Systematics of diatomic molecular transition moments

Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 765}774

Note

Systematics of diatomic molecular transition moments R. He!erlin *, L.A. Kuznetsova
Physics Department, Southern Adventist University, Collegedale, TN 37315, USA
Chemistry Department, Moscow State University, Moscow, Russia

Abstract Highly accurate intensity constants for bands in diatomic-molecular spectra are now available in the RADEN data bank. This paper uses them to improve earlier systematics of D(1R2) for A}X and related (0, 0) bands. The data are plotted to determine trends and wherever possible a heuristic equation is used to smooth them. Isoelectronic, dimer, hydride, and halide molecules are considered. Isoelectronic series containing a total of 28 molecules are plotted on "Z !Z ", the absolute di!erence of the atomic numbers of the two   atoms in the molecules. The data become rapidly larger as rare-gas molecules are approached, behaving in a qualitative way like the internuclear separation r . Dimers from periods R"2 and 3 behave similarly  to each other, which is evidence for periodicity in diatomic data. The ad hoc function D (1R2)"0.33#0.054cosh[1.5(C!4.2)], where 14 C46 or 7 is the group number of either atom,  describes their behavior reasonably well (C"1, 2, 13,216 or 17 in the IUPAC scheme). For unipositive dimers, D (1R2)"0.25#0.016cosh[(1.5(C!5)] tracks the behavior as well but with one outlier (C>). 16 >  neutral hydride and 11 unipositive hydride molecular data show clear periodicity when plotted versus Z #Z . The hydride data have nearly linear behavior, with increasing slopes for R"2}4, when plotted on   a function suggested by the 1/Z dependence of atomic oscillator strengths, with one major exception (CaH). The function is F "[1/(n #n )]!0.14, where n is the valence-electron count in atom i. The unipositive    G hydride data have close to linear behavior, with increasing slopes for R"2 and 3, when plotted on the similar function F "[1/(n #n )]!0.10, also with one major exception (BeH>). Clearly evident trends >   for halide molecules allow making qualitative estimates for two unknown transition strengths.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Diatomic molecules; Transition moments; Systematics; Periodicity

1. Introduction Experiments and ab initio computations have not kept up with the needs in astrophysics for data for small molecular properties, especially intensity constants [1}3]. Previous attempts to "ll the ***** * Corresponding author. Fax: 001 423 238 2349; E-mail: [email protected] 0022-4073/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 2 2 - 4 0 7 3 ( 9 8 ) 0 0 1 2 8 - 9

766

R. Hewerlin, L.A. Kuznetsova / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 765}774

gap by systematic analysis of tabulated data for main-group diatomic molecules were imprecise [4, 5]. Now, however, the RADEN data bank for intensity constants of electronic transition molecules [6}8] provides results which allow an improved understanding of the trends and even permit prediction of data.

2. Method A global theory for trends in diatomic transition moments does not exist. However, the existence of predictably periodic behavior has been thoroughly demonstrated for potential curves and for tabulated data of spectroscopic and thermodynamic properties of ground state neutral (and to a lesser extent for charged molecules) [9}13], and it is expected to exist in transition strengths. The essential step is to designate the molecules by (R , R , C , C ), where R and C are the     G H period and group numbers of the atoms. (Confusion between the R as used in this sense and the R used in RADEN can be avoided by recalling that the latter are usually enclosed in brackets, are not integers, or have no subscripts.) Then it becomes clear that for any "xed (R , Rl), rare-gas I molecules usually have low or high ground-state spectroscopic data, depending on the property, compared to the data for the other molecules. The ridges or valleys, respectively, lie along the central portions of isoelectronic sequences with 10 valence electrons [N , CO, BF, and BeNe if  (R , R )"(2, 2); MgNe, AlF, SiO, PN, CS, BCl, and BeAr if (R , R )"(2, 3) or (3, 2); P , SiS, AlCl,      and MgAr if (R , R )"(3, 3); etc.] [10}13]. Minimum or maximum data lie along series of   molecules containing rare-gas atoms and at the location of alkali-earth pairs (Be , BeMg, etc.).  ¹hermodynamic data (for a given temperature), while they may have less variation in magnitude than other properties for "xed (R , Rl), share the same periodicities [10}13]. Furthermore, I transition moments are related to internuclear separations. All this leads one to expect that transition moments for molecules near rare-gas boundaries may be higher than those for other molecules in between, and that the minima would be near the centers of the 10-electron isoelectronic sequences between the boundaries.

