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Quantum mechanical computations of collision-induced absorption in the second overtone band of hydrogen
Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
Quantum mechanical computations of collision-induced absorption in the second overtone band of hydrogen Yi Fu , Chunguang Zheng , Aleksandra Borysow * Physics Department, Michigan Technological University, Houghton, MI 49931, USA Niels Bohr Institute for Astronomy, Physics, and Geophysics, Copenhagen University Observatory, Juliane Maries vej 30, DK-2100 Copenhagen, Denmark Received 14 August 1998
Abstract The second overtone band of hydrogen is important for studies of both planetary and stellar atmospheres. Until recently, only one experimental measurement existed, taken at 85 K (McKellar, Welsh. Proc Roy Soc London Ser A 1971;322:421). In this paper we present the "rst quantum mechanical computations of the collision-induced rotovibrational absorption spectra of H pairs in the second (3}0) overtone band of hydrogen. We compare our computations with the data by McKellar and Welsh. The second overtone band is very weak and thus it is extremely di$cult to measure it in the laboratory, as well as to compute it based on the "rst principles. As it appears, the collision-induced dipoles of H pairs, which give rise to this CIA band spectra are so weak, that the numerical results, at some particular mutual orientations, are almost at the level of numerical uncertainty. Our computations are based on an extension of a database of H }H collision induced dipoles which already exists (Meyer et al. Phys Rev A 1989;39:2434}48) but which is inadequate for computing CIA bands of hydrogen at overtones higher than the "rst overtone. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Collision-induced spectroscopy; Infrared absorption; Quantum mechanical computations of lineshapes
1. Introduction Due to its symmetry a single hydrogen molecule does not possess a dipole moment. The intermolecular interactions between H molecules can, nevertheless, give rise to induced dipole * Corresponding author. Tel.: #45-35-325981; fax: #45-35-325989. E-mail address: [email protected] (A. Borysow). On professional leave of absence from Michigan Technological University. 0022-4073/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 2 4 0 - X
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moments. Here, we consider an isolated pair of H molecules, and therefore our results will be applicable only to binary collisions. Three mechanisms contribute to the induced dipole in such a molecular pair: the polarization of one H molecule in the multipolar "eld of another H molecule; an electron exchange in the H }H supermolecule at the near range; and the dispersion interaction. Due to these induced dipole moment contributions hydrogen gas absorbs in the far- and near-infrared regions [1], this process is called the collision-induced absorption (CIA). The CIA spectra are weak if compared with the spectra due to allowed dipole transitions. As the induced dipoles are extremely weak, they are detectable in the laboratory only at su$ciently high densities and/or long enough path lengths. The second overtone is very weak, causing considerable di$culties, especially if one wants to measure purely binary collisions. Only such spectra (i.e. in the low-density limit) can be compared at present with exact quantum mechanical theory. CIA spectra of H pairs are of considerable interest for the studies of stellar [2}5], as well as planetary atmospheres [6}8]. It may be of interest, that the "rst observation of the di!use feature around 0.827 lm by Kuiper [9] in spectra of Uranus and Neptune has been later on identi"ed by Herzberg [10] as the S (0) line of the CIA spectrum of H }H . The other lines accompanying the second overtone, clearly visible in the laboratory experimental spectra, were obscured by the allowed spectra of methane. This identi"cation has been used as a positive determination of the presence of hydrogen in the atmospheres of those outer planets. In order to compute the quantum mechanical collision-induced spectrum of H in the second overtone band, we needed to extend available database of interaction-induced, ab initio dipole moments [11]. In this work we compare our computations with the only available low-temperature experimental data of that band performed at the low-density limit [12]. We mention, that a paper appeared recently [13], which reports also low-temperature CIA measurements performed in the second overtone. In that paper, however, the spectra have been measured at densities between 500 and 1000 amagat, and no attempt has been made to extract binary absorption coe$cients of interest to this work. Instead, the work focuses on the "rst experimental evidence of the triple dipole transitions, which is a purely three-body e!ect. During the preparation of this work, new experimental data of the binary collision-induced absorption coe$cient in the second overtone band became available at temperatures 77.5 and 298 K. In a separate paper [14] we compare these new results, using the same procedures as outlined in the present paper.
2. Ab initio induced dipole moments The induced dipole moments are an essential input for accurate computations of the H }H CIA spectra. The ab initio computations of H }H collision-induced dipole moments pose problems similar to those known from the calculations of van der Waals potentials. For the most relevant molecular separations, i.e., those around the collisional diameter, the collision-induced dipole moments are rather small, but perturbation theory fails to adequately handle the exchange contribution which becomes large at such short distances. It therefore appears best to treat the H }H complex as a supermolecule and perform the computations in a self-consistent "eld (SCF) approximation and con"guration interaction (CI) calculations, provided that the basis set
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superposition error is e!ectively controlled and the CI excitation level is adequate for the long-range e!ects. Meyer et al. [11] developed H }H dipole moments from accurate ab initio SCF and CI calculations. In that work, the H }H collision-induced dipole moments k (R, r , r ) (i"x, y, z) G were computed for nine intermolecular distances R (between 3.5 and 9.0 a.u.), three internuclear H}H distances (denoted by: r "1.111 a.u., r "1.449 a.u., and r "1.787 a.u.), and 13 \ > nonequivalent relative orientations of two H molecules. The internuclear distance r corresponds to the ground state average of the H bond distance. With three internuclear distances r , r , and \ r , computations were carried out for four di!erent r r combinations, namely: r r , r r , r r , > \ > and r r . Next, functions A (R, r , r ) (see Section 3) had been obtained from the ab initio \ > H H * k (R, r , r ) data for fourteen leading j j "¸ terms. Using the symmetry relations for G A (R, r , r ), these functions were "tted by a quadratic form of r and r . By using rotovibra H H * tional wave functions "vj2 of H molecule, functions b (R), (i.e. the transition matrix elements H H * of A (R, r , r )), have been obtained for vibrational transitions with initial vibrational states H H * v "v "0, and "nal vibrational states: v "0, v "0 (rototranslational band) [15]; v "0, v "1 (fundamental band) (11); v "1, v "1 and v "0, v "2 ("rst overtone band) [16]. The rotational state dependences on variables j j j j were also given for the b (R) H H * functions, applicable for j , j 43 (i"1,2). G G The collision-induced dipole moments of H }H given in Ref. [11] are suitable for CIA computations of rototranslational, fundamental, and "rst overtone bands at low temperatures (below 300 K). Indeed, theoretical computations based on those dipoles agree very closely with laboratory CIA measurements, see [11,15,16]. For the second overtone larger internuclear distances (r'1.787 a.u.) are important. Therefore, the induced dipole data of H }H given in Refs. [11,15,16] are no longer adequate for our computations. To meet our goal of computing RV CIA of H }H at (3}0) band, we based our e!orts on the earlier work of Meyer et al. [11], and included one larger internuclear distance at r"2.150 a.u. First, we need to estimate which range of internuclear distances r is important for the second overtone. Since the last peak of the vibrational wave function "v"32 of the H molecule appears at about 2.1 a.u., internuclear distances larger than r "1.787 a.u. (the largest internuclear distance > in [11]) are necessary. Accordingly, we included one additional internuclear distance, r "2.150 a.u. in the ab initio computations of the induced dipole moments. Computations are >> performed for ten r r combinations, namely for r r , r r , r r , r r , r r , r r , \ > \ > >> > >> r r , r r , r r , and r r . For completeness, we mention that we also included two > > >> >> \ >> \ \ smaller intermolecular distances R (R"2.5 and 3.0 a.u.), not included in [11]. They are included in our new extended database, but their inclusion is not of relevance to this paper. These points are of great signi"cance if one wants to use our extended database for computing CIA of RT and RV bands at temperatures as high as 7000 K. In this work, we take the same thirteen nonequivalent mutual orientations of H }H as used in Ref. [11]. We express the orientation of H }H by three angles, namely h , h , and *u. h (h ) is the angle subtended by vectors r (r ) and R. *u is the dihedral angle between the planes r R and r R. In our computations the values for these angles are 0, 45, 90, and 1353. The Gaussian 92 program [17] is used for our ab initio computations, and the CCSD(T) method (a coupled cluster calculation with single and double excitations, with non-iterative inclusion of
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triple excitations) is chosen. The basis set of Gaussian-type functions used here is taken from [18], which is identical to that used in [11]. For a given con"guration, the energy of the H }H system is computed with a small electri cal "eld d applied in the relevant direction, i (i"x, y, z). The energy increase for the H }H system due to the presence of the electrical "eld, is *E "!k d. The dipole moment k is therefore given > G G by *E k "! > . G d
(1)
To reduce the computational noise, the electrical "eld !d is then applied, and the energy increase is now *E "k d. By taking the average of k from these two calculations, we get \ G G *E !*E \. k "! > G 2d
(2)
The relative deviations of k between our results and those of Ref. [11] are found to be within a few G percent for orientations with large k . For several orientations with the smallest k , while the G G absolute deviations remain the same in magnitude, the relative deviations become larger. This fact does not a!ect the accuracy of our "nal results signi"cantly, because smaller components contribute less to CIA spectra. We have taken all existing data from Ref. [11], and computed only the additional con"gurations discussed above. To avoid lengthy repetitions of data already presented [11], we collect all ab initio Cartesian dipole moment components in Table 1 on our WWW? page, where we list components k and k , k "0 for all con"gurations. V X W 3. Spherical expansion of the dipole moment The spherical components k (l"0,$1) of the induced dipole moment k can be expressed by J their Cartesian components as k "k , and k "G(k $ik )/(2. We expand k as [19]: J X ! V W (4p) (R, r , r )>J () , ) , )), A k (R, r , r , ) , ) , ))" H H * J (3 H H * H H *
(3)
where r and r are the internuclear (H}H) distances of the two H molecules, R is the inter molecular distance of H }H , and ) , ) and ) are their orientations, respectively. A (R, r , r ) are real expansion coe$cients. Functions >J are expressed in terms of H H * H H * spherical harmonics as given, for example, in Ref. [11]. The parameters j , j , ", and ¸ are non-negative integers. Due to the inversion symmetry of H molecules, the parity consider ations, and the selection rules imposed by Clebsch}Gordan coe$cients, the following conditions WWW?: http://www.astro.ku.dk/&aborysow/data/index.html
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apply to j , j , ", and ¸: j and j are even numbers, and ¸ is odd; "j !j "4"4"j #j " and "¸!1"4"4"¸#1". Based on these data, we obtain 14 leading expansion coe$cients, namely A (R, r , r ) of Eq. (3) H H * with j j "¸"0001, 2023, 0223, 2021, 0221, 2233, 2211, 4045, 0445, 2245, 2243, 2223, 2221, 2201. Higher-order terms can be safely neglected. The coe$cients A (R, r , r ) are given in Table 2 H H * at the same WWW? site. Table 2 deposited at WWW? allows the comparison of our extended database with that of [11]. We remind the reader that even though we based our computations on previous work [11] and used many already existing Cartesian dipole components, for our ab initio computations of the induced dipole moments at the missing con"gurations, unlike in [11], we used a di!erent, commercial program, Gaussian 92. Our "nal results are also a!ected by an inclusion of many new H }H con"gurations. 4. Functions b 1 2 KL (R) kk Functions b
(R) are the radial transition matrix elements of A (R, r , r ), i.e. H H * H H * (4) b (R)"1v j v j "A (R, r , r )"v j v j 2. H H * H H * They are essential inputs in the computations of CIA spectra. Functions bQ (R) (the subscript (c) A stands for j j "¸, and the superscript (s) for v j v j v j v j , the initial and "nal vibrational and rotational states of both H molecules) follow the symmetry relation: (5) bTHT H T H T H (R)"(!1)>*bT HT H T H T H (R). H H * H H * Functions bQ (R) can be obtained by a two-dimensional integration of A (R, r , r ) over H H * H H * r and r . The wave functions "vj2 of the H molecule are obtained by solving the SchroK dinger equation using the Numerov procedure, with the interaction potential energy for the H molecule in the ground electronic state [20}23]. With a small integration step size of dr+10\ a.u., the numerical values of inner products of the normalized wave functions, 1vj"v j 2, are not exactly equal to zero for vOv , but instead they are typically of the magnitude of 10\. In order to be able to integrate the expansion coe$cients A (R, r , r ) over r and r , Eq. (4), H H * we need to approximate them as functions of r and r . The quadratic form A (R, r , r )"a #a r #a r #a r #a r #a r r (6) H H * is used for this purpose for two reasons: "rst, it can closely represent the smooth nature of the r , r dependence of A (R, r , r ); second, higher-order polynomials with more parameters are H H * di$cult to control and sometimes produce oscillations.
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Fig. 1. The r r integration plane.
