Theory of pressure-induced vibrational and rotational absorption of diatomic molecules at high temperatures

1. Quant. Spectrosc. R&r.

Trm~fer. Vol. I 1, pp. 131 l-1 330. Pergamon Rcss 1971. Printed in Great Britain

THEORY OF PRESSURE-INDUCED VIBRATIONAL i’ ROTATIONAL ABSORPTION OF DIATOMIC MOLECl AT HIGH TEMPERATURES R. W. PATCH Lewis Research Center, Cleveland, Ohio 44135 (Received 12 November 1970) Abslraet-A derivation is given for the integrated absorption coefficient of pressure-induced pu and vibrational transitions in binary collisions of homonuclear diatomic molecules of the same chet The previously neglected effects of excited vibrational states, mechanical anharmonicity, and vibrat interaction are taken into account to obtain more accurate absorption coefficients at high tempen region of the fundamental wave number the excited vibrational states make more of a contrib absorption than their relative population would lead one to expect.

LIST

e

G, (i = 1,2,3,45) h fi (i = 1 2 . . ,9) J, (i = i, 2j J; (i = 1,2) K IICICl
OF SYMBOLS

function for molecule k involving integral of normal&d associated Legend linear absorption coefficient for wave number B excluding stimulated emisr integrated Einstein coefficient of absorption, m2 J- ’ set- ’ spectral Einstein coefficient of absorption, m3 J-i set-’ expansion coefficient for rf, C m speed of light, m see-’ expansion coefficient for pi, C product of two expansion coefficients, units vary charge of electron, C functions of Q, A and I, C2 ms Planck’s constant, J set h/2n, J set dipole moment integral, units vary total angular momentum quantum number for initial state of molecule i total angular momentum quantum number for final state of molecule i expansion coe5cient for K, C m Boltxmann constant, J K-i functions of J’ and J functions of Q, A, C and D, C m quantum number for component of total angular momentum along .r 81 state of molecule i quantum number for component of total angular momentum along z axis I of molecule i mass of hydrogen atom, kg reduced mass of oscillator, kg number of molecules number density of molecules, mm3 1311

1312 OAJi)(i = 1,2) P(v,, Ji, m,) (i = 1,2) P

Q,(k = 1.2) Q1(JiHi= I,21

qi (i = 1,2) 4xX94rr 7422 R r,(k = 1,2) SAJ,)(i = 1, 2) s T V vi (i = 1,2) vi (i = 1,2) W

x y,z x9 Y, z

Xi Y ZY zi 2 (i = 1,2) Ak (k = 1,2) Si (i = 1,2) a(J’, J)

PC,, P’y9Per

pi (i = 1,2)

_

vo

2 zi(i = 1,2) z,,,,, (k = 172)

Superscripts 0 I,

R. W. PATCH 0 branch for molecule i (Ji -Ji = - 2) probability of molecule i being in state with quantum numbers having values oi, Ji, mi number of pairs of molecules vibrational overlap integral for molecule k Q branch for molecule i (Ji - Ji = 0) scalar quadrupole moment of molecule i, C m2 elements of quadrupole moment tensor, C m* intermolecular distance, m internuclear distance of molecule k, m S branch for molecule i (J; - Ji = 2) integrated absorption coe5cient of transition with wave number removed, m- ’ temperature, “K time, set volume, m3 vibrational quantum number for initial state of molecule i vibrational quantum number for final state of molecule i transitiondependent factor in s, Cz m* Cartesian coordinates with origin at the midpoint of the line connecting two nuclei of molecule, with Z axis running along the internuclear axis, m Cartesian coordinates fixed in space (Fig. 1), m position coordinates of electrons, m spherical harmonic number of elemental charges on the jth nucleus z coordinate of ith electron, m z coordinate of jth nucleus,~m average polarixability of molecule i, Cz m2 J- ’ vibrational matrix element for molecule k, m anisotropy of polarixability of molecule i, C? m* J- 1 Kronecker delta function force constant for Lennard-Jones potential, J electric permittivity of free space, C2 N- 1m- * normalized associated Legendre function polar angle of molecule i, rad components of electric dipole moment, C m &@ri evaluated at rr = ry and r2 = ri, C pz for configuration 4, C m components of electric dipole moment matrix elements, C m photon wave number, m-’ fundamental wave nurnber of vibration, m- ’ radiation energy density per unit wave number, J m-’ element of volume in configuration space, m3 intermolecular potential averaged over orientations, J azimuthal angle of molecule i, rad wave function, units vary vibrational wave function of molecule k, m- ‘P

1

equilibrium internuclear distance of diatomic molecule alar, evaluated at r, = ry reduced

Subscripts 0 I

electronic nuclear

5 <’ P

B’

vl, vz , J, , J, collectively v; , II;, J; , J2 collectively vr , v2,J1, J, , m, , m, collectively , , , I v, , v2,J1 , J2, rn; , m2collectrvely

Pressure-induced vibrational and rotational absorption of diatomic molecules at higb temperatures