3. Data The RADEN data bank contains recommended intensity constants for A}X (and related) transitions based on exhaustive studies of the literature to 1996. This paper uses RADEN transition moments D(1R2), in atomic units (a e), where 1R2 is the average of the equilibrium M internuclear separations of the "rst excited and the ground states in a.u.: 1R2"(1/2)(r #r ). (1)   No data for rare-gas molecules are included. There are cases where the transition strength is evaluated at some values other than the mean value of R (entries like &&0.214 for R"2.7'') and also cases where the transition strength is essentially constant for all values of R (entries like &&1D2"0.71''). RADEN data are available from either author.

R. Hewerlin, L.A. Kuznetsova / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 765}774

767

4. Results Data for some isoelectronic sequences are su$ciently numerous that they invite "rst inspection (Table 1, which includes one or more rare-gas molecules in each sequence only as "ducial marks). Fig. 1 shows the data for the 14-electron sequence (N "14). Fig. 2 shows the data for N "22; the   Table 1 Transition moments for isoelectroninc series, by electron number and atomic-number di!erence Molecule

Transition

n 

(R , R )  

"Z !Z "  

9$ "Z1-Z2" for n "12, 14 

D(1R2)

C  BN BeO BeF> [HeNe] NaH

A % }X&>    1%}1&> A%}X&> A%}X&>

12 12 12 12 12 12

(2,2) (2,2) (2,2) (2,2) (1,2) (3,1)

0 2 4 5 8 10

9 7, 11 5, 13 (4,14)

0.34 0.46 1.68 0.38

N  NO# CO BF [BeNe] [MgHe] SiH>

b% }X&>   A%}X&> A%}X&> A%}X&>

14 14 14 14 14 14 14

(2,2) (2,2) (2,2) (2,2) (2,2) (3,1) (3,1)

0 1 2 4 6 10 13

9 8, 10 7, 11 5, 13

Na 

B%}X&> 

22 22 22 22 22 22 22

(3,3) (3,2) (3,2) (3,2) (3,2) (2,3) (2,3)

0 2 4 6 8 10 12

2.93

13 13 13 13 13 13 13 13 13

(2,2) (2,2) (2,2) (2,2) (2,2) (2,2) (2,2) (3,1) (3,1)

0 0 1 2 3 5 7 9 11

0.25 0.424 0.30 0.27 0.20 1.35

21 21 21 21 21 21 21

(3,2) (3,2) (2,3) (2,3) (2,3) (4,1) (4,1)

1 6 9 12 15 17 19

[MgNe]

A%}X&>

A%}X&>

AlF SiO PN CS BCl

A%}X&> A%}X&> A%}X&> A%}X&> A%}X&>

N>  C\  CN CO> BO BeF

A% }X&>   A% }X&>   A%}X&> A%}X&> A%}X&> A%}X&> [LiNe] [NaHe]

MgH

A%}X&> [NaNe]

SiO> CP CS>

A%}X&> A%}X&> A%}X&> [LiAr] [KHe]

CaH

A%}X&>

2.67 0.46 0.22 0.57 1D2"1.2

0.35

1.775 0.74 0.19 0.22 1D2"0.71

1.2 0.72 0.31 0.18

2.15

768

R. Hewerlin, L.A. Kuznetsova / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 765}774

Fig. 1. Transition moments for the sequence N "14  plotted on "Z !Z ". The sequence begins with N and    ends with SiH> at "Z !Z ""13; the vertical lines are at   the positions of BeNe and MgHe. (䉱) N , NO>, CO, BF,  and SiH>; (䉲 䉱 ) CN\. The transitions are all A%}X&> except for N , which has the transition b% }X&>.   

