Quantum mechanical computations of collision-induced absorption in the second overtone band of...

*Corresponding author. Tel.: #45-35-325981; fax: #45-35-325989.E-mail address: [email protected] (A. Borysow).1On professional leave of absence from Michigan Technological University.

Journal of Quantitative Spectroscopy &Radiative Transfer 67 (2000) 303}321

Quantum mechanical computations of collision-inducedabsorption in the second overtone band of hydrogen

Yi Fu!, Chunguang Zheng!, Aleksandra Borysow",*,1

!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 SocLondon Ser A 1971;322:421). In this paper we present the "rst quantum mechanical computations of thecollision-induced rotovibrational absorption spectra of H

2pairs in the second (3}0) overtone band of

hydrogen. We compare our computations with the data by McKellar and Welsh. The second overtone bandis very weak and thus it is extremely di$cult to measure it in the laboratory, as well as to compute it based onthe "rst principles. As it appears, the collision-induced dipoles of H

2pairs, which give rise to this CIA band

spectra are so weak, that the numerical results, at some particular mutual orientations, are almost at the levelof numerical uncertainty. Our computations are based on an extension of a database of H

2}H

2collision-

induced dipoles which already exists (Meyer et al. Phys Rev A 1989;39:2434}48) but which is inadequate forcomputing 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. Theintermolecular interactions between H

2molecules can, nevertheless, give rise to induced dipole

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

moments. Here, we consider an isolated pair of H2

molecules, and therefore our resultswill be applicable only to binary collisions. Three mechanisms contribute to the induced dipolein such a molecular pair: the polarization of one H

2molecule in the multipolar "eld of

another H2

molecule; an electron exchange in the H2}H

2supermolecule at the near range;

and the dispersion interaction. Due to these induced dipole moment contributions hydrogen gasabsorbs in the far- and near-infrared regions [1], this process is called the collision-inducedabsorption (CIA). The CIA spectra are weak if compared with the spectra due to allowed dipoletransitions. As the induced dipoles are extremely weak, they are detectable in the laboratory only atsu$ciently high densities and/or long enough path lengths. The second overtone is very weak,causing considerable di$culties, especially if one wants to measure purely binary collisions. Onlysuch spectra (i.e. in the low-density limit) can be compared at present with exact quantummechanical theory.

CIA spectra of H2

pairs are of considerable interest for the studies of stellar [2}5], as well asplanetary atmospheres [6}8]. It may be of interest, that the "rst observation of the di!use featurearound 0.827 lm by Kuiper [9] in spectra of Uranus and Neptune has been later on identi"ed byHerzberg [10] as the S

3(0) line of the CIA spectrum of H

2}H

2. The other lines accompanying the

second overtone, clearly visible in the laboratory experimental spectra, were obscured by theallowed spectra of methane. This identi"cation has been used as a positive determination of thepresence of hydrogen in the atmospheres of those outer planets.

In order to compute the quantum mechanical collision-induced spectrum of H2

in the secondovertone band, we needed to extend available database of interaction-induced, ab initio dipolemoments [11]. In this work we compare our computations with the only available low-temper-ature experimental data of that band performed at the low-density limit [12]. We mention, thata paper appeared recently [13], which reports also low-temperature CIA measurements performedin the second overtone. In that paper, however, the spectra have been measured at densitiesbetween 500 and 1000 amagat, and no attempt has been made to extract binary absorptioncoe$cients of interest to this work. Instead, the work focuses on the "rst experimental evidence ofthe triple dipole transitions, which is a purely three-body e!ect.

During the preparation of this work, new experimental data of the binary collision-inducedabsorption coe$cient in the second overtone band became available at temperatures 77.5 and298 K. In a separate paper [14] we compare these new results, using the same procedures asoutlined in the present paper.

2. Ab initio induced dipole moments

The induced dipole moments are an essential input for accurate computations of the H2}H

2CIA spectra. The ab initio computations of H

2}H

2collision-induced dipole moments pose

problems similar to those known from the calculations of van der Waals potentials. For the mostrelevant molecular separations, i.e., those around the collisional diameter, the collision-induceddipole moments are rather small, but perturbation theory fails to adequately handle the exchangecontribution which becomes large at such short distances. It therefore appears best to treatthe H

2}H

2complex 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

304 Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321

superposition error is e!ectively controlled and the CI excitation level is adequate for thelong-range e!ects.

Meyer et al. [11] developed H2}H

2dipole moments from accurate ab initio SCF and CI

calculations. In that work, the H2}H

2collision-induced dipole moments k

i(R, r

1, r

2) (i"x, y, z)

were computed for nine intermolecular distances R (between 3.5 and 9.0 a.u.), three internuclearH}H distances (denoted by: r

~"1.111 a.u., r

0"1.449 a.u., and r

`"1.787 a.u.), and 13

nonequivalent relative orientations of two H2

molecules. The internuclear distance r0

correspondsto the ground state average of the H

2bond distance. With three internuclear distances r

~, r

0, and

r`

, computations were carried out for four di!erent r1r2

combinations, namely: r0r0, r

0r~

, r0r`

,and r

~r`

. Next, functions Aj1j2"L(R, r

1, r

2) (see Section 3) had been obtained from the ab initio

ki(R, r

1, r

2) data for fourteen leading j

1j2"¸ terms. Using the symmetry relations for

Aj1j2"L(R, r

1, r

2), these functions were "tted by a quadratic form of r

1and r

2. By using rotovibra-

tional wave functions DvjT of H2

molecule, functions bj1j2"L(R), (i.e. the transition matrix elements

of Aj1j2"L(R, r

1, r

2)), have been obtained for vibrational transitions with initial vibrational states

v1"v

2"0, and "nal vibrational states: v@

1"0, v@

2"0 (rototranslational band) [15];

v@1"0, v@

2"1 (fundamental band) (11); v@

1"1, v@

2"1 and v@

1"0, v@

2"2 ("rst overtone band)

[16]. The rotational state dependences on variables j1j@1j2j@2

were also given for the bj1j2"L(R)

functions, applicable for ji, j@i43 (i"1,2).

The collision-induced dipole moments of H2}H

2given 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 withlaboratory 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

2}H

2given in Refs. [11,15,16] are no longer adequate for our

computations. To meet our goal of computing RV CIA of H2}H

2at (3}0) band, we based our

e!orts on the earlier work of Meyer et al. [11], and included one larger internuclear distance atr"2.150 a.u.

First, we need to estimate which range of internuclear distances r is important for the secondovertone. Since the last peak of the vibrational wave function Dv"3T of the H

2molecule 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 areperformed for ten r

1r2

combinations, namely for r0r0, r

0r~

, r0r`

, r~

r`

, r0r``

, r`

r``

,r`

r`

, r``

r``

, r~

r``

, and r~

r~

. For completeness, we mention that we also included twosmaller intermolecular distances R (R"2.5 and 3.0 a.u.), not included in [11]. They are included inour new extended database, but their inclusion is not of relevance to this paper. These points are ofgreat signi"cance if one wants to use our extended database for computing CIA of RT and RVbands at temperatures as high as 7000 K.

In this work, we take the same thirteen nonequivalent mutual orientations of H2}H

2as used in

Ref. [11]. We express the orientation of H2}H

2by three angles, namely h

1, h

2, and *u. h

1(h

2) is

the angle subtended by vectors r1

(r2) and R. *u is the dihedral angle between the planes r

1R and

r2R. 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

Y. Fu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 67 (2000) 303}321 305

2WWW(a): http://www.astro.ku.dk/&aborysow/data/index.html

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 H2}H

2system is computed with a small electri-

cal "eld d applied in the relevant direction, i (i"x, y, z). The energy increase for the H2}H

2system

due to the presence of the electrical "eld, is *E`"!k

id. The dipole moment k

iis therefore given

by

ki"!