1313

INTRODUCTION

RELIABLEestimates of pressure-induced absorption coefficients for high temperature, high pressure gases containing diatomic molecules are needed for radiative transfer calculations for gaseous-core nuclear rockets, late-type stars, and entry into certain planetary atmospheres. Pressure-induced transitions occur in the infrared and microwave regions and are classed as translational, rotational, or vibrational according to what degree of freedom absorbs most of the energy of the photon. This paper is restricted to rotational and vibrational transitions because these are the only important pressure-induced transitions at high temperature. It is further restricted to transitions involving binary collisions between two homonuclear diatomic molecules. A number of theories of pressure-induced rotational and vibrational absorption of colliding diatomic molecules have been published in the past. Only the more significant ones are discussed here. VAN KRANENDONKand BIRD(~) gave a theory for the vibrational absorption of pure hydrogen and deuterium. However, except for their calculations involving the electronic wave function, it is also applicable for other homonuclear diatomic molecules at other than cryogenic temperatures. BRITTON and CRAWFORD(‘) refined VAN KRWEND~NK and BIRD’S(~)calculation by including components of the electric dipole moment perpendicular to the intermolecular axis, as well as other refinements not of interest here. VAN KRWENDONK(~*~)extended his vibrational theory to cryogenic temperatures by considering quantum effects in the translational motion of the gas molecules. COLPAand KETELAAR(‘)gave a theory for the pressure-induced rotational absorption of diatomic molecules for other than cryogenic temperatures and for only the quadrupole-induced part of the dipole moment. VAN KRANENDONK and KISS@)presented a theory of the rotational absorption of diatomic molecules for any gas temperature and included all significant contributions to the dipole moment. The highest temperature in any of these calculations was 370”K, so vibrationally excited states, mechanical anharmonicity, and vibration-rotation interaction were neglected. However, at temperatures an order of magnitude higher these effects would be expected to be significant. This paper presents a derivation taking previously neglected vibrationally excited states, mechanical anharmonicity, and vibration-rotation interaction into account. It is a revision and extension of VAN KRANEND~NKand BIRD’S(‘) and BRI~TONand CRAWFORD’S(‘)theory. The most common current definition of the integrated absorption coefficient is used. Stimulated emission, which has frequently been omitted or included incorrectly, is included. The anisotropy of the polarizability is included correctly for vibrational transitions for the first time. Quantum effects in the translational motion of the molecules are neglected, so the theory is only applicable(4’ for reduced temperatures T’ = kT/c greater than approximately 5 (symbols are given in the List of Symbols on pp. 1311-1312. The ratios c/k have been tabulated by HIR~CHFELDER et al.“’ For H, this paper should be valid for temperatures of 185-7000°K.

ANALYSIS In this section an expression for the integrated absorption coefficient of a rotational or vibrational transition is given for the cases with the chemical species of the two molecules alike or different. The problem is then simplified by assuming that the two molecules are

1314

R. W. PATCH

the same chemical species in the same electronic states. Final results for the integrated absorption coefficient for like chemical species are given in terms of the intermolecular potential, temperature, number densities, expansion coefficients of the dipole moment and its derivative, vibrational overlap integrals, and vibrational matrix elements. Absorption coejficient in terms of Einstein coejficient Consider a gas containing homonuclear diatomic molecules. We are interested in the pressure-induced rotational or vibrational absorption occurring during binary collisions of pairs of these molecules. There are three possible cases : (1) two molecules of different chemical species, (2) two molecules of the same chemical species with all four isotopes the same, and (3) two molecules of the same chemical species but with one molecule containing one isotope and the other molecule containing a different isotope. At the beginning all three cases are considered collectively. In specifying an Einstein coefficient we do as customary by neglecting electron spin and assuming neither molecule has any electronic orbital angular momentum along its internuclear axis. Let vi, Ji and mi be the vibrational quantum number, total angular momentum quantum number, and quantum number for the component of total angular momentum along the intermolecular axis, respectively, for the ith molecule. A state p of the pair thus has the quantum numbers ul, u2, J1, J2, m, and mz provided the molecules are far enough apart. We define a spectral Einstein coefficient of absorption B,,,, in such a way that the probability of a pair of molecules in state p, exposed to radiation of wave number i;, absorbing a quantum hci; in time interval dt and wave number interval dii and making a transition to state p’ is given by B,,,,pS dt dv”,where pJ is the radiation energy density per unit wave number. The relations between different Einstein coefficients and absorption coefficients are t8) the linear absorption coefficient a, obtained next. According to CHANDRASEKHAR, (excluding stimulated emission) is related to the spectral Einstein coefficient by a, = P~B~~TWV

(1)

where V is the volume of homogeneous gas considered, pP is the number of pairs in state p and h is Planck’s constant. The integrated Einstein coefficient of absorption for the transition p + p’ is BP,. =

J

B,,., dv”.

(2)

0

The integrated absorption

coefficient with the wave number removed is defined by

(3) Combining equations (IH3), %f = pphBppJV.

