Dipole moment of ArH+X1Σ+ from analysis of pure rotational and vibration-rotational spectra

13 July 2001

Chemical Physics Letters 342 (2001) 293±298

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Dipole moment of ArH‡X1R‡ from analysis of pure rotational and vibration-rotational spectra Marcin Molski * Department of Theoretical Chemistry, Faculty of Chemistry, Adam Mickiewicz University, ul. Grunwaldzka 6, PL 60-780 Pozna n, Poland Received 2 April 2001; in ®nal form 11 May 2001

Abstract By a direct ®t to 431 experimental wave numbers of pure rotational and vibration-rotational transitions of ArH‡ X1 R‡ in six isotopic variants, 16 coecients of radial functions de®ning the Born±Oppenheimer potential energy, adiabatic and nonadiabatic e€ects have been determined at r^ ˆ 0:885 and F ˆ 4:07  1015 . Using a relation between nonadiabatic rotational e€ects and electric and magnetic properties of the molecule, a permanent dipole moment of 40 Ar1 H‡ is estimated as l0 ˆ 2:12…55† D which agrees both with the ab initio value 2.2(1) D and the experimental result l0 ˆ 3:0…6† D within quoted error limits. Ó 2001 Elsevier Science B.V. All rights reserved.

The conventional method for determination of dipole moment via observation of the Stark e€ect is dicult to apply to molecular ions because of the translational motion of net charges induced by an electric ®eld. Laughlin et al. [1±3] have developed a method for experimental determination of the dipole moment of molecular ions by measuring the rotational g-factors g…R†, g…R†0 for two di€erent isotopomers of a given molecule. Having determined them the electric dipole moment l…R† of a diatomic ion can be calculated from the relation [4] 0

m0 g…R† ˆ mg…R†

0

jmp ‰l…R† ‡ l…R† Š=…eR†;

…1†

in which l…R†0 ˆ l…R†

jeRQ

…2†

is the dipolar moment of an isotopic variant with respect to the centre of mass as origin [4], where

*

Fax: +48-61-865-8008. E-mail address: [email protected] (M. Molski).

j ˆ n0a =…n0a ‡ n0b †

na =…na ‡ nb †:

…3†

Here na…b† are nuclear masses, m reduced atomic mass, R internuclear separation, mp and e protonic mass and charge, whereas Q is the net charge on the molecule. Nuclear masses and reduced atomic mass of an isotopic variant are denoted by primed quantities. Introducing (2) into (1) one obtains the equation used by Laughlin et al. [3] to calculate the dipolar moment of a diatomic ion i  eR h 0 l…R† ˆ mg…R† m0 g…R† ‡ j2 mp Q : …4† 2jmp Application of this method to ArH‡ X1 R‡ yields [3] the dipole moment: (a) l0 ˆ 2:84…59† D calculated from the ground vibrational state g-factors, (b) l0 ˆ 2:95…59† D for extrapolated equilibrium g-factors using the method of Gruebele et al. [5], and (c) l0 ˆ 3:35…59† for extrapolated equilibrium g-factors using the method proposed by Ramsey [6]. Of them the method (b) is recommended [3] as yielding the most satisfactory value l0 ˆ 3:0…6† D.

0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 6 0 1 - 7

294

M. Molski / Chemical Physics Letters 342 (2001) 293±298

A secondary source of dipole moment for ArH‡ X1 R‡ was the high quality ab initio calculation by Rosmus [7]. Application of the coupled electron pair approximation method provided the value l0 ˆ 2:2…1† D. The numerical Hartree±Fock calculation repeated by Pyykko et al. [7] has proved that basis set limitations do not generate a substantial error in Rosmus' calculation. The experimental results by Laughlin et al. [3] are slightly outside the quoted errors from the ab initio value of Rosmus. To assess this situation the dipolar moment of ArH‡ is determined here by the direct inversion of highly resolved pure rotational and vibration-rotational spectra, particularly those obtained with Fourier-transform spectrometers and diode lasers. This method, ®rst proposed by Sauer and Ogilvie [8], by Ogilvie [9], and then generalized by Molski [10,11], has been applied to neutral molecules GaH [10] and LiH [11] in the X1 R‡ ground electronic state to yield rotational g-factors [10,11] and dipolar moment [11], which conforms acceptably with their values obtained from experiment or quantum-chemical computations. In the present work we apply this method to the molecular ion ArH‡ X1 R‡ to evaluate its dipole moment from pure rotational and vibration-rotational spectra. In the calculation, we employ a Herman±Ogilvie wave equation [4]   d2 B0 2 ‡ UvJ …x† EvJ wvJ …x† ˆ 0; …5† dx

