What is measured by hyper-Rayleigh scattering from a liquid?

Second-harmonic or hyper-Rayleigh scattering (HRS) is a nonlinear light scattering technique widely used to measure the first hyperpolarizability β of molecules in solution,^1,2 and HRS has largely replaced the previous alternative electric-field-induced second harmonic generation (ESHG or EFISH) technique.³ It is usually assumed that only incoherent scattering from the individual molecules contributes to HRS, and this is a good approximation for the contribution of a strong nonlinear optical chromophore in dilute solution, but it is not a good approximation for the solvent itself. The intensity, angle, and polarization dependence for incoherent HRS is modified by molecular orientation correlation extending to long range in the liquid.^4–9 Fluctuations in β due to intermolecular interactions produce an additional spectral contribution [collision-induced (CI) wing], wider than the spectral component due to reorientation, and of comparable intensity.^9–14 The dissolved ions in polar solvents also produce a coherent ESHG contribution with a narrow spectrum, which is due to the orientation correlations induced by the electric field of the ions in solution.^15,16 Some or all of these overlapping spectral contributions may fall within the frequency band for HRS measurements, which is determined by the laser spectral width and the scattered light spectral filter employed in the measurement. Lasers producing fs pulses with spectral width >100 cm⁻¹ are often used for HRS experiments since the high peak power gives a large HRS signal. In this case, all three contributions are summed in the measurement. The analysis often considers only the incoherent individual molecule contribution, and this will lead to errors for the hyperpolarizability of the solvent molecules and for hyperpolarizabilities determined by the internal reference method, where the solvent hyperpolarizability is used for calibration.² In polar liquids, there is often an extrinsic ion-induced contribution due to ionic contamination, which is difficult to avoid or remove, and there is also an intrinsic ion-induced contribution in liquids such as water or ionic liquids that dissociate into ions.

All the above-mentioned effects enter the quantitative relation between molecular hyperpolarizability and HRS intensity, and the effects of correlation are distinctly different for the HRS contributions from the irreducible first rank (vector) and third rank (octupolar) parts of the third rank hyperpolarizability tensor β. Recently, theoretical expressions have been developed which account for HRS including the effects of ion-induced¹⁵ and intrinsic orientation correlation⁸ in dipolar liquids. One of the aims of the present work is to test these expressions which show that vector β with long-range orientation correlation produces HRS with different intensities for components polarized parallel [longitudinal (L)] or perpendicular [transverse (T)] to the scattering wavevector.

HRS intensity from vector β can be expressed in terms of correlation functions for lab-frame components of the molecule-fixed vectors on pairs of molecules.⁸ In a homogeneous, isotropic liquid, the orientation correlation between unit vectors on pairs of molecules is a tensor function of the intermolecular displacement vector. This correlation tensor is diagonal in the coordinate frame with one axis aligned along the intermolecular vector, where the unit vector on each molecule has one longitudinal component (projection onto the intermolecular vector) and two transverse components (in the plane perpendicular to the intermolecular vector). The average products of corresponding longitudinal or transverse components of the two molecular unit vectors are the longitudinal (L) or transverse (T) orientation correlation functions, which are scalar functions of the intermolecular distance. The intensity of vector HRS as a function of the scattering wavevector is given by the spatial Fourier transform of this correlation tensor. The resulting spectral tensor is diagonal in the frame aligned with the scattering vector, with diagonal components longitudinal and transverse with respect to the scattering vector. The L and T spectral components produce HRS with distinctively different polarization and angle dependence. Expressing the L and T components of the spectral tensor in terms of the L and T components of the correlation tensor, one can show that unequal intensity for the L and T spectral components is the result of long-range orientation correlation. Dissection of the HRS spectrum into its L and T contributions is the main experimental result of this work.

Two dipolar liquids, acetonitrile and dimethyl sulfoxide (DMSO), are studied in the present work. In dipolar liquids, the dipole-dipole interaction results in long-range molecular orientation correlation, and theory predicts that this correlation in all cases produces a transverse HRS spectrum. Transverse HRS is observed for acetonitrile, in agreement with the theory, whereas longitudinal HRS is observed for DMSO, apparently contradicting the theory. This study seeks to understand the origin of longitudinal HRS from DMSO. The angle between the dipole and β vectors is not uniquely determined by symmetry in DMSO, and this angle affects the coherent HRS contribution and may also account for much weaker ion-induced HRS observed in DMSO than in acetonitrile. This study examines the quantitative accuracy of the theory for ion-induced HRS for both liquids. A broad collision-induced spectral wing makes another large contribution to the HRS intensity. This spectral wing results from rapid fluctuations induced by short-range intermolecular interactions, and in previous studies, it has been assumed that the intermolecular vibrations and librations mediated by these short-range interactions are local.^9–14 However, hyper-Raman spectra have demonstrated that intra-molecular vibrational modes in liquids and glasses are non-local,^17–19 and the present work finds evidence that the inter-molecular modes contributing to the collision-induced HRS spectrum are non-local as well.

In the following, the experimental methods are described and the experimental results are presented. Then the theoretical expressions for the angle and polarization dependence of intrinsic and ion-induced HRS are presented, followed by analysis and discussion of the results. The results of molecular dynamics (MD) simulations for the liquids are used in the analysis.

The HRS experimental apparatus and methods are similar to those previously employed^{5,7,15,16,20–22} and are described and discussed in detail in Ref. 21. Linearly polarized pulses from an injection-seeded single-longitudinal-mode Nd:YAG (yttrium aluminium garnet) laser (operating at λ₀ = 1064 nm, 4.3 kHz repetition rate, 100 ns pulse duration) are focused to a 4.5 μm beam waist radius in the liquid sample. Scattered light is collected and collimated by an aspheric lens (f = 13.8 mm), analyzed by a linear polarizer, focused into an optical fiber, and fiber-coupled to a spectral filter or spectrometer followed by the photon counting detector. The sample temperature was T = 25.0 °C, and the laser beam average power in the sample was typically P_av = 1.5 W for acetonitrile and 2 W for DMSO-d6. Strong thermal lens effects due to absorption of the laser beam by overtone H stretching vibrations in DMSO are avoided by using DMSO-d6.

