Vibrational–rotational spectra of GaF and global multi-isotopologue analysis

Journal of Molecular Spectroscopy 325 (2016) 20–28

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Vibrational–rotational spectra of GaF and global multi-isotopologue analysis Hiromichi Uehara ⇑, Koui Horiai, Shunsuke Katsuie Department of Chemistry, Faculty of Science, Josai University, Keyakidai 1
1, Sakado, Saitama 350-0295, Japan

a r t i c l e

i n f o

Article history: Received 21 March 2016 In revised form 19 May 2016 Accepted 19 May 2016 Available online 24 May 2016 Keywords: GaF Vibrational–rotational Multi-isotopologue analysis Dunham coefficients Breakdown Born–Oppenheimer approximation

a b s t r a c t In total, 521 vibrational–rotational spectral lines of the Dv = 1 transitions of 69GaF and 71GaF up to bands v = 5–4 and 4–3, respectively, were recorded in emission with a Fourier-transform spectrometer at unapodized resolution 0.010 cm
1 in range 625–660 cm
1. The response of a HgCdTe detector enforced the lower limit, 625 cm
1. To calibrate accurately the spectral lines, the absorption spectrum of CO2 was simultaneously recorded, using dual sample cells, to serve as wavenumber standards. A set of 782 spectral lines comprising all present vibrational–rotational spectra of 69GaF and 71GaF, the reported laser-diode measurements of the Dv = 1 band sequence and the reported rotational spectra was subjected to a global multi-isotopologue analysis through fitting with 11 isotopically invariant, irreducible molecular parameters in a single set. Normalized standard deviation 1.093 indicates a satisfactory fit. For the effects of the breakdown of the Born–Oppenheimer approximation on GaF, the values of non-Born– Oppenheimer parameters DBGa, DxGa and r1qGa(=r1qF) are experimentally determined for the first time. To facilitate the calculations or predictions of spectral frequencies, the values of the Dunham coefficients of 24 Yij and 81 band parameters for both 69GaF and 71GaF were back-calculated with uncertainties using the 11 evaluated molecular parameters. To date, various types of effective Be, re, xe, and k have been reported for GaF. Because, in the present work, Dunham coefficients Yij are algebraically expressed with the genuine Be, xe, ai (i = 1, . . .) and the non-Born
Oppenheimer correction parameters, the exact expressions for the physical significance of effective quantities are derivable. The various effective quantities of Be, re, xe and k calculated with these expressions for the physical significance and the determined values of the fitted parameters of GaF agree satisfactorily with the reported values. The physical significance of Ga the conventional treatments of adiabatic and nonadiabatic corrections for DGa 01 and D10 is discussed. Ó 2016 Elsevier Inc. All rights reserved.

1. Introduction The first high-resolution spectra for the ground electronic state of GaF, a monohalide of an element in group 13, were the microwave measurements of 69GaF and 71GaF that Hoeft et al. [1] reported in 1970; they observed the hyperfine structure of rotational transitions J = 1–0 (v = 0, 1, 2, 3 for both 69GaF and 71GaF) and J = 2–1 (v = 0, 1, 2 for 69GaF and v = 0 and 1 for 71GaF) and measured the Stark effect, which revealed the electric dipole moment of 69GaF at level v = 0. Honerjäger and Tischer [2] obtained the rotational g factor of 69GaF on measuring the Zeeman effect for transition J = 1–0 (v = 0) of 69GaF. Tiemann et al. [3] reported correction parameter DGa 01 for the Born–Oppenheimer approximation for GaF. Although they evaluated this value after measuring the pure rotational transitions, ⇑ Corresponding author. E-mail address: [email protected] (H. Uehara). http://dx.doi.org/10.1016/j.jms.2016.05.005 0022-2852/Ó 2016 Elsevier Inc. All rights reserved.

neither the measured line frequencies nor the details of the analysis were reported. Hoeft and Nair [4] reported rotational transitions J = 5–4 and 13–12 (v = 0, 1, 2 for 69GaF and 71GaF) and J = 12–11 (v = 0 for 71GaF); they reported no observable breakdown of the Born–Oppenheimer approximation within the limits of their experimental errors. Wasylishen et al. [5] measured the hyperfine components of rotational transitions J = 1–0 (v = 0, 1, 2 for 69GaF and v = 0, 1 for 71GaF) with improved accuracy; they obtained improved nuclear quadrupolar coupling coefficients and rotational parameters with the fluorine spin–rotation parameter. Using an FTIR spectrometer of a moderate resolution, Uehara et al. [6] measured the vibrational–rotational transitions of 69GaF and 71GaF in emission; they recorded vibrational–rotational bands v = 1–0 to 5–4 for 69GaF and to 4–3 band for 71GaF. Ogilvie et al. [7] reported measurements of the vibrational–rotational spectra of 69 GaF and 71GaF in absorption with a laser-diode spectrometer; in total, 229 spectral lines for transitions v = 1–0 to 8–7 for both 69 GaF and 71GaF were recorded.

21

H. Uehara et al. / Journal of Molecular Spectroscopy 325 (2016) 20–28

In this paper we report the first observation of the highresolution FTIR spectra of 69GaF and 71GaF. Bands v = 1–0 to 5–4 of 69GaF and v = 1–0 to 4–3 for 71GaF were observed in emission. Due to the cutoff of the HgCdTe detector at 625 cm
1, only the Rbranch sides of the bands were measured. A set of spectral data comprising the present vibrational–rotational and reported vibrational–rotational and rotational transitions of 69GaF and 71GaF was subjected to a multi-isotopologue analysis using a nonBorn
Oppenheimer Hamiltonian [8–11] with 11 fitting parameters of Ux (1 + meDxF/MF), UB (1 + meDBF/MF), ai (i = 1, 2, . . ., 6), DxGa, DBGa, and r1qGa(=r1qF) that are based on the traditional concept of molecular parameters; note that these molecular parameters for GaF are irreducible because 19F is a single stable nuclide. NonBorn
Oppenheimer parameters DxGa, DBGa and r1qGa(=r1qF), for the effects of the breakdown of the Born
Oppenheimer approximation, are first determined experimentally in this work. The multi-isotopologue analysis employed herein is comprehensively reviewed in Ref. [8] and has been applied to LiH [9], HCl [12,13], HF [11,14], and CS [15]. In the latter work on CS [15], only 22 molecular parameters in a single set to fit 3974 spectral lines for seven isotopologues generated 42 Yij and 351 band parameters for each of the 12 isotopologues of CS. The generated values of Yij and band parameters agree satisfactorily with the reported values. For GaF, effective Be, re, xe and k of various kinds have been reported [3–7]. As Dunham’s coefficients Yij in the present work are algebraically expressed using the genuine Be, xe, ai (i = 1, . . .), and the non-Born
Oppenheimer correction parameters, the exact expressions for the physical significance of these effective quantities are derivable. The values of the various effective quantities of Be, re, xe and k evaluated using these expressions and the determined values of the fitted parameters of GaF agree satisfactorily with the reported effective values. The present method of analysis therefore simultaneously reproduces the micro-structures (i.e., various kinds of effective equilibrium quantities) and macro-structures (i.e., values of Yij and band parameters of many isotopologues) of diatomic molecules.

