Vibration overtone hyperpolarizability measured for H2

The second hyperpolarizability (γ) of the H2 molecule was measured by gas-phase electric field induced second harmonic generation at the frequencies of the one-photon resonance for the 3–0 Q(J) overtone transitions (v, J = 0, J → 3, J for J = 0, 1, 2, and 3). The magnitude of the resonant contribution to γ was measured with 2% accuracy using the previously determined non-resonant γ for calibration. Pressure broadening and frequency shift for the transitions were also measured. A theoretical expression for the resonant vibrational γ contribution in terms of transition polarizabilities is compared to the observations. The measured γ resonance strength is 4%–14% larger than the results obtained from this theoretical expression evaluated using ab initio transition polarizabilities. Published under license by AIP Publishing. https://doi.org/10.1063/5.0005233., s


I. INTRODUCTION
The nonlinear optical (NLO) response of a centrosymmetric molecule is described by the second hyperpolarizability tensor γ αβγδ (−ωσ; ω 1 , ω 2 , ω 3 ), which is a complicated function of the incident field frequencies ω 1 , ω 2 , ω 3 and polarizations β, γ, δ due to the electronic, vibrational, and rotational excitations of the molecule. 1 The electronic hyperpolarizability contribution can be simply described by a universal dispersion formula that applies for all NLO processes, 1,2 but the expressions for the vibrational hyperpolarizability are more complicated and differ significantly from one NLO process to the next. 3,4 For the special case of a homonuclear diatomic molecule, a simple expression for the vibrational hyperpolarizability has been derived, which gives the vibrational hyperpolarizability in terms of the vibrational transition polarizabilities for the molecule, and applies for all NLO processes. 5 For a homonuclear diatomic molecule, using this expression to calculate the vibrational hyperpolarizability, the electronic hyperpolarizability can be determined from hyperpolarizability dispersion measurements for one NLO process. This then allows the dispersion curve for any other NLO processes to be determined. Such an analysis has been applied to hyperpolarizability measurements for H 2 , D 2 , N 2 , and O 2 . 6 However, some approximations are made in the derivation of this expression for the vibrational hyperpolarizability, and a direct experimental test of the expression has not been performed since the vibrational and electronic contributions are not distinguished in the usual experiments measuring the total molecular hyperpolarizability. Since the electronic and vibrational contributions are defined by their resonance frequencies, the resonant response of the vibrational hyperpolarizability can be used to unequivocally distinguish it from the electronic hyperpolarizability. The Δv = 3 overtone transitions in H 2 fall in the frequency range accessible with an available Ti:sapphire laser, and for H 2 there is no dipole-allowed absorption to impede a measurement of the resonant vibrational hyperpolarizability. In the following, the theoretical expression for the resonant vibrational hyperpolarizability is presented, the experiment to measure the resonant overtone vibrational hyperpolarizability in H 2 is described, and the experimental results are used to test the theoretical expression for the vibrational hyperpolarizability.

II. THEORY
The second hyperpolarizability of a non-polar molecule is given by the following sum-over-states expression: 5,[7][8][9] The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp where ωσ = ω 1 + ω 2 + ω 3 , |g⟩ is the initial (ground) state of the molecule, μα is the α Cartesian component of the electric dipole operator, and ∑P denotes the summation of the 24 terms obtained by permuting the frequencies −ωσ, ω 1 , ω 2 , ω 3 along with the associated spatial indices α, β, γ, δ that specify the molecular frame Cartesian components of the induced dipole and the applied electric fields.