(4)

Pressure-inducedvibrational and rotational absorption of’diatomic molecules at high temperatures 1315

However, BP,,, actually depends on the intermolecular distance R of the pair of molecules (Fig. l), so equation (4) must be corrected by averaging over R. When only the interactions of a single pair of molecules at a time are important, the averaging may be accomplished byt9’

s

PPh B,,,

SPP’= p

(5)

e-“lhT4nR2 dR

Molecule 2

Molecule 1

‘--x-z

plane

FIG. 1. Cartesian and polar coordinates for collisions of two homonuclear diatomic molecules. Nuclei are located at 4, b, c and d. The z-axis passes through the midpoints of the molecules. The angles O1 and 0, are polar angles. The angles cpl and cpz are azimuthal angles measured in the x-y plane.

where Q, is the intermolecular potential averaged over orientations so that it depends only on R. For neutral molecules @ decreases rapidly with increasing R, so the integration may be extended to R = 00 without loss of accuracy. More useful versions of equation (5) are obtained by replacing pP by number densities and summing. The number of pairs of molecules of the same chemical species and same isotopes is N(N- 1)/2, where N is the number of such molecules in volume V. For N >> 1 this is essentially N2/2. However, if the molecules in a pair differ chemically or isotopically, the number of pairs is N,N,, where N1 and N, are the numbers of the respective kinds of molecules. The quantity pP is thus PP +

(N2/2)P(~~,J1,~1)~(~2,J2,~2)

(64

pp =

N,N=P(~1,J1,ml)P(vz,Jz,mz)

W

depending on whether the molecules are alike or different, where P(v, J, m) is the probability of a diatomic molecule having quantum numbers with values v, J and m. States with the same vl, v2, .I1 and J2 but any values of mt and m, alI have the same energy. Likewise, states with the same vi, vi, J; and J; but any values of ;ri; arid m; all have the same energy. Consequently, transitions between these primed and unprinmd states all

1316

R.

W. PATCH

have the same wave number. Thus for convenience their integrated absorption coefficients [equation (5)] may be added.

X

s

BP,* e-“lkT4nR2 dR

(74

0

sc5,

=

n,n,h

z

$

5

z

P(v1,J,,m,)P(v2,J2,m2)

m

X

I

B,,. e-“1kT41tR2 dR

VW

0

where subscript [ stands for the quantum numbers ul, J1, v2 and J, ; a prime indicates quantum numbers after the transition ; and number densities are given by n = N/V, nr = N,/V and n2 = N,/V. Wave functions and expansions

The Einstein coefficient and hence s,,r.may be expressed in terms of the wave functions and expansion coefficients of the dipole moment. First a set of coordinates are needed. We neglect translation of the molecules and choose Cartesian coordinates fixed in space half way between the molecules (Fig. 1). The integrated Einstein coefficient is related to the components &&, etc. of the electric dipole matrix elements by(“)

where so is the electric permittivity of free space, c is the velocity of light, and the equation is in SI units. The x and y contributions to equation (8) can be expected to be considerably smaller than the z contribution for two reasons: (1) The electric quadrupole moment of each molecule may induce a dipole in the other molecule. The average magnitudes of the x and y components of the electric fields due to these quadrupole moments are much less than the z components. (2) The distortion of the orbitals due to overlap is principally in the z direction. Therefore the x and y contributions to equation (8) are neglected here. If equation (8) is substituted into equations (7a) and (7b), we get S[[. =

7c2n2P(vl, J1, O)P(v,, J,, 0) 3hce,

Pressure-induced vibrational and rotational absorption of diatomic molecules at high temperatures

S&

=

1317

2~2~l~2~(~1,J1,0)P(~2,32,0) 3hceo

o

ml=-Ji

mm2=--J~ mj=-Ji

mi=-Ji

where the fact that P(u, J, m) is independent of m has been used. The remainder of the derivation concerns the evaluation of the quadruple sum and the integral in equations (9a) and (9b). The relation between the matrix element @JpVp and the system wave functions $, and I& is

where Zi is the z coordinate of the ith electron, Zj is the z coordinate of the jth nucleus, e is the charge of an electron, eZj is the charge of the jth nucleus, and the summations cover all electrons and nuclei. If the relative motion of the two molecules is slow compared to the motion of the electrons, the BORN-OPPENHEIMER approximation’“’ may be applied to the pair of molecules collectively. where II/, is the electronic wave function, and $. is the nuclear wave function. In equation (11) Xi stands for the position coordinates of the electrons, rr for the internuclear distance of molecule 1, and r2 for the internuclear distance of molecule 2. It has been observed that the wave numbers of lines of pressure-induced transitions can be calculated from term values for isolated diatomic molecules. Consequently, it is reasonable to approximate J/, by”’

where is the vibrational wave function. In the past, harmonic oscillator wave functions have been used for but in this paper mechanical anharmonicity is included by using a wave function calculated from a RYDBWGKLEIN-REES(‘~*‘~)or similar potential energy for an isolated diatomic molecule. In addition, vibration-rotation interaction in the diatomic molecule is included by adding h’J(J+ 1)/8n2m,r2 to the potential energyo4) before calculating the vibrational wave function. In equation (12) &,,, is the spherical harmonic given by (13) @“J

$“J,

YJ&

4)

=

(24-

For m 2 0 the normalized associated and WILSON. For m < 0 they are 0

J&OS

6)

“2@Jm(cos

6)

eh4.

Legendre functions OJ,,, are given by PAULING

=

(-

i)m@

JlmJ(COS

0).

