004 907 −0.004 666 −0.022 026 −0.019 664 
5.0 −0.003 638 −0.003 073 −0.011 002 −0.009 779 
6.0 −0.002 025 −0.001 575 −0.002 654 −0.002 450 
7.0 −0.001 156 −0.001 016 −0.000 662 −0.000 638 
8.0 −0.000 689 −0.000 633 −0.000 210 −0.000 210 
9.0 −0.000 430 −0.000 405 −0.000 101 −0.000 100 
10.0 −0.000 282 −0.000 269 −0.000 063 −0.000 065 
θ = 90° 
3.0 −0.170 623 −0.171 036 
3.5 −0.096 544 −0.096 864 
4.0 −0.052 405 −0.052 610 
4.5 −0.028 028 −0.028 156 
5.0 −0.015 011 −0.015 089 
6.0 −0.004 580 −0.004 606 
7.0 −0.001 690 −0.001 696 
8.0 −0.000 805 −0.000 808 
9.0 −0.000 473 −0.000 472 
10.0 −0.000 308 −0.000 308 
Rμxμx(GMF)μzμz(GMF)
θ = 0° 
3.0 −0.068 855 −0.069 411 
3.5 −0.034 992 −0.035 530 
4.0 −0.011 906 −0.012 267 
4.5 −0.001 161 −0.001 367 
5.0 0.002 905 0.002 792 
6.0 0.003 829 0.003 781 
7.0 0.002 683 0.002 662 
8.0 0.001 689 0.001 684 
9.0 0.001 064 0.001 061 
10.0 0.000 693 0.000 688 
θ = 30° 
3.0 −0.014 908 −0.013 912 −0.092 080 −0.092 580 
3.5 −0.010 327 −0.009 463 −0.050 876 −0.049 865 
4.0 −0.007 211 −0.006 546 −0.022 959 −0.020 867 
4.5 −0.005 195 −0.004 811 −0.008 620 −0.006 361 
5.0 −0.003 811 −0.003 491 −0.002 064 −0.000 860 
6.0 −0.002 101 −0.001 746 0.001 544 0.001 708 
7.0 −0.001 194 −0.001 065 0.001 524 0.001 506 
8.0 −0.000 708 −0.000 654 0.001 038 0.001 013 
9.0 −0.000 442 −0.000 414 0.000 669 0.000 649 
10.0 −0.000 290 −0.000 274 0.000 438 0.000 421 
θ = 60° 
3.0 −0.012 731 −0.011 852 −0.143 668 −0.142 866 
3.5 −0.009 222 −0.008 541 −0.081 857 −0.080 270 
4.0 −0.006 677 −0.006 297 −0.043 229 −0.040 993 
4.5 −0.004 907 −0.004 666 −0.022 026 −0.019 664 
5.0 −0.003 638 −0.003 073 −0.011 002 −0.009 779 
6.0 −0.002 025 −0.001 575 −0.002 654 −0.002 450 
7.0 −0.001 156 −0.001 016 −0.000 662 −0.000 638 
8.0 −0.000 689 −0.000 633 −0.000 210 −0.000 210 
9.0 −0.000 430 −0.000 405 −0.000 101 −0.000 100 
10.0 −0.000 282 −0.000 269 −0.000 063 −0.000 065 
θ = 90° 
3.0 −0.170 623 −0.171 036 
3.5 −0.096 544 −0.096 864 
4.0 −0.052 405 −0.052 610 
4.5 −0.028 028 −0.028 156 
5.0 −0.015 011 −0.015 089 
6.0 −0.004 580 −0.004 606 
7.0 −0.001 690 −0.001 696 
8.0 −0.000 805 −0.000 808 
9.0 −0.000 473 −0.000 472 
10.0 −0.000 308 −0.000 308 

As mentioned above, we have carried out two sets of calculations with larger basis sets, one with an aug-cc-pV6Z basis (A6Z) and the other with a d-aug-cc-pV5Z basis (D5Z). Results from these calculations for r = 1.449 a.u., eight R values, and 19 angles θ are listed in Table S15 in the supplementary material. The results from these larger basis sets agree well with the results from the aug-cc-pV5Z basis—the only exceptions are found very near to the points where μz changes sign. Yet even including those points, the absolute values of the results from the A6Z basis agree with the A5Z results to ∼0.25%, and the results from the D5Z basis agree with the A5Z results to ∼0.23%.

In two of the cases where the GFM results20 and ours differ significantly, r = 1.449 a.u. and R = 5.0 a.u., with θ = 30° or θ = 60°, several additional independent calculations were run with Molpro 2015, the aug-cc-pV5Z basis, the tighter convergence criteria, and the base field strength f = 0.001 a.u. The results from RHF/RCCSD(T) and RHF/UCCSD(T) calculations are listed in Table III. Increasing f to 0.01 a.u. changed only the final digit in the values listed. The new calculations showed the same differences from GFM’s results20 as our previous work had shown.

TABLE III.

Calculated Cartesian dipole components (in a.u.) for two geometrical configurations of H2–H. The bond length of H2 is r = 1.449 a.u., and the separation between the centers of mass of H2 and H is R = 5.0 a.u. along the z axis in both cases. The angle between the H2 bond axis and the z axis is θ. Results from different ab initio methods and implementations are listed. The base field f is described in the text.

