Study of the H2O dipole moment and polarisability vibrational dependence by the analysis of rovibrational line shifts

Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 210 (2019) 275–280

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Study of the H2O dipole moment and polarisability vibrational dependence by the analysis of rovibrational line shifts V.I. Starikov a,b, T.M. Petrova c,⁎, A.M. Solodov c, A.A. Solodov c, V.M. Deichuli c,d a

Department of Advanced Mathematics, Tomsk State University of Control System and Radio Electronics, Lenina Av. 40, 634050 Tomsk, Russia Department of Informational Systems, National Research Tomsk Polytechnic University, Lenina Av. 30, 634050 Tomsk, Russia Department of Spectroscopy, V.E. Zuev Institute of Atmospheric Optics, Siberian Branch, Russian Academy of Sciences, 1, Academician Zuev Square, 634021 Tomsk, Russia d Department of Photonics and Informatics, National Research Tomsk State University, Lenina Av. 36, 634050 Tomsk, Russia b c

a r t i c l e

i n f o

Article history: Received 13 June 2018 Received in revised form 16 October 2018 Accepted 13 November 2018 Available online 14 November 2018 Keywords: Water Dipole moment and polarisability Line shift

a b s t r a c t The study of the H2O dipole moment μ and polarisability α vibrational dependence is based on the comparison of experimental and calculated line shifts induced by argon, nitrogen, and air pressure in different H2O vibrational bands. Obtained dependence α on the stretching vibrations is in good agreement with the existing ab initio calculations in the literature, but the dependence α on the bending vibration is quite different. To clarify the dependence of μ and α on the bending vibration, the shifts of selected H2O lines of the 4ν2, 5ν2, and 6ν2 bands induced by argon, hydrogen and helium pressure are measured with the help of a Bruker IFS HR 125 spectrometer at room temperature with a spectral resolution of 0.01 cm−1. The comparison of experimental and calculated results with different values of μ and α line shifts is given. © 2018 Published by Elsevier B.V.

1. Introduction Knowledge of the pressure-broadened line widths and line shifts in molecular gas vibration-rotation spectra is important for atmospheric applications as well as in the study of intermolecular forces between gas constituents. Line width and shift calculations require the vibrationrotation matrix elements of different physical quantities such as the dipole moment μ, polarisability α, etc. (see, for example, [1]). Line shifts mainly are determined by the difference of the isotropic interaction potential in different vibrational states of the active molecule. The H2O vibrational state (n) ≡ (v1,v2,v3) is defined by the three vibrational quantum numbers v1, v2, and v3, where v1 and v3 are associated with stretching vibrations and v2 is associated with the bending vibration. The vibrational dependence of a long-range part of the isotropic interaction potential for the system “H2O-perturber” is determined by the vibrational dependence of the dipole moment μ(n) and polarisability α(n). According to Shostak and Muenter [2], μ 1 ðnÞ ¼ −1:8570−0:0051ðv1 þ 1=2Þ þ 0:0317ðv2 þ 1=2Þ−0:0225ðv3 þ 1=2Þ  ðDÞ

ð1Þ

Then Mengel and Jensen [3] calculated dipole moments μ2(n) for many vibrational states (n), including highly excited bending state (0, ⁎ Corresponding author at: 1, Academician Zuev square, V.E. Zuev Institute of Atmospheric Optics SB RAS, Tomsk 634055, Russia. E-mail address: [email protected] (T.M. Petrova).

https://doi.org/10.1016/j.saa.2018.11.032 1386-1425/© 2018 Published by Elsevier B.V.

v2 = 11,0). The calculations were performed with an accurate potential energy surface and high-quality ab initio dipole moment surface. The calculations have been improved by refining the dipole moment surface in least-squares fit of experimental high-precision Stark data. There are no significant differences between μ1(n) and μ2(n), if in the state (n) the bending quantum number v2 = 0. The comparison of μ1(0, v2, 0) and μ2(0, v2, 0) is given in Fig. 1. This figure shows that the differences between μ1(0, v2, 0) and μ2(0, v2, 0) begin with v2 N 2. Note that calculated in Ref. [3], μ2(0, v2, 0) is in good agreement with μ(0, v2, 0) calculated in Ref. [4] for v2 b 4. The effective H2O polarisabilities α(0, 0, 0) = 1.4613, α(0, 1, 0) = 1.4656, α(0, 2, 0) = 1.4701, α(1, 0, 0) = 1.5042, and α(0, 0, 1) = 1.5027 (in Å3) for the five vibration states were obtained by Luo et al. [5] in ab initio calculations. This result may be presented as

3

α 1 ðnÞ ¼ 1:4613 þ 0:0429v1 þ 0:0044v2 þ 0:0415v3 Å

ð2Þ

Eqs. (1) and (2) are widely used in the literature for calculations of the broadening and shift coefficients of H2O lines perturbed by different foreign gases. In our analysis of the widths and shifts [6,7] of water vapour lines perturbed by the helium and argon pressures, the dipole moments μ(n) from Ref. [3] were used, and as a consequence the different dependence of α on the vibrational quantum numbers vi(i = 1, 2, 3) was found. This applies in particular to the bending quantum number v2.