Due to the symmetry relation: A (R, r , r )"(!1)>*A (R, r , r ), (7) H H * H H * we have values of A (R, r , r ) at all sixteen (r r ) points of the four by four grid over r and r , H H * as shown in Fig. 1. If we "t A (R, r , r ) at all sixteen points using Eq. (6) the "tting error is not H H * acceptable. We thus regroup the sixteen points into four groups, namely I, II, III, and IV as shown in Fig. 1. The r r plane is divided into four overlapping parts accordingly: I. r 4r 4r and \ > r 4r 4r ; II. r 4r 4r and r 4r 4r ; III. r 4r 4r and r 4r 4r ; IV. \ > \ > >> >> >> r 4r 4r and r 4r 4r , where variable r runs horizontally, and r runs vertically in >> \ > Fig. 1. For each part, the corresponding nine data points of group I, II, III, or IV are used for the "tting, and the "tted surface is used for the integration, Eq. (4), in this part. The total of the four parts gives the results for an analytic representation of A (R, r , r ) used to obtain the H H * functions bQ (R) of various bands. H H * The functions bQ (R) depend on all v , j , v , j , v , j , v , j parameters. To facilitate the CIA H H * computations and to reduce the amount of data, for a given vibrational transition v v v v , the dependence of bQ (R) on the rotational states j j j j is expressed by an expansion in terms of H H * j ( j #1) as follows: G G bQ+bQ #b j ( j #1)#b [ j ( j #1)] # b j ( j #1)#b [ j ( j #1)]#b j ( j #1) # b [ j ( j #1)]#b j ( j #1)#b [ j ( j #1)]#2 . (8) In the above equation, the intermolecular distance R and the subscripts j j "¸ are dropped for convenience for all b (R) and b (R) functions. The superscript (s) stands for L_H H * H H * v j v j v j v j as before, while (s ) stands for the same v v v v values, but with j "j "j "j "0.
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For a given rotovibrational transition, bQ (R) and b (R) (n"1,2,8) of bQ(R) can be obtained L by means of the least mean squares "tting of Eq. (8) for j , j , j and j values. The multipole-induced dipole components (j j "¸"2023, 0223, 2233, 4045, 0445) have well known long-range asymptotic forms, given in the appendix of Ref. [15]. The expressions below describe the classical, purely multipole-dipole induction, which, at su$ciently long intermolecular distances, are strictly satis"ed: bQ (R)P!(31v j "a"v j 21v j "q "v j 2/R, bQ (R)P(31v j "a"v j 21v j "q "v j 2/R, bQ (R)P(2/15[1v j "c"v j 21v j "q "v j 2 #1v j "c"v j 21v j "q "v j 2]/R, bQ (R)P!(51v j "a"v j 21v j "q "v j 2/R, bQ (R)P(51v j "a"v j 21v j "q "v j 2/R, (9) where 1vj"a"v j 2, 1vj"c"v j 2, and 1vj"q "v j 2 are the radial matrix elements of the trace and anisotH ropy of the polarizability tensor and of the multipole moments of order j. The asymptotic values are often used as a test of the accuracy of the ab initio computations. In Table 1, comparisons between these asymptotic values, results of Refs. [11,16], and results of this work are given for four Table 1 Functions bQ (R"9.0 a.u.) (in 10\ a.u.) for multipole-induced dipole comH H * ponents from: (a) asymptotic values; (b) Meyer et al. 1989 and 1993; (c) this work 2023
0223
2233
4045
0445
v "v "0, v "0, v "0 (a) 691 !691 (b) 692 !692 (c) 696 !696
!109 !104 !106
8 9 9
!8 !9 !9
v "v "0, v "0, v "1 (a) 94 !126 (b) 93 !131 (c) 93 !131
!26 !25 !25
1 1 1
!3 !5 !4
v "v "0, v "1, v "1 (a) 17 !17 (b) 17 !17 (c) 17 !17
!6 !5 !5
v "v "0, v "0, v "2 (a) !9 (b) !9 (c) !9
16 14 13
2 2 2
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known vibrational transitions. In that table, the dipole components of Eqs. (9) are computed at R"9.0 a.u. using matrix elements 1vj"a"v j 2, 1vj"c"v j 2, and 1vj"q "v j 2 from Refs. [24,25]. The H terms with j j "¸"4045 and 0445 of the "rst overtone band are very small and therefore are a!ected by larger relative numerical noise, and are not shown in Table 1. Generally, good agreement between the ab initio and the asymptotic values justi"es the satisfactory accuracy of our results. In Table 4 (our WWW? site), the results of bQ (R) and b (R) (n"1 to 8) are given for L_H H * H H * the "rst "ve terms (which are the most important induced dipole transition elements of the H pairs) with j j "¸"0001, 2023, 0223, 2021 and 0221, and for the four vibrational transitions (v "v "0): E v "0, v "0 (rototranslational band); E v "0, v "1 (fundamental band); E v "1, v "1 ("rst overtone band, double vibrational transitions); E v "0, v "2 ("rst overtone band, single vibrational transition). The parameters of "tting b (R) for v , v '0 are also available in our WWW? site. All H H * parameters are the result of "tting b (R) using j values up to 10, and may be useful for H H * computation of CIA spectra at high temperatures. Due to the exchange symmetry of vibrational transitions with v "0, v "0 and with v "1, v "1, bQ "0, bQ "!bQ , and bQ "!bQ , these terms are therefore not shown in Table 4. Functions bQ (R) listed in Table 4 are plotted in Figs. 2}5 on WWW@, for H H * the four vibrational transitions listed above. When compared with data from Refs. [11,16] we note a remarkable agreement between both sets of results. For v "0, v "2 (Fig. 5) the relative discrepancies become larger as we expected, because larger internuclear distances (where our new ab initio data take e!ect) become more important when higher vibrational states are involved. Another fact is that the relative discrepancies are larger at near intermolecular range (close to R values smaller than those in [11]) than at the far range. This fact indicates that larger internuclear distances are more important at near intermolecular range. It remains for future investigation to show whether the dramatic change of curve shapes of bQ (R) at near range H H * invalidates the commonly used analytic form of the b function. The form of the b funcH H * H H * tions we use is a slight generalization of the form used by van Kranendonk [26], and it describes the numerical results of previous works quite accurately [11,16,18]. The analytical form reads: B(R)"B /RL#B exp[B (R!R )#B (R!R )], (10) where B stands for bQ or b (n"1,2,2,8) given in Eq. (8), and (c) is again dropped for L convenience. B (i"1, 2, 3, 4) are the parameters which are obtained by "tting Eq. (10) to the data G of bQ (R) and b (R). The parameter R "6.0 a.u. is chosen to be close to the collisional diameter, L so that the parameter B gives the approximate size of the exchange and distortion dipole contribution at the collisional diameter [11]. In all cases, n"4 for j j "¸"2023 and 0223, n"7 for j j "¸"2021, 0221, and 0001, and n"6 for 4045 and 0445.
WWW@: http://www.astro.ku.dk/&aborysow/H2}dipole}CIA}LT/index.html
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As a test, we computed the "rst overtone band (v "v "0 and v "1, v "1 and v "0, v "2) H }H CIA spectra at temperatures below 300 K using our new results of bQ (R) and b (R). The computed spectra (not shown here) are almost identical with those H H * L_H H * obtained using the data from Ref. [16]. This is due to the fact that at temperatures below 300 K the most relevant intermolecular distances are those at R'4.0 a.u., where the discrepancies between our data and the data of Ref. [16] are very small for all major terms (see Figs. 4 and 5 on WWW@). All the data presented here are available at our World Wide Web site. This set of data makes possible the computations (or revisions) of H }H CIA spectra at all temperatures below 7000 K, and for many rotovibrational bands signi"cant for planetary and astrophysical applications for atmospheric radiative transfer models. 5. The line shape theory The absorption coe$cient arising from collision-induced dipole moments in H }H molecular pairs is written as follows [15]: 2p nu[1!exp(! u/k¹)]
(11)
where n is the number density of the gas and < is the volume. Constants and c denote Planck's constant and the speed of light, respectively. u denotes angular frequency (s\), and ¹ is temperature (K). The gas density is usually expressed in units of amagat as . "n/N , where N is the number of gas molecules per cubic centimeter at standard temperature and pressure. For hydrogen, N is almost equal to Loschmidt's number N "2.68676;10 cm\ amagat\, which * is the N for ideal gases. The spectral density function g(u;¹) is computed from the transition matrix elements of the induced electric dipole moment k (l"0,$1) as given in [11]. J When an isotropic intermolecular potential is assumed, the translational, and rotovibrational states of the molecular pair become separable. For H molecules, this assumption is usually nearly true, at least at low temperatures, and it is therefore also used throughout this work. In such a case, the spectral density function g(u;¹) will be equal to P C( j j j ; 000)P C( j j j ; 000) g(u;¹)" TH TH H H * T H T H T H T H ;G (u!u !u ; ¹). THTH THTH H H *
(12)
The total spectral density is thus a superposition of basic line pro"les, which we refer to as the pure translational components G (u;¹). G (u;¹) are shifted by molecular rotovibrational H H * H H * frequencies u and u , which may be positive, zero or negative. For details concerning THTH THTH the quantum mechanical computation of the spectral density functions G (u;¹) we refer the H H * reader to Ref. [11]. The computational techniques used in this work are identical to those described therein. For completeness, we mention, that each spectral density function G(u;¹) depends uniquely on two input functions: the induced dipole, b (R) of a particular RV band, and on the H H * intermolecular initial and "nal states.
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6. Computation of CIA spectra of H2 }H2 in the second overtone band at low temperature Using our newly developed H }H collision-induced dipole moments for the second overtone band (*v"3), we computed quantum mechanical H }H CIA spectra for this band for the "rst time, and compared our results with the existing low-density measurements. Only the ground vibrational state (v"0) of H needs to be considered as the initial state at low temperatures. With v "v "0, four di!erent vibrational transitions contribute to the second overtone band, centered around 0.83 lm: I. v "0, v "3; II. v "1, v "2; III. v "2, v "1; IV. v "3, v "0. Due to the exchange symmetry of the two hydrogen molecules, Eq. (5), we compute absorption spectra only for the "rst two transitions and multiply the results by two. For a single vibrational transition, v "0 and v "3, "ve terms with j j "¸"0001, 2023, 0223, 2021, and 0221 are included in our computations; and for double vibrational transitions, v "1 and v "2, four terms with j j "¸"2023, 0223, 2021, and 0221 are included. Other terms contribute less to the total spectral intensity and can be safely neglected. Table 2 lists the parameters bQ (R) and b (R) (n"1, 2,2, 8) of Eq. (8) for these terms. H H * L_H H * For the purpose of this work, we selected to "t only the results up to j , j , j , and j 43. For consistency, we would like to point out that the data deposited by us at our WWW? site are all obtained from "ts up to j"10, for all vibrational bands, and are user-ready for high-temperature applications. However, in order to preserve the best attainable accuracy, we restricted the range of the "t, just for the second overtone band at low temperatures, since higher values of j are irrelevant at temperatures below 300 K. In Figs. 2 and 3 we present our new functions bQ (R) for the second overtone band. As usual, H H * the functions bQ (R) are "tted by a second-order polynomial, Eq. (10), and coe$cients B are G H H * given in Table 3. Intermolecular distances R"2.5 and 3.0 a.u. are not included in our "t, because they are irrelevant at low