θ = 30°Wave functionBase fieldμxμz
GFM   −0.003 491 −0.000 860 
Molpro 2006 RHF/UCCSD(T) f = 0.002 −0.003 811 −0.002 065 
Molpro 2012 RHF/UCCSD(T) f = 0.002 −0.003 811 −0.002 064 
Molpro 2015 RHF/RCCSD(T) f = 0.001 −0.003 810 −0.002 022 
 RHF/UCCSD(T) f = 0.001 −0.003 811 −0.002 064 
θ = 30°Wave functionBase fieldμxμz
GFM   −0.003 491 −0.000 860 
Molpro 2006 RHF/UCCSD(T) f = 0.002 −0.003 811 −0.002 065 
Molpro 2012 RHF/UCCSD(T) f = 0.002 −0.003 811 −0.002 064 
Molpro 2015 RHF/RCCSD(T) f = 0.001 −0.003 810 −0.002 022 
 RHF/UCCSD(T) f = 0.001 −0.003 811 −0.002 064 
θ = 60°Wave functionBase fieldμxμz
GFM   −0.003 073 −0.009 779 
Molpro 2006 RHF/UCCSD(T) f = 0.002 −0.003 638 −0.011 002 
Molpro 2012 RHF/UCCSD(T) f = 0.002 −0.003 638 −0.011 002 
Molpro 2015 RHF/RCCSD(T) f = 0.001 −0.003 637 −0.010 969 
 RHF/UCCSD(T) f = 0.001 −0.003 638 −0.011 002 
θ = 60°Wave functionBase fieldμxμz
GFM   −0.003 073 −0.009 779 
Molpro 2006 RHF/UCCSD(T) f = 0.002 −0.003 638 −0.011 002 
Molpro 2012 RHF/UCCSD(T) f = 0.002 −0.003 638 −0.011 002 
Molpro 2015 RHF/RCCSD(T) f = 0.001 −0.003 637 −0.010 969 
 RHF/UCCSD(T) f = 0.001 −0.003 638 −0.011 002 

Differences with the earlier results may be due to the choice of basis set and/or the computational method. The GFM calculations20 were carried out in a Gaussian basis derived from the Huzinaga 10s basis,250 augmented by p and d functions. The s functions with the three smallest exponents in the Huzinaga set250 were placed at the center of the H2 bond. The remaining seven s functions from the 10s basis were associated with each H nucleus individually, but the five with the largest exponents were contracted and then allowed to float off the protons. A set of p functions with exponent 1.2 was assigned to each H center, and p sets with exponents 0.3 and 0.1 were located at the bond center. Additionally, two sets of d functions with exponents 0.4 and 0.13 were located at the bond center. The basis used for the separate H atom itself is not explicitly specified in Ref. 20; however, if 3s, 2p, and 2d functions were placed at the midpoint of each H–H segment, and 3s and one set of p functions were assigned to each H center, this would give a total of 75 contracted functions in the basis, vs 165 in our work.

The earlier calculations were run in a multistep process, starting with self-consistent field (SCF) calculations, then generating localized orbitals from the molecular orbitals, in order to make it possible to separate intramolecular and intermolecular correlation.251 Double excitations, from the 1σg2 configuration of H2 obtained at the SCF level to 1σu2, 1πu2, and 2σg2 configurations, were included to produce a multiconfigurational SCF (MCSCF) wave function. Then, all single and double excitations from the MCSCF function were included in the coupled-electron pair approximation (CEPA), which is size consistent.20,251 To minimize basis-set superposition errors, in previous calculations for H2 interacting with an inert gas atom, intra-H2 correlation had been treated separately from the correlations between electrons in orbitals localized to H2 and electrons in orbitals localized on the inert gas atom. The separation of correlation effects was implemented via the self-consistent coupled electron pair (SCEP) technique.20,251 This approach has yielded highly accurate results for the interaction-induced dipoles of H2–He158,159,161 and H2–H2.251 Our previous work on the collision-induced dipole of H2–He at the CCSD(T) level169 gave results for the Cartesian dipole components that agreed very well with those of Borysow, Frommhold, and Meyer.164 It is possible that differences between the wave function obtained in Ref. 20 and our RHF/UCCSD(T) function contribute to the observed differences in the dipole components for the open-shell system H2–H.

The breakdown of the Hellmann-Feynman theorem26,252–254 for various approximate wave functions255 means that the dipole obtained as an expectation value need not agree exactly with the dipole obtained from finite-field calculations. In general, the error in the expectation value of the dipole moment is of first-order in the error in the wave function, while the error in the energy is of second-order in the error in the wave function. This may make the finite-field results preferable.256 

For applications in computing line shapes for collision-induced spectra, the calculated dipole needs to be represented as a series in the spherical harmonics of the orientation angles Ωr and ΩR for the bond axis r and the intermolecular vector R, respectively,22–25 

(1)

In this equation, M designates the spherical-tensor component of the dipole. The M = 0 component is identical to μz, μ+1 = −(1/2)1/2x + iμy), and μ−1 = (1/2)1/2x − iμy). The values of DλL(r, R) depend only on λ, L, and the magnitudes of r and R. The quantity ⟨λ m L M − m|1M⟩ is a Clebsch-Gordan coefficient. Due to the symmetry of the H2 molecule, λ is always even, and for the Clebsch-Gordan coefficient to be nonzero, L = λ ± 1. Also, because R is oriented along the z axis, M − m = 0.

