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Dipole moment induction in homonuclear diatomic molecules—I. The Coulomb case
J. Quant.
Spectrosc.
Radiat.
Transfer.
Vol.
2, pp.
7~25~239.
PergamonPressLtd. Printedin GreatBritain
DIPOLE MOMENT INDUCTION IN HOMONUCLEAR DTATOMIC MOLECULES--I. THE COULOMB CASE* R. G. BREENE Physical Studies, Inc., 48 Maple Avenue, Centerville 59, Ohio (Received 7 February 1962) Abstract-We here consider the induction of an electric dipole moment in a homonuclear diatomic molecule by the short range forces due to an atom or a properly aligned molecule. We first develop a general form for the distorted molecular orbital and a simple perturbation method of obtaining the distortion coefficients. In applying this to the oxygen molecule we then take the distorting forces as those central Coulomb forces due to the perturber whose charge cloud we suppose undistorted by the Perturbed molei cule. This means that we neglect polartition and exchange. Further we restrict our numerical calculation to those cases where the perturber is either on the molecular axis or at 90” to it. This merely allows us to avoid the use of electronic computing machinery. Finally, the separation-dependent induced moment is averaged over a random distribution of perturber separations, the result being a linear density dependence in the moment. I.
INTRODUCTION
THE
symmetrical distributions of charge in homonuclear diatomic molecules implies that they possess neither permanent nor vibration-induced electric dipole moments. We are interested in the latter for we shall concern ourselves primarily with radiation in t& fundamental bands. The lack of a vibration+luced dipole moment means that molecules such as oxygen and nitrogen will be infrared inactive. The manner in which dipole activity can be induced by external electric fields was first studied by CONDON( He considered the case where the electric field, acting through the molecular polarizability, induced an electric dipole moment. Some years later, HERZBERG@) obtained infrared absorption from hydrogen and somewhat later, CRAWFORD, WELSH and LOCKE(~) obtained infrared absorption from oxygen and nitrogen. Of course, these absorptions could at first be attributed to quadrupole radiations; however, the latter authors demonstrated through pressure variations the probability that they arose as a result of inter-molecular forces. It is rather a short step then from the external electric fields of Condon to the internal electric fields proposed by MIZUSHIMA@). Now what Mizushima did was to assume in, say, oxygen that an electric field existed at the emitting oxygen molecule due to the quadrupole moments of the surrounding oxygen molecules. It was rather a straightforward matter to utilize a method originally developed by Holtsmark(5) and somewhat more succinctly stated by CHANDRASEKHAR@) to determine what this electric field would be. Having done this, Mizushima was able to &nnpute the induced electric dipole moments appealing to the :molecular polarizability. * The research reported in this paper has been sponsored by the Geophysics Research Directorate of the Air Force Cambridge Research Laboratories, Office of Aerospace Research, under Contract Number AF-19(604)-8417. 225
R. G. BREENE
226
Very little improvement on Mizushima’s treatment would appear to be called for, but later considerations indicated that there were additional contributions to the pressureinduced radiation. Specifically, Mizushima’s radiation is due to what might be called long-range forces. Extensive considerations of the short-range forces have been carried out by VAN KRANENDONKand BIRD(7~8) and later, in a more general fashion, by VAN KRANENDONK@~ 1%11). In addition, BRITTON(~~)has considered exchange forces. In the short-range force studies attention has been restricted primarily to the smaller molecules such as hydrogen and helium. In what follows, we consider the case for the larger diatomic molecules. Oxygen will be our specific example. II. THE PERTURBED WAVEFUNCTION FOR A GENERAL ATOM ORIENTATION Our problem is about as follows: We have a homonuclear diatomic molecule such as oxygen or nitrogen. A neutral atom is presumed to be at a relatively close separation from this molecule. We remark that another diatomic molecule aligned away from our original molecule is well approximated by such an atom, This atom now distorts the electronic charge cloud of the molecule through its short-range field. The problem is to describe this distorted wavefunction and, hence, the induced electric dipole moment. Now one could certainly attempt some method of building up the distorted molecular orbital by combining a number of unperturbed orbitals. In so doing, however, one (1) enters on a maze of a calculation in which (2) one rapidly loses sight of the physical conditions of the problem. We have therefore developed a simple wavefunction capable of describing a distorted orbital and adopted some perturbation theory to its obtention. We begin then by choosing the following general form for all of our distorted molecular orbitals :
xt = p+&%
h, ga)lxr’o’
(1)
wherein x&O)is our unperturbed orbital; at is a parameter, andArt,&, cpg)is some function of the spatial coordinates. Let us consider Fig. 1 which serves primarily to establish our coordinates. Suppose our perturber to be located on the molecular figure axis, say, to the right of nucleus “b”. Then if one assumes the field of the perturber to be (1) radial and (2) sourced at the perturber center one can picture the molecular charge cloud distorted along these radials. Further, one can picture the distortion of the charge cloud to be largest along the axis (8 = r) and decreasing both with decreasing “r” and with decreasing+. Further, we can allow the distorting effect to disappear for8 = 42, for this will mean that the cloud to the left of a vertical through the molecular center of symmetry will be undisturbed, an obviously reasonable assumption. This and the general nature of the forces then tell us that, or the on-axis case: OforO < 8 < 3. f(r, 8994 =
(2) r cos 8
for 2 Q 8 S rr. 2
Next, consider Fig, 2. Here we have depicted the general off-axis orientation
of the
Dipole moment induction in homonuclear diatomic molecules---I. The Coulomb case
227
perturbing atom. Again we work on the precept that only the charge cloud hemisphere on the side of the perturber will be affected, and this, as in the preceding paragraph. On this basis we are able to determine the following expression: Ofor-r’G8<~-l? f(r, 8, Y) = r cos 92/(1 -cos2g, sin2 P)r sin 8 cos 9, sin I?
@a) rr = sin-r(cos q sin To) dr
W-0
FIG. 1. The coordinates for the oxygen molecule and its perturher.
FIG. 2. The general off-axis orientation of the perturbing atom.
Equations (1) and (3) yield the perturbed orbitals, and we note that equation (3a) reduces to equation (2) for Ye = 0. The perturbation theory is now developed for the determination of the parameter “a”. We begin by assuming that, as inferred by equation (1): X$ = x’r
Xr’O) + atX’c
= f(r,
s,
p)xP
(W WI
228
R. G. BREENE
from which a first-order equation follows immediately: H’XP + H%IX’Z = E’axz’O) + E~%zsX’$.
(5)
We multiply through on the left by ~$0) and integrate over all space to obtain: (Xi’s’/H’lxr’O’)+ at(xa’O’lHOf(r, 8, ~)/XP’ = ci&O(xP’lf(r, 3, V)lXP’) + E’I.
(6)
Now note that : (xa(O)IHOf(r, 3, ‘PIXP)) = 2 (Xr(O)IHOIX3(O))(X~(O)IflXi(O)) = Jwxcq.f(r,
3, ~)I(XP’)
(7)
since Ito is surely diagonal in the ~a(0). Equations (6) and (7) yield: Efr = (xi’O’IH’[x(‘O’).