temperatures. By neglecting these two values, we obtain much better "t to Eq. (10). We compute line shapes GQ (u;¹) for j j j j "0000 with induced dipole functions H H * bQ (R) and the same intermolecular potential, <(R) [11], as the initial and the "nal states, H H * using rigorously quantum mechanical treatment. All, free}free, bound}bound, bound}free and free}bound transitions are accounted for. It is, however, highly impractical to compute line shapes GQ (u;¹) for each possible rotational transition j j j j , which are, in practice, very similar in H H * shape to GQ (u;¹). Therefore, we apply a method similar to the one employed earlier [11,16]. H H * We start with computing GQ (u;¹) for each set of j j "¸, using bQ (R). We compute the H H * H H * low-resolution spectra, for which bound}bound features are not discernible. Then we rescale the intensity of each spectral line shape corresponding to bQ (R) function. Zeroth translational H H * spectral moments M for any j j j j are much easier to compute than spectral densities _H H * GQ (u;¹), based on the existing quantum mechanical sum rules [27] and using the j j j j H H * dependent bQ (R) functions. Contributions from both free and bound states were included in H H * the computations of the zeroth moments MQ and MQ . Line shapes GQ (u;¹) are _H H * H H * H H * then obtained by simple rescaling of GQ (u;¹) by a factor MQ /MQ . The H }H H H * _H H * _H H * CIA spectra a(u;¹) are calculated using Eqs. (11) and (12). Figs. 4 and 5 show the comparison between our computations and the measurements of the hydrogen second overtone band recorded within the binary collisional regime, by McKellar and Welsh [12] at 85 K at two di!erent ortho to para hydrogen ratios. Due to the very weak spectral
!233.3 !152.1 !100.6 !63.2 !39.0 !25.0 !16.7 !11.6 !5.9 !3.4 !2.0
16.1 64.1 82.6 57.7 36.8 23.9 16.1 11.3 6.0 3.4 2.2
!126.4 !8.1 16.2 14.8 9.6 5.8 3.6 2.3 1.0 0.4 0.2
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
R(a.u.) bQ b
!0.43060[!2] !0.15807[!2] !0.46071[!3] !0.18493[!3] !0.10931[!3] !0.77925[!4] !0.55677[!4] !0.39593[!4] !0.19605[!4] !0.11934[!4] !0.71759[!5]
!0.98228[!3] !0.33815[!3] 0.48581[!4] 0.51784[!4] 0.66699[!4] 0.64953[!4] 0.53936[!4] 0.42947[!4] 0.26776[!4] 0.17622[!4] 0.14783[!4]
0.16806[#1] 0.15908[#1] 0.10577[#1] 0.53230 0.22799 0.87473[!1] 0.28676[!1] 0.62414[!2] !0.43358[!2] !0.37480[!2] !0.29959[!2] !0.29517[!2] !0.13991[!2] !0.81449[!3] !0.38496[!3] !0.14429[!3] !0.35400[!4] 0.83359[!5] 0.21819[!4] 0.21473[!4] 0.13900[!4] 0.91763[!5]
v "v "0; v "1, v "2
0.50319[!2] 0.18267[!1] 0.44269[!1] 0.27547[!1] 0.16454[!1] 0.10311[!1] 0.65382[!2] 0.43977[!2] 0.20713[!2] 0.12777[!2] 0.10894[!2]
v "v "0; v "1, v "2
!0.57494 !0.14629[#1] !0.13933[#1] !0.89164 !0.54589 !0.34523 !0.23001 !0.15982 !0.84361[!1] !0.48407[!1] !0.31534[!1]
v "v "0; v "1, v "2
b
!0.15540[#1] !0.59468 !0.34243 !0.19403 !0.10320 !0.47833[!1] !0.16895[!1] !0.27349[!2] 0.39379[!2] 0.31401[!2] 0.18870[!2]
!0.86795 !0.14779[#1] !0.13577[#1] !0.95303 !0.60414 !0.38134 !0.24599 !0.16508 !0.81370[!1] !0.45845[!1] !0.26887[!1]
!0.18757[#1] !0.79487 !0.26485 !0.77905[!1] !0.13255[!1] 0.35918[!2] 0.66205[!2] 0.62666[!2] 0.49994[!2] 0.33526[!2] 0.27676[!2]
b
!0.62982[!2] !0.25613[!2] !0.20518[!2] !0.13825[!2] !0.79656[!3] !0.39863[!3] !0.16683[!3] !0.51672[!4] 0.16664[!4] 0.18608[!4] 0.12615[!4]
!0.39464[!2] 0.70682[!3] 0.31489[!2] 0.23114[!2] 0.14493[!2] 0.92760[!3] 0.62127[!3] 0.44005[!3] 0.23358[!3] 0.13545[!3] 0.86487[!4]
!0.12587[!1] !0.50704[!2] !0.15669[!2] !0.26388[!3] 0.75584[!4] 0.11712[!3] 0.99051[!4] 0.74982[!4] 0.46110[!4] 0.27342[!4] 0.16576[!4]
b
!0.15588[#1] !0.15357[#1] !0.10587[#1] !0.54267 !0.23744 !0.93755[!1] !0.32077[!1] !0.77080[!2] 0.45576[!2] 0.41464[!2] 0.33479[!2]
j j "¸"2021
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
j j "¸"0223
0.33370 0.13441[#1] 0.13156[#1] 0.85659 0.52721 0.33371 0.22254 0.15469 0.81805[!1] 0.46826[!1] 0.30157[!1]
j j "¸"2023
b
0.11244[!2] 0.51456[!3] 0.34292[!3] 0.18916[!3] 0.93598[!4] 0.41609[!4] 0.13887[!4] 0.17483[!5] !0.36077[!5] !0.17894[!5] !0.15011[!5]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.20401[!2] 0.67879[!3] 0.25492[!3] 0.78756[!4] 0.18602[!4] 0.24182[!5] !0.26102[!5] !0.29486[!5] !0.33757[!5] !0.23861[!5] !0.25217[!5]
b
Table 2 Functions bQ (R) and b (R) (in 10\ a.u.) for the second overtone band; numbers in the square brackets stand for the power of L_H H * H H * 10. The data present the results of "ts of bQ (R) for j values up to 3 H H *
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
!0.68106 0.79753 0.11998[#1] 0.84721 0.51625 0.31395 0.19648 0.13001 0.62707[!1] 0.35115[!1] 0.20951[!1]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
b
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
!0.32850[!2] !0.33021[!2] !0.30285[!2] !0.21043[!2] !0.13842[!2] !0.92193[!3] !0.62980[!3] !0.44457[!3] !0.23584[!3] !0.13625[!3] !0.84437[!4]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
b
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321 313
b
!0.36449[!2] !0.14559[!2] !0.57248[!3] !0.21634[!3] !0.82399[!4] !0.24823[!4] 0.54994[!6] 0.76385[!5] 0.49385[!5] 0.19750[!5] 0.14204[!6]
!0.10398[!2] 0.44584[!3] 0.53562[!3] 0.35331[!3] 0.19028[!3] 0.97573[!4] 0.46160[!4] 0.22625[!4] 0.64062[!5] 0.17894[!5] 0.63804[!6]
!0.24629[!2] !0.71185[!3] 0.23931[!3] 0.27787[!3] 0.18619[!3] 0.10811[!3] 0.58395[!4] 0.32096[!4] 0.12144[!4] 0.55785[!5] 0.49276[!5]
v "v "v "0, v "3
0.46904 0.45361 0.18176 0.65038[!1] 0.23626[!1] 0.70763[!2] 0.61022[!3] 0.19579[!3] 0.17102[!2] 0.16987[!2] 0.19575[!2]
v "v "v "0, v "3
!0.11140 !0.50188[!1] !0.18858[!1] !0.91329[!2] !0.27024[!2] !0.26164[!4] 0.48393[!3] 0.33140[!3] !0.11955[!3] !0.16156[!3] !0.19526[!3]
v "v "0; v "1, v "2
b
347.81 !0.37617 226.49 !0.11504 152.06 0.19934[!1] 82.39 0.35751[!1] 43.74 0.30018[!1] 23.69 0.21443[!1] 13.21 0.14802[!1] 7.69 0.10425[!1] 3.10 0.56291[!2] 1.54 0.32947[!2] 1.16 0.22741[!2]
!666.72 !193.60 !43.97 !6.43 8.77 12.75 10.99 7.68 2.82 0.83 0.09
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
!132.3 !100.8 !70.1 !39.0 !19.0 !8.3 !3.3 !1.1 0.2 0.3 0.2
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
R(a.u.) bQ
Table 2. (continued )
0.30973[#1] 0.11361[#1] 0.33815 0.13423 0.64880[!1] 0.31307[!1] 0.12768[!1] 0.31791[!2] !0.22971[!2] !0.24376[!2] !0.13627[!2]
!0.27057[#1] !0.23904[#1] !0.13694[#1] !0.71293 !0.35463 !0.17103 !0.80424[!1] !0.37411[!1] !0.76325[!2] !0.11598[!2] !0.36152[!3]
!0.55470 0.82786 0.10393[#1] 0.72139 0.43837 0.24626 0.13309 0.70627[!1] 0.20434[!1] 0.59190[!2] 0.30147[!2]
b
0.18474[!2] 0.51529[!3] 0.37844[!3] 0.25261[!3] 0.16328[!3] 0.89215[!4] 0.38218[!4] 0.11624[!4] !0.30896[!5] !0.39765[!5] 0.49021[!6]
!0.74116[!3] !0.14929[!2] !0.13418[!2] !0.85945[!3] !0.47974[!3] !0.25826[!3] !0.13889[!3] !0.67948[!4] !0.78451[!5] 0.49202[!5] 0.81315[!5]
!0.98116[!2] !0.59284[!2] !0.37548[!2] !0.19561[!2] !0.84057[!3] !0.29469[!3] !0.71325[!4] 0.11081[!4] 0.40800[!4] 0.34700[!4] 0.19412[!4]
b
!0.39790 !0.11870 0.26194[!1] 0.40729[!1] 0.33119[!1] 0.23239[!1] 0.15803[!1] 0.11004[!1] 0.58683[!2] 0.34156[!2] 0.23803[!2]
j j "¸"2023
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
j j "¸"0001
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
j j "¸"0221
b
!0.11237[!2] !0.64108[!3] !0.37854[!3] !0.17964[!3] !0.92483[!4] !0.51860[!4] !0.30868[!4] !0.19472[!4] !0.91287[!5] !0.51265[!5] !0.42133[!5]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
b
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
!0.12828[#1] !0.16122[#1] !0.13832[#1] !0.83812 !0.45401 !0.22833 !0.10986 !0.50680[!1] !0.87326[!2] 0.12178[!2] 0.87556[!3]
b
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.16780[!2] 0.21238[!2] 0.17394[!2] 0.10773[!2] 0.58623[!3] 0.29341[!3] 0.14070[!3] 0.65171[!4] 0.12324[!4] !0.18499[!6] !0.53413[!6]
b
314 Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
!0.12907[!1] !0.43125[!2] !0.20479[!2] !0.15994[!2] !0.13264[!2] !0.10591[!2] !0.82209[!3] !0.63117[!3] !0.37350[!3] !0.22518[!3] !0.14437[!3]
!0.93964[!3] !0.58441[!3] !0.48244[!3] !0.33435[!3] !0.16507[!3] !0.65150[!4] !0.18782[!4] !0.46025[!6] 0.71964[!5] 0.64866[!5] 0.41139[!5]
!0.54014[!2] !0.59403[!2] !0.42762[!2] !0.24168[!2] !0.13050[!2] !0.68716[!3] !0.34848[!3] !0.17034[!3] !0.37175[!4] !0.97664[!5] !0.39011[!5]
v "v "v "0, v "3
!0.41939 !0.19661 !0.12019 !0.69713[!1] !0.35657[!1] !0.15883[!1] !0.57506[!2] !0.12557[!2] 0.94744[!3] 0.91817[!3] 0.54639[!3]
v "v "v "0, v "3
!0.71340 !0.38863 !0.16654 !0.97414[!1] !0.60044[!1] !0.36738[!1] !0.22409[!1] !0.13699[!1] !0.56384[!2] !0.28536[!2] !0.14481[!2]
158.02 !0.87853 !114.54 !0.26079 !120.73 !0.72423[!1] !74.38 0.13315[!2] !45.20 0.22161[!1] !27.12 0.23072[!1] !15.96 0.18437[!1] !9.43 0.13355[!1] !3.70 0.67348[!2] !1.91 0.36829[!2] !1.00 0.19938[!2]
313.23 69.36 22.17 8.26 5.67 3.29 1.16 !0.15 !0.80 !0.51 !0.28
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
!15.01 102.17 58.31 14.88 !3.64 !10.00 !11.01 !10.13 !7.06 !4.37 !2.86
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
v "v "v "0, v "3
!0.11297[#2] !0.54361[#1] !0.21550[#1] !0.98767 !0.42766 !0.17975 !0.80167[!1] !0.41143[!1] !0.17057[!1] !0.96676[!2] !0.57195[!2]
0.38371[#1] 0.17567[#1] 0.92197 0.41790 0.17108 0.58233[!1] 0.94581[!2] !0.92744[!2] !0.14305[!1] !0.96775[!2] !0.59390[!2]
0.17833[#2] 0.88968[#1] 0.45992[#1] 0.25567[#1] 0.14984[#1] 0.92198 0.59376 0.40011 0.20393 0.11598 0.72423[!1]
!0.88370[!3] !0.16749[!1] !0.14104[!1] !0.82969[!2] !0.47534[!2] !0.26720[!2] !0.14606[!2] !0.78769[!3] !0.24498[!3] !0.10579[!3] !0.52865[!4]
0.39689[!2] 0.19941[!2] 0.10082[!2] 0.37272[!3] 0.12260[!3] 0.31438[!4] !0.83046[!6] !0.12852[!4] !0.15193[!4] !0.91850[!5] !0.60693[!5]
!0.41623[!1] !0.12224[!1] !0.65076[!2] !0.59892[!2] !0.52130[!2] !0.42557[!2] !0.33481[!2] !0.25983[!2] !0.15528[!2] !0.93504[!3] !0.59624[!3]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
j j "¸"0221
!0.43366 !0.20593 !0.12658 !0.73698[!1] !0.37463[!1] !0.16510[!1] !0.58751[!2] !0.12093[!2] 0.10373[!2] 0.99247[!3] 0.59161[!3]
j j "¸"2021
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
j j "¸"0223
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
!0.19467[!3] 0.39989[!4] !0.21289[!4] !0.38060[!4] !0.31147[!4] !0.19336[!4] !0.95628[!5] !0.33475[!5] 0.12597[!5] 0.13556[!5] 0.95175[!6]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.15704[#2] 0.75736[#1] 0.33824[#1] 0.16117[#1] 0.72080 0.30520 0.12500 0.49217[!1] 0.52780[!2] !0.13174[!2] !0.87266[!3]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
!0.14550[#2] !0.78635[#1] !0.44388[#1] !0.25749[#1] !0.15315[#1] !0.94861 !0.61499 !0.41719 !0.21435 !0.12166 !0.75244[!1]
!0.22676[!2] 0.12079[!1] 0.10907[!1] 0.65784[!2] 0.38489[!2] 0.21975[!2] 0.12151[!2] 0.66074[!3] 0.20757[!3] 0.89204[!4] 0.44419[!4]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.33007[!1] 0.92324[!2] 0.47720[!2] 0.43177[!2] 0.37662[!2] 0.30926[!2] 0.24443[!2] 0.19056[!2] 0.11458[!2] 0.69137[!3] 0.44329[!3]
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321 315
316
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
Fig. 2. Functions bQ (R) for v "0, v "0, H H * v "1, v "2. Labels denote the expansion parameters j j "¸; the minus sign indicates negative bQ values. H H * The markers represent the long-range, classical purely quadrupole-induced contribution, circles: j j "¸" !2023; and boxes: j j "¸"0223.
Fig. 3. Functions bQ (R) for v "0, v "0, H H * v "0, v "3. Labels denote the expansion parameters j j "¸; the minus sign indicates negative bQ values. H H * The markers describe the long-range, classical quadrupole-induced contribution, circles: j j "¸"2023; and boxes: j j "¸"!0223.