The coefficients DλL(r, R) have been determined by least-squares fits to the Cartesian components of the dipole moment as functions of the orientation angle θ of H2, at fixed r and R. We fit the 17 nonzero values of μy for the various angles θ, together with all 19 values of μz. In the first study, dipole coefficients through λ = 24 and L = 25 were determined from the Cartesian dipole components. The coefficients beyond D89(r, R) are virtually negligible for r ≤ 1.787 a.u. For r = 2.125 a.u.–2.801 a.u., the higher coefficients start to grow at small R values, but they are still substantially smaller than the leading coefficients. Also, the higher coefficients tend to be erratic as functions of R, suggesting that while they do help to determine a least-squares fit to the data, they are not physically meaningful. In the second set of calculations, we fit the coefficients DλL(r, R) only through D89(r, R). In Table IV, we list the coefficients D01(r, R), D21(r, R), D23(r, R), D43(r, R), D45(r, R), D65(r, R), D67(r, R), D87(r, R), and D89(r, R) for r = 1.449 a.u. as obtained from fits to both sets of results with the A5Z basis, from a fit through D89 to the results from the A6Z basis and from a fit through D89 to the results from the D5Z basis. The A5Z results for the coefficients DλL(r, R) over the full range of r values from r = 0.942 a.u. to r = 2.801 a.u. are listed in Tables S16–S24 of the supplementary material. The dipole coefficients from the second set of A5Z calculations (with tighter convergence criteria) are indicated by a superscript . Generally, the values of the coefficients from D01 to D89 from all four sets of calculations at r = 1.449 a.u. and from the two sets of calculations at the other r values agree well. This is noteworthy, considering that the first set of results comes from fits up to λ = 24 and L = 25, while in the second, third, and fourth sets, no dipole coefficients beyond D89 were included. The differences indicate the level of uncertainty in the results. The coefficient D23 appears to be best determined overall; relative to the A5Z results, the average absolute value of the difference in the A5Z results is 0.064%; for the A6Z results, 0.11%; and for the D5Z results, 0.14%. Excluding the range from R = 8.0 a.u. to R = 10.0 a.u. (where D01 is typically single-digit), the average absolute value of the difference in D01 relative to A5Z is 0.011% for A5Z, 0.16% for A6Z, and 1.57% for D6Z. Differences in D21 among the results with different basis sets are less than 1%; differences in D45 are less than 2%, and differences in D43 are less than 3% (again, excluding R values where the coefficients are single-digit).

TABLE IV.

Coefficients for the spherical harmonic expansion of the dipole of H2–H in Eq. (1). Results in a.u. multiplied by 106 are listed as obtained with the A5Z basis and the default convergence criteria, with the A5Z basis and tighter convergence criteria (indicated by a superscript ), with the A6Z basis, and with the D5Z basis. Results from a fit limited to θ = 0°, 30°, 60°, and 90° are listed in the rows labeled (45), for the most direct comparison with the results of GFM.20 The H2 bond length is 1.449 a.u.

R (a.u.)D01D21D23D43D45D65D67D87D89
3.0 −135 447 −10 736 31 156 1079 59 121 −80 −2 
A6Z −135 448 −10 740 31 150 1079 56 119 −79 −2 
D5Z −135 455 −10 733 31 150 1082 57 118 −80 −4 
(45) −135 421 −10 687 31 100 1144     
GFM −135 348 −11 260 30 574 1281 −217     
3.4 −86 532 −5 943 21 022 248 466 56 −22 −2 
3.5 −76 474 −5 269 19 272 138 486 44 −14 −2 
(45) −76 466 −5 254 19 257 162 471     
GFM −75 599 −5 914 19 070 576 −100     
3.6 −67 364 −4 697 17 716 55 490 34 −8 −1 
3.7 −59 173 −4 201 16 319 −8 483 25 −3 −1 
3.8 −51 853 −3 764 15 055 −51 467 20 −1 −1 
3.9 −45 342 −3 372 13 905 −84 449 13 −1 
4.0 −39 575 −3 019 12 853 −103 425 10 
4.0 −39 575 −3 019 12 853 −102 425 10 
A6Z −39 581 −3 026 12 848 −104 422 
D5Z −39 584 −3 021 12 849 −104 425 
(45) −39 574 −3 017 12 852 −98 424     
GFM −38 087 −3 649 12 984 482 −392     
4.5 −19 590 −1 686 8 744 −113 295 
(45) −19 588 −1 687 8 746 −113 296     
GFM −17 976 −2 214 9 002 516 −492     
5.0 −9 378 −890 6 022 −78 190 −2 
5.0 −9 378 −890 6 023 −78 189 −2 
A6Z −9 381 −894 6 020 −77 189 −2 
D5Z −9 378 −890 6 018 −78 190 
(45) −9 379 −891 6 024 −79 191     
GFM −8 538 −1 384 5 992 347 −141     
6.0 −1 903 −219 3 004 −25 74 
6.0 −1 904 −219 3 004 −26 74 
A6Z −1 903 −221 3 002 −25 74 −1 
D5Z −1 892 −217 2 999 −25 73 
(45) −1 904 −219 3 004 −26 74     
GFM −1 778 −516 2 813 61 32     
7.0 −279 −51 1 622 −7 30 
7.0 −279 −51 1 622 −7 30 
A6Z −277 −51 1 620 −8 30 
D5Z −259 −49 1 621 −5 30 
(45) −279 −51 1 622 −7 30     
GFM −276 −137 1 546 28     
8.0 −12 942 −2 14 
8.0 −12 942 −2 14 
A6Z −12 942 −2 13 
D5Z 27 −12 945 −1 13 
(45) −12 942 −2 13     
GFM −2 −45 909 −4 17     
9.0 25 −5 586 −1 
9.0 31 −4 585 
A6Z 32 −4 583 −1 
D5Z 46 −5 586 
(45) 31 −5 585     
GFM 26 −18 567 −3     
10.0 13 −1 383 −1 
10.0 22 −2 382 
A6Z 21 −1 381 
D5Z 32 −4 383 
(45) 22 −2 382     
GFM 16 −8 371 −2     
R (a.u.)D01D21D23D43D45D65D67D87D89
3.0 −135 447 −10 736 31 156 1079 59 121 −80 −2 
A6Z −135 448 −10 740 31 150 1079 56 119 −79 −2 
D5Z −135 455 −10 733 31 150 1082 57 118 −80 −4 
(45) −135 421 −10 687 31 100 1144     
GFM −135 348 −11 260 30 574 1281 −217     
3.4 −86 532 −5 943 21 022 248 466 56 −22 −2 
3.5 −76 474 −5 269 19 272 138 486 44 −14 −2 
(45) −76 466 −5 254 19 257 162 471     
GFM −75 599 −5 914 19 070 576 −100     
3.6 −67 364 −4 697 17 716 55 490 34 −8 −1 
3.7 −59 173 −4 201 16 319 −8 483 25 −3 −1 
3.8 −51 853 −3 764 15 055 −51 467 20 −1 −1 
3.9 −45 342 −3 372 13 905 −84 449 13 −1 
4.0 −39 575 −3 019 12 853 −103 425 10 
4.0 −39 575 −3 019 12 853 −102 425 10 
A6Z −39 581 −3 026 12 848 −104 422 
D5Z −39 584 −3 021 12 849 −104 425 
(45) −39 574 −3 017 12 852 −98 424     
GFM −38 087 −3 649 12 984 482 −392     
4.5 −19 590 −1 686 8 744 −113 295 
(45) −19 588 −1 687 8 746 −113 296     
GFM −17 976 −2 214 9 002 516 −492     
5.0 −9 378 −890 6 022 −78 190 −2 
5.0 −9 378 −890 6 023 −78 189 −2 
A6Z −9 381 −894 6 020 −77 189 −2 
D5Z −9 378 −890 6 018 −78 190 
(45) −9 379 −891 6 024 −79 191     
GFM −8 538 −1 384 5 992 347 −141     
6.0 −1 903 −219 3 004 −25 74 
6.0 −1 904 −219 3 004 −26 74 
A6Z −1 903 −221 3 002 −25 74 −1 
D5Z −1 892 −217 2 999 −25 73 
(45) −1 904 −219 3 004 −26 74     
GFM −1 778 −516 2 813 61 32     
7.0 −279 −51 1 622 −7 30 
7.0 −279 −51 1 622 −7 30 
A6Z −277 −51 1 620 −8 30 
D5Z −259 −49 1 621 −5 30 
(45) −279 −51 1 622 −7 30     
GFM −276 −137 1 546 28     
8.0 −12 942 −2 14 
8.0 −12 942 −2 14 
A6Z −12 942 −2 13 
D5Z 27 −12 945 −1 13 
(45) −12 942 −2 13     
GFM −2 −45 909 −4 17     
9.0 25 −5 586 −1 
9.0 31 −4 585 
A6Z 32 −4 583 −1 
D5Z 46 −5 586 
(45) 31 −5 585     
GFM 26 −18 567 −3     
10.0 13 −1 383 −1 
10.0 22 −2 382 
A6Z 21 −1 381 
D5Z 32 −4 383 
(45) 22 −2 382     
GFM 16 −8 371 −2     

