- Email: [email protected]
Dipole moment function of HI
JOURNAL
OF MOLECULAR
SPECTROSCOPY
Dipole
39,
65-72 (1971)
Moment
Function of HI’
R. H. TIPPING AND
A. FORBES Department of Physics, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada Quantum mechanical transition probabilities are presented for the O-3 vibration-rotation band correct through two orders in y = Z&/W, . To lowest order, these agree with previous perturbation theory calculations but disagree with the analogous semiclassical results. Application is made to an analysis of the experimental intensity data of HI, and the dipole moment function M(z)
= [.445 f -
.02 -
(.0765 f
.0029)s + (.49G f
.040)z2
(1.94 & .20)zc3]D,
where z = (r - r.)/re is deduced. I. INTRODUCTION
The increasing availability of reliable experimental intensity data has sparked a renaissance in the calculation of accurate vibration-rotation matrix elements. Specifically, within the past year, at least three papers have treated t.he problem of vibration-rotation interaction effects on line intensities for diatomic molecules. Toth, Hunt and Plyler (1) have extended Herman and Wallis’ (9) quantum mechanical perturbation approach through third-order utilizing a quintic anharmanic potential, and have calculated dipole-moment transition probabilities for the O-l, O-2 and O-3 bands. On a different tack, Herman, Tipping and Short (3) have deduced simple yet accurate analytic wavefunctions for vibration-rotation levels of ‘C-state diatomic molecules based on an iterative solution of the radial Schradinger equation for the Dunham oscillator. Having obtained the wavefunctions, the calculation of transition probabilities is a straightforward procedure. This has been carried out for the O-l, O-2 and l-2 bands accurate to the lowest two powers in the expansion parameter y (= 2&/o, N 10-3) (4). Consistency in y, as pointed out, previously, is necessitated by substantial can1This research was supported by the National Research Council of Canada through Grant No. A-6025. 65
TIPPING
66
AND FORBES
cellation among the various correction terms (those down by a factor y). Their results for the O-l and O-2 bands agree with those of Toth et al. with the exception of small terms mMq which the latter authors neglected through an arbitrary truncation of the dipole-moment expansion. Recently, Jacobi (5) has extended the semiclassical treatment of vibration-rotation interaction outlined by Benesch (6) to include the first and second overtone bands. For comparison with the results of the above investigators for the O-3 band, we report here expressions derived in a manner identical to that described in Ref. (4): (Mo3(m))” = (Mo3(0))2 Fo3(m) = A0 + Aim + A2m2 +
(1)
- - -,
where
-_39 2 a22 +
_32 al4 909
+ 14y(3a13 -
+ M4[-24a1yl
123ata 4
)I+M&m
Gala2 + au,)] + Md4 + 12-d--at
-
(2)
ad]
+ M5[20rl , I
and F:(m)
-T+
= 1 -I- ‘7
-8&]+r[Mo(-
+M1
4al -
ia2
-
$
•k 6M1(1 -
a~)
40 f 32a2 - 40al - 4~112 -I- 28~ - 96ala2
81
(
45+34al-~a~-Tu2+~a~az-3%z~-~aal
- 82~~ - 28a12 + y
az -I- Md18
f
123
~a,-~a,a3-~a:+~u,l-sU~~
54d
-
24M4
1
+ M~[ZSU~+ 42~: - 84al czz]+ M3[- 12~12 - 12~21 - 24~1 M4 -;+h&-$--32 -
8M2}])+%(
2
1-I-
Mc,[-12 + 4~11+ MI
M21--6a1-k 61 13 - 6al - y a2
(3)
l>IPOLi’,
MOMKNT
FUNCTION
OF HI
(ii
Here, 0 = [Ali (s/4 ui2 + u2) + 4df2ul + 4Ala], the remaining notation being identical to that defined in Ref. (4). These results agree with those of Toth et al. t#olowest order. The small correction terms, reported here for the first time, are of marginal significance at present] in view of the lack of experimental intensity data for the 03 and O-5 bands crucial for an accurate determination of A/ and Afj . It is worthwhile mentioning, however, that the correction terms are larger in t,he O-3 band (-10 ‘ib of t,he leading terms) than in tile bands previously considered (-2 ‘Z). For comparison with the semiclassical results, we note that there is an error in Jacobi’s expression for the slope (linear tjr-dependence) of the O-3 band transition probabilities. His Eq. (I 9) should read
where f& = M,rk-‘/Al1 . From Eqs. (1) and (3), the corresponding mechanical result, (t,o loivext order) is
TABLI~: POTENTIAL HP al
-2.2537
a2
3.1880 -4.4987 4 7058
as ad 9 Reference b Reference c Reference d SW test.
