Majumdar, K

Varshni, Y P

English

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SPECTRO SCO PIC CO NSTAN TS OF M OLECULES I ON T H E  GRO UND ST A T E  FREQUENCIES OF DIATOM S OF TH E TYPE X XBv K ,  M A JU M D A R  and Y .  P .  V A K S H N I Department of Ph ysic s, Auahabad University, Ai,i,4habai){Received /or publication, December ig, igiy^ )ABSTRACT, The theoretical evaluation of the (Spectroscopic couslants of molecules by quantum mechanics has been retarded due to the didicultics in calculations. In the present and the succeeding papers the authors propose to discuss the existing relations and to suggest new ones—empirical or semi-theoretical—between the spectroscopic constants of molecules. A new unit for *io* dynes/cm/’ viz. “ Des.” after the pioneer worker Deslandrc is proposed.An improved form of Matuyama-Cla'k relation connecting the ground state frequencies of diatoms (diatomic molecules) formed of the same element has been sugges­ted. The relation is log «,=g-// log where n is the total quantum number of thevalence electrons, V the ionisation potential and g and /i are constants for each group. Values for FrFr, GeGe, SnSn, PoPo and AtAt have been predicted.0 K N E R A L IN  T R 0 D V C 'i‘ T O NThe explanation of liydrogun molecule on wave mechanics by Heiller and London (1927), and the refinements by later workers, placed the valence theories on a solid foundation and it became possible to calculate theoretically the spectioscoiiic constants of Hj. However, except for the simplest cases the theoretical discussion of molecules deals with the problem of a system consisting of lather large number of particles ^electrons and uucletj. There is hardly any possibility of solving such problems rigorous­ly. To arrive at an approximate solution various factors have to be neglected. According to the type of approximation used, w*e are led to different valence theories. Often the approximations are very questionable and the results obtained can only be called ‘daring appicximations . So far as numerical niagniludes are concerned, the calculations are almost hopelessly complicated for the heavier molecules. As Loulson (1952) remarks *‘It has laughingly been said that calculating the dissociation energy of a heavy molecule is like weighing the captain of a ship by deter­mining the difference in displacement of his ship when he is, or is not, onboard/'Attempts have been made in the past to obtain empirical cprrelations between different molecular constants, such as force constant, internuclear distance etc* Some of the well-known empirical formulae are due to Brige (1925), Mecke (1925), Morse (1929), Clark and his collaborators (1934-1952),11104 K . Majumdar and Y. P , VarshniAllen and Lonj^ a^ir (1935)* Badger ( i93'1. i 935)» Huggins ^I935i ^936), Sutherland (1938, 1940), Liiinett (J940, 1942), Oordy (1946), Guggenheimer (1946), Wu and Chao (1947) and others.It is the pur]>ose of the present and the succeeding papers to discuss the existing relations and to suggest new ones“-einpiiical or seini*theoretical. The methods will be applied to the prediction of undetermined constants wherever possible.It is expected that the present method of appioaching the problem of spectroscopic constants may prove useful, in encouraging experimental and theoretical developments. From the i)ractical standpoint, some incentive may be given to the task of verification or otherwise of predicted values. Fuither, a more careful determination of such constants which are found to be enif)irically anomalous would be advisable fe.g. vibration frequencies of F F , PN, AsN etc.) It is interesting to note that often two independent empirical methods lead to the same results.On the theoretical side, as mentioned above, progress by the rigorous methods of quantum mechanics has lieen retarded by difficulties of calcula­tion, and empirical methods may, therefore, be useful. The present trend indicates that the theory will advance more along a semi-empirical, rather than a purely theoretical approach. Some of the empirical relations may serve as hand holds by which the theoretician can climb to more command­ing positions.N o  A T T N A N D  8 Y  M H () L 8Clark (1935 b) has suggested the use of the term ' ‘di-atom’ ' as a short form of the somewhat cumbersome “ diatomic molecule" and he has advanced seveial