Evidence for a Double-Minimum Potential in an Excited State of C102 BY J. €3. COON, F. A. CESANI AND C. M. LOYD Physics Dept., Texas A. and M. University College Station, Texas, U.S.A. Received 21st January, 1963 Evidence is presented which suggests the existence of a double-minimum potential in the anti- symmetrical vibrational co-ordinate of an excited electronic state of C102. A previously reported vibrational analysis of the 36008, absorption system of C102 vapour proposes the excited state assignments 2 v j = (1+-Of) = 1559 cm-* and 4~5-2~5 = (2+- 1 +) = 16405 cm-1. In the present work, a two-parameter doubleminimum potential function is adjusted to fit these levels of the QS mode. The resulting function has a barrier of 2520 cm-1, and at each potential minimum one bond is 0.065 A shorter and one bond is 0.065 8, longer than the average bond of 1.620 A.The iso- tope shift calculated for the interval (1+-0+) agrees with the shift observed; however, the shift calculated for (2'-1+) does not. A Franck-Condon calculation based on the vibrational wave functions of the double-minimum potential yields intensity ratios in approximate agreement with the experimental ratios. The isotope shifts and intensity ratios calculated assuming a harmonic potential in the Qj mode do not agree with experiment. Recently 1 an intense progression of vibronic bands in the 3600 A absorption system of ClOz has been given the assignment (ui, 0,2)+(0,0,0), and a less intense progression has been given the assignment (pi, 0,4)+(0,0,0). These progressions are labelled (c) and (d) respectively.The observed intensities of these progressions relative to the (a) progression (ui, O,O)c(O, 0,O) is much greater than is expected on the basis of a harmonic potential in the Q; co-ordinate, but a double-minimum potential explains the observed intensities qualitatively.1 Furthermore, the isotope shift of 2vi as obtained from progressions (c) and (d) is about 25 % less than that expected for a harmonic potential. The purpose of this paper is to explain quan- titatively the anomalous intensity and isotope shift in terms of a double-minimum potential function in the Q; co-ordinate. Mulliken 2~ and Walsh 26 have discussed the probability that not only C102 but some excited states of SO2 and NO2 differ slightly from C2v symmetry.Mulliken suggests that the 2bl molecular orbital is responsible for these slightly unsymmetrical states. 1 . THE POTENTIAL FUNCTION3 It is assumed that the potential function in the Q; co-ordinate has the form V(Q) = ?Q2 + A exp (- a2Q2) (1.1) 1 where Q is a co-ordinate satisfying the condition 27'= Q2. The minima of this three-parameter potential function are located at & Qm given by A parameter p is introduced by writingJ . B . COON, F. A. CESANI AND C. M. LOYD 119 A frequency vo is defined by L = (2ncvo)2, and a dimensionless parameter B is intro- duced such that the barrier height is Bhcvo. From eqn. (l.l), (1.2) and (1.3), (eP-p- 1) ep B ~ c v ~ = V(0)- V(Q,) = A (1 -4) The barrier height in cm-1 is and the minima are located at b =Bvo, Qm where We may replace the parameters 2, A and a2 by the parameters VO, B and p.In the present application p is set equal to 1.5. For this value of p the minima are para- bolic. The energy levels and wave functions of the potential function (1.1) have been determined 3 for 60 different barrier heights ranging from B = 0.0 to B = 6.0 in intervals of 0.1. Machine calculations based on the secular equation of the linear variation method lead to wave functions of the form 2 3 where the 4 k are harmonic oscillator wave functions corresponding to the frequency YO. Numerical values of the dimensionless energy levels G(O+)/vo, G(O-)/vo, . . ., G(4+)/vo, G(4-)/Vo are given for each value of the dimensionless barrier height B. Here G is the energy above the potential minima in cm-1.The coefficients ak are tabulated against B for each of ten levels Of, . . ., 4-. This requires ten tables of coefficients. The energy level tables are available 3 but the tables of ak are too extensive to publish. 