(7). (8). (9).
CONST.\NTS
yuantum
I
OF HYDROGEN
H.\LIDES
HCI”
HBre
HI”
-2.364510
-2.13733
-2.5564
:(.66470 -4.6783 4.944
3.8645 -5.151 5 .23
.
4.076 -5.43 5.48
Al
AZ
All A1
-0.0044
i
f
0.0010
0.0038
0.029
1.968 f
0.0108
0.079
f
16.129
(O-l) band; Renesch et al. data”.
0.0574
2.15
16.129
(Cubic Jit)
0.0067
0.12
0 (10-Z)
f
f
f
0.079
0.0508
0.0060 0 (10-S)
f
16.129 f 2.15 f
0.079 0.12
0.19
f
-1.76
0.20
-1.94
M3
f
0.044
0.0030
0.549 f
f
0.040
-0.0799
0.496 f
-0.0765
Benesch
M,
.028
Meyer
Theoretical value
0.0029
0.445 f
Experimental value
II
f
Ml
MO
Coefficient (Debye)
TABLE
Comments
Fitted. Uncertainty primarily Uncertainty primarily Theoretical
in the theoretical value arises from that in MO m given above. in the theoretical value arises from that in MO and Ml . order of magnitude estimate.
Deduced from microwave spectroscopy of DI; zero point correction is negligible. Deduced primarily from A, and the sign of A1 as measured for the (O-l) band; principle theoretical error arises from uncertainties associated wit,h MP and Ma . Deduced primarily from A0 and the approximate magnitude of Al as measured for the (O-2) band; principle theoretical error arises from the lack of knowledge of M,. Deduced primarily from A0 and the approximate magnitude of A1 m measured for the (O-3) band; principle theoretical error arises from the lack of knowledge of MI and MS.
COMPARISON BETWEEN THE PRESENT THEORY AND EXPERIMENT
s
[0.0049
A2
0.0047 (2.5
3.224 0.082
0.082 0.03@]
f f
4.48)
x
10-d]
$1)
(-1.8
1.255 0.0132
0.047 0.009
f
f f X lo-”
(14). IICoefficieuts
0.9)
0.020 0.0035
f 0.0002 O(lo-4)
f f
Blieferenre (Id). b Referencr (f3)., c Reference see footnote on p. 5. e Reference (16).
f
[(O.@l
A2
Al
0.020 0.0922
1.255 f 0.0198 f
A0
3.224 0.105
-0.0011
(Quadratic
0.0911 1.9) X 10-4
f 0.047 AZ 0.013
et al. datae. (Cubic fit)
z!z 0.0151]
* f
--_
datac. (Q1dadratic fit)
(O-S) band; Meyer et al. data’.
‘43
A2
A1
A0
(O-d) band; Meyer
Al
4.260 [-0.033
Benesch’s
A0
(O-.2) band;
-
0.0002
0.082 0.010
f
X lo-”
st,atistical
the
t,he
In this respect,
primarily from M, and MS. primarily from 1114. significance.
Fit,ted. Theoret,ical error arises lack of knowledge of Theoretical error arises lack of knowledge of
Fitted. Theoretical error arises primarily from the lack of knowledge of Md, and from the uncertainty in M, . Principle uncertainty in the theoretical values arises from that in MS . Theoretical estimate.
