Hyper-Rayleigh spectral intensities of gaseous Kr–Xe mixture

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Hyper-Rayleigh spectral intensities of gaseous Kr–Xe mixture

W. Głaz, T. Bancewicz, and J. L. Godet

Citation: The Journal of Chemical Physics 122, 224323 (2005); doi: 10.1063/1.1925267 View online: http://dx.doi.org/10.1063/1.1925267

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/122/22?ver=pdfcov Published by the AIP Publishing

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Hyper-Rayleigh spectral intensities of gaseous Kr–Xe mixture

W. Głaz^a兲and T. Bancewicz

Nonlinear Optics Division, Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland

J. L. Godet

Laboratoire des Propriétés Optiques des Matériaux et Applications, Université d’Angers, 2 boulevard Lavoisier, 49045 Angers, France

共Received 25 February 2005; accepted 7 April 2005; published online 16 June 2005兲

Binary collision-induced hyper-Rayleigh共CIHR兲spectra of Kr–Xe gaseous system are computed quantum mechanically and classically within the frequency range up to 380 cm⁻¹. The intensities are expressed in absolute units. The details of the theory developed for the CIHR spectra are given and the properties of the profiles as well as the depolarization ratio frequency dependence are discussed. The contributions to the spectra related to the vector b˜

10共r兲 and the septor b˜

关DOI: 10.1063/1.1925267兴

I. INTRODUCTION

Among the numerous nonlinear optics phenomena caused by light of high intensity, scattering is one of the most widely studied despite the difficulty that besets its observation. This is a result of the fact that the light scattering is an elementary process whereby the reaction of the medium to the incident light wave is apparent in the simplest way. The possibility of the nonlinear scattering of light had been considered long before the coming of lasers. In the phenomena of this kind the nonlinear response of the medium to the incident light beam leads to the emergence of scattered higher harmonic component in addition to the linear component at␻L, the incident laser frequency. These processes are referred to as multiharmonic scattering共MS兲. One of the best known among the MS effects and widely studied currently, both theoretically and experimentally, is the hyper-Rayleigh 共HR兲 scattering which is a three-photon event in which a system is excited with two photons of the laser frequency␻L

and spontaneously emits one photon at a frequency of about 2␻L.^1,2 Kielich^3,4 proposed a compact and deep-reaching theory of multipolar harmonic scattering within both classical and quantum treatments共see also Refs. 5–8兲.

According to the theory, at the double frequency 2␻L, the main quantity responsible for the nonlinear response of the system is the first hyperpolarizability tensor b. This is a third rank tensor, odd under coordinate inversion, whereas the inversion leaves a centrosymmetric molecule and, conse- quently the b tensor. Hence, b must vanish identically for centrosymmetric systems. Nevertheless, the light scattered at frequencies around 2␻Lin systems composed of centrosymmetric molecules has been observed; this effect being attributed to the so-called collision-induced 共CI兲 components of the b tensor which result from the interactions between the constituent molecules of the scattering media.

A pair of atoms of the same type exhibits a center of

symmetry, so the collision-induced pair hyperpolarizability can be expected only for mixtures of unlike atoms.^8–12 Re- cently, several studies have been performed^13–17reporting the hyperpolarizability computations for heterodiatomic super- molecules. In our previous work¹² we considered the HR spectrum of Ne-Ar mixture. In this paper, we present a theory and numerical calculations of the hyper-Rayleigh spectrum of Kr–Xe mixture. The details of the theory of the CIHR spectrum have been given and computations have been performed in absolute units. As far as we are aware, this is the first result of this kind. In the numerical calculations of the profiles, the CI hyperpolarizabilities obtained on the basis of quantum-chemistry methods have been applied.¹⁶ The frequency-dependent depolarization ratio D²^␯^L共␯兲 of the hyper-Rayleigh scattered light is also computed.

II. THEORETICAL CONSIDERATIONS

A pair of unlike atoms forms a short-time 共of order 10⁻¹³s兲noncentrosymmetric supermolecule of the C_⬁␷symmetry. Its leading nonlinear optical properties are given by the collision-induced hyperpolarizability tensor b. Hence, the hyper-Rayleigh spectrum of such a system is due to the diatom collision 共interaction兲-induced hyperpolarizability and quantum and/or classical dynamics of the diatom.

The wave functions of the relative motion of the two atoms of the supermolecule separated by the distance r are given by^18,19

兩i典=兩nlm典= Y_lm共rˆ兲R_i共r兲

r , 共1兲

where the angular part of the product is given by the spheri- cal harmonic Y_lm共rˆ兲, whereas R_i共r兲 is the radial wave func- tion for state i.

