Moments of hyper-Rayleigh spectra of selected rare gas mixtures

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Moments of hyper-Rayleigh spectra of selected rare gas mixtures

Tadeusz Bancewicz, Waldemar Głaz, and Jean-Luc Godet

Citation: The Journal of Chemical Physics 127, 134308 (2007); doi: 10.1063/1.2772262 View online: http://dx.doi.org/10.1063/1.2772262

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/127/13?ver=pdfcov Published by the AIP Publishing

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Moments of hyper-Rayleigh spectra of selected rare gas mixtures

Tadeusz Bancewicz^a兲 and Waldemar Głaz^b兲

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

Jean-Luc Godet^c^兲

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

共Received 4 May 2007; accepted 24 July 2007; published online 3 October 2007兲

In this work we have analyzed spectral moments characterizing properties of the collisionally hyper-Rayleigh scattered light. This is a supplementary study undertaken in order to complete the series of our previously published papers on the collisional hyper-Rayleigh scattering spectral profiles. In order to evaluate the moments we have extended the theory so that it could embrace the 共hyper兲polarizabilities of higher rank. Using the expressions developed on the grounds of the theoretical principles and applying appropriate computational methods with ab initio hyperpolarizability values as an input, we have obtained desirable moment values for three diatomic noble gas systems: HeNe, HeAr, and NeAr, at several temperature points. The semiclassical and the quantum treatments have been taken into account, and the moments were calculated both from the sum rule method as well as from the spectral profiles. The results were compared and discussed.

©2007 American Institute of Physics.关DOI:10.1063/1.2772262兴

I. INTRODUCTION

If a system is illuminated with a high density flux of laser light photons of a frequency of␻Land scatters photons of frequency of about 2␻L, it is said to exhibit hyper- Rayleigh scattering or nonlinear 共second-harmonic兲 scattering.^1,2The light scattering process depends on the first hyperpolarizability tensor bijk. For centrosymmetric scattering microsystems the first hyperpolarizability tensorb_ijkvan- ishes identically; therefore, no hyper-Rayleigh signal is expected in the single-molecule case. Nevertheless, during a molecular collision the electron clouds of the molecules共at- oms兲 repeal, attract, and distort one another, which lead to experimentally observable changes in the properties of the molecules.^3–5 Intermolecular interactions often break the symmetry 共for instance that of dissimilar atomic pairs兲^6–8 and the process, forbidden in a monomer description, is ob- served even at a level as low as that of the binary regime.

Therefore, in a short time共⬇10⁻¹³s兲of a fly-by encounter of the colliding atoms forming the supermolecule, the collision- induced interatomic distance dependent hyperpolarizability b共R兲emerges. As a result, the collision-induced double frequency dipole moment␮^2␻^L共R兲is induced in the supermolecule being the origin of the collision hyper-Rayleigh scat- tering共CHRS兲.

As small as it is, when compared with its monomer counterparts, this effect is worth giving attention to since, in view of the tremendous progress in the experimental methods of molecular spectroscopy,^9–15hopefully, in the near fu-

ture it may find itself within the reach of measurement capa- bilities. Coincidently, this development in the experimental techniques has been accompanied by increasing power of the quantum chemistry theoretical and numerical methods, which have been providing a plethora of microscopic molecular properties during the recent decades. Among them, the values of the hyperpolarizabilities of different orders and for a large collection of molecules/atoms have been reported by Maroulis and Haskopoulos^16,17 and Haskopoulos et al.¹⁸ Owing to these two factors, an incentive has arisen to formu- late a framework for a theoretical description of CHRS. As a result we have published a series of works dealing with CHRS processes in systems consisting of a variety of noble gas pairs.^19–23 In those papers we have mainly concentrated on different aspects of the theoretical and numerical descrip- tions of the hyper-Rayleigh spectral distribution functions. It is well known, however, that although such profiles are prob- ably the richest source of information available from spectroscopic measurements, there are also other quantities de- scribing scattering processes, even though they convey less detailed knowledge. They are, nevertheless, quite convenient in many situations and widely used. In this paper we are going to supplement the series of the previous spectral stud- ies by discussing a kind of such parameters—the CHRS spectral moments 共up to the fourth order兲.