*E`

d. (1)

To reduce the computational noise, the electrical "eld !d is then applied, and the energy increaseis now *E

~"k

id. By taking the average of k

ifrom these two calculations, we get

ki"!

*E`!*E

~2d

. (2)

The relative deviations of kibetween our results and those of Ref. [11] are found to be within a few

percent for orientations with large ki. For several orientations with the smallest k

i, while the

absolute deviations remain the same in magnitude, the relative deviations become larger. This factdoes not a!ect the accuracy of our "nal results signi"cantly, because smaller components contrib-ute less to CIA spectra.

We have taken all existing data from Ref. [11], and computed only the additional con"gurationsdiscussed above. To avoid lengthy repetitions of data already presented [11], we collect all ab initioCartesian dipole moment components in Table 1 on our WWW(a) page,2 where we list componentskx

and kz, k

y"0 for all con"gurations.

3. Spherical expansion of the dipole moment

The spherical components kl (l"0,$1) of the induced dipole moment k can be expressed bytheir Cartesian components as k

0"k

z, and k

B1"G(k

x$ik

y)/J2. We expand kl as [19]:

kl (R, r1, r

2, )

1,)

2, ))"

(4p)3@2

J3+

j1j2"L

Aj1j2"L(R, r

1, r

2)>1lj1j2"L

()1, )

2, )), (3)

where r1

and r2

are the internuclear (H}H) distances of the two H2

molecules, R is the inter-molecular distance of H

2}H

2, and )

1, )

2and ) are their orientations, respectively.

Aj1j2"L(R, r

1, r

2) are real expansion coe$cients. Functions >1lj1j2"L

are expressed in terms ofspherical harmonics as given, for example, in Ref. [11]. The parameters j

1, j

2, ", and ¸

are non-negative integers. Due to the inversion symmetry of H2

molecules, the parity consider-ations, and the selection rules imposed by Clebsch}Gordan coe$cients, the following conditions


apply to j1, j

2,", and ¸:

j1

and j2

are even numbers, and ¸ is odd;

Dj1!j

2D4"4Dj

1#j

2D and D¸!1D4"4D¸#1D.

Based on these data, we obtain 14 leading expansion coe$cients, namely Aj1j2"L(R, r

1, r

2) of Eq. (3)

with

j1j2"¸"0001, 2023, 0223, 2021, 0221, 2233, 2211,

4045, 0445, 2245, 2243, 2223, 2221, 2201.

Higher-order terms can be safely neglected. The coe$cients Aj1j2"L(R, r

1, r

2) are given in Table 2

at the same WWW(a) site. Table 2 deposited at WWW(a) allows the comparison of our extendeddatabase with that of [11].

We remind the reader that even though we based our computations on previous work [11] andused many already existing Cartesian dipole components, for our ab initio computations of theinduced 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 newH

2}H

2con"gurations.

4. Functions bk1k2KL(R)

Functions bj1j2"L(R) are the radial transition matrix elements of Aj1j2"L

(R, r1, r

2), i.e.

bj1j2"L(R)"Sv

1j1v2j2DAj1j2"L

(R, r1, r

2)Dv@

1j@1v@2j@2T. (4)

They are essential inputs in the computations of CIA spectra. Functions b(s)(c)

(R) (the subscript (c)stands for j

1j2"¸, and the superscript (s) for v

1j1v2j2v@1j@1v@2j@2, the initial and "nal vibrational and

rotational states of both H2

molecules) follow the symmetry relation:

bv1j1v2 j2v@1 j

@1v

@2j

@2j1j2"L(R)"(!1)"`Lbv2 j2v1 j1v

@2 j

@2v

@1j

@1j2j1"L(R). (5)

Functions b(s)j1j2"L(R) can be obtained by a two-dimensional integration of Aj1j2"L

(R, r1, r

2) over

r1

and r2. The wave functions DvjT of the H

2molecule are obtained by solving the SchroK dinger

equation using the Numerov procedure, with the interaction potential energy for the H2

moleculein the ground electronic state [20}23]. With a small integration step size of dr+10~3 a.u., thenumerical values of inner products of the normalized wave functions, SvjDv@j@T, are not exactly equalto zero for vOv@, but instead they are typically of the magnitude of 10~9.

In order to be able to integrate the expansion coe$cients Aj1j2"L(R, r

1, r

2) over r

1and r

2, Eq. (4),

we need to approximate them as functions of r1

and r2. The quadratic form

Aj1j2"L(R, r

1, r

2)"a

0#a

1r1#a

2r2#a

3r21#a

4r22#a

5r1r2

(6)

is used for this purpose for two reasons: "rst, it can closely represent the smooth nature of the r1, r

2dependence of Aj1j2"L

(R, r1, r

2); second, higher-order polynomials with more parameters are

di$cult to control and sometimes produce oscillations.


Fig. 1. The r1r2

integration plane.

Due to the symmetry relation:

Aj1j2"L(R, r

1, r

2)"(!1)"`LAj2j1"L

(R, r2, r

1), (7)

we have values of Aj1j2"L(R, r

1, r

2) at all sixteen (r

1r2) points of the four by four grid over r

1and r

2,

as shown in Fig. 1. If we "t Aj1j2"L(R, r

1, r

2) at all sixteen points using Eq. (6) the "tting error is not

acceptable. We thus regroup the sixteen points into four groups, namely I, II, III, and IV as shownin Fig. 1. The r

1r2

plane is divided into four overlapping parts accordingly: I. r~4r

14r

ànd

r~4r

24r

`; II. r

~4r

14r

ànd r

04r

24r

``; III. r

04r

14r

`ànd r

04r

24r

``; IV.

r04r

14r

`ànd r

~4r

24r

`, where variable r

1runs horizontally, and r

2runs 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 fourparts gives the results for an analytic representation of Aj1j2"L

(R, r1, r

2) used to obtain the

functions b(s)j1j2"L(R) of various bands.

The functions b(s)j1j2"L(R) depend on all v

1, j1, v

2, j2, v@

1, j@1, v@

2, j@2

parameters. To facilitate the CIAcomputations and to reduce the amount of data, for a given vibrational transition v

1v2v@1v@2, the

dependence of b(s)j1j2"L(R) on the rotational states j

1j2j@1j@2

is expressed by an expansion in terms ofji( j

i#1) as follows:

b(s)+b(s0)#b1j1( j

1#1)#b

2[ j

1( j

1#1)]2

#b3j2( j

2#1)#b

4[ j

2( j

2#1)]2#b

5j@1( j@

1#1)

#b6[ j@

1( j@

1#1)]2#b

7j@2( j@

2#1)#b

8[ j@

2( j@

2#1)]2#2 . (8)

In the above equation, the intermolecular distance R and the subscripts j1j2"¸ are dropped for

convenience for all bj1j2"L(R) and b

n_j1j2"L(R) functions. The superscript (s) stands for

v1j1v2j2v@1j@1v@2j@2

as before, while (s0) stands for the same v

1v2v@1v@2

values, but withj1"j@

1"j

2"j@

2"0.


Table 1Functions b(s0 )j1j2"L

(R"9.0 a.u.) (in 10~6 a.u.) for multipole-induced dipole com-ponents from: (a) asymptotic values; (b) Meyer et al. 1989 and 1993; (c) this work

2023 0223 2233 4045 0445

v1"v

2"0, v@

1"0, v@

2"0

(a) 691 !691 !109 8 !8(b) 692 !692 !104 9 !9(c) 696 !696 !106 9 !9

v1"v

2"0, v@

1"0, v@

2"1

(a) 94 !126 !26 1 !3(b) 93 !131 !25 1 !5(c) 93 !131 !25 1 !4

v1"v

2"0, v@

1"1, v@

2"1

(a) 17 !17 !6(b) 17 !17 !5(c) 17 !17 !5

v1"v

2"0, v@

1"0, v@

2"2

(a) !9 16 2(b) !9 14 2(c) !9 13 2

For a given rotovibrational transition, b(s0 )(R) and bn(R) (n"1,2,8) of b(s)(R) can be obtained

by means of the least mean squares "tting of Eq. (8) for j1, j

2, j@

1and j@

2values.

