- Email: [email protected]
A comparative study of RKRV and reduced potential curves of the ground states of alkaline-earths chlorides
ARTICLE IN PRESS
Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9 www.elsevier.com/locate/jqsrt
A comparative study of RKRV and reduced potential curves of the ground states of alkaline-earths chlorides S.H. Beherea, M.D. Saksenab,, M.N. Deoc, A.S. Jadhavd a Department of Physics, Dr. Babasaheb Ambedkar University, Aurangabad – 431 004, India Spectroscopy Division, Mod. Labs., Bhabha Atomic Research Centre, Mumbai – 400 085, India c Synchrotron Radiation Section, Mod. Labs., Bhabha Atomic Research Centre, Mumbai – 400 085, India d Department of Physics, New Arts, Commerce and Science College, Ahmednagar – 414 001, India b
Received 11 June 2004; accepted 16 November 2004
Abstract In light of the recording and analyzing of high resolution FT spectrum of MgCl molecule, a comparative study of the RKRV and RP curves of the ground states of alkaline earth chloride viz. BeCl, MgCl, CaCl, SrCl and BaCl have been made using latest molecular constants. The H–H and extended Rydberg potential functions compete each other and have excellent agreement with RKRV curves for all the molecules. Because of the use of improved constants the Potential Energy curves in present study are better as compared to earlier workers. r 2005 Elsevier Ltd. All rights reserved. Keywords: Comparison of RKRV & RP curves; Alkaline-earths chlorides
1. Introduction There has been increased interest in the study of rotational structure of Gr. IIA alkaline-earths monohalides [1–22]. In particular, recently, the structure of MgCl molecule has been investigated by several research groups [1–3,8,10,11,14,21,22]. We have been successful in exciting the MgCl spectra in a hollow cathode (300 mA, 400 V). The A2 Pr X2 Sþ system of MgCl has been Corresponding author. Tel.: +91 22 25592990; fax: +91 22 25505151 & +91 22 25519613.
E-mail address: [email protected] (M.D. Saksena). 0022-4073/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.11.009
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
2
recorded on BOMEM DA8 Fourier Transform Spectrometer at an apodized resolution of 0:035 cm1 and the (0–0), (1–1) and (1–0) bands rotationally analyzed. The details of the experiment and the analysis are given in Ref. [22]. To determine the molecular constants, along with the latest data of our F.T. Spectrometer experiment we have also included the data of (0–1) and (0–2) bands [1] in a simultaneous least squares fit [22]. For drawing RKRV and reduced potential curves of the MgCl the constants thus determined have been used. The results of the comparative study of all the Gr IIA alkaline earths chlorides is presented in the following section.
2. RKRV and reduced potential curves The established procedure to calculate the turning points from the observed experimental data is Rydberg Klien Rees [23–25] usually known as RKR method. If the correction by Vanderslice [26] is included, then the method is named as RKRV method. A brief account of this method is given by Castano et al. [27]. A programme for computing the turning points is developed by Le Roy [28]. 2.1. Hulbert Hirschfelder function Several empirical potentials have also been used to draw the potential energy curves. Amongst them the Hulbert–Hirschfelder (H–H) [29] potential function gives best fit to RKR potential energy curves for a number of electronic states of various molecules. Bharate [30] has surveyed the utility of this function which is employed by different workers to calculate dissociation energy, etc. This function is UðrÞ ¼ De f½1 ex 2 þ cx3 ð1 þ bxÞe2x g,
(1)
where x ¼ x1 ðr re Þ
and x1 ¼ ðoe xe =Be Þ1=2 =re ;
c ¼ 1 þ a1 ðDe =a0 Þ2 ; a0 ¼ o2e =4Be ;
7 b ¼ 2 f½ð12 Þ ðDe ða2 =a0 ÞÞ=cg;
a1 ¼ 1 foe xe =ð6Be Þ2 g
and a2 ¼ ð54Þa1 ð23Þðoe xe =Be Þ. All constants have their usual meaning. The function, UðrÞ; is used to draw the potential energies of the molecules under study by substituting the r-values calculated from RKRV curves. 2.2. The extended Rydberg potential function Murrell and Sorbie [31] and Huxley and Murrel [32] have recently suggested a potential function which is based on the force field parameters and is similar to Rydberg potential function.
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
3
It has the form UðrÞ ¼ De De ½1 þ a1 r þ a2 r2 þ a3 r3 ea1 r ,
(2)
where r is ðr re Þ and a1 ; a2 and a3 are the constants defined through further discussions. They should not be confused with the constants appearing in H–H function described earlier. The constant a1 is determined from the solutions of the following quartic equation: De a41 6f 2 a21 4f 3 a1 f 4 ¼ 0.