reasons in sup])ort of his suggestion. Though his sugges­tion has not gained general currency, it is found that it is very convenient and will be used.Similarly following Clark a hoinonuclear diatomic molecule will be denoted by X X  instead ot the conventional Xo. Walsh <1951) has also used the same symbol.Clark has divided the molecules into ^periods’ uhich arc designated as K K , K L , KM , L L , LM etc., the letters indicating the closed electronic shells of the component atoms. Thus CO is (KK), AlO is (K Lj etc. Tins notation has since been followed by several authors and it will also be followed here.The symbols used for the molecular constants are usually those W'hicb have been used by Herzberg in his well-known book “ Molecular Spectra and Molecular Structure" Part 1 (1950).<•>*., and CD,,3;^  are defined by—internuclear distance at equilibrium . Mvt—reduced mass in atomic weight unitsSpectroscopic Constants of Moleculesf etc. 105f e e ~  5 *^ 8  ^X fiA wf dynes/ctn.Do—dissociation enerRy for the ground slate( ) constants and symbols in i)arcntheses are uncertain.The new unit :The force constant is usually expressed in terms of lo"’ dynes/cm. It seems better to have some short symbol for this unit as the force cons tantis used very frequently. The authors suggest tlie symbol “ Des’* , after- Deslandre, the ]>ioneeer worker in Molecular Spectra. It is hoped that the suggestion will meet the approval of otlier workers in the field.D A T AThe data has been taken in most cases from the following sources :1. O. Heizbc-rg: ‘ 'Molecular Spectra and Molecular Structure”  Part 195^1 (D. Vail Nostrand Co. Inc., New York).2. O. Herzberg : ' ‘Atomic vSpectra and Atomic Structure” , 1944 (Dover Publications, New York).3. A, G. Gaydon : “ Dissociation Knergies and vSpectra of Diatomic Molecules” , 1950 (Dover Publications Inc., New York).4. B. Rosen (editor) : Dounces vSpectroscopiques Conccrriant lesMolecules Diatomiques” , 1951 (Hermann and Cie, Depositaires, Paris V e)5. W. Jevotis: “ Report on Band-Spectra of Diatomic Molecules" (Appendix II), 1932 (Cambridge University Press).6. C. D. Hodgman (editor): “ Handbook of Chemistry and Physics” , T951 (Chemical Rubber Publishing Co., Ohio).7. C. U. iMoore : “ Atomic Phiergy Levels” , 1949 (Wasliington).i n t r o d u c t i o nMatuyaina (1934,' proposed that the logarithms of the ground state vibrational frequencies of diatoms X X  formed of the same element were linear with the logarithms of the atomic weights of X  for each group in the Periodic Table. The relation may be expressed aslog to,. + p  log .4 =  a constant ... (i)where P is a constant ('^ i) for a given group. It was found however, that discrepancies occurred in the LL  period, containing NaNa, PP, SS, ClCl. It may be added that this cannot hold for the different isotopes, because, for the isotopes k e is constant i.e.4;:^ -^  r®rof =  constantor, log o)e + I log .1 =  constantwhile in (i), p i.Clark (1937) modified the above relation as follows :log log 2/ ... (2)where n and P arc constants and 2Z is the number of extra nuclear electrons (twice the nuclear charge) of X X . He showed that equation (2) was more successful than Maiuyama*s, though the anomalies persisted in L L  period.106 K . Majumdar and Y. P. VarshniA  similar relation is found to hold good if, instead of 2Z, the effective charge Z^, defined as follows, is used.Rwhere V is the ionisation potential of X , n the total quantum number of the valence electrons, and R the Rydberg constant. We can write the final 1 elation aslog log n"V  ... (3)where g and h are constants for each group. The relationship is shown graphically figure i.  To avoid, overlapping the ordinates of the points of the fourth in fifth groups have been decreased by .6. The values of the constants g  and h for the vaiious groups are given in Table I.TABUi IGroup j /r !1 hI 4.059-1 1 . 1349IV 4.9401 1.0431V 6.4827 1.7.338V and VTT 5-2353 1.2015The results are tabulated in Table II . represents the valuescalculated from equation (3) and from the Matuyrna-Clark relation (2).Fig. X. The log values for the foltha nd fifth groups have been decreased by .6  to ji void overlappingSpectroscopic Constants oj Molecules etc. 107I  (experimental data for the ionisation potentials of Francium, Polonium and Astatine are available. However, Finkelnberg and Stern (igso) and Kohlrausch ^1949) have esthiiated them Ly seini-einpirical methods. The ionisation potential of Bi has been taken from Riclitmyer and Kennard (1947). Other data have been taken from general references.T a b i .k I IGroup Molecule n V in volt 1 (3) (A>, (2) observed11L iL i ; 5 -39 ^^ 3 5 1  4 ' 3 5 1 - 1 3S I .4NaNa 1 3 5 - 1 3 8 M 7 9 1 3 6 1 5 9  2K K 4 4 31 93 ^ 9 1 . 