(8)
A similar approach yields : (Xj’O’IH’IXP’) ai = (~+~‘lf(r, 9, @lx~‘~‘)(Et~- Ej”j
(9)
Equations (l), (3), and (9) now furnish us with our perturbed wavefunction. In the following sections we apply these equations to a determination of the perturbation of an oxygen molecule by an oxygen atom for the atom (1) on the molecular figure axis and (2) ninety degrees to the figure axis. For more general cases use of electronic computing machinery would be required, and we have attempted to avoid this. III. THE LCAO-MO’s FOR OXYGEN AND THE OXYGEN ATOM POTENTIAL We shall use the molecular orbitals for oxygen as developed by KOTANIet al.@) Thus our primitive symmetry orbitals will bk: 0~1s = Is,+ 1~ = 4
0~1s = Is,-
z@ ---e-a5 cash a~ J ?7 @~ -e-at J 77
lsb = -4
@ a,2s = 2Sa+2sb = Ro FTe-b+f J oU2s= 2&-2sb
os2p =
2pC%a+&ab
= Ro
J
=
(loa)
sinh a~
cash bq -7 sinh bv]
83 --ge-bc[v cash bv - 6 sinh b?]
g Ro -e-bf;[cosh J 7r
7r
uw
(104
bv - .fv sinh bv]
WeI
bq]
Wf)
q
cU2p = 2pCTa-2pCTb= RO ---e-bE[& J
(lob)
cash br)-sinh
Dipole moment induction
in homonuclear
diatomic
625
J
0u2p+ = 2pQ* + 2pPb * = Q&,
---e*~“~([s7r
7rg2p* = 2prra’ - 2pQ* = *Ro
-e*@1/(@-
825
J
7r
molecules-I.
The Coulomb case
l)(l -+)e--b5
cash b7
l)(l -Tz)e-bc
sinh by
229
ww
In these equations ROis the internuclear separation, 2.282 a.u., and b is 2.596. We shall utilize the single determinant form for the molecular wavefunction, although it is, of course, true that the linear combination of as many as fifteen determinants obtained by Kotani et al. is a great deal more accurate. Their use, however, is merely a repetition of the calculation we shall carry out. The orbitals may be obtained from Table 1 as linear combinations of the primitive symmetry orbitals above. The ground state of oxygen is a sCfB state arising from the following electron configuration: (1~s)2(1~~)~(2~9)2(2~U)2(3~~)2(1~~-)2(1~~+)2(1~~-)2(l~~+). This constitutes the orbital information requisite to the calculation. potential for an oxygen atom has been developed by ~~(14915) as: V= -
15.322 + 2 e-15 mr( r)*(
The Coulomb
2 38.330rs-k 18*368r+ 8.930 + - e-s.7isr r
14.092rsf 19*264r+ 13*168+ fl e-+s*sr. r
TABLE~.THE
MOLECULARWAVEFUNCTIONSANDORBITALENERGIESAS GIVEN BY K~TANI et al.
Primitive symmetry
SCF MO
(02&, U 0.5641 -0.3117 0.7500 0.5345
2%? 30s 2% 3OU
Orbital energy (a.u.) --- 1.5219 -0.5564 - 0.9782 0.7392
orbital -
(02P),* Y
0.1677 0.6148 -0.1617 0.9275
This potential contains (1) no non-central terms, (2) no polarization of the perturbing atom by the perturbed, and (3) no exchange terms. Even so, investigation shows it contains more than need reasonably be handled, so we approximate it by: H’ = I’ = - 108.3&-5; (11) This potential appears to reproduce the earlier one reasonably well. IV.
THE
ON-AXIS
CASE
FOR
02-O
The problem is the evaluation of equation (9) where equation (3) has reduced to equation (2). We first obtain the matrix elements offlr, 9, 9). In doing so we consider Fig. 3. The coordinates of the function in equation (2) are referred to the center-of-masse
230
R. G. BREENE
of the molecule. We first transform atom “b”.
the function so that its center-of-coordinates
is an
We now set limits on the integration of the center-of-mass system.
such that it is carried out over the (+ +) quadrant
FIG. 3. The regions of integration involved in the evaluation of the matrix elements of ‘f”.
A typical molecular orbital is of the form:
We consider the exponential centered on “a” as small compared to that centered on “b” throughout the (+ +) quadrant of the center-of-mass system. Thus, .we have simplified the MO’s of equations (10) and Table 1 considerably. Also we ignore all 1s contributions to the orbitals as well as the lcrg and 10, MO’s themselves. We give the following example: (20g1f(r, S))130g) = 0*1758(bg2.vlf(r, 9)10g2s)
+0.1031(crg2~lf(r,
S)10g2p)+0~2945(ag2sJf(r,
(CTg2s)f(r, %))lCTg2s)= ?[/rb’(a-rb ++ . rb2 Sill &drbd&,
8))log2p)
(12a)
cos &&?-26rB etc.
Wb)
As can be seen from Fig. 3 analytic integration is possible over part of the region while numerical integration over the remainder is necessary. We finally obtain the only required matrix elements :
’
(20g,lf(r, 8)130g) = O-26617
(13a)
(2aulf(r,
(13b)
3)130,) = O-11498
We are expressing everything in atomic units.
Dipole moment induction in homonuclear diatomic molecules--I. The Coulomb case
231
The matrix elements of H’ as given by equation (11) are now required. Here again we work on the premise that the coordinates in the MO referred to the nucleus remote from the perturber may be neglected. Then after integration over the azimuthal coordinate one obtains, for example : (14) Equation (14) indicates the desirability formation :
of choosing
the following
coordinate
5 c = -s 2 rb-CT rl= ---=
(15)
rb-r’ -. Rb
Rb
In these coordinates,
trans-
for example, equation (14) becomes:
(c@IH’Ic$2p)
= -
-~~[(1+ac)+(l-ec)]-4c~
1 --cZ)qZ(1 -c2>
[exp(-2SR~)I(~-~)2b3(~-~2)d~d~.
(16)
The general result for the first of the three required matrix elements is now directly obtainable as : (u&slH’~ CJS) =
-1d‘kt$%b4 eXp(-28Rb)
6 45 1 21 +6_ ___ ~ ____ 5 (2SRb)3 + (26Rb)4 + (2SRb)5 + (2aRz?
+
1 1 -5 (26R#’
45 +(26Rb)7. 1
(17)
We relegate the more lengthy expressions for ( crg2SJH’J ~62p) and (o,2pJH’I 0,2p) to Appendix I in which Rb=2b. The information required for the evaluation of the “a(” is now at hand. We have only to determine the a2;#, azgu. It is apparent from equation (9) that: aufl = -mug which arises due to the energies in the denominators of equation (9). We note the absence of any coe5cient for the m-orbitals and hence of any distortion of the s-orbitals. Doubtless the ?r-orbitals will be distorted, and the fact that we do not describe such distortion is of mathematical rather than physical import. It is apparent that in order to distort an orbital we must have another orbital in our basis set over which matrix elements may be taken.
R. G. BREENE
23.2
The set developed by Kotani et al. does not include any orbitals with which our m may interact. Were such orbitals present, the problem would simply be a repetition of the present work. There is some justification for our present neglect in that we should not anticipate too much overlap between the a’s in our set and those which might be added. In order to consider the wavefunction distortion let us write down only the first term in the equation (17): @R b4
A- c2
eXp( -26Rb)
= A(5)2S3Rt,4 exp( -26Rb)
The three factors of importance here are (1) the perturber separation, Rb, (2) the effective nuclear charge (ENC), 6, and (3) the potential exponent (PE), “5”. The perturbation fades out exponentially with Rb, thus assuring us that we are indeed dealing with short-range forces. Now note that 6, the effective nuclear charge, is what basically governs the spatial extension of the molecular orbital. It is the spatial extension of this orbital, rather than of the perturbing molecule which, through 6, strongly affects the decay of the perturbing effect. This is further borne out in the multiplicative factors in the previous equation, the ENC being to a higher power than the PE. One can then conclude that in the distortion of homonuclear diatomic molecules the type of molecule being distorted is a good deal more important than is the type of molecule doing the distorting. We have never actually normalized our distorted wavefunction. This is not of much importance for small values of “at”. Nevertheless, one may normalize through C, and obtain upon dropping a matrix element infs as of second order: c = [l + 2hr(x$J’lf(r, S))l~Jo’)]-*1s.
V.
