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Upper limits of rotational states of H2, H+2, CH and CH+
J. Quonr.
Specmsc.
Rodhr.
Transfer,
Vol. 12,pp. 1147-l 150.PergamonPress1972.Printed in Great Britain
NOTE UPPER LIMITS OF ROTATIONAL STATES OF H,, H;, CH AND CH+ S.
SUNDARAM
Department of Physics, University of Illinois, Chicago, Illinois 60680, U.S.A. (Received 31 March 1911)
Abstract-The theoretical considerations developed by the author in a previous study have been used to obtain the upper limits of rotational states of the systems of H,, H:, CH and CH+. The potential-energy functions due to Morse and Hulburt-Hirschfelder have been considered for this study. In general, the limiting values for the Morse function are higher than for the Hulburt-Hirschfelder potential in the systems discussed here.
INTRODUCTION STUDY of the effective potential curves and the upper limits of rotational states of diatomic molecules is of very great importance for applications to high-temperature properties, the stability of the molecules, and the predissociation by rotation. In a previous study by the author W)the correct general theory relating to the determination of limits of rotational states has been discussed using the various commonly applicable potentialenergy functions and taking into account the interaction of rotation and vibration in the molecules. It is the purpose of this study to apply these theoretical considerations and obtain the maximum values of the rotational quantum numbers for the electronic states of H, , H:, CH and CH+ from available spectral data. In this study, the Morse potential function and the Hulburt-Hirschfelder function have been considered and the corresponding J,,, values have been derived. THE
THEORETICAL
CONSIDERATIONS
As was stated in the previous article’@ in view of the importance of the vibrationalrotational energy limits for various applications, it was necessary to correct the erroneous expressions in the earlier theoretical study by KHACHKURUZOV(~*‘) when using the HulburtHirschfelder potential function. Taking into account the interaction of rotation and vibration, the effective potential energy of the diatomic molecule is given by(‘) U,(R) = U,(R) + B,J(J + 1)/R’,
(1)
where R = r/r, and B, is the rotational constant. The limiting value of R (= Rlim = R,) is given by the point of inflexion in the effective potential curve. Thus the conditions U;(R) = 0 and U;(R) = 0 define R, . As noted in the previous study, the equation 1147
1148 determining
S. SUNDARAM R, is 3 Ub(R,) + R, U;(R,)
It has been shownC6’ earlier function in the form
that
when
= 0.
one considers
(2) the reduced
Morse
potential
uO(x) = [ 1 - exp( -x)]‘, where x = a(R - 1) and a = w,/2J(B,D,),
equation
(3)
(2) is simplified
to
3ub(x,) +(a + X,)z&X~) = 0.
(4)
Also, $)(x0) where II = 1,2,3.. in the form
= (_l)n+‘2(e”0_2”~‘)e-‘“”
for the first, second,
third..
a = [3(e”“Therefore, Following
(5)
derivatives.
Hence, equation
l)/(e*~l - 2)] -x0.
(6)
knowing the value of a, one calculates the value of x,, satisfying KHACHKURUZOV(5) the limiting value of the rotational state is Jlim = @Jo,){
(4) gives x,,
1K&,4 Y. - 3)1/t % - 312);
with Y, = u + x0 = 3(eX0- l)/(e”” - 2). The quantity
equation
(6). (7)
c( is given by
cx = 0.52{[(G,,(0)/D,)(2Y,-3)2/Y,(Y,-3)]2~
11/2-(9/4Y,)+(3/Y;)I}1’3
(8)
and J:, CZJi,(l
--2X).
Hence, for the Morse potential function, the quantities J,, can be evaluated knowing the molecular constants. The HULBURT-HIRSCHFELDER’~) potential function function involving five constants. It is given by Q(X) = (1 -em”)2+Cx3
(9) x0, Y, and therefore is a modified
Jlim, CYand
form of the Morse
eC’*(l +/IX),
(10)
where b = 2+[7-(l/u2){15d2-(8~,,x,/B,)}],‘12~, (’ = 1 -d/u,
(11) (12)
and d = l-cc,w,/6B;.
(13)
It is worth noting that the relation for b given by KHACHKURUZOV~‘) is in error. In order to obtain the value of x0 corresponding to the point of inflexion, the condition expressed by equation (2) is used. The method outlined by KHACHKURUZOV~~) gives
Jlim= @D,l~,)y~ e -““JW3)
le-""-2-(~/2)f2(xo)ll,
(14)
Upper
limits of rotational
states of
1149
H, ,Hi, CH and CH’
where f(x) = x3 e-‘“(1 + bx),
(15)
and fi(x), fi(x), f3( x ) are the successive derivatives off(x). The quantity o! is given by x = 0.52[{G,,(O) eZ”PAl(x)Y
1+3(x)/4&4-
l2/yW3
(16)
where 4i(x) = 2(e”- l)+cf(x),
(17)
$3(x) = 2(e”- 1) + cf3(x).
(18)
and
Thus, one can calculate J, from Ji = Ji,(l -2a).