(14)

From equations (10) to (12)

x

sin 8, sin e2 dr, dr, df?, d4, de, d42

(1%

1318

R. W. PATCH

where the z component

of the electric dipole moment is given by -Fezi+zeZjzj i

Because zi is a linear, Hermitian operator, pZ must be real. The dipole moment ,uZmay be expanded in a Taylor series about the equilibrium internuclear distances I’: and r8 of the molecules. C(p(R,rl,rz,e1,~1,e2,~2)

= &R&,&,&,&)

+~L(R,e1,~1,e2,~2)(r1-r~)+~z(R,e,,~,,e2,~z)(r,-r~)

(17)

where pp is the value of u, for p1 is &/ar, , and ~1~is &/drZ, both evaluated for ry and r!. The quantities ,u,”and p1 can be expanded in spherical harmonics.

are expansion coefficients to be determined solely from whereCc,5,~2~2 and Dc1~,~2~2 symmetry and values for cl,”and ~1~for various configurations. Molecular symmetry and conjigurations

Considerable simplification of equations (18) and (19) is possible. Some of the expansion coefficients are zero because of molecular symmetry. The operation

ei + A - ei which is equivalent to interchanging the two identical nuclei, should not change p,” or pi for i = 1 or 2. In addition, ,up and pi should have azimuthal dependence solely on & - & because the orientation of the x and y axes should have no effect. Thus the only nonzero Cd151~Zc2 are Coo~o, GOOO, Coo20, GOZO, Cz--121, GIZ-1, G--222, G-2 and with 4, 2 4 or e, r 4. Also, CZ_121 = CZIZ_l because c(,”must not depend other G1cIG2tz on what direction about the z axis 4r and & are measured. Similarly C,_ZzZ = C222_2. All the above findings apply to the DG,r,d252as well. Further simplification of equations (18) and (19) is necessary because calculations of pZ are time consuming even on a high-speed digital computer. This is especially true if small values of R are involved, which occurs at high temperature. Thus it is necessary to limit the number of configurations 8,) 4,) 8,) q& to as few as possible. The ones selected are shown in Fig. 2. The O,, and O,_, are zero for all these configurations, so C,_iZl, GIZ-I 9 Dz-m 9 and Dz12_1 cannot be determined and are assumed to be zero. There are just enough configurations to determine all remaining nonzero CGIClheland DG1C,G2t2 with 8, < 4 and L’, < 4. Neglect of expansion coefficients with 8, 2 4 and/or /, 2 4 is reasonable because, if such coefficients were nonzero, they would allow transitions with .Ji or JZ changing by f4 or more, which have never been observed.

Pressure-induced vibrational and rotational absorption of diatomic molecules at high temperatures 1319

Configuration

X t

lo----o

End view

Side view

X

Y +A

ZM

a %

%

5

0 Molecule 1

FIG.

Molecule 2

2. Contigurations of two molecules for which p:(R) and p,(R) are required by the theory in this paper.

Additional simplification is obtained by limiting the derivation from this point to the end of the paper to molecules of the same chemical species. For such molecules in the same electronic states, the electronic wave functions are the same to the Born-Oppenheimer approximation(’ ” even if different isotopes are involved. Hence for rI = r! and r2 = rz, pu,is zero for configurations 1,2, and 5 and differs only in sign for configurations 3 and 4. Hence equation (18) gives 1 J5 J5 0= ZC,,,,+~C,,,o+~c,o~o+~~2010+~~I--Z11

J5 J5 o='c ~ctooo-~~,o~,+~zo*o+~~2-2~2 2 ooooJ5 ~oo*o-~Go20+~G-222 -& = ~0000+_zc20004 l4.4=

(20)

J5looo+~oo~o--ICzozo+oCZ--2 ~oooo-y 4

Js

J5

0 = ~oooo-~~ooo-+oozo+~~o~o-~~-~~~. This has the solution Coooo = Czozo = C2_-222= 0, Czooo = -Coo,,.

1320

R. W. PATCH

The derivative pL2can be expressed in terms of p1 because of the assumption of the same chemical species in the same electronic state. To do this, an arbitrary configuration is reflected in a plane bisecting the intermolecular axis, with the result PZ(R,~I,&,&,&)

= -~1(R,~-8,,~,,~-_8,,~,)

(21)

where the order of the arguments of p1 has the significance given in equation Substituting equation (21) into equation (17) gives

(19).

CL,= ~zO(R,e,,~,,e2,~Z)+~L1(R,e1,~1,e2,~2)(T1-10) (22)

-~,(R,~-e,,~,,n-e,,~,)(r,-r”). Final results

The final equation for the integrated absorption coefficient is obtained from equations (9a) and (9b) in two steps. The first step is to evaluate the quadruple sum in equations (9a) and (9b) by means of equations (15), (18), (19), (22), and the solution of equation (20). This step is carried out in the Appendix. The second step is to substitute the expression for the quadruple sum [equation (A.lO)] into equations (9a) and (9b) with the result

(234 scg’ =

2n2n,n2P(o,,J,,0)P(Y2,J2,0)W

(23’3

3hceo

where

(24) (25) G,

=

Q:Q~I~+A:Q~~,+Q:A~Z~+~Q~A~Q~~Z,-~Q:QZA~I~-~Q~A~Q~A~Z~

G, = Q:Q&

+

A:Q%

+ Q:A% -2QAQth

+2Q:QAb

-2QAQzAJ,

co

Ii =

s

(27)

(28)

= &Qz--QdU2(b+2W

6,

(26)

e-‘ikTEi(R)41rR2 dR

(i = 1,9).