The moments characterize certain aspect of spectral profiles in an important way²⁴ and they may be used so as to evaluate the feasibility of CHRS experiments.²¹These quantities make it possible, for instance, to perform comparisons between the integrated intensities of the CHRS spectrum and of the hyper-Rayleigh signal of a noncentrosymmetric molecule used as a reference. Besides, their values calculated by means of the sum rule expressions—recalled and developed hereafter—can be compared with these obtained from ex-

a兲Electronic mail: [email protected] URL: http://www.zon8.physd.amu.edu.pl/

b兲Electronic mail: [email protected]

c兲Electronic mail: [email protected]

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perimental and/or theoretical spectra and thus they constitute convenient criterions of checking and evaluating the reliability and quality of models used 共e.g., the hyperpolarizability and the potential surfaces兲.

The theoretical and numerical considerations having been developed so far provide a number of techniques of the spectral CHRS calculations—the most rigorous and accurate seemingly being the one strictly based on the quantum- mechanical theory. Practically though, it is often the case that this sort of treatment may leave us with lengthy and cumber- some procedures which involve time-consuming computations. Luckily, in many instances such a puristic attitude is not required as, depending on a scattering system properties, less refined routines may be applied with a small or even next to none loss of precision. However, as far as our study is considered, this statement seems not applicable for the purely classical calculations; it can be shown that for such atom pairs of relatively lesser masses such as those we study, this kind of method is prone to produce results of an accuracy that sometimes is not sufficient enough. Therefore, a more recommended routine to replace the quantum approach is the coarse classical description supplemented with a set of amendments which are to reflect the quantum aspect of the phenomena studied. This treatment known as the semiclassical approach, e.g., Ref. 4, apparently turns out to be really handy and accurate tool of the spectral calculations, yet it may happen that its feasibility meets limitations imposed mainly by the masses of the systems considered as well as the temperatures in which the scattering processes take place.

Generally, the less massive the colliding atoms and the lower the temperature, the more quantumlike situation is generated.

Anyway, it is often assumed to be desirable that the abovementioned methods of computing be compared and/or tested with regard to their applicability for a particular system, which is also going to be one of the goals of the reported research.

In what follows we shall present an analysis of the spectral CHRS moments computed for diatomic supermolecules composed of noble gas atoms: He–Ne, He–Ar, and Ne–Ar.

Analytical formulas will be given in a closed form and a brief theoretical description of CHRS followed by computational details will be provided. The calculations applying relatively simple expressions 共sum rules兲 are supplemented by those based on the line shape computations.^20–22The temperature dependence of the CHRS moments for all atomic pairs is discussed. According to the rationale stated in the paragraph above, all the quantities are obtained by having recourse to both the semiclassical as well as the quantum approach rules and eventually the results are compared.

II. THEORY

A pair of unlike atoms during their fly-by encounter forms a supermolecule of theC_⬁␯ symmetry. For this point group, assuming Kleinman’s full permutational symmetry of indices, the first hyperpolarizability tensor b has only two independent components expressed in the molecular frame of reference, namely,˜b

333and˜b

113=˜b

223. The irreducible Carte- sian tensor formula,

bijk=¹₅共b˜

333+ 2b˜

113兲共␦ijSk+␦jkSi+␦ikSj兲 +¹₅共b˜

333− 3b˜

113兲关5SiSjSk−共␦ijSk+␦jkSi+␦ikSj兲兴, 共1兲 withS designating the unit interatomic vector, immediately shows that this tensor has only two rotational invariants, namely,

共a兲 the vectorlike part,

˜b

1=³₅共b˜

333+ 2b˜

133兲, 共2兲

共b兲 and the septor part,

˜b

3=˜b

333− 3b˜

133. 共3兲

The tilde indicates the hyperpolarizability tensor in the molecular reference frame. For the collision-induced hyperpolarizability these invariants are distanceR dependent共hy- perpolarizability surface兲. These collision-induced invariants have been computed using advanced methods²⁵ of quantum chemistry 共MP2 method applied in this case兲 by Maroulis and co-workers for several rare gas atoms pairs: He–Ne,²¹ He–Ar,²² Ne–Ar.^16,17

The irreducible spherical tensor representation of the hyperpolarizability tensor b共R兲 happens to be very useful^26,27 as well. Due to the symmetry reason, for linear supermolecules, in the molecular reference frame only the irreducible hyperpolarizability spherical components with magnetic quantum number M= 0 survive.^28,29 Consequently, the labo- ratory frame spherical irreducible hyperpolarizability components bLM can be expressed by their molecular reference frame components˜b

