Murthy, N Sreedhara

Tawde, N R

English

IACS Institutional Repository

ON THE CALCULATIONAL PROCEDURES IN THE  DERIVATION OF THE VIBRATIONAL TRANSI­TION PROBABILITIES OF TH E FIRST  NEGATIVE (b‘S-,-»a‘i g  BANDS OF OJN. B. TAWDE AND N. SBEEDHARA MURTHYD epaktm ent op P hysics, K a# natak U n iv ersity , D hahwah {UecGivcd for publication, November 18, 1957)ABSTRACT. HeconlJy, R ao  and R anude have published some calculations of vibrational tra n sitio n  p robabilities of Ihu bands of O 2+ (first negative) system  A study  of th a t  paper lias rovoalod tw o very serious errors m  the  calculations and in  the  usage of th e  motliods apjiliod by  th e  authors The flis t error is duo to  th e  w rong usage of m a th em atica l tebliiuque o f M annoback for th e  calculation of v ib ra tiona l transitiona l jirobabiljtioa, and th e  o ther error iS 'iii th e  in to i'p retation  and  d.orivation of tran sitio n  p robabilities by the appIication\ of the m ethod  ol M annoback and R ahm an.Rao and Raiiado (R & R) (1957,) gavt^ calculations of what, according to them, are transition moments and vibrational transition probf^hilities of the hands of the first negative system of -> a^n j. They-have apphed the methods of(i) Manneback (1951) and (ii) Manneback and Rahman (1954) to compute these values. These data have been studied by them m terms of the available values of transition probabilities of Jarmain, Fraser and Nicholls (1955) bn this system As a result of this comparative study, they have emphasised the earlier conclusions of Shuler (1950, 1952) on OH (.4—fX) system, th a t howsoever one may take the mechanical anliarmonicity of the wave function into account in any theory of vibrational transition probability, the contribution at the same time, of the variation of electronic transition moment with internuclear separation R, could not be neglected, as it  is an important factor for consideration.The application of the theory of Mannoback and Rahman to such problems requii'es the knowledge of the exact relation of electronic transition moment with R  for the particular band system. For without it, the contribution of this varia­tion to the vibrational transition probability in relation to th a t of inechanicaJ anharmonicity cannot bo judged. To our knowledge, neither this relation, nor the experimental vibrational intensities for any of the bands of this system, from which this relation could be obtained, are on record. This aspect aroused our interest in the ealculations of (R & R). A careful scrutiny of their paper brought out certain very fundamental points and issues which nullify their results and the conclusions based on them. I t  was therefore thought imperative to put these findings on record, in order to see the work of (R & R) in its proper perspective,■ 1 6  ■ • •128Derivation of the Vibrational Transition Probabilities, etc. 129I t  has been shown already by Tawde, Patil, Sreodhara Murtljy and K atti (1957) in the case of OV\{A—>X) system that Hutchisson’s and Manncback’s analytical treatments, although dilfeniig in complexity, lead to exactly identical results. This test has been applied here to the first negative band system of Og"*" under consideration, employing the same constants as those used by Rao and Ranado (1957). The transition probabilities are calculated by Hiitchissoii's and Mamieback’s methods and these results are presented in Table 1 m columns 2 and 3 Por c.omparison, the values of (R & R) by Maniieback’s method are reproduced side by side in column 4.TABLE ICalculated transition probabilities.0 .2+(Fii'st negative) 6^ 2^ ,Btuidv', v"Hut.ohiHson (p]oaont Q.UtllOVH)M aiuiebaok(fire.sentaulhors)M aiinobark (R & R)0,0 0  2.f).5 0.255 0 2550 ,1 0 373 0.373 0 3230,2 0 .24(i 0.246 0.1810,3 0 006 0 .096 0.0501 ,0 0.323 0 323 0.3731 , 1 0 033 0.033 0 0331 , 2 0 092 0 092 0 0821,3 0 .25H 0 258 0.1052 ,0 0 228 0.228 0..3002 , 1 0 050 0 050 0,0612 ,2 0.159 0 150 0 1642,3 0 OOO3 0 0003 0 OOO23,0 0.118 0.118 0.1753,1 0 . J 68 0  168 0.2263,2 0 009 0.009 0 0 103,3 0.127 0 127 0.135I t  is evident from the table th a t oven in the case of this