Collision-induced hyper-Rayleigh spectrum of octahedral molecules: The case of SF<sub>6</sub>

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Collision-induced hyper-Rayleigh spectrum of octahedral molecules: The case of SF 6

Tadeusz Bancewicz, Jean-Luc Godet, and George Maroulis

Citation: The Journal of Chemical Physics 115, 8547 (2001); doi: 10.1063/1.1410979 View online: http://dx.doi.org/10.1063/1.1410979

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/115/18?ver=pdfcov Published by the AIP Publishing

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Collision-induced hyper-Rayleigh spectrum of octahedral molecules:

The case of SF

₆

Tadeusz Bancewicz

Nonlinear Optics Division, Institute of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan´, Poland

Jean-Luc Godet

Laboratoire des Proprie´te´s Optiques des Mate´riaux et Applications, Universite´ d’Angers, 2 boulevard Lavoisier, 49045 Angers, France

George Maroulis

Department of Chemistry, Physical Chemistry Laboratory, University of Patras, GR-26500 Patras, Greece 共Received 25 May 2001; accepted 23 August 2001兲

A theoretical expression giving the collision-induced hyper-Rayleigh共CI HR兲spectrum due to pairs of centrosymmetric molecules of octahedral symmetry has been derived. The dipole²-quadrupole hyperpolarizability light scattering mechanism of collision-induced hyperpolarizability ⌬␤L M is discussed in detail and proposed to explain the CI HR spectrum. Numerically we have applied our analytical formulas to binary CI HR spectrum of sulfur-hexafluoride. We have normalized our SF₆ CI HR spectrum to the monomer HR spectrum of CCl₄. The spectral contribution due to second hyperpolarizability-permanent hexadecapole HR light scattering mechanism has been estimated and showed to be negligible. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1410979兴

I. INTRODUCTION

Hyper-Rayleigh 共HR兲 scattering is a three-photon process in which a system is excited with two photons of laser frequency␻Land emits one photon at a frequency of about 2

␻L. The first theoretical inkling of this effect goes back to Dirac¹and Mayer.²The full theoretical treatment of HR phe- nomenon was given at the same period by several authors.^{3– 6} Terhune, Maker and Savage⁷ succeeded in observing HR scattering in H₂O, CCl₄, CH₃CN and fused quartz. For fur- ther studies see Refs. 8 –16. In his pioneering work^3,4,17 Kielich indicated that HR process arises in a molecule mainly due to its third-order dipolar hyperpolarizability tensor ␤i jk.

Recent technological demands of new efficient materials for nonlinear optics have provoked great interest in nonlinear and multipolar polarizabilities of molecules. Hyper-Rayleigh scattering has been developed as the best experimental technique for determination of the first hyperpolarizability tensor

␤i jkof molecules.^{18 –23}

Search for a precise value of the dipolar first hyperpolarizabilities␤i jkof molecules raised interest in the literature in the collision-induced hyper-Rayleigh scattering.^{24 –26} These studies indicate that collision 共interaction兲-induced contributions account for more than half of the hyper- Rayleigh intensity from the liquid phase of CCl₄. For centrosymmetric systems the␤i jktensor vanishes identically so no HR scattering is expected. Nevertheless, intensities at fre- quencies around the double incident frequency 2␻Land due to collision-induced共CI兲interactions have been observed for molecules with a center of inversion.^27–29 In this paper we discuss theoretical foundations of the collision-induced HR scattering spectral signal resulting from a pair of centrosymmetric molecules of octahedral symmetry. We note that

sometimes in linear regime the kind of a spectrum considered by us is called as collision-induced rotational Raman scattering.³⁰Numerically we apply our analytical formulas to HR spectrum of gaseous sulfur-hexafluoride. Our study indi- cates that the collision-induced HR method opens a new way to study molecular interactions and dynamics as well as is the only technique allowing a measurement of several kinds of dipole-multipole hyperpolarizabilities of a system.

II. HYPER-RAYLEIGH TIME CORRELATION FUNCTION In a dense medium, collisional effects as well as time- and space-fluctuations will in general lead to changes in the molecular hyperpolarizability tensor. In the notation of the irreducible spherical tensors the lowest order multipolar collision-induced pair hyperpolarizability tensor reads³¹

⌬␤L M关共11兲a1兴⫽3共1⫹P^AB兲all but L, M ,a

兺

共⫺1兲^j^B

⫻

冉

^共^2l^A^兲²^!^共^N^2l^B^兲^!