The adiabatic term re¯ects the dependence of internuclear potential energy on not only the distance between the nuclei but also their relative momenta; nonadiabatic rotational e€ects re¯ect the fact that electrons fail to follow perfectly the nuclei rotating about the centre of molecular mass, whereas nonadiabatic vibrational e€ects appear due to vibrational inertia of electrons. All radial functions in (6) are expanded into a series of the Ogilvie's variable [13] z ˆ x=…1 ‡ x=2† that remains ®nite 2 6 z < 2 throughout the range of molecular existence 0 6 R < 1. Additionally we include, according to the Herman± Ogilvie theory [4], a correction term j2 Qme =m valid for molecular ions. To calculate j we use the nuclear masses na and nb for 40 Ar and 1 H whereas nuclear masses n0a and n0b of an isotopic variant have been inserted to j automatically during ®tting procedure. The radial functions for a diatomic ion take the form ! X 2 i V …x† ˆ c0 z 1 ‡ ci z ; …7†

with an e€ective potential [11]

where me , ma and mb are masses of electron and atoms. The presence in (9) and (10) of the term j2 Qme =m ensures mass-independence of the coef®cients tia;b and sia;b . To solve the wave equation (5) we express all terms in (5) with an e€ective vibrational variable dependent on …v; J † [11]

UvJ …x† ˆ

B0 J …J ‡ 1†‰1 ‡ a…x† …1 ‡ x† ‡ V …x†‰1

b…x†Š

2 0

b…x†Š ‡ V …x† ‡ EvJ b…x† 0

…6†

including adiabatic V …x† , nonadiabatic rotational a…x† and vibrational b…x† corrections to Born± Oppenheimer energy levels. In the above equations, B0 ˆ  h2 =…2mR20 † denotes the rotational parameter for the Born±Oppenheimer equilibrium internuclear separation R0 , x ˆ …R R0 †=R0 is the Dunham's variable [12] whereas J and v are rotational and vibrational quantum numbers, respectively.

iˆ1

V …x†0 ˆ me

X X iˆ1 lˆa;b

a…x† ˆ me

X X iˆ0 lˆa;b

b…x† ˆ me

X X iˆ0 lˆa;b

g ˆ …R

uli ml 1 zi ;

…8†

til ml 1 zi ‡ j2 Qme =m;

…9†

sli ml 1 zi ‡ j2 Qme =m;

…10†

RvJ †=RvJ

…11†

to which, we apply a linear transformation x ˆ xvJ ‡ g…1 ‡ xvJ †;

xvJ ˆ …RvJ

R0 †=R0 :

…12†

The quantity xvJ in an analytical form is calculated from the criterion for a minimum

M. Molski / Chemical Physics Letters 342 (2001) 293±298



dUvJ …x† dx

 ˆ 0;

…13†

xˆxvJ

using the Maple processor [11]. The …v; J †-dependence of the dynamical reference conformation RvJ is a result of centrifugal deformation and nonadiabatic vibrational e€ects a…b† of high order (si>0 ) responsible for the appearance of the term EvJ b…x† in the e€ective potential energy (6) and an additional v-dependence of RvJ . Taking advantage of (12) we obtain an e€ective Schr odinger equation [11] " ! X d2 vJ 2 vJ s as g BvJ 2 ‡ a0 g 1 ‡ dg sˆ1 # …EvJ vJ avJ 0 ˆ b2 ;

bvJ 0 † wvJ …g† ˆ 0;

…14†

vJ vJ avJ s>0 ˆ bs‡2 =a0 ;

…15†

1

n n bvJ n ˆ …n!† ‰d UvJ …g†=dg Šgˆ0 ;