3. Spectral calibration All spectral line centers were determined through fitting the measured line profiles to pseudo-Voigt functions with software (OPUS, Bruker software supplied with the spectrometer); a pseudo-Voigt function is a weighted sum of Gaussian and Lorentzian contributions. The spectral positions of bands v = 1–0 to 5–4 for 69GaF and v = 1–0 to 4–3 for 71GaF were calibrated with the simultaneously recorded CO2 spectrum of which the wavenumbers were adopted from Guelachvili and Rao [16]. The calibrated spectral lines of 69GaF and 71GaF, in total 521, are listed in Table S1 (a) in Supplementary material. The uncertainties are estimated to be ±0.001 cm
1, but ±0.002 cm
1 for weaker lines for which ⁄ is added to the wavenumbers listed in Table S1(a). The influence of the hyperfine structure on line width due to the nuclear spins of Ga and F is 0.001 cm
1 for the R(5) and P(6) lines, which include the smallest J number of the present observations. Since the present measurements were made with an unapodized resolution of 0.010 cm
1, we ignored the influence of the hyperfine structure on line positions throughout. The diode-laser measurements of vibrational–rotational transitions reported by Ogilvie et al. [7] are listed in Table S1(b) in Supplementary material; the microwave measurements of the rotational transitions reported by Hoeft et al. [1], by Hoeft and Nair [4] and by Wasylishen et al. [5] are listed in Table S1(c) in Supplementary material. The reported uncertainty in the diode-laser measurements is ±0.0005 cm
1. The uncertainties of the microwave measurements are indicated in Table S1(c). 4. Analysis The multi-isotopologue analysis employed herein is based on a non-Born–Oppenheimer effective Hamiltonian expressed with determinable molecular parameters [8–11,14] as follows, 2

H ¼
Be ð1 þ dDB Þ þ

2. Experiments

d

dn02

þ

Be ð1 þ dDB Þ 2

ð1 þ n0 Þ

1þ

X

! 0 i

dr iq ðn Þ

JðJ þ 1Þ

i¼1

! X ½xe ð1 þ dDx Þ2 02 i ai ð1 þ dDaiq Þðn0 Þ ; n 1þ 4Be ð1 þ dDB Þ i¼1

ð1Þ

in which The spectra were measured for GaF in a high-temperature cell made of an alumina tube (inner diameter 42 mm, length 600 mm). The center portion of the tube (length 200 mm), was heated with a SiC spiral heater; both ends of the tube were cooled with water jackets. To generate GaF, a mixture of Ga (10 g) and AlF3 (10 g) was charged inside the cell and heated to 1870 K. Argon buffer gas (pressure 8 kPa) was admitted to minimize the migration of the sample vapour from the hot zone of the cell. At one end of the tube, a KRS-5 (thallium bromoiodide) window was mounted to transmit the infrared radiation of GaF from the heated gas. The radiation was focused on the emission port of a Fouriertransform spectrometer (Bruker IFS-125HR). Using a KCl beam splitter and a HgCdTe detector (77 K), we recorded the vibrational–rotational transitions of the Dv = 1 bands of GaF at unapodized resolution 0.010 cm
1 in range 625– 660 cm
1; the response of the HgCdTe detector defined the lower limit, 625 cm
1. To calibrate the wavenumbers of the GaF spectrum, we recorded simultaneously the spectra of GaF and CO2 as a calibration standard. A gas sample cell (length 15 cm) containing CO2 (160 Pa) was located in the sample compartment of the spectrometer to absorb the incident emitted radiation, generating the CO2 spectrum of band 0110–0000. Fig. 1 shows a portion of the simultaneously recorded spectra of GaF and CO2. Bands v = 1–0 to 5–4 for 69GaF and v = 1–0 to 4–3 for 71GaF were observed. During integration of the measurements 297 scans were added.

n0 ¼ ð1 þ dDB =2Þn þ dDB =2 ¼ ð1 þ dDB =2Þ

r
re þ dDB =2; re

ð2Þ

with Be = h/(8p2clr2e ) and xe = (1/2pc) (k/l)1/2 (both in unit cm
1). Quantity l denotes the reduced mass of a molecule, (1/l) = (1/Ma) + (1/Mb), with masses Ma and Mb of atoms A and B, respectively. All molecular parameters Be, xe, a1, . . . are those of the traditional concept in the Born–Oppenheimer scheme whereas non-Born–Oppenheimer effects are all included in the correction parameters; re is the equilibrium internuclear distance and k is the force coefficient at re of the Born–Oppenheimer potential. Notations Be, re and k in this work imply the physical significance of the BO BO commonly used parameters BBO e , re and k , respectively. The correction parameters dDB, dDx, dDaiq (i = 1, 2, . . .), and driq (i = 1, 2, . . .) are for the corrections of the breakdown of the Born–Oppenheimer approximation for the reduced equilibrium rotational constant UB, where UB = lBe, for the reduced harmonic vibrational wavenumber Ux, where Ux = l1/2xe, for the Dunham potential constants, a1, a2, . . ., and for the radial part of the rotational energy, respectively. For a pair of quantities xai and xbi , the symbol dxi denotes dxi = (me/Ma) xia + (me/Mb) xbi , in which xi stands for a non-Born– Oppenheimer correction parameter such as DB, Dx, riq or Daiq, (e.g., dDB = (me/Ma) DBa + (me/Mb) DbB, in which me is the mass of the electron). The non-Born–Oppenheimer correction parameters are combinations of qia,b, ria,b and sia,b, which are the expansion

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H. Uehara et al. / Journal of Molecular Spectroscopy 325 (2016) 20–28

Fig. 1. A portion of the vibrational-rotational spectra of 69GaF and 71GaF. The spectra of GaF and CO2 were simultaneously recorded in emission and absorption, respectively.