THE
NINETY
We now suppose the perturbing
DEGREE
CASE
FOR
02-O
oxygen atom to be somewhere
on the perpendicular
to the molecular ‘6gure axis passing thrdugh the motecular center of symmetry. change is, of course, induced in equation 0
The basic
(3a) which now becomes:
“<,<_” 2
2
r cos 9 sin y -rsinBcosp,
-“<,,
2
tlg)
which is directly obtainable from the earlier equation by setting P” equal to ninety degrees. We remark that integration over the azimuthal coordinate eliminates the first term on the right of equation (18). Furthermore, we may integrate over both terms to obtain: f(r, 3) = -2~. sin 8 = -2ad(l
-[2)($-
1)‘
(1%
and there is now no restriction on the polar angle integration. The matrix elements off will not be analytically evaluable unless the factors under the
Dipole moment induction in homonuclear diatomic molecules-I. The Coulomb case
233
square-root sign of the operator itself are approximated in some fashion. In the “71” range, which is from - 1 to 1, one may approximate this radical quite nicely while the exponentials involved in the “P’ coordinate allow one to also arrive at a quite reasonable approximation to this radical. We have utilized the’following expressions: d/(1 -7s
= (l/1-381)(1*381 -sinher])
2/t2 - 1 = 1.2085‘- 0.766.
(20a) (20b)
We now have the quite straightforward matter of evaluating the matrix elements of equations (19) using the approximations of equations (20) and the molecular orbitals as given by equations (IO). The results of these calculations are: (2o,lf(r,
8, r$J)134 = -0~01000
(2 la)
(2c4f(r,
9,434
(21b)
= -0.05151
We now evaluate the matrix elements of the perturbing potential. In general our molecular orbital will, of course, have the form: Cr@,-6Y.+ C*- 8rb where we suppose the constants and factors quite arbitrary. This will mean that our matrix element of the perturbing potential will have the general form: CisJF(r)e-s”e’e-s*~~dT+ C22JF(r)e-sre-2brbdr + 2CiC2JF(r)e-5r exp[ - 6(r, + rb)]dT. In this equation we have represented the perturbing portion of the Hamiltonian by the function in the exponential. It is apparent that, because of the location of the perturber, on the x-axis of Fig. 1, either the first or the second integrals in the above equation will generally contribute the maximum from any given spatial region. Under almost all circumstances, however, the overlap, whose contribution is given by the third integral, will be comparatively smaller because one or the other of the radial coordinates will be large. ,We then suppose that the matrix element be approximated by the sum of the first two integrals; these integrals corresponding precisely to the matrix elements evaluated for the on-axis case. In effect then, for a given separation from one of the molecular nuclei, we may now simply take twice the matrix element value for the on-axis case. However, the actual separation of the perturber from the oxygen molecule is: R = 1/(Rb2-a2).
(22)
We now have the requisite information for the evaluation of the orbital distortion coefficients, and hence of the electronic wavefunction, for the ninety degree case. The general comments which were made in connection with the on-axis case hold here with obvious modifications. VI. THE INDUCED ELECTRIC DIPOLE MOMENTS In general the expression for the electric dipole moment may be written as:
(23)
234
R. G. BREENE
Since we have supposed our molecular wavefunction equation (23) becomes, still in atomic units :
where the xr are the molecular moment is then:
to be a single determinant,
orbitals. The orbital contribution
to the electric dipole
= J~,c”)raxt’o’dT~ + 2arJ J~a’~‘]2f(r, 9, y),)rgdTr +cz~~SIX~(~‘/~S~(~, 8, &r
(25)
The Crst term in equation (25) will be zero in first approximation, and we shall so consider it here. One could probably use this term in introducing the long range distorting forces. We also neglect the third term in equation (25) as of second order in at. Next we recognize the components of the dipole moment as: x = a&?--
l)(l -T#+Qcos 9
(26a)
Y = a&s-
l)(l -7s)
(26b)
sin 9
Z = ah
(264
where “a” is one-half the nuclear separation in the distorted molecule. We are considering now both our on-axis and ninety degree off-axis cases at the same time. It is apparent that in the on-axis case the center of negative charge would be moved along the z-axis and toward the perturber. This means that only a z-component of the electric dipole moment would be induced. Such is found to be the case. In the ninety degree off-axis case we would anticipate that the center of negative charge would be moved up and again toward the perturber thus inducing only an x-component of the electric dipole moment. Such we also find to be the case. The result is that we obtain the following expressions for the electric dipole moment :
Mt = RoW[A+ B+ C- A’- B’ - C’ Jaz, + Ro7S5[D+ E + F]azr,,
GW
The terms A, B, etc. are set forth as Appendix II. The electric dipole moment of oxygen as a function of perturber separation for both cases considered is displayed as Fig. 4. Now in computing the absolute intensity of the dipole radiation emitted by this distorted molecule in its fundamental vibrational band, we consider the molecule as a simple harmonic oscillator. Furthermore, we suppose that its electric dipole moment may be given as the familiar series: M = Mo+M’(R-Ro)+
...
(2’9
where the first term is, of course, the permanent dipole moment responsible for pure rotational spectra and obtained by us as equations (27). The second term will be responsible for the radiation in the fundamental band. We are supposing a total lack of vibronic
Dipole moment induction in homonuclear diatomic molecules-I.
The
Coulomb ease
interaction so that we assume the dipole moment derivative obtainable of equation (27) with respect to Re: M,’
by differentiation
= ;MZZ
M,’ =
235
(294
$4~.
Wb)
This effectively defines the absolute intensity in the induced fundamental-vibrational band of oxygen.(l@
1
’ 2
’
3 Perturber
’
’
’
4
b
5
separation,
’
6
atomic
s units
FIG. 4. The induced electric dipole moment for both cases as a function of perturber separation. In the on-axis case the separation is from the atom in the molecule nearest the perturber. In the off-axis case the separation is from the molecular center-of-mass.
VII.
THE
OBSERVED
ELECTRIC
DIPOLE
MOMENT
FOR
02-O
We have computed the electric dipole moment for various perturber separations. It remains to determine the moment actually observed. In order to do this we must average
236
R. G. BREENE
the moment over some distribution of perturber separations. There is every justification for a random distribution of spatial coordinates, so that we choose dv sin 9dvd9dp, - = ya op(r)dr V (4/3)rrR3
= %dr. R3
Before carrying out this average we must decide what lower limit, that is, what lower separation we are going to use in the averaging integration. This question is automatically decided for us by our perturbation scheme. Surely our perturbation theory-which is indeed of the order-of-magnitude variety-will break down when the magnitude of the perturbation energy approaches the separation of the electronic energy levels. The nearest electronic level in oxygen is alAg which is at about an electron volt, that is, at about 0.04 atomic units from the ground state. From equation (8) we obtain the amount of the energy perturbation as a function of perturber separation. We find that at 3 a.u. of separation the perturbation is just about equal to the level separation while at 3.5 a.u. it has fallen to about one-fifth*. On this basis we take 3.5 a.u. as the lower limit of the averaging integration. Now note that this corresponds to a molecular separation of 5.78 or, about 6 a.u. when we add the internuclear separation in the oxygen molecule. The reader will note the relationship to the cut-off separation used by MARGENAU~~)in his pioneer work on van der Waals’ forces. Next we require a value for “R” in equation (30). This we take as the average intermolecular separation, that is to say, (4/3)77Rs = l/N, where “N” is the density in a gas of pure oxygen. If we now let Mi(u) be the dipole moment component as a function of perturber separation we may write equation (30) as: m Mt = 4rrN
s
Mi(v)rsdr.