The acetonitrile (CH₃CN, 99.98%, Aldrich) and DMSO-d6 ((CD₃)₂SO, 99.9 at. % D, Aldrich) samples were de-ionized by continuous flow in a closed loop containing a PTFE (polytetrafluoroethylene)-tube peristaltic pump, an ion-exchange resin column (Dowex Monosphere MR-450 UPW), a 0.2 μm PTFE filter, a sample cell, a conductivity cell, and a reservoir, in that order. Samples with controlled, larger ion concentration were obtained by adding LiClO₄ (>99%, Fluka) or KClO₄ (>99%, Aldrich) to the de-ionized circulating fluid, with the ion-exchange column by-passed. The ion concentration was determined from the measured conductivity using the published results for LiClO₄-CH₃CN and KClO₄–(CH₃)₂SO solutions.^23,24 The de-ionized sample conductivity was typically in the range 1–5 nS/cm (estimated ion concentration 6–30 nM) for acetonitrile and 3–6 nS/cm (estimated ion concentration 75–150 nM) for DMSO-d6.

HRS measurements were made with linearly polarized light at scattering angles θ_s in the range from 0° to 180°. The sample cell was either a special 8-window cell for 45°, 90°, or 135° measurements or a standard square 10 mm fluorimeter cuvette for measurements at or near θ_s = 0°, 90°, or 180°. The laser beam was normally incident on the entrance window in all cases, but off-normal incidence at the exit window of the cuvette was required for some of the measurements. Scattering configurations with incident and scattered light polarized either perpendicular or parallel to the horizontal scattering plane are denoted VV, HV, VH, and HH, where V denotes the vertical polarization, H denotes the horizontal polarization, and the first and second letters refer to the incident and scattered light, respectively. Ratios of HRS intensities were measured (I_VV/I_HV, I_HV/I_VH, and I_HH/I_VH). The effect of intensity drift was canceled by using several hundred alternate 10 s measurements of the two polarization configurations for each ratio. Rapid switching between polarization configurations was enabled using a liquid crystal variable wave plate (LCVWP) to control the laser polarization and a fast rotator²² to control the analyzing polarizer for the scattered light. The collection numerical aperture (NA = n sin θ) was controlled by a circular aperture following the collection lens, and the HRS intensity ratio at NA = 0 was obtained by extrapolating measurements in the range 0.04 < NA < 0.11, made using several apertures, to zero collection aperture. Ratios were corrected for polarization dependent reflection at non-normal incidence.

The main HRS polarization ratio measurements were made with an interference filter selecting a 60 cm⁻¹ spectral band (2 nm full width at half maximum, FWHM), centered on the 532 nm second-harmonic wavelength. Since the ion-induced HRS signal is difficult to eliminate, it was separately measured and then subtracted. The ion-induced HRS contribution was measured by inserting a fiber-coupled confocal Fabry–Pérot interferometer (10 cm mirror spacing, 750 MHz free spectral range) and scanning the spectrum with 13 MHz resolution. The spectral broadening of the ion-induced HRS signal due to the diffusive motion of the ions is typically <1 MHz, as compared to spectral broadening >30 GHz for the other components of the HRS spectrum, so the ion-induced signal appears as a sharp spike riding on the flat background produced by the multiple overlapping orders of the much broader HRS spectrum. The relative ion-induced contribution is given by the ratio of integrated intensities for the spike and background in the Fabry–Pérot spectrum. Typically, 10⁴ alternate scans of the reference spectrum (laser second-harmonic produced by a potassium titanyl phosphate [KTP] crystal) and the HRS spectrum were recorded, where the reference scans were used to lock the center of the Fabry–Pérot scans to the laser second-harmonic and to provide the instrument response function used for de-convolution of the ion-induced HRS spectrum.

Additional HRS polarization measurements were made at 90° scattering angle with NA = 0.15, using a wider filter to select a 336 cm⁻¹ spectral band (9.5 nm FWHM, 532 nm center wavelength) and also using a fiber-coupled grating spectrometer (Jobin-Yvon Ramanor tandem, 1 m focal length, 1800 groove/mm gratings) to select a 2 cm⁻¹ or 8 cm⁻¹ spectral band. The spectrometer is polarization sensitive, so accurate HRS polarization measurements require an effective depolarizer placed before the spectrometer. The length scale for polarization mode mixing in a step index optical fiber is about 10 m,^25,26 so the 2 m long spectrometer input fiber only partially depolarizes the light. The residual output polarization with linear polarized input light is measured by r = I_max/I_min, the ratio of maximum and minimum intensity transmitted through a linear polarizer placed at the fiber output. For the 2 m long, 200 μm core fiber, r < 1.15 was measured. A micro-patterned liquid crystal polymer depolarizer (Thorlabs DPP25-A) was placed after the fiber, and for this depolarizer, r < 1.15 was measured. For the combination of optical fiber followed by a micro-patterned depolarizer, r < 1.01 was measured, so the error in I_HV/I_VH measurements due to the polarization sensitivity of the spectrometer was less than 1%.

Measurements of the HRS intensity ratios I_VV/I_HV, I_HV/I_VH, and I_HH/I_VH versus scattering angle θ_s in the horizontal plane are given in Table I for acetonitrile and Table II for DMSO-d6 and are plotted in Fig. 1. The measurements have been corrected for the ion-induced HRS contribution and extrapolated to zero collection aperture. Photon counting statistics account for the stated error bars on most of the data points. The expressions fit to the HRS intensity ratio data in Fig. 1 are^7–9 

where A₀, A_T, and A_L are the intensity coefficients for HRS with the polarization and angle dependence for uncorrelated molecules (A₀) or for molecules with vector orientation correlation (A_T, A_L). Predominantly transverse HRS with A_T > A_L is indicated by polarization ratio I_HV/I_VH > 1, and longitudinal HRS with A_T < A_L is indicated by I_HV/I_VH < 1. The fit parameters are P² = 2.59 ± 0.04, R = 2.953 ± 0.005, A_T/A₀ = 5.22 ± 0.07, and A_L/A₀ = 0 for CH₃CN, and the fit parameters are P² = 6.020 ± 0.011, R = 2.974 ± 0.006, A_T/A₀ = 0, and A_L/A₀ = 1.137 ± 0.006 for (CD₃)₂SO. Since sums of terms with A_T = A_L have polarization and angle dependence identical to those of the A₀ terms and can be incorporated into the A₀ terms, the fitted curves obtained using Eqs. (1)–(4) are invariant for changes to alternative parameters $P′2$⁠, $AT′/A0′$⁠, $AL′/A0′$⁠, where⁷ 

The curves in Fig. 1 are a good fit to the data and show that HRS is predominantly transverse for CH₃CN and longitudinal for (CD₃)₂SO. The transverse and longitudinal intensity coefficients, A_T and A_L, account for coherent HRS from that part of the third-rank first hyperpolarizability tensor β which transforms as a vector and is affected by long-range orientation correlation of the molecules.