2. BARRIER HEIGHT A N D BOND LENGTHS The data used in this section are taken from Coon and Ortiz,l 2 4 = G(1')-G(O+) = 1559 cm-l, 4vi-2~; = G(2+)-G(1+) = 1640.5 cm-l. Examination of the table giving the levels G/VO for a barrier B described in 6 1 reveals that only for B = 2-050 does the ratio agree with the experimental value, 1640.511559. This establishes the value of B. The value of vo is determined from the identity The result is vo = 1229 cm-1. value of vo yields the levels Multiplication of tabulated values of G/VO by this G(O+) = 945 cm-', G(1') = 2504 cm-', G(2') = 4143 cm-l, G(O-) = 1014 cm-', G(l-) = 3034 cm-', G(23 = 5128 cm-l.(2.4) Accordingly the (O+-0-) separation is 69 cm-1. It also follows that the barrier height is (2.5) b = Bv, = 2520 crn-l.120 DOUBLE-MINIMUM POTENTIAL I N cloz In order to calculate the configuration of the molecule at the potential minima, consider fig. 1. Let ro be the bond length at the symmetrical configuration and let r be the change of bond length corresponding to an antisymmetrical displacement Q. For small displacements the kinetic- e n e r i T 2mM ' = 2m sin2 O+M' For the excited state 26 is 107" 24'.4 Using 2T = Q: = p& M is given by 2T = p32, where (2.6) @2 and setting Q = em, (2.7) FIG. 1. Since p = 1-5, eqn. (1.6) gives (2; = 0.1414 x 10-40 g cm2.Consequently by eqn. (2.7), r, = 0.0651 A. Since the average bond length 4 for the excited state is 1.620 A, the bond lengths at the potential minima are The displacement of the potential minima from the symmetrical configuration is considered to be within the range of small displacements. bond lengths = 1.620 A & 0-065 A. (2.8) 3. THE ISOTOPE SHIFT The values of B and v g determined in Q 2 along with the assumed value of p are sufficient to determine a specific double-minimum potential function having the form of eqn. (1.1). The isotope shift calculated on the basis of this potential function and that calculated on the basis of a harmonic potential function may be compared with the isotope shift spectroscopically observed. Table 1 gives the isotope shift ACT = CT (C13502) -a(C13702) observed for the three band progressions, a(u) ( U ' , , O , O)+(O, o,o>, 40) <u;,o, 2)-+(0,0, 01, (3.1) 42.4 <u;,o, 4)+, 0,O).TABLE 1 . 4 % ~ ~ EXTINCTION COEFFICIENT E IN (l./mole cm) AND THE ISOTOPE SHIFT ACJ = g(C135O2) -~~(c1370~) FOR THREE BAND PROGRESSIONS IN THE SPECTRUM OF CHLORINE DIOXIDE 0: 2 3 4 5 6 7 8 9 (a) ucm-1 2242 5-5 231 19.2 23806.9 24488.2 25 1 64.3 25835.7 26502-4 27164-1 Aa cm-1 8 Aa cm-1 E Aa cm-1 e 89 23960.6 20.9 25571.5 32.0 139 233 24636.7 24.8 180 26232.2 35.9 168 384 25307.6 31.6 295 26888.8 40-4 147 24.1 491 25973.8 35.4 409 27540-7 44-9 106 29.6 638 26635.5 41.2 495 28187.9 50.1 37 34.5 900 27292.0 45.9 470 41-0 1076 44-9 1108J . B. COON, F. A. CESANI AND C. M. LOYD 121 These isotope shifts are obtained by averaging the shift measured by Urey and Johnston 5 with that measured by Ku.6 For each progression Ao may be plotted against v; and a best straight line may be drawn through the points.The interval between the straight lines for progressions a(v) and c(u) gives an approximate value for the isotope shift of 2v;, (experimental) A(2v;) = 11.6 cm? (3 -2) (2vJ21(2v3 5 = pi/P* (3.3) To calculate the isotope shift for a harmonic potential we may use The subscript i refers to isotopic quantities. The reduced mass p is given by eqn. (2.6) and /ii contains the mass of C137 instead of CW. Substituting 2v; = 1559 cm-1 into eqn. (3.3) leads to (harmonic) A(2v;) = 15.8 cm-', (3.4) which deviates considerably from the observed value. To calculate the isotope shift for the double-minimum potential it is noted that the barrier height BVO and the position of the potential minima +rm do not change with isotopic substitution.Consequently eqn. (1.6) and (2.7) yield Using these relations the parameters Bg and vb for the isotopic molecule may be determined and hence the energy levels may be obtained from the table of (Glvo). It follows that [G(l+)- G(O+)]i = 1547.6 cm-1 which compared to 1559 cm-1 corresponds to (3.6) in agreement with the observed isotope shift. However, the success of the double- minimum potential function in explaining the isotopic shift of the second observed interval is less marked : (PilP)+ = volvao = &/B. (3.5) (double min.)A(2v3) = 11.4 cm-', (expt.)A(4v; - 2 4 ) = 10.4 cm- ' (double &.)A[G(2+)- G(l')] = 26.0 cm-'.