Results for the dipole coefficients D01, D21, D23, D43, and D45 based on the work of GFM20 are also listed in Table IV and in Tables S16–S24. These values were obtained from a least-squares fit of the dipole coefficients DλL to the Cartesian dipole components reported in Ref. 20. The ab initio calculations in Ref. 20 gave a total of 6 nonzero Cartesian dipole components, from which we have derived the coefficients up to D45. From the tables, the dipole coefficients obtained from Ref. 20 agree reasonably well with our results in terms of the overall pattern of the coefficients, but in terms of the numerical values, discrepancies with Ref. 20 are evident in cases where the results from our four basis sets (A5Z, A5Z, A6Z, and D5Z) agree well. We have carried out further calculations to separate effects on the dipole coefficients DλL(r, R) that are due to differences in the Cartesian components of the dipoles vs the effects of working with four angles in Ref. 20 and 19 angles in the current study. We determined the dipole coefficients D01, D21, D23, D43, and D45 from the Cartesian dipole components in the A5Z calculations but restricted to the angles 0°, 30°, 60°, and 90°. These results are listed on the lines labeled (45) in Tables IV and S16–S24. The results from the (45) fits are generally quite close to the results from the full 19-angle fits, and they are closer to those results than to the earlier results from Ref. 20. Our results suggest that the leading dipole coefficients can usually be determined quite well from calculations at a smaller number of angles.

In Fig. 5, we plot D01, D21, D23, D43, and D45 vs the H2–H separation R, for bond length r = 1.449 a.u., to show how the magnitudes of the coefficients compare. For small R, the isotropic overlap and exchange coefficient D01 is larger in magnitude than all of the other coefficients, and it is negative. The crossover of |D01| with D23 occurs between R = 5.0 a.u. and 6.0 a.u. In Fig. 6, we show D01, D21, D23, D43, and D45 vs R for r = 2.125 a.u. The coefficients are generally larger at 2.125 a.u., and D43 and D45 become noticeably different from zero on the same plot with the other coefficients.

FIG. 5.

Dipole coefficients D01, D21, D23, D43, and D45 for H2–H as functions of R for r = 1.449 a.u.

FIG. 5.

Dipole coefficients D01, D21, D23, D43, and D45 for H2–H as functions of R for r = 1.449 a.u.

Close modal
FIG. 6.

Dipole coefficients D01, D21, D23, D43, and D45 for H2–H as functions of R for r = 2.125 a.u.

FIG. 6.

Dipole coefficients D01, D21, D23, D43, and D45 for H2–H as functions of R for r = 2.125 a.u.