coefficients of the non-Born–Oppenheimer correction functions Qa, b(n), Ra,b(n) and Sa,b(n) for nonadiabatic vibrational, nonadiabatic rotational and adiabatic effects, respectively. Further details of the effective Hamiltonian (1) are given in a review article [8]. The explicit expressions for the correction parameters in terms of the expansion coefficients are given in Refs. [9,11,14]. The Dunham-like treatment of the Schrödinger equation of the Hamiltonian (1) generates algebraic expressions for the vibrational–rotational term values as [9,11,14]:

FvJ ¼

X  Y ij ðv þ 1=2Þi ½JðJ þ 1Þ j ;

ð3Þ

ij

in which Yij⁄(=Yij⁄(0) + Yij(2) +   ) include non-Born
Oppenheimer corrections to the Dunham coefficients. The Yij⁄(0) coefficients are expressed with included correction parameters dDB, dDx, dDa1q, dDa2q, dDa3q, dr1q, dr2q, dr3q and dr4q. Notably, the correction terms in the Yij⁄(0) coefficients are quantities of order Yij(2), i.e., the Dunham correction [17]. An analysis was made connecting these correction parameters with the vibrational–rotational energy levels through Y⁄ij(0) + Yij(2) and Yij(0). The present method of analysis is applicable to a spectral data set containing isotopologues AB in any combination with physically meaningful parameters; that is, it is applicable to combination AB for which both atoms A and B have multiple isotopic nuclides, only one atom A or B has multiple nuclides, or both atoms A and B have a single nuclide. As the fluorine atom has only a single stable nuclide, isotopically invariant quantities UB, Ux, k, re, a1, a2, . . . cannot be evaluated from a simultaneous analysis of 69GaF and 71 GaF, but the quantities invariant to isotopic nuclide Ga, i.e., the irreducible molecular parameters for GaF, can be determined. The level of the non-Born–Oppenheimer correction of a set of three Y⁄ij(0) including correction parameters {dDB, dDx,dr1q} [12] and the Dunham potential constants up to a6 sufficed for the present analysis. A set of 24 Yij coefficients for each of 69GaF and 71GaF that connect the fitting parameters with the energy levels is given by Y⁄ij(0)+Yij(2) for ij = 01, 02, and 10 and Yij(0) for ij = 03, 04, 05, 06, 07, 08, 11, 12, 13, 14, 15, 16, 20, 21, 22, 23, 24, 30, 31, 32 and 40.

⁄(0) ⁄(0) The algebraic expressions for Y⁄(0) 01 , Y02 , and Y10 (Eqs. (36), (37), (2) (2) (2) (0) (0) (0) and (40) of Ref. [9], respectively), Y01 , Y02 , Y10 , Y(0) 03 , Y04 , Y11 , Y12 , (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) Y(0) , Y , Y , Y , Y , Y , and Y [17], Y , Y , Y , and Y 13 20 21 22 30 31 40 05 14 23 32 (0) (0) (0) (0) (0) [18], Y(0) 06 , Y15 , and Y24 [19], and Y07 , Y08 , and Y16 [20], including the potential constants up to a6, are given in the cited references. In total, 782 spectral lines listed in Tables S1(a)–S1(c) were simultaneously fitted with 11 irreducible molecular parameters for GaF: UB (1 + meDBF/MF), Ux (1 + meDxF/MF), a1, a2, a3, a4, a5, a6, DBGa, DxGa and r1qGa(=r1qF) and parameters ai (i = 1, 2, . . ., 6) are the Dunham potential constants. The set of 782 spectral lines comprised the Dv = 1 band sequence of 280 transitions with v00 = 0–4 for 69GaF and 241 transitions with v00 = 0–3 for 71GaF from the present measurements; the laser-diode measurements of band sequence Dv = 1 with 115 and 114 transitions for v00 = 0–7 for 69GaF and 71GaF, respectively, of Ogilvie et al. [7]; 13 microwave spectral lines of transitions J = 1–0 (for v = 0, 1, 2 and 3) for 69GaF and 71GaF, J = 2–1(for v = 0, 1 and 2) for 69GaF, and J = 2–1 (for v = 0 and 1) for 71 GaF reported by Hoeft et al. [1]; 14 microwave and millimeterwave spectral lines of transitions J = 2–1 (for v = 2) for 71GaF, J = 5–4 (for v = 0, 1 and 2) for 69GaF and 71GaF, J = 12–11 (for v = 0) for 71GaF, and J = 13–12 (for v = 0, 1 and 2) for 69GaF and 71 GaF by Hoeft and Nair [4], and five rotational lines of transitions J = 1–0 (for v = 0, 1 and 2) for 69GaF and J = 1–0 (for v = 0 and 1) for 71 GaF by Wasylishen et al. [5]. The weights for the fit of the spectral data were assumed to be proportional to (1/dobs)2. The spectral uncertainties, dobs, were those given in Tables S1(a)–S1(c). The fundamental physical constants are taken from recommended values [CODATA 2006 21]. The fit of a single set of data comprising the reported transitions of 69GaF and 71GaF was satisfactory; with 11 parameters the normalized standard deviation was 1.093. Because the level of the non-Born–Oppenheimer corrections of a set of three Yij⁄(0), including correction parameters {dDB, dDx, dr1q}, sufficed for the present analysis, the values of the parameters DaiqGa for i = 1, 2, . . . and riqGa(=riqF) for i = 2, 3, . . . were zero within the experimental error; however, the fit provides no information on the parameters DaiqF for i = 1, 2, . . .. Although the notations a1, a2, . . . were used for the fitting parameters for the Dunham potential constants because

23

H. Uehara et al. / Journal of Molecular Spectroscopy 325 (2016) 20–28 Table 1 Molecular parameters of GaF. Parameters

Ref. [4]a

This work Fit 1 F

a b c


1

Fit 2 b

69

71


3.29753(33) 7.498(24)
13.90(19)


3.29773(52) 7.480(37)
13.75(30)

GaF

UB(1 + meDB /MF)/cm u Ux(1 + meDxF/MF)/cm
1 u1/2 a1 a2 a3 a4 a5 a6 DBGa DxGa r1qGa(=r1qF)

5.35439592(868) 2401.78112(813)
3.2985609(292) 7.471838(230)
13.64372(450) 20.7537(464)
27.298(207) 40.58(150)
0.662(206) 1.240(432)
3.19(224)