The ion-induced HRS that has been removed from the data in Fig. 1 is the longitudinal contribution to the coherent HRS intensity produced when dipolar molecules surrounding an ion are oriented by the radial electric field of the ion. Measurements of this ion-induced HRS signal are shown in Fig. 2 for acetonitrile and in Fig. 3 for DMSO-d6. Figure 2(a) shows I_HV/I_VH measured for LiClO₄-CH₃CN solutions as a function of ionic strength $c=(1/2)∑iZi2ρi$⁠, where Z_i is the charge and ρ_i is the concentration of ion species i. Figure 3(a) shows I_HV/I_VH measured for KClO₄-(CD₃)₂SO solutions. Ion-induced HRS is much weaker for DMSO-d6 than for acetonitrile, so the DMSO-d6 measurements were made at 16.6° to increase the ion-induced HRS signal. The curves fit to the data points in Figs. 2(a) and 3(a) have the form¹⁵ 

where x = c/c₀, $A=(IHV/IVH)c=0$⁠, and $B=(IVHi)c→∞/(IVH)c=0$ = $IVH,∞i/IVH,0$⁠. Equation (6) is the ratio of HRS intensity $IHV$⁠, which is independent of ion concentration, and $IVH=IVH,0+IVHi$⁠, which has an ion-induced contribution $IVHi$ that saturates at high ion concentration. The ion-induced HRS contribution $IVHi$ is determined more accurately from the ratio I_HV/I_VH because I_HV serves as an ion-independent internal reference for the measurements of the ion concentration dependent intensity I_VH. The polarization and angle dependence for ion-induced HRS is given by $IVVi=IHVi=0$⁠,^9,15

where $IVH,∞,90i$ is the maximum value for $IVHi$ at θ_s = 90°, $2⁡sin2(θ0/2)=(ni−ns)2/(2nins)$⁠, and n_i (n_s) is the liquid refractive index at the incident (scattered) wavelength (θ₀ = 0.15° for acetonitrile and 0.33° for DMSO-d6). The ion-induced HRS spectrum at high ionic strength, measured using the confocal Fabry–Pérot interferometer, is shown in Fig. 2(b) for acetonitrile and in Fig. 3(b) for DMSO-d6 (with HH polarization to maximise the signal for DMSO-d6). The ion-induced HRS contribution relative to the intrinsic HRS contribution is simply determined from such spectra; it is given by the integrated intensity ratio for the ion-induced peak and the flat intrinsic background in the spectrum. The ion-induced HRS contribution can be measured even at low ionic strength from such spectra. The ion-induced HRS intensity at low ionic strength, such that x ≪ 1, is given by $IVHi∝x/tan2(θs/2)$ and is largest at small forward angles.

The ion-induced HRS contributions from residual ions in the de-ionized samples were subtracted to obtain the results plotted in Fig. 1. The residual ion-induced HRS contribution for acetonitrile in this work was <1% of the intrinsic HRS contribution for all scattering angles >45°, ≈1% for θ_s = 45°, and ≈10% for θ_s = 18°. Although Fabry–Pérot scans were used to directly measure the ion-induced contribution, the accuracy of the correction for ion-induced HRS at θ_s = 18° was limited by uncontrolled variations ≈10 nM in the residual ion concentration during those measurements. The residual ion-induced HRS corrections for DMSO-d6 were smaller, <1% at 16.6° and <0.1% at all larger scattering angles.

The third contribution to the measured HRS intensity is due to collision-induced inter-molecular HRS, which has a spectrum much wider than the orientational HRS spectrum that it overlaps.^12–14 The intensity ratios I_VV/I_HV and I_HV/I_VH for just this HRS contribution can be determined from the differences of the HRS intensities measured with 60 and 336 cm⁻¹ bandwidth (BW) filters, as shown in Table III. The collision-induced HRS for frequency shift Δν > 30 cm⁻¹ is transverse for acetonitrile and longitudinal for DMSO-d6. This is shown by the polarization ratio $IHV/IVH2−1$ for Δν > 30 cm⁻¹ in the last line of Table III, which is 1.46 for acetonitrile, indicating predominantly transverse HRS, and 0.50 for DMSO-d6, indicating predominantly longitudinal HRS. The intensity ratios at smaller frequency shift can be measured using the grating spectrometer to select the measurement spectral band. Table IV gives measurements with an 8 cm⁻¹ band centered at 20 cm⁻¹ shift which excludes the central orientational HRS component (partially for acetonitrile and completely for DMSO-d6). This measurement of collision-induced HRS is also transverse for acetonitrile and longitudinal for DMSO-d6. Also shown in Table IV are DMSO-d6 intensity ratios measured in a narrow band (2 cm⁻¹) at 0 cm⁻¹ shift, which selects orientational HRS and effectively excludes collision-induced HRS for DMSO-d6. This measurement is consistent with incoherent orientational HRS (I_HV/I_VH = 1) and dominantly vector β (⁠$IVV/IHV≈9$⁠) for DMSO-d6.

Molecular hyperpolarizabilities are usually determined assuming incoherent HRS, using HRS relative intensity measurements for different samples in the same apparatus, where $I(2ω)∝Fβ2P2$⁠. The sample dependent factor F is^27,28

where ρ is the molecular number density, $L=(n2+2)/3$ is the Lorentz local field factor, $t=[1−(nw−n)2/(nw+n)2]$ is the Fresnel transmission factor, α is the sample absorption factor for the incident laser beam or the HRS light, and n (n_w) is the sample (window) refractive index. Molecular data and the calculated values for F are given in Table V. The absorption of the incident laser beam is α_ω < 0.1%, which is negligible, but the thermal defocusing of the incident beam due to this absorption is not negligible. The effect of the thermal lens is eliminated by using $I(2ω)/P2$ extrapolated to P = 0. Relative HRS intensity $[IDMSO(2ω)/P2]P=0/[ICH3CN(2ω)/P2]P=0$ = 0.482 ± 0.013 is measured for DMSO-d6 and acetonitrile. The corresponding relative hyperpolarizability obtained using^27,28

is $βDMSO/βCH3CN$ = 0.640 ± 0.010, slightly smaller for DMSO-d6 than for acetonitrile.²⁸ A better estimate would account for orientation correlation and other spectral components included in the measured intensity.