L0together with the Wigner rotation matrix D_M0^L 共Rˆ兲. For our systems the specific Wigner rotation matrix finally reduces to spherical harmonics YL共Rˆ兲. As a result we arrive at

bLM共R兲=D^L_M0^*共Rˆ兲b˜

L0共R兲=

冑

2L⁴^␲+ 1YLM共Rˆ兲b˜

L0共R兲. 共4兲 Consequently, for the dipolar collision-induced hyperpolarizability component we have

b_1M共R兲=

冑

⁴3^␲Y_1M共Rˆ兲b˜

10共R兲, 共5兲

whereas for the octopolar configuration interaction hyperpolarizability part we obtain

b_3M共R兲=

冑

⁴7^␲Y_3M共Rˆ兲b˜

30共R兲. 共6兲

The spherical irreducible hyperpolarizability tensor rotational invariantsb˜

10共R兲 andb˜

30共R兲 are related to their˜b

1共R兲 andb˜

3共R兲Cartesian counterparts关given by Eqs.共2兲and共3兲兴 by the following relations:

˜b

10共R兲= −

冑

⁵3˜b

1共R兲, ˜b

30共R兲=

冑

²5˜b

3共R兲. 共7兲

134308-2 Bancewicz, Głaz, and Godet J. Chem. Phys.127, 134308共2007兲

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The hyper-Rayleigh light scattering signal is composed of two parts: the dipolar共vector兲part and the octopolar共sep- tor兲 part.^21,26,30 To deal with each part we consider the vector-共L= 1兲 and the septorlike共L= 3兲correlation function,

HR共L兲F共t兲=具bL共0兲· bL共t兲典. 共8兲

Eventually, we calculate the hyper-Rayleigh light scattering vector and septor spectral moments by^4,31–33

Mn

L=i⁻ⁿ

冏冓

^b^L^共0兲^·^dⁿ^dt^b^Lⁿ^共t兲

冔 ^冏

^t=0^, ^共9兲

where the time derivatives are evaluated either quantum mechanically or classically.

III. CHRS SPECTRAL MOMENTS: OUTLINE OF PROCEDURES

There are several well-established methods of spectral moment evaluation, both analytically and numerically. Obvi- ously, the choice of one of them to be applied is mostly governed by the question of to what extent the classical nature of the physical system studied prevails and, last but not the least, what is the computing precision required in each particular case. The existing theories offer a number of quan- titative criterions for judging which kind of behavior is to be expected. More precisely, when we consider, for example, the angular momentum parameter defined in Ref.4,

បlav⬇␮

冉

^8k␲␮^B^T

冊

^1/2^b^av ^共10兲

共whereb_av is the mean range of the interaction of two colliding molecules/atoms兲, its value considerably exceeding unity indicates that the laws of classical physics are appli-

cable. In other words, this statement can also be expressed on more qualitative grounds by referring to the masses and the temperature of collisional microscopic species—the lighter the molecule/atom and the lower the temperature, the more pronounced might be the share of quantum properties in the overall picture of the process studied. It should be stated though, that in the case, when more rigorous requirements concerning the computing precision are assumed for diatomic systems such as the less massive pairs considered here, even relatively large values of l_avmay not exclude the quantumlike treatment. In order to find out the importance of the quantum nature of the systems studied we shall apply and compare, with regard to results they render, two methods of computing the CHRS moments: the semiclassical and the purely quantum approach.

Afterward we shall present briefly the basic principles on which the methods mentioned are based and the details of the numerical procedures they bring forth; the discussion of the results will follow.

A. CHRS classical and semiclassical moments Using the formulas derived in Ref.34we calculate the HR classical zeroth, second, and fourth moments, namely, 共a兲 the zeroth moment reads