…16†

amenable to direct solution in semi-classical [12] scheme producing eigenvalues X EvJ ˆ bvJ Yk0 …v ‡ 1=2†k ; …17† 0 ‡ kˆ0

in which Yk0 are Dunham's vibrational coecients, in analytical expressions [12], into which, we substitute fR0 ; an g ! fRvJ ; avJ n g. The eigenenergies (17) are applied to evaluate the e€ective potential (6). The coecients of radial functions (7)±(10) are directly ®tted to the measured rotational and vibration-rotational spectra using a weighted nonlinear least-square routine with weights taken as inverse squares of uncertainties of experimental data. Eigenenergies EvJ which appear in (6) are calculated numerically using the iterative procedure described in [11]. As input data we use 431 pure rotational Dv ˆ 0 up to v ˆ 4 and J ˆ 25 lines and vibration-rotational lines for Dv ˆ 1 up to v ˆ 7 and J ˆ 36 of 40 Ar1 H; 40 Ar2 H; 38 Ar1 H, 38 Ar2 H; 36 Ar1 H; 36 Ar2 H from various sources [2,3,5] and [14±19], including also unpublished data cited in [20]. To obtain the best set of parameters ®tted from the spectra, we use as criteria the minimum number of ®tted parameters consistent with a mini-

295

mum value of normalized standard deviation r^  1, a maximum value of F-statistic, and optimum values of estimated standard error ri of each ®tted parameter i and of correlation coecient cc…i; j† between parameters i and j. The results of the calculations are presented in Table 1. The uncertainty in parentheses is one estimated standard deviation in units of the last quoted digit of values of ®tted parameters. Detailed analysis of the results obtained indicates that the application of the proposed procedure enables a reduction of wave numbers of 431 pure rotational and vibration-rotational transitions of ArH‡ in six isotopic variants to 16 free adjustable radial parameters and 1 (or 2) constrained ones. For comparison, the application of the semiclassical approach by Ogilvie [21] to 331 lines, and the numerical method by Gruebele et al. [5] to 312 lines of ArH‡ spectrum provided 17 radial parameters in di€erent sets. The ®t converges within ®ve iterations. Despite the parameter t0Ar ˆ 0:180…178† being poorly evaluated, after ®ve iterations its value ¯uctuates (during additional iterations) insigni®cantly at the third decimal point in the range h 0:002; 0:002i, and does not in¯uence the value of the calculated dipole moment. In preliminary ®ts of the data, we discovered that parameter c6 ˆ 0:003…26† had a relatively small magnitude and large standard error. Consequently it was constrained to zero during the ®t. To ensure maximum signi®cance of all ®tted parameters, we constrained the parameter sAr 0 representing nonadiabatic vibrational e€ect [10] H Ar sAr 0 ˆ s0 ‡ t 0

t0H ‡ t1Ar ;

…18†

and the potential-energy parameter c9 in (7) ! 8 X De 2 9‡ 2 9‡i ci ; c9 ˆ 11 2 c0 …19† iˆ1 De ˆ 4:24 eV to conform to relation [10] De ˆ lim V …R†R!1 ˆ 4c0 1 ‡

X iˆ1

! i

2 ci

…20†

296

M. Molski / Chemical Physics Letters 342 (2001) 293±298

Table 1 Radial parametersa of ArH‡ X1 R‡ c0 cm 1 c1 c2 c3 c4 c5 c6 c7 c8 c9 t0Ar t0H t1H sH0 sAr 0 sH1 4 1 uAr 1 =10 cm uH1 =104 cm 1 R0 =1010 m r^ F =1015 lArH D 0 g0ArH g0ArD a

175700.38 (54) )1.563771 (20) 1.1933592 (45) )0.58232 (16) 0.21070 (89) )0.1163 (25) [0.0] )0.1521 (96) 0.321 (13) [)0.1177] )0.180 (178) 0.507 (12) 0.160 (34) 0.603 (42) [)0.2460] 1.042 (97) 4.16 (146) )6.28 (21) 1.2803730 (11) 0.885 4.07 2.12 (55) 0.502 (13) 0.2493 (75)

175700.16 (56) )1.563767 (20) 1.193592 (45) )0.58232 (16) 0.21066 (89) )0.1162 (25) [0.0] )0.1523 (96) 0.321 (13) [)0.1178] )0.163 (127) 0.509 (12) 0.175 (35) 0.610 (42) [0.0] 1.074 (99) 3.14 (89) )6.33 (21) 1.2803727 (10) 0.884 4.07 2.07 (40) 0.505 (12) 0.2509 (66)

Values of parameters that were held ®xed in the ®t are given in square brackets.