(31)
3.5
The value of Mt for the on-axis case is 0.01386 x lo-21A where “A” is the density in atmospheres. This means there will be an electric dipole moment of magnitude 0.000139 x 1O-18e.s.u. induced in oxygen at a density of 10 atm. The value of Mt for the off-axis case is O-57898 x lo-a1 A where “A” is the density in atmospheres. This means a moment of 0.00579 x 10-l* e.s.u. at 10 atm. If we apply the absorption coefficient formulas of our earlier work(is) we get, at 1595 cm-r, an absorption coefficient of 0.14809 x 10-s As cm-l for the on-axis case. At 50 atmospheres this yields an absorption coefficient of 0*000185 cm-l. CRAWFORD,WELSH and LOCKE(~)obtained a quadratic dependence of the absorption coefficient on the density experimentally while MIZUSHIMA(~) obtained such a dependence in his theory of long range forces. The fact that no cubic dependence, such as we predict here, has been observed would indicate that (1) higher than binary collisions and (2) long range forces are influencing the dipole induction down to as low densities as have been observed. VIII. REMARKS In our considerations of this problem we have tacitly a.ssumed a great deal more quiescence than can be justified; specifically, we have ignored rotation on the part of our * We take 4 a.u. for the off-axis case.
Dipole moment induction
in homonuclear
diatomic
molecules-I. The Coulomb case
237
distorted molecule and translation on the part of our perturbing molecule or atom. The !atter approximation holds quite well for the high densities of interest here@. However let us consider the supposition that no rotation of the molecule is taking place. Suppose for a moment that the molecule does rotate in the plane established by the three nuclei. In this case, we can say with some degree of justification, that very roughly, something like half of the time the perturbing atom is on-axis, or approximately so, and half of the time the perturbing atom is ninety degrees to the axis, or approximately so. The result of this, of course, would be that we could treat the oxygen molecule as having just one half the magnitude of the induced z-component of the electric dipole moment while at the same time having just one half the induced component of the x-component. It is apparent that this. is only an approximation, but it should be a reasonable one. Now if we orient matters somewhat differently, we can see that we will approach the same condition only now the molecule will possess a z- and a y-component of the electric dipole moment. In general then and with more or less random orientation one could approximate the situation reasonably well by this assumption : The molecule has induced one half the indicated x-component of the dipole moment and one half the indicated z-component of the dipole moment. Now let us go back and consider the approximations which were made in our selection of the potential. We recall that these had to do with (1) non-central forces, (2) exchange interactions, and (3) the effect of polarizing the perturbing atom. ‘We have shown in a somewhat different context(i5)--and MIZUSHIMA c4) has also remarked-that the noncentral portion of the Coulomb field is probably of very little importance. Perhaps then it would be well to maintain ~this particular central approximation. .-‘. The exchanges of electrons between the two- molecules which give rise to these spin forces have an upper limit. Whether or not we have reached it at our cut-off of 6 a.u., we should have approached it closely enough for the neglect of exchange forces not to be a serious approximation. We are left then with the effect on the potential of the polarization of the perturber’s charge cloud by the distorted molecule. Now our potential is negative, which is to say that it will be attractive to the negative charge cloud in our distorted atom. In like fashion we may expect the potential due to the distorted atom to be attractive rather than repulsive to the charge cloud of the perturbing atom. This means that the perturber’s charge cloud is displaced in the direction of the molecule; which in turn will mean that the attractive potential in which the distorted molecule had existed will be reduced. In our work on the free electron in the field of an oxygen atom, we demonstrated the reverse situation(i*). In that case, the field of the electron is completely repulsive to the atomic charge cloud, so that the charge cloud is pushed away from the electron. This means in turn that the potential will be more attractive which indeed it proves to be. It is apparent that in the present case we encounter the opposite situation. The net result of the introduction of polarization will be to reduce the distortion of the charge cloud and as a result, to reduce the magnitude of the electric dipole moment. Acknowledgements-The author wishes to express his appreciation portant assistance in the extensive numerical calculations.
to NANCY M. BREENEfor her im-
R. G. BREENE
238
1. 2. 3. 4. 5. 6.
7. 8. 9.
10. 11.
12. 13. 14.
15. 16. 17. 18.
REFERENCES E. U. CQNDON,Phys. Rev. 41, 759 (1932). G. HERZBERG,Nature 163, 170 (1949). M. F. CRAWFORD,H. L. WELSHand J. L. LOCKE,Phys. Rev. 75, 1607 (1949). M. MIZUSHIMA,Phys. Rev. 76, 1268 (1949). R. G. BREENE,JR., The Shif and Shape of Spectral Lines. Pergamon Press, London (1961). S. CHANDRASEKHAR, Rev. Mod. Phys. 15, 1 (1943). J. VAN KRANENDONKand R. B. BIRD, Physica 17, 953 (1951). J. VAN KRANENDONKand R. B. BIRD, Physica 17, 968 (1951). J. VAN KRANENWNK, Physica 23, 825 (1957). J. VAN KRANENDONK,Physica 24, 347 (1958). J. VAN KRANENDONK,Physicu 25, 337 (1959). F. R. BRITTON,Thesis, University of Toronto (unpublished). M. KOTANI, Y. MIZIJNO,K. KAYAMA and E. ISHIGURO,J. Phys. Sot., Japan 12, 707 (1957). R. G. BREENE,JR., Phys. Rev. 111, 1111 (1958). R. G. BREENE,JR. Phys. Rev. 115, 93 (1959). R. G. BREENE,JR. J. Chem. Phys. 29, 512 (1958). H. MARGENAU,Phys. Rev. 40, 387 (1932). R. G. BREENJZ, JR., Phys. Rev. 123, 1718 (1961).
APPENDIX
I
s5Rb7
+so40 +
(2/3)(1+3c+)-8c
(2sRb)5+5(2sRb)4+20(2sRbr
GW1[ (*js)‘] [(28Rb) +I]) 1 [(2’7);;;;;; =0.211 (49’1 ff’1 C&4 I[
I[
(BRb)'
+60(26Rb)2+120(26Rb)+120
5-3c2)+(16/3)c
+
(2aRd8
(28&d*
I[
[ (’ ---zb;“)]
[ (26R#
+3(2sR1~)~+6(2S&,)+6 +
52ga5
+
‘=p( - 28&)(
+ 8(26&,)7
5f-$=R# + 336(28Rb)5 + 1680(28&,)* + 6720(2tjRb)3
I c
2 (llc4-lOc2+7) +20,160(2~R~)2+40,320(26R~)+40,320 + j I (2aRd7
1(2aRb>’+ 6(26Rd5+
30(26&)* + 120(26Rb)3 +
360(26~~)2
+‘t(26Rb)3 I[(2%)4 I [(-p-gc2) +12(2aR b)2 +24(26R 6) +1 [ I -I26R+26R (28Rd2+2(2aRa)+2 C I-26RL
+720(26&)+720
+f
3
(26Rb)5
2 (201c4+242c2-35) 24 + 35 (2aRa)'
8_9079c* 2.7301~2 2.0635 b
b
b
Dipole moment induction in homonuclear diatom& molecules-I.
+ 2sczP
)
Where
B=-
e-2bE df 2 og =
Cl(q72s) + c2(q72p)
~((~+~+$)sinh2b-(~+~)axh2b-$+~
+ 2c1cap
-
(
The Coulomb case
)
e-2bE df --
1
1 - +
4 it b
;+;+~~tmh2b-;]+[
(c&?+c22p
15+ 2b3
~++hzb 2b3
>
(c12t2+~2~if~++c1c2f3)
e-266
d{
+ clc25fJ]e-2bEdf +
&Cl',
ca')
A(Cl', cd)
Wl',
c2') =
WI',
~2') =
@(cl,' c2')Ii
C(Cl', c2')
D(Cl", ca")
&l",
c2")
&l",
c2")
Wl",
C2") =
C2")
C(d,
c27
ml",
>
3a, = c1'(4.s)+c2'(~g2p)
>
2% = c1”(a&) +c2”(a&)
239
Spectrosc.