Table 3 Fitting parameters, of bQ K (R) (in a.u.) for the second overtone band HH * j j K¸
n
B
B
v "0, v "0, v "0, v "3 0001 7 !0.324727 2023 4 0.005854 0223 4 0.019459 2021 7 0.171489 0221 7 !3.175620
0.94377 0.31768 0.49026 !0.14823 1.78181
v "0, v "0, v "1, v "2 2023 4 0.009561 0223 4 !0.029781 2021 7 !0.410787 0221 7 0.519855
!0.185631 0.337341 0.365088 !0.293648
10\ 10\ 10\ 10\ 10\
10\ 10\ 10\ 10\
B
B
!0.91567 !1.54759 !1.46861 !3.15735 !1.95641
!0.31102 !0.04182 !0.05335 !0.34385 0.07598
!0.71801 !0.69671 !1.01774 !1.65565
0.06218 0.06055 0.08488 !0.03110
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
317
Fig. 4. The CIA spectra of normal H (C "0.25) in the second overtone band at 85 K. Solid line: theoretical data without j dependence; dashed line: theoretical data with j dependence; circles: experimental data by McKellar and Welsh (1971).
Fig. 5. CIA spectra of H (C "0.57) in the second overtone band at 85 K. Solid line: theoretical data without j dependence; dashed line: theoretical data with j dependence; circles: experimental data by McKellar and Welsh (1971).
intensity of this band only those measurements are available in the literature up to this day. Most recently, measurements at 77.5 and 298 K have been performed, see [14], and the experimental results have been compared with the theory outlined here. The analysis presented by McKellar and Welsh [12] of their measurements [12], is based on the purely multipole-induced dipole theory. The authors point out that the unexplained experimental
318
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
spectral intensity in the Q ( j"0,1) region may be arising from the unknown quantum overlap term, j j "¸"0001. They further speculate that Q lines (arising from the pure overlap, j j "¸"0001, contribution) exist for 0P1, 0P3, 0P5, etc., rotovibrational bands and are absent in the 0P2, 0P4, etc., rotovibrational bands. An alternative explanation given in [12] is that the j j "¸"0001 term is absent for only in 0P2 band for some particular reason. In our work we "nd out that j j "¸"0001 is strong in the second overtone band for single vibrational transition 0P3. Q lines exist for all RV bands for all single vibrational transitions, but the j j "¸"0001 spectral component may sometimes be weak, as for example for 0P2 (for details, see [16]), and also for the double transition (0P2 & 0P1) in the second overtone. In the experimental data we observe "ve distinctive peaks at frequencies below 13,000 cm\. In order to understand better the "ve peaks in Figs. 4 and 5, in Table 4 below we make assignments of the rotovibrational transitions contributing to each peak. As we expected, the relative discrepancies between theoretical and experimental spectra are larger for the second overtone than was observed for lower overtone bands. Our computational results are in much better agreement for double vibrational transitions, v v v v "0102 (the last two peaks), than for the single vibrational transition, v v v v "0003 (the "rst two peaks). In Figs. 4 and 5 we present theoretical results obtained by assuming, or neglecting the j-dependence. It turns out that whereas the j-dependence is especially weak for double vibrational transition, it is non-negligible for a single vibrational transition and should not be ignored. In order to understand better the reasons for the observed discrepancy between our theoretical results and the measured data, we attempted to estimate the uncertainty of our computations of a(u;¹) at various frequency ranges, by relating it to the inaccuracy introduced by the functions b (R). Accordingly, we tried to estimate the uncertainties of functions b (R) for this band. H H * H H * We compute asymptotic values of the two main terms bQ (R) and bQ (R) at R"9.0 a.u. from Eqs. (9), using matrix elements 1vj"a"v j 2 and 1vj"q "v j 2 from Ref. [24]. The comparison between Table 4 Assignment of peaks, counting from left to right Peaks
Roto-vibrational transitions
Vibrational bands
First
Q ( j"0,1,2)#Q ( j"0,1,2)
(3}0)
Second
Q ( j"0,1,2)#S ( j"0) S ( j"0)#Q ( j"0,1,2)
(3}0) (3}0)
Third
Q ( j"0,1,2)#S ( j"1) S ( j"1)#Q ( j"0,1,2) Q ( j"0,1,2)#Q ( j"0,1,2)
(3}0) (3}0) (1}0)#(2}0)
Fourth
S ( j"0)#Q ( j"0,1,2) Q ( j"0,1,2)#S ( j"0)
(1}0)#(2}0) (1}0)#(2}0)
Fifth
Q ( j"0,1,2)#S ( j"1) S ( j"1)#Q ( j"0,1,2)
(1}0)#(2}0) (1}0)#(2}0)
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
319
Table 5 Leading functions bQ K (R"9.0 a.u.) for the second overtone H H * band (in 10\ a.u.). Upper line: asymptotic values, lower line: our results v "0, v "0, v "0, v "3
v "0, v "0, v "1, v "2
b
b
b
b
1.27 1.2
!2.28 !2.9
!1.65 !2.0
2.18 2.2
the asymptotic values and our results are given in Table 5. The terms with j j "¸"4045, 0445 and 2233 of the second overtone band are negligible, and therefore are not included in Table 5. For bQ (R) (v "0, v "3) and bQ (R) (v "1, v "2), the relatively larger discrepancies between our results and the asymptotic values than those observed for lower overtones (compare with Table 1) indicate somewhat less accurate b (R) functions. This is, in our opinion, mainly H H * due to the very small b (R) values of this band, about one order of magnitude smaller than H H * those of the "rst overtone band and therefore resulting in larger relative errors. For example, at R"9.0 a.u., A (R, r , r ) is of the order 1000;10\ a.u., while its radial matrix element (R, r , r )"v j v j 2 becomes only about 1;10\ a.u. for vibrational bQ (R)"1v j v j "A transition v "0Pv "0, and v "0Pv "3. It is also interesting to compare the intensity of the leading (quadrupole-induced) dipoles of RT, fundamental, "rst overtones, with those of the second overtone, for the single vibrational bands. At the collisional diameter (approx. 6. a.u.) the ratios of the dipoles are: &400, &65, &6.5, respectively, the approximate values correspond to averages between 2023 and 0223 terms. Having in mind that the absorption intensity is proportional to the square of the dipole, we can see that the RT spectra are roughly 1}2;10 more intense than the spectra of the second overtone band (for details, see [14]). Even compared to the "rst overtone, it appears that the second overtone is about 30}36 times less intense. That, of course, a!ects the accuracy of the "nal results. For such small values of bQ the major contributions to the uncertainty are the "tting in H H * Eq. (6), and the two-dimensional integration procedures. Another reason for the smaller accuracy of the b (R) functions of the second overtone, than what was obtained for lower overtones, is H H * the inclusion of higher vibrational states (v"2 and 3) in the computations of this band. For higher vibrational states of H , the internuclear distances of H beyond r "2.15 a.u. become more >> important. Here the uncertainties in the "tting of A (R, r , r ) in Eq. (6), become especially H H * large. Very good agreement between our results and the asymptotic values for bQ (R) (v "0, v "3) and bQ (R) (v "1, v "2) must be considered somewhat coinciden tal because of the above-mentioned computational uncertainties. It may, however, well be that at the largest intermolecular distance considered here, R"9.0 a.u., the computed dipoles do not yet reach their asymptotic values (see Figs. 2 and 3). There is no unique value of the relative error for b (R), because it depends upon the absolute H H * value of b (R) at various intermolecular distances R, though the absolute error of H H * A (R, r , r ) and the orthogonality of the wave functions 1vj"v j 2 can be estimated. This H H *
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implies that the relative errors for di!erent b (R) are not even the same (but depend upon the H H * induced dipole intensity and intermolecular distance R). Therefore, it is very di$cult to estimate the theoretical error of G (u), as well as of the theoretical spectra a(u;¹). H H * Furthermore, it is known that the intermolecular potential
Acknowledgements Support by NASA, Planetary Atmospheres Division, and by NASA, Astrophysics Theory Program, are gratefully acknowledged by the authors. Two of the authors (A.B. and Y.F.) would like to thank The Niels Bohr Institute, University Observatory, for the generous hospitality they experienced while working on this paper. We acknowledge Gaussian Inc. for the license agreement of Gaussian 92. C.Z. would like to thank Dr. Mark Cybulski for his patience and help in using the Gaussian 92 program, as well as for all his valuable comments.
References [1] Welsh HL. Pressure induced absorption spectra of hydrogen. In: Buckingham AD, Ramsay DA, editors. MTP International Review of Science-Physical Chemistry, Series 1, vol. 3: Spectroscopy, London: Butterworths, 1972. p. 33}71. [2] Linsky JL. On the pressure-induced opacity of molecular hydrogen in late-type stars. Astrophys J 1969;156:989}1005. [3] Lenzuni P, Cherno! DF, Salpeter EE. Rosseland and Planck mean opacities of a zero-metallicity gas. Astrophys J 1991;76 (Suppl):759}801. [4] Borysow A. Pressure-induced molecular absorption in stellar atmospheres. In: J+rgensen UG. editor. Molecules in the Stellar Environment, Lecture Notes in Physics, 1st ed., Berlin: Springer, 1994. p. 209}22.
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[5] Borysow A, J+rgensen UG, Zheng C. Model atmospheres of cool, low metallicity stars: the importance of collision-induced absorption. Astron Astrophys 1997;324:185}95. [6] Conrath BJ, Hanel RA, Samuelson RE, Origin and evolution of planetary and satellite atmospheres. Tucson: University of Arizona Press, 1989; p. 513}38. [7] Hanel RA, Conrath BJ, Jennings DE, Samuelson RE. Exploration of the solar system by infrared remote sensing. Cambridge, NY: University Press, 1992. [8] Trafton LM. Induced spectra in planetary atmospheres. In: Tabisz G, Neuman MN, editors. Collision- and Interaction-Induced Spectroscopy. NATO Advanced Research Workshops. Dordrecht: Kluwer, 1995. 517}28. [9] Kuiper GP. Astrophys J 1949;109:540. [10] Herzberg G. Spectroscopic evidence of molecular hydrogen in the atmospheres of Uranus and Neptune. Astrophys J 1952;115:337}40. [11] Meyer W, Borysow A, Frommhold L. Absorption spectra of H }H pairs in the fundamental band. Phys Rev A 1989;40:6931}49. [12] McKellar ARW, Welsh HL. Collision-induced spectra of hydrogen in the "rst and second overtone regions with applications of planetary atmospheres. Proc Roy Soc London Ser A 1971;322:421. [13] Reddy SP, Xiang F, Varghese G. Observation of the new triple transitions Q (J ) # Q (J ) # Q (J ) in molecular hydrogen in its second overtone region. Phys Rev Lett 1995;74:367}70. [14] Brodbeck C, Bouanich J-P, Nguyen-Van-Thanh Fu Y, Borysow A. Collision-induced absorption by H pairs in the second overtone band at 298 and 77.5 K: comparison between experimental and theoretical results. J Chem Phys 1999;110:4750. [15] Meyer W, Frommhold L, Birnbaum G. Rototranslational absorption spectra of H }H pairs in the far infrared. Phys Rev A 1989;39:2434}48. [16] Meyer W, Borysow A, Frommhold L. Collision-induced "rst overtone band of gaseous hydrogen from "rst principles. Phys Rev A 1993;47:4065}77. [17] Frisch MJ, Trucks GW, Head-Gordon M, Gill PMW, Wong MW, Foresman JB, Johnson BG, Schlegel HB, Robb MA, Replogle ES, Gomperts R, Andres JL, Raghavachari K, Binkley JS, Gonzalez C, Martin RL, Fox DJ, Defrees DJ, Baker J, Stewart JJP, Pople JA. Gaussian 92, Revision g.2. 1992. [18] Meyer W, Frommhold L. Collision-induced rototranslational spectra of H }He from an accurate ab initio dipole moment surface. Phys Rev A 1986;34:2771. [19] Hunt JL, Poll JD. Can J Phys 1978;56:950. [20] Kolos W, Szalewicz K, Monkhorst H. New Born}Oppenheimer energy curve and vibrational energies for the electronic ground state of the hydrogen molecule. J Chem Phys 1986;84:3278. [21] Kolos W, Wolniewicz L. J Mol Spectra 1965;43:2429. [22] Kolos W, Wolniewicz L. J Chem Phys 1968;49:404. [23] Kolos W, Wolniewicz L. J Mol Spectra 1975;54:303}11. [24] Hunt JL, Poll JD, Wolniewicz L. Can J Phys 1984;62:1719. [25] Karl G, Poll JD, Wolniewicz L. Can J Phys 1975;53:1781. [26] van Kranendonk J. Physica 1958;24:347. [27] Moraldi M, Borysow A, Frommhold L. Quantum sum formulae for the collision induced spectroscopies: Molecular systems as H }H . Chem Phys 1984;86:339}47. [28] Moraldi M, Borysow A, Frommhold L. Rotovibrational collision-induced absorption by nonpolar gases and mixtures (H }He pairs): about the symmetry of line pro"les. Phys Rev A 1988;38:1839}47. [29] Moraldi M, Borysow J, Frommhold L. Spectral moments for the collision-induced rotovibrational absorption bands of nonpolar gases and mixtures (H }He). Phys Rev A 1987;36:4700}3.