5.354368102(292) 2401.804101(819)
3.2985512(293) 7.471856(232)
13.64689(444) 20.7813(458)
27.357(205) 40.01(150) 0c 0c 0c

Reduced standard deviation

1.093

1.106

GaF

Only molecular parameters that can be compared directly are listed. The uncertainty (one standard error) in the last digits is given in parentheses. Fixed.

the present non-Born–Oppenheimer treatments were confined within a level set of three Yij⁄(0), these fitting parameters imply a1(1 + meDa1qF/MF), a2(1 + meDa2qF/MF), . . ., respectively. The molecular parameters determined in the fit appear in Fit 1 in Table 1. To demonstrate the need for these 11 parameters to fit the present set of the spectral data, we set the three non-Born– Oppenheimer parameters DBGa, DxGa, and r1qGa(=rF1q) equal to zero. The result of fitting with eight adjustable parameters is shown in Fit 2 in Table 1. The normalized standard deviation is 1.106, which is larger than that of Fit 1. Although the evaluated values of the parameters UB(1 + meDBF/MF) and Ux(1 + meDxF/MF) have uncertainties smaller than those in Fit 1, these smaller values are due to the setting of the three parameters to zero without uncertainties. Another fit in which the values of these three parameters were fixed to those evaluated in Fit 1 generated uncertainties for UB(1 + meDBF/MF) and Ux(1 + meDxF/MF) that were slightly smaller than those in Fit 2, i.e., 2.89  10
7 cm
1 u and 8.09  10
4 cm
1 u1/2 for UB(1 + meDBF/MF) and Ux(1 + meDxF/MF), respectively. Therefore, the values of the 11 parameters determined in Fit 1 were established and have been used throughout this paper. 5. Discussion The values of the non-Born–Oppenheimer parameters for the effects of the breakdown of the Born–Oppenheimer approximation of GaF, DBGa, DxGa and r1qGa(=r1qF), in Table 1 are the first determined experimentally. Although Tiemann et al. [3] reported that Ga DGa + DDunham = DBGa + D01) =
0.60 (30) without reporting 01 (=DB 01 measured line frequencies, the subsequent paper of Hoeft and Nair [4] indicated that there is no observable breakdown of the Born– Oppenheimer approximation for GaF within the accuracy of their microwave measurements [3,4]. Wasylishen et al. [5] measured rotational transitions J = 1–0 of 69GaF and 71GaF with an improved accuracy, but they did not examine the breakdown effects. Tiemann et al. reported DDunham =
0.093 for GaF. Our notation of 01 the isotopically invariant Dunham correction D01 [15] is equal to DDunham and is given by 01

ham correction is sensitive to values of potential constants a1, a2, and a3. The value of D01 calculated from the values of a1, a2 and a3 given by Hoeft and Nair [4] is
0.0036. As potential constants ai were evaluated with smaller standard errors in the present work, our value of the Dunham correction is more accurate. The value DBGa + D01 =
0.662(206) + 0.005359(320) =
0.657(206) in the present work can be compared with the value of DGa 01 reported by Tiemann et al. [3]. The present determination of the value of DBGa results from the inclusion of the rotational transitions given by Wasylishen et al. [5] in the data set used for fitting. On fitting the data set excluding the rotational transitions reported by Wasylishen et al. [5], we confirmed the conclusion of Hoeft and Nair [4], who observed no breakdown effects in their microwave measurements. To facilitate calculations or predictions of the spectral frequencies, the values of 24 Yij for both 69GaF and 71GaF, which connect 11 molecular parameters with the energy values of 69GaF and 71GaF, were back-calculated using the values of the parameters given in Table 1. The calculated values of Yij listed in Table 2 together with those error limits that were calculated with the laws of error propagation are those of Yij⁄ given in Eq. (3); we use Yij instead of Yij⁄ for convenience. The uncertainties in Table 1 were given in three digits in order to accurately reproduce the values of Yij and the band parameters. The evaluated values listed in the tables in this paper have three-digit uncertainties throughout. With the exception of Y24, all Yij values for each isotopologue of GaF were determined with smaller standard errors. The values of Yij reported by Hoeft and Nair [4] and by Hoeft et al. [1] are also listed in Table 2. 5.1. Inner structure of the conventionally used molecular parameters Be, re, xe and k Molecular parameters Be, ae, De, xe and xexe, conventionally used, are effective parameters [22]; that is, apart from sign, they correspond to coefficients Yij in the Dunham expansion for the term values,

FvJ ¼

X Y ij ðv þ 1=2Þi ½JðJ þ 1Þ j :

ð5Þ

ij

D01 ¼

 lB2e  15 þ 14a1
9a2 þ 15a3
23a1 a2 þ 21ða21 þ a31 Þ=2 : 2 2me xe ð4Þ D01(=DDunham ) 01

The value of Dunham correction = 0.005359(320) obtained from the parameter values of the present analysis differs from the value given by Tiemann et al. [3]. The value of the Dun-

For example, Y01, Y11 and Y10 are the effective parameters Be,
ae and xe. In our work, however, as Dunham coefficients Yij are expressed explicitly as physical quantities Be, xe, ai and nonBorn–Oppenheimer correction parameters DBa,b, Dxa,b, riqa,b and Daiqa,b, which are defined by the effective Hamiltonian (1), we use notation Yij⁄ instead of Yij in Eq. (3).

24

H. Uehara et al. / Journal of Molecular Spectroscopy 325 (2016) 20–28

Table 2 Values of Dunham coefficients Yij/cm
1 for two isotopologues of GaF back-calculated from 11 molecular parameters given in Table 1. 69

Coeff.

a

10 Y10 10
1Y20 10Y30 104Y40 Y01

0.622360023(220)
0.3305070(109) 0.127341(196)
0.4318(113) 0.3595159075(238) 0.359515885(50) 0.35951614(26)
0.28642080(373)
0.2864078(50)
0.286424(26) 0.102293(167) 0.10137(14) 0.1017(26)
0.1553(182)
0.4799341(395)
0.48027(13)
0.500(50)

102Y11

104Y21

107Y31 106Y02

a

71

GaF


3

GaF

Coeff.