Radiat.
Transfer.
Vol.
2, pp.
7~25~239.
PergamonPressLtd. Printedin GreatBritain
DIPOLE MOMENT INDUCTION IN HOMONUCLEAR DTATOMIC MOLECULES--I. THE COULOMB CASE* R. G. BREENE Physical Studies, Inc., 48 Maple Avenue, Centerville 59, Ohio (Received 7 February 1962) Abstract-We here consider the induction of an electric dipole moment in a homonuclear diatomic molecule by the short range forces due to an atom or a properly aligned molecule. We first develop a general form for the distorted molecular orbital and a simple perturbation method of obtaining the distortion coefficients. In applying this to the oxygen molecule we then take the distorting forces as those central Coulomb forces due to the perturber whose charge cloud we suppose undistorted by the Perturbed molei cule. This means that we neglect polartition and exchange. Further we restrict our numerical calculation to those cases where the perturber is either on the molecular axis or at 90” to it. This merely allows us to avoid the use of electronic computing machinery. Finally, the separation-dependent induced moment is averaged over a random distribution of perturber separations, the result being a linear density dependence in the moment. I.
INTRODUCTION
THE
symmetrical distributions of charge in homonuclear diatomic molecules implies that they possess neither permanent nor vibration-induced electric dipole moments. We are interested in the latter for we shall concern ourselves primarily with radiation in t& fundamental bands. The lack of a vibration+luced dipole moment means that molecules such as oxygen and nitrogen will be infrared inactive. The manner in which dipole activity can be induced by external electric fields was first studied by CONDON( He considered the case where the electric field, acting through the molecular polarizability, induced an electric dipole moment. Some years later, HERZBERG@) obtained infrared absorption from hydrogen and somewhat later, CRAWFORD, WELSH and LOCKE(~) obtained infrared absorption from oxygen and nitrogen. Of course, these absorptions could at first be attributed to quadrupole radiations; however, the latter authors demonstrated through pressure variations the probability that they arose as a result of inter-molecular forces. It is rather a short step then from the external electric fields of Condon to the internal electric fields proposed by MIZUSHIMA@). Now what Mizushima did was to assume in, say, oxygen that an electric field existed at the emitting oxygen molecule due to the quadrupole moments of the surrounding oxygen molecules. It was rather a straightforward matter to utilize a method originally developed by Holtsmark(5) and somewhat more succinctly stated by CHANDRASEKHAR@) to determine what this electric field would be. Having done this, Mizushima was able to &nnpute the induced electric dipole moments appealing to the :molecular polarizability. * The research reported in this paper has been sponsored by the Geophysics Research Directorate of the Air Force Cambridge Research Laboratories, Office of Aerospace Research, under Contract Number AF-19(604)-8417. 225
R. G. BREENE
226
Very little improvement on Mizushima’s treatment would appear to be called for, but later considerations indicated that there were additional contributions to the pressureinduced radiation. Specifically, Mizushima’s radiation is due to what might be called long-range forces. Extensive considerations of the short-range forces have been carried out by VAN KRANENDONKand BIRD(7~8) and later, in a more general fashion, by VAN KRANENDONK@~ 1%11). In addition, BRITTON(~~)has considered exchange forces. In the short-range force studies attention has been restricted primarily to the smaller molecules such as hydrogen and helium. In what follows, we consider the case for the larger diatomic molecules. Oxygen will be our specific example. II. THE PERTURBED WAVEFUNCTION FOR A GENERAL ATOM ORIENTATION Our problem is about as follows: We have a homonuclear diatomic molecule such as oxygen or nitrogen. A neutral atom is presumed to be at a relatively close separation from this molecule. We remark that another diatomic molecule aligned away from our original molecule is well approximated by such an atom, This atom now distorts the electronic charge cloud of the molecule through its short-range field. The problem is to describe this distorted wavefunction and, hence, the induced electric dipole moment. Now one could certainly attempt some method of building up the distorted molecular orbital by combining a number of unperturbed orbitals. In so doing, however, one (1) enters on a maze of a calculation in which (2) one rapidly loses sight of the physical conditions of the problem. We have therefore developed a simple wavefunction capable of describing a distorted orbital and adopted some perturbation theory to its obtention. We begin then by choosing the following general form for all of our distorted molecular orbitals :
xt = p+&%
h, ga)lxr’o’
(1)
wherein x&O)is our unperturbed orbital; at is a parameter, andArt,&, cpg)is some function of the spatial coordinates. Let us consider Fig. 1 which serves primarily to establish our coordinates. Suppose our perturber to be located on the molecular figure axis, say, to the right of nucleus “b”. Then if one assumes the field of the perturber to be (1) radial and (2) sourced at the perturber center one can picture the molecular charge cloud distorted along these radials. Further, one can picture the distortion of the charge cloud to be largest along the axis (8 = r) and decreasing both with decreasing “r” and with decreasing+. Further, we can allow the distorting effect to disappear for8 = 42, for this will mean that the cloud to the left of a vertical through the molecular center of symmetry will be undisturbed, an obviously reasonable assumption. This and the general nature of the forces then tell us that, or the on-axis case: OforO < 8 < 3. f(r, 8994 =
(2) r cos 8
for 2 Q 8 S rr. 2
Next, consider Fig, 2. Here we have depicted the general off-axis orientation
of the
Dipole moment induction in homonuclear diatomic molecules---I. The Coulomb case
227
perturbing atom. Again we work on the precept that only the charge cloud hemisphere on the side of the perturber will be affected, and this, as in the preceding paragraph. On this basis we are able to determine the following expression: Ofor-r’G8<~-l? f(r, 8, Y) = r cos 92/(1 -cos2g, sin2 P)r sin 8 cos 9, sin I?
@a) rr = sin-r(cos q sin To) dr
W-0
FIG. 1. The coordinates for the oxygen molecule and its perturher.
FIG. 2. The general off-axis orientation of the perturbing atom.
Equations (1) and (3) yield the perturbed orbitals, and we note that equation (3a) reduces to equation (2) for Ye = 0. The perturbation theory is now developed for the determination of the parameter “a”. We begin by assuming that, as inferred by equation (1): X$ = x’r
Xr’O) + atX’c
= f(r,
s,
p)xP
(W WI
228
R. G. BREENE
from which a first-order equation follows immediately: H’XP + H%IX’Z = E’axz’O) + E~%zsX’$.
(5)
We multiply through on the left by ~$0) and integrate over all space to obtain: (Xi’s’/H’lxr’O’)+ at(xa’O’lHOf(r, 8, ~)/XP’ = ci&O(xP’lf(r, 3, V)lXP’) + E’I.
(6)
Now note that : (xa(O)IHOf(r, 3, ‘PIXP)) = 2 (Xr(O)IHOIX3(O))(X~(O)IflXi(O)) = Jwxcq.f(r,
3, ~)I(XP’)
(7)
since Ito is surely diagonal in the ~a(0). Equations (6) and (7) yield: Efr = (xi’O’IH’[x(‘O’).