冊

^1/2^X^j^A^j^B^Nx

⫻

再

^j^lâÂÂ ^l^j¹^B^B ^N^L^x

冎

^兵^T^N^(AB)^丢^关^␤^jÂ^关共¹¹^兲âlÂ^兴

丢␣j_B关1l_B兴兴^(x)其^{L M}^, ^共¹^兲 where ␣J_pM_p关(1)l_p兴 is the irreducible dipole-2^l^p-pole共mul- ti兲polarizability tensor of a molecule p and␤J_pM_p关(11)al_p兴 denotes the irreducible J_p-rank spherical tensor of the first order dipole²⫺2^l^p-pole hyperpolarizability of a molecule p in a coupling scheme where two dipoles are first connected

8547

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and, subsequently, the 2^l^p-pole multipolar moment. More- over T_N^(AB) is the spherical interaction tensor and 兵^{. . .}其 stands for the 9-j Wigner symbol whereasP^ABpermutes the indices A and B. In our considerations we assume the Kleinman³² 共full index兲 symmetry of the collision-induced pair hyperpolarizability tensor⌬␤i jkand the same symmetry of polarizabilities and hyperpolarizabilities of monomers. In our formulas䉺 and _丢 symbols stand for scalar and tensor products, respectively.

We apply Eq.共1兲 to a pair of octahedral molecules and we consider the ␣0T₃␤j关(11)a2兴共first nonzero兲 light scattering mechanism. To describe this mechanism, let’s consider a pair of octahedral molecules A and B. A linearly polarized laser light E^L is incident upon the pair. The laser light induces in each molecule a dipole moment at a frequency

␻L:␮_␣⫽␣_␣␤^E_␤^L and a quadruple moment at a frequency 2␻L of the form

⌰_␣␤²^␻^L⫽ ¹2␤␥,␦^,␣␤E_␥^LE_␦^L. 共2兲 Then the dipole moment at a frequency 2␻L induces in the molecule A in two-fold complementary ways:共1兲through the dipole²-quadrupole hyperpolarizability␤tensor of molecule A, laser field E^Ltogether with the field gradient of induced dipole moment of molecule B 共2兲due to the molecular field 共at the frequency 2␻L) of the quadrupole moment of Eq.共2兲 induced in molecule B by laser field and the dipole²polarizability tensor ␣ of molecule A now acting at frequency 2␻L.¹⁵Analogous considerations apply to molecule B. Then the rank one and three components of the pair hyperpolarizability tensor are of the form:

⌬␤1 M关共11兲21兴⫽⫺3

冑

10³ 兵␣^(B)关T₃^(AB)_丢␤4

(A)关共11兲22兴兴

⫺␣^(A)关T₃^(AB)_丢␤4

(B)关共11兲22兴兴其^{1 M}^, ^共³^兲

⌬␤3 M关共11兲21兴⫽⫺

冑

³³70兵␣^(B)关T₃^(AB)_丢␤4

(A)关共11兲22兴兴

⫺␣^(A)关T₃^(AB)_丢␤4

(B)关共11兲22兴兴其^{3 M}^. ^共⁴^兲 For the octahedral symmetry molecules considered, the spherical irreducible dipole²polarizability tensor␣j

(i)关11兴 of Eq. 共1兲 is isotropic 共j⫽0兲 and the following relation holds

␣0

(i)关11兴⫽⫺)␣⁽ⁱ⁾^{, where} ␣⁽ⁱ⁾ is the conventional dipole polarizability of molecule i. In Eq. 共1兲␤j

(i)关(11)22兴 stands for dipole²-quadrupole hyperpolarizability tensor of mol- ecule i. In Buckingham³³and Kielich¹³convention this ten- sor is denoted by B.³⁴ In order to make our formulas more compatible with previous papers we follow this notation. For the octahedral symmetry molecules, only the irreducible ten- sors B_j of ranks j⫽0 and j⫽4 are nonzero.³¹ Generally our formulas共1兲–共4兲apply to a pair of different molecules A and B. However, when we consider scattering from one compo- nent fluid (A⫽B) we easily note that in this case the contri- bution to the pair hyperpolarizability⌬␤L M关(11)a1兴 due to B₀ vanishes. Moreover we note that the discussed here mechanism ␣^T3B can be considered as nonlinear analogue of the ␣^T3A light scattering mechanism, so successfully used in the collision-induced linear light scattering.^30,35,36

The autocorrelation function

F_LL共t兲⫽具⌬␤L共0兲䉺⌬␤L共t兲典共5兲 is the key quantity required in the spectral analysis of the scattered light. We calculate the autocorrelation function Eq.

