Electric dipole moment function of H35Cl

Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 35} 40

Electric dipole moment function of HCl F. Kiriyama *, B.S. Rao , V.K. Nangia
Department of Physics, University of North Dakota, Grand Forks, ND 58202-7129, USA
Division of Science and Mathematics, University of Minnesota Morris, Morris, MN 56267, USA Received 21 July 1999; accepted 19 May 2000

Abstract The radial SchroK dinger wave equation was solved numerically using an anharmonic potential function with 11 parameters to generate wavefunctions and vibration}rotation energy levels of the hydrogen chloride molecule HCl. Measured line strengths available in the literature cover transitions up to v"7. Using the measured values of the line strengths along with the computed wavefunctions, a set of 135 dipole moment matrix elements MTY(m) were calculated. Finally, the dipole moment coe$cients, M , M ,2, M , were     evaluated from the matrix elements.  2001 Elsevier Science Ltd. All rights reserved.

1. Introduction During the past few decades, extensive study of the hydrogen chloride molecule HCl (hereafter HCl) has been conducted due to demands for precise understanding of this molecule in "elds such as astrophysics, meteorology, physical chemistry, and environmental science. HCl is found in terrestrial and stellar atmospheres and in the interstellar medium. In particular, HCl, resulting from various chemical reactions in the earth's atmosphere, is considered to be one of the important molecules in the mechanism of ozone depletion. In addition, the molecule plays an important role in the operation of the near-infrared chemical laser. The dipole moment of a diatomic molecule depends on its internuclear distance r and is conveniently expanded in a power series about the equilibrium internuclear distance r in the  form M(r)" M mL, L L * Corresponding author. E-mail address: [email protected] (F. Kiriyama). 0022-4073/01/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 0 ) 0 0 0 5 9 - 5

(1)

36

F. Kiriyama et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 35}40

where m"(r!r )/r and n"0,1,2,2. Its value at a given internuclear distance cannot be   measured directly. However, it can be obtained from the dipole moment matrix elements MTY(m) T which are given by MTY(m)"1v, J"M(r)"v, J2" M 1v, J"mL"v, J2, (2) T L L where v and J are the vibration and rotation quantum numbers, "v, J2 and "v, J2 are the state vectors of the lower and upper vibration}rotation levels respectively, and m"J#1 for the R-branch and !J for the P-branch. The dipole moment matrix elements can be calculated from measured line strengths STY(m) using the relation T 4pN T( uTY(Y"m" "MTY(m)", STY(m)" T T 3c (2J#1) T(

(3)

where N is the number density of molecules in the state (v, J) and uTY(Y is the frequency of the T( T( transition between "v, J2 and "v, J2 expressed in wavenumbers (cm\). Many investigators have reported line strengths for vibration}rotation bands up to v"7. In this investigation, the line strengths of the 1-0 band reported by Pine et al. [1], the 2-0 band by Toth et al. [2], the 3-0 band by Ogilvie and Lee [3], and the 4-0 through 7-0 bands by Zughul [4] were chosen for calculating the dipole moment matrix elements. From these matrix elements, the electric dipole moment coe$cients, M , M ,2, M , were evaluated and compared with those obtained    earlier by Ogilvie and Lee [3] and Kobayashi and Suzuki [5].

2. Methods of calculation Physical constants used throughout the present work were obtained from Cohen and Taylor [6]. The masses of the constituent atoms were obtained from [7]. The radial wavefunctions R (r) were calculated by numerically solving the radial SchroK dinger T( wave equation




 



dR J(J#1) 2k 1 d r T( # ! # (E !;(r)) R "0, T( dr r

 T( r dr

(4)

where k is the reduced mass of the molecule. An anharmonic potential function






;(r)"hc a m 1# a mG , (5)  G G where c is speed of light and h is Planck's constant, was chosen. The equilibrium internuclear distance was calculated from the equilibrium rotational constant h . B "  8pckr 

(6)

F. Kiriyama et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 35} 40

37

Solutions to Eq. (4) were obtained using the method described by Loucks [8] that employs a fourth order Runge}Kutta method in conjunction with the Milne method. This approach, being iterative, requires starting values of E and ;(r). The values for E were obtained from the standard T( T( Dunham formula [9] for rovibrational energies E " > (v#)J[J(J#1)]H, T( JH  JH