In order to develop quantities describing spectral properties of the HR scattered light we extend the Gordon²⁰for- mula for the linear scattering double differential cross section 共DDCS兲to the hyper-Rayleigh scattering case. However, for

a兲Electronic mail: [email protected]

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nonlinear processes, it is not easy to determine it. Therefore, for the hyper-Rayleigh scattering we have decided to define and calculate a double differential intensity per supermol- ecule共diatom兲scattered into a frequency d␻ within a solid angle d⍀and normalized to the square of the intensity of the incident radiation I₀共which will be from here on denoted as NDDI—normalized double differential intensity兲. To begin with, we consider a general scattering geometry when the incident radiation is polarized in the arbitrary direction e and the scattered light is measured behind an analyzer of arbi- trary polarization n. Then, for the quantum transition i→i⬘ we obtain

冉

⳵^⳵⍀²^Iⁿ²⳵␻^␻^L

冊

HR

冒

^I⁰²⁼^2c^␲^k^s⁴

^兺

^i,i_⬘^␳ⁱ^兩具ⁱ^⬘^兩^{n · b:ee}^兩ⁱ^典兩²^␦^共^␻⁻^␻ⁱ^⬘ⁱ^兲^,

共2兲 whereប␻ii⬘= E_i_⬘− E_i,␳iis the density-matrix element of the initial state i, and k_s stands for the wave vector of the scattered light.

Now, we simplify our light scattering geometry to the usually used right angle case. We assume the incident light propagating along the Y axis with the electric vector vibrat- ing along the Z axis. The scattered light is observed along the X direction axis. As a consequence, the total scattered inten- sity can be divided into two components: a vertical 共polar- ized兲component ZZ and a horizontal one 共depolarized兲 YZ, defined in relation to the plane of observation. On applying Eq. 共2兲, the corresponding components of NDDI are of the following form:

冉

⳵^⳵⍀²^I^AZ²⳵␻^␻^L

冊

HR

冒

^I⁰²⁼^2c^␲^k^s⁴

^兺

^i,i_⬘^␳ⁱ^兩具ⁱ^⬘^兩^b^AZZ^兩ⁱ^典兩²^␦^共^␻⁻^␻ⁱ^⬘ⁱ^兲^,

共3兲 where A = Z for the polarized component and A = Y for the depolarized component. According to Eq.共1兲we notice that the matrix element in Eq.共3兲can be separated into an angular and a translational part. From here on, we shall discuss these parts separately.

A. Angular part

In order to deal with the angular part we perform a trans- formation of the hyperpolarizability tensor b, which in Eq.

共3兲is defined in laboratory coordinates, to molecular coordinates. The calculation of the matrix element between the two spherical harmonics is greatly simplified by using a spherical representation of the hyperpolarizability tensor since, for such a case, the transformation coefficients of spherical tensors for linear molecules are expressed in terms of the spheri- cal harmonics Y_l,m共␣^,␤兲关the Wigner functions D_km^j 共␣^,␤^,␥兲 in general兴:

b_m^l =

兺

m⬘

D_mm^共^l^兲_⬘共␣^,␤^,␥兲˜b

m⬘

l . 共4兲

For linear molecules m⬘= 0. Here the tilde denotes the tensor properties referred to molecular fixed coordinates. In our considerations here we assume that the tensor b_AZZ²^␻ is totally

symmetric with respect to its indices 共Kleinman symmetry兲.²¹ Consequently, the Cartesian elements of b_AZZ²^␻ appearing in Eq.共3兲can be expressed as the following linear combination of some spherical tensors共see, e.g., Ref. 2兲:

共5兲

where i stays for the imaginary unit.

For a heterodiatomic supermolecule the totally symmet- ric hyperpolarizability tensor b˜ has seven nonzero components in the molecular reference frame. However, only two of them are independent, namely, b˜

zzz and b˜

xxz= b˜

xzx= b˜

zxx

= b˜

yyz= b˜

yzy= b˜

zyy. In the reducible spherical representation²² the b˜ tensor is of the form^23,24

˜b

111 000= b˜

zzz,

共6兲

˜b

1 1 1 0±1⫿1= b˜

1 1 1

±10⫿1= b˜

1 1 1

±1⫿10= − b˜

xxz.

The irreducible polarizability tensors are constructed from 共6兲by the standard coupling methods:^25,26

˜b

l0关共11兲a1兴=

兺

m₁,m₂

C_1m

11m₂ a−共m₁+m₂兲C_a−_共_m

1+m₂兲1m₁+m₂ l0

⫻˜b

1 1 1

m₁m₂−共m₁+m₂兲, 共7兲

where C_j

1n₁j₂n₂

jn stands for the Clebsch–Gordan coefficient.