(8)
A similar approach yields : (Xj’O’IH’IXP’) ai = (~+~‘lf(r, 9, @lx~‘~‘)(Et~- Ej”j
(9)
Equations (l), (3), and (9) now furnish us with our perturbed wavefunction. In the following sections we apply these equations to a determination of the perturbation of an oxygen molecule by an oxygen atom for the atom (1) on the molecular figure axis and (2) ninety degrees to the figure axis. For more general cases use of electronic computing machinery would be required, and we have attempted to avoid this. III. THE LCAO-MO’s FOR OXYGEN AND THE OXYGEN ATOM POTENTIAL We shall use the molecular orbitals for oxygen as developed by KOTANIet al.@) Thus our primitive symmetry orbitals will bk: 0~1s = Is,+ 1~ = 4
0~1s = Is,-
z@ ---e-a5 cash a~ J ?7 @~ -e-at J 77
lsb = -4
@ a,2s = 2Sa+2sb = Ro FTe-b+f J oU2s= 2&-2sb
os2p =
2pC%a+&ab
= Ro
J
=
(loa)
sinh a~
cash bq -7 sinh bv]
83 --ge-bc[v cash bv - 6 sinh b?]
g Ro -e-bf;[cosh J 7r
7r
uw
(104
bv - .fv sinh bv]
WeI
bq]
Wf)
q
cU2p = 2pCTa-2pCTb= RO ---e-bE[& J
(lob)
cash br)-sinh
Dipole moment induction
in homonuclear
diatomic
625
J
0u2p+ = 2pQ* + 2pPb * = Q&,
---e*~“~([s7r
7rg2p* = 2prra’ - 2pQ* = *Ro
-e*@1/(@-
825
J
7r
molecules-I.
The Coulomb case
l)(l -+)e--b5
cash b7
l)(l -Tz)e-bc
sinh by
229
ww
In these equations ROis the internuclear separation, 2.282 a.u., and b is 2.596. We shall utilize the single determinant form for the molecular wavefunction, although it is, of course, true that the linear combination of as many as fifteen determinants obtained by Kotani et al. is a great deal more accurate. Their use, however, is merely a repetition of the calculation we shall carry out. The orbitals may be obtained from Table 1 as linear combinations of the primitive symmetry orbitals above. The ground state of oxygen is a sCfB state arising from the following electron configuration: (1~s)2(1~~)~(2~9)2(2~U)2(3~~)2(1~~-)2(1~~+)2(1~~-)2(l~~+). This constitutes the orbital information requisite to the calculation. potential for an oxygen atom has been developed by ~~(14915) as: V= -
15.322 + 2 e-15 mr( r)*(
The Coulomb
2 38.330rs-k 18*368r+ 8.930 + - e-s.7isr r
14.092rsf 19*264r+ 13*168+ fl e-+s*sr. r
TABLE~.THE
MOLECULARWAVEFUNCTIONSANDORBITALENERGIESAS GIVEN BY K~TANI et al.
Primitive symmetry
SCF MO
(02&, U 0.5641 -0.3117 0.7500 0.5345
2%? 30s 2% 3OU
Orbital energy (a.u.) --- 1.5219 -0.5564 - 0.9782 0.7392
orbital -
(02P),* Y
0.1677 0.6148 -0.1617 0.9275
This potential contains (1) no non-central terms, (2) no polarization of the perturbing atom by the perturbed, and (3) no exchange terms. Even so, investigation shows it contains more than need reasonably be handled, so we approximate it by: H’ = I’ = - 108.3&-5; (11) This potential appears to reproduce the earlier one reasonably well. IV.
THE
ON-AXIS
CASE
FOR
02-O
The problem is the evaluation of equation (9) where equation (3) has reduced to equation (2). We first obtain the matrix elements offlr, 9, 9). In doing so we consider Fig. 3. The coordinates of the function in equation (2) are referred to the center-of-masse
230
R. G. BREENE
of the molecule. We first transform atom “b”.
the function so that its center-of-coordinates
is an
We now set limits on the integration of the center-of-mass system.
such that it is carried out over the (+ +) quadrant
FIG. 3. The regions of integration involved in the evaluation of the matrix elements of ‘f”.
A typical molecular orbital is of the form:
We consider the exponential centered on “a” as small compared to that centered on “b” throughout the (+ +) quadrant of the center-of-mass system. Thus, .we have simplified the MO’s of equations (10) and Table 1 considerably. Also we ignore all 1s contributions to the orbitals as well as the lcrg and 10, MO’s themselves. We give the following example: (20g1f(r, S))130g) = 0*1758(bg2.vlf(r, 9)10g2s)
+0.1031(crg2~lf(r,
S)10g2p)+0~2945(ag2sJf(r,
(CTg2s)f(r, %))lCTg2s)= ?[/rb’(a-rb ++ . rb2 Sill &drbd&,
8))log2p)
(12a)
cos &&?-26rB etc.
Wb)
As can be seen from Fig. 3 analytic integration is possible over part of the region while numerical integration over the remainder is necessary. We finally obtain the only required matrix elements :
’
(20g,lf(r, 8)130g) = O-26617
(13a)
(2aulf(r,
(13b)
3)130,) = O-11498
We are expressing everything in atomic units.
Dipole moment induction in homonuclear diatomic molecules--I. The Coulomb case
231
The matrix elements of H’ as given by equation (11) are now required. Here again we work on the premise that the coordinates in the MO referred to the nucleus remote from the perturber may be neglected. Then after integration over the azimuthal coordinate one obtains, for example : (14) Equation (14) indicates the desirability formation :
of choosing
the following
coordinate
5 c = -s 2 rb-CT rl= ---=
(15)
rb-r’ -. Rb
Rb
In these coordinates,
trans-
for example, equation (14) becomes:
(c@IH’Ic$2p)
= -
-~~[(1+ac)+(l-ec)]-4c~
1 --cZ)qZ(1 -c2>
[exp(-2SR~)I(~-~)2b3(~-~2)d~d~.
(16)
The general result for the first of the three required matrix elements is now directly obtainable as : (u&slH’~ CJS) =
-1d‘kt$%b4 eXp(-28Rb)
6 45 1 21 +6_ ___ ~ ____ 5 (2SRb)3 + (26Rb)4 + (2SRb)5 + (2aRz?
+
1 1 -5 (26R#’
45 +(26Rb)7. 1
(17)
We relegate the more lengthy expressions for ( crg2SJH’J ~62p) and (o,2pJH’I 0,2p) to Appendix I in which Rb=2b. The information required for the evaluation of the “a(” is now at hand. We have only to determine the a2;#, azgu. It is apparent from equation (9) that: aufl = -mug which arises due to the energies in the denominators of equation (9). We note the absence of any coe5cient for the m-orbitals and hence of any distortion of the s-orbitals. Doubtless the ?r-orbitals will be distorted, and the fact that we do not describe such distortion is of mathematical rather than physical import. It is apparent that in order to distort an orbital we must have another orbital in our basis set over which matrix elements may be taken.
R. G. BREENE
23.2
The set developed by Kotani et al. does not include any orbitals with which our m may interact. Were such orbitals present, the problem would simply be a repetition of the present work. There is some justification for our present neglect in that we should not anticipate too much overlap between the a’s in our set and those which might be added. In order to consider the wavefunction distortion let us write down only the first term in the equation (17): @R b4
A- c2
eXp( -26Rb)
= A(5)2S3Rt,4 exp( -26Rb)
The three factors of importance here are (1) the perturber separation, Rb, (2) the effective nuclear charge (ENC), 6, and (3) the potential exponent (PE), “5”. The perturbation fades out exponentially with Rb, thus assuring us that we are indeed dealing with short-range forces. Now note that 6, the effective nuclear charge, is what basically governs the spatial extension of the molecular orbital. It is the spatial extension of this orbital, rather than of the perturbing molecule which, through 6, strongly affects the decay of the perturbing effect. This is further borne out in the multiplicative factors in the previous equation, the ENC being to a higher power than the PE. One can then conclude that in the distortion of homonuclear diatomic molecules the type of molecule being distorted is a good deal more important than is the type of molecule doing the distorting. We have never actually normalized our distorted wavefunction. This is not of much importance for small values of “at”. Nevertheless, one may normalize through C, and obtain upon dropping a matrix element infs as of second order: c = [l + 2hr(x$J’lf(r, S))l~Jo’)]-*1s.