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V.I. Starikov et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 210 (2019) 275–280

The ATC and RB methods require the calculations of interruption functions of the first and second orders (S1 and S2). The exact forms of the S2 term in the ATC and RB methods are given in Refs. [1,8], and Refs. [10,13], respectively. The S1 term is determined by the difference ~ ðnÞ ðRÞ−V ~ ð0Þ ðRÞ of the effective isotropic potential in the excited (n) V isot

isot

and ground vibrational state (n = 0) = (0, 0, 0), respectively. It is the term much affected on the calculated shift coefficients. For the effective isotropic potential ~ ðnÞ =R6 þ C ~ ðnÞ =R12 ~ ðnÞ ðRÞ ¼ −C V isot 6 12

ð3Þ

the function S1 is written as [6] S1 ¼

Fig. 1. Variation of H2O dipole moment μ in the vibrational state (0, v2, 0) according to Ref. [2], (line 1), Ref. [3] (line 2) and Ref. [4] (line 3).

Thus at present there is an ambiguity in the values of the dipole moments and polarisability of the H2O molecule, especially for the highly excited vibrational states with v2 ≠ 0. Knowledge of the dipole moments μ(n) and polarisabilities α(n) in different vibrational states (n) is very important to determine the derivatives of the dipole moment and polarisability of H2O molecule with respect to normal coordinates. These derivatives are necessary to calculate the line intensities of IR and Raman spectra. The aim of the present paper is to study which of the functions μ(n) and α(n) are more acceptable in the calculation of shifts in H2O lines induced by the pressures of different foreign gases. The study is based on the comparison of experimental and calculated shifts of H2O lines. In an initial step of the study, we compare the lines perturbed by argon, nitrogen, and air pressure. These collision partners were selected because, firstly, in the literature there are a lot of shift coefficients measured for many H2O vibrational bands (0, 0, 0) → (v1, v2, v3) with v1 ≤ 3, v2 ≤ 3, v3 ≤ 2; secondly, for these partners, calculated coefficients δ strongly depend on the values of μ(n) and α(n). The broadening coefficients are not highly sensitive to small variations in μ(n) and α(n). In the second step of the study, we compare the measured and calculated shift coefficients for the selected lines of 4ν2, 5ν2 and 6ν2 vibrational bands perturbed by argon, hydrogen and helium pressures. For these lines the broadening coefficients are measured using a highresolution Fourier transform spectrometer Bruker IFS 125HR.


 3π 21 ΔC 12 −a ΔC þ a 6 6 12 8vℏr 5c 32 r 6c

ð4Þ

where v is the relative velocity of interacting particles, rc is the closest ~ ðnÞ −C ~ ðn¼0Þ and the constants approach distance between them, ΔC p ¼ C p

p

~ ðnÞin Eq. (3) and ΔCp ap are defined in Ref. [14], p = 6, 12. The quantities C p in Eq. (4) depend on the vibrational (n) and rotational (J, K ≡ Ka) quantum numbers of the H2O molecule [17]. For H2O–N2, H2O–air and H2O– Ar systems, the vibrational dependence of the short-range part of the potential (3) can be neglected, and thus for these systems ΔC12 = 0 in Eq. (4). The vibrotational dependence of the long-range part of potential (3) is determined by the vibrotational dependence of the effective induction +dispersion potential and may be presented as. h ~ ind−disp ðRÞ noneVÞ isot;long

¼−

i ~ ðnÞ  α 2 μ~ ðnÞ2 þ 3=2u  α R6

ð5Þ

~ ðnÞ are the effective dipole moment and mean here μ~ ðnÞ and α polarisability of the H2O molecule, u = u1u2/(u1 + u2), α2 is the polarisability of the buffer gas, and u1 and u2 are the ionization energies of the active and buffer molecule, respectively. In our calculations, the effective dipole moment μ~ ðnÞ was taken as μ~ ðnÞ ¼ μ ðnÞ þ h200 ð0; v2 ; 0Þ  J ð J þ 1Þ þ h020 ð0; v2 ; 0Þ  K 2a :