Molecular dynamics (MD) simulations for each liquid provide information about molecular correlations needed to go beyond the incoherent approximation in the analysis of the HRS observations. In particular, the observed HRS polarization and angle dependence can be explained by long-range orientation correlation of the molecular dipole vectors and the vector part of the β hyperpolarizability tensor.⁸ Molecular dynamics simulations were performed using the GROMACS software package (version 5.1.4).²⁹ All-atom, non-polarizable molecules were simulated in a cubic box with periodic boundary conditions; long-range Coulomb interactions were treated by the particle mesh Ewald (PME) method with conducting boundary conditions, while Lennard-Jones (LJ) 6-12 interactions were treated using a 1.0 nm cutoff radius and analytical dispersion correction.³⁰ Temperature and pressure were controlled using a modified velocity rescaling thermostat³¹ and a Berensden barostat. Equilibrium density at T = 298 K and p = 101 kPa was determined using NPT simulations, while NVT simulations at the equilibrium density were used to determine the correlation functions. Simulations for acetonitrile with N = 27 000 molecules in a 13.33 nm box were equilibrated for 2 ns before a 20 ns production run with 2 fs time steps. Simulations for DMSO with 8000 molecules in a 9.8 nm box were run for 2 ns with 2 fs time steps. Further details of the MD simulation models are given in the supplementary material.

The full, longitudinal, and transverse dipole-dipole correlation functions computed from the simulation trajectory are^32,33

where $μ^i$ is the dipole unit vector on molecule i and $r^ij$ is the unit vector in the direction from molecule i to j. Corrections $δF=3δL=3δT=−(ε−1)2/(9yεN)$^9,34,35 are applied to the MD simulation results to obtain results for an infinite homogeneous system, where³⁴ 

is the dimensionless dipole strength, ρ is the molecular number density, and μ is the molecular dipole moment. The dielectric constant ε is calculated using the correlation function expression³⁴ 

for the Kirkwood correlation factor g_K and then solving the Kirkwood relation³⁴ 

for ε. The correction is about δL = −5 × 10⁻⁶ (−4 × 10⁻⁵) for the acetonitrile (DMSO) MD simulation. The correlation functions at large r for a non-polarizable dipole fluid have the asymptotic values^9,34

with the correlation strength parameter^9,34

The MD simulation results for acetonitrile correlation functions are shown in Fig. 4. The radial pair distribution function g(r) plotted in Fig. 4(a) shows that other molecules are excluded for r < 0.27 nm and that position correlations vanish for r > 2 nm. The orientation correlation functions r³L(r) and r³T(r) plotted in Figs. 4(b) and 4(c) reach constant long-range limiting values for r > 5 nm, with limiting values in good agreement with the values calculated using Eqs. (17) and (18). The dipole moment of polarizable molecules in the liquid is larger than the gas phase molecular dipole. A combined Monte Carlo/quantum mechanical (QM) calculation finds that μ = 3.94 D calculated for an isolated acetonitrile molecule is increased to 4.65 ± 0.19 D for a molecule in liquid acetonitrile,³⁶ consistent with μ = 4.5 ± 0.1 D obtained from a measurement of the integrated intensity of far-infrared absorption.³⁷ MD simulation results are shown for the two models, the first (MD1) with dipole moment near the measured gas phase value and the second (MD2) with dipole moment close to the measured liquid phase dipole. The correlation strength increases with the magnitude of the molecular dipole moment. MD simulation results for the two acetonitrile models are summarized in Table VI.

The MD simulation results for the DMSO dipole correlation functions shown in Fig. 5 are qualitatively similar to those for acetonitrile. In contrast to acetonitrile, the dipole vector direction for DMSO is not determined by symmetry, and the β vector may not be parallel to the dipole vector. The β vector correlations that result from the dipole correlations will be reduced by the projection factor $cos2θμβ$⁠, where θ_μβ is the angle between the dipole and β vectors. The DMSO dipole lies along the O-S bond,³⁸ and the β vector may lie in the direction of the S lone-pair. The angle θ_μβ = 104° is obtained from the MD simulation assuming that the β vector is in the direction of the S lone-pair, which is estimated as the direction given by the vector sum of the bond vectors for the C–S, C–S, and O–S bonds. Figure 5(c) shows the β vector orientation correlation functions calculated from the MD trajectory, for the β vector in the direction of the S lone-pair. The ratio of β correlation functions in Fig. 5(c) and the corresponding dipole correlation functions in Fig. 5(b), averaged over 0.5 < r < 2.5 nm, is 0.10 ± 0.03 for the longitudinal correlations and 0.04 ± 0.04 for the transverse correlations, as compared to the projection factor $cos2θμβ$ = 0.06 for θ_μβ = 104°. MD simulation results for DMSO with θ_μβ = 0° and 104° are summarized in Table VI.

The terms with intensity coefficients A_T and A_L in Eqs. (1)–(4) are due to vector β HRS for molecules with long-range orientation correlations given by the transverse and longitudinal vector correlation functions B_T(r) and B_L(r). The intensity coefficients A_T and A_L are proportional to the diagonal components S_T(K) and S_L(K) of the spatial Fourier transform of the correlation tensor, given by the expressions^8,9,33

where j_n(x) are the spherical Bessel functions.

These integrals are evaluated piecewise over three regions,^8,9,33

The delta function self-correlation in region 1 (r < r₁, where r₁ is the radius of the excluded volume) gives⁸ 

where $4πr03/3=ρ−1$ is the volume per molecule. In region 2 (r₁ < r < r₂), the correlation functions B_L(r) = L(r) and B_T(r) = T(r) obtained from the MD simulation are integrated to give S_T,2 and S_L,2. At short range where Kr ≪ 1, it is a good approximation to take the K = 0 limit for the integrand in Eqs. (19) and (20), which gives⁸ 

This integral also appears in Eq. (15) for g_K. The results in Table VI for acetonitrile (DMSO) are obtained using r₁ = 0.27 (0.34) nm and r₂ = 5.0 (4.5) nm.

