Computational study on Kerr constants of neutral and ionized gases M. Sato, A. Kumada, and K. Hidaka Citation: Appl. Phys. Lett. 107, 084102 (2015); View online: https://doi.org/10.1063/1.4929455 View Table of Contents: http://aip.scitation.org/toc/apl/107/8 Published by the American Institute of Physics APPLIED PHYSICS LETTERS 107, 084102 (2015) Computational study on Kerr constants of neutral and ionized gases M. Sato,a) A. Kumada, and K. Hidaka Department of Electrical Engineering and Information Systems, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo 113-8656, Japan (Received 1 June 2015; accepted 12 August 2015; published online 25 August 2015) In order to quantitatively examine the measurement capability of Poisson’s field using electro-optic Kerr-effect (EOKE), Kerr constants of neutral molecules and ions are examined by means of first principle calculations. We have systematically computed Kerr constants of neutral molecules and ions of several molecular symmetry groups, with consistent theory level and basis sets. Computed Kerr constants of neutral molecules (N2, CO2, SF6, and CF3I) ranging across two orders of magnitudes are within 50% error of the experimental values, which are comparable to the scattering between experimental values itself. The results show that SF6 has smaller Kerr constant due to its high molecular symmetry compared to those of N2 and CO2. In contrast, CF3I has large Kerr constant due to its permanent dipole. Computed Kerr constants for anions are larger by two orders of magnitude than those of neutral molecules, probably due to the shielding effect. For cations, the opposite holds true; however, due to anisotropic polarizability, computed Kerr constants for some cations are comparable to neutral molecules, while others show smaller values. The ratio of Kerr constants of ions to those of neutral molecules are at most 102; EOKE is valid for measuring electric field in C 2015 AIP Publishing LLC. weakly ionized gas whose ionization degree is smaller than 103. V [http://dx.doi.org/10.1063/1.4929455] One of the most fundamental parameters required to understand discharges in gas is the electric field profile during discharges (Poisson’s field), which is different from the static electric field (Laplace field) due to the existence of electrical charges. Therefore, measurement of Poisson’s field has been desired. It is well known that electric field in liquid has been measured by electro-optic Kerr effect (EOKE).1 However, it was assumed that EOKE cannot be applied to measure electric field in gas, since the number density of gas molecules is smaller by three orders of magnitude than that of liquid. Despite the fact, recently, we have developed a high voltage measuring apparatus based on EOKE in gas and measured the Laplace field.2,3 Since electric field measurement by EOKE has multiple advantages such as high temporal and spatial resolution, and low disturbance by measurements, it is desirable to apply this technique to measure the Poisson’s field. In this research, in order to quantitatively examine the measurement capability of Poisson’s field using EOKE, Kerr constants of molecules and ions were computed by means of quantum chemical calculation. P Under the external electric field a;n Ea ðxn Þ ðV=mÞ whose angular frequency is xn and the direction is a, the dipole moment of a molecule can be written as4 l0a dðxr ¼ 0Þ þ aab ðxr ; x1 ÞEb ðx1 Þ X 1 b ðxr ; x1 ; x2 ÞEb ðx1 ÞEc ðx2 Þ þ 2! x1 ;x2 abc 1 X þ c ðxr ; x1 ; x2 ; x3 Þ 3! x1 ;x2 ;x3 abcd l a ðx r Þ ¼ E b ðx 1 ÞE c ðx 2 ÞE d ðx 3 Þ þ ; a) Electronic mail: sato@hvg.t.u-tokyo.ac.jp 0003-6951/2015/107(8)/084102/5/$30.00 (1) where l0 (Cm), aab (Cm2/V or C2m2/J), babc (Cm3/V2 or C3m3/J2), and cabcd (Cm4/V3 or C4m4/J3) are permanent dipole, polarizability, first hyperpolarizability and second hyperpolarizability tensors, respectively, and xr is the angular frequency of polarization. It should be noted that there are different definitions for polarizability and hyperpolarizabilities5 such as 1~ 1 0 la ¼ la þ e0 ~a ab Eb þ b abc Eb Ec þ ~c abcd Eb Ec Ed þ ; 2! 3! (2) or ^ Eb Ec þ ^c Eb Ec Ed þ Þ; (3) la ¼ l0a þ e0 ð^a ab Eb þ b abc abcd where e0 (C/(Vm)) is the vacuum permittivity. The angular frequencies are omitted for simplicity. The relations among different definitions are as follows: aab ¼ e0 ~a ab ¼ e0 ^a ab ; ~ ¼ 2! e0 b ^ ; babc ¼ e0 b abc abc (4) cabcd ¼ e0~c abcd ¼ 3! e0^c abcd : Relations between different conventions for hyperpolarizabilities are discussed in detail in Refs. 6 and 7. We will adopt the convention defined in Eq. (1), which is often used to describe polarizabilities and hyperpolarizabilities of molecules.4,8,9 For DC-Kerr effect, the terms that one should consider is permanent dipole, frequency dependent polarizability aeab aab ðxe ; xe Þ, static polarizability aab aab ð0; 0Þ, and frequency dependent hyperpolarizabilities, babc ðxe ; xe ; 0Þ and dabcd ðxe ; xe ; 0; 0Þ, where xe denotes the angular frequency of the laser. Making use of quantum chemical calculation, these values can be computed either by 107, 084102-1 C 2015 AIP Publishing LLC V 084102-2 Sato, Kumada, and Hidaka Appl. Phys. Lett. 107, 084102 (2015) perturbative approach10 or by Finite Field method11 at Hartree-Fock (H-F) level, second order Møller-Plesset (MP2) level12 and density functional method (DFT). It is known that DFT methods can reproduce second hyperpolarizability computed with accurate electron correlation techniques for various small molecules with much less computational cost.9,13 The theoretical expression of molar Kerr constant in terms of the above mentioned values is derived by Buckingham in his seminal work.4 The expression in SI units can be found in Ref. 14. Molar Kerr constant mK (m5/(V2 mol)) is defined as mK ðn2 6nðnk n? ÞM ; 2 þ 2Þ ðer þ 2Þ2 dE2 (5) where n and er are the isotropic refractive index and relative permittivity of the gas, respectively, (nk n?) is the refractive index difference for light polarized parallel and perpendicular to the electric field, and d (kg/mol) and M (kg/m2 are density and molar weight of the gas, respectively.15 Note that n and er goes to unity in dilute gas. On the other hand, Kerr constant B (m/V2), which is used in measurement of electric field in liquid1 and gas2,3 is defined as B ðnk n? Þ ; k0 E2 (6) where k0 (m) is the vacuum wavelength of laser. Thus, if B is given, the electric field can be evaluated by measuring the phase shift between the light field polarized parallel and perpendicular to the applied electric field. Taking into consideration Eqs. (5) and (6), for axisymmetric molecules, the expression of molar Kerr constant derived by Buckingham can be simplified as " 2 # N 2l0 b aeaniso aaniso aeaniso l0 þ B¼ þ cþ 2 5kT 6e0 k0 3kT 5ðkT Þ N ½K1 þ K2 þ K3 þ K4 ; 6e0 k0 (7) where