ð6Þ

The terms with the constants h200(0, v2, 0) and h020(0, v2, 0) from Ref. [10,18] determine the rotational dependence of the dipole moment. The effective mean polarisability was taken in the form ~ ðnÞ ¼ α ðnÞ þ Δα ð J Þ  J ð J þ 1Þ þ Δα ðK Þ K 2a : α

ð7Þ

The values of the parameters Δα(J) and Δα(K) are presented in Refs. [10,19]. Thus, in our approach in Eq. (4) 2. Calculation Method The line broadening and the line shift coefficients were calculated using the framework of semi-classical methods ATC [8] and RobertBonamy (RB) [9]. The ATC method was employed for H2O–N2 and H2O–air systems, and the RB method was employed for H2O–He, H2O– Ar, and H2O–H2 systems. Both methods are well described in Refs. [1,8–10]. Here we briefly discuss a few details. For H2O–N2 and H2O–air systems, the potential consists of the leading electrostatic components (the dipole and quadrupole moments of H2O with the quadrupole moments of N2 or O2) and the induction potential. The quadrupole moments of H2O are taken as qxx = −0.13, qyy = −2.5, qzz = 2.63 (in 10−26 esu) [11]. For the N2 molecule, we used q = −1.4 × 10−26 esu [11], and polarisability α = 1.74 Å3 [12], and for O2 molecule we used q = −0.4 × 10−26 esu Å [11] and α = 1.571 Å3 [12]. The interaction potentials for H2O–He and H2O–Ar systems were chosen in the form of atom-atom potential [13] with potential parameters from Refs [14,15]. The interaction potential for the H2O–H2 system is described in Ref. [16].

~ ð f Þ−α ~ ðiÞα 2 ; ΔC 6 ¼ ½μ~ ð f Þ−μ~ ðiÞ þ 3=2uðα 2

2

ð8Þ

~ ðf Þ, α ~ ðiÞ are the dipole moments and mean where μ~ ðf Þ, μ~ ðiÞ and α polarisabilities of the H2O molecule in the final (f) = (v1 v2 v3)[Jf Kaf Kcf] and initial (i) = (000)[Ji Kai Kci] vibration-rotational states, respectively, in the transition (i) → (f). According to Eqs. (6), (7), and (8), the quantity ΔC6 may be presented as ΔC6 = ΔC6vib + ΔC6rot with ðnÞ−C 6 ð0Þ ΔC vib 6 ¼C 6  ¼ μ 2 ðv1 ; v2 ; v3 Þ−μ 2 ð0; 0; 0Þ þ 3=2uðα ðv1 ; v2 ; v3 Þ−α ð0; 0; 0ÞÞ α 2 ð9Þ h i h   i ð JÞ ðK Þ 2 2 ΔC rot 6 ≈ C 6  J f J f þ 1 − J i ð J i þ 1Þ þ C 6  K af −K ai In the last equation h i ð JÞ C 6 ¼ 2μ ðv1 ; v2 ; v3 Þ  h200 þ 3=2u  Δα ð J Þ α 2

ð10Þ

V.I. Starikov et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 210 (2019) 275–280

h i ðK Þ C 6 ¼ 2μ ðv1 ; v2 ; v3 Þ  h020 þ 3=2u  Δα ðK Þ α 2

277

For H2O–air system with N = 11, rmsα = 1.5 × 10−3 Å3 and

ð11Þ

α H2 O−air ðv1 ; v2 ; v3 Þ ¼ 1:4613 þ ð0:045  0:006Þv1

(

2

ðδi ð expÞ−δi ðcalÞÞ N i¼1 N

)1=2

∑

ð15Þ

In Eqs. (13)–(15) α(0) ≡ α(0, 0, 0) was fixed to the ab initio value α(0, 0, 0) = 1.4613 Å3 [5]. In all Eqs. (13)–(15), the dependence α(v1, v2, v3) on the bending quantum number v2is significantly different from the ab initio dependence (2). Finally, averaged over N = 30, the vibrational bands of H2O perturbed by Ar, N2, and air

The quality of calculations of shift coefficients was characterized by the quantity rmsδ defined as