For l = 3 the a parameter must be equal 2; whereas for l = 1 we also have a = 0 and a = 1. For a = 1 the sum in Eq. 共7兲 vanishes. To get the totally symmetric part of the b˜ tensor, the following transformation using genealogical coefficients is necessary:^2,23

˜b

10=¹₃兵

冑

5b˜

10关共11兲01兴+ 2b˜

10关共11兲21兴其. 共8兲 Consequently, for the irreducible spherical tensor compo- nents of b˜ we obtain

˜b

10= −

冑

₅³共b˜

zzz+ 2b˜

xxz兲, 共9兲

˜b

30=

冑

²5共˜b

zzz− 3b˜

xxz兲. 共10兲

In the theoretical considerations presented here we use the spherical irreducible b˜

10and b˜

30components of the hyperpolarizability tensor since they appear in a natural way, as, e.g., in Eq.共4兲. In some instances, however, other expressions are more convenient. In our previous work as well as in the numerical section below we refer rather to the tensorial in- variants b₁ and b₃,¹² closely related both to the notation b₁

= b¯ and b₃=⌬␤ used by Buckingham⁸ and Maroulis and Haskopoulos¹⁵ and to the above-mentioned quantities:

224323-2 Głaz, Bancewicz, and Godet J. Chem. Phys. 122, 224323共2005兲

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˜b

10= −

冑

⁵₃^{¯ = −}^b

冑

⁵₃^b1,

共11兲

˜b

30=

冑

²₅⌬␤⁼

冑

²₅^b3.

On applying transformation rules for the spherical tensors in Eqs.共5兲together with the Wigner–Eckart theorem and appropriate orthogonality conditions for the 3 − j Wigner co- efficients, the angular part can be expressed via the terms of the shape:

mm

兺

⬘

冏

^具^l^⬘^m^⬘^兩^coeff

^冑

^{2l + 1}⁴^␲ ^Y^km^共^rˆ^兲兩^lm^典

冏

²

=兩coeff兩²2l + 1 2k + 1H共k兲l

l⬘, 共12兲

where the numerical factors coeff stem from the spherical- Cartesian tensor transformation in Eq. 共5兲, while for the H共k兲l

l⬘we obtain H共k兲l

l⬘=共2l⬘^{+ 1}兲

冉

^l^{0 0 0}^⬘ ^k ^l

冊

²^, ^共¹³^兲

the coefficients that can be directly compared to the b_l

⬘

l ones introduced by Placzek and Teller²⁷ for the linear scattering 共k = 2兲.

By Eqs. 共3兲, 共5兲,共11兲, and 共12兲the appropriate expres- sion for the NDDI of the hyper-Rayleigh spectra for the polarized component reads

冉

⳵^⳵⍀²^I⳵␻^ZZ²^␻

冊冒

^I⁰²⁼^2c^␲^k^s⁴

^兺

i,i⬘␳i

兵

^共^{2l + 1}^兲

关

¹⁵^H^共¹^兲^l^l^⬘^兩共^˜^b¹⁰^兲ⁱⁱ^⬘^兩²

+₃₅²H共3兲l l⬘兩共˜b

30兲i

i⬘兩²

兴其

^␦^共^␻⁻^␻ⁱⁱ⬘兲, 共14兲 whereas for the depolarized component we obtain

冉

⳵^⳵⍀²^I⳵␻^YZ²^␻

冊冒

^I⁰²⁼^2c^␲^k^s⁴

^兺

i,i⬘␳i

兵

^共^{2l + 1}^兲

关

⁴⁵¹^H^共¹^兲^l^l^⬘^兩共^˜^b¹⁰^兲ⁱⁱ^⬘^兩²

+₁₀₅⁴ H共3兲l l⬘兩共˜b

30兲i

i⬘兩²

兴其

^␦^共^␻⁻^␻ⁱⁱ_⬘^兲^, ^共¹⁵^兲

where 共˜b

k0兲i i⬘=

冕

0

⬁

R_i^*_⬘共r兲˜b

k0共r兲R_i共r兲dr 共16兲

is the radial matrix element of the b˜

k0共r兲 component of the collision-induced hyperpolarizability tensor. R_i共r兲 is the ra- dial wave function with energy E and the angular momentum quantum number l; whereas for R_i_⬘共r兲the wave function cor- responds, respectively, to E⬘ ^{and l}⬘^.

When unpolarized detectors are used, the scattered intensity is a sum of two signals with orthogonal polarizations.