(3)
The parameters f 2 ; f 3 and f 4 are called the force field parameters and are defined as f 2 ¼ 4p2 mo2e c2 f 3 ¼ ð3f 2 =re Þ½1 þ ae oe =6B2e f 4 ¼ ðf 3 =re Þ2 f15½1 þ ðae oe =6B2e Þ ð8oe xe =Be Þg usually the largest positive root of Eq. (3) gives the best results [32]. Once a1 is calculated, other parameters viz. a2 and a3 can be calculated from the following equations: a2 ¼ ð12Þ½a21 ðf 2 =De Þ a3 ¼ ½a1 a2 ða31 =3Þ ðf 3 =6De Þ. The extended Rydberg potential function was investigated in detail and was compared with Dimitreva-Zinevich potential by Bhartiya and Behere [33]. Birajdar [34] has applied this potential function for constructing the PE curves of large number of molecules. This potential function also was used to calculate the potential energies of molecules under study and were compared with RKR and H–H potentials. 2.3. Reduced potential energy curves (RPCs) This novel technique of constructing the potential energy curves was introduced by Jenc [35]. To understand the concept of RPC, let us define two parameters u and r: [This r is not to be confused with r re in extended Rydberg function described earlier]. u ¼ U=De
and r ¼ ½r ð1 er=rij Þrij =½re ð1 ere =rij Þrij ,
where rij ¼ ½re ðkDe =ke Þ1=2 =½ð1 ere =rij Þrij k ¼ 3:96 and ke is force constant. It can be easily seen that 0orij ore : The RPC satisfies the following conditions: at r ¼ re ;
r¼1
at r ¼ 0;
r¼0
at 1 ¼ re ;
r ¼ 1.
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
4
Secondly, uX0 for UX0 u¼0
for U ¼ 0
u ¼ 1 for U ¼ 1 u ¼ 1 for U ¼ De . The coordinates of the minimum of this RPC are r ¼ 1 and u ¼ 1 if the zero of the energy is taken as the dissociation limit. The RPCs are plotted as r on the x-axis and u þ 1 on y-axis and now the coordinates of minima will be r ¼ 1 and u ¼ 0: Rules obeyed by RPC (i) In general, the RPCs of different diatomic molecules do not intersect each other. (ii) The ordering of the quasi parallel RPCs is the same in the attractive and repulsive limb. (iii) With the exception of some of the special cases the left hand boundary of admissible RPC region is RPC of Li2 molecule. The right hand boundary is represented by the almost coinciding RPCs of rare gas molecules. (iv) The ordering rule: in a chemically related group of molecules (like in the present study, the monochlorides of alkaline earths) with the increasing atomic numbers the RPCs turn around the common minimum and become broadened. There are various applications of RPC method and can be found in Ref. [35]. 2.4. Computation of the potential energy curves The spectroscopic constants given in Table 1 for the ground states of BeCl, MgCl, CaCl, SrCl and BaCl have been used to calculate the potential energies and the turning points for given Table 1 Molecular constants of the ground states of BeCl, MgCl, CaCl, SrCl and BaCl molecules (in cm1 ) Constant/parameter
BeCl
MgCl
CaCl
SrCl
BaCl
Reduced mass ðmÞ oe oe xe oe ye oe ze Be ae ( r (in A)
7.165490 846.7 4.85 — — 0.7285 0.0069 1.7971
14.226871 466.08 2.00 — — 0.24561 0.00161 2.19615
18.6496606 370.210 1.3732 9:3 103 4:8 104 0.15223017 0.00079896 2.43693
27.895376 302.448 0.950 — — 0.10155929 0.0004524 2.57603
27.895377 279.89095 0.814137 1:1308 104 —
De (DE) References
36800 [43]
27145 [1]
33172 Varma [36] Berg [37] Ernst [38] Klynning [39]
33708 Schroder [41]
36840 Hafid [42]
e
8:39682 102 3:34832 104 2.68291
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
5
Table 2 Parameters of the H–H, extended Rydberg and RPC potentials for the ground states of BeCl, MgCl, CaCl, SrCl and BaCl molecules Constant/parameters
BeCl
MgCl
CaCl
SrCl
BaCl
H–H parameters C b ( 1 x ðAÞ
9:63477 102 0:649206 1.4387609
7:67863 102 0.548802 1.299572
0:200564 0.344567 1.068878
0.24234 0.298208 1.003329
0.277854 0.427783 0.937854
1
Extended Rydberg parameters ( 1 2.77942 a1 ðAÞ 1 ( 1.791889 a ðAÞ
2.43764
1.70504
1.64067
1.44750
1.28461
0.309984
0.33914
0.167708
( 1 a3 ðAÞ
1.08952
0.66313
0.343771
0.33920
0.286328
0.91761
1.409872
1.3356744
1.392095
1.379578
[43]
[1]
Varma [36] Berg [37] Ernst [38] Klynning [39]
Schroder [41]
Hafid [42]
2
( RPC parameters rij ðAÞ References
number of vibrational levels. The data for turning points then obtained is substituted in the H–H and extended Rydberg potential functions given by the equations. Prior to this the respective constants of these molecules were calculated (Table 2). The potential energies thus calculated for the same r values obtained from RKR were plotted. The comparative diagrams for different molecules are shown in Fig. 1, respectively, for BeCl, MgCl, CaCl, SrCl and BaCl molecules. The RPCs drawn for the alkaline-earths chlorides are shown in Fig. 2. Since both the parameters of RPC namely, r and u þ 1 differ from RKR viz. U and r, the RPC cannot be compared with the RKR. To study the ordering effect all the RPCs viz. for BeCl to BaCl are drawn together in Fig. 3 and the different parameters are presented in Table 2.