rmsδ ¼

3

þ ð0:021  0:004Þv2 þ ð0:043  0:007Þv3 Å

3. Line Shift Calculations for Ar, N2, and Air Collisions

ð12Þ

α ðv1 ; v2 ; v3 Þ ¼ 1:4613 þ ð0:039  0:002Þv1 þ ð0:022  0:002Þv2 3

þ ð0:041  0:003Þv3 Å

where N is the number of used data. For H2O lines perturbed by Ar, N2, and air pressure, we analysed 12, 7, and 11 vibrational bands, respectively, they are shown in Table 1. For each band (0, 0, 0) → (v1, v2, v3) and each collisional partner Ar, N2, and air, the α(v1, v2, v3) value for ΔC6vib in Eq. (9) was selected by hand to minimize rmsδ (12). These values α(v1, v2, v3) may be called optimal polarisabilities of the H2O molecule, which is in a given vibrational state (v1, v2, v3). The dipole moments μ(v1, v2, v3) in Eq. (9) were taken from Ref. [3], because for all investigated bands with v2 ≤ 3, and for these bands the differences in the dipole moments μ1(v1, v2, v3) from Ref. [2] and μ2(v1, v2, v3) from Ref. [3] are not significant (see Fig. 1), and do not affect the calculated shift coefficients δ(cal). The optimal polarisabilities α(v1, v2, v3) obtained for the investigated H2O vibrational bands are given in Table 1. The values of α(v1, v2, v3) obtained for each collisional partner were used in root-squared analysis and the following polynomial functions were determined. For H2O–Ar system

Calculated rmsα = 1.26 × 10−3 Å3. This function may be considered as an optimal function, given the best description of the line shift of 30 vibrational bands of the H2O molecule (v1 ≤ 3, v2 ≤ 3, v3 ≤ 2) perturbed by Ar, N2, and air pressure. In Table 2, we give the comparison of experimental and calculated data, with function α(n) (16) shift coefficients for the transitions of H2O vibrational bands with v2 ≠ 0. The results for other bands for H2O–N2 and H2O–O2 collisions are given as Supplementary Materials. For H2O–air collisions δ(H2O–air) may be calculated as δ(H2O–air) = 0.79 δ(H2O–N2) + 0.21δ(H2O–O2). For H2O–Ar collisions the results are published in Ref. [6]. For N = 38 lines from Table 2 with known δ (exp) rmsδ (12) is equal 2.5·10−3 cm−1/atm that is the average agreement of experimental and calculated data is rather good. But for 7 lines there is N100% difference between experimental and calculated shifts. These big differences may be attributed to the uncertainties in the observations as well as in the theory. Note that δ(exp) for the hot bands (0, 2, 0) ← (0, 1, 0) and (0, 2, 1) ← (0, 1, 0) were not used in the determination of the function α(n) (16).

α H2 O−Ar ðv1 ; v2 ; v3 Þ ¼ 1:4613 þ ð0:036  0:001Þv1 þ ð0:0207  0:0014Þv2 3

þ ð0:039  0:002Þv3 Å

ð16Þ

ð13Þ

4. Highly Excited Bending States The parameters in this equation were found earlier in Ref. [6] and only the confidence intervals for them are added in the present study. With function (13) rmsα = 4.2 × 10−3 Å3. The rmsα is defined by Eq. (12), where the symbol δ has been substituted by the symbol α, and N = 12 is the number of investigated bands. For H2O–N2 system with N = 7,

Performed line shift calculations confirm that function α(n) (16) and dipole moments μ1(n) and μ2(n) from Ref. [2,3] may be well used in the shift coefficient calculations for H2O vibrational bands (0, 0, 0) → (v1, v2, v3) with v2 ≤ 3. For v2 N 3, dipole moments μ1(n) from Ref. [2] and μ2(n) [3] are different, according to Fig. 1. To clarify the dependence of μ and α on the bending vibrational quantum number v2 we measured shifts of selected H2O lines for the 4ν2, 5ν2 and 6ν2 bands perturbed by argon, hydrogen, and helium pressures. The water vapour absorption spectra in the 7000–8700 cm−1 region perturbed by the pressure of Ar, He and H2 were recorded using a highresolution Fourier transform spectrometer Bruker IFS 125HR located in

α H2 O−N2 ðv1 ; v2 ; v3 Þ ¼ 1:4613 þ ð0:038  0:006Þv1 3

þ ð0:026  0:008Þv2 þ ð0:042  0:013Þv3 Å

ð14Þ

Calculated rmsα = 1.2 × 10−2 Å3.

Table 1 Optimal polarisabilities α(v1, v2, v3) (in Å3) of H2O molecule obtained from the line shift coefficients measured for different vibrational bands perturbed by Ar, N2 and air. H2O–Ar α (0,1,0) (1,0,0) (0,2,0) (0,3,0) (0,0,1) (1,1,0) (0,1,1) (1,1,1) (0,2,1) (2,0,0) (1,0,) (0,0,2) (1,2,0) (3,0,1) (2,2,1)

– 1.50

1.503 1.52 1.52 1.56 1.54 1.53 1.535 1.543 1.55 1.615 1.61

H2O–N2 Ref. [6]

[6] [6] [6] [15] [6] [6] [6] [6] [6] [20] [20]

N 16

45 33 66 85 50 38 48 8 5 78 32

H2O-air

rmsδ

α

Ref.