The multipole-induced dipole components (j1j2"¸"2023, 0223, 2233, 4045, 0445) have well-

known long-range asymptotic forms, given in the appendix of Ref. [15]. The expressions belowdescribe the classical, purely multipole-dipole induction, which, at su$ciently long intermoleculardistances, are strictly satis"ed:

b(s)0223

(R)P!J3Sv1j1DaDv@

1j@1TSv

2j2Dq

2Dv@2j@2T/R4,

b(s)2023

(R)PJ3Sv2j2DaDv@

2j@2TSv

1j1Dq

2Dv@1j@1T/R4,

b(s)2233

(R)PJ2/15[Sv1j1DcDv@

1j@1TSv

2j2Dq

2Dv@2j@2T

#Sv2j2DcDv@

2j@2TSv

1j1Dq

2Dv@1j@1T]/R4,

b(s)0445

(R)P!J5Sv1j1DaDv@

1j@1TSv

2j2Dq

4Dv@2j@2T/R6,

b(s)4045

(R)PJ5Sv2j2DaDv@

2j@2TSv

1j1Dq

4Dv@1j@1T/R6, (9)

where SvjDaDv@j@T, SvjDcDv@j@T, and SvjDqj Dv@j@T are the radial matrix elements of the trace and anisot-ropy of the polarizability tensor and of the multipole moments of order j. The asymptotic valuesare often used as a test of the accuracy of the ab initio computations. In Table 1, comparisonsbetween these asymptotic values, results of Refs. [11,16], and results of this work are given for four


3WWW(b): http://www.astro.ku.dk/&aborysow/H2}dipole}CIA}LT/index.html

known vibrational transitions. In that table, the dipole components of Eqs. (9) are computed atR"9.0 a.u. using matrix elements SvjDaDv@j@T, SvjDcDv@j@T, and SvjDqj Dv@j@T from Refs. [24,25]. Theterms with j

1j2"¸"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, goodagreement between the ab initio and the asymptotic values justi"es the satisfactory accuracy of ourresults.

In Table 4 (our WWW(a) site), the results of b(s0 )j1j2"L(R) and b

n_j1j2"L(R) (n"1 to 8) are given for

the "rst "ve terms (which are the most important induced dipole transition elements of theH

2pairs) with j

1j2"¸"0001, 2023, 0223, 2021 and 0221, and for the four vibrational transitions

(v1"v

2"0):

f v@1"0, v@

2"0 (rototranslational band);

f v@1"0, v@

2"1 (fundamental band);

f v@1"1, v@

2"1 ("rst overtone band, double vibrational transitions);

f v@1"0, v@

2"2 ("rst overtone band, single vibrational transition).

The parameters of "tting bj1j2"L(R) for v

1, v

2'0 are also available in our WWW(a) site. All

parameters are the result of "tting bj1j2"L(R) using j values up to 10, and may be useful for

computation of CIA spectra at high temperatures.Due to the exchange symmetry of vibrational transitions with v@

1"0, v@

2"0 and with

v@1"1, v@

2"1,b(s0 )

0001"0, b(s0 )

0223"!b(s0 )

2023, and b(s0)

0221"!b(s0 )

2021, these terms are therefore not

shown in Table 4. Functions b(s0 )j1j2"L(R) listed in Table 4 are plotted in Figs. 2}5 on WWW(b),3 for

the four vibrational transitions listed above. When compared with data from Refs. [11,16] we notea remarkable agreement between both sets of results. For v@

1"0, v@

2"2 (Fig. 5) the relative

discrepancies become larger as we expected, because larger internuclear distances (where our newab 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 toR values smaller than those in [11]) than at the far range. This fact indicates that largerinternuclear distances are more important at near intermolecular range. It remains for futureinvestigation to show whether the dramatic change of curve shapes of b(s0 )j1j2"L

(R) at near rangeinvalidates the commonly used analytic form of the bj1j2"L

function. The form of the bj1j2"Lfunc-

tions we use is a slight generalization of the form used by van Kranendonk [26], and it describesthe numerical results of previous works quite accurately [11,16,18]. The analytical form reads:

B(R)"B1/Rn#B

2exp[B

3(R!R

0)#B

4(R!R

0)2], (10)

where B stands for b(s0 ) or bn

(n"1,2,2,8) given in Eq. (8), and (c) is again dropped forconvenience. B

i(i"1, 2, 3, 4) are the parameters which are obtained by "tting Eq. (10) to the data

of b(s0 )(R) and bn(R). The parameter R

0"6.0 a.u. is chosen to be close to the collisional diameter,

so that the parameter B2

gives the approximate size of the exchange and distortion dipolecontribution at the collisional diameter [11]. In all cases, n"4 for j

1j2"¸"2023 and 0223, n"7

for j1j2"¸"2021, 0221, and 0001, and n"6 for 4045 and 0445.


As a test, we computed the "rst overtone band (v1"v

2"0 and v@

1"1, v@

2"1 and

v@1"0, v@

2"2) H

2}H

2CIA spectra at temperatures below 300 K using our new results of

b(s0 )j1j2"L(R) and b

n_j1j2"L(R). The computed spectra (not shown here) are almost identical with those

obtained using the data from Ref. [16]. This is due to the fact that at temperatures below 300 K themost relevant intermolecular distances are those at R'4.0 a.u., where the discrepancies betweenour data and the data of Ref. [16] are very small for all major terms (see Figs. 4 and 5 on WWW(b)).

All the data presented here are available at our World Wide Web site. This set of data makespossible the computations (or revisions) of H

2}H

2CIA spectra at all temperatures below 7000 K,

and for many rotovibrational bands signi"cant for planetary and astrophysical applications foratmospheric radiative transfer models.

5. The line shape theory

The absorption coe$cient arising from collision-induced dipole moments in H2}H

2molecular

pairs is written as follows [15]:

a(u;¹)"2p2

3+cn2u[1!exp(!+u/k¹)]<g(u;¹), (11)

where n is the number density of the gas and < is the volume. Constants + and c denote Planck'sconstant and the speed of light, respectively. u denotes angular frequency (s~1), and ¹ istemperature (K). The gas density is usually expressed in units of amagat as ."n/N

A, where N

Ais

the number of gas molecules per cubic centimeter at standard temperature and pressure. Forhydrogen, N

Ais almost equal to Loschmidt's number N

L"2.68676]1019 cm~3 amagat~1, which

is the NA

for ideal gases. The spectral density function g(u;¹) is computed from the transitionmatrix elements of the induced electric dipole moment kl (l"0,$1) as given in [11].

When an isotropic intermolecular potential is assumed, the translational, and rotovibrationalstates of the molecular pair become separable. For H

2molecules, 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

g(u;¹)" +j1j2"L

+v1j1v

@1j

@1v2j2v

@2j

@2

Pv1j1

C( j1j1j@1; 000)2P

v2 j2C( j

2j2j@2; 000)2

]Gj1j2"L(u!u

v1j1v@1j

@1!u

v2j2v@2j

@2;¹). (12)

The total spectral density is thus a superposition of basic line pro"les, which we refer to as the puretranslational components Gj1j2"L

(u;¹). Gj1j2"L(u;¹) are shifted by molecular rotovibrational

frequencies uv1 j1v

@1j

@1

and uv2j2v

@2j

@2, which may be positive, zero or negative. For details concerning

the quantum mechanical computation of the spectral density functions Gj1j2"L(u;¹) we refer the

reader to Ref. [11]. The computational techniques used in this work are identical to those describedtherein. For completeness, we mention, that each spectral density function G(u;¹) dependsuniquely on two input functions: the induced dipole, bj1j2"L

(R) of a particular RV band, and on theintermolecular initial and "nal states.