where ωng is a rovibrational transition frequency, ωmg and ωpg are electronic transition frequencies, and the factors in braces have the form of transition polarizabilities. An expression for the vibrational hyperpolarizability of a gas of homonuclear diatomic molecules for any third order nonlinear optical process [Eq. (17) in Ref. 5] is obtained from Eq. (2) by making the approximation that optical frequencies are negligible compared to electronic transition frequencies in the transition polarizability factors, summing over permutations, and evaluating the isotropic average. The isotropic averaged hyperpolarizability tensor for electric field induced second harmonic generation (ESHG) has two independent components γ || = ⟨γ(−2ω; ω, ω, 0)⟩ XXXX and γ = ⟨γ(−2ω; ω, ω, 0)⟩ XYYX , with optical and static electric fields polarized either parallel or perpendicular, respectively. From Eq. (17) in Ref. 5, the one-photon resonance contribution to the ESHG vibrational hyperpolarizability due to a Q branch vibration transition, near resonance where |ω 0J,vJ − ω| ≪ ω, is given by where α = (α || + 2α )/3 and Δα = (α || − α ) are the mean and anisotropy of the transition polarizabilities for light polarized || or to the molecular axis for the selected Q branch transition with resonance frequency, ω 0J ,vJ , and ρ(J) is the fractional population of rotational level J. Similar expressions are obtained for the O and S branch transitions with frequencies ω 0J ,vJ ±2 , Equations (3)-(6) are obtained by neglecting optical frequencies in the polarizability factors in braces in Eq. (2). A better estimate of frequency dispersion, without this approximation, can be made if one considers only the terms in Eq. (2) near one-photon resonance for a single vibration transition frequency ωng. Selecting just those terms, ignoring spatial indices αβγδ, writing out the 24 frequency permutations for ESHG, relabeling dummy indices, and collecting terms gives the following expression: The tensor γ v αβγδ for H 2 ESHG has seven independent components (αβγδ = zzzz, zxxz, xzzx, zzxx, xxzz, xyyx, and xxxx, where z is the molecular axis). Equation (7) is the exact result for the zzzz and xxxx components of γ v αβγδ , where all permutations of αβγδ are the same. The other components have more complicated expressions with the same resonance denominators. The polarizability factors in braces in Eq. (7) can be compared to the expressions for the static polarizability α 0 , and the Raman polarizability αR, 10 where ωP is the pump frequency and ωS = ωP − ωng is the Stokes frequency for Raman scattering. The effective polarizability αγ for γ v res is the square root of the product of the two transition polarizabilities appearing in Eq. (7). The dispersion for αγ is given by the geometric mean of factors with resonance denominators (ωmg + 2ω) and (ωmg − ω), whereas the dispersion of αR is given by the arithmetic mean of terms with resonance denominators (ωmg + ωS) and (ωmg − ωP). The resonance denominators in Eqs. (7) and (9) indicate that the Raman polarizability αR will increase with increasing frequency ωP since ωS < ωP, whereas αγ will decrease with increasing ω (when ωmg > 4ω) since 2ω > ω.

ARTICLE scitation.org/journal/jcp
To deal with zero detuning from resonance, the usual phenomenological imaginary damping iΓ term is added to the resonance denominators in Eqs. (3)-(6), giving complex hyperpolarizabilities The observed second harmonic signal near a vibrational resonance is S (2ω) ∝ |γ| 2 where the total hyperpolarizability γ is the sum of the complex resonant vibrational hyperpolarizability and the slowly varying non-resonant real hyperpolarizability due to all other contributions, The signal variation near resonance is determined by the relative size of the resonant and non-resonant contributions, so the resonant hyperpolarizability γ v res can be calibrated in terms of γnr.

III. EXPERIMENT
The electric field induced second harmonic generation (ESHG) apparatus used in the present gas phase H 2 vibration overtone hyperpolarizability measurements is similar to that previously described. 6,[11][12][13] A coherent beam of second harmonic light is generated by the laser beam in a H 2 gas sample when a transverse static electric field is applied to the gas. The static electric field distorts the molecule and breaks centrosymmetry, allowing optical second harmonic generation even from spherically symmetric systems such as an atom or H 2 in the J = 0 rotational state. The spatially alternating static electric field applied to the gas is produced by a periodic array of N electrode pairs, and the ESHG signal is enhanced by a factor N 2 when the coherence length in the gas is adjusted to match the longitudinal period of the electrode array. To make an overtone signal measurement, the gas density ρ is first adjusted to set the coherence length equal to the array period when the laser frequency is at the vibration overtone frequency. Then, with the gas density held constant at this value, the second harmonic signal is measured as the laser frequency is scanned over a 2 cm −1 range around the vibration overtone frequency. Four Q-branch Δv = 3 transitions near λ = 850 nm were measured. Figure 1 shows a schematic diagram of the apparatus. The cw ring Ti:sapphire laser (Ti:S) produces a frequency tunable single longitudinal mode (SLM) output beam with power about 400 mW in the 850 nm wavelength range. The laser beam is collimated by lens L1, the desired linear polarization state is prepared by the half wave plate (HWP) and prism polarizer (POL), lens L2 focuses the beam to a waist in the center of the gas cell with 20 cm confocal parameter, and the red glass filter (RG645) blocks any light at the second harmonic wavelength that may be present in the incident beam before it enters the gas cell (CELL). ESHG occurs as the laser beam passes between the electrodes in the gas cell (Fig. 1, inset). The transmitted fundamental beam and coaxial second harmonic beam generated in the gas cell are collimated by lens L3 and then separated by using the tandem Brewster prism spectrometer (P1, P2). The narrow second harmonic beam is expanded by lens L4 before reaching the photocathode of the photon counting photomultiplier tube (PMT). The quantum efficiency of the bialkali photocathode at the 850 nm laser wavelength is so low that sufficient rejection of the fundamental laser beam is achieved using just the double prism spectrometer without any additional spectral filters.