0.620461723(215)
0.3284941(108) 0.126179(194)
0.4266(112) 0.3573263292(236) 0.357326250(73) 0.35732650(26)
0.28380827(370)
0.2837879(73)
0.283809(26) 0.101051(165) 0.09991(21) 0.1004(26)
0.1529(179)
0.4741059(388)
0.47433(21)
0.493(50)

9

10 Y12 10

10 Y22 1012Y32 1012Y03 1014Y13 1016Y23 1018Y04 1019Y14 1022Y24 1025Y05 1025Y15 1029Y06 1031Y16 1035Y07 1040Y08

69

71

0.925233(605) 1.030(76) 0.20576(262)
0.1866(305)
0.1912574(197)
0.19088(23) 0.45702(341)
0.5838(737)
0.275381(157) 0.12218(281) 0.630(796) 0.4015(518)
0.1289(435) 0.13338(248) 0.558(282) 0.3067(318) 0.10402(956)

0.911209(596) 0.95(12) 0.20203(257)
0.1827(299)
0.1877843(193)
0.18746(36) 0.44735(334)
0.5697(719)
0.268735(153) 0.11887(273) 0.611(772) 0.3895(503)
0.1246(420) 0.12858(239) 0.537(271) 0.2938(304) 0.09906(910)

GaF

GaF

The uncertainty (one standard error) in the last digits is given in parentheses. The second- and third-row entries are the values reported in Refs. [4,1], respectively.

Watson [23] expressed Yij with empirical correction parameters Dija,b for the breakdown of the Born–Oppenheimer approximation as

Y ij ¼ l
ðiþ2jÞ=2 U ij ½1 þ ðme =MA ÞDaij þ ðme =M B ÞDbij :

ð6Þ

The inner structure of Dij was discussed by Watson [24] and by Tiemann et al. [3] only for Da,b 01 ; that is,

  me a me b Y 01 ¼ l
1 U 01 1 þ D01 þ D01 Ma Mb  o me n nonad Dunham
1 ðD01 Þa þ ðDad Þa ¼ l U 01 1 þ 01 Þa þ ðD01 Ma o me n nonad Dunham ðD01 Þb þ ðDad Þ þ ð D Þ þ 01 b 01 b Mb

ð7Þ

in which Dnonad and Dad 01 01 are the nonadiabatic and adiabatic corrections, respectively, and DDunham is the Dunham correction [17]. The 01 nonadiabatic correction Dnonad is related to the rotational g factor 01 and the permanent electric dipole moment. Eq. (7) is equivalent to our expression [9,15]: " B2 ð0Þ ð2Þ Y 01 þ Y 01 ¼ Be ð1 þ dDB Þ þ e 2 2xe  ð15 þ 14a1
9a2 þ 15a3
23a1 a2 þ 21ða21 þ a31 Þ=2Þ   me a me b me ¼ Be 1 þ DB þ DB þ D01 Ma Mb l       me a 4Be a me b 4Be b me ¼ Be 1 þ r 0 þ 2 s1 þ r 0 þ 2 s1 þ D01 ; Ma xe Mb xe l

ð8Þ (Dnonad )a,b 01

ra,b 0 ,

(Dad 01)a,b

2 e)

sa,b 1 ,

i.e., = = (4Be/x and Similarly, our expression for Y⁄10 [9,15] is

"

ð0Þ Y 10

þ

ð2Þ Y 10

¼ xe ð1 þ dDx Þ þ

(DDunham )a,b 01

= D01.

B2e ð25a4
95a1 a3 =2
67a22 =4 4x2e #

þ459a21 a2 =8
1155a41 =64Þ   me a me b me Dx þ Dx þ D10 ¼ xe 1 þ Ma Mb l    me Be a ¼ xe 1 þ q0 =2
2 ð3a1 sa1
2sa2 Þ Ma xe    me Be me b þ q0 =2
2 ð3a1 sb1
2sb2 Þ þ D10 Mb xe l

Expressions Yij⁄, not only Y⁄01 and Y⁄10, include explicit expressions of the nonadiabatic vibrational and rotational, adiabatic 2 a,b and Dunham corrections. For Y⁄10, terms qa,b 0 /2,
(Be/xe ) (3a1s1
2sa,b ) and D are the nonadiabatic vibrational, adiabatic and 2 10 Dunham corrections, respectively. Note that individual parameters qia,b, ria,b and sia,b are not determinable, whereas the combinations DBa,b, Dxa,b, riqa,b and Daiqa,b are determinable through spectral fitting. Values of four kinds have been reported so far for Be and re as indicated below. The following discussion is summarized in Table 3. The values of Be and xe were calculated for only 69GaF because the purpose of the comparison below is to show the physical significance of each reported effective value and to assess the accuracies of those values and of the present work. The errors in re and k in the present work include no uncertainties in the fundamental constants. (1) Conventionally used effective constants that correspond to coefficient Y01 as reported by Wasylishen et al. [5], i.e., effective Be = 10778.0159(12)/29979.2458 = 0.3595159121(400) cm
1 for 69GaF and effective re = 177.43698(8) pm. Our corresponding Y01 values in Table 2 agree satisfactorily, i.e., 0.3595159075(238) cm
1 and 177.43696789(587) pm, respectively. The physical significances of these effective Be and re values are

  me Ga me F me Be 1 þ DB þ DB þ D01 and MGa MF l    1 me Ga me F me DB þ DB þ D01 ; re 1
2 M Ga MF l

ð10Þ

respectively. The value of Y01 for 69GaF given by Hoeft and Nair [4] agrees satisfactorily with the Be value of Wasylishen et al. [5] and with our Y01 value. (2) The value re = 177.43907(61) pm reported by Hoeft and Nair [4] was obtained on applying Dunham correction
1.39  10
7 [=(me/l) D01, in our notation] to Y01. The physical significances of these Be and re values are

     me Ga me F 1 me Ga me F Be 1 þ DB þ DB and r e 1
DB þ DB ; 2 MGa MGa MF MF ð11Þ

ð9Þ

respectively. Our calculated value of (me/l) D01 and the corresponding values of Be and re given by Eq. (11) are 1.974 (118)  10
7 and 0.3595158365(241) cm
1 for 69GaF and

25

H. Uehara et al. / Journal of Molecular Spectroscopy 325 (2016) 20–28 Table 3 Various effective Be, re, xe and k so far reported. Physical significance
1

Be/cm of GaF  me F me Be 1 þ MmGae DGa B þ M F DB þ l D01  me F Be 1 þ MmGae DGa B þ M F DB  me F Be 1 þ M DB , (irreducible) F


1

Other reported values of Be/cm  me 4Be Ga me 4Be F Be 1 þ M x2 s1 þ MF x2e s1 h Ga e i me F me DB þ M D01 re 1
12 M F F  me 2Be Ga me 2Be F r e 1
M x 2 s1
M x 2 s1 Ga

F

e



me F me xe 1 þ MmGae DGa x þ MF Dx þ l D10



me F xe 1 þ MmGae DGa x þ MF Dx

a b c d e f

The The Ref. Ref. Ref. Ref. Ref.