2 92.6Rbkb 1 5 4 . 1 7 6 58 .9 5 6 .1 57-3CsCs 1 FrPr11673 893  4 - o ±  I*'4 -’3 '> 7 -f 1 .2423 0 .142TVVIVIIccvSiSiGeGevSnSnPbPbNN I ’P As As SbSb liiHi00SvSSeSeTeTePoPoPPCJClPrTlrIIAtAt11 i6.s 8 1 IQ 8 1^ 6 7 .^32 7 P 515-545m 977 in..S 8 f)A8 n"13JO.357<;-75y .0 0 78.4 ±. 3"i 6 m 3 9S7 543 38<' *’ 5 >^ 51 :\S7 420.9 2 72  3 i^ ')S-31412 73 )^ 3 3y8 3 25f>-3 200 2J (  >2f't43^ -5269 8 157-4j ^ 8^.. 4 709-8 380.2251 15^1 31641.3(750)256.52350-6780.44294269 S 1 7 2 . 71580.4725.7391.817.422 1049 TOSo32.959 56s < 604.511.844 31b 3 312.9if).44 214.9 214.69-5 ± -2" 172.9 ±4  3 1139-4(892)564.9 323'2 214.6fl—Richtmyer and Kennard (1947) . „  ^b-Finkclnburg and Stern (i95o)- Kohlrausch’.s (1949) estimate.s foi Fr. I*o and At are 4.11. 8.3 and 9.4 respectively. Richter (195°) estimates 3 6 for francium.d i s c u s s i o nIt will be observed that the discrepancies in the L L  perit^ have been much reduced. While the calculated results for NaNa, SS, ClCl are satis­factory, there are still large errors in case of SiSi and PP. However, it must be remembered that the observed value of SiSi is uncertain, in as- much as the ideiitificalioii of the bands as due to SiSi is tentative (Downic108 K . Majumdar and Y, P. Varshniand Barrow, 1947). It is also not certain that the lower state is the ground state. It is significant to note that there are strong vibrational and rota­tional pertiiihations in case of PP IHerzbeig, 1932 ; Marais, 1946). The calculated value of C)() is also much in error. The points for the V lth  and V llth  groups are seen to lie on the same straight line. No determination of the for F rl'r , GcOe, SnSn, PoPo and AtAl has been made. The pre­dicted values are given in the table. The predicted value for F'rFr is close to Clark's i)redicted value, but there is much dilTerence for PoPo and AtAt.A C K N 0 VV Iv H D C M F. N TThe authors arc thankful to the Council of Scientific and Industrial Reseaich for the financial assistance and the permission to publish the results.R  F  F  K R K N C K vSAllen, II. vS. and Longair, A. K , Phil. Maj;., 19, lus?.Hadgcr, R . M., 193.1, ]. ( hnu Phyi .^, 2, i2S. hadgei, R . M., 1935, J. Chew. Phy^ , 3 . 71c B ad ge i.R . M., 1935, Phys. R ev  , 48, .;S4,Birge, R . T.. 1925, Phys. Rev.,  25, 240.Clark, C\ II. D., 1035 a, Phys. Rev.,  47, 238.Clark, C. II. D 1935b, Phil. IMay^ ., 19, 476Clark, C. II . I) , 19^7, Trans. Fanniny ,Sne , 198, i^cjSClark, C. II. I), and Webb, K.  R ., 1941, 'I'huis. Fa>aday S < u 37, 293.Coiilsoii, C. A., 1952, Valence, (Oxfortl Fiii\ersit\ I ’ress) p. vS8.Downie, A. R. and Barrow, R. IC, 1947, Nalute,  160, i()8 Fiiikelnburg, W. and vStern, F ., 1950, Phy^. Rev  , 77, 303.Gordy, W., 1946 ].  Chem. Phys.^ 14, 305.Guggenhcinier, K. M , 19-16, Ptoc Phys. Soe., 58, 150 Heitler. W. and Ivondon, F. ig.»7. Z. Phys.  44, 445 Herzberg, G ., 1932, Ann. Physih\ 15, 077 Huggins, M. h ., 1935, /. Chan. Phys  , 3, 473.Huggins, M I/., i <)36, J .  Chan. I^hys., 4 , 30S Kohlrauscb, K . W. F ., 1949, Phys. Justtiaca,  3, 29 Linnett, J .  W., 1940, Tians. Faiaday Soe , 36, 1123.Linnett, J  W., 1942, Trans. Fataday Soc , 38, 1.Marais, E . J . ,  1946, Phys. Rev., 70, 499 Matuyaina, E ., 1934, Nature,  133, 567.Mecke, R .. 1925, Z. Phys.,  31, 709.Morse, r .  M., 1929, phys. Rev.,  34, 57.Richter A. F ., 1950. Casopisu Ccshelio Lekarnictva, 63, 280.Richtniyer, V. K . and Kennaid, K. H ., 1947, Introduction of IN'odern. Physics, (McGraw Hill Book Co. Inc ).Sutherland, G. B. B. M., 193S, Ptoc. Jnd. Acad. Set.. 9, 34:Sutherland, G. B. B. M., 1940, J ,  Chan, Phys , 8, i6i.Walsh, A. D , 1951, Ptoc. Roy. Soc  , 207, A, 13.Wu, C. K. and Chao, S. C., 1947, Phys. Rev.,  71, nS.

Spectroscopic Constants of Molecules I on the Ground State Frequencies of Diatoms of the Type XX

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Spectroscopic Constants of Molecules I on the Ground State Frequencies of Diatoms of the Type XX

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