The laser frequency, power, and mode were continuously monitored during the ESHG measurements. About 1% of the laser beam is reflected by each of three beam samplers (BS1, BS2, and BS3). The beam sample from BS1 is divided by using beam splitter BS4 and then fiber-coupled to a Burleigh WA-20 wavemeter (WM) to measure the laser frequency and to a scanning Fabry Perot interferometer (FP) to monitor the laser mode. The beam sample from BS2 is focused by using lens L5 onto a 5 GHz InGaAs photodiode (PD) feeding a RF spectrum analyzer monitoring the beat note spectrum of the laser output. Since second harmonic generation (SHG) measurements are sensitive to mode composition, 14,15 measurements were rejected if the appearance of an inter-mode beat note indicated that the laser was not producing SLM output. The final beam sample from BS3 goes to an integrating sphere (IS) and photodetector, to produce an output which is proportional to the laser power P actually reaching the gas cell during the ESHG measurement. For each ESHG measurement, P was determined from the average of multiple power measurements and S (2ω) /P 2 was computed to account for the effect of beam power fluctuations. BS3 is placed after HWP and POL so that etalon effects in these components, which can cause uncontrolled changes in beam transmission, do not corrupt the power measurement. To prevent etalon effects in the beam path after BS3, all of the following components are tilted or wedged so that no reflected beams overlap and L4 is inserted to diverge and expand the signal beam onto the photocathode, disrupting possible etalon fringes at the PMT entrance window.
Second harmonic generation by the laser beam at the surfaces of prism P1 is not forbidden by symmetry, and such a coherent SHG background could cause ESHG measurement errors. The observed signal in the presence of coherent background is S± ∝ (±Es + E b cos φ) 2 , where ±Es and E b cos ϕ are the electric field amplitudes of the ESHG signal and the coherent SHG background, respectively, and ± indicates the two choices for the polarity of the voltage applied to the electrode array. The electric field between every pair of electrodes changes sign when the array polarity is reversed. This changes the sign but not the magnitude of the ESHG signal Es. The relative amplitude of the coherent background was determined from the ratio of signals S+/S−, measured with forward and reversed array voltage, using the expression E b cos φ/Es = (1/4)(S+/S− − 1). The measured amplitude for the coherent background is negligible, |E b cos φ/Es| < 0.07%. Figure 2 shows a scan of the ESHG signal spectrum for the H 2 Q(1) overtone transition, with the sharp vibration resonance riding on the much broader phase match peak. The functional form of the broad peak can be calculated by numerical integration of the phase match integral given in Refs. 6 and 12 using the electrode array and laser beam parameters for this experiment. The electrode array has N = 82 pairs of electrodes with period 5.08 mm, similar to array 1 in Ref. 13, and the confocal parameter of the focused laser beam is 20 cm. The normalized phase match peak obtained by numerical integration is approximated by the function where x = (ν − νm)/w and w ≈ 54 cm −1 . The peak position νm and width w are parameters determined by fitting Eq. (13) to the phase match peak measurements, and the broad curve in Fig. 2 is the function Smg(ν) fit to the off-resonance data. The gas density at phase match varies as ρm ∝ ν −3 m , 12 and for each experiment, the gas density is adjusted to place νm within about 1 cm −1 of the molecular resonance frequency. The phase match function g(ν) multiplies the resonance signal function, and when νm is placed within 1 cm −1 of the resonance frequency, the variation of g(ν) over the resonance measurement range is <0.2%. Phase match at λ = 850 nm is obtained with H 2 gas density ρ = 406 mol m −3 (9.11 amagat). The typical offresonance photon count signal was 300 s −1 for laser power 400 mW and static electric field 6 kV/mm, and the 0.6 s −1 PMT background count rate measured with zero static field was subtracted from this signal.