Quantum mechanical computations of collision-induced absorption in the second overtone band of hydrogen Yi Fu , Chunguang Zheng , Aleksandra Borysow * Physics Department, Michigan Technological University, Houghton, MI 49931, USA Niels Bohr Institute for Astronomy, Physics, and Geophysics, Copenhagen University Observatory, Juliane Maries vej 30, DK-2100 Copenhagen, Denmark Received 14 August 1998
Abstract The second overtone band of hydrogen is important for studies of both planetary and stellar atmospheres. Until recently, only one experimental measurement existed, taken at 85 K (McKellar, Welsh. Proc Roy Soc London Ser A 1971;322:421). In this paper we present the "rst quantum mechanical computations of the collision-induced rotovibrational absorption spectra of H pairs in the second (3}0) overtone band of hydrogen. We compare our computations with the data by McKellar and Welsh. The second overtone band is very weak and thus it is extremely di$cult to measure it in the laboratory, as well as to compute it based on the "rst principles. As it appears, the collision-induced dipoles of H pairs, which give rise to this CIA band spectra are so weak, that the numerical results, at some particular mutual orientations, are almost at the level of numerical uncertainty. Our computations are based on an extension of a database of H }H collision induced dipoles which already exists (Meyer et al. Phys Rev A 1989;39:2434}48) but which is inadequate for computing CIA bands of hydrogen at overtones higher than the "rst overtone. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Collision-induced spectroscopy; Infrared absorption; Quantum mechanical computations of lineshapes
1. Introduction Due to its symmetry a single hydrogen molecule does not possess a dipole moment. The intermolecular interactions between H molecules can, nevertheless, give rise to induced dipole * Corresponding author. Tel.: #45-35-325981; fax: #45-35-325989. E-mail address: [email protected] (A. Borysow). On professional leave of absence from Michigan Technological University. 0022-4073/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 2 4 0 - X
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moments. Here, we consider an isolated pair of H molecules, and therefore our results will be applicable only to binary collisions. Three mechanisms contribute to the induced dipole in such a molecular pair: the polarization of one H molecule in the multipolar "eld of another H molecule; an electron exchange in the H }H supermolecule at the near range; and the dispersion interaction. Due to these induced dipole moment contributions hydrogen gas absorbs in the far- and near-infrared regions [1], this process is called the collision-induced absorption (CIA). The CIA spectra are weak if compared with the spectra due to allowed dipole transitions. As the induced dipoles are extremely weak, they are detectable in the laboratory only at su$ciently high densities and/or long enough path lengths. The second overtone is very weak, causing considerable di$culties, especially if one wants to measure purely binary collisions. Only such spectra (i.e. in the low-density limit) can be compared at present with exact quantum mechanical theory. CIA spectra of H pairs are of considerable interest for the studies of stellar [2}5], as well as planetary atmospheres [6}8]. It may be of interest, that the "rst observation of the di!use feature around 0.827 lm by Kuiper [9] in spectra of Uranus and Neptune has been later on identi"ed by Herzberg [10] as the S (0) line of the CIA spectrum of H }H . The other lines accompanying the second overtone, clearly visible in the laboratory experimental spectra, were obscured by the allowed spectra of methane. This identi"cation has been used as a positive determination of the presence of hydrogen in the atmospheres of those outer planets. In order to compute the quantum mechanical collision-induced spectrum of H in the second overtone band, we needed to extend available database of interaction-induced, ab initio dipole moments [11]. In this work we compare our computations with the only available low-temperature experimental data of that band performed at the low-density limit [12]. We mention, that a paper appeared recently [13], which reports also low-temperature CIA measurements performed in the second overtone. In that paper, however, the spectra have been measured at densities between 500 and 1000 amagat, and no attempt has been made to extract binary absorption coe$cients of interest to this work. Instead, the work focuses on the "rst experimental evidence of the triple dipole transitions, which is a purely three-body e!ect. During the preparation of this work, new experimental data of the binary collision-induced absorption coe$cient in the second overtone band became available at temperatures 77.5 and 298 K. In a separate paper [14] we compare these new results, using the same procedures as outlined in the present paper.
2. Ab initio induced dipole moments The induced dipole moments are an essential input for accurate computations of the H }H CIA spectra. The ab initio computations of H }H collision-induced dipole moments pose problems similar to those known from the calculations of van der Waals potentials. For the most relevant molecular separations, i.e., those around the collisional diameter, the collision-induced dipole moments are rather small, but perturbation theory fails to adequately handle the exchange contribution which becomes large at such short distances. It therefore appears best to treat the H }H complex as a supermolecule and perform the computations in a self-consistent "eld (SCF) approximation and con"guration interaction (CI) calculations, provided that the basis set
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superposition error is e!ectively controlled and the CI excitation level is adequate for the long-range e!ects. Meyer et al. [11] developed H }H dipole moments from accurate ab initio SCF and CI calculations. In that work, the H }H collision-induced dipole moments k (R, r , r ) (i"x, y, z) G were computed for nine intermolecular distances R (between 3.5 and 9.0 a.u.), three internuclear H}H distances (denoted by: r "1.111 a.u., r "1.449 a.u., and r "1.787 a.u.), and 13 \ > nonequivalent relative orientations of two H molecules. The internuclear distance r corresponds to the ground state average of the H bond distance. With three internuclear distances r , r , and \ r , computations were carried out for four di!erent r r combinations, namely: r r , r r , r r , > \ > and r r . Next, functions A (R, r , r ) (see Section 3) had been obtained from the ab initio \ > H H * k (R, r , r ) data for fourteen leading j j "¸ terms. Using the symmetry relations for G A (R, r , r ), these functions were "tted by a quadratic form of r and r . By using rotovibra H H * tional wave functions "vj2 of H molecule, functions b (R), (i.e. the transition matrix elements H H * of A (R, r , r )), have been obtained for vibrational transitions with initial vibrational states H H * v "v "0, and "nal vibrational states: v "0, v "0 (rototranslational band) [15]; v "0, v "1 (fundamental band) (11); v "1, v "1 and v "0, v "2 ("rst overtone band) [16]. The rotational state dependences on variables j j j j were also given for the b (R) H H * functions, applicable for j , j 43 (i"1,2). G G The collision-induced dipole moments of H }H given in Ref. [11] are suitable for CIA computations of rototranslational, fundamental, and "rst overtone bands at low temperatures (below 300 K). Indeed, theoretical computations based on those dipoles agree very closely with laboratory CIA measurements, see [11,15,16]. For the second overtone larger internuclear distances (r'1.787 a.u.) are important. Therefore, the induced dipole data of H }H given in Refs. [11,15,16] are no longer adequate for our computations. To meet our goal of computing RV CIA of H }H at (3}0) band, we based our e!orts on the earlier work of Meyer et al. [11], and included one larger internuclear distance at r"2.150 a.u. First, we need to estimate which range of internuclear distances r is important for the second overtone. Since the last peak of the vibrational wave function "v"32 of the H molecule appears at about 2.1 a.u., internuclear distances larger than r "1.787 a.u. (the largest internuclear distance > in [11]) are necessary. Accordingly, we included one additional internuclear distance, r "2.150 a.u. in the ab initio computations of the induced dipole moments. Computations are >> performed for ten r r combinations, namely for r r , r r , r r , r r , r r , r r , \ > \ > >> > >> r r , r r , r r , and r r . For completeness, we mention that we also included two > > >> >> \ >> \ \ smaller intermolecular distances R (R"2.5 and 3.0 a.u.), not included in [11]. They are included in our new extended database, but their inclusion is not of relevance to this paper. These points are of great signi"cance if one wants to use our extended database for computing CIA of RT and RV bands at temperatures as high as 7000 K. In this work, we take the same thirteen nonequivalent mutual orientations of H }H as used in Ref. [11]. We express the orientation of H }H by three angles, namely h , h , and *u. h (h ) is the angle subtended by vectors r (r ) and R. *u is the dihedral angle between the planes r R and r R. In our computations the values for these angles are 0, 45, 90, and 1353. The Gaussian 92 program [17] is used for our ab initio computations, and the CCSD(T) method (a coupled cluster calculation with single and double excitations, with non-iterative inclusion of
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triple excitations) is chosen. The basis set of Gaussian-type functions used here is taken from [18], which is identical to that used in [11]. For a given con"guration, the energy of the H }H system is computed with a small electri cal "eld d applied in the relevant direction, i (i"x, y, z). The energy increase for the H }H system due to the presence of the electrical "eld, is *E "!k d. The dipole moment k is therefore given > G G by *E k "! > . G d
(1)
To reduce the computational noise, the electrical "eld !d is then applied, and the energy increase is now *E "k d. By taking the average of k from these two calculations, we get \ G G *E !*E \. k "! > G 2d
(2)
The relative deviations of k between our results and those of Ref. [11] are found to be within a few G percent for orientations with large k . For several orientations with the smallest k , while the G G absolute deviations remain the same in magnitude, the relative deviations become larger. This fact does not a!ect the accuracy of our "nal results signi"cantly, because smaller components contribute less to CIA spectra. We have taken all existing data from Ref. [11], and computed only the additional con"gurations discussed above. To avoid lengthy repetitions of data already presented [11], we collect all ab initio Cartesian dipole moment components in Table 1 on our WWW? page, where we list components k and k , k "0 for all con"gurations. V X W 3. Spherical expansion of the dipole moment The spherical components k (l"0,$1) of the induced dipole moment k can be expressed by J their Cartesian components as k "k , and k "G(k $ik )/(2. We expand k as [19]: J X ! V W (4p) (R, r , r )>J () , ) , )), A k (R, r , r , ) , ) , ))" H H * J (3 H H * H H *
(3)
where r and r are the internuclear (H}H) distances of the two H molecules, R is the inter molecular distance of H }H , and ) , ) and ) are their orientations, respectively. A (R, r , r ) are real expansion coe$cients. Functions >J are expressed in terms of H H * H H * spherical harmonics as given, for example, in Ref. [11]. The parameters j , j , ", and ¸ are non-negative integers. Due to the inversion symmetry of H molecules, the parity consider ations, and the selection rules imposed by Clebsch}Gordan coe$cients, the following conditions WWW?: http://www.astro.ku.dk/&aborysow/data/index.html
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apply to j , j , ", and ¸: j and j are even numbers, and ¸ is odd; "j !j "4"4"j #j " and "¸!1"4"4"¸#1". Based on these data, we obtain 14 leading expansion coe$cients, namely A (R, r , r ) of Eq. (3) H H * with j j "¸"0001, 2023, 0223, 2021, 0221, 2233, 2211, 4045, 0445, 2245, 2243, 2223, 2221, 2201. Higher-order terms can be safely neglected. The coe$cients A (R, r , r ) are given in Table 2 H H * at the same WWW? site. Table 2 deposited at WWW? allows the comparison of our extended database with that of [11]. We remind the reader that even though we based our computations on previous work [11] and used many already existing Cartesian dipole components, for our ab initio computations of the induced dipole moments at the missing con"gurations, unlike in [11], we used a di!erent, commercial program, Gaussian 92. Our "nal results are also a!ected by an inclusion of many new H }H con"gurations. 4. Functions b 1 2 KL (R) kk Functions b
(R) are the radial transition matrix elements of A (R, r , r ), i.e. H H * H H * (4) b (R)"1v j v j "A (R, r , r )"v j v j 2. H H * H H * They are essential inputs in the computations of CIA spectra. Functions bQ (R) (the subscript (c) A stands for j j "¸, and the superscript (s) for v j v j v j v j , the initial and "nal vibrational and rotational states of both H molecules) follow the symmetry relation: (5) bTHT H T H T H (R)"(!1)>*bT HT H T H T H (R). H H * H H * Functions bQ (R) can be obtained by a two-dimensional integration of A (R, r , r ) over H H * H H * r and r . The wave functions "vj2 of the H molecule are obtained by solving the SchroK dinger equation using the Numerov procedure, with the interaction potential energy for the H molecule in the ground electronic state [20}23]. With a small integration step size of dr+10\ a.u., the numerical values of inner products of the normalized wave functions, 1vj"v j 2, are not exactly equal to zero for vOv , but instead they are typically of the magnitude of 10\. In order to be able to integrate the expansion coe$cients A (R, r , r ) over r and r , Eq. (4), H H * we need to approximate them as functions of r and r . The quadratic form A (R, r , r )"a #a r #a r #a r #a r #a r r (6) H H * is used for this purpose for two reasons: "rst, it can closely represent the smooth nature of the r , r dependence of A (R, r , r ); second, higher-order polynomials with more parameters are H H * di$cult to control and sometimes produce oscillations.
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Fig. 1. The r r integration plane.