Then the numerical coefficients in formulas defining the intensity for unpolarized detectors关analogical to Eqs.共14兲and 共15兲兴 read, respectively, ²₉ and ₂₁² for the polarized compo- nent I_⬜共the incident light polarization perpendicular to the scattering plane兲, while for the depolarized component I_储共the incident light polarization parallel to the scattering plane兲 they are equal; ₃₅⁴ and ₁₀₅⁸ .¹² From Eqs. 共14兲 and 共15兲 we

notice that in the hyper–Rayleigh case there are a dipolar/

vector type 共for k = 1兲 and an octopolar/septor type 共for k

= 3兲 contributions to the scattered intensity. For the dipole type components of b we have

H共1兲l

l+1= l + 1

2l + 1, ⌬l = + 1,

共17兲 H共1兲l

l−1= l

2l + 1, ⌬l = − 1,

which are identical with those calculated in the infrared absorption theory.¹⁹For the octopolar type components of b we obtain

H共3兲l l+1=3

2

l共l + 1兲共l + 2兲

共2l + 5兲共2l − 1兲共2l + 1兲, ⌬l = + 1,

共18兲 H共3兲l

l−1=3 2

l共l − 1兲共l + 1兲

共2l + 3兲共2l − 3兲共2l + 1兲, ⌬l = − 1,

H共3兲l l+3=5

2

共l + 1兲共l + 2兲共l + 3兲

共2l + 1兲共2l + 3兲共2l + 5兲, ⌬l = + 3,

共19兲 H共3兲l

l−3=5 2

l共l − 1兲共l − 2兲

共2l + 1兲共2l − 1兲共2l − 3兲, ⌬l = − 3.

B. Radial part

The basic quantity to be counted when dealing with the radial part of NDDI is the matrix element of the radial- dependent component of the hyperpolarizability tensor:

B_k0共␻兲=

兺

i,i⬘共2l + 1兲H共k兲l l⬘e⁻^␤^Eⁱ

Z

⫻

冕

0

⬁

兩R_E

i⬘^lⁱ⬘

* 共r兲b˜

k0共r兲R_E

il_i共r兲dr兩²␦共␻⁻␻i⬘ⁱ兲. 共20兲 Following the usual path of calculation of such integrals, we arrive at

B_k0共␻兲=L₀³

V

冉

ប²␲^␮²

冊 ^兺

l,⌬l

共2l + 1兲H共k兲l l⬘

⫻

冕

0

⬁ dE

共EE⬘兲^1/2e⁻^␤^E兩共˜b

k0兲l

l⬘共E,␻兲兩², 共21兲

where

兩共˜b

k0兲l

l⬘共E,␻兲兩²=

冏 ^冕

⁰^⬁^R^ˆ^*^共^r;E^⬘^,l^⬘^兲^˜^b^k0^共^r^兲^R^ˆ^共^r;E,l^兲^dr

冏

²^,

共22兲 E⬘^{= E +}ប␻

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and ␮ is the reduced mass of the supermolecule, L₀ is the thermal De Broglie wavelength of the relative motion of two atoms and V is the active scattering volume. After changing the normalization conditions, Eq.共21兲finally leads to

B_k0共␻兲=បL₀³ V

兺

l,⌬l

共2l + 1兲H共k兲l l⬘共B˜

k0兲l

l⬘共␻兲, 共23兲

where

共B˜

k0兲l

l⬘共␻兲=

冕

0

⬁

dEe⁻^␤^E兩共˜b

k0兲l

l⬘共E,␻兲兩². 共24兲

Thus, according to Eqs.共14兲and共23兲, for the polarized component of the the hyper-Rayleigh scattering the following relation holds:

JZZ

2␯L共␯兲= V

冉

^⳵⳵²⍀^I^ZZ²⳵␯^␯^L

冊冒

^I⁰²⁼^␲²^k^s⁴^hL⁰³

^兺

^l ^共^{2l + 1}^兲

^兵

¹⁵^关^H^共¹^兲^l^l+1^共B^˜¹⁰^兲^l^l+1^共^␯^兲^{+ H}^共¹^兲^l^l−1^共B^˜¹⁰^兲^l^l−1^共^␯^兲兴⁺³⁵²^关^H^共³^兲^l^l+1^共B^˜³⁰^兲^l^l+1^共^␯^兲

+ H共3兲l l−1共B˜

30兲l

l−1共␯兲+ H共3兲l l+3共B˜

30兲l

l+3共␯兲+ H共3兲l l−3共B˜

30兲l

l−3共␯兲兴

其

^. ^共²⁵^兲

Bandwidth are now measured in reciprocal centimeters. In the following we shall use the notation:

V

冉

^⳵⳵²⍀^I^ZZ²⳵␯^␯^L

冊冒

^I⁰²⁼^J^ZZ²^␯^L^共^␯^兲^. ^共²⁶^兲

Then, for the spectral distribution of the polarized component of the HR scattered light per unit volume irradiated, we obtain

共27兲

where n_idenotes the partial number density of a component i and similarly for the depolarized component.