3. Results and discussion The H–H and extended Rydberg potential functions compete each other and have shown an excellent agreement with RKRV curves for all the molecules. The percentage deviation is hardly 3% to 5% for BeCl and CaCl molecules but it is less than 1% for MgCl, SrCl and BaCl molecules. Because of the improved constants the PE curves in present study are better compared to those from Varma et al. [36], who reported PE curves of monochlorides of Be, Mg and Ca. In case of CaCl the molecular constants of Berg et al. [37] and Ernst et al. [40] have been used which they derived from laser spectroscopic studies. Klynning and Martin [38] reported the turning points of X2 Sþ state of CaCl to which the present data of turning points match up to the
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
6
18000 16000
25000
Potential energy (cm-1)
Potential energy (cm-1)
30000
20000 15000 10000 5000
1.7
2.2
8000 6000 4000
3.2
MgCl 2
(b)
14000
12000
12000
10000
Potential energy (cm-1)
Potential energy (cm-1)
2.7
10000
0 1.6
r (Å)
(a)
12000
2000
BeCl 0 1.2
14000
10000 8000 6000 4000 2000
2.4 2.8 r (Å)
3.2
8000 6000 4000 2000
CaCl 0 1.8 (c)
3.6
SrCl 0
2.4
3
2
3.6
r (Å)
2.4
2.8 r (Å)
(d)
3.2
3.6
Potential energy (cm-1)
12000 10000 8000 6000 4000 2000 0 2.2 (e)
BaCl 2.4
2.6
2.8 3 r (Å)
3.2
3.4
3.6
Fig. 1. The RKR, H–H, extended Rydberg potential energy curves for the ground state of alkaline earths chlorides: (a) BeCl, (b) MgCl, (c) CaCl, (d) SrCl, (e) BaCl.
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9 1.8
1.7
1.7
1.6
1.6
1.5 u+1
u+1
1.5 1.4
7
1.4 1.3
1.3 1.2
1.2
1.1
1.1 BeCl
1.0 0.5
1.0
1.5
2.5
0.5
3.0
1.0
1.5 ρ
(b)
1.4
1.4
1.3
1.3
1.3
1.3
1.2
1.2
u+1
u+1
2.0 ρ
(a)
MgCl
1.0
1.2 1.1
2.5
1.2 1.1
1.1
1.1 CaCl
SrCl
1.0
1.0 0.5
(c)
2.0
1.0
1.5
2.0
0.5
ρ
1.0
1.5 ρ
(d) 1.3 1.3
u+1
1.2 1.2 1.1 1.1 BaCl 1.0 0.5 (e)
1.0
1.5
2.0
ρ
Fig. 2. The reduced potential curves for the ground states of alkaline-earths chlorides: (a) BeCl, (b) MgCl, (c) CaCl, (d) SrCl, (e) BaCl.
ARTICLE IN PRESS 8
S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9 1.8 BeCl MgCl CaCl SrCl BaCl
1.7
1.6
u+1
1.5
1.4
1.3
1.2
1.1
1.0 0.6
1.1
1.6
2.1
ρ
Fig. 3. The combined reduced potential curves of the ground states of alkaline earths chlorides: (a) BeCl, (b) MgCl, (c) CaCl, (d) SrCl, (e) BaCl.
fourth digit after decimal up to v ¼ 11th level. Rao et al. [39] reported PE curves of the ground state of CaCl to which also the present calculations match very well. For SrCl and BaCl a very scarce data is available. In case of SrCl the molecular constants derived from Doppler free laser spectroscopy by Schroder et al. [41] are used whereas for BaCl the spectroscopic constants of the ground state are taken from the work of Hafid et al. [42] who used laser induced fluorescence technique. All these calculations show that the H–H and extended Rydberg potential functions show good agreement with RKR curves. The potential energies calculations for the vibrational levels are up to 69%, 57%, 30%, 32% and 27% of respective dissociation energies of BeCl, MgCl, CaCl, SrCl and BaCl molecules. The RP curves plotted for all these molecules show a similarity to RKR curves (Fig. 2). All other observations regarding RP curves validate the rules i.e. minima at (1,0). The ordering rule is verified as shown in Fig. 3. The RP curve of lightest molecule viz. BeCl show a deep minima whereas that of the heaviest molecule BaCl shows wider curve.