Fitted. Theoretical error arises primarily from the lack of knowledge of AR&, and from the uncertainties in MO and Mz . Uncertainty in the theoretical value arises primarily from that in MP and Ma .
are of doubtful
0.45)
f 0.020 & 0.0035
f
f f
iu brackets
(-0.89
1.255 0.0137
-0.0012
4.260 0.123
_.
70
TIPPING
AND FORBES
Identifying k’ = 3dal , kN = 2% and recognizing that the dipole-moment coefhcients appearing in Eq. (4) are equal to Mi/rei in the present notation, one can see that the semiclassical and quantum mechanical expressions differ somewhat: M. (- 16 + 12s) + 16M1 versus gMO (- 16 + 12a,) + 12M1. It is of interest to note that the two methods agree (to lowest order) for the fundamental band slope, while for the first overtone band they differ by a factor 3/4 in the 80 terms (5). II. APPLICATION
TO HI
The application of the above quantum mechanical results, together with those in Ref. (4) to an analysis of intensity measurements, requires accurate values for al through a4, B, and ae . In Table I, potential constants for HF, HCl and HBr are tabulated along with approximate values for HI used in subsequent computations. These were determined in the following way: al and a2 were computed from the (approximate) Dunham expressions YlO a’
=
Yll
6Yz1
-1
and
5
aa=4at$m1,
2Y20
employing the experimental data for Y10, Yll (10) and Yol (11). However, because of the nature of the expressions for a3 and a4 (small differences of relatively large numbers) whereby small experimental uncertainties can lead to large errors, it is felt that more accurate constants can be deduced through a linear least-squares extrapolation of the analogous parameters for the other hydrogen halides. To cast the experimental information into a form convenient for comparison with theory, the transition probabilities were fitted by a least-squares procedure to equations of the form (M,“(m))z
= Ao + Am
+ Am2 + Agn3 +
... .
(6)
The criterion used for fitting was to increase the degree of the polynomial until the standard deviation of at least one coefficient was significantly greater than the coefficient itself. If the experimental scatter is small, the first two coefficients are relatively insensitive to the degree of the fit.2 The degree of the polynomial adopted and the corresponding coefficients are listed in Table II, the quoted errors being the standard deviations. By assuming M4 = Ms = 0 and fitting A0 for all three bands using approximate values for the slopes (4), illI , M2 , and MB were determined uniquely (see Table II). MO is known from microwave measurements (12). This knowledge of 2The largest fluctuation occurred for Benesch’s O-2 band data where, for comparison, the coefficients for linear and cubic fit (in units of 10M6D)are: AC, = 4.278 & 0.053, AI = -0.023 f. 0.022; and A@ = 4.152 k 0.096, A1 = 0.015 f 0.043, Az = 0.047 f 0.030, Aa = -0.013 f
0.008.
DIPOLE MOMENT FUNCTION
OF HI
71
the dipole-momentexpansioncoefficientsenables one to compute the m-dependence of the transition probabilitieg for each band. These results are also tabulated in Table II along with the estimated theoretical errors. III. DISCUSSION OF RESULTS
As can be seen from Table II, the different experimental values for (M02(0))2 lead to slightly different dipole-moment coefficients. Both sets, however, imply a positive slope for the O-2 band transition moments, the agreement being in better accord with Meyer and co-workers’ data than those of Benesch. In any case, the sign of the slope cannot be unambiguously determined from Benesch’s measurements (see footnote), However, assuming a negative slope for the O-2 band and calculating the dipole-moment coefficients accordingly, leads to unsatisfactory results for t,he slopes of the fundamental and second overtone bands. Furthermore, Benesch, in analyzing his own data using vibration-rotation transition probabilities computed elect.ronically assuming Rlorse wavefunctions, has deduced M1 and Mz (- 0.082 and 0.41 D, respectively) in good agreement with the present values (16). From the data in Ref. (15), Jacobi (I?‘) has obtained a value MZ/M1 = -5.23 f .64 D in substantial agreement with the present value of -6.48 f .58 D, the discrepancies in large part being accounted for by the improved accuracy of the theoretical transition probabilities and different handling of the experimental data. On the other hand, l\leyer et al. (15), employing the “wavefunction approximation” of Trischka and Salwen (18) with iUorse wavefunctions, have calculated M1 = -0.088 D and M2 = +0.089 D. They assert that the coefficient M2 receives an appreciable contribution from the O-4 band vibrational matrix element. In bhe present work, while knowledge of the quantity jMo4(0)1 would allow the calculation of two possible M, values, M4 enters only in second-order in the determination of Mz (and MS). Nevertheless, their conclusion that the dipole-moment function for HI cannot be well represented by the first two or three terms in a truncated Taylor series is corroborated by the present investigation, in that the convergence is less rapid than in the case of HBr (4). The nature of the coefficients suggests that for HI the inflexion point (19) (as well as the maximum) of t’he dipole moment function occurs at an internuclear separation considerably smaller than re . RECEIVED:
November 19, 1970 REFERENCES
I. R. A. TOTH, R. H. HUNT (1969); 36, 110 (1970). 2. 8. 4. 5.
6.
R. It. R. N.
AND
E. K. PLYLER, J. Afol. Speclrosc.
33, 74 (1969); 32, 85
HERMAN AND R. F. WALLIS, J. Chem. Phys. 23, 637 (1955). M. HERMAN, R. H. TIPPING AND S. SHORT, J. Chem. Phys. 63, 595 (1970). H. TIPPING AND R. M. HERMAN, J. Mol. Spectrosc., 36, 404 (1970). JACOBI, J. Chem. Phys. 62, 2694 (1970). W. BENESCH, J. iVfoZ.Spectrosc. 16, 140 (1965).
72
TIPPING
AND
FORBES
7. D. U. WEBB AND K. N. RAO, J. Mol. Spectrosc. 26, 121 (1968). 8. D. H. RINK, B. S. R.40 AND T. A. WIGGINS, J. Mol. Spectrosc. 17, 122 (1965). 9. D. H. R.~NK, U. FINK AND T. A. WIGGINS, J. Mol. Spectrosc. 18, 170 (1965). 10. L. H. JONES, J. Mol. Xpectrosc. 1, 179 (1957). 11. P. ARCAS, C. HAEUSLER,C. JOFFRIN,C. MEYER, N. THANH, AND P. BARCHEWITZ,Appl Optics 2, 909 (1963). 12. C. A. BURRUS, J. Chem. Phgs. 31, 1270 (1959). 13. G. AMEER AND W. BENESCH,J. Chem. Phys. 37, 2699 (1962). 14. W. BENESCH,J. Chem. Phys. 39, 1048 (1963). 16. C. MEYER, C. HAEUSLERAND P. BARCHEWITZ,J. Phys. 26, 305 (1965). 16. W. BENESCH,J. Chem. Phys. 40, 422 (1964). i7. N. JACOBI,J. Mol. Spectrosc. 32, 76 (1967). 18. J. TRISCHKAAND H. SALWEN, J. Chem. Phys. 31, 218 (1959). 19. K. CASHION, J. Mol. Spectrosc. 10, 182 (1963).