We note that forJZZ

2␯L共␯兲, according to Eq.共25兲and having in mind that 关b兴=共cm⁹erg⁻¹兲^1/2, we obtain 关JZZ

2␯L共␯兲兴

= cm⁸s erg⁻¹. Taking into account that关I₀兴= erg cm⁻²s⁻¹, the double differential intensity of HR scattering is of the dimen- sion 关共⳵²^IZZ

2␯L/⳵⍀⳵␯兲兴= erg cm s⁻¹. We shall use these rela- tions in the numerical calculations presented in the following part of our work.

III. NUMERICAL CONSIDERATIONS

The quantum-mechanical theoretical approach outlined in Sec. II forms a firm basis for a convenient routine of numerical calculations of the hyper-Rayleigh spectra. The matrix elements in Eq.共22兲can be evaluated towards resulting spectral profiles provided that an appropriate quantum- mechanical numerical method of solving the Schroedinger equation for the molecular system of interest is at hand. A number of such procedures have been developed, one of

them successfully applied by the authors of this work for the hyper-Rayleigh Ne–Ar spectra in Ref. 12. It must be recalled here that the pairs of Kr and Xe, due to their relatively large masses, may be regarded as more classical and, conse- quently, they usually require long and time consuming calculations including a numerous set of the partial waves for a large range of the angular momentum l quantum numbers.

Therefore, if appropriate accuracy of the final results is to be achieved with a reasonable computer workload applied, one has to apply an efficient algorithm and a code including sub- stantially CPU time saving procedures. The so-called complex-plane method of solving differential equations and calculating necessary matrix elements has been proved to meet the requirement.²⁸ Another possible approach for the numerical evaluation of spectral profiles for systems consisting of relatively heavy components might be to treat the problem in the entirely classical manner. In this work we concentrate on developing a general quantum theory suitable also when considering systems of light atomic pairs 共which we take into account in our following work being currently in progress兲. Nevertheless, for comparison and benchmark- ing purposes, we shall present in the section below the profiles obtained by means of the classical approach.

In order to perform the numerical calculations of the spectral profiles, there are two raw datasets needed in addi- tion to the above computing procedure. First, it is inevitable to have a functional form of the hyperpolarizability tensor dependence on the interatomic distance and secondly, a reli- able potential curves describing interactions between constituent atoms of a supermolecule.

A. Collisional hyperpolarizabilities

A number of analytical and numerical methods of evalu- ating collisional properties have been developed. Treated as a first approach effect, the multipolar approximation diatomic tensorial invariants, though expressed in a convenient analytical form, are generally valid for systems consisting of rather complex molecules interacting at relatively long inter-

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molecular separations. Therefore, for the case considered in this work, with the noble gas diatoms involved, we have decided to develop the collisional spectral profiles using the pair hyperpolarizability values obtained by having recourse to the so-called ab initio procedures of quantum chemistry.

Fortunately, a vast number of such methods have been developed and applied in recent years.^14,15,17 Of those, the datasets obtained by Maroulis and Haskopoulos^14,15provide exceptionally wide range of r distances including very small interatomic separations. In the previous work of the series¹² we successfully applied their datasets for Ne–Ar systems.

Here we also use the results produced by the Maroulis’ group for Kr–Xe pairs.¹⁶ They have been computed by means of the second-order Møller–Plesset perturbation theory 共MP2兲 with the关8s7p6d5f / 9s8p7d5f兴basis set. These data are dis- crete sets of values and should be fitted to a functional analytical dependence, so that they could be used in our computing methods. It can be easily done by means of a method analogical to the one used in Ref. 12. Likewise, we apply here fitting functions of relatively simple form of a sum of some exponential terms and inverse powers of the interatomic distance. The fitting procedure used is the well- known nonlinear least-squares Marquardt–Levenberg²⁹ 共the reader should refer to Ref. 12 for the details of the calculations; functional formulas and fitted parameters for Kr–Xe can be obtained on demand form [email protected]兲. The di- polar b₁共r兲and octopolar b₃共r兲components of the hyperpolarizability tensor are shown in Figs. 1共a兲 and 1共b兲, respectively. One can easily notice that the fitted profiles are in a quite good agreement with the dataset points. The root- mean-square deviation 共RMSD兲 between the dataset points and the fitted curve is of order of⬃0.01 for b₁and⬃0.03 for b₃, compared with ⬃0.02 and ⬃0.06, respectively, for the RMSD due to the roundoff error of the original ab initio calculations.

Mentioning some specific properties of b共r兲 may be noteworthy here. First, let us notice that the b₁共r兲 profile shows a profound negative well for short interatomic separations, whereas in the case of the previously studied system of Ne–Ar all of the functions but one 共B3LYP model兲 are exclusively positive.¹² On the other hand, the b₃共r兲 component—unlike in the Ne–Ar case—does not reach negative values. Instead, it goes through a characteristic pattern of minimum/maximum shape within the short distance range. As we shall see, this kind of behavior of the b共r兲 functions reflects in some of the features of resulting spectral profiles. The multipole contribution to b₃共r兲 has been in- cluded in the fitting expressions, yet seemingly it does not play a significant role but for relatively long intermolecular separations.