(7)

and the initially chosen values of the parameters a in Eq. (5) were taken from the earlier work of G Zimprich [10]. The equilibrium rotational constant B and the Dunham coe$cients > needed in  JH Eqs. (6) and (7) were obtained from Clayton et al. [11]. The internuclear distance between a point r near the origin and a relatively far o! point r was   divided into small intervals, each of step size *r"0.007a where a is Bohr radius. The point r was chosen to be at 6.5a where the inter-atomic interaction is treated as negligible. With the  values of E and a chosen as stated above, R (r) was calculated at each grid point. To test the T( G T( continuity of R (r) and its derivative, two values of its logarithmic derivative were calculated at r , T(  one proceeding from r to r and the other from r to r . If the di!erence between these two values     exceeded an imposed criterion of 10\, the calculation was repeated by adjusting the E . This T( process was continued until the criterion was satis"ed. The value of E obtained in this manner T( was treated as an `eigenvaluea corresponding to the chosen values of a . A number of such energy G `eigenvaluesa was calculated, one for each pair of v and J. Transition frequencies calculated from these `eigenvaluesa were compared with the measured frequencies. If the measured and computed frequencies did not match, the potential parameters were changed and the procedure described above was repeated. The parameter adjustment was continued until the agreement between the two sets of frequencies, computed and measured, was within the accuracy of the experimental values. Adjustment of this large set of parameters was carried out utilizing the numerical routine DUMINF from the International Mathematical and Statistical Libraries (IMSL). This routine, based on a quasi-Newton algorithm, minimizes a function of N variables without the need of a user-supplied gradient. Details of this method are given by Kluckner [12]. The dipole moment coe$cients M were extracted from a least-squares "t of the right-hand side L of Eq. (2) to experimental values of the matrix elements MTY(m) by minimizing the expression T





MTY(m)! M 1v, J"mL"v, J2 T L L TY(YT(



.

(8)

The coe$cients, M , M ,2, M , were then obtained from a total of 136 vibration}rotation matrix    elements. The value 1.108 57 D of matrix element M (0) reported by Ogilvie and Tipping [13] was  used in the evaluation of the above expression. The usual convention for the signs of the matrix elements (#!#!!#!) as adopted by other investigators [3,5] was followed in the present calculation. In the past, other investigators determined these coe$cients based solely on rotationless dipole moment matrix elements. For purpose of comparison with their results, a similar calculation using rotationless matrix elements obtained in this work was carried out. Standard errors in dipole moment coe$cients were estimated using the method discussed in detail by Pugh and Winslow [14].

38

F. Kiriyama et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 35}40 Table 1 Potential parameters of HCl obtained in the present investigation (a in cm\; a , a ,2, a dimensionless)     a  a  a  a  a  a 

211 746.8 !2.360 605 3.644 507 !4.812 118 5.979 798 !6.223 529

a  a  a  a  a 

3.461 250 !0.120 547 6 0.058 869 34 !0.225 831 9 !0.004 801 804

3. Results and discussion The anharmonic potential parameters a determined by solving the radial SchroK dinger wave G equation are listed in Table 1. Su$ciently accurate results were obtained using potential parameters ranging from a to a .   With this "nal set, a large number of energy eigenvalues and associated wavefunctions were generated, which in turn provided the requisite number of transition frequencies. It was found that the calculated frequencies agreed with the measured ones within the standard deviation in the experimental data. These data have been measured by Clayton et al. [11] for the 1-0 band, Rank et al. [15] for the 2-0 band, Guelachvili et al. [16] for the 3-0 band, and Zughul [4] for the 4-0 through 7-0 bands. In addition, the orthogonalities of the wavefunctions were checked. These two criteria, namely the agreement with experimental frequencies and the check on orthogonalities led us to accept these potential parameters as the best choice. Using the frequencies obtained from energy eigenvalues and the measured line strengths, dipole moment matrix elements were evaluated. These matrix elements were "tted to a polynomial in m for each band. The degree of the polynomial was determined using Gauss criterion [17]. The rotationless matrix elements (which cannot be obtained from experimental data) as well as rovibrational matrix elements were determined from these polynomials for various values of m. Both vibration}rotation and rotationless matrix elements for each band are presented in Table 2. The dipole moment coe$cients obtained from all the 136 matrix elements from seven bands are listed in column 1 and those found using only eight rotationless matrix elements are listed in column 2 of Table 3. For comparison, the results of Ogilvie and Lee [3] and Kobayashi and Suzuki [5] are also included in columns 3 and 4, respectively. The number listed in parentheses is the standard error in the last digits of a given value. The four dipole moment functions presented in Table 3 are plotted against the internuclear distance in Fig. 1. From the evaluation of the radial wavefunctions, the classical turning points of the vibrational level v"7 were found to be roughly 0.8 and 2.0 As . Thus, the dipole moment functions evaluated in the present work as observed from Fig. 1 are most reliable in the region between these two limits. Within the range of standard uncertainties in each coe$cient, the last three sets in Table 3 are in reasonable agreement. It may be noted that the coe$cients, M ,2, M , obtained from 136 matrix  

F. Kiriyama et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 35} 40

39

Table 2 Matrix elements of electric dipole moment of HCl in units of D m

"M (m)";10 "M (m)";10 "M (m)";10 "M (m)";10 "M (m)";10 "M (m)";10 "M (m)";10       

!10 !9 !8 !7 !6 !5 !4 !3 !2 !1 0 1 2 3 4 5 6 7 8 9 10 11 12

0.71877 0.71026 0.70179 0.69336 0.68497 0.67662 0.66830 0.66003 0.65179 0.64359 0.63542 0.62730 0.61921 0.61116 0.60315 0.59518 0.58725 0.57935 0.57150 0.56368 0.55590 0.54815 *