OF MOLECULAR
SPECTROSCOPY
Dipole
39,
65-72 (1971)
Moment
Function of HI’
R. H. TIPPING AND
A. FORBES Department of Physics, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada Quantum mechanical transition probabilities are presented for the O-3 vibration-rotation band correct through two orders in y = Z&/W, . To lowest order, these agree with previous perturbation theory calculations but disagree with the analogous semiclassical results. Application is made to an analysis of the experimental intensity data of HI, and the dipole moment function M(z)
= [.445 f -
.02 -
(.0765 f
.0029)s + (.49G f
.040)z2
(1.94 & .20)zc3]D,
where z = (r - r.)/re is deduced. I. INTRODUCTION
The increasing availability of reliable experimental intensity data has sparked a renaissance in the calculation of accurate vibration-rotation matrix elements. Specifically, within the past year, at least three papers have treated t.he problem of vibration-rotation interaction effects on line intensities for diatomic molecules. Toth, Hunt and Plyler (1) have extended Herman and Wallis’ (9) quantum mechanical perturbation approach through third-order utilizing a quintic anharmanic potential, and have calculated dipole-moment transition probabilities for the O-l, O-2 and O-3 bands. On a different tack, Herman, Tipping and Short (3) have deduced simple yet accurate analytic wavefunctions for vibration-rotation levels of ‘C-state diatomic molecules based on an iterative solution of the radial Schradinger equation for the Dunham oscillator. Having obtained the wavefunctions, the calculation of transition probabilities is a straightforward procedure. This has been carried out for the O-l, O-2 and l-2 bands accurate to the lowest two powers in the expansion parameter y (= 2&/o, N 10-3) (4). Consistency in y, as pointed out, previously, is necessitated by substantial can1This research was supported by the National Research Council of Canada through Grant No. A-6025. 65
TIPPING
66
AND FORBES
cellation among the various correction terms (those down by a factor y). Their results for the O-l and O-2 bands agree with those of Toth et al. with the exception of small terms mMq which the latter authors neglected through an arbitrary truncation of the dipole-moment expansion. Recently, Jacobi (5) has extended the semiclassical treatment of vibration-rotation interaction outlined by Benesch (6) to include the first and second overtone bands. For comparison with the results of the above investigators for the O-3 band, we report here expressions derived in a manner identical to that described in Ref. (4): (Mo3(m))” = (Mo3(0))2 Fo3(m) = A0 + Aim + A2m2 +
(1)
- - -,
where
-_39 2 a22 +
_32 al4 909
+ 14y(3a13 -
+ M4[-24a1yl
123ata 4
)I+M&m
Gala2 + au,)] + Md4 + 12-d--at
-
(2)
ad]
+ M5[20rl , I
and F:(m)
-T+
= 1 -I- ‘7
-8&]+r[Mo(-
+M1
4al -
ia2
-
$
•k 6M1(1 -
a~)
40 f 32a2 - 40al - 4~112 -I- 28~ - 96ala2
81
(
45+34al-~a~-Tu2+~a~az-3%z~-~aal
- 82~~ - 28a12 + y
az -I- Md18
f
123
~a,-~a,a3-~a:+~u,l-sU~~
54d
-
24M4
1
+ M~[ZSU~+ 42~: - 84al czz]+ M3[- 12~12 - 12~21 - 24~1 M4 -;+h&-$--32 -
8M2}])+%(
2
1-I-
Mc,[-12 + 4~11+ MI
M21--6a1-k 61 13 - 6al - y a2
(3)
l>IPOLi’,
MOMKNT
FUNCTION
OF HI
(ii
Here, 0 = [Ali (s/4 ui2 + u2) + 4df2ul + 4Ala], the remaining notation being identical to that defined in Ref. (4). These results agree with those of Toth et al. t#olowest order. The small correction terms, reported here for the first time, are of marginal significance at present] in view of the lack of experimental intensity data for the 03 and O-5 bands crucial for an accurate determination of A/ and Afj . It is worthwhile mentioning, however, that the correction terms are larger in t,he O-3 band (-10 ‘ib of t,he leading terms) than in tile bands previously considered (-2 ‘Z). For comparison with the semiclassical results, we note that there is an error in Jacobi’s expression for the slope (linear tjr-dependence) of the O-3 band transition probabilities. His Eq. (I 9) should read
where f& = M,rk-‘/Al1 . From Eqs. (1) and (3), the corresponding mechanical result, (t,o loivext order) is
TABLI~: POTENTIAL HP al
-2.2537
a2
3.1880 -4.4987 4 7058
as ad 9 Reference b Reference c Reference d SW test.
(7). (8). (9).
CONST.\NTS
yuantum
I
OF HYDROGEN
H.\LIDES
HCI”
HBre
HI”
-2.364510
-2.13733
-2.5564
:(.66470 -4.6783 4.944
3.8645 -5.151 5 .23
.