M_0c^L =V具bL0共R兲²典, 共11兲共b兲 while for the classical second moment we have

M_2c^L =V

冉

^k␮^B^T

冊冓

^L共L^{+ 1兲}^˜^b^L0^R^共R兲² ²⁺^˜^b^L0^⬘ ^共R兲²

冔

^, ^共12兲

共c兲 and for the classical fourth moment we obtain

M_4c^L =V

冉

^k␮^B^T

冊

²

冓

^{L共2 + 5L}^{+ 6L}²^{+ 3L}³^兲^b^˜^L0^R^共R兲⁴ ²^{− 2L共L}^{+ 1兲}^˜^b^L0^R^共R兲³

^冋

^˜^b^L0^⬙ ^共R兲R⁺^b^˜^L0^⬘ ^共R兲

^冉

^{8 −}^U^k^⬘^B^共R兲^T ^R

^冊 ^册

+ 1

R²

冋

^3b^˜^L0^⬙ ^共R兲²^R²^{− 2b}^˜^L0^⬘ ^共R兲b^˜^L0^⬙ ^共R兲R

冉

^{− 2 +}^Uk^⬘B^共R兲T R

冊

⁺

冉

^{8 + 4L}^{+ 4L}²^{− 4}^Uk^⬘B^共R兲T R+

冉

^Uk^⬘B^共R兲T

冊

²^R²

冊

^b^˜^L0^⬘ ^共R兲²

册冔

^, ^共13兲

whereU共R兲 designates the isotropic intermolecular interaction potential, b˜

L0⬘ ⁼⳵^˜^bL0/⳵^R ^and ^b^˜L0⬙ ⁼⳵²^b^˜L0/⳵^R²^, ^U⬘

=⳵^U^/⳵^R,␮is the reduced mass,kBstands for the Boltzmann factor, and T for the temperature. Furthermore, 具F共R兲典 denotes the classical low-density mean value,

具F共R兲典=4␲ V

冕

0

⬁

F共R兲R²g₀共R兲dR, 共14兲

where g₀共R兲= exp共−U共R兲/kBT兲 is the classical pair correlation function. In order to shorten the notation, we will sup- press hereafter the R dependence of the hyperpolarizability

components as well as the tilde. For the fourth moment of the CHRS dipolar component we obtain

M_4c^L=1=V

冉

^k␮^B^T

冊

²

冓

^3b¹⁰^⬙²⁺^4b¹⁰^⬙^R^b¹⁰^⬘ ⁺ ⁴^共^4b¹⁰^⬘²^R⁻²^b¹⁰^⬙^b¹⁰^兲

− 32b₁₀⬘ b₁₀ R³ +16b₁₀²

R⁴ + 2U⬘ kBT

⫻

冉

^2b^R¹⁰^⬘²^b¹⁰⁻^2b^R¹⁰^⬘²⁻^b¹⁰^⬙ ^b¹⁰^⬘

冊

⁺^b¹⁰^⬘²

冉

k^UB^⬘T

冊

²

冔

^,

共15兲

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whereas for the CHRS octopolar component we have

M_4c^L=3=V

冉

^k␮^B^T

冊

²

冓

^3b³⁰^⬙²⁺^4b³⁰^⬙^R^b³⁰^⬘ ⁺^8共7b³⁰^⬘²^{− 3b}^R² ³⁰^⬙ ^b³⁰^兲

−192b₃₀⬘^b30

R³ +456b₃₀² R⁴ + 2U⬘

kBT

⫻

冉

^12b^R³⁰^⬘²^b³⁰⁻^2b^R³⁰^⬘²⁻^b³⁰^⬙ ^b³⁰^⬘

冊

⁺^b³⁰^⬘²

冉

^k^UB^⬘T

冊

²

冔

^.

共16兲 In order to take into account the low-order quantum corrections 共up toប²兲, we can consider the Wigner-Kirkwood ex- pansion of the pair-distribution function,³³

g₂共R兲=g₀共R兲

再

^{1 +}¹²^␮^ប^共k²^B^T兲²

^冉

^2k^U^⬘^B²^T⁻^2U^R^⬘⁻^U^⬙

^冊冎

^.

共17兲 Replacingg₀byg₂−g₀in Eqs.共11兲and共12兲, we thus obtain

⌬M_0c^L and⌬M_2c^L, quantum corrections to the zeroth and second classical moments, respectively. The semiclassical moments M₀^L, M₁^L, and M₂^Lcan be deduced from these corrections and from the classical moments. To the lowest order for quantum effects we have

M₀^L=M_0c^L +⌬M_0c^L, 共18兲

M₁^L=␤ប

2 M_2c^L, 共19兲

M₂^L=M_2c^L +⌬M_2c^L + 1

3

冉

^␤²^ប

冊

²^M^4c^L^. ^共20兲

B. Quantum spectral moments

Thereafter in this section we shall introduce the basic principles of the quantum treatment of the problem of CHRS on which the spectral moment evaluation may be founded.

Understandably, more thorough study should be rather performed on the systems in which the “quantumness” might pronounce to greater extent, i.e., the lightest of the pairs taken into account, He–Ne and He–Ar, yet the Ne–Ar supermolecule would also be given attention to.

Theoretical approach that we are going to apply here was developed in a number of works, e.g., Ref. 31. Since then the methods have proven to be a very efficient tool for interpreting the phenomena in the field of both the collisional absorption effect and the Rayleigh and Raman scattering processes. In what follows we shall present that the procedures may be easily extended to the CHRS case.