Due to the symmetry relation: A (R, r , r )"(!1)>*A (R, r , r ), (7) H H * H H * we have values of A (R, r , r ) at all sixteen (r r ) points of the four by four grid over r and r , H H * as shown in Fig. 1. If we "t A (R, r , r ) at all sixteen points using Eq. (6) the "tting error is not H H * acceptable. We thus regroup the sixteen points into four groups, namely I, II, III, and IV as shown in Fig. 1. The r r plane is divided into four overlapping parts accordingly: I. r 4r 4r and \ > r 4r 4r ; II. r 4r 4r and r 4r 4r ; III. r 4r 4r and r 4r 4r ; IV. \ > \ > >> >> >> r 4r 4r and r 4r 4r , where variable r runs horizontally, and r runs vertically in >> \ > Fig. 1. For each part, the corresponding nine data points of group I, II, III, or IV are used for the "tting, and the "tted surface is used for the integration, Eq. (4), in this part. The total of the four parts gives the results for an analytic representation of A (R, r , r ) used to obtain the H H * functions bQ (R) of various bands. H H * The functions bQ (R) depend on all v , j , v , j , v , j , v , j parameters. To facilitate the CIA H H * computations and to reduce the amount of data, for a given vibrational transition v v v v , the dependence of bQ (R) on the rotational states j j j j is expressed by an expansion in terms of H H * j ( j #1) as follows: G G bQ+bQ #b j ( j #1)#b [ j ( j #1)] # b j ( j #1)#b [ j ( j #1)]#b j ( j #1) # b [ j ( j #1)]#b j ( j #1)#b [ j ( j #1)]#2 . (8) In the above equation, the intermolecular distance R and the subscripts j j "¸ are dropped for convenience for all b (R) and b (R) functions. The superscript (s) stands for L_H H * H H * v j v j v j v j as before, while (s ) stands for the same v v v v values, but with j "j "j "j "0.
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For a given rotovibrational transition, bQ (R) and b (R) (n"1,2,8) of bQ(R) can be obtained L by means of the least mean squares "tting of Eq. (8) for j , j , j and j values. The multipole-induced dipole components (j j "¸"2023, 0223, 2233, 4045, 0445) have well known long-range asymptotic forms, given in the appendix of Ref. [15]. The expressions below describe the classical, purely multipole-dipole induction, which, at su$ciently long intermolecular distances, are strictly satis"ed: bQ (R)P!(31v j "a"v j 21v j "q "v j 2/R, bQ (R)P(31v j "a"v j 21v j "q "v j 2/R, bQ (R)P(2/15[1v j "c"v j 21v j "q "v j 2 #1v j "c"v j 21v j "q "v j 2]/R, bQ (R)P!(51v j "a"v j 21v j "q "v j 2/R, bQ (R)P(51v j "a"v j 21v j "q "v j 2/R, (9) where 1vj"a"v j 2, 1vj"c"v j 2, and 1vj"q "v j 2 are the radial matrix elements of the trace and anisotH ropy of the polarizability tensor and of the multipole moments of order j. The asymptotic values are often used as a test of the accuracy of the ab initio computations. In Table 1, comparisons between these asymptotic values, results of Refs. [11,16], and results of this work are given for four Table 1 Functions bQ (R"9.0 a.u.) (in 10\ a.u.) for multipole-induced dipole comH H * ponents from: (a) asymptotic values; (b) Meyer et al. 1989 and 1993; (c) this work 2023
0223
2233
4045
0445
v "v "0, v "0, v "0 (a) 691 !691 (b) 692 !692 (c) 696 !696
!109 !104 !106
8 9 9
!8 !9 !9
v "v "0, v "0, v "1 (a) 94 !126 (b) 93 !131 (c) 93 !131
!26 !25 !25
1 1 1
!3 !5 !4
v "v "0, v "1, v "1 (a) 17 !17 (b) 17 !17 (c) 17 !17
!6 !5 !5
v "v "0, v "0, v "2 (a) !9 (b) !9 (c) !9
16 14 13
2 2 2
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known vibrational transitions. In that table, the dipole components of Eqs. (9) are computed at R"9.0 a.u. using matrix elements 1vj"a"v j 2, 1vj"c"v j 2, and 1vj"q "v j 2 from Refs. [24,25]. The H terms with j j "¸"4045 and 0445 of the "rst overtone band are very small and therefore are a!ected by larger relative numerical noise, and are not shown in Table 1. Generally, good agreement between the ab initio and the asymptotic values justi"es the satisfactory accuracy of our results. In Table 4 (our WWW? site), the results of bQ (R) and b (R) (n"1 to 8) are given for L_H H * H H * the "rst "ve terms (which are the most important induced dipole transition elements of the H pairs) with j j "¸"0001, 2023, 0223, 2021 and 0221, and for the four vibrational transitions (v "v "0): E v "0, v "0 (rototranslational band); E v "0, v "1 (fundamental band); E v "1, v "1 ("rst overtone band, double vibrational transitions); E v "0, v "2 ("rst overtone band, single vibrational transition). The parameters of "tting b (R) for v , v '0 are also available in our WWW? site. All H H * parameters are the result of "tting b (R) using j values up to 10, and may be useful for H H * computation of CIA spectra at high temperatures. Due to the exchange symmetry of vibrational transitions with v "0, v "0 and with v "1, v "1, bQ "0, bQ "!bQ , and bQ "!bQ , these terms are therefore not shown in Table 4. Functions bQ (R) listed in Table 4 are plotted in Figs. 2}5 on WWW@, for H H * the four vibrational transitions listed above. When compared with data from Refs. [11,16] we note a remarkable agreement between both sets of results. For v "0, v "2 (Fig. 5) the relative discrepancies become larger as we expected, because larger internuclear distances (where our new ab initio data take e!ect) become more important when higher vibrational states are involved. Another fact is that the relative discrepancies are larger at near intermolecular range (close to R values smaller than those in [11]) than at the far range. This fact indicates that larger internuclear distances are more important at near intermolecular range. It remains for future investigation to show whether the dramatic change of curve shapes of bQ (R) at near range H H * invalidates the commonly used analytic form of the b function. The form of the b funcH H * H H * tions we use is a slight generalization of the form used by van Kranendonk [26], and it describes the numerical results of previous works quite accurately [11,16,18]. The analytical form reads: B(R)"B /RL#B exp[B (R!R )#B (R!R )], (10) where B stands for bQ or b (n"1,2,2,8) given in Eq. (8), and (c) is again dropped for L convenience. B (i"1, 2, 3, 4) are the parameters which are obtained by "tting Eq. (10) to the data G of bQ (R) and b (R). The parameter R "6.0 a.u. is chosen to be close to the collisional diameter, L so that the parameter B gives the approximate size of the exchange and distortion dipole contribution at the collisional diameter [11]. In all cases, n"4 for j j "¸"2023 and 0223, n"7 for j j "¸"2021, 0221, and 0001, and n"6 for 4045 and 0445.
WWW@: http://www.astro.ku.dk/&aborysow/H2}dipole}CIA}LT/index.html
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
311
As a test, we computed the "rst overtone band (v "v "0 and v "1, v "1 and v "0, v "2) H }H CIA spectra at temperatures below 300 K using our new results of bQ (R) and b (R). The computed spectra (not shown here) are almost identical with those H H * L_H H * obtained using the data from Ref. [16]. This is due to the fact that at temperatures below 300 K the most relevant intermolecular distances are those at R'4.0 a.u., where the discrepancies between our data and the data of Ref. [16] are very small for all major terms (see Figs. 4 and 5 on WWW@). All the data presented here are available at our World Wide Web site. This set of data makes possible the computations (or revisions) of H }H CIA spectra at all temperatures below 7000 K, and for many rotovibrational bands signi"cant for planetary and astrophysical applications for atmospheric radiative transfer models. 5. The line shape theory The absorption coe$cient arising from collision-induced dipole moments in H }H molecular pairs is written as follows [15]: 2p nu[1!exp(! u/k¹)]
(11)
where n is the number density of the gas and < is the volume. Constants and c denote Planck's constant and the speed of light, respectively. u denotes angular frequency (s\), and ¹ is temperature (K). The gas density is usually expressed in units of amagat as . "n/N , where N is the number of gas molecules per cubic centimeter at standard temperature and pressure. For hydrogen, N is almost equal to Loschmidt's number N "2.68676;10 cm\ amagat\, which * is the N for ideal gases. The spectral density function g(u;¹) is computed from the transition matrix elements of the induced electric dipole moment k (l"0,$1) as given in [11]. J When an isotropic intermolecular potential is assumed, the translational, and rotovibrational states of the molecular pair become separable. For H molecules, this assumption is usually nearly true, at least at low temperatures, and it is therefore also used throughout this work. In such a case, the spectral density function g(u;¹) will be equal to P C( j j j ; 000)P C( j j j ; 000) g(u;¹)" TH TH H H * T H T H T H T H ;G (u!u !u ; ¹). THTH THTH H H *
(12)
The total spectral density is thus a superposition of basic line pro"les, which we refer to as the pure translational components G (u;¹). G (u;¹) are shifted by molecular rotovibrational H H * H H * frequencies u and u , which may be positive, zero or negative. For details concerning THTH THTH the quantum mechanical computation of the spectral density functions G (u;¹) we refer the H H * reader to Ref. [11]. The computational techniques used in this work are identical to those described therein. For completeness, we mention, that each spectral density function G(u;¹) depends uniquely on two input functions: the induced dipole, b (R) of a particular RV band, and on the H H * intermolecular initial and "nal states.
312
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
6. Computation of CIA spectra of H2 }H2 in the second overtone band at low temperature Using our newly developed H }H collision-induced dipole moments for the second overtone band (*v"3), we computed quantum mechanical H }H CIA spectra for this band for the "rst time, and compared our results with the existing low-density measurements. Only the ground vibrational state (v"0) of H needs to be considered as the initial state at low temperatures. With v "v "0, four di!erent vibrational transitions contribute to the second overtone band, centered around 0.83 lm: I. v "0, v "3; II. v "1, v "2; III. v "2, v "1; IV. v "3, v "0. Due to the exchange symmetry of the two hydrogen molecules, Eq. (5), we compute absorption spectra only for the "rst two transitions and multiply the results by two. For a single vibrational transition, v "0 and v "3, "ve terms with j j "¸"0001, 2023, 0223, 2021, and 0221 are included in our computations; and for double vibrational transitions, v "1 and v "2, four terms with j j "¸"2023, 0223, 2021, and 0221 are included. Other terms contribute less to the total spectral intensity and can be safely neglected. Table 2 lists the parameters bQ (R) and b (R) (n"1, 2,2, 8) of Eq. (8) for these terms. H H * L_H H * For the purpose of this work, we selected to "t only the results up to j , j , j , and j 43. For consistency, we would like to point out that the data deposited by us at our WWW? site are all obtained from "ts up to j"10, for all vibrational bands, and are user-ready for high-temperature applications. However, in order to preserve the best attainable accuracy, we restricted the range of the "t, just for the second overtone band at low temperatures, since higher values of j are irrelevant at temperatures below 300 K. In Figs. 2 and 3 we present our new functions bQ (R) for the second overtone band. As usual, H H * the functions bQ (R) are "tted by a second-order polynomial, Eq. (10), and coe$cients B are G H H * given in Table 3. Intermolecular distances R"2.5 and 3.0 a.u. are not included in our "t, because they are irrelevant at