Fig. 2. Same as Fig. 1 for the 22-electron isoelectronic sequence. The series starts with Na ; has vertical lines for  the rare-gas molecules MgNe (shown), BeAr (at "Z !Z ""14), and CaHe (not shown); and goes to a last   possible member TiH> at "Z !Z ""21. (䉲) Na , AlF,    SiO, PN, CS, and BCl. The transitions are all A%}X&> except for Na , with transition B% }X&>.   

data form a rather symmetric pattern around the minimum at "Z !Z ""9. This minimum is   close to where it would be expected, at "Z !Z ""8; there the abscissa is half-way between the   rare-gas boundaries and there the molecule PN is isovalent to N . If the 04"Z !Z "45 data in    Fig. 1 are re#ected through the D(1R2) axis, then the resulting necessarily symmetric pattern strongly resembles the pattern of all but the upper left point in Fig. 2. Fig. 3 shows the patterns for N "14 (Fig. 1) and also for N "12, both superimposed with their vertices at "Z !Z ""9 on     the 44"Z !Z "414 portion of Fig. 2 (N "22). These "gures lead to Inference 1: D(1R2) for    these even-numbered series have minima at C "C (if R "R ) or C !C "1 [at PN, if       (R , R )"(2, 3) or (3, 2)] and rise as the nearest rare-gas molecule &&boundaries'' are approached.   The one counter-example to Inference 1 is BeF>. In Fig. 1, the datum for SiH> is consistent with a portion of a new cycle beyond MgHe for "Z !Z "'10; in Table 1, the datum for NaH is suggestive of a new portion beyond HeNe. The   general trend is similar to that seen for ground-state internuclear separations r ; however, the  correlation between D(1R2) and either r or 1R2 is poor.  Table 1 also includes two similar-transition series with odd numbers of electrons: N "13  (Fig. 4) and 21 (Table 1). These radicals share with the previously discussed molecules the large transition-strength values near rare-gas boundaries. However, the data in between the large values do not increase as smoothly and the minimum datum for N "13 is not discernable. Table 1 does  not include a series of radicals with N "15, all having transition B*!X%>. The data in  between the large values for this series also do not increase as smoothly as do those for molecules with even numbers of electrons. The data for N (Fig. 1), Na (Fig. 2), C (Fig. 3), and N> and C\ (Fig. 4) invite an analysis of      neutral and unipositive dimer molecules. The data are given in Table 2 and shown in Figs. 5 and 6.

R. Hewerlin, L.A. Kuznetsova / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 765}774

Fig. 3. Superposition of Fig. 1 going both ways from "Z !Z ""9 in the domain 44"Z !Z "414 of Fig. 2.     The SiH> datum of Fig. 1 and the Na datum of Fig. 2 do  not appear. All but one datum [BeF> (ordinate 0.38)] for N "12 are also superposed: (䊏) C , BN, BeO, and NaH.   The transitions are all A%}X&> except for C ,  A% }X&>.  

769

Fig. 4. Same as Fig. 1 but for the sequence N "13, which  starts with N>, has vertical lines for the rare-gas molecules  LiNe and NaHe, and goes to a last possible member MgH at "Z !Z ""11. (䊏) N>, CN, CO>, BeF. and MgH; (䉲)    C\. The transitions are all A%}X& except that the  homonuclear species both have additional symmetry elements as in A% }X&>.  

Table 2 Transition moments of neutral and unipositive dimer species, by atomic period and group numbers Period

Group

Dimer

Transition

D(1R2)

Dimer

Transition

D(1R2)

2

1 3 4 5 6 7

Li  B  C  N  O 

A&>}X&>   A&\}X&\   A% }X&>   b% }X&>   B&\}X&\  

3.34 0.76 0.34 0.46 0.77

Li> 

1%}X&>

1.9

Na  Al  Si  S 

A&>}X&>   A&\}X%   *&}b% B&\}X&\  

3.86 0.36 0.51 0.97

B&>}X%   A% }X&>   B% }X%   A% }X%   1%}X&> A%}X&

0.655 0.25 0.3 0.41

1 3 4 6 7

C>  N>  O>  F>  Na>  Al>  Cl> 

A% }X%  

1D2"0.27

3

2.04 0.2

From these data we obtain Inference 2: For the various resonance transitions of neutral and unipositive dimers, D(1R2) has a minimum when the atoms are from group 4 or 5 and rises somewhat symmetrically toward group 1 and perhaps group 7 dimers. The curve described by the ad hoc function D (1R2)"0.33#0.054cosh[1.5(C!4.2)] 

(2)

770

R. Hewerlin, L.A. Kuznetsova / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 765}774

Fig. 5. Neutral homonuclear diatomic molecules from period 2 (Li , B , C , N , and O , shown by 䊏) and period      3 (Na , Al , Si , and S , 䉲). The curve is given by Eq. (2).     The transitions vary along the sequences.