The CHRS zeroth quantum spectral moment assumes the form analogous to the classical expression,

M_0q^L =V具b˜

L0

2 典q, 共21兲

though the averaging procedure differs, 具F共R兲典q=4␲

V

冕

0

⬁

F共R兲R²g_q共R兲dR, 共22兲 as it includes gq共R兲 being the quantum radial pair- distribution function.

The first spectral moment does not appear in the classical description, while its quantum representation is described by

M_1q^L =V ប

2␮

冓

^L共L^{+ 1兲}^˜^b^R^L0²² ⁺^共b^˜^L0^⬘ ^兲²

冔

^q^. ^共23兲

As far as the second spectral moment is considered, its form is relatively involved, so it is given in its final form here,

M_2q^L = 4␲

␤␮

冉

⁸^L^␲⁰²

^冕

⁰^⬁^R²^dR

^冉

^3共b^˜^L0^⬙ ^兲²^{+ 2b}^˜^L0^⬘ ^˜^b^L0^⵮ ^{+ 2}^L^共^L^R^{+ 1}² ^兲^共b^˜^L0^⬘ ^兲²^{− 4}^L^共^L^R^{+ 1}³ ^兲^b^˜^L0^˜^b^L0^⬘ ⁺^L²^共^L^R^{+ 1}⁴ ^兲²^b^˜^L0²

^冊

^g^q^共R兲

−

冕

0

⬁

R²dR共b˜

L0⬘ 兲²2␤

冉

^⳵^g⳵␤^q^共R兲⁺^Ug^q^共R兲

冊

⁺

冕

0

⬁

R²dR

冉

^共b^˜^L0^⬘ ^兲²^共3g^q^共R兲^{− 2g}^r^共R兲兲⁺^L共LR^{+ 1兲}² ˜b

L0

2 gr共R兲

冊冊

^. ^共24兲

The pair correlation functions that appear in the above expressions, gq共R兲andgr共R兲, can be derived within the quantum approach from the first principles and in a “ready-to- compute” shape read, respectively,^4,33,35,36

gq共R兲=

兺

l=0

⬁

gl共R兲= L₀³␮^3/2 2^1/2␲²ប³

兺

l=0

⬁

共2l+ 1兲

⫻

冕

0

⬁

dEE^1/2exp共−␤^E兲REl*共R兲REl共R兲共25兲 and

gr共R兲= L₀² 4␲^R²

兺

l=0

⬁

l共l+ 1兲gl共R兲, 共26兲

where REl=⌿El共R兲/R in which⌿El共R兲 is a solution of the radial Schrödinger equation of the system considered related to the energy E, whereas L₀ is the well-known thermal de Broglie wavelength of the relative motion of two particles.

IV. NUMERICAL DETAILS

Here, we consider the He–Ne,¹¹ He–Ar,²² and Ne–Ar¹⁹ noble gas mixtures, with their ab initio hyperpolarizability

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surfaces ˜b

1共R兲 and ˜b

3共R兲 computed by Maroulis and Haskopoulos^16,17; and Haskopoulos et al.¹⁸ and interatomic potentials recently proposed by Lopez-Cacheiroet al.³⁷Ac- cording to Eqs.共11兲,共12兲,共15兲, and共16兲, the classical zeroth, second, and the fourth CHRS spectral moments have been directly computed for the dipolar 共L= 1兲 and octopolar 共L= 3兲parts of the CHRS intensity of each heterodiatom. The semiclassically corrected CHRS zeroth, first, and second moments were computed by using Eqs. 共18兲–共20兲. Quantum- mechanical calculations were performed at the same time.

Rather obviously, for the supermolecular collisions such as those of interest in this work, no general analytical solutions of the Schrödinger equations are expected to be at hand.

Hence, we are forced to bypass the difficulty of finding the necessary sets of wave functions by having recourse to one of the standard techniques of numerical treatment of the problem. We can, for example, deal with it on the grounds of the method utilizing the Numerov procedure of solving dif- ferential equations,³⁸with two initial points derived from the Wentzel-Kramers-Brillouin approximation³⁹ and the normalization applying an appropriate linear combination of free- particle solutions given in terms of the spherical Bessel functions. The procedure follows the path which is stated, for instance, in Ref. 40, where the details can be found if necessary. In so doing we arrive at the set of resulting values of the moments of the zeroth, first, and second orders for every system considered and at a variety of temperatures, yielded by both quantum and semiclassical codes. The results stem- ming from these two approaches will be discussed and compared hereafter. Besides, throughout the further course of the report, we choose to provide the reduced moments ⍀1

=M₁/M₀and⍀2=共M2/M₀兲^0.5instead ofM₁andM₂. Indeed, these characteristic frequencies give more readable information about the asymmetry 共⍀1兲 and the narrowness 共⍀2, mainly兲 of the spectrum. Tables I and II present the spectral moments for all pairs given solely at room tempera-

ture. Conclusions concerning the other possible diatom/

component combinations could be very closely compared to these abovementioned; therefore, for the sake of briefness, we would rather refrain from reporting the related data here.