Close modal

Long-range classical induction effects are reflected in D23 for quadrupolar induction and D45 for hexadecapolar induction.22–25 Through order R−7, D23 is the sum of a direct quadrupolar induction term that varies as R−4 and an R−7 back-induction term. From Ref. 23, for an atom A and molecule B, through order R−7 at long range,

(2)

where αA is the polarizability of the atom, ΘB(r) is the quadrupole of the molecule as a function of bond length r, αB,zz(r) is the molecular polarizability parallel to the bond axis as a function of r, and αB,xx(r) is the polarizability perpendicular to the bond axis. The quadrupole of H2 and the components of the polarizability of H2 are known accurately at the specific r values used here, from the work of Miliordos and Hunt.257 The polarizability of the H atom is αH = 4.5 a.u. In Fig. 7, we show that D23(r, R) converges to the quadrupolar induction form as R increases, for bond lengths r = 1.111 a.u., r = 1.449 a.u., and r = 1.787 a.u. Results for other bond lengths are similar. In Table V, we list the values of the quadrupole that would be deduced from D23(r, R) at R = 8.0, 9.0, and 10.0 a.u. for each bond length r, if the coefficient were entirely due to quadrupolar induction. At shorter range, the values of D23(r, R) reflect overlap damping of the classical induction effects, as well as exchange and orbital distortion, which tend to reduce D23(r, R) from its long-range limiting form. Two estimates of the quadrupole are shown for each r value, the first (a) obtained with the R−4 term in Eq. (2) alone and the second (b) from the full version of Eq. (2), with ab initio values used for the polarizability components.257 Agreement with the quadrupole values that have been calculated ab initio257 is very good, and the agreement improves as R increases, as shown in Fig. S2 in the supplementary material. The average error in the value of Θ derived from D23(r, R) at R = 10.0 a.u. is 1.20%, when the R−7 back-induction term is included in the analysis. The estimated quadrupole is always larger than the ab initio quadrupole.

FIG. 7.

Dipole coefficients D23 as functions of R, showing convergence of the ab initio values to the quadrupolar induction form at long range, for bond lengths r = 1.111, 1.449, and 1.787 a.u.

FIG. 7.

Dipole coefficients D23 as functions of R, showing convergence of the ab initio values to the quadrupolar induction form at long range, for bond lengths r = 1.111, 1.449, and 1.787 a.u.

Close modal
TABLE V.

Quadrupole moments (in a.u.) estimated from D23(r, R) compared with quadrupole moments Θcalc calculated ab initio.257 The tabulated D23 values have been multiplied by 106. The estimate Θest(a) is obtained by considering only the direct quadrupolar induction contribution to D23(r, R) of order R−4, while Θest(b) also includes back-induction effects of order R−7. The calculated quadrupole at r = 1.618 a.u. has been obtained by interpolation of the results in Ref. 257, as have the polarizability tensor components at r = 1.618 a.u. needed to obtain Θest(b).

R = 8.0 (a.u.)R = 9.0 (a.u.)R = 10.0 (a.u.)
r (a.u.)D23Θest(a)Θest(b)D23Θest(a)Θest(b)D23Θest(a)Θest(b)Θcalc
0.942 446.1 0.2344 0.2332 277.6 0.2337 0.2328 182.2 0.2338 0.2331 0.2302 
1.111 599.5 0.3150 0.3130 373.1 0.3140 0.3126 244.6 0.3137 0.3127 0.3086 
1.280 766.2 0.4027 0.3997 476.4 0.4011 0.3990 312.0 0.4004 0.3989 0.3934 
1.449 942.2 0.4951 0.4909 584.8 0.4923 0.4893 382.4 0.4907 0.4885 0.4823 
1.618 1123.0 0.5902 0.5843 695.3 0.5853 0.5812 454.1 0.5825 0.5795 0.5729 
1.787 1303.8 0.6852 0.6774 805.0 0.6776 0.6723 525.0 0.6735 0.6696 0.6624 
2.125 1642.9 0.8634 0.8513 1008.3 0.8488 0.8404 655.7 0.8412 0.8351 0.8266 
2.463 1911.7 1.0047 0.9880 1165.5 0.9811 0.9696 755.6 0.9694 0.9611 0.9507 
2.801 2063.7 1.0845 1.0643 1248.0 1.0506 1.0368 806.3 1.0344 1.0244 1.0126 
R = 8.0 (a.u.)R = 9.0 (a.u.)R = 10.0 (a.u.)
r (a.u.)D23Θest(a)Θest(b)D23Θest(a)Θest(b)D23Θest(a)Θest(b)Θcalc
0.942 446.1 0.2344 0.2332 277.6 0.2337 0.2328 182.2 0.2338 0.2331 0.2302 
1.111 599.5 0.3150 0.3130 373.1 0.3140 0.3126 244.6 0.3137 0.3127 0.3086 
1.280 766.2 0.4027 0.3997 476.4 0.4011 0.3990 312.0 0.4004 0.3989 0.3934 
1.449 942.2 0.4951 0.4909 584.8 0.4923 0.4893 382.4 0.4907 0.4885 0.4823 
1.618 1123.0 0.5902 0.5843 695.3 0.5853 0.5812 454.1 0.5825 0.5795 0.5729 
1.787 1303.8 0.6852 0.6774 805.0 0.6776 0.6723 525.0 0.6735 0.6696 0.6624 
2.125 1642.9 0.8634 0.8513 1008.3 0.8488 0.8404 655.7 0.8412 0.8351 0.8266 
2.463 1911.7 1.0047 0.9880 1165.5 0.9811 0.9696 755.6 0.9694 0.9611 0.9507 
2.801 2063.7 1.0845 1.0643 1248.0 1.0506 1.0368 806.3 1.0344 1.0244 1.0126 