system as in OH (/I—►A) system, there is complete agreement between the results emerging from both the methods, viz., Hutchisson’s and Manneback’s, and this is as it should be, as inade out in the work of Tawde, Patil, Sreedhara Murthy and Katti (1957), and will be further shown by mathematical equivalence between the two in a forth­coming paper. The fact th a t the calculated values of (R & R) by the same , mathematical treatm ent do not agree with ours indicates th at they have badly erred ip following the method of Mannoback (1951), and applying it to the problem in hand. In  arriving a t this conclusion, we have applied whatever internal checks were necessary a t each step of the calculations, leading to the results of columns 2 and 3 (Table I), and we have reproduced the values at every stage.130 R. N , Tmvde and N . Sreedhara MurthyAlthough, the luethod of Hiitchisson is cumbersome and roquii’es laborious cal­culations over jirotracted period, have, as an external check, employed it here only to be sure of the correcincss of t)iir procedures in working out the data from Manncback’s method.While attempting to locate the source of the values arrived at by Rao and Ranade it was found that they resulted from the misunderstanding on the part of the authors, of the use of the two general formulae I and I I  given by Manneback for the calculation of t ' {n'n") matrix These formulae contain the quantities. A, k, a and b, and are valid for any of these values including negatives as staled by Manneback. Correct values of transition probabilities would follow by the direct use of those quantities with their proper signs in the two formulae. The values happen to bo negative in the particular case of (first negative) ^stem , and are to be directly used with their signs in the general formulae for the evalua­tion of C'(a'y?/') matrix The square of C{n'ii") then gives the transition pro­babilities. Tins way have resulted our data ol column 3 in Table I Anlaltor- iiative but simpler way suggested by Manneback when negative values of the above are encountered, is to interchange the meaning of upper and lower states\ i.e. of n' and n", while retaining their iiositive values. This simpler way has been adopted by (R & R) but in using it, they have not correctly interpreted the meaning of it, which is obviously more than mere interchange of n' and n . (R & R) have taken the above remark of Manneback to mean merely the interchange of quantum numbers, n' and n'\ without considering the real meaning of it, viz., interchange aiso of the quantities involved in the uiiper and lower states along with the interchange of n' and n". The consequence is that wrong values have emerged for C {n' n") and hence the transition probabilities, and they are represented as such in their paper. We have also tried this alternative but simpler ivay with its correct moaning given above, and arrived at the values identical with those of column 3 (Tabic I). In following the correct use ol Manneback’s mathematical technique, the method of Manneback has to be understood in the context of Hutchissoii’s analytical treatmoiii.I IIn  their attem pt to include the contribution of variation of electronic transi­tion moment with B, to the transition probability against the background of the above results, the authors have also erred in their fundamental approach to this question This problem is sought to be investigated through the application ol Manneback and Rahman’s treatment. But, in doing so, they have made untenable assumptions. This can be understood from the following analysisFollowing Shuler (1950), Manneback and Rahman start from the integral involving the wave functions of the upper and lower states :( R ) f  (R)dB ... (1)where the electronic traiisitjon moment can be taken to be constant,or a variable of JR. The main object of Maiineback and Hah man’s treatment has been to take the latter into account in the theory, viz., the variability of elec­tronic transition moment with E, and they considered the variation according to the linear relation of Shuler viz., ^„{E) c{l-\ pE), p being the expansioncoefficient. W ith the introduction of this relation m equation (1 ), the integral I Mn'n" ! becomesDerivation of the Vibrational Transition Probabilities, etc. 1^1, I ^  C [ lAn" W E  +  p J {E)l/ln'iE)EdE\Ih is expression (2) is written in the fromMnfn'f I =  c\G{n' n") +  pl){n''n"\(2 )(3)where C[n' n") := J [E) ir^n{E)dE (3a)and Di:n' n") -  \ilj,,,Wf,,»{E)EdE ... (3b)The square of the quantity | 1 gives the transition probability, p. The ex­pression for ] Mfi-n" ! consists thcretorc of two terms. The fiist term ??