The resonance signal shown in Fig. 2 is measured at closely spaced points over an interval of about 2 cm −1 around resonance. The fit to the resonance data uses the following function based on Eq. (12): where a is the relative magnitude of the resonant and nonresonant hyperpolarizabilities, b is the resonance frequency, c is the resonance width, and a, b, and c are parameters to be determined from the experimental observations. The experimental units are cm −1 for ν, a, b, and c. The observed second harmonic signal is given by the product of Eqs. (13) and (14),

IV. RESULTS AND DISCUSSION
The results of measurements for the 3-0 Q(J) overtone transitions for J = 0, 1, 2, and 3 are shown in Fig. 3. Each graph shows the combined results of multiple complete scans for each resonance. Separate scans of a resonance were combined by first fitting Eq. (15) to the S (2ν) /P 2 data for each scan and then computing and combining the normalized and baseline corrected data f (ν) = [S (2ν) /P 2 ]/[Ag(ν)] for each scan. Figures 3(a)-3(e) show the combined data for 3, 3, 1, 3, and 2 separate scans, respectively. The frequency sequence in each scan was shuffled to reduce the effect of slow signal drift during the course of the measurements, and 2, 2, 0, 2, 1 of the scans in Figs. 3(a)-3(e) were also done by "triplets." For these scans, a point about 0.4-1.0 cm −1 offresonance was designated as the internal reference for the scan and measurements at other frequencies designated as the signal. Reference and signal measurements were alternated, and each signal measurement was normalized using the average of the reference measurements immediately before and after that signal measurement. of Chemical Physics  Table I gives the fit parameters obtained by fitting Eq. (14) to the combined data for each resonance. As outlined above, special care was taken to eliminate or account for any trend in the baseline underlying each resonance to prevent a systematic error in the resonance strength parameter a. The 0.001-0.003 cm −1 precision obtained for the resonance position and width parameters b and c determined by the fits to the data is 3-10 times better than the 0.01 cm −1 absolute frequency accuracy of the wavemeter. Photon counting statistics accounts for about half of the total variance of the data points around the fit curves, except for the Q(1) resonance with laser beam polarization perpendicular to the static field in the cell, as shown in Fig. 3(c), where the signal is 10 times smaller and the variance is dominated by photon counting statistics. For the fit to this spectrum, the position and width for the resonance were set to the values determined for the same resonance with parallel polarization and only the resonance strength parameter a was determined from the fit to the data. The O and S branch transitions were not measured since the signal for these transitions was expected to be too weak, comparable to, or smaller than the signal for the corresponding Q branch transitions with perpendicular polarization.
Hydrogen gas at the pressure used in the present measurements is well into the collision broadening regime, where the resonance frequency shift and width are proportional to pressure. 16 Table II gives the pressure shift and broadening coefficients calculated from the present measurements and a comparison with other previous measurements for H 2 gas. 17,18 The frequency shift is determined from the resonance frequency b measured in the present experiment using the ab initio result for the vibration transition frequency of an isolated H 2 molecule as the frequency in the limit of zero pressure broadening. 19 The ab initio result in Table II for the 1-0 Q(0) transition (v, J = 0, 0 → 1, 0) differs from the experiment by 0.002 cm −1 , and by just 0.0002 cm −1 when relativistic and QED effects are included. 20 A similar comparison of experiment and theory for the 3-0 S(3) transition gives a difference of 0.0003 cm −1 , 21 with collisional line shape effects accounting for 0.0001 cm −1 of the difference. 22 The pressure shift coefficients given in Table II for the 3-0 Q(J) transitions are determined assuming that the ab initio frequencies for these transitions are also accurate at the 0.002 cm −1 level, while the accuracy for the experimental frequencies from the present work is 0.01 cm −1 . Figure 4 shows the trend of the frequency shift and broadening coefficients as functions of the transition frequency. The pressure shift (3.8 ± 0.1) × 10 −3 cm −1 /amagat recently measured for where ρ(J) ∝ (2J + 1)gns(J)exp(−E 0J /kBT) is the fractional population of rotational level J of the ground state at temperature T and eH = 219 474.6307 cm −1 /hartree converts energy difference ̵ h(ω 0J ,vJ − ω) from atomic units to cm −1 . For Q branch transitions with parallel optical and static field polarization, and for Q branch transitions with perpendicular field polarization, The rotational level populations ρ(J) were calculated using the energy levels from the ab initio calculation in Ref. 19, and α 2 eff was calculated using the ab initio static transition polarizabilities from Ref. 23. A subsequent ab initio calculation of α 2 eff for the Raman 3-0 Q(1) transition 10 is in good agreement with the earlier static result. 