(3.7) 4. INTENSITY CALCULATION Another method of checking the validity of the double-minimum potential function of 92 is to test the ability of the wave functions of this potential to produce observed intensity ratios. Let &/a( 1 +, 0) designate the peak extinction coefficient divided by the frequency for a band of the progression c(u), and let e/a(O+, 0) designate the same for the corresponding band of progression a(v). According to the Franck- Condon principle, &/O(l+, 0) R(l+, 0) &/O(O+ -[-I y 0) - R(O+ 0) ' where R(u, 0) = Y'(v)'€"'(0)dQ3. (4.2) s In this overlap integral, Qj = Q;. The average value of the intensity ratio as determined from the data of table 1 is122 DOUBLE-MINIMUM POTENTIAL I N cloz The data for bands v; = 3,4,5 and 6 are averaged.This experimental intensity ratio is to be compared with the value calculated from eqn. (4.1). For the parameter B = 2.050 the tables described in 6 1 give the coefficients a k of the wave functions, ”’(1’) = 0.706 4; - 0.408 4; - 0.570 4; + . . . , Y’(0’) = 0-623 4h-tO.744 4;+0.241 4:+ . . . . (4.4) The 4; are harmonic oscillator wave functions corresponding to the frequency given by the double-minimum parameter vo = 1229 cm-1. The ground-state vibrational wave function is Y ( 0 ) = &, (4.5) where 4; is the harmonic oscillator wave function for 0’’ = 0 corresponding to the frequency v’j = 11 10.5 cm-1.1 The ratio R(1+, O)/R(O+, 0) is easily evaluated by use of the formulae, where r(v, 0) = J4&YQso [R(1+, O)/R(O+, 0)12 = 1.13, [R(2+, O)/R(O+, O)-y = 0.22, The result is (4.7) (4.8) which is in fair agreement with eqn.(4.3). A similar calculation yields which compares in order of magnitude to the intensity ratio of progression d(v) to progression a(v). These calculated intensities are a great improvement over those based on a harmonic potential in the Qj mode. Using vj = 1559/2 and v’j = 11 10.5 cm-1 the harmonic potential leads to 0.015 and 040036 in the place of values given in eqn. (4.7) and (4.8) respectively. DISCUSSION The double-minimum potential function specified by ( p = 1.5, B = 2-050, and YO = 1229 cm-I), having a barrier of 2520 cm-1, explains the two observed vibra- tional intervals 1559 cm-1 = If-Of and 1640-5 cm-1 = 2f- 1’. However, this potential explains the small isotope shift of only the first interval. Calculations not reported above show that a potential function with a barrier of about 3500 cm-1 is able to explain the small isotope shift observed for both intervals. For such a high barrier the second vibrational interval is reduced to 75 % of the value observed. The fact that both of the intervals and the isotope shifts of both intervals can not be explained by a single double-minimum potential function constitutes a serious difficulty.J . B. COON, F. A. CESANI AND C. M. LOYD 123 This work was supported by the United States Air Force under Contract No. AF49(638)-593 monitored by the AF Office of Scientific Research of the Office of Aerospace Research. 1 Coon and Ortiz, J. Mol. Spectr., 1957, 1, 81. ZaMulliken, Can. J. Chem., 1958, 36, 10. Ritchie, Walsh and Warsop, Spectroscopy (ed. Wells), (Pergamon Press, Oxford, 1962). 3 Coon, Naugle, Henderson and McKenzie, Report Air Force OSR (Contract 4 Coon, DeWames and Loyd, J. Mol. Spectr., 1962, 8, 285. 5 Urey and Johnston, Physic. Rev., 1931,38, 2131. 6 Ku, Physic. Rev., 1933, 44, 376. AF(638)-593), MU. 1963.