and the known dissociation energy De ˆ 4:24 eV of 40 Ar1 H [5]. Treating sAr 0 as a free ®tted parameter, we obtain sAr ˆ 1:73…89† which di€ers from sAr 0 0 ˆ 0:235…363† calculated from (18). Hence, we remove sAr 0 from the set of ®tted parameters and use it as a constrained one. The electric dipole moment of 40 Ar1 H‡ can be evaluated from the relationship [10,11] including parameters tia;b to represent nonadiabatic rotational e€ects X
l…R† ˆ …eR=2† tia tib zi ; …21† iˆ0

whereas for another isotopic variant using formula (2). Another way to determine it is the calculation of the rotational g-factor for 40 Ar1 H‡ using equation [10,11] X X l g…R† ˆ mp ti ml 1 zi ; …22† iˆ0 lˆa;b

and for an isotopic variant

g0 …R† ˆ mp

X X iˆ0 lˆa;b

til m0l 1 zi ‡ j2 Qmp =m0 ;

…23†

and then application (4). From relation (21) and parameters tiAr;H obtained for sAr 0 ˆ 0 we evaluate the equilibrium (R ˆ R0 ) value of the dipole moment of 40 Ar1 H l0 ˆ 2:07…40† D, whereas for sAr 0 constrained according to Eq. (18) l0 ˆ 2:12…55† D, each for the ‡ polarity ‰ ArH‡ Š . They conform acceptably with the ab initio value l0 ˆ 2:2…1† of Rosmus [7] and less precise with the experimental result l0 ˆ 3:0…6† D of Laughlin et al. [3]. From Eqs. (22) and (23) we calculate g0ArH ˆ 0:502…13† and g0ArD ˆ 0:2493…75† which di€er from the experimental values g0ArH ˆ 0:6638…34† and g0ArD ˆ 0:3295…16† about 32%. Introducing the former to (4) we obtain identical value of l0 ˆ 2:12  2:49 but the greater standard error propagating from errors of the determined rotational g-factors. Because the ratio g0ArH =g0ArD ˆ 2:015…79† agrees (within stated error) with the experimental ratio

M. Molski / Chemical Physics Letters 342 (2001) 293±298 Table 2 Dynamical reference conformation RvJ of

40

297

Ar1 H‡ X1 R‡ in the …v; J †-state

J

v

RvJ

J

v

RvJ

0 0 1 1 2 2 2 2 2 5 5 5 5 5 5 5

0 1 0 1 0 1 2 3 4 0 1 2 3 4 5 6

1.28049376 1.28048824 1.28064631 1.28064080 1.28095153 1.28094601 1.28094075 1.28093573 1.28093095 1.28278542 1.28277989 1.28277462 1.28276959 1.28276480 1.28276026 1.28275594

10 10 10 10 10 10 20 20 20 20 20 30 30 30 33 33

0 1 2 3 4 5 0 1 2 3 4 0 1 2 0 1

1.28893084 1.28892527 1.28891997 1.28891491 1.28891010 1.28890553 1.31323571 1.31323001 1.31322458 1.31321942 1.31321452 1.35508505 1.35507910 1.35507346 1.37150722 1.37150118

g0ArH =g0ArD ˆ 2:016…14† [3] we tried to ®nd a linear transformation of the type g0exp ˆ Ag0the ‡ B;

…24† g0exp

linking experimental and theoretical g0the values of the rotational g-factors. The calculated parameters A ˆ 1:322…79†

B ˆ 0:000…22†;

…25†

indicate that the experimental values of the rotational g-factors di€er from theoretical ones by the multiplicative factor A ˆ 1:322…79†. A source of this discrepancy is unknown at present. In Table 2 values of the dynamical reference conformations RvJ calculated from (13) and transformed Eq. (12) RvJ ˆ R0 …1 ‡ xvJ †

…26†

are presented. They clearly show that RvJ increases with J due to the action of centrifugal force and diminishes with v due to the nonadiabatic vibrational e€ects of high order represented by the parameter sH 1 . The obtained results show that from highly resolved infrared and microwave spectra of molecular ion ArH‡ one can derive a reliable value of electric dipolar moment which conforms acceptably with previous results. This value is evaluated from only spectra of samples measured in the absence of externally applied electric or magnetic

®eld, at the same accuracy r ˆ 0:6 D as achieved in the experiment [3]. The result obtained for ArH‡ indicate that not only magnitude but also sign of the dipolar moment may be deduced from the spectra; this information is dicult to acquire otherwise. We have a new source of meaningful information on electric properties of diatomic ions, derivable by direct inversion of highly resolved IR and MW spectra.

Acknowledgements I thank Prof. J. Konarski and Dr. J.A. Coxon for stimulating consultations, and Dr. J.F. Ogilvie for supplying data of ArH‡ including unpublished results.

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298 [6] [7] [8] [9] [10] [11] [12] [13] [14]

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