4.076 -5.43 5.48
Al
AZ
All A1
-0.0044
i
f
0.0010
0.0038
0.029
1.968 f
0.0108
0.079
f
16.129
(O-l) band; Renesch et al. data”.
0.0574
2.15
16.129
(Cubic Jit)
0.0067
0.12
0 (10-Z)
f
f
f
0.079
0.0508
0.0060 0 (10-S)
f
16.129 f 2.15 f
0.079 0.12
0.19
f
-1.76
0.20
-1.94
M3
f
0.044
0.0030
0.549 f
f
0.040
-0.0799
0.496 f
-0.0765
Benesch
M,
.028
Meyer
Theoretical value
0.0029
0.445 f
Experimental value
II
f
Ml
MO
Coefficient (Debye)
TABLE
Comments
Fitted. Uncertainty primarily Uncertainty primarily Theoretical
in the theoretical value arises from that in MO m given above. in the theoretical value arises from that in MO and Ml . order of magnitude estimate.
Deduced from microwave spectroscopy of DI; zero point correction is negligible. Deduced primarily from A, and the sign of A1 as measured for the (O-l) band; principle theoretical error arises from uncertainties associated wit,h MP and Ma . Deduced primarily from A0 and the approximate magnitude of Al as measured for the (O-2) band; principle theoretical error arises from the lack of knowledge of M,. Deduced primarily from A0 and the approximate magnitude of A1 m measured for the (O-3) band; principle theoretical error arises from the lack of knowledge of MI and MS.
COMPARISON BETWEEN THE PRESENT THEORY AND EXPERIMENT
s
[0.0049
A2
0.0047 (2.5
3.224 0.082
0.082 0.03@]
f f
4.48)
x
10-d]
$1)
(-1.8
1.255 0.0132
0.047 0.009
f
f f X lo-”
(14). IICoefficieuts
0.9)
0.020 0.0035
f 0.0002 O(lo-4)
f f
Blieferenre (Id). b Referencr (f3)., c Reference see footnote on p. 5. e Reference (16).
f
[(O.@l
A2
Al
0.020 0.0922
1.255 f 0.0198 f
A0
3.224 0.105
-0.0011
(Quadratic
0.0911 1.9) X 10-4
f 0.047 AZ 0.013
et al. datae. (Cubic fit)
z!z 0.0151]
* f
--_
datac. (Q1dadratic fit)
(O-S) band; Meyer et al. data’.
‘43
A2
A1
A0
(O-d) band; Meyer
Al
4.260 [-0.033
Benesch’s
A0
(O-.2) band;
-
0.0002
0.082 0.010
f
X lo-”
st,atistical
the
t,he
In this respect,
primarily from M, and MS. primarily from 1114. significance.
Fit,ted. Theoret,ical error arises lack of knowledge of Theoretical error arises lack of knowledge of
Fitted. Theoretical error arises primarily from the lack of knowledge of Md, and from the uncertainty in M, . Principle uncertainty in the theoretical values arises from that in MS . Theoretical estimate.
Fitted. Theoretical error arises primarily from the lack of knowledge of AR&, and from the uncertainties in MO and Mz . Uncertainty in the theoretical value arises primarily from that in MP and Ma .
are of doubtful
0.45)
f 0.020 & 0.0035
f
f f
iu brackets
(-0.89
1.255 0.0137
-0.0012
4.260 0.123
_.