共5兲assuming that we consider one component fluid. The correlation function Eq. 共5兲 generally deals with a situation when the rotational and translational degrees of freedom are coupled. However, when considering radiation scattered by low density gaseous systems we usually are justified in assuming that the molecules of the scattering volume are cor- related radially but uncorrelated orientationally.^30,35,36 We use the notation

S_N共t兲⫽具^D00

N共␦⍀12共t兲兲R₁₂共0兲^⫺^(N^⫹¹⁾R₁₂共t兲^⫺^(N^⫹¹⁾典^, ^共⁶^兲 R_j共t兲⫽具^Dnn

j 共␦⍀共t兲兲典^, ^共⁷^兲

where␦⍀(t) and␦⍀12(t) denote the reorientation angles of the molecule and intermolecular vector, respectively, at the time t. According to our model we assume that the molecules of the system rotate freely. In this case the reorientational correlation function R_j(t)⫽具^Dnn

j (␦⍀(t))典 is independent of the magnetic quantum number n of the Wigner function in- volved, D_nn^j (␦⍀(t)). With this notation we have:

具^TN

(AB)共0兲䉺T_N^(AB)共t兲典⫽共2N兲!

2^N S_N共t兲, 共8兲具^Bj共0兲䉺B_j共t兲典⫽R_j共t兲共B˜_j䉺B˜_j兲, 共9兲 where B˜

jdenotes the dipole²-quadrupole hyperpolarizability tensor in its molecular frame reference system.

Using the irreducible spherical tensors decoupling procedure³⁷ for isotropic one component fluid we have:

具␣关T₃^(AB)共0兲丢B₄共0兲兴L䉺␣关T₃^(AB)共t兲丢B₄共t兲兴L典

⫽共⫺1兲^(L^⫹¹⁾共2L⫹1兲 63 ␣²具^T3

(AB)共0兲䉺T₃^(AB)共t兲典

⫻具^B4共0兲䉺B₄共t兲典

⫽共⫺1兲^(L^⫹¹⁾10

7 共2L⫹1兲␣²共B˜₄䉺B˜₄兲S₃共t兲R₀共t兲R₄共t兲. 共10兲 We consider molecules of O_h symmetry in their nuclei equilibrium. Then the polarizability tensor ␣ is isotropic, whereas for the components of the dipole²-quadrupole tensor B, the following relations hold³⁸

B˜

x,x,xx⫽B˜

y ,y ,y y⫽B˜

z,z,zz⫽⫺2B˜

x,x,y y⫽⫺2B˜

y ,y ,xx

⫽⫺2B˜

x,x,zz⫽⫺2B˜

z,z,xx⫽⫺2B˜

y ,y ,zz

⫽⫺2B˜

z,z,y y⫽B₁, 共11兲

Cˆ B˜_{x,y ,xy}⫽Cˆ B˜_{y ,z,y z}⫽Cˆ B˜_x,z,xz⫽B₂, 共12兲

8548 J. Chem. Phys., Vol. 115, No. 18, 8 November 2001 Bancewicz, Godet, and Maroulis

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where the operator Cˆ permutes the dipolar indices ␣ ^and␤ and quadrupolar pair of indices ␥␦ of the tensor B˜_␣,␤,␥_␦. The permutation within the pair ␥␦ is also allowed. Taking into account Eqs. 共11兲and 共12兲we find the following relations between the molecule-frame irreducible spherical components of the dipole²-quadrupole tensor B˜_jm关(11)2a兴 and their Cartesian counterpart:

B˜

00关共11兲22兴⫽

冑

^6/5共B₁⫹2B₂兲, B˜₄₀关共11兲22兴⫽

冑

^7/15共3B₁⫺4B₂兲

2 ,

B˜₄₄关共11兲22兴⫽B˜₄_⫺₄关共11兲22兴⫽

冑

^5/14B^˜40关共11兲22兴. 共13兲 Using formulas 共5兲–共13兲 we calculate the respective time correlation functions of the hyper-Rayleigh scattered light:

F₁₁共t兲⫽具^⌬␤1关共11兲21兴共0兲䉺⌬␤1关共11兲21兴共t兲典

⫽162

35 ␣²共3B₁⫺4B₂兲²S₃共t兲R₀共t兲R₄共t兲, 共14兲 F₃₃共t兲⫽具⌬␤3关共11兲21兴共0兲䉺⌬␤3关共11兲21兴共t兲典

⫽66

35␣²共3B₁⫺4B₂兲²S₃共t兲R₀共t兲R₄共t兲. 共15兲

III. HYPER-RAYLEIGH LIGHT SCATTERING SPECTRUM

Consider a system of N molecules in a volume V on which is incident a linearly direction e polarized light wave with the electric vector

E^L共r,t兲⫽eE₀^Lexp兵ⁱ^共^k•r⫺␻Lt兲其^. ^共¹⁶^兲 Then the time correlation function of hyper-Rayleigh scat- tered light analyzed at the point R⫽k_s R共of the wave zone RⰇ␭0) with an analyzer transmitting waves with polariza- tion n (n⬜k_s) is of the form¹⁴

I_n共t兲⫽A²^␻

再

⁸^共^n•e⁴⁵^兲²^⫹¹^F¹¹^共^t^兲⫹²^共^n•e¹⁰⁵^兲²^⫹⁴^F³³^共^t^兲

冎

共17兲 when the factor A²^␻ contains␻s

4 共where␻s is the scattered frequency兲 and 具兩E^L(0)E^L(t)兩²典 in addition to other quantities.¹³ In order to get spectral distribution of HR scattered light we take the Fourier transform (Ft) of Eq.共17兲.

We assume right-angle light scattering geometry and incident beam polarized perpendicularly to the scattering plane. We calculate the frequency dependent depolarization ratio of HR scattering as a ratio of the horizontal component I_VH(␯⁾共when n⬜e兲of scattered spectrum to its vertical com- ponent I_VV(␯⁾ 共when n储e兲:

␩共␯兲⫽I_VH共␯兲

I_VV共␯兲. 共18兲

Taking into account Eqs. 共14兲–共17兲we obtain

␩共␯兲⫽Ft关45¹ F₁₁共t兲⫹ 105⁴ F₃₃共t兲兴 Ft关¹5 F₁₁共t兲⫹35² F₃₃共t兲兴 ⫽107

633⫽0.169. 共19兲 Since our spectrum results only from one light scattering mechanism the frequency dependent depolarization ratio is reduced to a constant. We note quite different value of this constant from the depolarization ratio equal to ²₃ for one- molecule HR scattering by molecules of tetrahedral symme- try共e.g., CCl₄兲.

It is interesting to note that in the case of octahedral molecules, another first order 共in T_N兲 hyper-Rayleigh light scattering mechanism, due to second hyperpolarizability ␥ and permanent SF₆hexadecapole (Q₄) moment is possible:³¹

⌬␤L M关共11兲a1兴⫽⫺共1⫹P^AB兲allbut L, M ,a

兺

共⫺1兲^j^A^⫹^L^⫹^N

⫻

冉

^共^2l^A^兲²^!^共^N^2l^B^兲^!

冊

^1/2^X^X^j^Ll^A^NA

⫻兵␥j_A关共共11兲a1兲Ll_A兴丢关T_N共R_AB兲

丢Q_l

B兴^(l^A⁾其^{L M}^. ^共²⁰^兲

From the formula Eq.共20兲, however, it results that this

␥^T5Q₄ light scattering mechanism and the pair induced hy- perpolarizability varies as R_AB^⫺⁶ being therefore smaller by R_AB^⫺²than the now discussed␣^T3B hyper-Rayleigh light scat- tering mechanism in R_AB^⫺⁴. Consult the figure for estimated

␥^T5Q₄ contributions to HR SF₆ spectrum.