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
9
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
Singh M, Ghodgaonkar GS, Saksena MD. Can J Phys 1987;65:1594–603. Singh M, Saksena MD, Ghodgaonkar GS. Can J Phys 1988;66:570–5. Bogey M, Demuynck C, Destombes JL. Chem Phys Lett 1989;155:265–8. Rostas J, Shafizadeh N, Taieb G, Bourguignon B. Chem Phys 1990;142:97–109. Leach CA, Waldeck JR, Noda C, McKillop JS, Zare RN. J Mol Spectrosc 1991;146:465–92. Zhao D, Vaccaro PH, Tsekouras AA, Leach CA, Zare RN. J Mol Specrosc 1991;148:226–42. Leach CA, Athanassions A, Tsekouras AA, Zare RN. J Mol Specrosc 1992;153:59–72. Uttam KN, Gopal R, Joshi MM. Indian J Phys B 1992;66:379–82. Launila O, Royen P, Zackrisson M. Mol Phys 1993;78:763–7. Ohshima Y, Endo Y. Chem Phys Lett 1993;213:95–100. Anderson MA, Ziurys LM. Chem Phys Lett 1994;24:381–90. Guo B, Zhang KQ, Bernath PF. 1995; 170:59–74. Tai G, Verma RD. J Mol Spectrosc 1995;170:59–74. Drakes JA. JQSRT 1995;54:1039. Colarusso P, Guo B, Zhang KQ, Bernath PF. J Mol Spectrosc 1996;175:158–71. Kaley A, Walker CL, Gerry MCL. J Chem Phys 1997;107:9835–41. Coxon JA, Dickinson CS. J Mol Spectrosc 1998;190:150–6. Bernard A, Effantin C, d’Incan J, Topuzkhanian A, Wannous G. J Mol Spectrosc 1999;195:29–44. Gutterres RF, Verges J, Amiot C. J Mol Spectrosc 1999;195:29–44. Hirao T, Pinchemel B, Bernath PF. J Mol Spectrosc 2000;202:213–22. Hirao T, Bernath PF, Fellows CE, Gutterres RF, Vervloet M. J Mol Spectrosc 2002;212:53–6. Jadhav AS. Spectroscopic studies of some diatomic molecules. Ph.D. thesis, 2002. Rydberg R. Z Phys 1933;80:514. Klien O. Z Phys 1932;76:226. Rees ALG. Proc Phys Soc London (Sect. A) 1947;59:998. Vanderslice JT, Mason EA, Maisch WG, Lippincott ER. J Chem Phys 1960;33:614–5. Castano De Juan J, Martinez E. J Chem Edn 1983;60:91. Le’Roy RJ. University of Waterloo chemical and physical research report, 1992, CP.425. Hulbert HM, Hirschfelder JO. J Chem Phys 1941;9:61–9. Bharate NS. Studies in molecular structures and spectra of diatomic molecules. Ph.D. thesis, 1998. Murrell JN, Sorbie KS. J Chem Soc Faraday Trans 1974;70:1152. Huxley P, Murrell JN. J Chem Soc Faraday Trans 1983;79:323. Bhartiya JB, Behere SH. JQSRT 1989;42:163–5. Birajdar SV. Comparative studies in diatomic molecular potential energy functions. Ph.D. thesis, 2001. Jenc F, Brandt BA. J Mol Spectrosc 1994;165:590. Varma MP, Ishwar NB, Jha BL. Ind J Pure Appl Phys 1982;20:828–9. Berg LE, Klynning L, Martin H. Phys Scripta 1980;22:216–20. Klynning LS, Martin H. Phys Scripta 1980;22:221–3. Rao TVR, Reddy RR, Reddy AS. J Mol Struct 1983;105:249–57. Ernst WE, Schroder JO, Buck U, Kesper J, Sealeman J, Berg LE, Martin H. J Mol Spectrosc 1986;117:342–54. Shroder JO, Zeller B, Ernst WE. J Mol Spectrosc 1988;127:255–71. Hafid M, Anuiot C, Verges J. Chem Phys Lett 1993;210:45. Huber KP, Herzberg G. Constants of diatomic molecules. New York: Van Nonstrand Reinhold Company; 1979.
Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9 www.elsevier.com/locate/jqsrt
A comparative study of RKRV and reduced potential curves of the ground states of alkaline-earths chlorides S.H. Beherea, M.D. Saksenab,, M.N. Deoc, A.S. Jadhavd a Department of Physics, Dr. Babasaheb Ambedkar University, Aurangabad – 431 004, India Spectroscopy Division, Mod. Labs., Bhabha Atomic Research Centre, Mumbai – 400 085, India c Synchrotron Radiation Section, Mod. Labs., Bhabha Atomic Research Centre, Mumbai – 400 085, India d Department of Physics, New Arts, Commerce and Science College, Ahmednagar – 414 001, India b
Received 11 June 2004; accepted 16 November 2004
Abstract In light of the recording and analyzing of high resolution FT spectrum of MgCl molecule, a comparative study of the RKRV and RP curves of the ground states of alkaline earth chloride viz. BeCl, MgCl, CaCl, SrCl and BaCl have been made using latest molecular constants. The H–H and extended Rydberg potential functions compete each other and have excellent agreement with RKRV curves for all the molecules. Because of the use of improved constants the Potential Energy curves in present study are better as compared to earlier workers. r 2005 Elsevier Ltd. All rights reserved. Keywords: Comparison of RKRV & RP curves; Alkaline-earths chlorides
1. Introduction There has been increased interest in the study of rotational structure of Gr. IIA alkaline-earths monohalides [1–22]. In particular, recently, the structure of MgCl molecule has been investigated by several research groups [1–3,8,10,11,14,21,22]. We have been successful in exciting the MgCl spectra in a hollow cathode (300 mA, 400 V). The A2 Pr X2 Sþ system of MgCl has been Corresponding author. Tel.: +91 22 25592990; fax: +91 22 25505151 & +91 22 25519613.
E-mail address: [email protected] (M.D. Saksena). 0022-4073/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.11.009
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
2
recorded on BOMEM DA8 Fourier Transform Spectrometer at an apodized resolution of 0:035 cm1 and the (0–0), (1–1) and (1–0) bands rotationally analyzed. The details of the experiment and the analysis are given in Ref. [22]. To determine the molecular constants, along with the latest data of our F.T. Spectrometer experiment we have also included the data of (0–1) and (0–2) bands [1] in a simultaneous least squares fit [22]. For drawing RKRV and reduced potential curves of the MgCl the constants thus determined have been used. The results of the comparative study of all the Gr IIA alkaline earths chlorides is presented in the following section.
2. RKRV and reduced potential curves The established procedure to calculate the turning points from the observed experimental data is Rydberg Klien Rees [23–25] usually known as RKR method. If the correction by Vanderslice [26] is included, then the method is named as RKRV method. A brief account of this method is given by Castano et al. [27]. A programme for computing the turning points is developed by Le Roy [28]. 2.1. Hulbert Hirschfelder function Several empirical potentials have also been used to draw the potential energy curves. Amongst them the Hulbert–Hirschfelder (H–H) [29] potential function gives best fit to RKR potential energy curves for a number of electronic states of various molecules. Bharate [30] has surveyed the utility of this function which is employed by different workers to calculate dissociation energy, etc. This function is UðrÞ ¼ De f½1 ex 2 þ cx3 ð1 þ bxÞe2x g,
(1)
where x ¼ x1 ðr re Þ
and x1 ¼ ðoe xe =Be Þ1=2 =re ;
c ¼ 1 þ a1 ðDe =a0 Þ2 ; a0 ¼ o2e =4Be ;
7 b ¼ 2 f½ð12 Þ ðDe ða2 =a0 ÞÞ=cg;
a1 ¼ 1 foe xe =ð6Be Þ2 g
and a2 ¼ ð54Þa1 ð23Þðoe xe =Be Þ. All constants have their usual meaning. The function, UðrÞ; is used to draw the potential energies of the molecules under study by substituting the r-values calculated from RKRV curves. 2.2. The extended Rydberg potential function Murrell and Sorbie [31] and Huxley and Murrel [32] have recently suggested a potential function which is based on the force field parameters and is similar to Rydberg potential function.
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
3
It has the form UðrÞ ¼ De De ½1 þ a1 r þ a2 r2 þ a3 r3 ea1 r ,
(2)
where r is ðr re Þ and a1 ; a2 and a3 are the constants defined through further discussions. They should not be confused with the constants appearing in H–H function described earlier. The constant a1 is determined from the solutions of the following quartic equation: De a41 6f 2 a21 4f 3 a1 f 4 ¼ 0.