6. Computation of CIA spectra of H2}H2 in the second overtone band at low temperature

Using our newly developed H2}H

2collision-induced dipole moments for the second overtone

band (*v"3), we computed quantum mechanical H2}H

2CIA 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

2needs to be considered as the initial state at low

temperatures. With v1"v

2"0, four di!erent vibrational transitions contribute to the second

overtone band, centered around 0.83 lm: I. v@1"0, v@

2"3; II. v@

1"1, v@

2"2; III. v@

1"2, v@

2"1;

IV. v@1"3, v@

2"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@

1"0 and v@

2"3, "ve terms with j

1j2"¸"0001, 2023,

0223, 2021, and 0221 are included in our computations; and for double vibrational transitions,v@1"1 and v@

2"2, four terms with j

1j2"¸"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 b(s0 )j1j2"L

(R) and bn_j1j2"L

(R) (n"1, 2,2, 8) of Eq. (8) for these terms.For the purpose of this work, we selected to "t only the results up to j

1, j

2, j@

1, and j@

243. For

consistency, we would like to point out that the data deposited by us at our WWW(a) site are allobtained from "ts up to j"10, for all vibrational bands, and are user-ready for high-temperatureapplications. However, in order to preserve the best attainable accuracy, we restricted the range ofthe "t, just for the second overtone band at low temperatures, since higher values of j are irrelevantat temperatures below 300 K.

In Figs. 2 and 3 we present our new functions b(s0 )j1j2"L(R) for the second overtone band. As usual,

the functions b(s0 )j1j2"L(R) are "tted by a second-order polynomial, Eq. (10), and coe$cients B

iare

given in Table 3. Intermolecular distances R"2.5 and 3.0 a.u. are not included in our "t, becausethey are irrelevant at low temperatures. By neglecting these two values, we obtain much better "t toEq. (10).

We compute line shapes G(s0 )j1j2"L(u;¹) for j

1j2j@1j@2"0000 with induced dipole functions

b(s0 )j1j2"L(R) and the same intermolecular potential, <00

0(R) [11], as the initial and the "nal states,

using rigorously quantum mechanical treatment. All, free}free, bound}bound, bound}free andfree}bound transitions are accounted for. It is, however, highly impractical to compute line shapesG(s)j1j2"L

(u;¹) for each possible rotational transition j1j2j@1j@2, which are, in practice, very similar in

shape to G(s0 )j1j2"L(u;¹). Therefore, we apply a method similar to the one employed earlier [11,16].

We start with computing G(s0 )j1j2"L(u;¹) for each set of j

1j2"¸, using b(s0 )j1j2"L

(R). We compute thelow-resolution spectra, for which bound}bound features are not discernible. Then we rescale theintensity of each spectral line shape corresponding to b(s)j1j2"L

(R) function. Zeroth translationalspectral moments M

0_j1j2"Lfor any j

1j2j@1j@2

are much easier to compute than spectral densitiesG(s)j1j2"L

(u;¹), based on the existing quantum mechanical sum rules [27] and using the j1j2j@1j@2

dependent b(s)j1j2"L(R) functions. Contributions from both free and bound states were included in

the computations of the zeroth moments M(s0 )0_j1j2"L

and M(s)0,j1j2"L

. Line shapes G(s)j1j2"L(u;¹) are

then obtained by simple rescaling of G(s0 )j1j2"L(u;¹) by a factor M(s)

0_j1j2"L/M(s0 )

0_j1j2"L. The H

2}H

2CIA 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 thehydrogen second overtone band recorded within the binary collisional regime, by McKellar andWelsh [12] at 85 K at two di!erent ortho to para hydrogen ratios. Due to the very weak spectral


Tab

le2

Func

tion

sb(

s0)

j1j2

"L(R

)and

b n_j1

j2"L

(R)(

in10

~6

a.u.)

forth

ese

cond

ove

rtone

ban

d;nu

mber

sin

thesq

uar

ebr

acket

sst

and

forth

epow

erof

10.T

hedat

apre

sent

the

resu

lts

of"ts

ofb(

s) j1j2

"L(R

)fo

rjva

lues

upto

3

R(a

.u.)

b(s0)

b 1b 2

b 3b 4

b 5b 6

b 7b 8

v 1"

v 2"

0;v@ 1

"1,

v@ 2"

2j 1

j 2"¸"

2023

2.5

!23

3.3

!0.

5749

4!

0.43

060[!

2]!

0.18

757[

#1]

!0.

1258

7[!

1]0.

3337

00.

2040

1[!

2]0.

0000

00.

0000

03.

0!

152.

1!

0.14

629[

#1]

!0.

1580

7[!

2]!

0.79

487

!0.

5070

4[!

2]0.

1344

1[#

1]0.

6787

9[!

3]0.

0000

00.

0000

03.

5!

100.

6!

0.13

933[

#1]

!0.

4607

1[!

3]!

0.26

485

!0.

1566

9[!

2]0.

1315

6[#

1]0.

2549

2[!

3]0.

0000

00.

0000

04.

0!

63.2

!0.

8916

4!

0.18

493[!

3]!

0.77

905[!

1]!

0.26

388[!

3]0.

8565

90.

7875

6[!

4]0.

0000

00.

0000

04.

5!

39.0

!0.

5458

9!

0.10

931[!

3]!

0.13

255[!

1]0.

7558

4[!

4]0.

5272

10.

1860

2[!

4]0.

0000

00.

0000

05.

0!

25.0

!0.

3452

3!

0.77

925[!

4]0.

3591

8[!

2]0.

1171

2[!

3]0.

3337

10.

2418

2[!

5]0.

0000

00.

0000

05.

5!

16.7

!0.

2300

1!

0.55

677[!

4]0.

6620

5[!

2]0.

9905

1[!

4]0.

2225

4!

0.26

102[!

5]0.

0000

00.

0000

06.

0!

11.6

!0.

1598

2!

0.39

593[!

4]0.

6266

6[!

2]0.

7498

2[!

4]0.

1546

9!

0.29

486[!

5]0.

0000

00.

0000

07.

0!

5.9

!0.

8436

1[!

1]!

0.19

605[!

4]0.

4999

4[!

2]0.

4611

0[!

4]0.

8180

5[!

1]!

0.33

757[!

5]0.

0000

00.

0000

08.

0!

3.4

!0.

4840

7[!

1]!

0.11

934[!

4]0.

3352

6[!

2]0.

2734

2[!

4]0.

4682

6[!

1]!

0.23

861[!

5]0.

0000

00.

0000

09.

0!

2.0

!0.

3153

4[!

1]!

0.71

759[!

5]0.

2767

6[!

2]0.

1657

6[!

4]0.

3015

7[!

1]!

0.25

217[!

5]0.

0000

00.

0000

0

v 1"

v 2"

0;v@ 1

"1,

v@ 2"

2j 1

j 2"¸"

0223

2.5

16.1

0.50

319[!

2]!

0.98

228[!

3]!

0.86

795

!0.

3946

4[!

2]0.

0000

00.

0000

0!

0.68

106

!0.

3285

0[!

2]3.

064

.10.

1826

7[!

1]!

0.33

815[!

3]!

0.14

779[

#1]

0.70

682[!

3]0.

0000

00.

0000

00.

7975

3!

0.33

021[!

2]3.

582

.60.

4426

9[!

1]0.

4858

1[!

4]!

0.13

577[

#1]

0.31

489[!

2]0.

0000

00.

0000

00.

1199

8[#

1]!

0.30

285[!

2]4.

057

.70.

2754

7[!

1]0.

5178

4[!

4]!

0.95

303

0.23

114[!

2]0.

0000

00.

0000

00.

8472

1!

0.21

043[!

2]4.

536

.80.

1645

4[!

1]0.

6669

9[!

4]!

0.60

414

0.14

493[!

2]0.

0000

00.

0000

00.