low temperatures. By neglecting these two values, we obtain much better "t to Eq. (10). We compute line shapes GQ (u;¹) for j j j j "0000 with induced dipole functions H H * bQ (R) and the same intermolecular potential, <(R) [11], as the initial and the "nal states, H H * using rigorously quantum mechanical treatment. All, free}free, bound}bound, bound}free and free}bound transitions are accounted for. It is, however, highly impractical to compute line shapes GQ (u;¹) for each possible rotational transition j j j j , which are, in practice, very similar in H H * shape to GQ (u;¹). Therefore, we apply a method similar to the one employed earlier [11,16]. H H * We start with computing GQ (u;¹) for each set of j j "¸, using bQ (R). We compute the H H * H H * low-resolution spectra, for which bound}bound features are not discernible. Then we rescale the intensity of each spectral line shape corresponding to bQ (R) function. Zeroth translational H H * spectral moments M for any j j j j are much easier to compute than spectral densities _H H * GQ (u;¹), based on the existing quantum mechanical sum rules [27] and using the j j j j H H * dependent bQ (R) functions. Contributions from both free and bound states were included in H H * the computations of the zeroth moments MQ and MQ . Line shapes GQ (u;¹) are _H H * H H * H H * then obtained by simple rescaling of GQ (u;¹) by a factor MQ /MQ . The H }H H H * _H H * _H H * CIA spectra a(u;¹) are calculated using Eqs. (11) and (12). Figs. 4 and 5 show the comparison between our computations and the measurements of the hydrogen second overtone band recorded within the binary collisional regime, by McKellar and Welsh [12] at 85 K at two di!erent ortho to para hydrogen ratios. Due to the very weak spectral
!233.3 !152.1 !100.6 !63.2 !39.0 !25.0 !16.7 !11.6 !5.9 !3.4 !2.0
16.1 64.1 82.6 57.7 36.8 23.9 16.1 11.3 6.0 3.4 2.2
!126.4 !8.1 16.2 14.8 9.6 5.8 3.6 2.3 1.0 0.4 0.2
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
R(a.u.) bQ b
!0.43060[!2] !0.15807[!2] !0.46071[!3] !0.18493[!3] !0.10931[!3] !0.77925[!4] !0.55677[!4] !0.39593[!4] !0.19605[!4] !0.11934[!4] !0.71759[!5]
!0.98228[!3] !0.33815[!3] 0.48581[!4] 0.51784[!4] 0.66699[!4] 0.64953[!4] 0.53936[!4] 0.42947[!4] 0.26776[!4] 0.17622[!4] 0.14783[!4]
0.16806[#1] 0.15908[#1] 0.10577[#1] 0.53230 0.22799 0.87473[!1] 0.28676[!1] 0.62414[!2] !0.43358[!2] !0.37480[!2] !0.29959[!2] !0.29517[!2] !0.13991[!2] !0.81449[!3] !0.38496[!3] !0.14429[!3] !0.35400[!4] 0.83359[!5] 0.21819[!4] 0.21473[!4] 0.13900[!4] 0.91763[!5]
v "v "0; v "1, v "2
0.50319[!2] 0.18267[!1] 0.44269[!1] 0.27547[!1] 0.16454[!1] 0.10311[!1] 0.65382[!2] 0.43977[!2] 0.20713[!2] 0.12777[!2] 0.10894[!2]
v "v "0; v "1, v "2
!0.57494 !0.14629[#1] !0.13933[#1] !0.89164 !0.54589 !0.34523 !0.23001 !0.15982 !0.84361[!1] !0.48407[!1] !0.31534[!1]
v "v "0; v "1, v "2
b
!0.15540[#1] !0.59468 !0.34243 !0.19403 !0.10320 !0.47833[!1] !0.16895[!1] !0.27349[!2] 0.39379[!2] 0.31401[!2] 0.18870[!2]
!0.86795 !0.14779[#1] !0.13577[#1] !0.95303 !0.60414 !0.38134 !0.24599 !0.16508 !0.81370[!1] !0.45845[!1] !0.26887[!1]
!0.18757[#1] !0.79487 !0.26485 !0.77905[!1] !0.13255[!1] 0.35918[!2] 0.66205[!2] 0.62666[!2] 0.49994[!2] 0.33526[!2] 0.27676[!2]
b
!0.62982[!2] !0.25613[!2] !0.20518[!2] !0.13825[!2] !0.79656[!3] !0.39863[!3] !0.16683[!3] !0.51672[!4] 0.16664[!4] 0.18608[!4] 0.12615[!4]
!0.39464[!2] 0.70682[!3] 0.31489[!2] 0.23114[!2] 0.14493[!2] 0.92760[!3] 0.62127[!3] 0.44005[!3] 0.23358[!3] 0.13545[!3] 0.86487[!4]
!0.12587[!1] !0.50704[!2] !0.15669[!2] !0.26388[!3] 0.75584[!4] 0.11712[!3] 0.99051[!4] 0.74982[!4] 0.46110[!4] 0.27342[!4] 0.16576[!4]
b
!0.15588[#1] !0.15357[#1] !0.10587[#1] !0.54267 !0.23744 !0.93755[!1] !0.32077[!1] !0.77080[!2] 0.45576[!2] 0.41464[!2] 0.33479[!2]
j j "¸"2021
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
j j "¸"0223
0.33370 0.13441[#1] 0.13156[#1] 0.85659 0.52721 0.33371 0.22254 0.15469 0.81805[!1] 0.46826[!1] 0.30157[!1]
j j "¸"2023
b
0.11244[!2] 0.51456[!3] 0.34292[!3] 0.18916[!3] 0.93598[!4] 0.41609[!4] 0.13887[!4] 0.17483[!5] !0.36077[!5] !0.17894[!5] !0.15011[!5]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.20401[!2] 0.67879[!3] 0.25492[!3] 0.78756[!4] 0.18602[!4] 0.24182[!5] !0.26102[!5] !0.29486[!5] !0.33757[!5] !0.23861[!5] !0.25217[!5]
b
Table 2 Functions bQ (R) and b (R) (in 10\ a.u.) for the second overtone band; numbers in the square brackets stand for the power of L_H H * H H * 10. The data present the results of "ts of bQ (R) for j values up to 3 H H *
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
!0.68106 0.79753 0.11998[#1] 0.84721 0.51625 0.31395 0.19648 0.13001 0.62707[!1] 0.35115[!1] 0.20951[!1]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
b
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
!0.32850[!2] !0.33021[!2] !0.30285[!2] !0.21043[!2] !0.13842[!2] !0.92193[!3] !0.62980[!3] !0.44457[!3] !0.23584[!3] !0.13625[!3] !0.84437[!4]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
b
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321 313
b
!0.36449[!2] !0.14559[!2] !0.57248[!3] !0.21634[!3] !0.82399[!4] !0.24823[!4] 0.54994[!6] 0.76385[!5] 0.49385[!5] 0.19750[!5] 0.14204[!6]
!0.10398[!2] 0.44584[!3] 0.53562[!3] 0.35331[!3] 0.19028[!3] 0.97573[!4] 0.46160[!4] 0.22625[!4] 0.64062[!5] 0.17894[!5] 0.63804[!6]
!0.24629[!2] !0.71185[!3] 0.23931[!3] 0.27787[!3] 0.18619[!3] 0.10811[!3] 0.58395[!4] 0.32096[!4] 0.12144[!4] 0.55785[!5] 0.49276[!5]
v "v "v "0, v "3
0.46904 0.45361 0.18176 0.65038[!1] 0.23626[!1] 0.70763[!2] 0.61022[!3] 0.19579[!3] 0.17102[!2] 0.16987[!2] 0.19575[!2]
v "v "v "0, v "3
!0.11140 !0.50188[!1] !0.18858[!1] !0.91329[!2] !0.27024[!2] !0.26164[!4] 0.48393[!3] 0.33140[!3] !0.11955[!3] !0.16156[!3] !0.19526[!3]
v "v "0; v "1, v "2
b
347.81 !0.37617 226.49 !0.11504 152.06 0.19934[!1] 82.39 0.35751[!1] 43.74 0.30018[!1] 23.69 0.21443[!1] 13.21 0.14802[!1] 7.69 0.10425[!1] 3.10 0.56291[!2] 1.54 0.32947[!2] 1.16 0.22741[!2]
!666.72 !193.60 !43.97 !6.43 8.77 12.75 10.99 7.68 2.82 0.83 0.09
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
!132.3 !100.8 !70.1 !39.0 !19.0 !8.3 !3.3 !1.1 0.2 0.3 0.2
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
R(a.u.) bQ
Table 2. (continued )
0.30973[#1] 0.11361[#1] 0.33815 0.13423 0.64880[!1] 0.31307[!1] 0.12768[!1] 0.31791[!2] !0.22971[!2] !0.24376[!2] !0.13627[!2]
!0.27057[#1] !0.23904[#1] !0.13694[#1] !0.71293 !0.35463 !0.17103 !0.80424[!1] !0.37411[!1] !0.76325[!2] !0.11598[!2] !0.36152[!3]
!0.55470 0.82786 0.10393[#1] 0.72139 0.43837 0.24626 0.13309 0.70627[!1] 0.20434[!1] 0.59190[!2] 0.30147[!2]
b
0.18474[!2] 0.51529[!3] 0.37844[!3] 0.25261[!3] 0.16328[!3] 0.89215[!4] 0.38218[!4] 0.11624[!4] !0.30896[!5] !0.39765[!5] 0.49021[!6]
!0.74116[!3] !0.14929[!2] !0.13418[!2] !0.85945[!3] !0.47974[!3] !0.25826[!3] !0.13889[!3] !0.67948[!4] !0.78451[!5] 0.49202[!5] 0.81315[!5]
!0.98116[!2] !0.59284[!2] !0.37548[!2] !0.19561[!2] !0.84057[!3] !0.29469[!3] !0.71325[!4] 0.11081[!4] 0.40800[!4] 0.34700[!4] 0.19412[!4]
b
!0.39790 !0.11870 0.26194[!1] 0.40729[!1] 0.33119[!1] 0.23239[!1] 0.15803[!1] 0.11004[!1] 0.58683[!2] 0.34156[!2] 0.23803[!2]
j j "¸"2023
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
j j "¸"0001
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
j j "¸"0221
b
!0.11237[!2] !0.64108[!3] !0.37854[!3] !0.17964[!3] !0.92483[!4] !0.51860[!4] !0.30868[!4] !0.19472[!4] !0.91287[!5] !0.51265[!5] !0.42133[!5]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
b
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
!0.12828[#1] !0.16122[#1] !0.13832[#1] !0.83812 !0.45401 !0.22833 !0.10986 !0.50680[!1] !0.87326[!2] 0.12178[!2] 0.87556[!3]
b
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.16780[!2] 0.21238[!2] 0.17394[!2] 0.10773[!2] 0.58623[!3] 0.29341[!3] 0.14070[!3] 0.65171[!4] 0.12324[!4] !0.18499[!6] !0.53413[!6]
b
314 Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
!0.12907[!1] !0.43125[!2] !0.20479[!2] !0.15994[!2] !0.13264[!2] !0.10591[!2] !0.82209[!3] !0.63117[!3] !0.37350[!3] !0.22518[!3] !0.14437[!3]
!0.93964[!3] !0.58441[!3] !0.48244[!3] !0.33435[!3] !0.16507[!3] !0.65150[!4] !0.18782[!4] !0.46025[!6] 0.71964[!5] 0.64866[!5] 0.41139[!5]
!0.54014[!2] !0.59403[!2] !0.42762[!2] !0.24168[!2] !0.13050[!2] !0.68716[!3] !0.34848[!3] !0.17034[!3] !0.37175[!4] !0.97664[!5] !0.39011[!5]
v "v "v "0, v "3
!0.41939 !0.19661 !0.12019 !0.69713[!1] !0.35657[!1] !0.15883[!1] !0.57506[!2] !0.12557[!2] 0.94744[!3] 0.91817[!3] 0.54639[!3]
v "v "v "0, v "3
!0.71340 !0.38863 !0.16654 !0.97414[!1] !0.60044[!1] !0.36738[!1] !0.22409[!1] !0.13699[!1] !0.56384[!2] !0.28536[!2] !0.14481[!2]
158.02 !0.87853 !114.54 !0.26079 !120.73 !0.72423[!1] !74.38 0.13315[!2] !45.20 0.22161[!1] !27.12 0.23072[!1] !15.96 0.18437[!1] !9.43 0.13355[!1] !3.70 0.67348[!2] !1.91 0.36829[!2] !1.00 0.19938[!2]
313.23 69.36 22.17 8.26 5.67 3.29 1.16 !0.15 !0.80 !0.51 !0.28
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
!15.01 102.17 58.31 14.88 !3.64 !10.00 !11.01 !10.13 !7.06 !4.37 !2.86
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0
v "v "v "0, v "3
!0.11297[#2] !0.54361[#1] !0.21550[#1] !0.98767 !0.42766 !0.17975 !0.80167[!1] !0.41143[!1] !0.17057[!1] !0.96676[!2] !0.57195[!2]
0.38371[#1] 0.17567[#1] 0.92197 0.41790 0.17108 0.58233[!1] 0.94581[!2] !0.92744[!2] !0.14305[!1] !0.96775[!2] !0.59390[!2]
0.17833[#2] 0.88968[#1] 0.45992[#1] 0.25567[#1] 0.14984[#1] 0.92198 0.59376 0.40011 0.20393 0.11598 0.72423[!1]
!0.88370[!3] !0.16749[!1] !0.14104[!1] !0.82969[!2] !0.47534[!2] !0.26720[!2] !0.14606[!2] !0.78769[!3] !0.24498[!3] !0.10579[!3] !0.52865[!4]
0.39689[!2] 0.19941[!2] 0.10082[!2] 0.37272[!3] 0.12260[!3] 0.31438[!4] !0.83046[!6] !0.12852[!4] !0.15193[!4] !0.91850[!5] !0.60693[!5]
!0.41623[!1] !0.12224[!1] !0.65076[!2] !0.59892[!2] !0.52130[!2] !0.42557[!2] !0.33481[!2] !0.25983[!2] !0.15528[!2] !0.93504[!3] !0.59624[!3]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
j j "¸"0221
!0.43366 !0.20593 !0.12658 !0.73698[!1] !0.37463[!1] !0.16510[!1] !0.58751[!2] !0.12093[!2] 0.10373[!2] 0.99247[!3] 0.59161[!3]
j j "¸"2021
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
j j "¸"0223
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
!0.19467[!3] 0.39989[!4] !0.21289[!4] !0.38060[!4] !0.31147[!4] !0.19336[!4] !0.95628[!5] !0.33475[!5] 0.12597[!5] 0.13556[!5] 0.95175[!6]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.15704[#2] 0.75736[#1] 0.33824[#1] 0.16117[#1] 0.72080 0.30520 0.12500 0.49217[!1] 0.52780[!2] !0.13174[!2] !0.87266[!3]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
!0.14550[#2] !0.78635[#1] !0.44388[#1] !0.25749[#1] !0.15315[#1] !0.94861 !0.61499 !0.41719 !0.21435 !0.12166 !0.75244[!1]
!0.22676[!2] 0.12079[!1] 0.10907[!1] 0.65784[!2] 0.38489[!2] 0.21975[!2] 0.12151[!2] 0.66074[!3] 0.20757[!3] 0.89204[!4] 0.44419[!4]
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.33007[!1] 0.92324[!2] 0.47720[!2] 0.43177[!2] 0.37662[!2] 0.30926[!2] 0.24443[!2] 0.19056[!2] 0.11458[!2] 0.69137[!3] 0.44329[!3]
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321 315
316
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
Fig. 2. Functions bQ (R) for v "0, v "0, H H * v "1, v "2. Labels denote the expansion parameters j j "¸; the minus sign indicates negative bQ values. H H * The markers represent the long-range, classical purely quadrupole-induced contribution, circles: j j "¸" !2023; and boxes: j j "¸"0223.
Fig. 3. Functions bQ (R) for v "0, v "0, H H * v "0, v "3. Labels denote the expansion parameters j j "¸; the minus sign indicates negative bQ values. H H * The markers describe the long-range, classical quadrupole-induced contribution, circles: j j "¸"2023; and boxes: j j "¸"!0223.