0.3595159075(238)a

Beb Y01c

0.3595159121(400) 0.3595158854(500)

0.359517731(582)

me MF

for

None

177.43696789(587)

reb

177.43698(8)

177.43698540(596)

rec

177.43697(61)

177.436517(143)

None

69

GaF and re/pm





Dx , (irreducible) F

k/N m
1 h  i me F me k 1 þ 2 MmGae DGa x þ MF Dx þ l D10 h  i me F k 1 þ 2 MmGae DGa x þ MF Dx  F e k 1 þ 2m M F Dx , (irreducible) *

Reported values

0.3595276074(524)

U01/le

0.3595275939(449)

177.436503(143)

rBOd e

177.43351(25)*

177.4340806(129)

Ree

177.43410(15)

622.360023(220)

Y10f Y10c

e

xe/cm
1 of 69GaF

xe 1 þ

Reported notation

0.3595158365(241)

re/pm h  i me F me r e 1
12 MmGae DGa B þ M F DB þ l D01 h  i me F re 1
12 MmGae DGa B þ M F DB  me re 1
2M DFB , (irreducible) F



This work

69

622.361375(231)


1/2

l

U10e

622.35523(210)

622.367(11) 622.104(88) 622.361101(246) None

339.878512(240) 339.879989(252)

kee

339.87327(230)

339.87998(34) None

origin of the difference is indeterminate. See text. uncertainty (one standard error) in the last digits is given in parentheses. [5]. [4]. [3]. [7]. [6].

177.43698540(596) pm, respectively. Although the value of re reported by Hoeft and Nair [4] differs from our corresponding value of re, the difference is attributed primarily to the differences in the values of the fundamental constants. If the value of their conversion factor is revised using the updated fundamental constants, the value of re from Hoeft and Nair [4] becomes 177.43697(61) pm. The contribution of the difference in the Dunham correction value is small. (3) From the value of UB (1 + meDBF/MF) = 5.35439592(868) cm
1 u in the present work listed in Table 1, we obtained 0.359517731(582) cm
1 for 69GaF and 177.436517(143) pm for the quantities

    me F me F Be 1 þ DB and r e 1
DB ; MF 2MF

ð12Þ

respectively. The physical significance of rBO e = 177.43351 (25) pm reported by Tiemann et al. [3] is re(1
meDBF/ 2MF
meD01/2MF). The corresponding value in the present work for rBO by Tiemann et al. calculated applying a correction e
meD01/2MF to Eq. (12) is 177.436503(143) pm. The differGa ence between DGa + 01 inTiemann et al. [3] and our DB Ga D01(=D01 ) contributes to the re value only 0.000040(257) pm. Based on the conversion factor reported by Tiemann et al. [3], the contribution from the fundamental constants is also small. As Tiemann et al. reported no Y01 value, the origin of the difference in re between the present work and that of Tiemann et al. is indeterminate.

As 19F is a single stable nuclide, the DFB values cannot be obtained from spectra; parameter UB(1 + meDBF/MF) that we chose for the spectral fit is hence irreducible. Note that terms higher than O(me/Ma,b) are ignored throughout this work. (4) Ogilvie et al. [7] reported values U01 = 5.35454281(67) cm
1 u and Re = 177.43410(15) pm. These values were obtained on correcting the nonadiabatic rotational part, tGa 0 and tF0, which was calculated from the experimental values of the electric dipole moment [1] and the rotational g factor [2] for 69GaF. The Dunham correction was naturally applied in their analysis of the algebraic approach. As shown in Eq. (7), the values of the nonadiabatic rotational part plus the adiabatic part for Ga and F are necessary to evaluate U01 from a spectral analysis. Ogilvie et al. [7] F assumed the adiabatic part, uGa 1 and u1, to equal zero, but, as discussed later, the contribution of the adiabatic part to U01 is considerable. F As is further elaborated below, the corrections of tGa 0 and t0 reported by Ogilvie et al. [7] are exactly the same as those of rGa and rF0 in this work; their values of U01/ 0 l = 0.3595275939(449) cm
1for 69GaF and Re are hence basically the same as

  me 4Be Ga me 4Be F Be 1 þ s1 þ s1 and 2 2 MGa xe M F xe   me 2Be Ga me 2Be F s1
s1 ; re 1
2 2 M Ga xe M F xe

ð13Þ

26

H. Uehara et al. / Journal of Molecular Spectroscopy 325 (2016) 20–28

respectively, in the present work. Applying corrections (me/ F MGa) rGa 0 , (me/MF) r0 and (me/l) D01 to Eqs. (8) and (10), we obtain the values of Eq. (13) as 0.3595276074(524) cm
1 and 177.4340806(129) pm; the agreement is thus satisfactory. For the corrections, we used values rGa 0 =
0.4382(12) and rF0 =
1.0131(30) calculated below. The values of xe and k reported so far follow. (1) The conventionally used effective parameters xe that correspond to coefficient Y10 and k have the physical significances



 me Ga me F me Dx þ Dx þ D10 and MGa MF l    me Ga me F me Dx þ Dx þ D10 ; k 1þ2 M Ga MF l

xe 1 þ

ð14Þ

respectively. In the present work, the values of these effective xe and k for 69GaF are 622.360023(220) cm
1 and 339.878512(240) N m
1, respectively. The reported Y10 values of the effective xe for 69GaF are 622.367(11) cm
1 by Uehara et al. [6] and 622.104(88) cm
1 by Hoeft and Nair [4]. The former value agrees satisfactorily with the value of this work. (2) Ogilvie et al. [7] claimed that U10 = 2401.80378(95) cm
1 u1/2, 69 i.e., xe = 622.361101(246) cm
1 for GaF, and ke = 339.87998(34) N m
1. To obtain the value of U10, Ogilvie Ga et al. assumed that expansion coefficients sGa 0 and u1 of the non-Born–Oppenheimer correction functions were equal to Ga Ga Ga zero and neglected uGa 2 ; their s0 , u1 and u2 correspond to Ga Ga Ga our q0 , s1 and s2 , respectively. Eq. (9) shows that their values of xe(=l
1/2 U10) and ke were obtained on applying only the Dunham correction (me/l) D10 to the quantities (14), i.e.,