5162

5!

0.13

842[!

2]5.

023

.90.

1031

1[!

1]0.

6495

3[!

4]!

0.38

134

0.92

760[!

3]0.

0000

00.

0000

00.

3139

5!

0.92

193[!

3]5.

516

.10.

6538

2[!

2]0.

5393

6[!

4]!

0.24

599

0.62

127[!

3]0.

0000

00.

0000

00.

1964

8!

0.62

980[!

3]6.

011

.30.

4397

7[!

2]0.

4294

7[!

4]!

0.16

508

0.44

005[!

3]0.

0000

00.

0000

00.

1300

1!

0.44

457[!

3]7.

06.

00.

2071

3[!

2]0.

2677

6[!

4]!

0.81

370[!

1]0.

2335

8[!

3]0.

0000

00.

0000

00.

6270

7[!

1]!

0.23

584[!

3]8.

03.

40.

1277

7[!

2]0.

1762

2[!

4]!

0.45

845[!

1]0.

1354

5[!

3]0.

0000

00.

0000

00.

3511

5[!

1]!

0.13

625[!

3]9.

02.

20.

1089

4[!

2]0.

1478

3[!

4]!

0.26

887[!

1]0.

8648

7[!

4]0.

0000

00.

0000

00.

2095

1[!

1]!

0.84

437[!

4]

v 1"

v 2"

0;v@ 1

"1,

v@ 2"

2j 1

j 2"¸"

2021

2.5

!12

6.4

0.16

806[

#1]

!0.

2951

7[!

2]!

0.15

540[

#1]

!0.

6298

2[!

2]!

0.15

588[

#1]

0.11

244[!

2]0.

0000

00.

0000

03.

0!

8.1

0.15

908[

#1]

!0.

1399

1[!

2]!

0.59

468

!0.

2561

3[!

2]!

0.15

357[

#1]

0.51

456[!

3]0.

0000

00.

0000

03.

516

.20.

1057

7[#

1]!

0.81

449[!

3]!

0.34

243

!0.

2051

8[!

2]!

0.10

587[

#1]

0.34

292[!

3]0.

0000

00.

0000

04.

014

.80.

5323

0!

0.38

496[!

3]!

0.19

403

!0.

1382

5[!

2]!

0.54

267

0.18

916[!

3]0.

0000

00.

0000

04.

59.

60.

2279

9!

0.14

429[!

3]!

0.10

320

!0.

7965

6[!

3]!

0.23

744

0.93

598[!

4]0.

0000

00.

0000

05.

05.

80.

8747

3[!

1]!

0.35

400[!

4]!

0.47

833[!

1]!

0.39

863[!

3]!

0.93

755[!

1]0.

4160

9[!

4]0.

0000

00.

0000

05.

53.

60.

2867

6[!

1]0.

8335

9[!

5]!

0.16

895[!

1]!

0.16

683[!

3]!

0.32

077[!

1]0.

1388

7[!

4]0.

0000

00.

0000

06.

02.

30.

6241

4[!

2]0.

2181

9[!

4]!

0.27

349[!

2]!

0.51

672[!

4]!

0.77

080[!

2]0.

1748

3[!

5]0.

0000

00.

0000

07.

01.

0!

0.43

358[!

2]0.

2147

3[!

4]0.

3937

9[!

2]0.

1666

4[!

4]0.

4557

6[!

2]!

0.36

077[!

5]0.

0000

00.

0000

08.

00.

4!

0.37

480[!

2]0.

1390

0[!

4]0.

3140

1[!

2]0.

1860

8[!

4]0.

4146

4[!

2]!

0.17

894[!

5]0.

0000

00.

0000

09.

00.

2!

0.29

959[!

2]0.

9176

3[!

5]0.

1887

0[!

2]0.

1261

5[!

4]0.

3347

9[!

2]!

0.15

011[!

5]0.

0000

00.

0000

0


Tab

le2.

(con

tinu

ed)

R(a

.u.)

b(s0)

b 1b 2

b 3b 4

b 5b 6

b 7b 8

v 1"

v 2"

0;v@ 1

"1,

v@ 2"

2j 1

j 2"¸"

0221

2.5

!13

2.3

!0.

1114

0!

0.36

449[!

2]!

0.55

470

!0.

9811

6[!

2]0.

0000

00.

0000

0!

0.12

828[

#1]

0.16

780[!

2]3.

0!

100.

8!

0.50

188[!

1]!

0.14

559[!

2]0.

8278

6!

0.59

284[!

2]0.

0000

00.

0000

0!

0.16

122[

#1]

0.21

238[!

2]3.

5!

70.1

!0.

1885

8[!

1]!

0.57

248[!

3]0.

1039

3[#

1]!

0.37

548[!

2]0.

0000

00.

0000

0!

0.13

832[

#1]

0.17

394[!

2]4.

0!

39.0

!0.

9132

9[!

2]!

0.21

634[!

3]0.

7213

9!

0.19

561[!

2]0.

0000

00.

0000

0!

0.83

812

0.10

773[!

2]4.

5!

19.0

!0.

2702

4[!

2]!

0.82

399[!

4]0.

4383

7!

0.84

057[!

3]0.

0000

00.

0000

0!

0.45

401

0.58

623[!

3]5.

0!

8.3

!0.

2616

4[!

4]!

0.24

823[!

4]0.

2462

6!

0.29

469[!

3]0.

0000

00.

0000

0!

0.22

833

0.29

341[!

3]5.

5!

3.3

0.48

393[!

3]0.

5499

4[!

6]0.

1330

9!

0.71

325[!

4]0.

0000

00.

0000

0!

0.10

986

0.14

070[!

3]6.

0!

1.1

0.33

140[!

3]0.

7638

5[!

5]0.

7062

7[!

1]0.

1108

1[!

4]0.

0000

00.

0000

0!

0.50

680[!

1]0.

6517

1[!

4]7.

00.

2!

0.11

955[!

3]0.

4938

5[!

5]0.

2043

4[!

1]0.

4080

0[!

4]0.

0000

00.

0000

0!

0.87

326[!

2]0.

1232

4[!

4]8.

00.

3!

0.16

156[!

3]0.

1975

0[!

5]0.

5919

0[!

2]0.

3470

0[!

4]0.

0000

00.

0000

00.

1217

8[!

2]!

0.18

499[!

6]9.

00.

2!

0.19

526[!

3]0.

1420

4[!

6]0.

3014

7[!

2]0.

1941

2[!

4]0.

0000

00.

0000

00.

8755

6[!

3]!

0.53

413[!

6]

v 1"

v 2"

v@ 1"

0,v@ 2

"3

j 1j 2

"¸"

0001

2.5

!66

6.72

0.46

904

!0.

1039

8[!

2]!

0.27

057[

#1]

!0.

7411

6[!

3]0.

0000

00.

0000

00.

0000

00.

0000

03.

0!

193.

600.

4536

10.

4458

4[!

3]!

0.23

904[

#1]

!0.

1492

9[!

2]0.

0000

00.

0000

00.

0000

00.

0000

03.

5!

43.9

70.

1817

60.

5356

2[!

3]!

0.13

694[

#1]

!0.

1341

8[!

2]0.

0000

00.

0000

00.

0000

00.

0000

04.

0!

6.43

0.65

038[!

1]0.

3533

1[!

3]!

0.71

293

!0.

8594

5[!

3]0.

0000

00.

0000

00.

0000

00.

0000

04.

58.

770.

2362

6[!

1]0.

1902

8[!

3]!

0.35

463

!0.

4797

4[!

3]0.

0000

00.

0000

00.

0000

00.

0000

05.

012

.75

0.70

763[!

2]0.

9757

3[!

4]!

0.17

103

!0.

2582

6[!

3]0.

0000

00.

0000

00.

0000

00.

0000

05.

510

.99

0.61

022[!

3]0.

4616

0[!

4]!