All these quantities which are originated from the sum rule principles are denoted by “sr” in the tables. On the other hand, moments and characteristic frequencies obtained from the integrated profiles of the computed classical and quantum-mechanical spectra given in Refs.20–22are distin- guished from the precedents by the abbreviation “cs”共stand- ing for “computed spectra”兲. Concerning classical spectra, cs first and second semiclassical moments can be obtained from two methods. First, they can be deduced from ^共cs兲M_2c^L,

共cs兲M_4c^L, and Eqs.共19兲 and共20兲as long as ⌬M_2c^L can be ne- glected or correctly evaluated. Second, they can be directly deduced from the desymmetrized semiclassical spectrum.

The results obtained with the two methods must be close together for the most appropriate desymmetrization procedure. This can be a test for the latter procedure and provide some order of magnitude of the uncertainties on the computed moments. In our case, the agreement is generally very satisfactory for the following desymmetrization formula given in Frommhold’s book,⁴

I_desy共␯兲= h␯^/kBT

1 − exp共−h␯^/kBT兲I_class共␯兲. 共27兲 Bound and metastable dimers should be considered too.

At room temperature, for example, portions of the zeroth- order moments due to the bound and metastable dimers, which are to be excluded from the semiclassical intensities, account for less than 0.4%共He–Ne兲to 3%共Ne–Ar兲, according to the calculations performed using the statistical methods developed by Stogryn and Hirschfelder⁴¹or by Levine.⁴² However, especially at low temperatures and for heavy systems, uncertainties of the values of the free dimer portions of the semiclassical moments can also be a source of error in

TABLE I. Quantum共Q兲and semiclassical共SC兲moments associated with b₁₀due to the free dimers of the heterodiatoms He–Ne, He–Ar, and Ne–Ar at T= 295 K: zeroth-order moments M₀ 共in 10⁻⁸⁸cm¹²erg⁻¹兲, and ratios

⍀1=M₁/M₀and⍀2=共M₂/M₀兲^0.5共in 10¹²s⁻¹兲. We give the values obtained from the computed spectra共cs兲and these are given by means of the sum rules共sr兲.

295 K

Diatoms

He–Ne He–Ar Ne–Ar

M₀ Q_cs 3.945 1.331⫻10³ 1.505⫻10³

Q_sr 3.928 1.340⫻10³ 1.493⫻10³ SC_cs 4.013 1.355⫻10³ 1.507⫻10³ SC_sr 3.945 1.341⫻10³ 1.504⫻10³

⍀1 Q_cs 3.149 2.796 0.822

Q_sr 3.167 2.797 0.813

SC_cs 3.111 2.772 0.844共±1 %兲

SC_sr 3.165 2.800 0.8450

⍀2 Q_cs 16.05 15.13 8.056

Q_sr 15.95 15.13 8.066

SC_cs 15.95 15.06 8.148

SC_sr 15.99 15.07 8.144

TABLE II. Quantum共Q兲and semiclassical共SC兲moments associated with b₃₀due to the free dimers of the heterodiatoms He–Ne, He–Ar, and Ne–Ar atT= 295 K: zeroth-order momentsM₀共in 10⁻⁸⁸cm¹²erg⁻¹兲, and ratios⍀1

and⍀2共in 10¹²s⁻¹兲.

295 K

Diatoms

He–Ne He–Ar Ne–Ar

M₀ Q_cs 0.123 1.128⫻10² 1.348⫻10²

Q_sr 0.123 1.129⫻10² 1.362⫻10² SC_cs 0.1250 1.146⫻10² 1.357⫻10² SC_sr 0.1244 1.130⫻10² 1.362⫻10³

⍀1 Q_cs 2.913 4.708 1.446

Q_sr 2.947 4.708 1.410

SC_cs 2.775 4.623 1.477

SC_sr 2.79共±1.5%兲 4.69共±1 %兲 1.472

⍀2 Q_cs 16.89 19.77 10.73

Q_sr 16.95 19.83 10.64

SC_cs 16.6共±1.5%兲 19.59 10.82

SC_sr 16.9共±1 %兲 19.66 10.79

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the sr computation procedure. The uncertainties of our semiclassical results can be kept at a level of 0.5%. Exceptions are mentioned in TablesIandII.