23 The non-resonant hyperpolarizability γnr is obtained using the expression fit to previous 0.5% accurate ESHG measurements made for H 2 over a wide frequency range with parallel polarization, 6 where   Table III, and a possible reason for the discrepancy between the calculated and experimental results is the neglect of the frequency dependence of the polarizabilities in the theoretical calculation. To estimate the frequency dispersion, one can approximate each sum over excited electronic states appearing in Eqs. (7)-(9) by a single term with an effective electronic excitation energy. The effective electronic excitation energy can be obtained by fitting approximate Eq. (9) to the polarizability dispersion calculated in Ref. 10 for the 1-0 Q(1) H 2 Raman transition. The ab initio Raman polarizability at ̵ hω = 0.10 a.u. is 1.094 times larger than the static polarizability, and the fit to this dispersion value gives an effective electronic transition frequency νeg = 82 000 cm −1 (0.374 a.u.). Using the single term approximation with νeg = 82 000 cm −1 for the polarizabilities in Eq. (7), one estimates that αγ at 11 765 cm −1 is reduced to 0.95 times its static value so that the values for a calc in Table III are overestimates by 10%, and the discrepancy between theory and experiment is actually 14% rather than 4%. This result from the present experiment indicates that the ab initio polarizabilities for the Δv = 3 transitions are 2%-7% too small. Also consistent with the present result, the previously measured value for the 3-0 Q(1) Raman transition polarizability 25 is 5% larger than the ab initio value. 10 Negative dispersion that is deduced for the polarizability αγ is unexpected, but the ab initio calculation for the Δv = 3 overtone Raman polarizability also exhibits small negative dispersion due to the combined effects of mechanical anharmonicity and strong frequency dependence of the polarizability second derivative. 10 Further support for the conclusion that the 3-0 Q(J) polarizabilities are larger that the ab initio calculated values comes from a Morse potential model calculation for H 2 that gives values for the high overtone polarizabilities that are orders of magnitude higher than the extrapolated results of ab initio calculations 26 and also from recent experiments selectively populating high vibrational states of H 2 that are fit by a polarizability more than an order of magnitude larger than the ab initio polarizability for the Δv = 4 transition. 27 The hyperpolarizability contribution from the 3-0 Q(1) transition reaches maximum magnitude |γ v | = 515 a.u. on resonance but decreases to ±0.26 a.u. just 100 cm −1 off-resonance. In comparison, the rovibrational contribution to the nonresonant hyperpolarizability is γ vr nr = −20.5 a.u. at the 3-0 Q(1) resonance frequency and is dominated by the far off-resonance contribution of the fundamental vibration transition. There is good agreement between calculations for the fundamental transition polarizabilities, 10,28 but the frequency dependence of the effective polarizabilities appearing in the expression for γ vr nr is uncertain. The expression for γ vr nr is based on Eq. (17) of Ref. 5, which was obtained by neglecting the frequency dependence of the polarizability factors in Eq. (2). A simple expression such as Eq. (7) is not obtained for γ v far off-resonance. The dispersion of the effective polarizability in γ vr nr may be significant, as suggested by the Raman polarizability at νP = 11 765 cm −1 for the 1-0 Q(1) transition, which is 1.044 times the static polarizability. 10 This dispersion would produce a 10% increase in the calculated value for γ vr nr at this frequency. An indication that the dispersion for the effective polarizability may be less than that for αR comes from the comparison of accurate experimental ESHG and ab initio results for γ of H 2 at ν = 19 430 cm −1 , 13 where the experiment and theory agree with ±0.1% uncertainty. This sets a bound on the possible error for the calculated vibrational hyperpolarizability γ vr nr , which contributes 0.74% of γ at this frequency. Agreement within 0.1% indicates that the increase in γ vr nr due to the effective polarizability dispersion is <14% at this frequency. The uncertainty for γ vr nr does not affect the accuracy of the vibrational resonance hyperpolarizability calibration in terms of γnr in the present work. The uncertainty for γ vr nr affects the partition of γnr into electronic and rovibrational contributions but not the total, since Eq. (19) for γnr is the sum of the two contributions adjusted to fit the experimental γnr data.

V. CONCLUSION
This experiment measured the second overtone vibrational contribution to the second hyperpolarizability of H 2 and found that the measured strength of the observed resonance is 4%-14% larger than predicted using the transition polarizabilities from high level ab initio calculations. This experiment determines the transition polarizability with accurate absolute calibration provided by the nonresonant hyperpolarizability previously determined for the molecule. This method could be used to measure overtone transition polarizabilities for other small molecules.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.