0.80

424[!

1]!

0.13

889[!

3]0.

0000

00.

0000

00.

0000

00.

0000

06.

07.

680.

1957

9[!

3]0.

2262

5[!

4]!

0.37

411[!

1]!

0.67

948[!

4]0.

0000

00.

0000

00.

0000

00.

0000

07.

02.

820.

1710

2[!

2]0.

6406

2[!

5]!

0.76

325[!

2]!

0.78

451[!

5]0.

0000

00.

0000

00.

0000

00.

0000

08.

00.

830.

1698

7[!

2]0.

1789

4[!

5]!

0.11

598[!

2]0.

4920

2[!

5]0.

0000

00.

0000

00.

0000

00.

0000

09.

00.

090.

1957

5[!

2]0.

6380

4[!

6]!

0.36

152[!

3]0.

8131

5[!

5]0.

0000

00.

0000

00.

0000

00.

0000

0

v 1"

v 2"

v@ 1"

0,v@ 2

"3

j 1j 2

"¸"

2023

2.5

347.

81!

0.37

617

!0.

2462

9[!

2]0.

3097

3[#

1]0.

1847

4[!

2]!

0.39

790

!0.

1123

7[!

2]0.

0000

00.

0000

03.

022

6.49

!0.

1150

4!

0.71

185[!

3]0.

1136

1[#

1]0.

5152

9[!

3]!

0.11

870

!0.

6410

8[!

3]0.

0000

00.

0000

03.

515

2.06

0.19

934[!

1]0.

2393

1[!

3]0.

3381

50.

3784

4[!

3]0.

2619

4[!

1]!

0.37

854[!

3]0.

0000

00.

0000

04.

082

.39

0.35

751[!

1]0.

2778

7[!

3]0.

1342

30.

2526

1[!

3]0.

4072

9[!

1]!

0.17

964[!

3]0.

0000

00.

0000

04.

543

.74

0.30

018[!

1]0.

1861

9[!

3]0.

6488

0[!

1]0.

1632

8[!

3]0.

3311

9[!

1]!

0.92

483[!

4]0.

0000

00.

0000

05.

023

.69

0.21

443[!

1]0.

1081

1[!

3]0.

3130

7[!

1]0.

8921

5[!

4]0.

2323

9[!

1]!

0.51

860[!

4]0.

0000

00.

0000

05.

513

.21

0.14

802[!

1]0.

5839

5[!

4]0.

1276

8[!

1]0.

3821

8[!

4]0.

1580

3[!

1]!

0.30

868[!

4]0.

0000

00.

0000

06.

07.

690.

1042

5[!

1]0.

3209

6[!

4]0.

3179

1[!

2]0.

1162

4[!

4]0.

1100

4[!

1]!

0.19

472[!

4]0.

0000

00.

0000

07.

03.

100.

5629

1[!

2]0.

1214

4[!

4]!

0.22

971[!

2]!

0.30

896[!

5]0.

5868

3[!

2]!

0.91

287[!

5]0.

0000

00.

0000

08.

01.

540.

3294

7[!

2]0.

5578

5[!

5]!

0.24

376[!

2]!

0.39

765[!

5]0.

3415

6[!

2]!

0.51

265[!

5]0.

0000

00.

0000

09.

01.

160.

2274

1[!

2]0.

4927

6[!

5]!

0.13

627[!

2]0.

4902

1[!

6]0.

2380

3[!

2]!

0.42

133[!

5]0.

0000

00.

0000

0


v 1"

v 2"

v@ 1"

0,v@ 2

"3

j 1j 2

"¸"

0223

2.5

!15

.01

!0.

7134

0!

0.12

907[!

1]0.

1783

3[#

2]!

0.41

623[!

1]0.

0000

00.

0000

0!

0.14

550[

#2]

0.33

007[!

1]3.

010

2.17

!0.

3886

3!

0.43

125[!

2]0.

8896

8[#

1]!

0.12

224[!

1]0.

0000

00.

0000

0!

0.78

635[

#1]

0.92

324[!

2]3.

558

.31

!0.

1665

4!

0.20

479[!

2]0.

4599

2[#

1]!

0.65

076[!

2]0.

0000

00.

0000

0!

0.44

388[

#1]

0.47

720[!

2]4.

014

.88

!0.

9741

4[!

1]!

0.15

994[!

2]0.

2556

7[#

1]!

0.59

892[!

2]0.

0000

00.

0000

0!

0.25

749[

#1]

0.43

177[!

2]4.

5!

3.64

!0.

6004

4[!

1]!

0.13

264[!

2]0.

1498

4[#

1]!

0.52

130[!

2]0.

0000

00.

0000

0!

0.15

315[

#1]

0.37

662[!

2]5.

0!

10.0

0!

0.36

738[!

1]!

0.10

591[!

2]0.

9219

8!

0.42

557[!

2]0.

0000

00.

0000

0!

0.94

861

0.30

926[!

2]5.

5!

11.0

1!

0.22

409[!

1]!

0.82

209[!

3]0.

5937

6!

0.33

481[!

2]0.

0000

00.

0000

0!

0.61

499

0.24

443[!

2]6.

0!

10.1

3!

0.13

699[!

1]!

0.63

117[!

3]0.

4001

1!

0.25

983[!

2]0.

0000

00.

0000

0!

0.41

719

0.19

056[!

2]7.

0!

7.06

!0.

5638

4[!

2]!

0.37

350[!

3]0.

2039

3!

0.15

528[!

2]0.

0000

00.

0000

0!

0.21

435

0.11

458[!

2]8.

0!

4.37

!0.

2853

6[!

2]!

0.22

518[!

3]0.

1159

8!

0.93

504[!

3]0.

0000

00.

0000

0!

0.12

166

0.69

137[!

3]9.

0!

2.86

!0.

1448

1[!

2]!

0.14

437[!

3]0.

7242

3[!

1]!

0.59

624[!

3]0.

0000

00.

0000

0!

0.75

244[!

1]0.

4432

9[!

3]

v 1"

v 2"

v@ 1"

0,v@ 2

"3

j 1j 2

"¸"

2021

2.5

313.

23!

0.41

939

!0.

9396

4[!

3]0.

3837

1[#

1]0.

3968

9[!

2]!

0.43

366

!0.

1946

7[!

3]0.

0000

00.

0000

03.

069

.36

!0.

1966

1!

0.58

441[!

3]0.

1756

7[#

1]0.

1994

1[!

2]!

0.20

593

0.39

989[!

4]0.

0000

00.

0000

03.

522

.17

!0.

1201

9!

0.48

244[!

3]0.

9219

70.

1008

2[!

2]!

0.12

658

!0.

2128

9[!

4]0.

0000

00.

0000

04.

08.

26!

0.69

713[!

1]!

0.33

435[!

3]0.

4179

00.

3727

2[!

3]!

0.73

698[!

1]!

0.38

060[!

4]0.

0000

00.

0000

04.

55.

67!

0.35

657[!

1]!

0.16

507[!

3]0.

1710

80.

1226

0[!

3]!

0.37

463[!

1]!

0.31

147[!

4]0.

0000

00.

0000

05.

03.

29!

0.15

883[!

1]!

0.65

150[!

4]0.

5823

3[!

1]0.

3143

8[!

4]!

0.16

510[!

1]!

0.19

336[!

4]0.

0000

00.

0000

05.

51.

16!

0.57

506[!

2]!

0.18

782[!

4]0.

9458

1[!

2]!

0.83

046[!

6]!

0.58

751[!

2]!

0.95

628[!

5]0.

0000

00.

0000

06.

0!

0.15

!0.

1255

7[!

2]!

0.46

025[!

6]!

0.92

744[!

2]!

0.12

852[!

4]!

0.12

093[!

2]!

0.33

475[!

5]0.

0000

00.

0000

07.

0!

0.80

0.94

744[!

3]0.

7196

4[!

5]!