The integrals for region 3 (r > r₂) are evaluated using the correlation functions^8,9,33

which give the correct asymptotic r⁻³ dependence and a solenoidal (zero divergence) vector field. Equations (19) and (20) with these correlation functions and g(r) = 1 give⁸ 

where K_n(x) is the modified Bessel function of the second kind, of order n. Again taking the K = 0 limit, the integrals over region 3 (r > r₂) for these correlation functions are⁸ 

where

The contributions to Eqs. (21) and (22) from each region, and the totals for S_T and S_L, are given in Table VI for acetonitrile and DMSO. Orientation correlation affects S_T and S_L in two ways.³³ The first effect is that S_T and S_L differ from the incoherent HRS value S_T,1 = S_L,1, by the intensity enhancement factors C_T and C_L given in Table VI. The second effect is that S_T and S_L are not equal. The second effect is entirely due to the long-range correlation in region 3, and it produces HRS polarization and angle dependence observably different from that for incoherent HRS. Orientation correlations due to the dipole-dipole interaction always produce transverse HRS with S_L ≤ S_T.³³ Table VI shows S_L ≪ S_T for both acetonitrile models and for the DMSO model with θ_μβ = 0°. Table VI shows S_L ≈ S_T for the DMSO model with θ_μβ = 104°, and for θ_μβ = 90°, the long-range correlation for vector β due to the dipole correlation will vanish, giving S_L = S_T. A value θ_μβ > 90° is indicated by liquid phase ESHG experiments which find $γ||+μβ||/3kBT$ is negative for DMSO.^39,40

HRS is mediated by the tensor β which is the direct sum of four irreducible spherical tensors⁴¹ 

where $βm[ν,l]$ is a spherical tensor of rank l with 2l + 1 components m, index ν labels the symmetry under permutation of the Cartesian tensor indices (ss is totally symmetric, while ms is non-symmetric for first index permutations), and the mixed symmetry ν = ms tensors vanish when Kleinman symmetry holds. Expressed in terms of spherical tensor components, the HRS intensities given by Eqs. (1)–(4) at θ_s = 90° are⁴¹ 

where $β[ν,l]2=∑mβm[ν,l]2$⁠. The β components with l = 1 transform as vectors under rotations, and R² = (I_VV/I_HV)_l=1 is the HRS intensity ratio due to just these terms. Similarly, P² = (I_VV/I_HV)_l=2,3 is the HRS intensity ratio due to just the terms with l = 2 and 3. Dipole orientation correlation affects only the l = 1 HRS contribution and is accounted for by the factors C_T and C_L. Short range quadrupolar and octupolar orientation correlation effects for the l = 2 and 3 terms are ignored since they can change the values for P² and A₀, but not the polarization and angle dependence.

The coefficients A₀, A_T, and A_L in Eqs. (33)–(35) will be directly comparable to the coefficients experimentally determined from the fits of Eqs. (1)–(4) to the HRS data in Fig. 1, provided that P² in the experimental fit includes only the l = 2 and 3 contributions. The transformation to the set of fit coefficients with the correct value for P² is given by Eq. (5). The correct value is P² = 3/2 in the case that Kleinman symmetry holds so that the l = 2 contribution vanishes, but this is not true in general. However, for acetonitrile which has C_3v symmetry, the nonvanishing independent Cartesian tensor components are β_zzz, β_zxx, β_xxz, and β_yyy, and the l = 2 contribution vanishes by symmetry.⁴¹ Transforming the parameters for the acetonitrile HRS fit in Fig. 1(a) using Eq. (5) gives P² = 1.500, R = 2.953 ± 0.005, A_T/A₀ = 6.33 ± 0.12, and A_L/A₀ = 0.179 ± 0.008. The ratio A_L/A_T = 0.0282 ± 0.0014 for these fit coefficients is directly comparable with the MD simulation results for S_L/S_T in Table VI. The result S_L/S_T = 0.029 for MD2 agrees with the experimental result, while S_L/S_T = 0.035 for MD1 does not. The MD simulation results are sensitive to the dipole moment of the model molecules, and the HRS measurement results indicate that the correct correlations are calculated by model MD2 which has molecular dipole moment close to the estimated liquid phase value. The best estimate for the dipole correlation strength parameter for acetonitrile is a³ = 6.29 × 10⁻³ nm³, the value for which the MD2 simulation gives S_L/S_T in exact agreement with the HRS result.

Equations (33)–(35) can be solved for ratios of the β tensor components. The β tensor components determine R² and P², dipole correlations determine C_T and C_L, and both combined determine A_T/A₀, I_VV/I_HV, and I_HV/I_VH. A particular solution for the acetonitrile β tensor is specified by giving three ratios of the nonvanishing independent Cartesian tensor components β_zzz, β_zxx, β_xxz, and β_yyy. There is a continuum of such solutions that exactly fit the HRS observations in Fig. 1 (e.g., using MD2, C_T = 1.39, C_L = 0.041, β_zxx/β_zzz = 0.178, β_xxz/β_zzz = 0.171, β_yyy/β_zzz = −0.114). Ab initio calculations of β tensor components for acetonitrile give β_zxx/β_zzz = 0.021 and β_yyy/β_zzz = −0.136 (Ref. 42, gas phase, static, electronic, CCSD [coupled cluster single double], t-aug-cc-pVTZ), β_xxz/β_zzz = 0.117 (Ref. 43, liquid, 1064 nm, CCSD(T) [perturbational triple], d-aug-cc-pVDZ), and β_zxx/β_zzz = 0.132, β_xxz/β_zzz = 0.096 (Ref. 44, liquid, 514.5 nm, MCSCF RAS [restricted active space]). Many β tensors with components similar to the ones from ab initio calculations fit the HRS observations for acetonitrile (using orientation correlations from the MD simulation). For DMSO, there is no solution of this type since dipole orientation correlation does not produce longitudinal HRS.