B. Potential

As far as the interaction potential is concerned, in our previous work¹² we apply—treated as a first approach approximation—the well-known, yet rather crude, Lennard- Jones 12-6 function. In this work, however, we shall refer to a more sophisticated model. Seemingly, there is a rather lim- ited number of appropriate potential surfaces for Kr–Xe proposed in literature,^30,31 the most suitable for our purposes being the concept developed by Pack et al.³⁰This potential is given in terms of analytical expressions and is based on the so-called Morse-spline-van der Waals model 共M3SV兲. All the formulas and parameters used by us are listed in Ref. 30.

The shape of the potential is shown in Fig. 2.

C. Calculations 1. Quantum spectrum

Much of the details of the quantum calculations is outlined in Refs. 12, 28, and 32 and in the articles cited there.

Customarily based on the Numerov method of solving differential equations adapted to computations performed in the complex plane, the procedure can cope successfully with the

FIG. 1. 共a兲The dipolar b1共r兲 hyperpolarizability component vs the interatomic distance.共b兲The octopolar b3共r兲hyperpolarizability component vs the interatomic distance共Ref. 16兲.

FIG. 2. The interaction potential for KrXe diatom after Ref. 30.

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numerical propagation/cancelation errors, especially notori- ous when high-frequency spectral profiles are to be treated.

Although, in general, the routine provides reasonable CPU time savings, for systems such as Kr–Xe even this approach suffers from lengthy and numerous computing steps, necessary so that desired convergence and acceptable accuracy might be achieved. Owing to the classical behavior of the supermolecule considered, the range of the quantum l num- bers and, consequently, the set of the partial functions that should be taken into account, are relatively large when compared with lighter diatoms. As a result, we have found that the uppermost value of l should reach as far as 350.

The characteristics of the other parameters involved are mainly determined by features such as the potential and the hyperpolarizability dependence on the interatomic distance, r. According to safe estimation, desirable convergence of the calculations can be gained with r ranging from the region closely approaching the steep repulsive wall of the potential to the distances of negligible potential strength. A thorough analysis based on these criteria leads to r_min= 0.77␴ ^and r_max= 10.0␴ 共␴^{= 3.73 Å}兲,³⁰ with up to 10 000 grid points in between.

Finally, initial energies of the colliding atoms rendering convergent matrix elements cover values from 0.02␧ up to 540␧共␧= 7.33⫻10⁻⁴ h a.u.兲.³⁰

In our calculations only the energies related to free pair transitions were taken into account. Obviously though, the bond and quasibond states are also possible. Yet, as it is known from other studies, they are apparently important mainly for the frequencies close to the central part of the profile around␯= 0, and therefore they do not affect the over- all wide remaining part of the spectrum. Moreover, these contributions can be calculated and added by well-known methods when necessary.

On applying the above-mentioned parameter values in the quantum computing routine, we arrive at the collisional spectral profiles shown in Figs. 3 and 4, of which the former figure illustrates a comparison between the classical and quantum features. The results will be discussed in more a detailed manner in the last section of the study.

2. Classical spectrum

A standard procedure^18,33,34 has been used to compute the classical trajectories of coliding atoms characterized by the polar coordinates关r共t兲,␪共t兲兴at a time t for a given Kr–Xe intermolecular potential U共r兲 and every possible value of b, the impact parameter and s, the relative velocity of the en- counter. Taking into account these free dimer trajectories, the Fourier transforms of the vector b₁₀共r兲 and the septor b₃₀共r兲 components of the hyperpolarizability tensor can be computed according to the Posch formulas:³³

B₁共␯^,b,s兲= 4

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⁰^⬁^b¹⁰^关^r^共^t,b,s^兲兴^cos^共²^␲␯^t^兲^cos^关^␪^共^t,b,s^兲兴^dt

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⁰^⬁^b¹⁰^关^r^共^t,b,s^兲兴^sin^共²^␲␯^t^兲^sin^关^␪^共^t,b,s^兲兴^dt

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B₃共␯^,b,s兲

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⁰^⬁^b³⁰^关^r^共^t,b,s^兲兴^cos^共²^␲␯^t^兲^cos^关^␪^共^t,b,s^兲兴^dt

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⁰^⬁^b³⁰^关^r^共^t,b,s^兲兴^sin^共²^␲␯^t^兲^sin^关³^␪^共^t,b,s^兲兴^dt

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FIG. 3. Comparison of the polarized quantum-mechanical and classical hyper-Rayleigh light scattering spectrum for Kr–Xe pairs at T = 295 K. The septor part of the HR spectrum is displayed here.