* * 0.73181 0.72626 0.72093 0.71585 0.71103 0.70649 0.70226 0.69837 0.69483 0.69167 0.68890 0.68656 0.68466 0.68323 0.68228 0.68185 0.68195 0.68261 0.68385 0.68569 *

* * 0.59557 0.58186 0.57157 0.56438 0.55999 0.55807 0.55832 0.56043 0.56409 0.56898 0.57480 0.58123 0.58796 0.59468 0.60108 0.60685 0.61168 0.61525 0.61726 0.61739 0.61533

* * * 0.28468 0.28629 0.28844 0.29108 0.29417 0.29768 0.30157 0.30580 0.31034 0.31514 0.32018 0.32541 0.33080 0.33631 0.34191 0.34754 0.35319 0.35881 0.36436 *

* * * 0.80718 0.79781 0.79697 0.80217 0.81128 0.82251 0.83439 0.84579 0.85591 0.86429 0.87081 0.87567 0.87943 0.88296 0.88747 0.89451 0.90597 0.92406 * *

* * * 0.58365 0.59607 0.60802 0.61956 0.63077 0.64170 0.65242 0.66300 0.67350 0.68399 0.69453 0.70518 0.71601 0.72709 0.73848 0.75024 0.76244 0.77514 0.78842 *

* * * * 0.28186 0.29014 0.29740 0.30409 0.31058 0.31713 0.32393 0.33106 0.33852 0.34622 0.35398 0.36152 0.36847 0.37439 0.37871 0.38081 * * *

Table 3 Electric dipole moment coe$cients of HCl in units of D Our results

M  M  M  M  M  M  M  M 

Ogilvie and Lee [3]

Kobayashi and Suzuki [5]

Using 136 matrix elements

Using 8 matrix elements

Using 8 matrix elements

Using 10 matrix elements

1.095 056(1) 1.076 259(152) !0.001 429(155) !1.252 768(94) !0.383 620(52) 0.104 152(100) !0.223 482(1 828) 2.024 332(4 263)

1.095 146(303) 1.075 400(1 443) 0.004 933(12 601) !1.252 165(36 023) !0.412 571(134 096) !0.009 595(248 292) !0.594 751(345 169) 2.480 779(846 447)

1.093 004(75) 1.236 14(44) 0.020 63(5 30) !1.531 8(13 1) !0.918 80(26 38) !0.351 9(42 5) !0.436 7(72 5) 0.351 3(206 8)

1.092 04(55) 1.249 9(11 0) 0.021 44(26 00) !1.658(76) !0.949 5(180 0) 0.227 4(280 0) !0.464 5(680 0) !2.276(2 200)

40

F. Kiriyama et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 35}40

Fig. 1. Electric dipole moment function of HCl (I, present work with 136 matrix elements; II, present work with 8 rotationless matrix elements; III, Ogilvie and Lee [3]; IV, Kobayashi and Suzuki [5]).

elements in the present work have smaller errors compared to those obtained from rotationless matrix elements only. This emphasizes the need to incorporate the rovibrational interaction to obtain a reliable dipole moment function. When additional experimental data on higher vibration}rotation transitions and line strengths become available, a more accurate dipole moment function valid over a larger range of internuclear distance can be obtained using the procedure described in the present investigation.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Pine AS, Fried A, Elkins JW. J Mol Spectrosc 1985;109:30. Toth RA, Hunt RH, Plyler EK. J Mol Spectrosc 1970;35:110. Ogilvie JF, Lee Y-P. Chem Phys Lett 1989;159:239. Zughul MB. Dirasat 1985;XII:51. Kobayashi M, Suzuki I. J Mol Spectrosc 1986;116:422. Cohen ER, Taylor BN. Phys Today 1994;47:9. Lide DR Editor-in-Chief. CRC handbook of chemistry and physics. 76th ed. Boca Raton, FL: CRC Press, 1995}1996. Loucks TL. Augmented plane wave method. New York: W.A. Benjamin, 1967. p. 56. Dunham JL. Phys Rev 1932;41:721. Zimprich TA. MS Thesis, University of North Dakota, 1989. Clayton CM, Merdes DW, Pliva J, McCubbin Jr TK, Tipping RH. J Mol Spectrosc 1983;98:168. Kluckner MH. PhD Dissertation, University of North Dakota, 1979. Ogilvie JF, Tipping RH. JQSRT 1985;33:145. Pugh EM, Winslow GH. The analysis of physics measurements. Reading, MA: Addison-Wesley, 1966. Rank DH, Rao BS, Wiggins TA. J Mol Spectrosc 1965;17:122. Guelachvili G, Niay P, Bernage P. J Mol Spectrosc 1981;85:271. Worthing AG, Ge!ner J. Treatment of experimental data. New York: Wiley and Sons, 1943. p. 265.