N (1/m3), k (J/K), and T (K) are the number density of gas molecules, Boltzmann constant, and temperature, respectively, and1 the following relations hold: aaniso ½ð3aab aab aaa abb Þ=22 ; b ðbk b? Þ=3; bk ðbzaa þ baza þ baaz Þ=5; b? ðbzaa 3baza þ 2baaz Þ=5; c 3ðck c? Þ=2, ck ðcaabb þcabba þ cabab Þ=15 and c? ð2cabba caabb Þ=15, where Einstein summation convention is applied. The z axis corresponds to the axis of molecular symmetry. Since the expression of molar Kerr constant is derived by classical statistical mechanics, averaging over all orientation of a molecule with a Boltzmann-type weighting factor, Kerr constants depend on temperature, except for spherically symmetric molecules. Prior to calculating Kerr constants of neutral molecules and ions, we have estimated the charged species produced by discharges. Rate coefficients were obtained with Bolsigþ.16 The electron collision cross section database in Refs. 17–21, 22, 23, and 24 were used for N2, CO2, SF6, and CF3I, respectively. The rate coefficients for inelastic collisions of N2 and SF6 are shown in Fig. 1. It is seen in Fig. 1 that there are excitation and ionization for N2, whereas electron attachment does not appear in the figure. Thus, the majority of charged molecules produced in N2 will be Nþ 2 . Likewise, we decided þ to consider COþ 2 for CO2, SF5 ; SF6 ; SF5 , and F for SF6 þ and CF3I and I for CF3I as primary ions. Though most types of collisions lead to excitation, de-excitation rate is usually sufficient especially in atmospheric pressure.25 Thus, effects of electronically excited species are neglected. In addition to reactions between electrons and molecules, there are molecule-molecule, radical-molecule, ion-molecule reactions, and so on.26 Although these reactions may be essential for discharges in gas mixtures,27 most ions produced by secondary reactions are neglected and ruled out from our computation of Kerr constants. Geometry of ions and neutral molecules was optimized with Gaussian 09 package28 at ground state electron configuration. For open-shell systems, unrestricted methods were used. Becke 3-Parameter, Lee, Yang, and Parr (B3LYP) DFT level and MP2 ¼ FULL level of theory was selected. In general, it is desirable to use large basis set to calculate high order polarizabilities. For a prominent example, t-aug-ccpV5Z basis are used to calculate the fourth-order hyperpolarizability of Argon.8 However, due to limitation of FIG. 1. Rate constants of inelastic collisions between electrons and N2 and SF6. 084102-3 Sato, Kumada, and Hidaka Appl. Phys. Lett. 107, 084102 (2015) computational resources, doubly augmented correlation on all atoms (daug-cc-pVTZ)29 was chosen as a basis set. In order to enable comparison of the computed values, diffuse functions were added even when computing neutral molecules and cations. Preliminary calculation showed that basis set errors of N20 s second-order polarizability was less than 2% when daug-cc-pVTZ was used. Since iodine has large atomic number, so as to reduce the computational cost and to account for relativistic effect, we used aug-cc-pVTZ basis with small-core relativistic Pseudo-Potential (aug-cc-pVTZPP),30 as an exception. Computed bond lengths and angles of each molecule were in good agreement with experimental values and Couple Cluster calculations;31–36 most errors were within 0.02 Å for bond length and 1 for angles, respectively. Static