(3)
The parameters f 2 ; f 3 and f 4 are called the force field parameters and are defined as f 2 ¼ 4p2 mo2e c2 f 3 ¼ ð3f 2 =re Þ½1 þ ae oe =6B2e f 4 ¼ ðf 3 =re Þ2 f15½1 þ ðae oe =6B2e Þ ð8oe xe =Be Þg usually the largest positive root of Eq. (3) gives the best results [32]. Once a1 is calculated, other parameters viz. a2 and a3 can be calculated from the following equations: a2 ¼ ð12Þ½a21 ðf 2 =De Þ a3 ¼ ½a1 a2 ða31 =3Þ ðf 3 =6De Þ. The extended Rydberg potential function was investigated in detail and was compared with Dimitreva-Zinevich potential by Bhartiya and Behere [33]. Birajdar [34] has applied this potential function for constructing the PE curves of large number of molecules. This potential function also was used to calculate the potential energies of molecules under study and were compared with RKR and H–H potentials. 2.3. Reduced potential energy curves (RPCs) This novel technique of constructing the potential energy curves was introduced by Jenc [35]. To understand the concept of RPC, let us define two parameters u and r: [This r is not to be confused with r re in extended Rydberg function described earlier]. u ¼ U=De
and r ¼ ½r ð1 er=rij Þrij =½re ð1 ere =rij Þrij ,
where rij ¼ ½re ðkDe =ke Þ1=2 =½ð1 ere =rij Þrij k ¼ 3:96 and ke is force constant. It can be easily seen that 0orij ore : The RPC satisfies the following conditions: at r ¼ re ;
r¼1
at r ¼ 0;
r¼0
at 1 ¼ re ;
r ¼ 1.
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
4
Secondly, uX0 for UX0 u¼0
for U ¼ 0
u ¼ 1 for U ¼ 1 u ¼ 1 for U ¼ De . The coordinates of the minimum of this RPC are r ¼ 1 and u ¼ 1 if the zero of the energy is taken as the dissociation limit. The RPCs are plotted as r on the x-axis and u þ 1 on y-axis and now the coordinates of minima will be r ¼ 1 and u ¼ 0: Rules obeyed by RPC (i) In general, the RPCs of different diatomic molecules do not intersect each other. (ii) The ordering of the quasi parallel RPCs is the same in the attractive and repulsive limb. (iii) With the exception of some of the special cases the left hand boundary of admissible RPC region is RPC of Li2 molecule. The right hand boundary is represented by the almost coinciding RPCs of rare gas molecules. (iv) The ordering rule: in a chemically related group of molecules (like in the present study, the monochlorides of alkaline earths) with the increasing atomic numbers the RPCs turn around the common minimum and become broadened. There are various applications of RPC method and can be found in Ref. [35]. 2.4. Computation of the potential energy curves The spectroscopic constants given in Table 1 for the ground states of BeCl, MgCl, CaCl, SrCl and BaCl have been used to calculate the potential energies and the turning points for given Table 1 Molecular constants of the ground states of BeCl, MgCl, CaCl, SrCl and BaCl molecules (in cm1 ) Constant/parameter
BeCl
MgCl
CaCl
SrCl
BaCl
Reduced mass ðmÞ oe oe xe oe ye oe ze Be ae ( r (in A)
7.165490 846.7 4.85 — — 0.7285 0.0069 1.7971
14.226871 466.08 2.00 — — 0.24561 0.00161 2.19615
18.6496606 370.210 1.3732 9:3 103 4:8 104 0.15223017 0.00079896 2.43693
27.895376 302.448 0.950 — — 0.10155929 0.0004524 2.57603
27.895377 279.89095 0.814137 1:1308 104 —
De (DE) References
36800 [43]
27145 [1]
33172 Varma [36] Berg [37] Ernst [38] Klynning [39]
33708 Schroder [41]
36840 Hafid [42]
e
8:39682 102 3:34832 104 2.68291
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
5
Table 2 Parameters of the H–H, extended Rydberg and RPC potentials for the ground states of BeCl, MgCl, CaCl, SrCl and BaCl molecules Constant/parameters
BeCl
MgCl
CaCl
SrCl
BaCl
H–H parameters C b ( 1 x ðAÞ
9:63477 102 0:649206 1.4387609
7:67863 102 0.548802 1.299572
0:200564 0.344567 1.068878
0.24234 0.298208 1.003329
0.277854 0.427783 0.937854
1
Extended Rydberg parameters ( 1 2.77942 a1 ðAÞ 1 ( 1.791889 a ðAÞ
2.43764
1.70504
1.64067
1.44750
1.28461
0.309984
0.33914
0.167708
( 1 a3 ðAÞ
1.08952
0.66313
0.343771
0.33920
0.286328
0.91761
1.409872
1.3356744
1.392095
1.379578
[43]
[1]
Varma [36] Berg [37] Ernst [38] Klynning [39]
Schroder [41]
Hafid [42]
2
( RPC parameters rij ðAÞ References
number of vibrational levels. The data for turning points then obtained is substituted in the H–H and extended Rydberg potential functions given by the equations. Prior to this the respective constants of these molecules were calculated (Table 2). The potential energies thus calculated for the same r values obtained from RKR were plotted. The comparative diagrams for different molecules are shown in Fig. 1, respectively, for BeCl, MgCl, CaCl, SrCl and BaCl molecules. The RPCs drawn for the alkaline-earths chlorides are shown in Fig. 2. Since both the parameters of RPC namely, r and u þ 1 differ from RKR viz. U and r, the RPC cannot be compared with the RKR. To study the ordering effect all the RPCs viz. for BeCl to BaCl are drawn together in Fig. 3 and the different parameters are presented in Table 2.