N

rmsδ

α

Ref

N

rmsδ

[21]

3

2.9·10−3

1.515

[21]

3

2.5·10−3

3.1·10−3

1.51 –

1.48 1.525

[23] [23]

4 13

1.0·10−3 2.3·10−3

1.545 1.535

[23]

63 176

2.6·10−3 3.7·10−3

2.1·10−3 2.6·10−3 2.4·10−3 2.6·10−3 2.8·10−3 4.0·10−3 3.3·10−3 1.9·10−3 2.1·10−3 3.6·10−3 2.4·10−3

– – 1.545 1.57 1.545 1.545 1.545 – – – 1.625

[22] [22] [22] [22] [22]

30 35 20 12 26

2.6·10−3 2.2·10−3 2.1·10−3 1.8·10−3 2.4·10−3

[20]

21

3.3·10−3

1.545 1.54 1.545 1.54 1.535

[23] [23] [23] [23] [23] [23]

77 77 93 49 48

2.4·10−3 3.1·10−3 3.6·10−3 2.6·10−3 3.4·10−3

1.65

[20]

22

2.0·10−3

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V.I. Starikov et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 210 (2019) 275–280

Table 2 Experimental and calculated shift coefficients δ (in cm−1/atm) of H2O lines perturbed by N2, O2 and air, T = 296 Ka. Transition

O2

N2 δ(exp)

ν2 [1129] ← [1156] [1138] ← [1249] [13212] ← [14113] 2ν2–ν2 [524] ← [633] [541] ← [652] [542] ← [651]

δ(cal)

Table 3 Experimental conditions.

Air

δ(exp)

δ(cal)

δ(exp)

δ(cal)

Spectral resolution, cm−1

Partial pressure of H2O, atm

Partial pressure of buffer gas, atm

Temperature, K

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

0.0099 0.0099 0.0099 0.0099 0.0099 0.0099 0.0099 0.0099 0.0099

0 0.258 0.594 1.014 1.494 1.494 1.995 2.478 3.087

294.0 293.9 294.0 294.0 294.1 294.1 294.0 293.9 294.0

0.01 0.01 0.01 0.01

0.0168 0.0168 0.0175 0.0178

0 0.278 0.585 0.878

295.0 295.1 294.9 295.0

0.01 0.01 0.01 0.01 0.01 0.01

0.0101 0.0101 0.0173 0.0173 0.0173 0.0173

0 0.409 0.179 0.330 0.481 0.675

296.0 296.0 295.8 296.0 295.9 295.7

H2O–He [14] −0.0125 −0.0101 −0.0084 −0.0069 −0.0127 −0.0094 −0.0119 −0.0051 −0.0061 −0.0038 −0.0110 −0.0048 −0.0037 −0.0021 −0.0039 −0.0039 −0.0039 −0.0024

−0.0017 −0.0022 −0.0036 −0.0032 −0.0018 −0.0024 0.0021 0.0004 −0.0022 −0.0026 0.0022 −0.0002 0.00057 0.0001 −0.0029 −0.0026 0.0005 −0.0004

2ν2 [321] ← [330] [624] ← [515] [524] ← [616] [835] ← [826]

– – – –

−0.0040 −0.0002 0.0009 0.0039

– – – –

−0.0035 −0.0040 −0.0030 −0.0007

−0.0021 0.0012 0.0013 0.0065

−0.0039 −0.0010 0.0000 0.0039

3ν2 [101] ← [110] [221] ← [110] [101] ← [212] [212] ← [303] [303] ← [414] [414] ← [505] [404] ← [515] [616] ← [707] [606] ← [717] [625] ← [716] [716] ← [827] [818] ← [909] [827] ← [918]

– – – – – – – – – – – – –

−0.0090 0.0031 −0.0066 0.0005 −0.0074 −0.0034 −0.0077 −0.0053 −0.0066 −0.0015 −0.0069 −0.0060 −0.0028

– – – – – – – – – – – – –

−0.0057 −0.0004 −0.0070 −0.0031 −0.0077 −0.0062 −0.0082 −0.0082 −0.0087 −0.0057 −0.0077 −0.0093 −0.0071

−0.0071 0.0015 −0.0061 −0.0009 −0.0081 −0.0033 −0.0092 −0.0054 −0.0082 −0.0006 −0.0145 −0.0065 −0.0020

−0.0084 0.0024 −0.0067 −0.0001 −0.0075 −0.0041 −0.0079 −0.0059 −0.0070 −0.0024 −0.0071 −0.0067 −0.0038

2ν2 + ν3–ν2 [111] ← [110] [211] ← [212] [422] ← [423]

– – –

−0.0074 −0.0025 −0.0062

– – –

−0.0071 −0.0076 −0.0073 −0.0070 −0.0031 −0.0034 −0.0094 −0.0020 −0.0068

a Eq. (16) and dipole moments μ2(n) from Ref. [3] are used; experimental data for (0, 1, 0) ← (0, 0, 0), (0, 2, 0) ← (0, 1, 0) and for (0,2,0) ← (0, 0, 0), (0, 3, 0) ← (0, 0, 0), (0, 2, 1) ← (0, 1, 0) bands are taken from Refs. [21,23], respectively.