IV. COLLISION-INDUCED HYPER-RAYLEIGH SPECTRUM OF SF₆

Now using Eqs.共14兲–共17兲and the sulphur-hexafluoride (SF₆) polarizability and hyperpolarizability values³⁹ as well as the SF₆ potential data⁴⁰ we are in a position to compute the collision-induced spectrum for gaseous SF₆. The data used in our computations are collected in a Table I. Again the right-angle light scattering geometry is used. We compute the polarized component (n储e) I_VV⁽²^␻^L⁾(⌬␻) of the CI HR scattered spectrum:

TABLE I. Numerical values of the sulphur-hexafluoride and carbon- tetrachloride polarizability and hyperpolarizabilities used in our calculations 共polarizability and hyperpolarizability values are in atomic units a.u.兲.

Sulphur-hexafluoride Carbon-tetrachloride

␣ 30.35^a

Bzz,zz ⫺134^b

Bxz,xz ⫺156^b

␤ 11.0^c

⌫(cm^⫺1) 4.6^d

aReference 39.

bReference 40.

cReference 26.

dReference 24.

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I_VV⁽²^␻^L⁾共⌬␻兲⫽A²^␻n

2Ft

冋

¹⁵^F¹¹^共^t^兲⫹³⁵² ^F³³^共^t^兲

册

^共²¹^兲

of SF₆fluid at the molecular number density n⫽25 Amagat and at T⫽294 K.

When using spherical top wave functions for the evalu- ation of transition matrix elements, the Fourier transform of R_j

1(t)R_j

2(t) has the form:³⁵ Ft关R_j

1共t兲R_j

2共t兲兴

⫽共2J₁⫹1兲共2J₂⫹1兲共2J₁⬘⫹1兲共2J₂⬘⫹1兲 Z₁Z₂

⫻exp

冋

^⫺^共^E^J¹^k^⫹^B^T^E^J²^兲

册

^␦^共^␻^⫺^␻^J¹^J²^J¹^⬘^J²^⬘^兲^, ^共²²^兲

where:

␻J₁J₂J

1⬘^J2⬘⫽⫺关J₁⬘共J₁⬘⫹1兲⫹J₂⬘共J₂⬘⫹1兲⫺J₁共J₁⫹1兲

⫺J₂共J₂⫹1兲兴B. 共23兲 B is the rotational constant and Z_i denotes the rotational partition function. The selection rules have the following form:

⌬J₁⫽0,⫾1,⫾2 . . .⫾j₁ J₁⫹J₁⬘⭓j₁, 共24兲

⌬J₂⫽0,⫾1,⫾2 . . .⫾j₂ J₂⫹J₂⬘⭓j₂. 共25兲 We calculate the rotational part of our spectrum using Eqs.

共22兲–共25兲. To deal with the translational part S_N(t) of our spectrum we compute the zeroth, second and fourth translational spectral moments for the ␣^T3B and ␥^T5⌽ HR light scattering mechanisms using the SF₆–SF₆intermolecular potentials discussed here. Then, we compute the Birnbaum–

Cohen model profiles⁴¹ for each HR light scattering mechanism involved and in turn convolute the rotational transitions with their corresponding translational ones. As a next step, we normalize our SF₆collision-induced HR spectrum to the Lorentzian monomer HR spectrum of CCl₄:²⁴

I_储⁽²^␻^L⁾共⌬␻兲⫽A²^␻12 35␤²␲¹

⌫

共⌬␻兲²⫹⌫² 共26兲

with⌫ denoting its half width at half maximum 共HWHM兲. For normalization we take the spectrum Eq. 共26兲 at ⌬␻

⫽40 cm^⫺¹. We note that our reference CCl₄spectrum should be rather considered as a model spectrum resulting only from the permanent hyperpolarizability ␤ ^{of CCl}4 molecule.

Moreover, in our considerations we assume that: 共1兲 both CCl₄ and SF₆ spectra are calculated per one molecule; 共2兲 local field corrections give the same contributions to both spectra considered.