Fig. 6. Unipositive homonuclear diatomic molecules from period 2 (Li>, C>, N>, O>, and F>, 䊏) and period 3 (Na>,       Al>, and Cl>, 䉲). The curve is given by Eq. (3). The   transitions vary along the sequences.

"ts the data in Fig. 5 reasonably well, as can be seen in Fig. 7. The somewhat similar curve given by D (1R2)"0.25#0.016cosh[(1.5(C!5)] (3) > "ts the data in Fig. 6 fairly well, as can be seen in Fig. 8, with one outlier (C>). The two functions  di!er only in translations on both axes and in the coe$cient of the Cosh. The trends of Inference 2 are remarkable because the electronic transitions between the lowest excited states and the ground states go between di!erent terms as the abscissa changes (Table 2). The appearance of NaH in Fig. 3 and MgH in Fig. 4 lead to a search for a trend among hydrides. This data are shown in Table 3 and it is clear that for the transitions considered we can make Inference 3: For neutral and unipositive hydrides with a given period, D(1R2) decreases with increasing atomic number of the atom bonded to hydrogen in spite of the fact that the data represent transitions having alternating multiplicities and di!erent angular momentum numbers ". Figs. 9 and 10 show the obvious periodicity when all the data are plotted on Z #Z . If the points   are translated on the Z #Z axes, they fall close together into curves that resemble hyperbolae.   But these curves do not have good 1/(Z #Z ) behavior which would mimic the 1/Z dependence of   oscillator strengths common to many atomic/ionic isoelectronic sequences [14]. Surprisingly linear trends for D(1R2) do result by heavy-handedly plotting the neutral hydride data on F "[1/(n #n )]!0.14, (4)    where n is the number of valence electrons of atom i, as the independent variable (Fig. 11). The G discrepant datum near abscissa 0.2 is CaH. Fig. 12 shows the same for ionic species except that the independent variable is F "[1/(n #n )]!0.10. (5)    The discrepant datum near abscissa 0.4 is BeH>; unfortunately, there is no datum for MgH> with which to evaluate the discrepancy.

R. Hewerlin, L.A. Kuznetsova / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 765}774

Fig. 7. The Data of Fig. 5 plotted versus Eq. (2).

771

Fig. 8. The Data of Fig. 6 plotted versus Eq. (3).

Table 3 Transition moments for "xed-row neutral and unipositive hydrides, by atomic period number and number of valence electrons n #n D(1R2)  

Ion

Transition

Z #Z  

n #n D(1R2)  

4

2

1.35

A%}X&> A*}X% A%}X&\ A&>}X%

6 7 8 9

4 5 6 7

0.58 0.3 0.22 0.11

BeH> BH> CH> NH> OH> HF>

A&>}X&> A%}X&> A%}X&> A&>}X% A%}X&\ A&>}X%

5 6 7 8 9 10

2 3 4 5 6 7

1.17 0.34 0.23 0.15 0.11 0.044

3 NaH MgH AlH SiH PH SH

A&>}X&> A%}X&> A%}X&> A*}X% A%}X&\ A&>}X%

12 13 14 15 16 17

2 3 4 5 6 7

2.1 1.2 0.826 0.27 0.23 0.12

AlH> SiH> PH> SH> HCl>

A%}X&> A%}X&> A*}X% A%}X&\ A&>}X%

14 15 16 17 18

3 4 5 6 7

0.623 0.35 0.214
0.15 0.098

4 KH CaH GeH SeH

A&>}X&> A%}X&> A*}X% A&>}X%

20 21 22 23

2 3 5 7

2.7 2.15 0.33 0.09

R Neutral

Transition

2 LiH

A&>}X&>

BH CH NH OH

Z #Z  

1D2"2.15.
0.214 for R"2.7.

Fixed column hydride data are presented in Table 4; they lead to Inference 4: For neutral and unipositive species consisting of group I, II, or III hydrides, D(1R2) increases with increasing period for the atom bonded with the hydrogen. This qualitative trend has already been seen in Figs. 9 and 10. (The terms in these comparisons are all the same except for additional symmetry elements pertaining to H .) 

772

R. Hewerlin, L.A. Kuznetsova / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 765}774

Fig. 9. Neutral hydride data plotted on Z #Z . The   points go from left to right in the same order as listed in Table 3.

Fig. 11. Neutral hydrides plotted against F . (䉲) (from right  to left) LiH, BH, CH, NH, OH; (䊏) NaH, MgH, AlH, SiH, PH, SH; (䉱) KH, CaH, GeH, SeH. The transitions vary along the series. The lines are eyeball "ts to the data.