V.E. Zuev Institute of Atmospheric Optics SB RAS. A detailed description of the experimental setup and fitting procedure have been given previously [6,14–16]. All measurements are summarized in Table 3. For the determination of the spectral line parameters, a program was used that allows for deriving line parameters from their simultaneous fitting to several spectra recorded under different conditions [24]. The quadratic speed dependent Voigt profile was used to retrieve the spectroscopic parameters from spectra. Measured broadening and shift coefficients are presented in the 3th and 4th columns of Table 4. First the calculations of the broadening coefficients were performed for the H2O–Ar system, because for H2O–H2 and H2O–He systems the calculated coefficients δ(cal), in general, depend not only on the pair (μ, α), but on the quantity ΔC12 in Eq. (4), which determines the vibrational dependence of the short range part of isotropic potential (3). For the vibrational state (0, 4, 0), (0, 5, 0) and (0, 6, 0) the dipole moments μ1(0, v2, 0) = {1.726, 1.694, 1.663} [2] and μ2(0, v2, 0) = {1.691, 1.620, 1.510} (in D) [3]. For a given band (0, 0, 0) → (0, v2, 0) and chosen μ1 or μ2 the value of α(0, v2, 0) were selected by hand to give the best description of experimental shift coefficients. Obtained pairs (μ1, α), (μ2, α) and corresponding calculated shift coefficients are presented in the 6th and 7th columns of Table 4. It is not possible to give the preference to any pair. Fig. 2 shows the variation of H2O polarisability α in the vibrational state (0, v2, 0) obtained according to Eqs. (2) and (16) and according to performed calculations. The use of Eq. (2) for α(0, v2, 0) (line 1 in Fig. 2) and dipole moments μ1(0, v2, 0) [2] or μ2(0, v2, 0) [3] leads to positive shift coefficients for investigated lines. This is not consistent with the experimental data.

H2O–Ar [6,15]

H2O–H2 [16]

Pairs (μ1, α) and (μ2, α) obtained for the H2O–Ar system then were used in the calculation of the broadening coefficients γ and δ for the lines of the 6ν2 band perturbed by H2 and He. In our previous analysis of the H2O–H2 system [16,25], we could not find the dependence of the quantity ΔC12(n) in Eq. (4) on the bending quantum number v2. Performed calculations showed that the agreement between δ(cal) and present experimental shift coefficients δ(exp) may be found for ΔC12 (0, 6, 0) = 700.0 Å9D2. In the case of perturbing by He gas, ΔC12(0, 6, 0) = 370.0 Å9D2. Comparison of δ(cal) and δ(exp) is given in Table 4. 5. Second Derivatives of Polarisability Obtained vibrational dependence α(n) (16) was used for the estimation of the second derivatives in the expansion of the function α(q) in power series α ¼ α0 þ

X

α i qi þ

i

1X α qq þ⋯ 2 i; j ij i j

ð17Þ

whereαi, αij are the first and second derivatives for α(q), i, j = 1, 2, 3, and qi are dimensional normal coordinates of H2O molecule. The relationship between α(n) and derivatives is defined by Eq. (19) α ðv1 ; v2 ; v3 Þ ¼ α 0 þ

!  X ϕ  αk 1X 1 ikk α ii − vi þ 2 i 2 ωk k

ð18Þ

where ωk and ϕijk are the harmonic frequencies and intra-molecular anharmonic cubic force constants [26] of the H2O molecule. The first derivatives α1 = 0.146, α2 = 0.0022 (in Å3) are found in Ref. [27,28]. The second derivatives (in Å3) α11 = 0.014, α33 = 0.0128 (proposed in Ref [7]) and α22 = 0.052 transform Eq. (18) into α ðv1 ; v2 ; v3 Þ ¼ α 0 þ 0:041v1 þ 0:021v2 þ 0:043v3

ð19Þ

The numerical coefficients in this equation are close to the coefficients of optimal function α(n) (16). The second derivatives α22 = 0.052 Å3 is rather large by comparison with α11 and α33. This derivative is greater than α22 = 0.038 Å3 proposed in Ref. [7] as the optimal value. This optimal value was found in Ref. [7] in the analysis of H2O line shifts only for two systems, H2O–Ar and H2O–He.