and frequency dependent polarizability, frequency dependent first and second hyperpolarizability of neutral molecules and ions were calculated with Gaussian 09 package at the same level or theory as geometry optimization. The optimized geometry obtained at B3LYP level was chosen for the input of atomic coordination. Since all molecules have Cn axis of molecular symmetry, where n is larger than 2, Kerr constants were calculated by Eq. (7). The computed values are shown in Table I. As shown in Table I, static polarizabilities of neutral molecules, anions, and cations calculated at B3LYP level are in reasonable agreement with those calculated at MP2 ¼ FULL level and other post H-F methods, except for Nþ 2 ; for most molecules and ions, the difference between them are within 5%. For neutral molecules whose experimental static polarizabilities are available, excellent TABLE I. Computed static and frequency dependent polarizability, hyperpolarizabilities, and Kerr constants of molecules. Kerr constants were calculated at T ¼ 300 K, k0 ¼ 632.8 nm: wavelength of He-Ne laser. The single digit number “0” is determined from the molecule’s point group. Molecule Point group 1040aisoa (Cm2/V) 1040 aeiso a (Cm2/V) 1080 aaniso aeaniso (C2m4/V2) 1050b (Cm3/V2) 1061c( ¼ K1) (Cm4/V3) 1061K2 (Cm4/V3) 1061K3 (Cm4/V3) 1061K4 (Cm4/V3) 1019B (m/V2) N2 (calc.) N2 (expt.) D1h D1h 1.97, 1.93b 1.94e 2.00, 1.94c 1.97e 0.604 0.599e 0.00 0 0.00 0 2.92 2.90 0.00 0 Nþ 2 (calc.) Nþ 2 (expt.) CO2 (calc.) CO2 (expt.) COþ 2 (calc.) SF6 (calc.) SF6 (expt.) D1h D1h D1h D1h D1h Oh Oh 2.39, 0.00b 2.65 6 0.06i 2.86, 2.93b 2.93e, 2.89j 2.46, 2.17b 5.27, 5.02b 5.06l 1.97 1.55 0.00 0.756, 0.686c, 0.701d 0.9e, 0.792f1 0.642 6 0.0075 f2 3.00, 6.71h1, 7.94h2 0.00 7.48 0.00 2.67 2.55e 5.2g 3.27 2.90, 2.91c 3.26e 2.58 5.32 4.66 6.15e 7.35 0.00 0 0.00 0 0.00 0.00 0 0.00 0 0.00 0.00 0 22.5 29.7e 3.55 0.00 0 0.00 0 0.00 0.00 0 17.1 23.8e 2.6 0.928 1.2m SFþ 5 (calc.) SF 6 (calc.) SF 5 (calc.) F (calc.) CF3I (calc.) CF3I (expt.)q CF3Iþ (calc.) I (calc.) D3h Oh C4v 4.19, 4.04o 8.61, 8.25o 7.42, 7.06o 2.99, 2.78p 8.63, 8.23o 7.73r, 9.89s 7.56, 7.06o 10.5, 10.9p 4.23 9.08 7.66 3.47 8.89 0.00 0.00 5.14 0.00 8.45 0.00 0.00 0.04 0.00 0.981 0.992, 0.748c 5.6e, 1.16k, 0.831f2 0.110 1.28 1.49 6 0.1m 0.985 6 0.007, 0.993 6 0.009n 0.523 105 20.1 67.5 10.8 0.00 0.00 33.1 0.00 47.3 0.345 0.00 24.8 0.00 40.8 0.00 0.00 6.75 0.00 106 7.78 11.5 20.6 0.00 0.595 0.00 2.92 45.1 29.7 0.00 99.5 0.00 166 0.00 0.632 76.5 37.8 49.1 149 189t 217 32.8 a C3v C3v C3v aiso ¼ aaa/3 with Einstein summation convention. 31 þ National Institute of Standards and Technology (N2: MP2/aug-cc-pVDZ, Nþ 2 :B2PLYP/ aug-cc-pVTZ, CO2: MP2/aug-cc-pVDZ, CO2 : B2PLYP /6-31G). 40 2 For Nþ , static polarizability was computed at HF/daug-cc-pVTZ and MP2/daug-cc-pVTZ level additionally; HF/daug-cc-pVTZ: 0.645 10 (Cm /V) and 2 MP2/daugcc-pVTZ: 3.24 1040. c Sekino and Bartlett.37 (CCSD(T) with polarization consistent basis set recommended by Sadlej, 644.3 nm, c: EOKE. The relations between atomic units and SI units are 1 a.u. ¼ 1.649 1041 (Cm2/V) and 1 a.u. ¼ 6.235 1065 (Cm4/V3) for a and c, respectively.) d Gubler and Bosshard.5 (Value in the table is converted through Eq. (4). k0 ¼ 1064 nm, c(3x; x, x, x): third harmonic generation (THG).) e Buckingham et al.15 (N2: 299 K, CO2: 301 K, k0 ¼ 632.8 nm). f Ward and Miller38 (f1: THG, f2: c(2x; x, x, 0): dc-induced