V.
THE
NINETY
We now suppose the perturbing
DEGREE
CASE
FOR
02-O
oxygen atom to be somewhere
on the perpendicular
to the molecular ‘6gure axis passing thrdugh the motecular center of symmetry. change is, of course, induced in equation 0
The basic
(3a) which now becomes:
“<,<_” 2
2
r cos 9 sin y -rsinBcosp,
-“<,,
2
tlg)
which is directly obtainable from the earlier equation by setting P” equal to ninety degrees. We remark that integration over the azimuthal coordinate eliminates the first term on the right of equation (18). Furthermore, we may integrate over both terms to obtain: f(r, 3) = -2~. sin 8 = -2ad(l
-[2)($-
1)‘
(1%
and there is now no restriction on the polar angle integration. The matrix elements off will not be analytically evaluable unless the factors under the
Dipole moment induction in homonuclear diatomic molecules-I. The Coulomb case
233
square-root sign of the operator itself are approximated in some fashion. In the “71” range, which is from - 1 to 1, one may approximate this radical quite nicely while the exponentials involved in the “P’ coordinate allow one to also arrive at a quite reasonable approximation to this radical. We have utilized the’following expressions: d/(1 -7s
= (l/1-381)(1*381 -sinher])
2/t2 - 1 = 1.2085‘- 0.766.
(20a) (20b)
We now have the quite straightforward matter of evaluating the matrix elements of equations (19) using the approximations of equations (20) and the molecular orbitals as given by equations (IO). The results of these calculations are: (2o,lf(r,
8, r$J)134 = -0~01000
(2 la)
(2c4f(r,
9,434
(21b)
= -0.05151
We now evaluate the matrix elements of the perturbing potential. In general our molecular orbital will, of course, have the form: Cr@,-6Y.+ C*- 8rb where we suppose the constants and factors quite arbitrary. This will mean that our matrix element of the perturbing potential will have the general form: CisJF(r)e-s”e’e-s*~~dT+ C22JF(r)e-sre-2brbdr + 2CiC2JF(r)e-5r exp[ - 6(r, + rb)]dT. In this equation we have represented the perturbing portion of the Hamiltonian by the function in the exponential. It is apparent that, because of the location of the perturber, on the x-axis of Fig. 1, either the first or the second integrals in the above equation will generally contribute the maximum from any given spatial region. Under almost all circumstances, however, the overlap, whose contribution is given by the third integral, will be comparatively smaller because one or the other of the radial coordinates will be large. ,We then suppose that the matrix element be approximated by the sum of the first two integrals; these integrals corresponding precisely to the matrix elements evaluated for the on-axis case. In effect then, for a given separation from one of the molecular nuclei, we may now simply take twice the matrix element value for the on-axis case. However, the actual separation of the perturber from the oxygen molecule is: R = 1/(Rb2-a2).
(22)
We now have the requisite information for the evaluation of the orbital distortion coefficients, and hence of the electronic wavefunction, for the ninety degree case. The general comments which were made in connection with the on-axis case hold here with obvious modifications. VI. THE INDUCED ELECTRIC DIPOLE MOMENTS In general the expression for the electric dipole moment may be written as:
(23)
234
R. G. BREENE
Since we have supposed our molecular wavefunction equation (23) becomes, still in atomic units :
where the xr are the molecular moment is then:
to be a single determinant,
orbitals. The orbital contribution
to the electric dipole
(25)
The Crst term in equation (25) will be zero in first approximation, and we shall so consider it here. One could probably use this term in introducing the long range distorting forces. We also neglect the third term in equation (25) as of second order in at. Next we recognize the components of the dipole moment as: x = a&?--
l)(l -T#+Qcos 9
(26a)
Y = a&s-
l)(l -7s)
(26b)
sin 9
Z = ah
(264
where “a” is one-half the nuclear separation in the distorted molecule. We are considering now both our on-axis and ninety degree off-axis cases at the same time. It is apparent that in the on-axis case the center of negative charge would be moved along the z-axis and toward the perturber. This means that only a z-component of the electric dipole moment would be induced. Such is found to be the case. In the ninety degree off-axis case we would anticipate that the center of negative charge would be moved up and again toward the perturber thus inducing only an x-component of the electric dipole moment. Such we also find to be the case. The result is that we obtain the following expressions for the electric dipole moment :
Mt = RoW[A+ B+ C- A’- B’ - C’ Jaz, + Ro7S5[D+ E + F]azr,,
GW
The terms A, B, etc. are set forth as Appendix II. The electric dipole moment of oxygen as a function of perturber separation for both cases considered is displayed as Fig. 4. Now in computing the absolute intensity of the dipole radiation emitted by this distorted molecule in its fundamental vibrational band, we consider the molecule as a simple harmonic oscillator. Furthermore, we suppose that its electric dipole moment may be given as the familiar series: M = Mo+M’(R-Ro)+
...
(2’9
where the first term is, of course, the permanent dipole moment responsible for pure rotational spectra and obtained by us as equations (27). The second term will be responsible for the radiation in the fundamental band. We are supposing a total lack of vibronic
Dipole moment induction in homonuclear diatomic molecules-I.
The
Coulomb ease
interaction so that we assume the dipole moment derivative obtainable of equation (27) with respect to Re: M,’
by differentiation
= ;MZZ
M,’ =
235
(294
$4~.
Wb)
This effectively defines the absolute intensity in the induced fundamental-vibrational band of oxygen.(l@
1
’ 2
’
3 Perturber
’
’
’
4
b
5
separation,
’
6
atomic
s units
FIG. 4. The induced electric dipole moment for both cases as a function of perturber separation. In the on-axis case the separation is from the atom in the molecule nearest the perturber. In the off-axis case the separation is from the molecular center-of-mass.
VII.
THE
OBSERVED
ELECTRIC
DIPOLE
MOMENT
FOR
02-O
We have computed the electric dipole moment for various perturber separations. It remains to determine the moment actually observed. In order to do this we must average
236
R. G. BREENE
the moment over some distribution of perturber separations. There is every justification for a random distribution of spatial coordinates, so that we choose dv sin 9dvd9dp, - = ya op(r)dr V (4/3)rrR3
= %dr. R3
Before carrying out this average we must decide what lower limit, that is, what lower separation we are going to use in the averaging integration. This question is automatically decided for us by our perturbation scheme. Surely our perturbation theory-which is indeed of the order-of-magnitude variety-will break down when the magnitude of the perturbation energy approaches the separation of the electronic energy levels. The nearest electronic level in oxygen is alAg which is at about an electron volt, that is, at about 0.04 atomic units from the ground state. From equation (8) we obtain the amount of the energy perturbation as a function of perturber separation. We find that at 3 a.u. of separation the perturbation is just about equal to the level separation while at 3.5 a.u. it has fallen to about one-fifth*. On this basis we take 3.5 a.u. as the lower limit of the averaging integration. Now note that this corresponds to a molecular separation of 5.78 or, about 6 a.u. when we add the internuclear separation in the oxygen molecule. The reader will note the relationship to the cut-off separation used by MARGENAU~~)in his pioneer work on van der Waals’ forces. Next we require a value for “R” in equation (30). This we take as the average intermolecular separation, that is to say, (4/3)77Rs = l/N, where “N” is the density in a gas of pure oxygen. If we now let Mi(u) be the dipole moment component as a function of perturber separation we may write equation (30) as: m Mt = 4rrN
s
Mi(v)rsdr.