The HRS intensity is increased by the factor C_T due to orientation correlation, and the polarization and angle dependence for HRS is also modified by the long-range part of the orientation correlation. This has also been studied in several other liquids including water and nitrobenzene.^5,7,9,33 The effect of the liquid environment on β itself has been investigated by QM calculations with reaction field models. Compared to the gas phase value, calculated $β||$ is increased in liquid acetonitrile by a factor of 2.0 (Ref. 44, 514.5 nm, MCSCF RAS) or 3.0 (Ref. 43, 1064 nm, CCSD(T), d-aug-cc-pVDZ), and recent QM/Monte Carlo calculations for nitrobenzene solutions indicate further small changes in β when nearest neighbour orientation correlations are explicitly included.⁴⁵ The intermolecular interactions produce both orientation correlation and molecular distortion. The HRS intensity for acetonitrile is increased by the factor C_T due to the correlations, further increased due to larger β for the distorted molecules in the liquid, and increased once more by the local field factors. The change in the HRS intensity due to the average molecular distortion is distinct from the HRS intensity change due to orientation correlation.

Slow fluctuation in molecular orientation is not the only motion contributing to the HRS signal measured in Fig. 1. A rapidly fluctuating increment Δβ is induced in each molecule during collisions with other molecules, and the resulting broad spectral component can also contribute to the measured HRS signal. The increment Δβ that is induced in one molecule, by the electric field of the permanent dipole, quadrupole, or octupole on a neighbouring molecule, is proportional to the second hyperpolarizability γ of the first molecule.^10,11 The Δβ tensor resulting from the isotropic part of γ transforms as a vector, so a polar mode results from Δβ induced during intermolecular vibration and libration. It is usually assumed that the correlations for Δβ are short-range since the induction mechanism for Δβ is short-range.

Orientation and collision-induced (CI) HRS contributions can be distinguished spectroscopically (e.g., see HRS measurements and analysis for CDCl₃ in Ref. 13). Accordingly, the HRS spectrum previously measured for acetonitrile was decomposed into the sum of a narrow Lorentzian reorientation component (⁠$[1+(Δν/νL)2]−1$ with ν_L = 1.68 cm⁻¹) and an overlapping broad exponential collision-induced component (⁠$exp[−|Δν/νE|]$ with ν_E = 35 cm⁻¹).¹⁴ The respective integrated intensities of these components are 0.70 and 0.30 of the total for the HV HRS spectrum. The intensity ratio (I_HV/I_VH)_L measured for the Lorentzian components of the HV and VH spectra was 2.0, but the ratio was not measured for the exponential component. The single Lorentzian in the fit combines the two Lorentzians with widths in the ratio 1:6, due to the dipolar and octupolar parts of β, predicted by a simple rotational diffusion model.¹³ The HRS spectrum of DMSO has not been measured, but ν_L = 0.5 cm⁻¹ is estimated for the narrow Lorentzian component based on the measured 21.1 ps dielectric relaxation time.⁴⁶ 

The entire reorientation spectrum and part of the collision-induced spectrum are included in the 60 cm⁻¹ measurement bandwidth (BW) used for most of the HRS measurements in the present work. The collision-induced contribution for $|Δν|$ >30 cm⁻¹ can be separately determined from the difference in the HRS results obtained using filters with 60 and 336 cm⁻¹ BW, as shown in Table III. The result of this measurement is that CI HRS is mainly transverse for acetonitrile (with I_HV/I_VH = 1.46) and CI HRS is mainly longitudinal for DMSO (with I_HV/I_VH = 0.50). This result is confirmed in Table IV by grating spectrometer measurements for a narrow band at Δν = 20 cm⁻¹, where the reorientation spectrum is weak compared to the CI spectrum, giving I_HV/I_VH = 1.28 for acetonitrile and 0.45 for DMSO. The DMSO measurement at Δν = 0 cm⁻¹ with 2 cm⁻¹ BW, which includes the reorientation spectrum and excludes the CI spectrum, gives I_HV/I_VH = 0.98 ± 0.03, which is consistent with θ_μβ ≈ 90° and the near absence of long-range orientation correlation for vector β. The non-local longitudinal HRS contribution for DMSO appears to be entirely collision-induced. The result I_VV/I_HV = 8.2 at Δν = 0 cm⁻¹ for DMSO in Table IV indicates that the vector part of β is dominant.

The HRS results for acetonitrile and DMSO show that the CI HRS is non-local and can be either transverse or longitudinal. The origin and functional form for the vector Δβ correlations are not known, but correlations for vector Δβ decreasing asymptotically as r⁻³ would produce HRS with the same form as given by Eqs. (1)–(4). Non-local transverse optical (TO) and longitudinal optical (LO) vibration modes have been previously observed by hyper-Raman scattering in crystals, glasses, and liquids.^17–19 The non-local CI HRS modes observed here may be similar, with both TO and LO components, each with its own spectrum. Polar non-local modes mediated by CI Δβ also can be present for nonpolar and centrosymmetric molecules and may account for $IHV/IVH≠1$ observed for HRS from liquids such as C₂Cl₄, C₆H₆, and CS₂.⁵ 

The intensity of ion-induced HRS increases with ion concentration and saturates at high ion concentration. The ion concentration dependence of the ion-induced HRS intensity in Eq. (6) is derived using the Debye-Huckel theory for the ion correlations, and the resulting expression for the ionic strength parameter c₀ in Eq. (6) is¹⁵ 

where K is the magnitude of the scattering wavevector, ε₀ is the vacuum permittivity, ε_s is the static relative dielectric constant, k_B is the Boltzmann constant, and e is the electronic charge. The scattering wavenumber K is given by⁹ 

where λ₀ is the vacuum wavelength of the incident light and n_i, n_s are refractive indices at the incident and scattered light frequencies. The experimental results in Figs. 2(a) and 3(a) for the ion-induced HRS concentration dependence can be compared with the theoretical result for c₀ given by Eqs. (36) and (37), evaluated using the data in Table V.^47–51 For the experiments in Figs. 2 and 3, 2π/K = 281.0 nm at θ_s = 90° for acetonitrile and 2π/K = 1252 nm at θ_s = 16.6° for DMSO-d6. The calculated value c₀ = 21.2 μM for acetonitrile from Eq. (36) is in good agreement with the experimental value c₀ = 21.1 ± 0.5 μM determined from the fit in Fig. 2(a). The calculated value c₀ = 1.38 μM for DMSO-d6 is used in the fit to the data shown in Fig. 3(a).