The coefficient D45(r, R) is determined by hexadecapolar induction at long range. For an atom A and molecule B, to leading order,22–25 

(3)

where ΦB(r) denotes the hexadecapole of the molecule as a function of bond length. In Fig. 8, we show that D45(r, R) from our calculations converges to the known hexadecapolar induction form for r = 1.111 a.u., r = 1.449 a.u., and r = 1.787 a.u. Results for other r values are similar, except for r = 0.942, where the hexadecapole is quite small. Hexadecapolar induction effects were not detectable in the study of the interaction-induced dipole of H2–H in Ref. 20. In Table VI, we compare the values of the hexadecapole as a function of bond length deduced from D45(r, R) at R = 8.0, 9.0, and 10.0 a.u. with the values that have been computed directly ab initio. Figure S3 in the supplementary material shows that the agreement increases with increasing R. The difference between the ab initio values of Φ257 and the estimates of Φ based on D45(r, R) at R = 10.0 a.u. is 40% for r = 0.942 a.u. and 9.5% for 1.111 a.u.; for the other bond lengths, the percent error in the hexadecapole estimated from D45(r, R) at R = 10.0 a.u. ranges from 4.7% to 7.2%. The estimated hexadecapole is always larger than the ab initio hexadecapole for r ≥ 1.280 a.u.

FIG. 8.

Dipole coefficients D45 as functions of R, showing convergence of the ab initio values to the hexadecapolar induction form at long range, for bond lengths r = 1.111, 1.449, and 1.787 a.u.

FIG. 8.

Dipole coefficients D45 as functions of R, showing convergence of the ab initio values to the hexadecapolar induction form at long range, for bond lengths r = 1.111, 1.449, and 1.787 a.u.

Close modal
TABLE VI.

Hexadecapole moments (in a.u.) estimated from D45(r, R) compared with hexadecapole moments Φcalc calculated ab initio.257 The tabulated D45 values have been multiplied by 106. The estimate Φest is obtained from the direct hexadecapolar induction contribution to D45(r, R) of order R−6. The calculated hexadecapole at r = 1.618 a.u. has been obtained by interpolation of the results in Ref. 257.

R = 8.0 (a.u.)R = 9.0 (a.u.)R = 10.0 (a.u.)
r (a.u.)D45ΦestD45ΦestD45ΦestΦcalc
0.942 1.83 0.05 0.99 0.05 0.38 0.04 0.0628 
1.111 4.39 0.11 2.31 0.12 1.07 0.11 0.1175 
1.280 8.52 0.22 4.34 0.23 2.10 0.21 0.1993 
1.449 13.77 0.36 6.81 0.36 3.36 0.33 0.3139 
1.618 20.50 0.53 9.91 0.52 4.97 0.49 0.4664 
1.787 29.13 0.76 13.89 0.73 7.00 0.70 0.6582 
2.125 52.15 1.36 24.65 1.30 12.42 1.23 1.1649 
2.463 83.19 2.17 38.58 2.04 19.36 1.92 1.8052 
2.801 117.46 3.06 53.74 2.84 26.88 2.67 2.4912 
R = 8.0 (a.u.)R = 9.0 (a.u.)R = 10.0 (a.u.)
r (a.u.)D45ΦestD45ΦestD45ΦestΦcalc
0.942 1.83 0.05 0.99 0.05 0.38 0.04 0.0628 
1.111 4.39 0.11 2.31 0.12 1.07 0.11 0.1175 
1.280 8.52 0.22 4.34 0.23 2.10 0.21 0.1993 
1.449 13.77 0.36 6.81 0.36 3.36 0.33 0.3139 
1.618 20.50 0.53 9.91 0.52 4.97 0.49 0.4664 
1.787 29.13 0.76 13.89 0.73 7.00 0.70 0.6582 
2.125 52.15 1.36 24.65 1.30 12.42 1.23 1.1649 
2.463 83.19 2.17 38.58 2.04 19.36 1.92 1.8052 
2.801 117.46 3.06 53.74 2.84 26.88 2.67 2.4912 

At long range, the coefficient D01(r, R) is determined by van der Waals dispersion and back-induction;23 it varies as R−7 to leading order. The polarization of the electronic charge distribution on each center (H2 or H) due to the H2–H interaction attracts the nuclei on the same center.26 The van der Waals dispersion force between two atoms in S states was explained by Feynman as a result of this electrostatic attraction.26 Hunt later proved Feynman’s “conjecture” analytically and proved that it also holds for molecules of arbitrary symmetry.32 Electron correlation allows electronic charge to accumulate between the nuclei, but the dispersion forces on the nuclei themselves are classical electrostatic forces.26,32

The dispersion dipole is derived from the change in dispersion energy for interacting molecules in a uniform applied electric field. The electric field alters the interaction energy in two ways: Each molecule is hyperpolarized by the concerted effects of the applied field and the fluctuating field from the neighboring molecule.31 Additionally, the applied field alters the correlations of the spontaneously fluctuating charge densities on each center.31 

Perturbation analyses of the dispersion dipole have been presented by Byers Brown and Whisnant27 and by Craig and Thirunamachandran.30 Hunt developed an approximation for the dispersion dipole in terms of static polarizabilities α and dipole-dipole quadrupole polarizabilities B of the interacting molecules, along with the C6 van der Waals coefficient for the pair.28 Subsequently, Galatry and Gharbi29 derived an exact expression for the dispersion dipole of a pair of atoms A and B in terms of the integrals of αA(iω) BB(0, iω) and αB(iω) BA(0, iω) over imaginary frequencies. Bohr and Hunt carried out a symmetry analysis to find the dispersion contributions to D01, D21, D23, and D43 for an atom A interacting with molecule B;23 the dispersion contribution to D45 vanishes.23 Combining the dispersion term in D01 with the back-induction effect23 gives