V. CONCLUSIONS AND FINAL REMARKS

In this work we analyze the properties of the hyper- Rayleigh collisional spectra expressed via sets of the first three spectral moments calculated semiclassically as well as quantum mechanically. To this end, we develop suitable theoretical and numerical tools. The nonlinear scattering effect of interest unlike its counterparts attributed to permanent molecular properties 共hyperpolarizabilities兲, which have been long studied, still requires a proper treatment and insight in view of new experimental possibilities becoming more precise still.

The kind of effects considered in the sections above, yet driven not by interaction-induced processes but appearing due to permanent nonlinear properties of monomer molecular systems, has been a subject of both experimental and theoretical researches for a last few decades. This is because the intensity of the light scattered in this case is high enough to be measured by means of the present day experimental techniques. On the other hand the collisional spectra are relatively less intense. Having taken into account the results obtained, we have found it useful to benchmark a collisional treatment signal against the one scattered by a monomer. To this end the CHRS moment data given in TablesIandIIcan be compared with the integrated monomer allowed HR intensities measured recently in gases composed of CX₄ molecules.¹²We have chosen the CF₄gas as a reference. The reduced monomer allowed hyper-Rayleigh intensity共per one CF₄ molecule兲 for the vertical-vertical 共VV兲 polarization reads¹

S_VV^CF⁴=¹²₃₅b_xyz² . 共28兲 Taking into account that¹² b_xyz^CF⁴= 5.4 a.u. we obtain S_VV^CF⁴= 7.64⫻10⁻⁶⁴cm⁹erg⁻¹. The respective reduced CHRS intensity for selected A–B pair reads 共e.g., for the vector part兲

S_VV^A–B=¹₅M₀^A–BNL, 共29兲 whereNL= 2.686 763⫻10¹⁹stands for Loschmidt’s number.

Then for the vector part of the He–Ar CHRS spectrum we obtainS_VV^He–Ar= 7.164⫻10⁻⁶⁷cm⁹erg⁻¹. Consequently for ar- bitrary density we have

共⳵^IVV2␻L/⳵⍀兲He–Ar

共⳵^I_VV^2␯^L^/⳵⍀兲CF₄

=␳He␳Ar

␳CF₄

9.37⫻10⁻⁴, 共30兲 where␳i denotes the density of the speciesiin amagats.

It comes as no surprise, obviously, that the ratio rendered by our estimation indicates that in the case of CHRS we deal with a phenomenon of much more subtle nature than in the monomer related scattering. Nevertheless, as it was mentioned before, due to the incredible progress in spectroscopic measurements, we may expect in several years to come a rapprochement between the theory and the experiment that will allow measurements with which our theoretical predic- tions and their numerical outcome could be confronted.

As far as these computational results of our method are considered, they are partially given in TablesIandIIand in Figs. 1–3, where only the most typical examples of the physical situations studied have been taken into account.

An interesting observation concerning the obtained data is that quite a number of features characteristic of different aspects of the calculated quantum and semiclassical moments for the variety of cases taken into account are very close to each other. Quantum and semiclassical results are in agreement within numerical precision of our computations.

Our quantum numerical procedures are sufficient for an accuracy of the final results of 1.5%. For instance, there is very little discrepancy between the semiclassical results and the quantum values obtained for the same physical systems and conditions. The results presented show that in many typical occurrences of CHRS the semiclassical treatment turns out to be a sufficient tool of the spectral moment analyses that can replace the strict quantum approach except for rather extreme physical conditions and high accuracy requirements.

The other obtained values, which tend not to differ no- ticeably between themselves, are the moments obtained by the sum rule techniques versus those computed from spectral profiles. From the data given in TablesIandIIa comparison can be made, from which it is clearly evident that both the semiclassical quantities as well as the quantum ones calcu-

FIG. 1. 共a兲 The quantum-mechanical 共crosses兲 and the semiclassical 共squares兲zeroth spectral moments for the He–Ar diatoms. Values related to two spectral components: b₁₀ and b₃₀ 共in 10⁻⁸⁶cm¹²erg⁻¹兲. 共b兲 The quantum-mechanical共crosses兲and the semiclassical共squares兲spectral moments of the zeroth order for the He–Ne and He–Ar diatoms,b₁₀component only共in 10⁻⁸⁸cm¹²erg⁻¹兲.