FIG. 4. The quantum-mechanical-polarized hyper-Rayleigh JZZ 2␯L共␯兲 light scattering spectrum for Kr–Xe pairs at T = 295 K共quantum-mechanical re- sults兲. The normalized double diferential intensity is given in the absolute units共see the text兲.

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⁰^⬁^b³⁰^关^r^共^t,b,s^兲兴^sin^共²^␲␯^t^兲^sin^关^␪^共^t,b,s^兲兴^dt

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共29兲 The corresponding spectral absolute intensities scattered by a scattering volume V may be derived analogically to those given for the linear scattering:¹⁸

I1共␯兲= ␲

2ck_s⁴g共␯兲c

冕

0

⬁

sf_MB共s,T兲4␲^s²^ds

⫻

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0

⬁

B₁共␯^,b,s兲2␲^bdb, 共30兲

I3共␯兲= ␲

2ck_s⁴g共␯兲c

冕

0

⬁

sf_MB共s,T兲4␲^s²^ds

⫻

冕

0

⬁

B₃共␯^,b,s兲2␲^bdb, 共31兲

where f_MB共s , T兲4␲^s²ds stands for the Maxwell–Boltzmann factor, k_s is the wave vector of the scattered light at ␭L/ 2, and g共␯兲= exp共h␯^/共2k_BT兲兲is the detailed balance correction.

It is worth noting that using a similar procedure for argon, we found the isotropic and anisotropic Rayleigh spectra in absolute units very close to those obtained by quantum mechanics.³⁵For the vector共k = 1兲and septor共k = 3兲compo- nents of the hyperpolarizability tensor we compute the zeroth moment of the respective parts of the HR spectrum according to the sum rule procedure:

V具b_k²典=

冕

0

⬁

b_k共r兲²exp

冉

⁻^U^kB^共^rT^兲

冊

⁴^␲^r²^dr. ^共³²^兲

To check our numerical results and procedures we compare the above data to the free dimer integrated intensities M_k resulting for vector and septor components of the HR spectral distributions:

M_k= 4c

␲^ks 4

冕

0

⬁Ik共␯兲g共␯兲⁻¹d␯^. 共33兲

We found that M_k⬇x_kV具b_k²典,共where x_kis the proportion of the momenta due to free dimers³⁶兲, which may be checked in Table I. Finally, for the polarizedJZZ

2␯L共␯兲and the depolarized JYZ

2␯L共␯兲the hyper-Rayleigh spectra considered quantum mechanically in the preceding section we obtain

JZZ

2␯L共␯兲=¹₅I1共␯兲+₃₅²I3共␯兲, 共34兲

JYZ

2␯L共␯兲=₄₅¹I1共␯兲+₁₀₅⁴ I3共␯兲. 共35兲 These spectra are provided in Fig. 3 共just the septor contribution of JZZ兲 together with the corresponding quantum- mechanical calculations and in Fig. 5—the depolarized component.

IV. RESULTS, DISCUSSION, AND CONCLUSION

We computed the hyper-Rayleigh binary collision- induced spectra for gaseous Kr–Xe mixture at 295 K. To this end, we have developed a quantum theory of the CIHR scattering, which provides a handy tool for numerical evaluation of quantum spectral profiles. In addition, a classical approach has been applied and the results of the two methods compared. They are shown in Fig. 3, where only the septor components of the total effect are illustrated as an example共the vector parts reveal similar properties兲. The figure presents remarkable agreement between the classical and quantum results: the two lines are barely separable except for several points around the very specific region close to ␯

⬃200 cm⁻¹ and then for the very far tail of the profile. It comes as no surprise, as—according to a numerical estimation—the root-mean-square deviation between the two spectral functions reaches no more than⬃0.025 for the wide range of the low midfrequencies共␯艋150 cm⁻¹兲, with points where the relative deviation does not exceed the value of 1%.

For higher frequencies 共␯艌180 cm⁻¹兲 the picture is sightly less satisfactory, with RMSD ⬃0.06. This discrepancy can be supposedly attributed to physical and/or numerical factors. If the latter is the case, a more refined and restrictive treatment of the computing routine might bring forth even better agreement if necessary. Nevertheless, we do believe that the discussed comparison between the classical and quantum results can serve as a solid proof of validity of both—the quantum formulas obtained as well as the quan-

TABLE I. Comparison between the integrated classical intensities M_kof the HR spectra due to free dimers and related to components b_k共i = 1 , 3兲of the hyperpolarizability tensor and the corresponding momenta xkV具b_k²典. The in- tegrals Mkand V具b_k²典are expressed in cm¹²erg⁻¹. The ratios xkgiving momentum portions due to free dimers are dimensionless.

i V具b_k²典 xk xkV具b_k²典 Mk

1 3.283⫻10⁻⁸⁴ 0.748 2.455⫻10⁻⁸⁴ 2.454⫻10⁻⁸⁴ 3 5.534⫻10⁻⁸⁴ 0.768 4.251⫻10⁻⁸⁴ 4.226⫻10⁻⁸⁴

FIG. 5. The same as in Fig. 3 but for the depolarized component. The classical profiles are displayed here.