(31)
3.5
The value of Mt for the on-axis case is 0.01386 x lo-21A where “A” is the density in atmospheres. This means there will be an electric dipole moment of magnitude 0.000139 x 1O-18e.s.u. induced in oxygen at a density of 10 atm. The value of Mt for the off-axis case is O-57898 x lo-a1 A where “A” is the density in atmospheres. This means a moment of 0.00579 x 10-l* e.s.u. at 10 atm. If we apply the absorption coefficient formulas of our earlier work(is) we get, at 1595 cm-r, an absorption coefficient of 0.14809 x 10-s As cm-l for the on-axis case. At 50 atmospheres this yields an absorption coefficient of 0*000185 cm-l. CRAWFORD,WELSH and LOCKE(~)obtained a quadratic dependence of the absorption coefficient on the density experimentally while MIZUSHIMA(~) obtained such a dependence in his theory of long range forces. The fact that no cubic dependence, such as we predict here, has been observed would indicate that (1) higher than binary collisions and (2) long range forces are influencing the dipole induction down to as low densities as have been observed. VIII. REMARKS In our considerations of this problem we have tacitly a.ssumed a great deal more quiescence than can be justified; specifically, we have ignored rotation on the part of our * We take 4 a.u. for the off-axis case.
Dipole moment induction
in homonuclear
diatomic
molecules-I. The Coulomb case
237
distorted molecule and translation on the part of our perturbing molecule or atom. The !atter approximation holds quite well for the high densities of interest here@. However let us consider the supposition that no rotation of the molecule is taking place. Suppose for a moment that the molecule does rotate in the plane established by the three nuclei. In this case, we can say with some degree of justification, that very roughly, something like half of the time the perturbing atom is on-axis, or approximately so, and half of the time the perturbing atom is ninety degrees to the axis, or approximately so. The result of this, of course, would be that we could treat the oxygen molecule as having just one half the magnitude of the induced z-component of the electric dipole moment while at the same time having just one half the induced component of the x-component. It is apparent that this. is only an approximation, but it should be a reasonable one. Now if we orient matters somewhat differently, we can see that we will approach the same condition only now the molecule will possess a z- and a y-component of the electric dipole moment. In general then and with more or less random orientation one could approximate the situation reasonably well by this assumption : The molecule has induced one half the indicated x-component of the dipole moment and one half the indicated z-component of the dipole moment. Now let us go back and consider the approximations which were made in our selection of the potential. We recall that these had to do with (1) non-central forces, (2) exchange interactions, and (3) the effect of polarizing the perturbing atom. ‘We have shown in a somewhat different context(i5)--and MIZUSHIMA c4) has also remarked-that the noncentral portion of the Coulomb field is probably of very little importance. Perhaps then it would be well to maintain ~this particular central approximation. .-‘. The exchanges of electrons between the two- molecules which give rise to these spin forces have an upper limit. Whether or not we have reached it at our cut-off of 6 a.u., we should have approached it closely enough for the neglect of exchange forces not to be a serious approximation. We are left then with the effect on the potential of the polarization of the perturber’s charge cloud by the distorted molecule. Now our potential is negative, which is to say that it will be attractive to the negative charge cloud in our distorted atom. In like fashion we may expect the potential due to the distorted atom to be attractive rather than repulsive to the charge cloud of the perturbing atom. This means that the perturber’s charge cloud is displaced in the direction of the molecule; which in turn will mean that the attractive potential in which the distorted molecule had existed will be reduced. In our work on the free electron in the field of an oxygen atom, we demonstrated the reverse situation(i*). In that case, the field of the electron is completely repulsive to the atomic charge cloud, so that the charge cloud is pushed away from the electron. This means in turn that the potential will be more attractive which indeed it proves to be. It is apparent that in the present case we encounter the opposite situation. The net result of the introduction of polarization will be to reduce the distortion of the charge cloud and as a result, to reduce the magnitude of the electric dipole moment. Acknowledgements-The author wishes to express his appreciation portant assistance in the extensive numerical calculations.
to NANCY M. BREENEfor her im-
R. G. BREENE
238
1. 2. 3. 4. 5. 6.
7. 8. 9.
10. 11.
12. 13. 14.
15. 16. 17. 18.
REFERENCES E. U. CQNDON,Phys. Rev. 41, 759 (1932). G. HERZBERG,Nature 163, 170 (1949). M. F. CRAWFORD,H. L. WELSHand J. L. LOCKE,Phys. Rev. 75, 1607 (1949). M. MIZUSHIMA,Phys. Rev. 76, 1268 (1949). R. G. BREENE,JR., The Shif and Shape of Spectral Lines. Pergamon Press, London (1961). S. CHANDRASEKHAR, Rev. Mod. Phys. 15, 1 (1943). J. VAN KRANENDONKand R. B. BIRD, Physica 17, 953 (1951). J. VAN KRANENDONKand R. B. BIRD, Physica 17, 968 (1951). J. VAN KRANENWNK, Physica 23, 825 (1957). J. VAN KRANENDONK,Physica 24, 347 (1958). J. VAN KRANENDONK,Physicu 25, 337 (1959). F. R. BRITTON,Thesis, University of Toronto (unpublished). M. KOTANI, Y. MIZIJNO,K. KAYAMA and E. ISHIGURO,J. Phys. Sot., Japan 12, 707 (1957). R. G. BREENE,JR., Phys. Rev. 111, 1111 (1958). R. G. BREENE,JR. Phys. Rev. 115, 93 (1959). R. G. BREENE,JR. J. Chem. Phys. 29, 512 (1958). H. MARGENAU,Phys. Rev. 40, 387 (1932). R. G. BREENJZ, JR., Phys. Rev. 123, 1718 (1961).
APPENDIX
I
s5Rb7
+so40 +
(2/3)(1+3c+)-8c
(2sRb)5+5(2sRb)4+20(2sRbr
GW1[ (*js)‘] [(28Rb) +I]) 1 [(2’7);;;;;; =0.211 (49’1 ff’1 C&4 I[
I[
(BRb)'
+60(26Rb)2+120(26Rb)+120
5-3c2)+(16/3)c
+
(2aRd8
(28&d*
I[
[ (’ ---zb;“)]
[ (26R#
+3(2sR1~)~+6(2S&,)+6 +
52ga5
+
‘=p( - 28&)(
+ 8(26&,)7
5f-$=R# + 336(28Rb)5 + 1680(28&,)* + 6720(2tjRb)3
I c
2 (llc4-lOc2+7) +20,160(2~R~)2+40,320(26R~)+40,320 + j I (2aRd7
1(2aRb>’+ 6(26Rd5+
30(26&)* + 120(26Rb)3 +
360(26~~)2
+‘t(26Rb)3 I[(2%)4 I [(-p-gc2) +12(2aR b)2 +24(26R 6) +1 [ I -I26R+26R (28Rd2+2(2aRa)+2 C I-26RL
+720(26&)+720
+f
3
(26Rb)5
2 (201c4+242c2-35) 24 + 35 (2aRa)'
8_9079c* 2.7301~2 2.0635 b
b
b
Dipole moment induction in homonuclear diatom& molecules-I.
+ 2sczP
)
Where
B=-
e-2bE df 2 og =
Cl(q72s) + c2(q72p)
~((~+~+$)sinh2b-(~+~)axh2b-$+~
+ 2c1cap
-
(
The Coulomb case
)
e-2bE df --
1
1 - +
4 it b
;+;+~~tmh2b-;]+[
(c&?+c22p
15+ 2b3
~++hzb 2b3
>
(c12t2+~2~if~++c1c2f3)
e-266
d{
+ clc25fJ]e-2bEdf +
&Cl',
ca')
A(Cl', cd)
Wl',
c2') =
WI',
~2') =
@(cl,' c2')Ii
C(Cl', c2')
D(Cl", ca")
&l",
c2")
&l",
c2")
Wl",
C2") =
C2")
C(d,
c27
ml",
>
3a, = c1'(4.s)+c2'(~g2p)
>
2% = c1”(a&) +c2”(a&)
239