0.14

305[!

1]!

0.15

193[!

4]0.

1037

3[!

2]0.

1259

7[!

5]0.

0000

00.

0000

08.

0!

0.51

0.91

817[!

3]0.

6486

6[!

5]!

0.96

775[!

2]!

0.91

850[!

5]0.

9924

7[!

3]0.

1355

6[!

5]0.

0000

00.

0000

09.

0!

0.28

0.54

639[!

3]0.

4113

9[!

5]!

0.59

390[!

2]!

0.60

693[!

5]0.

5916

1[!

3]0.

9517

5[!

6]0.

0000

00.

0000

0

v 1"

v 2"

v@ 1"

0,v@ 2

"3

j 1j 2

"¸"

0221

2.5

158.

02!

0.87

853

!0.

5401

4[!

2]!

0.11

297[

#2]

!0.

8837

0[!

3]0.

0000

00.

0000

00.

1570

4[#

2]!

0.22

676[!

2]3.

0!

114.

54!

0.26

079

!0.

5940

3[!

2]!

0.54

361[

#1]

!0.

1674

9[!

1]0.

0000

00.

0000

00.

7573

6[#

1]0.

1207

9[!

1]3.

5!

120.

73!

0.72

423[!

1]!

0.42

762[!

2]!

0.21

550[

#1]

!0.

1410

4[!

1]0.

0000

00.

0000

00.

3382

4[#

1]0.

1090

7[!

1]4.

0!

74.3

80.

1331

5[!

2]!

0.24

168[!

2]!

0.98

767

!0.

8296

9[!

2]0.

0000

00.

0000

00.

1611

7[#

1]0.

6578

4[!

2]4.

5!

45.2

00.

2216

1[!

1]!

0.13

050[!

2]!

0.42

766

!0.

4753

4[!

2]0.

0000

00.

0000

00.

7208

00.

3848

9[!

2]5.

0!

27.1

20.

2307

2[!

1]!

0.68

716[!

3]!

0.17

975

!0.

2672

0[!

2]0.

0000

00.

0000

00.

3052

00.

2197

5[!

2]5.

5!

15.9

60.

1843

7[!

1]!

0.34

848[!

3]!

0.80

167[!

1]!

0.14

606[!

2]0.

0000

00.

0000

00.

1250

00.

1215

1[!

2]6.

0!

9.43

0.13

355[!

1]!

0.17

034[!

3]!

0.41

143[!

1]!

0.78

769[!

3]0.

0000

00.

0000

00.

4921

7[!

1]0.

6607

4[!

3]7.

0!

3.70

0.67

348[!

2]!

0.37

175[!

4]!

0.17

057[!

1]!

0.24

498[!

3]0.

0000

00.

0000

00.

5278

0[!

2]0.

2075

7[!

3]8.

0!

1.91

0.36

829[!

2]!

0.97

664[!

5]!

0.96

676[!

2]!

0.10

579[!

3]0.

0000

00.

0000

0!

0.13

174[!

2]0.

8920

4[!

4]9.

0!

1.00

0.19

938[!

2]!

0.39

011[!

5]!

0.57

195[!

2]!

0.52

865[!

4]0.

0000

00.

0000

0!

0.87

266[!

3]0.

4441

9[!

4]


Fig. 2. Functions b(s0 )j1j2"L(R) for v

1"0, v

2"0,

v@1"1, v@

2"2. Labels denote the expansion parameters

j1j2"¸; the minus sign indicates negative b(s0 )j1j2"L

values.The markers represent the long-range, classical purelyquadrupole-induced contribution, circles: j

1j2"¸"

!2023; and boxes: j1j2"¸"0223.

Fig. 3. Functions b(s0 )j1j2"L(R) for v

1"0, v

2"0,

v@1"0, v@

2"3. Labels denote the expansion parameters

j1j2"¸; the minus sign indicates negative b(s0 )j1j2"L

values.The markers describe the long-range, classical quadru-pole-induced contribution, circles: j

1j2"¸"2023; and

boxes: j1j2"¸"!0223.

Table 3Fitting parameters, of b(s0 )j1j2KL

(R) (in a.u.) for the second overtone band

j1j2K¸ n B

1B2

B3

B4

v1"0, v

2"0, v@

1"0, v@

2"3

0001 7 !0.324727 0.94377 10~5 !0.91567 !0.311022023 4 0.005854 0.31768 10~5 !1.54759 !0.041820223 4 0.019459 0.49026 10~5 !1.46861 !0.053352021 7 0.171489 !0.14823 10~6 !3.15735 !0.343850221 7 !3.175620 1.78181 10~5 !1.95641 0.07598

v1"0, v

2"0, v@

1"1, v@

2"2

2023 4 0.009561 !0.185631 10~4 !0.71801 0.062180223 4 !0.029781 0.337341 10~4 !0.69671 0.060552021 7 !0.410787 0.365088 10~5 !1.01774 0.084880221 7 0.519855 !0.293648 10~5 !1.65565 !0.03110


Fig. 4. The CIA spectra of normal H2

(C1!3!

"0.25) in the second overtone band at 85 K. Solid line: theoretical datawithout j dependence; dashed line: theoretical data with j dependence; circles: experimental data by McKellar and Welsh(1971).

Fig. 5. CIA spectra of H2

(C1!3!

"0.57) in the second overtone band at 85 K. Solid line: theoretical data withoutj 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. Mostrecently, measurements at 77.5 and 298 K have been performed, see [14], and the experimentalresults have been compared with the theory outlined here.

The analysis presented by McKellar and Welsh [12] of their measurements [12], is based on thepurely multipole-induced dipole theory. The authors point out that the unexplained experimental


Table 4Assignment of peaks, counting from left to right

Peaks Roto-vibrational transitions Vibrational bands

First Q3( j"0,1,2)#Q

0( j"0,1,2) (3}0)

Second Q3( j"0,1,2)#S

0( j"0) (3}0)

S3( j"0)#Q

0( j"0,1,2) (3}0)

Third Q3( j"0,1,2)#S

0( j"1) (3}0)

S3( j"1)#Q

0( j"0,1,2) (3}0)

Q1( j"0,1,2)#Q

2( j"0,1,2) (1}0)#(2}0)

Fourth S1( j"0)#Q

2( j"0,1,2) (1}0)#(2}0)

Q1( j"0,1,2)#S

2( j"0) (1}0)#(2}0)

Fifth Q1( j"0,1,2)#S

2( j"1) (1}0)#(2}0)

S1( j"1)#Q

2( j"0,1,2) (1}0)#(2}0)

spectral intensity in the Q3

( j"0,1) region may be arising from the unknown quantum overlapterm, j

1j2"¸"0001. They further speculate that Q lines (arising from the pure overlap,

j1j2"¸"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] isthat the j

1j2"¸"0001 term is absent for only in 0P2 band for some particular reason. In our

work we "nd out that j1j2"¸"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 thej1j2"¸"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 theexperimental data we observe "ve distinctive peaks at frequencies below 13,000 cm~1. In order tounderstand better the "ve peaks in Figs. 4 and 5, in Table 4 below we make assignments of therotovibrational transitions contributing to each peak.

As we expected, the relative discrepancies between theoretical and experimental spectra arelarger for the second overtone than was observed for lower overtone bands. Our computationalresults are in much better agreement for double vibrational transitions, v

1v@1v2v@2"0102 (the last

two peaks), than for the single vibrational transition, v1v@1v2v@2"0003 (the "rst two peaks). In

Figs. 4 and 5 we present theoretical results obtained by assuming, or neglecting the j-dependence. Itturns out that whereas the j-dependence is especially weak for double vibrational transition, it isnon-negligible for a single vibrational transition and should not be ignored.

In order to understand better the reasons for the observed discrepancy between our theoreticalresults and the measured data, we attempted to estimate the uncertainty of our computations ofa(u;¹) at various frequency ranges, by relating it to the inaccuracy introduced by the functionsbj1j2"L

(R). Accordingly, we tried to estimate the uncertainties of functions bj1j2"L(R) for this band.