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lated 共at 295 K兲 by the two methods mentioned are very close to each other, the only set of data showing larger dis- crepancies being the first rank moments for the Ne–Ar pairs.

An important conclusion may be derived from this observation: the agreement between the results yielded by the independent sum rule routine and those provided by the spectral shape calculations gives proof of reliability of our previously evaluated CHRS spectral lines.^19–23

Finally a closer look should be cast at Figs.1–3. They illustrate the dependence of the moments 共and/or reduced moments兲 on temperature. First and foremost it must be pointed out that the points related to the quantum and the semiclassical values of the moments are barely distinguish- able, if at all, so the temperature dependence shows the same pattern for these two types of the computational approach.

Moreover, the tendency illustrated in the figures is pretty clear in the case of M₀ and⍀2, as these quantities can be unambiguously tied to well-defined physical properties: M₀ represents the integrated intensity of a spectrum, whereas⍀2

is closely related to the mean width of it. These two parameters grow with temperature for all systems under consider- ation, which resemble properties of spectra of the collisional scattering of the other kind 共Rayleigh兲. Regretfully, such a conclusion cannot be valid as far as the first CHRS moments

are considered. Generally, the first moment is to some extent dependent on the asymmetry of the spectral line shape; in particular, it reduces to zero for the totally symmetrical classical profiles. Here, however, the picture is more entangled since there are also other factors involved that shape the⍀1

temperature dependence, i.e., the absolute intensities of the spectra and the collisional mechanism contributing to the hyperpolarizability tensor values. As a consequence⍀1 may both decrease with temperature 共toward more symmetrical profiles兲, yet it may well be increasing in other circum- stances 共Fig. 2兲. In the systems discussed in this work the first kind of behavior is typical of almost all diatomic pairs and collisional effects except for the b₃ contribution to the He–Ne spectral component. Thus, although carefully applied it gives some insight into spectral shapes, by no means can

⍀1 be treated as universal indicator of the CHRS spectral line asymmetry. When in doubt, some additional measures are to be taken so as to deduce about it共e.g., normalization to the maximum value of the intensity at peak of spectral lines兲.⁴⁰

To sum up, in this work we have extended our previous theoretical and numerical treatments of the hyper-Rayleigh collisional effect in the molecular light scattering to an analysis of the spectral moments—parameters characterizing

FIG. 3. 共a兲 The quantum-mechanical 共crosses兲 and the semiclassical 共squares兲reduced second spectral moments for the He–Ar diatoms. Values related to two spectral components: b₁₀ and b₃₀ 共in 10¹³s⁻¹兲. 共b兲 The quantum-mechanical共crosses兲and the semiclassical共squares兲second spectral moments for the He–Ne and He–Ar diatoms,b₁₀component only共in 10¹³s⁻¹兲.

FIG. 2. 共a兲 The quantum-mechanical 共crosses兲 and the semiclassical 共squares兲 reduced spectral first moments for the He–Ar diatoms. Values related to two spectral components:b₁₀andb₃₀共in 10¹²s⁻¹兲. Additionally the temperature dependence of theb₃₀first reduced moment for the He–Ne pairs.共b兲The quantum-mechanical共crosses兲and the semiclassical共squares兲 zeroth spectral moments for the He–Ne and He–Ar diatoms,b₁₀component only共in 10¹²s⁻¹兲.

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important properties of the scattered signal and allowing for benchmarking estimation of both experimental results and theoretical models of the processes considered. We show the possibility of forming theoretical and numerical routines that enable us to evaluate the quantities of interest. On the basis of the procedures developed we performed calculations yielding the moment values for different diatomic noble gas systems at a variety of temperature points. Within the discussion we also have compared alternative procedures of calcu- lating the moments: the semiclassical and the quantum formulas, on the one hand, and the sum rule approach versus spectral line computing, on the other. As a result, we have found a good agreement between the values of different ori- gins for the lowest rank moments共zeroth, first, and second兲 indicating the validity of the assumptions and procedures formulated in our earlier work as well as the correctness of the resulting spectral profiles. Additionally, it has confirmed a satisfactory capability of the semiclassical approximation of providing reliable moment values within a wide range of possible physical situations.

ACKNOWLEDGMENTS

This work has been supported by Grant No. 1 P03B 082 30 of the Polish Ministry of Education and Sciences. We wish to thank Professor George Maroulis for many valuable discussions.

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Moments of hyper-Rayleigh spectra of selected rare gas mixtures

Moments of hyper-Rayleigh spectra of selected rare gas mixtures

Moments of hyper-Rayleigh spectra of selected rare gas mixtures

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