xe 1 þ

me Ga me F D þ D MGa x MF x



   me Ga me F and k 1 þ 2 Dx þ Dx ; MGa MF ð15Þ

respectively. An expression for D10 same as that in Eq. (4) is found in Eq. (9). On correcting the value (me/l) D10 =
0.000002173(114) calculated with the parameters given in Table 1, to Y10 in this work, we obtain values of corresponding xe = 622.361375(231) cm
1 for 69GaF and k = 339.879989(252) N m
1 of Eq. (15); the agreement is satisfactory. (3) As 19F is a single stable nuclide, parameter Ux (1 + meDxF/ MF) is irreducible. From the value Ux (1 + meDxF/MF) = 2401.78112(813) cm
1 u1/2 in the present work (Table 1), we obtain 622.35523(210) cm
1 for 69GaF and 339.87327 (230) N m
1 for quantities



xe 1 þ

me F D MF x



  2me F and k 1 þ Dx ; MF

ð16Þ

respectively. These values are exactly the same as those obtained on applying the determined correction parameter DxGa = 1.240(432) to xe given in Eq. (15). In the present work, we obtained also a value of correction parameter r1qGa(=r1qF) =
3.19(224). The parameter r1qGa(=r1qF) contributes to neither Y01 nor Y10 but contributes to the Dunham coefficients Yij (j P 2). 5.2. Band parameters The band parameters for 69GaF and 71GaF listed in Tables 4a and 4b, respectively, were calculated from the Yij values in Table 2. The notations for the vibrational term values and the rotational param-

eters in Tables 4a and 4b are given by the equation for the energy levels as

F v ðJÞ ¼ Gv þ Bv JðJ þ 1Þ
Dv ½JðJ þ 1Þ2 þ Hv ½JðJ þ 1Þ3 þ Lv ½JðJ þ 1Þ4 þ Mv ½JðJ þ 1Þ5 þ Nv ½JðJ þ 1Þ6 þ Ov ½JðJ þ 1Þ7 þ Pv ½JðJ þ 1Þ8 : ð17Þ Term Y00 = (Be/8) (3a2
7a21/4), not listed in Table 2, was included in the calculation of Gv. In the present analysis, potential constants up to a6 that generated a set of 24 Yij coefficients [9,17– 20] given in Table 2 were necessary to reproduce the spectral transitions. Because the maximum value of j in 24 Yij is 8, the terms up to Pv[J(J + 1)]8 are necessary to calculate Fv(J) using expression (17). Values of Bv reported by Wasylishen et al. [5] are also listed in Tables 4a and 4b; their Bv values agree satisfactorily with those of this work. The values of the band parameters with error limits are readily obtained from Gv up to Pv in this work from the values of the molecular parameters in Table 1 via the values of Yij in Table 2. Through the merit of the algebraic approach, the numerical values of the present potential function and the eigenvalues are easily reproduced, including the effect of the breakdown of the Born
Oppenheimer approximation. The 11 molecular parameters determined experimentally generated the values of the 48 Yij coefficients in Table 2 and the 162 vibrational-rotational parameters in Tables 4a and 4b with great accuracy. 5.3. Evaluation of the individual expansion coefficients of the nonBorn–Oppenheimer correction functions Herman and Ogilvie [25], Uehara and Ogilvie [9] and Uehara [8] reported relations among rotational g factor gJ(n), permanent electric dipole moment M(n), and nonadiabatic expansion coefficients ra,b i :

g J ðnÞ ¼

X X fðmp =M a Þr ai þ ðmp =M b Þr bi gni ¼ ðmp =me Þ dr i ni ; i¼0

ð18Þ

i¼0

and

2MðnÞ=fere ð1 þ nÞg ¼

X ðr bi
r ai Þni :

ð19Þ

i¼0

Eq. (19) is defined for a molecule AB of relative polarity
AB+. If functions gJ(n) and M(n) for a molecule AB are available from experiments with external fields or from calculations, one can estimate the values of ra,b (i = 0, 1, . . .) from Eqs. (18) and (19). The fiti a ,b ting parameters Dax,b, Da,b and Daaiq,b (i = 1, 2, . . .) can then B , riq become resolved with these values to yield expansion coefficients qa,b (i = 0, 1, . . .) and sa,b (i = 1, 2, . . .). Parameters riGa,F (i = 0, 1, . . .), i i qiGa,F (i = 0, 1, . . .) and siGa,F (i = 1, 2, . . .) are the expansion coefficients for the functions of the nonadiabatic rotational RGa,F(n), nonadiabatic vibrational QGa,F(n) and adiabatic SGa,F(n) corrections, respectively, which are given in Eqs. (3)–(5) of Ref. [9]. Although function gJ(n) of GaF is unknown, an experimental value of the rotational g factor of 69Ga19F for rotational transition J = 1–0 in state v = 0 was reported [2] with an experimental electric dipole moment value of 69Ga19F for rotational transition J = 1–0 in F state v = 0 [1]. Only expansion coefficients sGa 1 and s1 can be evaluated from these values. The reported experimental values gJ =
0.06012(12) and dipole moment M = 2.45(5) D [1,2] are for state v = 0 of 69Ga19F. The values of gJ(0) and M(0), i.e., gJ and M at re, were approximated with the above experimental values. From the relations

 g J ð0Þ ¼ mp

rGa rF 0 þ 0 MGa M F

 ð20Þ

27

H. Uehara et al. / Journal of Molecular Spectroscopy 325 (2016) 20–28 Table 4a Band parameters (unit cm
1) for v

a b

a

GaF calculated from 42Yij given in Table 2.

Gv a,b

Gv
G0

Bv

107Dv

1013Hv

1019Lv

0.0

4.794663(395) 4.79755 4.785006(395) 4.78724 4.774953(396) 4.77693 4.764517(397) 4.753709(401) 4.742539(408) 4.731020(421) 4.719161(443) 4.706976(479)


1.889869(261)


2.69256(211)


1.845334(573)


2.56911(484)


1.801967(990)


2.44441(875)


1.75976(151)
1.71873(215)
1.67887(292)
1.64017(383)
1.60264(487)
1.56628(607)


2.3184(139)
2.1912(205)
2.0627(286)
1.9329(383)
1.8019(495)
1.6696(623)

0

310.506988(118)

1

926.298042(418)

615.791053(435)

2

1535.592310(931)

1225.085321(939)

3 4 5 6 7 8

2138.46412(176) 2734.98678(305) 3325.23253(491) 3909.27261(752) 4487.1772(110) 5059.0154(157)

1827.95713(177) 2424.47979(305) 3014.72554(491) 3598.76562(752) 4487.1772(110) 4748.5084(157)