(29)

0

Ei

is Diooo,

Diooo,

%OZO,

Di020,

Dk-222,

C$OOO,

CZOOODZOOO~

C200oDoo20

and

D200oDoo20

for i = 1 to 9, respectively. The functions Lo(J’, J) and L,(J’, J) are given by equations (A.ll) and (A.12) and limit changes of Jr and .I2 to 0 or +2 during a transition. The vibrational overlap integrals Q1 and Q2 are given by equation (A.4). The vibrational matrix elements A1 and A2 are given by equation (A.5). Equation (23a) applies to homonuclear diatomic molecules of the same chemical species with all isotopes the same. Equation (23b) applies to the absorption that may occur when homonuclear diatomic molecules of the same chemical species but with different isotopes collide. Different

Pressure-induced vibrational and rotational absorption of diatomic molecules at high temperatures 1321

isotopes will not only have different P(u, J, 0) but also will have different Qk and Ak. The method one would follow to derive sc5,for homonuclear diatomic molecules of different chemical species should be obvious from this paper. In applying equations (23)-(29) there are several subtleties to note. There is no restriction on o1 or v2, so excited vibrational states may be included. There is no restriction on which u changes or how much it changes. In fact, neither, one, or both may change in a transition. Also, there is no restriction on whether u1 or u2 increases or decreases provided that the wave number of the resulting transition is positive. Neither, one, or both J’s may change. If equation (23a) is applied to a fundamental vibrational transition, the transitions with v1 changing are considered distinct from those with u2 changing. The integrated absorption coefficient sc6*does not include stimulated emission. In accordance with Kirchhoff’s law, stimulated emission should be taken into account by multiplying the linear absorption coefficient by 1 -exp(- hc?/lkT) after considering the line shape. (16)Here v”is the wave number of the photon, which is not, in general, the same as the wave number calculated from the diatomic vibration-rotation energy levels involved. DISCUSSION

In this section equations for specific types of transitions are considered as well as possible simplifications. Comparisons with previous derivations are made, and some omitted effects are discussed. Sources of fundamental data are pointed out. Specific types of transitions

The designation of pressure-induced transitions is not standardized. Here we designate the vibrational part of a transition by (0; -or, I.$ -vJ. An example is a (2-0,1-O) transition. The rotational part of a transition is designated S,(J,)+O,(J,), etc., where subscripts indicate which molecule is involved and the two terms of the designation are otherwise each the same as for nonpressure-induced transitions. The first half of the rotational designation is omitted if neither v1 or Jr changes; the second half of the designation is omitted if neither o2 or J, changes. Pure rotational transitions. These are (vi -v 1, v2 - u2) transitions and may be single or double. For a single S,(J,) transition equations (A.ll), (A.12) and (24) give w

=

3

VI

+

WI

8 For a single S,(J,)

+2w2

+

PJ1+ 3)

(J2

1) G

+

1)J2

1*

(30)

1

(31)

2 + (2J2 + 3)(2J2 - 1)G4s

transition

w= ~
w,

+ 3)

G+ 3

(Ji + OJI (2J,+3)(W1-1)G45

*

For a double S,(J,)+ S,(J,) transition, equations (A.1 l), (A.12), (23) and (24) give a result which strongly disagrees with experiment(“) and we therefore omit the equation for W. This is an unusual case where the x and y components of the dipole moment are quite important. However, double rotational transitions are weak compared to single rotational transitions.

R. W. PATCH

1322

Vihtiod transitions. Single vibrational transitions in general may involve single or double rotational transitions. For single vibrational transitions of the type (0; - uI , v2 - v2) there are nine types of rotational transitions as follows. For a Q1(J1) transition ,=(2J,+WJ2+1)

G

(Jl+wl

+

1 (2J1 + 3)(2J1 - l)Gz

4

(J,+UJ, G3+(2J,+3)(2J,-1)G45

(Jz + 1)Jz +(25,+3)(25,-l)

’

For a S,(J,) transition equation (30) applies. For a Q&J&+S,(J,) applies. For an O,(J,) transition

w= ~(JI-WIPJZ+~) 8

For a Q1(J1)+02(J2)

8

25,-l

- l)J2

(JI + ~)(JI+2)(32

+ l)(Jz + 2)G

(w,+3)(2J2+3)

450

1.

(34)

(35)

transition (36)

transition w = 2 (JI + ~)(JI+2)(J, - lYzG 45’ 16 (W, + 3)(2J2 - 1)

For an 0 l(J1) + S,(J,)

(33)

’

(J,+l)J, G 3 + (25, + 3)(2J, - 1)G45

W = 2 (JI - l)J,(J, - UJ2, 16 (2J1 - 1)(2J2- 1) ti45’

For a S,(J,)+O,(J,)

1

transition W = 9 16

For an O,(J,)+O,(J,)

(25, + 3)(2J2 - l)G45

transition

w = 2 @JI+ l)(J2 For a S,(J,)+S,(J,)

transition equation (31)

(Jz+ UJz

G+ ’

2J1-1

(32)

(37)

transition WE

~(J,-~)JI(J,+~)(J,+~)~ 16 (2J1 - l)(w, + 3)

45’

(38)

Vibrational transitions of the rotational type corresponding to equations (35) to (38) have never been observed, probably because G45 is small. Single vibrational transitions of the type (tlI -vl, o; -uJ also have nine types of rotational transitions. The equations for W are the same as for (Y; - vl, u2 - u2) transitions, but the rotational designations of course differ slightly. For double vibrational transitions of the type (a; - v1, vi -u2) there are again nine types of rotational transitions. However, for Q1(Ji)+Q2(J2) transitions Q1 = Q2 = 0 so the G1 and G,, terms in equation (32) can be omitted. Otherwise W is calculated just as for other vibrational transitions.