Fig. 10. Ionized hydride data plotted on Z #Z . The   points go from left to right in the same order as listed in Table 3.

Fig. 12. Positively ionized hydrides plotted against F .  (䉲) (from right to left) BeH>, BH>, CH>, NH>, OH>, HF>; (䊏) AlH>, SiH>, PH>, HS>, HCl>. The transitions vary along the series. The lines are eyeball "ts to the data.

Among halogens, the consistent behavior shown in Table 5 leads to Inference 5: for a given non-halide atom, D(1R2) decreases as the halogen group number increases, and to Inference 6: for a given halogen, D(1R2) increases as the period of the other atom increases. The data for CaCl and SrCl are not necessarily in con#ict with this inference because D(1R2) is being compared with 1D2. It may be estimated that D(1R2) and 1D2 for both MgF and AlBr will be close to 2.0 and 1.3, respectively. Although RADEN has data for many oxide molecules, no trend has as yet been determined. Neither ionized molecules nor homonuclear species have data which show consistent di!erences compared to other data.

R. Hewerlin, L.A. Kuznetsova / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 765}774

773

Table 4 Transiton moments for "xed-column neutral hydrides, by group and period number of atom to which hydrogen is bonded C 

R 

Molecule

Transition

D(1R2)

C 

R 

Molecule

Transition

D(1R2)

1

1 2 3 4

H  LiH NaH KH

B& }X&   A&>}X&> A&>}X&> A&>}X&>

1.27 1.35 2.1 2.7

4

2 3

CH SiH

A*}X% A*}X%

0.3 0.27

5

2 3

NH PH

A%}X&\ A%}X&\

0.22 0.23

2

3 4

MgH CaH

A%}X&> A%}X&>

1.2 2.15

6

3

2 3

BH AlH

A%}X&> A%}X&>

0.58 0.83

2 3 4

OH SH SeH

A&>}X% A&>}X% A&>}X%

0.11 0.12 0.09

Table 5 Transition strengths of halogen molecules, by period and group number of halogenated atom Group

Transiion

Halogenated atom

Halogen atom F

2

3

Cl

Br

I

1D2"1.76 1D2"1.87

A%}X&> A%}X&> A%}X&> A%}X&>

Be Mg Ca Sr

1.35 1D2"2.34 1D2"2.45

2.26 1D2"2.2

1D2"1.9 1D2"2.1

A%}X&> A%}X&>

B Al

1D2"1.2 1.775

1D2"0.71 1.367

1D2"0.71

References [1] Smith PL. Atomic and molecular data required for ultraviolet and optical astrophysics. Cambridge, MA: Harvard College Observatory, 1989. [2] Jorgensen UG. Rev Mexicana de Astron y Astro"s 1991;23:49}62. [3] Grevesse N, Noels A, Sauval AJ. Laboratory and astronomical high resolution spectra. ASP Conf Ser 1995;81:74}87. [4] He!erlin R. JQSRT 1976;16:1101. [5] He!erlin R. JQSRT 1978;19:335. [6] Kuznetsova LA, Pazyuk EA, Stolyarov AV. The data bank RADEN. Proc Colloq 146 Int Astronomical Union, May 24}19. Copenhagen, Denmark: Niels Bohr Institute and Nordita, 1993:231}7. [7] Kuznetsova LA, Pazyuk EA, Stolyarov AV. Russ J Phys Chem 1993;67(11):2046}9. [8] http://www.chem.msu.ru/eng/raden/index.html. [9] Jenc\ F. Int Rev Phys Chem 1996;15:467}523. [10] He!erlin R, Campbell R, Gimble D, Kuhlman H, Cayton T. JQSRT 1979;21:315,337. [11] He!erlin R, Innis W. JQSRT 1983;29:97}112.

774

R. Hewerlin, L.A. Kuznetsova / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 765}774

[12] He!erlin R. Periodic systems of molecules and their relation to the systematic analysis of molecular data. Lewiston, NY: Edwin Mellen Press, 1984:xxiii}xxxiv and 134}329. [13] He!erlin R, Zhuvikin G, Caviness K, Duerksen P. JQSRT 1984;32:257}68. [14] Smith MW, Wiese WL. Astrophys J Suppl 1971;23 Suppl 196:103}92.