/') in till! bracket is the same integral as referred to m Maniieback’s Iheorotical ap}3lica- tioii disi'iissed in Section 1 above The second term is the additional one brought about as a result of the electronic transition moment being ('onsideroil to vary with E  Tt consists of tw’^o factors, (a) p, the expansion coefficient and {b) D{n' n"), another integral stated above As a close parallel to the treatment for C(n' n") matrix, Manneback and Rahman have given the following formula for computing the D{ti' n") matrixD{n n") =  XC{n' n")-{-Yn'G{n' — i, n")-\-Zn"G{n', n"— I) (4)where the quantities X , Y  and Z  have the moaning defined by them. D[n' n") matrix can therefore bo computed if C{n' n") martrix is knownThe quantity | Mn n” 1 equation (3) is therefore calculable from G{n' n") and D{n' )h"), provided the value of p is known. The value of constant, c, could be eliminated by considering the relative values of transition probabilities for a Iiair of bands. But p for this particular system of Og+(first negative) is not known, nor have (R & R) derived any value for it. Hence | Mn'^n ] and therefore p  cannot be estimated. I t  is therefore not understood how' (R & R) could get the values of 1 il/n'n" 1 and hence p. I t  has, hoAvever, been noticed that (R & R) have assumed l>{n' n”) to be itself the “overlap integral” and have taken the square of it to be the value of p  for the condition under which the electronic transition moment is supposed to vary with E. This is obviously wrong, since it involves the assump­tions that G{n' n ”) =  0, c =  1 and p =  1, which are untenable. As has been shown,132 R. N . Tawde and N . Sreedhara MurthyD{n’ n") taken alon^ with the factor p is only an additional term to C(n' n"), to take account of the variation of electronic transition moment with H in the cal­culation ol transition probabilities. Further, the numerical values given by (11 & R) for D(n' n'') are also wrong since the values of C(?i' n") on which they de­pend have been shown to be erroneous.Apart from the two fundamental defects from which the calculations of (R & R) suffer, there arc some minor corrections recpiired which are not so significant as to need a particular note. As a consequence of these defects the main results of their calculations are erroneous. Hence, the conclusions drawn by them from the comxiarative study of those results in terms of the values of Jarmain, Fraser and Nicholls (1955) cannot sustain.A C K N O W L E D G M E N TOne of the authors (N.8 M.) acknoAvledges his gi-atitude to the MiniH^-y of Education, Government of India, for the award of a Senior tResearch Traming Scholarshij) which enabled him to participate in this particular piece of work in the general programme m hand.It IC F  E  R  E  N  C E  HJarm a in , W . it . ,  Fmaor, P . A., and  NichoUa R  W ., 1955, Astrophys. J., 122, 55i M annoback, C., 1951, Physica, 17, 1001.M annobaok, C. and  R ahm an , A., 1954, Physica, 20, 497.R ao, P . S. and  Raiiado, .1. D., 1957, Ind. J . Phy., 31, 491.Shuler, K . E ., 1960 , .7. Ghem. Phys., 18, 1221.,, 1952, Proo. Phys. Hoc. {London), A65, 70.Tawde, N . R ., P a id , B S., S reedhara M urihy , N . and  K a tt i ,  M. R ., 1967, Physica, 23, 151.

On the Calculational Procedures in the Derivation of the Vibrational Transition Probabilities of the First Negative (b4Σ-g→a4Πu) Bands of O2+

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On the Calculational Procedures in the Derivation of the Vibrational Transition Probabilities of the First Negative (b4Σ-g→a4Πu) Bands of O2+

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