Pressure-induced vibrational and rotational absorption of diatomic molecules at high temperatures 1323

Simplifications and their ualidity For the harmonic-oscillator

rigid-rotator

model (lo) for U; 2 vk

Qr = 6(4,4

(39)

Ar = [h(vk+ 1)/47~n,cB,]“26(u;,, uk+ 1)

(40)

where M, is the reduced mass, and v”,is the fundamental wave number of vibration. Hence all overtone and double vibrational transitions are forbidden. Experimentally these are much weaker than the fundamental at all temperatures at which they have been observed. In equation (40) Cois proportional to m; ‘I2 so Ak is proportional to m,““. Thus for a given chemical species Akhas a weak dependence on what isotope the species contains. According to equations (23a)-(28) and (40), the integrated absorption coefficient of single vibrational transitions of type [(vl + l)- vl, v2 - v2] is proportional to v1 + 1, so excited vibrational states should be much more important for wave numbers near the fundamental than their relative populations would indicate. Deviations from the harmonic-oscillator rigid-rotator model are classed as mechanical anharmonicity and vibration-rotation interaction. The effects of these on the energy levels and hence on the wave number of a transition are too well known to warrant discussion. The effects on the vibrational matrix elements and the vibrational overlap integrals are of more interest. The effect of mechanical anharmonicity on the vibrational matrix elements of H, can be seen in Figs. 3 and 4 by comparing the harmonic-oscillator rigid-rotator lines with the vibrating-rotator curves for J’-.l = 0 at J = 0 and is indiscernible in Fig. 3 and moderate in Fig. 4. The effect of mechanical anharmonicity on A is generally larger for larger values of o and o’ so can be expected to be important when excited vibrational states are important. Mechanical anharmonicity has no effect on the vibrational overlap integrals because nondegenerate eigenfunctions of the same potential are orthogonal. This is shown in Figs. 5 and 6 (SPINDLW’S(~~) H2 potential was used for the .5-

- --_

Vibrating rdator (J’ - J as noted) Harmonic oscillator rigid rotator (all J’ - J)

.4-

10 15 20 25 T&alangular momentum quantum number, J

30

I 35

FIG. 3. Vibrational matrix elements for H, for I) = 0, v’ = 1 computed for two models to show the effect of vibration-rotation interaction.

1324

E

“i”” I- ---

R.

w. PATCH

Vibrating rotator (J’ - J as noied) Harmonic oscillator rigid rotator (all J’ - J)

5

10 15 Total angular momentum

I 20 25 quantum number,

J

30

35

FIG. 4. Vibrational

matrix elements for Hz for u = 0, u’ = 0 computed for two models to show the effect of mechanical anharmonicity and vibration-rotation interaction.

vibrating-rotator curves in Figs. 3-6). For ground electronic states of diatomic molecules it is doubtful if larger anharmonicity effects exist for the lowest two vibrational states because H, is relatively anharmonic. The effect of vibration-rotation interaction can be seen in Figs. 3-6 by comparing the vibrating-rotator curves with J-J = - 2,0, and 2 with the J’-J = 0 curve at J = 0.

-.31 0

FIG. 5. Vibrational

overlap

effect of vibration-rotation

I 5

I

1

10 15 20 25 Tdal angular momentum quantum number, J

integrals

30

35

for Hz for u = 0, u’ = 1 computed for two models to show the interaction. The harmonic-oscillator rigid-rotator line coincides with the vibrating-rotator line for J’ - J = 0.

Pressure-induced vibrational and rotational absorption of diatomic molecules at high temperatures

1325

J’-J

----

Vibrating rdator (J’ -J as noted1 Harmonic oscillator rigid rotator (all J’ -J)

20 25 lo 15 Total angular momentum quantum number, J FIG. 6. Vibrational overlap integrals for H, for u = 0, u’ = 0 computed for two models to show the effect of vibration-rotation interaction. The harmonic-oscillator rigid-rotator line coincides with the vibrating-rotator line for .I’- J = 0.

This effect is large in Figs. 3-5, but small in Fig. 6. The effect for J’ -J = 0 is due solely to adding h’J(J+ 1)/87r2m,r2 to the potential energy. For molecules with large m,, the maximum value of J is generally higher, so, if the rotationless potential curve is similar to H,, the effect of vibration-rotation interaction will still be roughly the same but will be shifted to higher J values. The differences between the Y-J = -2 and 2 curves compared to the J’-.J = 0 curves are appreciable in Figs. 3 and 5 and small in Figs. 4 and 6. These differences will tend to become smaller for molecules with larger m,. In any case, the vibration-rotation interaction effects shown in Figs. 3-6 are sufhcient to cast serious doubt on the high-temperature applicability of any pressure-induced absorption theory that neglects these effects. Another possible simplification is to assume that the electric dipole moment that occurs when two homonuclear diatomic molecules collide is due solely to the electric quadrupole moment of each molecule polarizing the other molecule. To obtain the C ~lW2C2and DC1tlclt2 needed in equation (29), p, is expanded in spherical harmonics.