We compute asymptotic values of the two main terms b(s0 )2023

(R) and b(s0 )0223

(R) at R"9.0 a.u. fromEqs. (9), using matrix elements SvjDaDv@j@T and SvjDq

2Dv@j@T from Ref. [24]. The comparison between


Table 5Leading functions b(s0 )j1j2KL

(R"9.0 a.u.) for the second overtoneband (in 10~6 a.u.). Upper line: asymptotic values, lower line:our results

v1"0, v

2"0, v@

1"0, v@

2"3 v

1"0, v

2"0, v@

1"1, v@

2"2

b2023

b0223

b2023

b0223

1.27 !2.28 !1.65 2.181.2 !2.9 !2.0 2.2

the asymptotic values and our results are given in Table 5. The terms with j1j2"¸"4045, 0445

and 2233 of the second overtone band are negligible, and therefore are not included in Table 5.For b(s0 )

0223(R) (v@

1"0, v@

2"3) and b(s0 )

2023(R) (v@

1"1, v@

2"2), the relatively larger discrepancies

between our results and the asymptotic values than those observed for lower overtones (comparewith Table 1) indicate somewhat less accurate bj1j2"L

(R) functions. This is, in our opinion, mainlydue to the very small bj1j2"L

(R) values of this band, about one order of magnitude smaller thanthose of the "rst overtone band and therefore resulting in larger relative errors. For example, atR"9.0 a.u., A

2023(R, r

1, r

2) is of the order 1000]10~6 a.u., while its radial matrix element

b(s0 )2023

(R)"Sv1j1v2j2DA

2023(R, r

1, r

2)Dv@

1j@1v@2j@2T becomes only about 1]10~6 a.u. for vibrational

transition v1"0Pv@

1"0, and v

2"0Pv@

2"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. Atthe collisional diameter (approx. 6. a.u.) the ratios of the dipoles are: &400, &65, &6.5, respec-tively, the approximate values correspond to averages between 2023 and 0223 terms. Having inmind that the absorption intensity is proportional to the square of the dipole, we can see that theRT spectra are roughly 1}2]105 more intense than the spectra of the second overtone band (fordetails, see [14]). Even compared to the "rst overtone, it appears that the second overtone is about30}36 times less intense. That, of course, a!ects the accuracy of the "nal results.

For such small values of b(s0)j1j2"Lthe major contributions to the uncertainty are the "tting in

Eq. (6), and the two-dimensional integration procedures. Another reason for the smaller accuracyof the bj1j2"L

(R) functions of the second overtone, than what was obtained for lower overtones, isthe inclusion of higher vibrational states (v"2 and 3) in the computations of this band. For highervibrational states of H

2, the internuclear distances of H

2beyond r

``"2.15 a.u. become more

important. Here the uncertainties in the "tting of Aj1j2"L(R, r

1, r

2) in Eq. (6), become especially

large. Very good agreement between our results and the asymptotic values forb(s0 )2023

(R) (v@1"0, v@

2"3) and b(s0 )

0223(R) (v@

1"1, v@

2"2) must be considered somewhat coinciden-

tal because of the above-mentioned computational uncertainties. It may, however, well be that atthe largest intermolecular distance considered here, R"9.0 a.u., the computed dipoles do not yetreach their asymptotic values (see Figs. 2 and 3).

There is no unique value of the relative error for bj1j2"L(R), because it depends upon the absolute

value of bj1j2"L(R) at various intermolecular distances R, though the absolute error of

Aj1j2"L(R, r

1, r

2) and the orthogonality of the wave functions SvjDv@j@T can be estimated. This


implies that the relative errors for di!erent bj1j2"L(R) are not even the same (but depend upon the

induced dipole intensity and intermolecular distance R). Therefore, it is very di$cult to estimate thetheoretical error of Gj1j2"L

(u), as well as of the theoretical spectra a(u;¹).Furthermore, it is known that the intermolecular potential <vv{

0(R) depends on the vibrational

states v, v@ of both of the hydrogen molecules. Thus, for the second overtone band we are dealingwith, potentials <03

0(R) and <12

0(R) need to be known for the "nal states (the superscripts &03' and

&12' denote the "nal vibrational states v@1"0, v@

2"3 and v@

1"1, v@

2"2 for single and double

vibrational transitions). Unfortunately, at present, such v-dependent potentials as <030

(R) and<12

0(R) are not available for our purpose. By using <00

0(R) instead, the integrated spectral

intensities M(s0 )0_j1j2"L

, which depend only on the initial state potential, remain una!ected, while thespectral line shapes (related to M(s0)

1_j1j2"Land M(s0 )

2_j1j2"L) become slightly inaccurate [28,29].

Finally, whether the isotropic potential approximation is justi"ed when vibrational states ashigh as v"3 are involved remains for future investigation. It may well be, that for collisions withH

2molecules in higher vibrational states, the intermolecular potential becomes signi"cantly more

anisotropic, due to a sizeably larger internuclear stretch between the two hydrogen atoms.As a conclusion, the relatively smaller accuracy of the bj1j2"L

(R) functions is the main source ofthe apparent sizeable discrepancy. The neglect of the vibrational state dependence of the interac-tion potential, which causes slight distortions of spectral pro"les, may play a less signi"cant role. Itis worth mentioning that also the experimental uncertainties become larger when spectral inten-sities become smaller, though they are not explicitly given in Ref. [12]. Despite the relatively largerdiscrepancies, our results agree with the experimental data within 20}30% at most frequencies.Further discussions about the possible sources of numerical inaccuracies are given in [14].

Acknowledgements

Support by NASA, Planetary Atmospheres Division, and by NASA, Astrophysics TheoryProgram, are gratefully acknowledged by the authors. Two of the authors (A.B. and Y.F.) wouldlike to thank The Niels Bohr Institute, University Observatory, for the generous hospitality theyexperienced while working on this paper. We acknowledge Gaussian Inc. for the license agreementof Gaussian 92. C.Z. would like to thank Dr. Mark Cybulski for his patience and help in using theGaussian 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. MTPInternational 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. AstrophysJ 1969;156:989}1005.

[3] Lenzuni P, Cherno! DF, Salpeter EE. Rosseland and Planck mean opacities of a zero-metallicity gas. AstrophysJ 1991;76 (Suppl):759}801.

[4] Borysow A. Pressure-induced molecular absorption in stellar atmospheres. In: J+rgensen UG. editor. Molecules inthe Stellar Environment, Lecture Notes in Physics, 1st ed., Berlin: Springer, 1994. p. 209}22.


[5] Borysow A, J+rgensen UG, Zheng C. Model atmospheres of cool, low metallicity stars: the importance ofcollision-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- andInteraction-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

2}H

2pairs 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

1(J

1) # Q

1(J

2) # Q

1(J

3) 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

2pairs in the

second overtone band at 298 and 77.5 K: comparison between experimental and theoretical results. J Chem Phys1999;110:4750.

[15] Meyer W, Frommhold L, Birnbaum G. Rototranslational absorption spectra of H2}H

2pairs 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, DefreesDJ, Baker J, Stewart JJP, Pople JA. Gaussian 92, Revision g.2. 1992.

[18] Meyer W, Frommhold L. Collision-induced rototranslational spectra of H2}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 H2}H

2. Chem Phys 1984;86:339}47.

[28] Moraldi M, Borysow A, Frommhold L. Rotovibrational collision-induced absorption by nonpolar gases andmixtures (H

2}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 absorptionbands of nonpolar gases and mixtures (H

2}He). Phys Rev A 1987;36:4700}3.


Quantum mechanical computations of collision-induced absorption in the second overtone band of...

Documents

Transcript of Quantum mechanical computations of collision-induced absorption in the second overtone band of...