70
TIPPING
AND FORBES
Identifying k’ = 3dal , kN = 2% and recognizing that the dipole-moment coefhcients appearing in Eq. (4) are equal to Mi/rei in the present notation, one can see that the semiclassical and quantum mechanical expressions differ somewhat: M. (- 16 + 12s) + 16M1 versus gMO (- 16 + 12a,) + 12M1. It is of interest to note that the two methods agree (to lowest order) for the fundamental band slope, while for the first overtone band they differ by a factor 3/4 in the 80 terms (5). II. APPLICATION
TO HI
The application of the above quantum mechanical results, together with those in Ref. (4) to an analysis of intensity measurements, requires accurate values for al through a4, B, and ae . In Table I, potential constants for HF, HCl and HBr are tabulated along with approximate values for HI used in subsequent computations. These were determined in the following way: al and a2 were computed from the (approximate) Dunham expressions YlO a’
=
Yll
6Yz1
-1
and
5
aa=4at$m1,
2Y20
employing the experimental data for Y10, Yll (10) and Yol (11). However, because of the nature of the expressions for a3 and a4 (small differences of relatively large numbers) whereby small experimental uncertainties can lead to large errors, it is felt that more accurate constants can be deduced through a linear least-squares extrapolation of the analogous parameters for the other hydrogen halides. To cast the experimental information into a form convenient for comparison with theory, the transition probabilities were fitted by a least-squares procedure to equations of the form (M,“(m))z
= Ao + Am
+ Am2 + Agn3 +
... .
(6)
The criterion used for fitting was to increase the degree of the polynomial until the standard deviation of at least one coefficient was significantly greater than the coefficient itself. If the experimental scatter is small, the first two coefficients are relatively insensitive to the degree of the fit.2 The degree of the polynomial adopted and the corresponding coefficients are listed in Table II, the quoted errors being the standard deviations. By assuming M4 = Ms = 0 and fitting A0 for all three bands using approximate values for the slopes (4), illI , M2 , and MB were determined uniquely (see Table II). MO is known from microwave measurements (12). This knowledge of 2The largest fluctuation occurred for Benesch’s O-2 band data where, for comparison, the coefficients for linear and cubic fit (in units of 10M6D)are: AC, = 4.278 & 0.053, AI = -0.023 f. 0.022; and A@ = 4.152 k 0.096, A1 = 0.015 f 0.043, Az = 0.047 f 0.030, Aa = -0.013 f
0.008.
DIPOLE MOMENT FUNCTION
OF HI
71
the dipole-momentexpansioncoefficientsenables one to compute the m-dependence of the transition probabilitieg for each band. These results are also tabulated in Table II along with the estimated theoretical errors. III. DISCUSSION OF RESULTS
As can be seen from Table II, the different experimental values for (M02(0))2 lead to slightly different dipole-moment coefficients. Both sets, however, imply a positive slope for the O-2 band transition moments, the agreement being in better accord with Meyer and co-workers’ data than those of Benesch. In any case, the sign of the slope cannot be unambiguously determined from Benesch’s measurements (see footnote), However, assuming a negative slope for the O-2 band and calculating the dipole-moment coefficients accordingly, leads to unsatisfactory results for t,he slopes of the fundamental and second overtone bands. Furthermore, Benesch, in analyzing his own data using vibration-rotation transition probabilities computed elect.ronically assuming Rlorse wavefunctions, has deduced M1 and Mz (- 0.082 and 0.41 D, respectively) in good agreement with the present values (16). From the data in Ref. (15), Jacobi (I?‘) has obtained a value MZ/M1 = -5.23 f .64 D in substantial agreement with the present value of -6.48 f .58 D, the discrepancies in large part being accounted for by the improved accuracy of the theoretical transition probabilities and different handling of the experimental data. On the other hand, l\leyer et al. (15), employing the “wavefunction approximation” of Trischka and Salwen (18) with iUorse wavefunctions, have calculated M1 = -0.088 D and M2 = +0.089 D. They assert that the coefficient M2 receives an appreciable contribution from the O-4 band vibrational matrix element. In bhe present work, while knowledge of the quantity jMo4(0)1 would allow the calculation of two possible M, values, M4 enters only in second-order in the determination of Mz (and MS). Nevertheless, their conclusion that the dipole-moment function for HI cannot be well represented by the first two or three terms in a truncated Taylor series is corroborated by the present investigation, in that the convergence is less rapid than in the case of HBr (4). The nature of the coefficients suggests that for HI the inflexion point (19) (as well as the maximum) of t’he dipole moment function occurs at an internuclear separation considerably smaller than re . RECEIVED:
November 19, 1970 REFERENCES
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