Also we imply the same laser intensity for the SF₆ as well as for the CCl₄HR light scattering experiments. In other words when calculating the normalized intensity factors, A²^␻ in Eqs.共21兲and共26兲mutually cancel. We then plot our normalized Stokes wing of the hyper-Rayleigh collision-induced spectrum

(CI)Iˆ_储⁽²^␻^L⁾共⌬␻兲⫽

(CI)I_储⁽²^␻^L⁾共⌬␻兲

(CCl₄)I_储⁽²^␻^L⁾共40 cm^⫺¹兲共27兲

in dimensionless units. Figure 1 shows the results of our computations. We note that for Aziz et al.⁴² potential the computed HR spectral shape is quite different than for the other SF₆ potentials. When calculated per one molecule the HR collision-induced SF₆spectrum at zero frequency is one order of magnitude smaller than our model the monomer HR spectrum 共due to permanent hyperpolarizability of CCl₄兲 at

⌬␻⫽40 cm^⫺¹. Moreover we estimated the CI HR spectral contributions due to ␥^T5⌽(Q₄⫽⌽) light scattering mechanism. For the same potential the ␥^T5⌽contribution is more than 100 times smaller than␣^T3B HR light scattering spectrum. However when we consider the potential leading to a very steep spectrum共e.g., Pleich one兲and the very flat spec- trum given by the potential of Aziz et al.共Ref. 43兲potential the ␣^T3B contribution and the ␥^T5⌽ contribution cross at high frequency 共about 180 cm^⫺¹兲共see Fig. 1兲.

We computed as well the integrated normalized CI HR intensities. For Aziz et al.⁴² potential the CI HR integrated intensity normalized to integrated CCl₄intensity of Eq.共26兲 amounts to 0.0060, for McCoubrey and Singh potential⁴³ to 0.0054, for Zarkova potential to 0.0045 and for Pleich potential to 0.0048. The normalized CI HR integrated intensity due to␥^T5⌽light scattering mechanism for Aziz et al. po- tential equals 0.000 013 whereas the McCoubrey and Singh potential equals 0.000 011. We note that the integrated␥^T5⌽ HR intensity amounts less that 0.2 percent of the intensity due to ␣^T3B HR light scattering mechanism considered in this work.

FIG. 1. Normalized collision-induced ␣T3B hyper-Rayleigh spectra

(CI)Iˆ_储^(2␻^L⁾(⌬␻^{) of SF}6for different intermolecular potentials. From top at 200 cm^⫺¹we plot the SF6spectra for the respective potentials, Aziz et al.

共Ref. 43兲, McCoubrey and Singh共Ref. 44兲, Zarkova共Ref. 45兲, and Pleich 共Ref. 46兲. Estimated normalized HR spectra due to ␥^T5⌽light scattering mechanism are plotted as well, respectively, for Aziz et al. and McCoubrey and Singh potentials.

8550 J. Chem. Phys., Vol. 115, No. 18, 8 November 2001 Bancewicz, Godet, and Maroulis

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V. CONCLUSION

This paper presents a spectral theory for the pair collision-induced hyper-Rayleigh spectrum of centrosymmetric octahedral molecules. The long-range contribution to the pair hyperpolarizability, the dipole²-quadrupole B⁽¹^⫹¹^⫹²⁾ term in R_AB^⫺⁴ is discussed in detail and attributed to CI HR spectrum. We apply the results of our theoretical considerations to the case of SF₆. Using our spectral hyper-Rayleigh light scattering formulas and the sulphur-hexafluoride polarizability and hyperpolarizability values as well as the SF₆ potential data, the sulphur-hexafluoride CI HR spectrum is computed. For the hyper-Rayleigh light scattering mechanism ␣^T3B discussed, the frequency dependent depolarization ratio␩⁽␯) is reduced to a constant 107/633. We normalize our computed CI hyper-Rayleigh spectrum to the monomer HR spectrum of CCl₄at 40 cm^⫺¹. Figure 1 shows an important potential’s dependence of the collision-induced hyper-Rayleigh spectrum. The spectral contribution due to

␥^T5⌽ HR light scattering mechanism is estimated as well and showed to be negligible. Contemporary success in de- tecting very weak spectral signals共see e.g., Refs. 42 and 47兲 as well as the results of our computations throws a promising light on gaseous collision-induced experimental spectros- copy.

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Collision-induced hyper-Rayleigh spectrum of octahedral molecules: The case of SF&lt;sub&gt;6&lt;/sub&gt;

Collision-induced hyper-Rayleigh spectrum of octahedral molecules: The case of SF 6

Collision-induced hyper-Rayleigh spectrum of octahedral molecules:

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Collision-induced hyper-Rayleigh spectrum of octahedral molecules: The case of SF<sub>6</sub>