V.I. Starikov et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 210 (2019) 275–280

279

Table 4 Experimental and calculated broadening γ (in cm−1/atm) and shift δ (in cm−1/atm) coefficients of H2O molecule perturbed by Ar, H2, and He, T = 296 Ka. ν, cm−1

Transition

γ (exp)

δ(exp)

γ (cal)

δ(cal)

δ(cal)

(μ1; α)

(μ2; α) (1.691; 1.551) −0.0103 −0.0128 (1.62;1.572) −0.0121 (1.51;1.610) −0.0110 −0.0135 −0.0123

Argon 7026.17856 7031.47330 7479.09163 8954.13020 8847.73212 8618.23518

4ν2 [651] ← [524] [945] ← [818] 5ν2 [505] ← [414] 6ν2 [616] ← [523] [616] ← [625] [616] ← [725]

0.0246(3) 0.0236(2)

−0.0095(1) −0.0137(2)

0.036 0.032

0.0390(2)

−0.0132(1)

0.033

0.0411(3) 0.0351(2) 0.0404(3)

−0.0152(1) −0.0133(1) −0.0132(1)

0.0406 0.0315 0.0368

(1.726; 1.544) −0.0103 −0.0129 (1.694;1.558) −0.01240 (1.66;1.582) −0.0111 −0.0135 −0.0123

0.0748(2) 0.0585(3) 0.0688(3)

−0.0067(1) −0.0055(1) −0.0033(1)

0.0712 0.0592 0.0659

(1.66;1.582) −0.0069 −0.0074 −0.0069

(1.51;1.61) −0.0068 −0.0073 −0.0067

0.0031(1) 0.0046(1)

0.0173 0.0149 0.0162

(1.66;1.582) 0.0036 0.0036 0.0037

(1.51;1.61) 0.0032 0.0032 0.0033

Hydrogen 8954.13020 8847.73212 8618.23518

6ν2 [616] ← [523] [616] ← [625] [616] ← [725]

Helium 8954.13020 8847.73212 8618.23518

6ν2 [616] ← [523] [616] ← [625] [616] ← [725]

0.0152(1) 0.0166(1)

a Dipole moments μ1 and μ2 (in D) are taken from Ref. [2], Eq. (1), and Ref. [3], respectively; polarizability α is given in Å3; calculated broadening coefficients γ (cal) (with the accuracy in 1%) do not depend on the pair (μ; α); for H2O–Ar system we used σ(0, 4, 0) = 3.538, σ(0, 5, 0) = 3.50, σ(0,6, 0) = 3.480 Å and other potential parameters from Ref. [6]; for H2O–H2 and H2O–He systems we used σ(0, 6, 0) = 2.80, ΔC12 = 700.0 (in Eq. (4)) and σ(0, 6, 0) = 2.60 (in Å), ΔC12 = 360.0 (in Å9 D2), respectively. Other potential parameters for these systems are taken from Ref. [6,14].

6. Discussion and Conclusion In the present paper the study of the vibrational dependence of the H2O dipole moment μ and polarisability α is performed through the analysis of molecular H2O line shifts induced by pressures of different foreign gases such as Ar, N2, air, H2, and He. Calculated shift coefficients δ strongly depend on the values of μ(n) and α(n), where (n) = (v1, v2, v3) are the vibrational states of the H2O molecule. In Refs. [2,3], two different results for μ(n) (with v2 N 3) are found (see Fig. 1). The α(n) values for (n) = (0, 0, 0), (1, 0, 0), (0, 0, 1), (0, 1, 0), and (0, 2, 0) vibrational states of H2O are obtained in the ab initio calculations of Ref. [5]. These values are used to present α(n) in form (2). The comparison of calculated values for different H2O vibrational bands (0, 0, 0) → (v1, v2, v3) and the shift coefficients for different buffer gases (Ar, N2, air) with experimental data clearly shows that Eq. (2) is not correct for the bands with v2 ≠ 0. The dipole moments μ(n) from Refs [2,3]. (where v2 ≤ 3) coincide with and lead to the same calculated shift coefficient. The vibrational dependence of polarisability α (16)

Fig. 2. Variation of H2O polarisability α in the vibrational state (0, v2, 0) obtained according to Eq. (2) (line 1), Eq. (16) (line 2, present calculations of shift coefficients for the bands with v2 ≤ 3) and according to calculations of shift coefficients for 4ν2, 5ν2 and 6ν2 vibrational bands perturbed by Ar (lines 3, 4). The lines 3 and 4 were obtained with dipole moments μ1 [2] and μ2 [3], respectively.