second harmonic generation (dc-SHG), k0 ¼ 644.3 nm). g Breazeale39 (T ¼ 293 K, k0 ¼ 650 nm). h h1: This work, czzzz: 6.71 (k0 ¼ 632.8 nm), h2: Tarazkar et al.13 (EOKE, czzzz, MCSCF/t-aug-cc-pV5Z, k0 ¼ 1101 nm). i McCormack et al.40 (Inferred value: see Ref. 40 for details.) j Spackman.41 (These values from analysis of several different experiments.) k Ward and Elliott.42 (THG, see Ref. 37 for conversion between conventions.) l Watson and Ramaswamy.43 See also Ref. 44. (1 (Å3) ¼ 1.113 1040 (Cm2/V)). m Buckingham and Dunmur45 (T ¼ 300 K, k0 ¼ 632.8 nm, 1 esu (statvolt2 cm5) ¼ 1.238 1025 (Cm4/V3)). n Shelton and Mizrahi.46 (dc-SHG, k0 ¼ 620 and 650 nm, see Ref. 46 for details.) o MP2 ¼ full/daug-cc-pVTZ, I: aug-cc-pVTZ-PP (B3LYP/daug-cc-pVTZ, I: aug-cc-pVTZ-PP). p Bichoutskaia and Pyper (MP2, see Ref. 47 for basis sets). q Dipole moment of CF3I is obtained by Stark effect36 (l: 3.50 1030 (Cm)) and heterodyne beat apparatus48 (l: 3.07 1030), see also Ref. 49 for experimental details. This work: 3.03 1030. (1 a.u. ¼ 8.48 1030 (Cm), 1 D ¼ 1018 esu cm ¼ 3.336 1030 (Cm).) r Giacomo and Smith.48 (Obtained from estimated electronic polarization. See Ref. 48 for details.) s Marienfeld et al.50 (Estimated by interpolation between different molecules. See Ref. 50 for details.) t Kamiya et al.2 (k0 ¼ 632.8 nm). b 084102-4 Sato, Kumada, and Hidaka agreement is seen between experimental and computed values. In addition, frequency dependent polarizabilities of N2 and CO2 show good agreement with experimental values. Although polarizability of CF3I is not measured directly, computed dipole moment of CF3I shows good agreement with experimental values, as shown in the footnote (q) in Table I. In order to account for the discrepancy between static polarizabilities of Nþ 2 computed at different theory levels, we have additionally computed the static polarizability of Nþ 2 at H-F level and MP2 level for comparison. As denoted in the footnote (b) of Table I, MP2 value shows large deviation from other values; moreover, the sign is incorrect. These results show that electron correlation has a great effect on polarizability of Nþ 2 as discussed in Ref. 13. This is a wellknown fact for open-shell systems. In addition, it is known that MP2 level of theory works well for computing polarizability and hyperpolarizabilities of closed-shell system; however, it often fails to compute those of open-shell system.13,51 In contrast to MP2 values, B3LYP/daug-ccpVTZ level shows reasonable agreement with experimental values. Unlike Nþ 2 , it is easier to account for electron correla tion for SFþ ; SF 5 ; F , and I , whose ground state is a sin5 glet state. This is consistent with the fact that post H-F results and B3LYP results agrees well for these ions. Computed values of frequency dependent second hyperpolarizabilities (hereinafter c) of N2 and SF6 are in good agreement with experimental values and other computational values, and the errors are within 20%. It is notable that c of CO2 obtained by Buckingham is larger compared with other values. This is presumably due to the drawbacks of the determination of second hyperpolarizability by EOKE. Since molar Kerr constant contains all the contribution from l, a, b, and c, determination of hyperpolarizabilities by EOKE requires fitting experimental molar Kerr constants to a function of reciprocal of the absolute temperature, in order to separate each