0.3580863588(305) 0.3580863775(166) 0.3552425590(718) 0.3552425608(200) 0.352419078(145) 0.3524190828(176) 0.349615822(256) 0.346832700(414) 0.344069616(626) 0.341326479(901) 0.33860319(124) 0.33589967(168)

v

1026Mv

1030Nv

1036Ov

1041Pv

0 1 2 3 4 5 6 7 8

3.371(562) 2.081(833) 0.79(120)
0.49(160)
1.78(202)
3.07(244)
4.36(287)
5.65(330)
6.94(373)

1.3617(285) 1.4176(490) 1.4735(747) 1.529(101) 1.585(129) 1.641(157) 1.697(185) 1.752(213) 1.808(241)

3.067(318) 3.067 3.067 3.067 3.067 3.067 3.067 3.067 3.067

1.0402(956) 1.0402 1.0402 1.0402 1.0402 1.0402 1.0402 1.0402 1.0402

The uncertainty (one standard error) in the last digits is given in parentheses. The second-row entries are the values reported in Ref. [5]. Term Y00 was included in the calculation of Gv. See text.

Table 4b Band parameters (unit cm
1) for

b

69

71

GaF calculated from 42Yij given in Table 2. Bv

107Dv

1013Hv

1019Lv

4.736453(388) 4.73854 4.726943(388) 4.72903 4.717045(389) 4.706770(390) 4.696130(393) 4.685135(400) 4.673797(413) 4.662125(435) 4.650132(471)


1.855617(256)


2.62776(206)


1.812021(561)


2.50766(471)

1221.407714(927) 1822.49889(175) 2417.27881(302) 3005.81908(487) 3588.19029(745) 4164.4620(109) 4734.7027(155)

0.3559098122(303) 0.3559098191(166) 0.3530918900(712) 0.3530919180(166) 0.350294040(143) 0.347516171(253) 0.344758191(409) 0.342020009(618) 0.339301532(889) 0.33660266(123) 0.33392332(165)


1.769565(969)
1.72824(147)
1.68807(210)
1.64903(285)
1.61113(374)
1.57437(476)
1.53875(592)


2.38634(850)
2.2638(135)
2.1400(199)
2.0150(278)
1.8888(371)
1.7614(480)
1.6327(604)

1026Mv

1030Nv

1036Ov

1042Pv

3.271(545) 2.024(807) 0.77(116)
0.46(155)
1.71(195)
2.96(236)
4.21(278)
5.45(319)
6.70(361)

1.3126(274) 1.3663(471) 1.4200(719) 1.4737(979) 1.527(124) 1.581(151) 1.634(177) 1.688(204) 1.742(231)

2.938(304) 2.938 2.938 2.938 2.938 2.938 2.938 2.938 2.938

9.906(910) 9.906 9.906 9.906 9.906 9.906 9.906 9.906 9.906

v

Gv

0

309.561932(116)a,b

1

923.494568(412)

2 3 4 5 6 7 8

1530.969646(920) 2132.06082(175) 2726.84074(302) 3315.38101(487) 3897.75222(745) 4474.0239(109) 5044.2646(155)

v 0 1 2 3 4 5 6 7 8

Gv
G0 0.0 613.932636(428)

The uncertainty (one standard error) in the last digits is given in parentheses. The second-row entries are the values reported in Ref. [5]. Term Y00 was included in the calculation of Gv. See text.

ð4Be =x2e ÞsGa 1 ¼
0:224ð206Þ

and

2Mð0Þ F ¼ r Ga 0
r0 ; ere

ð21Þ

the values of

and 5
1 sGa 1 ¼
0:602ð555Þ  10 cm :

Relations

r Ga 0 ¼
0:4382ð121Þ

nonad rGa ÞGa ; 0 ¼ ðD01

ð22Þ

and

ad ð4Be =x2e ÞsGa 1 ¼ ðD01 ÞGa

ð23Þ

r F0

¼
1:0131ð30Þ

are obtained for the nonadiabatic rotational corrections assuming the relative polarity
FGa+. Experimental value DBGa[=
0.662(206)] that is listed in Table 1 with the value of rGa 0 generated the following adiabatic correction,

and

D01 ¼ ðDDunham ÞGa 01

ð24Þ DGa 01

hold for empirical quantities [3] for the nonadiabatic, adiabatic and Dunham corrections. The present value of (4Be/x2e ) sGa 1 can be

28

H. Uehara et al. / Journal of Molecular Spectroscopy 325 (2016) 20–28

compared with value (Dad 01)Ga =
0.07(30) given by Tiemann et al. [3]. The difference comes from the difference in the Dunham correction; Tiemann et al. used (DDunham )Ga =
0.093, whereas the value 01 evaluated in the present work is 0.00535(32). Adiabatic correction (4Be/x2e ) sGa 1 is smaller than the nonadiabatic correction rGa 0 but considerably larger than the adiabatic correction reported by Tiemann et al. [3]. It is noted that the relations F between nonadiabatic corrections rGa 0 and r0 and the g factor and electric dipole moment in Eqs. (20) and (21) are exactly the same F as those for tGa 0 and t0 given by Ogilvie et al. [7]. For this reason, we were able to compare directly the values of U01 and Re reported by Ogilvie et al. with the above quantities given by Eq. (13). It should be noted that the above treatments to evaluate rGa 0 and F r0 include additional errors because the values of gJ(0) and M(0) were taken from those determined from rotational transition J = 1–0 in state v = 0. According to Kirschner et al. [26], the expected variation in gJ between adjacent vibrational states is of order (4Be/xe) gJ(0), and an additional error approximately 0.0023 is expected in the present discussion regarding gJ(0). The assumed M(0) value similarly includes an additional error. The value of DxGa obtained in the present work plus Dunham correction D10 =
0.05900(310) yields a conventional empirical quantity DGa 10 : Ga DGa 10 ¼ Dx þ D10 ¼ 1:180ð432Þ:

Relations similar to Eqs. (23) and (24) are expressible as nonad qGa ÞGa 0 =2 ¼ ðD10

ð25Þ

and ad Ga
ðBe =x2e Þð3a1 sGa 1
2s2 Þ ¼ ðD10 ÞGa :

ð26Þ

Although a value of sGa 1 was obtained above from the experimental values of gJ and M, (Dnonad )Ga and (Dad 10 10)Ga cannot be determined separately from the observed value DxGa, because DxGa includes Ga two unknowns qGa 0 and s2 , as Watson [23] found.

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