(4)

where

(5)

and

(6)

The integrals in Eq. (4) have been evaluated very accurately for H2–H by Bishop and Pipin, using explicitly correlated wave functions for H2 and an H2 bond length r of 1.449 a.u.34 

The coefficient D01(r, R) typically changes sign with increasing H2–H separation near R = 8.0 a.u., where there are still substantial exchange/overlap contributions to D01(r, R). Therefore, in order to make comparisons with the known long-range analytical form of D01(r, R) as a function of R for r = r0 = 1.449 a.u., we have determined D01(r0, R) for R = 10.0–15.0 a.u. in intervals of 1.0 a.u. Placing H2–H in the yz plane, we calculated the dipole in the y and z directions at angles θ = 0°, 15°, 30°, 45°, 60°, 75°, and 90° using a base field strength f = 0.01 a.u. and the A5Z basis and then determined the coefficients D01, D21, D23, D43, and D45 from the results. We have found that corrections for the basis set superposition error (BSSE)258 are small with basis sets of the size used here, in previous work on interaction effects on the polarizability of H2–H2.172 At very long range, however, the collision-induced dipole is quite small, and the BSSE corrections become appreciable relative to the values of the dipole. Values of D01(r0, R) obtained with and without BSSE corrections are given in Table S25. A log-log plot of D01 vs R in the range from 10.0 to 15.0 a.u. has a slope of −6.06, based on values without BSSE corrections. When the BSSE corrections are added, the slope is −7.31, vs the expected value −7 at very long range. The dispersion contribution to D01 for H2–H at r = 1.449 a.u. is 186.39 R−7 from the work of Bishop and Pipin.34 The back-induction contribution is appreciably smaller at 5.17 R−7, based on αzz = 6.7179 a.u., αxx = 4.7319 a.u., Θ = 0.4823 a.u. (from Ref. 257), and αH = 4.5 a.u. Figure 9 shows the ab initio results for D01 with BSSE corrections vs R for r = 1.449 a.u. compared with the analytical long-range form. The level of agreement is striking, given the challenges in obtaining accurate values for this numerically sensitive property.

FIG. 9.

Dipole coefficient D01(r0, R) as a function of R, for bond length r0 = 1.449 a.u. Points plotted in blue are the ab initio results with BSSE corrections. The red curve shows the accurate R−7 term in D01 due to van der Waals dispersion and back-induction.

FIG. 9.

Dipole coefficient D01(r0, R) as a function of R, for bond length r0 = 1.449 a.u. Points plotted in blue are the ab initio results with BSSE corrections. The red curve shows the accurate R−7 term in D01 due to van der Waals dispersion and back-induction.

Close modal

The results from both full sets of calculations for the Cartesian components of the dipole moment with an aug-cc-pV5Z basis set are given in Tables S1–S9 of the supplementary material, for H2 bond lengths of 0.942 through 2.801 a.u., for separations R between the centers of mass of H2 and H from 3.0 a.u. to 10.0 a.u., and for the angle θ between the H2 bond vector r and the vector R from the center of mass of H2 to the nucleus of the H atom ranging from 0° to 90° in intervals of 5°. The H2–H system lies in the yz plane and R points along z, with y to the right. In general, the results obtained with the default convergence criteria agree very well with the results obtained with tighter convergence criteria, marked with a superscript . Differences are observed in some cases where the dipole is small, either where R is large or where the dipole function crosses zero as R increases. In those cases, the results obtained with the tighter convergence criteria are to be preferred. Cartesian components of the dipole are compared in detail with the results from Ref. 20 in Tables S10–S14. Table S15 compares components of the dipole obtained with the aug-cc-pV5Z, aug-cc-pV6Z, and d-aug-cc-pV5Z basis sets, for r = 1.449 a.u.

In Tables S16–S24, the results from multiple sets of calculations for the spherical-tensor dipole coefficients D01, D21, D23, D43, D45, D65, D67, D87, and D89 are listed. Again, the results obtained with the tighter convergence criteria (marked by ) should be considered more accurate when the results differ. The results obtained by truncating the series in Eq. (1) at the D89 term (in the calculations marked by ) generally agree quite well with the results obtained by continuing the series to DλL with λ = 24 and L = 25, in the original calculations where the default convergence criteria were used. Differences are apparent between the DλL coefficients derived from the values of the Cartesian components of the dipole in Ref. 20 vs our values. By contrast, the results for D01, D21, D23, D43, and D45 obtained from our Cartesian dipole components at angles θ = 0°, 30°, 60°, and 90° agree rather well with the results from the full set of angles in this work. The results from the four-angle fits are listed in Tables S16–S24 in the rows labeled (45).

Recent work by Miliordos and Hunt257 has given the polarizability tensor components, the quadrupole moment, and the hexadecapole moment of H2 at the specific bond lengths used in this work, except for r = 1.618 a.u. The values of αzz and αxx in Ref. 257 agree well earlier calculations of the polarizability of H2259–262 and show quite close agreement with interpolated values based on the work of Rychlewski261 and of Raj, Hamaguchi, and Witek.262 The quadrupoles agree well with interpolated values based on earlier accurate work,263–265 and similarly the hexadecapoles agree well with interpolated values based on accurate calculations completed earlier.263–265 With the values from Ref. 257, we have confirmed the convergence of D23 and D45 from our calculations to the known long-range forms, the first time this has been observed for the hexadecapolar induction term D45. Convergence of D23 and D45 to the long-range forms has been found for each of the bond lengths in this work, except for D45 at r = 0.942 or 1.111 a.u. where the hexadecapole is quite small.