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tum numerical calculations applied. It is noteworthy here to mention that the computations have been performed in the absolute units.

In Fig. 4 the polarizedJ_ZZ²^␯^L共␯兲hyper-Rayleigh quantum spectrum is shown. The separate contributions due to the vector b˜

10共r兲and the septor b˜

30共r兲components of the hyperpolarizability tensor are given together with the resulting profile. Interestingly, the shape of the b₃ contribution to the spectrum exhibits an intriguing behavior, which can be shown to be a result of the characteristic pattern of the b₃共r兲 dependence for short interatomic distances. From the figure it is also evident that the septor component gives noticeable contribution at high frequencies共␯艌220 cm⁻¹兲.

Figure 5 presents the depolarized spectrum obtained by means of the classical approach共the quantum graphs are al- most identical兲. In this case the predominance of the vector part of the spectrum is less significant due to more favorable values of the geometrical weight coefficients for the septor contribution in Eq. 共15兲 when compared with those in Eq.

共14兲共1 / 45 and 4 / 105 vs 1 / 5 and 2 / 35兲. It results in a stron- ger influence of the septor component on the final spectrum.

It is clearly visible in the low-frequency region in Fig. 5 and finally in the frequency dependence of the depolarization ratio.

In Fig. 6 we show the frequency-dependent depolariza- tion ratio D²^␯^L共␯兲 calculated for the linearly polarized incident light and for collecting light scattered optics applying analyzer. We note that in the frequency range of 30– 220 cm⁻¹, where the vector component predominates, the depolarization ratio is close to the theoretical value 1 / 9.¹² Interestingly though, quite visibly the function exceeds this value, especially within the convex part of the graph. This effect, with no doubt, can be accounted for by the influence of the b₃共r兲 contribution, which therefore proves to be sig- nificant. An even more pronounced role of b₃共r兲 is observed in the lower- and higher-frequency regions, where strong deviation from the value of 1 / 9 occurs.

In the study of Haskopoulos et al.¹⁶the distance dependence of the linear dipole polarizability␣共r兲is computed as well. In Fig. 7 we show the distance dependence of its anisotropy ⌬␣共r兲. Figure 8 illustrates differences between the

HR spectral distribution and the collision-induced Rayleigh scattering case. The polarizability anisotropy used for calculation of the latter is obtained by having recourse to the same ab initio methods as for the hyperpolarizability.¹⁶Quite similarly to what we found in our previous study concerning the Ne–Ar system, the linear feature is less diffuse with the peak-to-wing ratio of an order 10¹³compared to 10¹⁰for the nonlinear effect.

To sum up, in this work the hyper-Rayleigh binary spectra for a Kr–Xe gaseous mixture are computed both quantum mechanically and classically. The frequency-dependent HR depolarization ratio D²^␯^L共␯兲is calculated as well. The purely collision-induced depolarized Kr–Xe Rayleigh light scattering spectrum is also obtained for the same conditions.

To the best of our knowledge, for the first time, the CIHR spectra are given in the absolute units, which—we do believe—should offer a helpful contribution to the field of the gaseous collision–induced experimental spectroscopy.

FIG. 6. The frequency-dependent depolarization ratio D²^␯^L共␯兲computed for the linearly polarized incident light and for the collection light scattered optics applying analyzer. The strong influence of the septor共b3兲component is visible at low frequencies共about 0 – 30 cm⁻¹兲and for the high-frequecy values共about 220– 300 cm⁻¹兲. The results of quantum 共—兲and classical 共---兲computations are shown.

FIG. 7. The anisotropy of the linear dipole polarizability of KrXe diatom vs the interatomic distance after Ref. 16.

FIG. 8. The depolarized Rayleigh共R兲purely collision-induced light scattering spectrum for the Kr–Xe pair computed with the polarizability anisotropy presented in Fig. 7. The hyper-Rayleigh共HR兲spectrum computed for the same temperature and for the same model is also presented. The R spectrum has been adjusted to fit the HR one at the initial intensity. For comparison, the Ne–Ar spectra are added共Ref. 12兲.

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ACKNOWLEDGMENTS

We would like to thank Professor George Maroulis for sending us their results prior to publication and for many interesting discussions.

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Hyper-Rayleigh spectral intensities of gaseous Kr–Xe mixture

Hyper-Rayleigh spectral intensities of gaseous Kr–Xe mixture

Hyper-Rayleigh spectral intensities of gaseous Kr–Xe mixture

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