The width of the ion-induced HRS spectrum is determined by ion diffusion, increases with ion concentration, and is usually much narrower than the width of the reorientation and collision-induced HRS spectra. An expression for the width of the Lorentzian ion-induced HRS spectrum is^15,52

where x = c/c₀, $D−1=(D1−1+D2−1)/2$⁠, and D₁ (D₂) is the diffusion coefficient for ion species 1 (2). The results calculated using this expression can be compared to spectral widths experimentally determined from the spectra in Figs. 2(b) and 3(b). Diffusion coefficients determined from the limiting conductance for acetonitrile-electrolyte solutions²³ are 1.863 (2.759) × 10⁻⁹ m² s⁻¹ for Li⁺ (ClO₄⁻), which gives D = 2.224 × 10⁻⁹ m² s⁻¹ and ν_K = 0.177 MHz. The degree of dissociation is 0.96 for the 2.0 mM LiClO₄-acetonitrile solution, so c = 1.92 mM, x = 90.7, and $νLi$ = 16.6 MHz, in good agreement with 16.0 ± 0.6 MHz measured for the spectrum in Fig. 2(b). For DMSO, the diffusion coefficients determined from the limiting conductances for DMSO-electrolyte solutions²⁴ are 0.386 (0.652) × 10⁻⁹ m² s⁻¹ for K⁺ (ClO₄⁻), which gives D = 0.485 × 10⁻⁹ m² s⁻¹ and ν_K = 1.95 kHz. The 0.60 mM KClO₄-DMSO solution is fully dissociated, so c = 0.60 mM, x = 433, and $νLi$ = 0.85 MHz, in good agreement with 0.84 ± 0.17 MHz measured for the spectrum in Fig. 3(b).

Figure 2 shows two complementary measurements of the ion-induced HRS contribution for acetonitrile, the first using I_HV/I_VH versus ionic strength in Fig. 2(a) and the second using a high resolution VH HRS spectrum in Fig. 2(b). The relative ion-induced contribution 0.88 ± 0.04 in Fig. 2(b) is given by the ratio of the integrated intensity of the ion-induced HRS peak to the integrated intensity of the flat background due to intrinsic HRS. This result from Fig. 2(b) is in good agreement with the relative contribution Bx/(1 + x) = 0.90 calculated from the fit in Fig. 2(a). Ion-induced HRS is much weaker for DMSO-d6, so Fig. 3(b) shows the spectrum for HH HRS which is more intense than VH HRS by a factor about R² at θ_s = 16.6°. The HH integrated intensity ratio is 0.0630 ± 0.0014 for the ion-induced HRS peak to intrinsic HRS background in Fig. 3(b). Using this result, with Eqs. (7) and (8) and the intrinsic HRS results from Fig. 1(b), one obtains B = 0.0533 at θ_s = 16.6°, which is used in the fit to I_HV/I_VH shown in Fig. 3(a), and also B = 0.0367 at θ_s = 90°.

The parameter $B=(IVHi)c→∞/(IVH)c=0$ = $IVH,∞i/IVH,0$ in Eq. (6) is a measure of the maximum ion-induced HRS intensity relative to the intrinsic HRS intensity. The ion-induced HRS intensities are given by¹⁵ 

where

is the Debye-Huckel screening parameter, c is the ionic strength, $f(0)=εs(ε∞+2)/(ε∞+2εs)$ is the Onsager local field factor, $β⊥=(1/5)∑α(2βzαα−βααz)$⁠, $γ⊥=(1/15)∑αβ(2γαββα−γααββ)$⁠, $γαβγδ(−2ω;0,ω,ω)$ is the second hyperpolarizability,⁵³ and the factor C includes all other geometric, optical field, and instrumental factors.

A simple expression is obtained for B at θ_s = 90° in the case that there are no intrinsic orientation correlations, Kleinman symmetry applies, θ_μβ = 0, and both γ and octupolar β are negligible so that $IVH=(1/45)|β[ss,1]|2=βXZZ2$ and $β⊥2/βXZZ2$ = 3. The expression is

where y is the dimensionless dipole strength given by Eq. (14). Table V shows calculated B₀ values which are 2.4× and 37× the respective experimental B values for acetonitrile and DMSO.

A better theoretical estimate for B is obtained by including the effects of γ, orientation correlation, and collision-induced HRS. Including γ increases the ion-induced intensity $IVH,∞i$ for acetonitrile by the factor $[1+3kBTγ⊥/(μβ⊥)]2$ = 1.23, calculated using the ab initio results μ = 4.59 D, $β||$ = 67.7 au, and $γ||$ = 4776 au for liquid phase ESHG at λ = 1064 nm.⁴³ Including orientation correlations increases I_VH,0 by a factor C_T = 1.4, and including collision-induced HRS increases I_VH,0 by a further factor 1/0.70. The revised estimate is then B = 1.35, which is still 1.5× larger than the experimental value for acetonitrile. Quantitative agreement is not obtained for B, which may be due to uncertainties in the parameters entering the calculation and also due to theoretical uncertainty about the Onsager local field factor in Eq. (41). For DMSO, the difference between B₀ and measured B is mainly due to the large angle θ_μβ between μ and vector β, which reduces B by the factor $cos2θμβ$⁠. Neglecting γ, orientation correlation, octupolar β, and collision-induced HRS, and fitting $B0⁡cos2θμβ$ to the experimental value B = 0.0367, one obtains θ_μβ = 99.5°. Including γ will increase the estimated angle θ_μβ since $IVH,∞i$ is decreased by the cancelation of the γ and μβ terms in Eq. (41), which have opposite sign for DMSO.^39,40

There are several sources of uncertainty for the ion-induced HRS intensity. The Onsager local field factor in Eq. (41) expresses the relation between the macroscopic electric field of the ions and the microscopic field that orients and distorts the molecules. Although the Onsager local field factor is commonly used, its accuracy is uncertain. An independent determination of the local field factor by ion-induced HRS may be possible⁹ and could have implications for surface studies that use second harmonic generation as a probe.⁵⁴ The ion-induced HRS intensity is measured relative to the intrinsic HRS for ion-free liquids. The relative intensity of ion-induced HRS and reorientation HRS due to just vector β, given by Eq. (44), is a simple expression where the poorly known hyperpolarizabilities cancel out, but this expression omits several contributions that are too large to ignore. The intrinsic HRS intensity is the sum of several different overlapping components, and additional spectral measurements and analysis are needed to disentangle the components and express them in terms of the vector β HRS contribution. Ion-induced HRS also has a significant contribution from γ in polar liquids. In non-polar liquids, ion-induced HRS is entirely due to γ, but in this case the intensity will be limited by the low solubility and dissociation of ionic compounds in non-polar solvents.