3. Results and discussion The H–H and extended Rydberg potential functions compete each other and have shown an excellent agreement with RKRV curves for all the molecules. The percentage deviation is hardly 3% to 5% for BeCl and CaCl molecules but it is less than 1% for MgCl, SrCl and BaCl molecules. Because of the improved constants the PE curves in present study are better compared to those from Varma et al. [36], who reported PE curves of monochlorides of Be, Mg and Ca. In case of CaCl the molecular constants of Berg et al. [37] and Ernst et al. [40] have been used which they derived from laser spectroscopic studies. Klynning and Martin [38] reported the turning points of X2 Sþ state of CaCl to which the present data of turning points match up to the
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
6
18000 16000
25000
Potential energy (cm-1)
Potential energy (cm-1)
30000
20000 15000 10000 5000
1.7
2.2
8000 6000 4000
3.2
MgCl 2
(b)
14000
12000
12000
10000
Potential energy (cm-1)
Potential energy (cm-1)
2.7
10000
0 1.6
r (Å)
(a)
12000
2000
BeCl 0 1.2
14000
10000 8000 6000 4000 2000
2.4 2.8 r (Å)
3.2
8000 6000 4000 2000
CaCl 0 1.8 (c)
3.6
SrCl 0
2.4
3
2
3.6
r (Å)
2.4
2.8 r (Å)
(d)
3.2
3.6
Potential energy (cm-1)
12000 10000 8000 6000 4000 2000 0 2.2 (e)
BaCl 2.4
2.6
2.8 3 r (Å)
3.2
3.4
3.6
Fig. 1. The RKR, H–H, extended Rydberg potential energy curves for the ground state of alkaline earths chlorides: (a) BeCl, (b) MgCl, (c) CaCl, (d) SrCl, (e) BaCl.
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9 1.8
1.7
1.7
1.6
1.6
1.5 u+1
u+1
1.5 1.4
7
1.4 1.3
1.3 1.2
1.2
1.1
1.1 BeCl
1.0 0.5
1.0
1.5
2.5
0.5
3.0
1.0
1.5 ρ
(b)
1.4
1.4
1.3
1.3
1.3
1.3
1.2
1.2
u+1
u+1
2.0 ρ
(a)
MgCl
1.0
1.2 1.1
2.5
1.2 1.1
1.1
1.1 CaCl
SrCl
1.0
1.0 0.5
(c)
2.0
1.0
1.5
2.0
0.5
ρ
1.0
1.5 ρ
(d) 1.3 1.3
u+1
1.2 1.2 1.1 1.1 BaCl 1.0 0.5 (e)
1.0
1.5
2.0
ρ
Fig. 2. The reduced potential curves for the ground states of alkaline-earths chlorides: (a) BeCl, (b) MgCl, (c) CaCl, (d) SrCl, (e) BaCl.
ARTICLE IN PRESS 8
S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9 1.8 BeCl MgCl CaCl SrCl BaCl
1.7
1.6
u+1
1.5
1.4
1.3
1.2
1.1
1.0 0.6
1.1
1.6
2.1
ρ
Fig. 3. The combined reduced potential curves of the ground states of alkaline earths chlorides: (a) BeCl, (b) MgCl, (c) CaCl, (d) SrCl, (e) BaCl.
fourth digit after decimal up to v ¼ 11th level. Rao et al. [39] reported PE curves of the ground state of CaCl to which also the present calculations match very well. For SrCl and BaCl a very scarce data is available. In case of SrCl the molecular constants derived from Doppler free laser spectroscopy by Schroder et al. [41] are used whereas for BaCl the spectroscopic constants of the ground state are taken from the work of Hafid et al. [42] who used laser induced fluorescence technique. All these calculations show that the H–H and extended Rydberg potential functions show good agreement with RKR curves. The potential energies calculations for the vibrational levels are up to 69%, 57%, 30%, 32% and 27% of respective dissociation energies of BeCl, MgCl, CaCl, SrCl and BaCl molecules. The RP curves plotted for all these molecules show a similarity to RKR curves (Fig. 2). All other observations regarding RP curves validate the rules i.e. minima at (1,0). The ordering rule is verified as shown in Fig. 3. The RP curve of lightest molecule viz. BeCl show a deep minima whereas that of the heaviest molecule BaCl shows wider curve.