The equations KJJTELAAR.(~)

for the expansion coefficients KdltlC2e2 have been given by COLPA and Obviously CdlCldzt2= KGICIC2G(R,ry, r$, which yields C 20o0 = 3(5)“2q,a2/101rsoR4.

(42)

R. W. PATCH where a2 is the average polarizability moment of molecule 1 defined by 41 =

of molecule 2, and q1 is the scalar quadrupole

--xx

=

-qrr

=

f4z.z.

(43)

Here X, Y and 2 are Cartesian coordinates with origin at the midpoint of the line connecting the two nuclei, with the Z axis running along the internuclear axis. The quantities qxx, qyy and qzz are elements of the quadrupole moment tensor of the diatomic molecule. To find the Ddlcldzr2we differentiate the KG,C,GZCZ with respect to rr giving D,,,, = D,_,,, = 0 and D 2000 = 3(5)“2q;a2/107t&,R4

(4)

D 0020 = - 3(5)“2q2a’;/10rr.s,R4

(45)

D2020

(46)

=

GiP2-~2W/5~~,R4

where a double prime indicates a first derivative with respect to rr , and a2 is the anisotropy of the polarizability of molecule 2. Unfortunately, equations (42) and (44)--(46) neglect the dipole moment resulting from the overlap of the orbitals of the two molecules. This neglect causes a small error in the pressure-induced pure rotational absorption but a large error in the fundamental vibrational absorption. Comparison with previous work Equations (A.ll), (A.12) and (23)-(29) may be compared to previous derivations for pressure-induced rotational and vibrational absorption. For a comparison with COLPA and KETELAAR 's(')derivation of pressure-induced rotational absorption, we assumed a harmonic oscillator rigid rotator, giving Qr = Q2 = 1 and Ai = A2 = 0. This yields G, = G,, = 0 and G2 = G, = I,. Our srr, is then summed over all transitions. This gives COLPAand KJSTELAAR’S result for the pz contribution with three exceptions : (1) COLPA and KETELAARincluded stimulated emission incorrectly, in violation of Kirchhoff’s law for wave numbers different from the line center (the correct treatment was given by POLL(M)). (2) COLPAand KETELAARdefined the integrated specific absorption differently from our c sssS. They did not include f in the denominator as was done in equation (3). This is not necessarily wrong but makes it impossible to apply recent experimental line shapes (l*) to their results in a straight-forward manner. (3) This paper does not have C2_r2r and C2r2-, terms because they could not be determined from the configurations used. For the quadrupole-induced dipole moment in collisions of molecules of the same chemical species, these terms are zero. For a comparison with VAN KRANEND~NKand BIRD’S(‘) derivation of pressureinduced fundamental vibrational absorption, we assume a harmonic oscillator rigid rotator and a (1-0,0-O) transition, giving Qr = 0, Q2 = 1, A1 = (h/2nmHcici;o)“2,and A2 = 0. Van Kranendonk and Bird found D,_,,, negligible so neglected it. Therefore we neglect it in making a comparison. This gives G1 = A:Zr , G2 = A:Z,, G, = A:Z3, and G45 = A:Z,. Next set. is summed over the branch. The sum must be multiplied by two if (l-0, O-O) and (O-O, 1-O) transitions are to be considered collectively, as VAN KRAN~~D~NKand BIRD did. This gives VAN KRANEND~NKand BIRD’S result with two

Pressure-induced vibrational and rotational absorption of diatomic molecules at bigb temperatures 1327

exceptions : (1) VAN KRANENDONK and BIRD defined the integrated absorption coefficient differently from our 2 C sr,.,. They did not include i; in the denominator, as discussed in the previous paragraph. (2) There is a typographical error in their c(J).

E$ects omitted At least two effects were omitted from this paper : electrical anharmonicity and ternary collisions. Inclusion of electrical anharmonicity would mean including second derivatives of p, with respect to r1 and r2 and would greatly complicate calculations. However, HARE and WELSH believe this effect is important in the overtone region. Ternary collisions have been shown experimentally to be important at high densities(‘g’ that have not as yet occurred in any practical radiative transfer problem where the temperature is well above room temperature but low enough that pressure-induced absorption is important.

Sources of fundamental data Fundamental data required to apply the method of this paper include term values, vibrational overlap integrals, vibrational matrix elements, electric dipole moments for colliding molecules, and derivatives of electric dipole moments for colliding molecules. Term values are readily available from numerous sources. Vibrational overlap integrals and matrix elements may be calculated with Franck-Condon factor computer programs such as SPINDL,ER's('~) or ZARE’s.(‘~)The dipole moments and their derivatives can be obtained for large R values by adding the overlap contribution(‘*2’ to the quadrupoleinduced contribution. For small R values ab initio molecular structure calculations are advisable, although it might be possible to fit experimental data obtained at various temperatures for various transitions. Of the fundamental data, the dipole moments and their derivatives are the most diflicult to obtain.