Orientation correlations for homogeneous, isotropic, random vector fields have been considered in this work and are described by the orientation correlation functions B_L(r) and B_T(r). Coherent HRS contributions from β⁽¹⁾ are directly related to the spectral functions S_T(K) and S_L(K) given by Eqs. (19) and (20), obtained from the Fourier transform of the correlation tensor. The coefficients A_T and A_L for the fits to the HRS data in Fig. 1 are constants, independent of scattering wavevector magnitude $K∝sin(θs/2)$⁠, and this observation places a strong constraint on the possible correlation functions. Spectral functions $ST≠SL$ which are slowly varying near K = 0, and constant coefficients $AT≠AL$ in Eqs. (1)–(4), are the result of correlation functions B_L and B_T with r⁻³ long range asymptotic form.⁸ 

Ion-induced HRS in the limit of low ionic strength, which is governed by radial orientation correlation with r⁻² asymptotic form at long range, gives a different result. In this case, the orientation vector field is not homogeneous since it is radial around each isolated ion. Equations (40)–(43) can be put in the form of Eqs. (1)–(4) with A₀ = A_T = 0, but with $AL∝K−2$ when c → 0.

For ion-induced HRS at high ionic strength, the field of the screened ions becomes homogeneous and the coefficient $AL∝SL(K)∝(K2+KD2)−1$ given by Eq. (41) tends to a constant. In this case, the orientation correlation function obtained from the inverse Fourier transform of $SL(K)∝(K2+KD2)−1$ and S_T(K) = 0 is

At high ionic strength and at long range such that $KDr≫1$⁠, the correlation function has r⁻³ long-range asymptotic form.

The presence of correlations with r⁻³ long range asymptotic form is indicated by HRS data, such as that in Fig. 1, which is fit by Eqs. (1)–(4) with constant coefficients $AT≠AL$⁠. Longitudinal HRS with A_T < A_L is observed for DMSO-d6 and is entirely due to the collision-induced HRS contribution. This indicates that collision-induced Δβ for DMSO has long-range correlation with r⁻³ asymptotic form. Dipole interaction for collision-induced polar modes of intermolecular vibration and libration may be the source of the long-range correlation, but contrary to the observation that dipole-dipole orientation correlation gives only transverse orientation HRS, the HRS contribution for collision-induced modes may be either transverse (acetonitrile) or longitudinal (DMSO). Contributions from correlated molecules up to at least r = π/K must be included to observe $AT≠AL$⁠, so the correlations between molecules over distances r > 100 nm are required to account for the HRS observations.³³ 

The HRS results for acetonitrile and DMSO illustrate the main features present in the HRS spectra of a pure liquid. HRS is often thought of as incoherent second harmonic light scattering from randomly oriented, uncorrelated, and undistorted molecules. The spectral component due to molecular reorientation most closely matches this picture, but this contribution is modified by the effect of both short-range and long-range molecular orientation correlations and by the average distortion of the molecule by its environment. The long-range correlation due to the dipole-dipole interactions in a polar liquid acting on the first rank irreducible (vector) part of the molecular first hyperpolarizability tensor β results in a transverse HRS spectrum, with a clear observational signature in its polarization and angle dependence. The intensity of the coherent HRS contribution depends on the size of the vector and octupolar parts of β and on the angle between vector β and the molecular dipole vector. The spectral width of the reorientation HRS component is typically 0.1–10 cm⁻¹, inversely proportional to the orientation relaxation time for molecules in the liquid. This is often but not always the most intense HRS contribution. Although it is often the goal, there is usually not enough information in the HRS measurements to uniquely determine the β tensor components for the molecule. However, the long-range dipole correlation strength a³ can be precisely determined from combined MD and reorientation HRS results.

The fluctuating environment for each molecule results in a distortion of the molecule which fluctuates on the collision time scale. An increment $Δβ∝γE$ is induced by the multipole electric field E of the neighbouring molecules acting on the molecular second hyperpolarizability γ, and the intensity of collision-induced HRS due to Δβ can be comparable to the intensity of HRS due to β. The multipole field varies rapidly in space, so the increment Δβ also varies rapidly in time during molecular collisions, producing a broad HRS spectrum with typical width 10–100 cm⁻¹. Explicit calculation of Δβ using MD simulation has recently become feasible and may lead to quantitative calculations of CI HRS.⁵⁵ The increment Δβ ∝ γE has a part that transforms as a vector, so the intermolecular vibrations and librations associated with Δβ are polar modes. The observed polarization dependence for collision-induced HRS shows that these modes are non-local and that they can be either transverse or longitudinal. Analogous to the hyper-Raman scattering observations seen for intra-molecular vibrations, each mode may be split into transverse and longitudinal components with different spectra. The correlation function for Δβ appears to have r⁻³ long-range asymptotic form.

The third distinct HRS component is induced by the electric field of dissolved ions. The ion field orients the dipoles, correlates β, and induces correlated Δβ on the surrounding molecules. Coherent second harmonic light scattering by these molecules is the result of the correlations. In a dilute electrolyte solution, most molecules are far from the ions, so the net electric field seen by a molecule varies slowly in time as the ions diffuse through the liquid. The resulting HRS spectrum is narrow, typically 10⁻⁶–10⁻³ cm⁻¹. The intensity of ion-induced HRS increases with ion concentration, saturating at a maximum intensity comparable to the other two HRS contributions, at ionic strength above about 0.1 mM. Theoretical predictions for the ion concentration dependence of the intensity and spectral width are in quantitative agreement with the experiment for ion-induced HRS. However, the result of the comparison of theory and experiment for the ion-induced HRS intensity is less satisfactory and more uncertain.

See supplementary material for details about the molecular dynamics simulation models for acetonitrile and DMSO.

This work was supported by the National Science Foundation (NSF) through Grant No. CHE-1212114.