Table 3 Fitting parameters, of bQ K (R) (in a.u.) for the second overtone band HH * j j K¸
n
B
B
v "0, v "0, v "0, v "3 0001 7 !0.324727 2023 4 0.005854 0223 4 0.019459 2021 7 0.171489 0221 7 !3.175620
0.94377 0.31768 0.49026 !0.14823 1.78181
v "0, v "0, v "1, v "2 2023 4 0.009561 0223 4 !0.029781 2021 7 !0.410787 0221 7 0.519855
!0.185631 0.337341 0.365088 !0.293648
10\ 10\ 10\ 10\ 10\
10\ 10\ 10\ 10\
B
B
!0.91567 !1.54759 !1.46861 !3.15735 !1.95641
!0.31102 !0.04182 !0.05335 !0.34385 0.07598
!0.71801 !0.69671 !1.01774 !1.65565
0.06218 0.06055 0.08488 !0.03110
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
317
Fig. 4. The CIA spectra of normal H (C "0.25) in the second overtone band at 85 K. Solid line: theoretical data without j dependence; dashed line: theoretical data with j dependence; circles: experimental data by McKellar and Welsh (1971).
Fig. 5. CIA spectra of H (C "0.57) in the second overtone band at 85 K. Solid line: theoretical data without j dependence; dashed line: theoretical data with j dependence; circles: experimental data by McKellar and Welsh (1971).
intensity of this band only those measurements are available in the literature up to this day. Most recently, measurements at 77.5 and 298 K have been performed, see [14], and the experimental results have been compared with the theory outlined here. The analysis presented by McKellar and Welsh [12] of their measurements [12], is based on the purely multipole-induced dipole theory. The authors point out that the unexplained experimental
318
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
spectral intensity in the Q ( j"0,1) region may be arising from the unknown quantum overlap term, j j "¸"0001. They further speculate that Q lines (arising from the pure overlap, j j "¸"0001, contribution) exist for 0P1, 0P3, 0P5, etc., rotovibrational bands and are absent in the 0P2, 0P4, etc., rotovibrational bands. An alternative explanation given in [12] is that the j j "¸"0001 term is absent for only in 0P2 band for some particular reason. In our work we "nd out that j j "¸"0001 is strong in the second overtone band for single vibrational transition 0P3. Q lines exist for all RV bands for all single vibrational transitions, but the j j "¸"0001 spectral component may sometimes be weak, as for example for 0P2 (for details, see [16]), and also for the double transition (0P2 & 0P1) in the second overtone. In the experimental data we observe "ve distinctive peaks at frequencies below 13,000 cm\. In order to understand better the "ve peaks in Figs. 4 and 5, in Table 4 below we make assignments of the rotovibrational transitions contributing to each peak. As we expected, the relative discrepancies between theoretical and experimental spectra are larger for the second overtone than was observed for lower overtone bands. Our computational results are in much better agreement for double vibrational transitions, v v v v "0102 (the last two peaks), than for the single vibrational transition, v v v v "0003 (the "rst two peaks). In Figs. 4 and 5 we present theoretical results obtained by assuming, or neglecting the j-dependence. It turns out that whereas the j-dependence is especially weak for double vibrational transition, it is non-negligible for a single vibrational transition and should not be ignored. In order to understand better the reasons for the observed discrepancy between our theoretical results and the measured data, we attempted to estimate the uncertainty of our computations of a(u;¹) at various frequency ranges, by relating it to the inaccuracy introduced by the functions b (R). Accordingly, we tried to estimate the uncertainties of functions b (R) for this band. H H * H H * We compute asymptotic values of the two main terms bQ (R) and bQ (R) at R"9.0 a.u. from Eqs. (9), using matrix elements 1vj"a"v j 2 and 1vj"q "v j 2 from Ref. [24]. The comparison between Table 4 Assignment of peaks, counting from left to right Peaks
Roto-vibrational transitions
Vibrational bands
First
Q ( j"0,1,2)#Q ( j"0,1,2)
(3}0)
Second
Q ( j"0,1,2)#S ( j"0) S ( j"0)#Q ( j"0,1,2)
(3}0) (3}0)
Third
Q ( j"0,1,2)#S ( j"1) S ( j"1)#Q ( j"0,1,2) Q ( j"0,1,2)#Q ( j"0,1,2)
(3}0) (3}0) (1}0)#(2}0)
Fourth
S ( j"0)#Q ( j"0,1,2) Q ( j"0,1,2)#S ( j"0)
(1}0)#(2}0) (1}0)#(2}0)
Fifth
Q ( j"0,1,2)#S ( j"1) S ( j"1)#Q ( j"0,1,2)
(1}0)#(2}0) (1}0)#(2}0)
Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
319
Table 5 Leading functions bQ K (R"9.0 a.u.) for the second overtone H H * band (in 10\ a.u.). Upper line: asymptotic values, lower line: our results v "0, v "0, v "0, v "3
v "0, v "0, v "1, v "2
b
b
b
b
1.27 1.2
!2.28 !2.9
!1.65 !2.0
2.18 2.2
the asymptotic values and our results are given in Table 5. The terms with j j "¸"4045, 0445 and 2233 of the second overtone band are negligible, and therefore are not included in Table 5. For bQ (R) (v "0, v "3) and bQ (R) (v "1, v "2), the relatively larger discrepancies between our results and the asymptotic values than those observed for lower overtones (compare with Table 1) indicate somewhat less accurate b (R) functions. This is, in our opinion, mainly H H * due to the very small b (R) values of this band, about one order of magnitude smaller than H H * those of the "rst overtone band and therefore resulting in larger relative errors. For example, at R"9.0 a.u., A (R, r , r ) is of the order 1000;10\ a.u., while its radial matrix element (R, r , r )"v j v j 2 becomes only about 1;10\ a.u. for vibrational bQ (R)"1v j v j "A transition v "0Pv "0, and v "0Pv "3. It is also interesting to compare the intensity of the leading (quadrupole-induced) dipoles of RT, fundamental, "rst overtones, with those of the second overtone, for the single vibrational bands. At the collisional diameter (approx. 6. a.u.) the ratios of the dipoles are: &400, &65, &6.5, respectively, the approximate values correspond to averages between 2023 and 0223 terms. Having in mind that the absorption intensity is proportional to the square of the dipole, we can see that the RT spectra are roughly 1}2;10 more intense than the spectra of the second overtone band (for details, see [14]). Even compared to the "rst overtone, it appears that the second overtone is about 30}36 times less intense. That, of course, a!ects the accuracy of the "nal results. For such small values of bQ the major contributions to the uncertainty are the "tting in H H * Eq. (6), and the two-dimensional integration procedures. Another reason for the smaller accuracy of the b (R) functions of the second overtone, than what was obtained for lower overtones, is H H * the inclusion of higher vibrational states (v"2 and 3) in the computations of this band. For higher vibrational states of H , the internuclear distances of H beyond r "2.15 a.u. become more >> important. Here the uncertainties in the "tting of A (R, r , r ) in Eq. (6), become especially H H * large. Very good agreement between our results and the asymptotic values for bQ (R) (v "0, v "3) and bQ (R) (v "1, v "2) must be considered somewhat coinciden tal because of the above-mentioned computational uncertainties. It may, however, well be that at the largest intermolecular distance considered here, R"9.0 a.u., the computed dipoles do not yet reach their asymptotic values (see Figs. 2 and 3). There is no unique value of the relative error for b (R), because it depends upon the absolute H H * value of b (R) at various intermolecular distances R, though the absolute error of H H * A (R, r , r ) and the orthogonality of the wave functions 1vj"v j 2 can be estimated. This H H *
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Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321
implies that the relative errors for di!erent b (R) are not even the same (but depend upon the H H * induced dipole intensity and intermolecular distance R). Therefore, it is very di$cult to estimate the theoretical error of G (u), as well as of the theoretical spectra a(u;¹). H H * Furthermore, it is known that the intermolecular potential
Acknowledgements Support by NASA, Planetary Atmospheres Division, and by NASA, Astrophysics Theory Program, are gratefully acknowledged by the authors. Two of the authors (A.B. and Y.F.) would like to thank The Niels Bohr Institute, University Observatory, for the generous hospitality they experienced while working on this paper. We acknowledge Gaussian Inc. for the license agreement of Gaussian 92. C.Z. would like to thank Dr. Mark Cybulski for his patience and help in using the Gaussian 92 program, as well as for all his valuable comments.
References [1] Welsh HL. Pressure induced absorption spectra of hydrogen. In: Buckingham AD, Ramsay DA, editors. MTP International Review of Science-Physical Chemistry, Series 1, vol. 3: Spectroscopy, London: Butterworths, 1972. p. 33}71. [2] Linsky JL. On the pressure-induced opacity of molecular hydrogen in late-type stars. Astrophys J 1969;156:989}1005. [3] Lenzuni P, Cherno! DF, Salpeter EE. Rosseland and Planck mean opacities of a zero-metallicity gas. Astrophys J 1991;76 (Suppl):759}801. [4] Borysow A. Pressure-induced molecular absorption in stellar atmospheres. In: J+rgensen UG. editor. Molecules in the Stellar Environment, Lecture Notes in Physics, 1st ed., Berlin: Springer, 1994. p. 209}22.
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[5] Borysow A, J+rgensen UG, Zheng C. Model atmospheres of cool, low metallicity stars: the importance of collision-induced absorption. Astron Astrophys 1997;324:185}95. [6] Conrath BJ, Hanel RA, Samuelson RE, Origin and evolution of planetary and satellite atmospheres. Tucson: University of Arizona Press, 1989; p. 513}38. [7] Hanel RA, Conrath BJ, Jennings DE, Samuelson RE. Exploration of the solar system by infrared remote sensing. Cambridge, NY: University Press, 1992. [8] Trafton LM. Induced spectra in planetary atmospheres. In: Tabisz G, Neuman MN, editors. Collision- and Interaction-Induced Spectroscopy. NATO Advanced Research Workshops. Dordrecht: Kluwer, 1995. 517}28. [9] Kuiper GP. Astrophys J 1949;109:540. [10] Herzberg G. Spectroscopic evidence of molecular hydrogen in the atmospheres of Uranus and Neptune. Astrophys J 1952;115:337}40. [11] Meyer W, Borysow A, Frommhold L. Absorption spectra of H }H pairs in the fundamental band. Phys Rev A 1989;40:6931}49. [12] McKellar ARW, Welsh HL. Collision-induced spectra of hydrogen in the "rst and second overtone regions with applications of planetary atmospheres. Proc Roy Soc London Ser A 1971;322:421. [13] Reddy SP, Xiang F, Varghese G. Observation of the new triple transitions Q (J ) # Q (J ) # Q (J ) in molecular hydrogen in its second overtone region. Phys Rev Lett 1995;74:367}70. [14] Brodbeck C, Bouanich J-P, Nguyen-Van-Thanh Fu Y, Borysow A. Collision-induced absorption by H pairs in the second overtone band at 298 and 77.5 K: comparison between experimental and theoretical results. J Chem Phys 1999;110:4750. [15] Meyer W, Frommhold L, Birnbaum G. Rototranslational absorption spectra of H }H pairs in the far infrared. Phys Rev A 1989;39:2434}48. [16] Meyer W, Borysow A, Frommhold L. Collision-induced "rst overtone band of gaseous hydrogen from "rst principles. Phys Rev A 1993;47:4065}77. [17] Frisch MJ, Trucks GW, Head-Gordon M, Gill PMW, Wong MW, Foresman JB, Johnson BG, Schlegel HB, Robb MA, Replogle ES, Gomperts R, Andres JL, Raghavachari K, Binkley JS, Gonzalez C, Martin RL, Fox DJ, Defrees DJ, Baker J, Stewart JJP, Pople JA. Gaussian 92, Revision g.2. 1992. [18] Meyer W, Frommhold L. Collision-induced rototranslational spectra of H }He from an accurate ab initio dipole moment surface. Phys Rev A 1986;34:2771. [19] Hunt JL, Poll JD. Can J Phys 1978;56:950. [20] Kolos W, Szalewicz K, Monkhorst H. New Born}Oppenheimer energy curve and vibrational energies for the electronic ground state of the hydrogen molecule. J Chem Phys 1986;84:3278. [21] Kolos W, Wolniewicz L. J Mol Spectra 1965;43:2429. [22] Kolos W, Wolniewicz L. J Chem Phys 1968;49:404. [23] Kolos W, Wolniewicz L. J Mol Spectra 1975;54:303}11. [24] Hunt JL, Poll JD, Wolniewicz L. Can J Phys 1984;62:1719. [25] Karl G, Poll JD, Wolniewicz L. Can J Phys 1975;53:1781. [26] van Kranendonk J. Physica 1958;24:347. [27] Moraldi M, Borysow A, Frommhold L. Quantum sum formulae for the collision induced spectroscopies: Molecular systems as H }H . Chem Phys 1984;86:339}47. [28] Moraldi M, Borysow A, Frommhold L. Rotovibrational collision-induced absorption by nonpolar gases and mixtures (H }He pairs): about the symmetry of line pro"les. Phys Rev A 1988;38:1839}47. [29] Moraldi M, Borysow J, Frommhold L. Spectral moments for the collision-induced rotovibrational absorption bands of nonpolar gases and mixtures (H }He). Phys Rev A 1987;36:4700}3.