ARTICLE IN PRESS S.H. Behere et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 1–9
9
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
Singh M, Ghodgaonkar GS, Saksena MD. Can J Phys 1987;65:1594–603. Singh M, Saksena MD, Ghodgaonkar GS. Can J Phys 1988;66:570–5. Bogey M, Demuynck C, Destombes JL. Chem Phys Lett 1989;155:265–8. Rostas J, Shafizadeh N, Taieb G, Bourguignon B. Chem Phys 1990;142:97–109. Leach CA, Waldeck JR, Noda C, McKillop JS, Zare RN. J Mol Spectrosc 1991;146:465–92. Zhao D, Vaccaro PH, Tsekouras AA, Leach CA, Zare RN. J Mol Specrosc 1991;148:226–42. Leach CA, Athanassions A, Tsekouras AA, Zare RN. J Mol Specrosc 1992;153:59–72. Uttam KN, Gopal R, Joshi MM. Indian J Phys B 1992;66:379–82. Launila O, Royen P, Zackrisson M. Mol Phys 1993;78:763–7. Ohshima Y, Endo Y. Chem Phys Lett 1993;213:95–100. Anderson MA, Ziurys LM. Chem Phys Lett 1994;24:381–90. Guo B, Zhang KQ, Bernath PF. 1995; 170:59–74. Tai G, Verma RD. J Mol Spectrosc 1995;170:59–74. Drakes JA. JQSRT 1995;54:1039. Colarusso P, Guo B, Zhang KQ, Bernath PF. J Mol Spectrosc 1996;175:158–71. Kaley A, Walker CL, Gerry MCL. J Chem Phys 1997;107:9835–41. Coxon JA, Dickinson CS. J Mol Spectrosc 1998;190:150–6. Bernard A, Effantin C, d’Incan J, Topuzkhanian A, Wannous G. J Mol Spectrosc 1999;195:29–44. Gutterres RF, Verges J, Amiot C. J Mol Spectrosc 1999;195:29–44. Hirao T, Pinchemel B, Bernath PF. J Mol Spectrosc 2000;202:213–22. Hirao T, Bernath PF, Fellows CE, Gutterres RF, Vervloet M. J Mol Spectrosc 2002;212:53–6. Jadhav AS. Spectroscopic studies of some diatomic molecules. Ph.D. thesis, 2002. Rydberg R. Z Phys 1933;80:514. Klien O. Z Phys 1932;76:226. Rees ALG. Proc Phys Soc London (Sect. A) 1947;59:998. Vanderslice JT, Mason EA, Maisch WG, Lippincott ER. J Chem Phys 1960;33:614–5. Castano De Juan J, Martinez E. J Chem Edn 1983;60:91. Le’Roy RJ. University of Waterloo chemical and physical research report, 1992, CP.425. Hulbert HM, Hirschfelder JO. J Chem Phys 1941;9:61–9. Bharate NS. Studies in molecular structures and spectra of diatomic molecules. Ph.D. thesis, 1998. Murrell JN, Sorbie KS. J Chem Soc Faraday Trans 1974;70:1152. Huxley P, Murrell JN. J Chem Soc Faraday Trans 1983;79:323. Bhartiya JB, Behere SH. JQSRT 1989;42:163–5. Birajdar SV. Comparative studies in diatomic molecular potential energy functions. Ph.D. thesis, 2001. Jenc F, Brandt BA. J Mol Spectrosc 1994;165:590. Varma MP, Ishwar NB, Jha BL. Ind J Pure Appl Phys 1982;20:828–9. Berg LE, Klynning L, Martin H. Phys Scripta 1980;22:216–20. Klynning LS, Martin H. Phys Scripta 1980;22:221–3. Rao TVR, Reddy RR, Reddy AS. J Mol Struct 1983;105:249–57. Ernst WE, Schroder JO, Buck U, Kesper J, Sealeman J, Berg LE, Martin H. J Mol Spectrosc 1986;117:342–54. Shroder JO, Zeller B, Ernst WE. J Mol Spectrosc 1988;127:255–71. Hafid M, Anuiot C, Verges J. Chem Phys Lett 1993;210:45. Huber KP, Herzberg G. Constants of diatomic molecules. New York: Van Nonstrand Reinhold Company; 1979.