contribution. However, the contribution of c is usually small, and thus, the accuracy is greatly reduced.14 This is consistent with the fact that K1 of CO2 is an order of magnitude smaller than K3, i.e., the contribution of c to molar Kerr constant is small, while K1 and K3 of N2 are comparable. Regarding SF6, since c is the sole contributor to molar constant, EOKE measurement should be valid. Although there are few studies on c of ions, especially for 13 polyatomic ions, c of Nþ 2 is studied in detail by Tarazkar. þ0 Counterintuitive negative value of N2 s c is discussed in detail in Ref. 13. Computed Kerr constants of neutral molecules are in good agreement with experimental results. CF3I has the largest Kerr constants followed by CO2, N2, and SF6. Large Kerr constant of CF3I is due to its permanent dipole and small Kerr constant of SF6 is a result of its high molecular symmetry; K2, K3, and K4 goes to zero. The term K3, which is due to anisotropy of the polarizability, is dominant for CO2 and N2. This is probably the result of their anisotropic molecular geometry. It can be seen that I shows large polarizabilities. This is a well-known fact. Iodine has large shielding effect due to its large atomic radius. Shielding effect is enhanced when iodine is negatively charged, resulting in large polarizability. Appl. Phys. Lett. 107, 084102 (2015) Likewise, polarizabilities of anions tend to be larger than neutral molecules. Thus, anions have larger Kerr constant than neutral molecules. For cations, the opposite holds true; it is known that polarizabilities of atomic cations are smaller by more than an order of magnitude than those of neutral atoms.52 However, this does not always hold true for molecular cations. In addition, values such as anisotropic polarizabilities or hyperpolarizabilities are not necessarily smaller compared to those of neutral molecules. As a result, the calculated values of Kerr constants of cations tend to be smaller or comparable to those of neutral molecules. These results show that at least with respect to N2, CO2, SF6, and CF3I, Kerr constants of anions and cations are at most 102 times larger than those of neutral molecules. When birefringence induced by ions is negligible compared to that induced by neutral molecules, EOKE measurement using Kerr constants of neutral molecules is valid. Taking into account that Kerr constants are proportional to number density of molecules for dilute gas, EOKE measurement of Poisson’s field seems to be valid for measuring electric field in weakly ionized gas whose ionization degree is smaller than 103. 1 R. E. Hebner, R. A. Malewski, and E. C. Cassidy, Proc. IEEE 65, 1524–1548 (1977). 2 T. Kamiya, S. Matsuoka, A. Kumada, and K. Hidaka, IEEE Trans. Dielectr. Electr. Insul. 22, 760–765 (2015). 3 A. Kumada, A. Iwata, K. Ozaki, M. Chiba, and K. Hidaka, J. Appl. Phys. 92, 2875 (2002). 4 A. D. Buckingham and J. A. Pople, Proc. Phys. Soc. A 68, 905 (1955). 5 U. Gubler and C. Bosshard, Phys. Rev. B 61, 10702 (2000). 6 A. Willetts, J. E. Rice, D. M. Burland, and D. P. Shelton, J. Chem. Phys. 97, 7590 (1992). 7 H. Reis, J. Chem. Phys. 125, 014506 (2006). 8 M. Tarazkar, D. A. Romanov, and R. J. Levis, J. Chem. Phys. 140, 214316 (2014). 9 P. Calaminici, K. Jug, and A. M. K€ oster, J. Chem. Phys. 109, 7756 (1998). 10 D. Bishop, J. Chem. Phys. 100, 6535 (1994). 11 H. A. Kurtz, J. J. P. Stewart, and K. M. Dieter, J. Comput. Chem. 11, 82–87 (1990). 