obtained in the present paper is optimal for calculations of shift coefficients of H2O transitions for the bands with v1 ≤ 3, v2 ≤ 3, and v3 ≤ 2. For these vibrational quantum numbers, this formula is tested in the case of the perturbation of H2O transitions by Ar, N2, and air. The problem of a selection of the functions μ(n) andα(n) which are optimal for the shift calculations is more complex when v2 N 3. In Ref. [3], the irregular behaviour of the dipole moments with bending excitation (Fig. 1) is explained by the large anharmonicity in the bending potential of the water molecule. The change in behaviour of the dipole moment at v2 = 7 in Fig. 1 originates in the fact that at v2 = 7, the bending vibrational energy becomes larger than the barrier in linearity in the potential energy function. Such μ(0, v2, 0) behaviour is physically justified. From the formal point of view, both functions μ1(n) [2] and μ2(n) [3] may be used in the calculations of shift coefficients. But for different selected functions μ(n), the different values of α(n) must be taken to obtain appropriate results for calculated line shifts.

Fig. 3. Experimental (dark symbols) and calculated (empty symbols) shift coefficient for the lines of 4ν2 (lines with N = 1,2), 5ν2 (N = 3) and 6ν2 bands (N = 5,6,7) of H2O molecule perturbed by He, H2 and Ar pressure. The lines are numbered according to Table 4.

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This is confirmed by the calculations of the coefficients for 4v2, 5ν2, and 6ν2 vibrational bands of H2O perturbed by Ar, H2, and He. For each band, we selected pairs (μ1,α) and (μ2,α), which give an accurate description of the measured shift coefficients. These pairs are presented in Table 4. Fig. 3 shows the comparison of experimental and calculated data. In the calculations pairs (μ1,α) were used. In the case of the shifts of H2O lines by H2 and He pressures, the vibrational dependence of the short range part was taken into account. We believe that because the dipole moments μ2(n) [3] are physically justified, then the corresponding α(n) are physically justified as well. For example, the values α(0, v2, 0) which correspond to μ2(0, v2, 0) [3] are presented by line 4 in Fig. 2. Acknowledgments The authors acknowledge support from the Russian Foundation for Basic Research (RFBR, grants 17-52-16022). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.saa.2018.11.032. References [1] R.P. Leavitt, Pressure broadening and shifting in microwave and infrared spectra of molecules of arbitrary symmetry: an irreducible tensor approach, J. Chem. Phys. 73 (1980) 5432–5450, https://doi.org/10.1063/1.440088. [2] S.L. Shostak, J.S. Muenter, The dipole moment of water. II. Analysis of the vibrational dependence of the dipole moment in terms of a dipole moment function, J. Chem. Phys. 94 (1991) 5883–5890, https://doi.org/10.1063/1.460472. [3] M. Mengel, P. Jensen, A theoretical study of the Stark effect in triatomic molecules: application to H2O, J. Mol. Spectrosc. 169 (1995) 73–91, https://doi.org/10.1006/ jmsp.1995.1007. [4] V.I. Starikov, S.N. Mikhailenko, Effective dipole-moment operator for nonrigid H2Xtype molecules. Application to H2O, J. Mol. Struct. 271 (1992) 119–131, https://doi. org/10.1016/0022-2860(92)80215-4. [5] Y. Luo, H. Agren, O. Vahtras, P. Jorgensen, V. Spirko, H. Hettema, Frequencydependent polarizabilities and first hyperpolarizabilities of H2O, J. Chem. Phys. 98 (1993) 7159–7164, https://doi.org/10.1063/1.464733. [6] T.M. Petrova, A.M. Solodov, A.A. Solodov, V.M. Deichuli, V.I. Starikov, Measurements and calculations of Ar-broadening parameters of water vapour transitions in a wide spectral region, Mol. Phys. 115 (2017) 1642–1656, https://doi.org/10.1080/ 00268976.2017.1311422. [7] V.I. Starikov, T.M. Petrova, A.M. Solodov, A.A. Solodov, V.M. Deichuli, Eur. Phys. J. D 71 (2017), 108. https://doi.org/10.1140/epjd/e2017-70685-9. [8] C.J. Tsao, B. Curnutte, Line-widths of pressure-broadened spectral lines, J. Quant. Spectrosc. Radiat. Transf. 2 (1962) 41–91, https://doi.org/10.1016/0022-4073(62) 90013-4. [9] D. Robert, J. Bonamy, Short range force effects in semiclassical molecular line broadening calculations, J. Physiol. Paris 40 (1979) 923–943, https://doi.org/10.1051/ jphys:019790040010092300.

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