The coefficient D01 gives the contribution to the dipole that is isotropic in the orientation of H2. At short range where D01 reflects overlap and exchange effects as well as dispersion, D01 is negative. At long range, both the long-range dispersion contribution to D01 and the much smaller back-induction contribution are positive. After correcting for basis set superposition error, we have obtained good agreement with the leading term in the long-range series for D01, as shown in Fig. 9.

A comparison of the results for H2–H with those for H2–He shows that D23 and D45 are positive in both cases, as expected. For bond lengths r = 1.111, 1.449, and 1.787 a.u., we have confirmed that the ratio of D23 for H2–H to D23 for H2–He converges at large R values to the ratio of the polarizabilities of H and He, 4.5/1.383, as expected. The convergence is illustrated in Fig. S4 in the supplementary material. Figure S5 in the supplementary material shows that the ratio of D45 for H2–H to D45 for H2–He converges to the ratio of H and He polarizabilities for r = 1.449, 1.787, and 2.125 a.u. For R < 7.0 a.u., both ratios drop below the long-range limits and both become smaller as R decreases, showing that overlap effects on D23 and D45 are greater for H2–H than for H2–He.

The most striking difference between the dipoles of H2–H and H2–He is found for the coefficient D01. As noted above, D01 is negative at short range for H2–H and positive at long range, indicating that the polarity averaged over H2 orientations corresponds to H2δ+Hδ− at short range—where overlap and exchange effects predominate—and to H2δ−Hδ+ at long range—where the van der Waals dispersion term is most significant. By contrast, for H2–He, D01 is positive at short range and negative at long range, so after averaging over the orientations of H2, the polarity is H2δ−Heδ+ at short range and H2δ+Heδ− at long range. The signs of D01 at long range obtained from the ab initio results in this work and earlier work on the H2–He dipole agree with the signs of the dispersion dipole calculated directly by Bishop and Pipin34 for H2–H and H2–He. In both cases, D23 > D01 at long range, but |D01| > D23 at short range. The crossover of |D01| and D23 occurs for R values somewhat smaller than Re, the location of the potential minima. The coefficients D21 and D43, which carry information about the anisotropic overlap effects on the dipole, have the same signs for H2–H and H2–He.

Our earlier results for the interaction-induced dipoles of H2–H2 and H2–He169 at the CCSD(T) level in an aug-cc-pV5Z basis have been used to calculate the binary collision-induced absorption spectra for H2 gas and to determine infrared and far infrared absorption during H2–He collisions in an H2/He mixture, over a range of temperatures.180,181 Excellent agreement with experimental measurements from 77 K to 300 K has been found in both cases.169,180,181 The calculations in this work on H2–H have been carried out with basis sets of similar size, at the UCCSD(T) level for the wave function. While the open-shell character of H2–H causes differences from the earlier work, we expect the values of the dipole moment presented here to be comparable in accuracy to our results for the H2–H2 and H2–He dipoles.169,180,181

Tables S1–S9 give the interaction-induced dipole moments μy and μz from the first and second set of calculations for H2 bond lengths r of 0.942, 1.111, 1.280, 1.449, 1.618 (second set only), 1.787, 2.125, 2.463, and 2.801 a.u., for angles θ from 0° to 90°, and for separations R between the centers of mass of H2 and H from 3.0 a.u. to 10.0 a.u. Default convergence criteria were used in the first set of calculations, and tighter convergence criteria were used in the second set (marked by ). In Tables S10–S14, our Cartesian dipole components are compared with the results obtained in Ref. 20, and in Table S15, Cartesian dipole components obtained with aug-cc-pV5Z, aug-cc-pV6Z, and d-aug-cc-pV5Z basis sets are compared for r = 1.449 a.u., the averaged internuclear distance in the ground rovibrational state of H2. Tables S16–S24 list the spherical tensor expansion coefficients D01, D21, D23, D43, D45, D65, D67, D87, and D89 from our two full calculations, along with D01, D21, D23, D43, and D45 from the work of GFM20 and from our calculations at θ = 0°, 30°, 60°, and 90°. Table S25 lists the values of D01(r, R) for r = 1.449 a.u. and R from 10.0 to 15.0 a.u. as obtained from calculations with and without corrections for basis set superposition error. The dependence of the Cartesian components of the dipole on r, R, and θ is discussed in detail in a section of the supplementary material. In Fig. S1, the dipole component μz is plotted vs θ and R at r = 1.449 a.u., for ΔE/kB less than or equal to 30 K, 300 K, 750 K, 1050 K, and 2600 K, where ΔE is the difference between the energy of a specific configuration and the minimum on the potential energy surface. In Figs. S2 and S3, we show that the quadrupoles and hexadecapoles derived from D23(r, R) and D45(r, R), respectively, converge to the ab initio values with increasing R, for the various bond lengths in this work. A plot of the ratios of D23 for H2–H to D23 for H2–He vs R for r = 1.111, 1.449, and 1.787 a.u. is included, along with a plot of the ratios of D45 for H2–H to D45 for H2–He, for r = 1.449, 1.787, and 2.125 a.u.

This research was supported in part by NSF Grant No. 1300063 from the program in Chemical Theory, Models, and Computational Methods. E.M. is indebted to Auburn University for financial support of this research. Part of the calculations reported here were completed with resources provided by the Auburn University Hopper Cluster.

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Supplementary Material