CONCLUDING

REMARKS

The integrated absorption coefficient was derived for pressure-induced pure rotational and vibrational transitions in binary collisions of homonuclear diatomic molecules of the same chemical species. Excited vibrational states, mechanical anharmonicity, and vibration-rotation interaction were taken into account. It was found that excited vibrational states should be much more important in determining absorption at wave numbers near the fundamental than their relative populations would indicate. The effects of mechanical anharmonicity and vibration-rotation interaction on integrated absorption coefficients were not explicitly evaluated, but these effects on H, overlap integrals and matrix elements (which help determine the integrated absorption coefficients) were evaluated. The results of the study were such as to cast serious doubt on the high-temperature validity of pressure-induced absorption theories that do not take excited vibrational states, mechanical anharmonicity, and vibration-rotation interaction into account. Application of this theory to pressure-induced absorption of H2 resulting from vibrational transitions with a net change of + 1 vibrational quanta is given in another paper in this issue.

1328

R. W. PATCH APPENDIX

Evaluation of integrals and sums

The quadruple sum of the squares of the dipole matrix elements in equations (9a) and (9b) can be expressed simply in terms of vibrational overlap integrals, vibrational matrix elements, expansion coefficients, and Kronecker delta functions. The first step is to combine equations (15), (18), (19), (22) and the solution of equation (20) with the result

where

(A-3)

The quantities Qr and Ak are the vibrational overlap integral and vibrational matrix element, respectively, for the isolated diatomic molecule k. The quantity 6(m;, j+ mk) is the Kronecker delta function. The A,,(K) are readily evaluated by using the propertiesC2’) and orthogonality of the normalized associated Legendre functions. A,,(k) = 2- 1’26(m’, m)6(J, J) A2o(k)

=

,/lO

4 -N-

3

[(J+1)2-m21[(J+2)2--21

(25+ 1)(W+3)2(2J+5)

04.6)

1 ’

1’26(JJ+2)+W(J+1)-6m2 (W+3)(251) 64.7)

Pressure-induced vibrational and rotational absorption of diatomic molecules at high temperatures

1329

1

(J+m+4)(J+m+3)(J+m+2)(J+m+l) (25 + 5)(25 + 3)2(W + 1)

I”

(J-m+4)(5-m+3)(5-m+2)(5-m+l) (W+5)(U,+3)2(2J+1)

where the subscript k is understood to apply to all quantities on the right sides of equations (A.6HA.9). Squaring and summing equation (A.l) and making use of equations (A.6HA.9) gives

ml=-J1mi=-Jinl=-J=mi=-Ji + %%J;

> JlK,(J;

9 J2) +wi

+ 2W~,Vl,

J&2&,

J2)

(A.lO)

where I@,

J) = [(25 + 1)/2]6(5’, .I)

(A.1 1)

(A.12)

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

J. VAN KRANENDONK and R. B. BIRD, Physica 17,953 (1951). F. R. BRITM~Nand M. F. CRAWPORD,Con. J. Phys. 36,761 (1958). J. VAN KRANENDONK, Physica 23,825 (1957). J. VAN KRANHWONK, Physica 24,347 (1958). J. P. COLPAand J. A. A. KKIXLMR, Molec. Phys. 1,343 (1958). J. VAN KRANBNDONK and Z. J. Kiss, Can. .I. Phys. 37, 1187 (1959). J. 0. HJRXHPBLDBR,C. F. CUR- and R. B. BIRD, Moleculur Theory of Gases and Liquids, pp. 1110-l 113, Wiley, New York (1954). S. CHANDB, An Introduction to the Study of Stellar Structure, p. 191, Dover, New York (1939). N. R. DAVIDSON,SmtiEtical Mechanics, pp. 471475, McGraw-Hill, New York (1%2). E. B. WILSON,JR., J. C. DLUUS and P. C. CROSS,Molecular Vibrations; the Theory of Infrared and Raman Vibrational Spectra., pp. 38-290, McGraw-Hill, New York (1955). M. BORN and J. R. OPPENHBQ(BR, Ann. Physik 84,457 (1927).

1330 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

R. W. PATCH

A. L. G. Ram, Proc. Phys. Sot. 59,998 (1947). R. J. SPINDLER,JR., JQSRT9,597 (1%9). R. W. PATCHand B. J. MCBRIDE, NASA TN D-4523 (1968). L. PAULINGand E. B. WILSON,JR., Introduction to Quantum Mechanics, pp. 134-135, McGraw-Hill, New York (1935). J. D. POLL, Theory of the translational effects in induced infrared spectra, Ph.D. Thesis, Univ. of Toronto, Canada (1960). Z. J. Krss and H. L. WELSH,Can. J. Phys. 37,1249 (1959). J. W. MACTAGGARTand J. L. HUNT, Cm. J. Phys. 47,65 (1969). W. F. J. HARE and H. L. Wem, Can. J. Phys. 36,88 (1958). T. C. JAMES,Astrophys. J. 146,572 (1966). H. A. Bmm and E. E. SALPETER, Quantum Mechanics of One- and Two-electron Atoms, pp. 344-347, Springer-Verlag. Berlin (1957).