12 J. E. Rice and N. C. Handy, J. Quantum Chem. 43, 91–118 (1992). 13 M. Tarazkar, D. A. Romanov, and R. J. Levis, J. Phys. B: At., Mol. Opt. Phys. 48, 094019 (2015). 14 D. P. Shelton and J. E. Rice, Chem. Rev. 94, 3–29 (1994). 15 A. D. Buckingham, M. P. Bogaard, D. A. Dunmur, C. P. Hobbs, and B. J. Orr, Trans. Faraday Soc. 66, 1548 (1970). 16 G. J. M. Hagelaar and L. C. Pitchford, Plasma Sources Sci. Technol. 14, 722–733 (2005). 17 See http://jila.colorado.edu/~avp/collision_data/ for collision data base for N2. 18 Y. Itikawa, J. Phys. Chem. Ref. Data 35, 31–53 (2006). 19 P. C. Cosby, J. Chem. Phys. 98, 9544 (1993). 20 L. J. Kieffer, Information Center Report 13, JILA, University of Colorado, Boulder (1973). 21 I. Shimamura, JILA Report No. 21092 (1989). 22 A. V. Phelps and R. J. Van Brunt, J. Appl. Phys. 64, 4269 (1988). 23 L. G. Christophorou and J. K. Olthoff, J. Phys. Chem. Ref. Data 29, 267–330 (2000). 24 S. Kawaguchi, K. Satoh, and H. Itoh, Eur. Phys. J. D 68, 100 (2014). 25 F. Valk, M. Aints, P. Paris, T. Plank, J. Maksimov, and A. Tamm, J. Phys. D: Appl. Phys. 43, 385202 (2010). 26 J. S. Chang, Plasma Sources Sci. Technol. 17, 045004 (2008). 27 A. Bekstein, M. Yousfi, M. Benhenni, O. Ducasse, and O. Eichwald, J. Appl. Phys. 107, 103308 (2010). 28 M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson et al., Gaussian, Inc., Wallingford, CT (2010). 29 T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989). 30 K. A. Peterson, D. Figgen, E. Goll, H. Stroll, and M. Dolg, J. Chem. Phys. 119, 11113 (2003). 084102-5 Sato, Kumada, and Hidaka Appl. Phys. Lett. 107, 084102 (2015) 31 42 32 43 See http://nist.gov/ for National Institute of Standard Technology. L. E. Sutton and D. Phil, Tables of Interatomic Distances and Configurations in Molecules and Ions (The Chemical Society, London, 1965), Special Publication No. 18, pp. 7s–21s. 33 K. K. Irikura, J. Chem. Phys. 102, 5357 (1995). 34 G. L. Gutsev and R. J. Bartlett, Mol. Phys. 94, 121–125 (1998). 35 T. Ziegler and G. L. Gutsev, J. Chem. Phys. 96, 7623 (1992). 36 A. P. Cox, G. Duxbury, J. A. Hardy, and Y. Kawashima, J.C.S. Faraday II 76, 339–350 (1980). 37 H. Sekino and R. Bartlett, J. Chem. Phys. 98, 3022 (1993). 38 J. F. Ward and C. K. Miller, Phys. Rev. A 19, 826 (1979). 39 W. M. Breazeale, Phys. Rev. 48, 237–240 (1935). 40 E. F. McCormack, S. T. Pratt, J. L. Dehmer, and P. M. Dehmer, Phys. Rev. A 44, 3007–3015 (1991). 41 M. A. Spackman, J. Phys. Chem. 93, 7594 (1989). J. F. Ward and D. S. Elliott, J. Chem. Phys. 69, 5438 (1978). H. E. Watson and K. L. Ramaswamy, Proc. R. Soc. A 156, 144 (1936). S. M. El-Sheikh, N. Meinander, and G. C. Tabisz, Chem. Phys. Lett. 118, 151 (1985). 45 A. D. Buckingham and D. A. Dunmur, Trans. Faraday Soc. 64, 1776 (1968). 46 D. P. Shelton and V. Mizrahi, Chem. Phys. Lett. 120, 318 (1985). 47 E. Bichouskaia and N. C. Pyper, J. Phys. Chem. C 111, 9548 (2007). 48 A. D. Giacomo and C. P. Smyth, J. Am. Chem. Soc. 77, 774 (1955). 49 K. B. McAlpine and C. P. Smyth, J. Am. Chem. Soc. 55, 453 (1933). 50 S. Marienfeld, I. I. Fabrikant, M. Braun, M-W. Ruf, and H. Hotop, J. Phys. B: At., Mol. Opt. Phys. 39, 105 (2006). 51 M. Wergifosse, F. Wautelet, B. Champagne, R. Kishi, K. Fukuda, H. Matsui, and M. Nakano, J. Phys. Chem. A 117, 4709 (2013). 52 J. Mitroy, M. S. Safronova, and C. W. Clark, J. Phys. B: At., Mol. Opt. Phys. 43, 202001 (2010). 44

1/--страниц