However, approximate
(19)
values for G,,(O) have to be used from known values of G(0). RESULTS
In the present study, for the systems H,, H:, CH and CH+, the molecular constants given by BOND et al.(‘) were used. Two sets of calculations corresponding to the Morse function and the Hulburt-Hirschfelder function, respectively, were made using the spectral data. The results are given in Table 1 for the various electronic states. The approximate values of G,,(O) are such that a is not much less than unity either for the Morse function or for the Hulburt-Hirschfelder function. As a consequence, one cannot use the approximate relation J, N Jli,(l -a) (20) TABLE 1. UPPERLIMITSOFROTATIONALSTATESOF
System HZ
Electronic state X1x+ B’Z.+p
Cl l-I: AI, a3Z+ d31-I; k31-I, nsrt,
H,,H:,CH
AND CH+*
n
JiE
a”
JM 0
JEll
1.4398 0.8983 1.5266 1.4112
0.3596 0.2329 0.3634 0.3312
0.1671 0.0861 0.1676 0.1247
16 24 15 21
0.3454 0.3359 0.3354 0.3337
20 53 20 25 23 24 24 25
19 26 19 24
1.4532 1.4176 1.4304 1.4301
38 72 38 43 40 42 42 43
23 27 24 26
0.1038 0.1411 0.1272 0.1119
20 21 23
a”
J!f
H:
X*x+g
1.4022
42
0.3352
24
26
0.1065
23
CH
X’l-I A28 B2ZCCZZ-
2.1755 2.4620 5.3857 2.4673
62 53 36 52
0.2922 0.3442 1.2329 0.3478
40 30 32 29
38 36 34 42
0.1222 0.1466 0.3229 0.1631
33 30 23 35
CH+
‘TI
3.3491
31
0.6303
16
23
0.1795
19
* The superscripts M and H for J,i,, a and J, in the above table refer to the calculated function and Hulburt-Hirschfelder function respectively.
values using Morse
1150
S. SUNDARAM
in these studies. It is easily seen that, when G,,(O) is very small compared with D,, J, 1: Jlim as a is negligible. Further, it is obvious that the approximation proposed by KHACHKURUZOV(~) for G,,(O) will not influence the values of Jlim obtained in this investigation for the various electronic states of the different systems. The values of Jlim depend on D,, o,, and x0 only for Morse potential function or the function proposed by HulburtHirschfelder. Acknowledgement-The
author wishes to thank Mrs. A. SHADAC~PAN for assistance
with computations
REFERENCES 1. J. W. BOND, K. M. WATSON and J. A. WELCH, Alomic Theory qf’Gas Dynamic.), Addison-Wesley, Mass. (1965). 2. G. HERZBERG, Spectra qf Diatomic Molecules, 2nd Ed, D. Van Nostrand, New York (1950). 3. H. M. HULBURT and J. 0. HIRSCHFELDER,J. them. Phys. 35, 1901 (1961). 4. G. A. KHACHKURUZOV, Opt. Spectrosc. 21, 91 (1966). 5. G. A. KHACHKURUZOV, Opt. Spectrosc. 22, 1I (1967). 6. S. SUNDARAM, Spectroscopia Molecular 21, I (1972).
Reading,
Specmsc.
Rodhr.
Transfer,
Vol. 12,pp. 1147-l 150.PergamonPress1972.Printed in Great Britain
NOTE UPPER LIMITS OF ROTATIONAL STATES OF H,, H;, CH AND CH+ S.
SUNDARAM
Department of Physics, University of Illinois, Chicago, Illinois 60680, U.S.A. (Received 31 March 1911)
Abstract-The theoretical considerations developed by the author in a previous study have been used to obtain the upper limits of rotational states of the systems of H,, H:, CH and CH+. The potential-energy functions due to Morse and Hulburt-Hirschfelder have been considered for this study. In general, the limiting values for the Morse function are higher than for the Hulburt-Hirschfelder potential in the systems discussed here.
INTRODUCTION STUDY of the effective potential curves and the upper limits of rotational states of diatomic molecules is of very great importance for applications to high-temperature properties, the stability of the molecules, and the predissociation by rotation. In a previous study by the author W)the correct general theory relating to the determination of limits of rotational states has been discussed using the various commonly applicable potentialenergy functions and taking into account the interaction of rotation and vibration in the molecules. It is the purpose of this study to apply these theoretical considerations and obtain the maximum values of the rotational quantum numbers for the electronic states of H, , H:, CH and CH+ from available spectral data. In this study, the Morse potential function and the Hulburt-Hirschfelder function have been considered and the corresponding J,,, values have been derived. THE
THEORETICAL
CONSIDERATIONS
As was stated in the previous article’@ in view of the importance of the vibrationalrotational energy limits for various applications, it was necessary to correct the erroneous expressions in the earlier theoretical study by KHACHKURUZOV(~*‘) when using the HulburtHirschfelder potential function. Taking into account the interaction of rotation and vibration, the effective potential energy of the diatomic molecule is given by(‘) U,(R) = U,(R) + B,J(J + 1)/R’,
(1)
where R = r/r, and B, is the rotational constant. The limiting value of R (= Rlim = R,) is given by the point of inflexion in the effective potential curve. Thus the conditions U;(R) = 0 and U;(R) = 0 define R, . As noted in the previous study, the equation 1147
1148 determining
S. SUNDARAM R, is 3 Ub(R,) + R, U;(R,)
It has been shownC6’ earlier function in the form
that
when
= 0.
one considers
(2) the reduced
Morse
potential
uO(x) = [ 1 - exp( -x)]‘, where x = a(R - 1) and a = w,/2J(B,D,),
equation
(3)
(2) is simplified
to
3ub(x,) +(a + X,)z&X~) = 0.
(4)
Also, $)(x0) where II = 1,2,3.. in the form
= (_l)n+‘2(e”0_2”~‘)e-‘“”
for the first, second,
third..
a = [3(e”“Therefore, Following
(5)
derivatives.
Hence, equation
l)/(e*~l - 2)] -x0.
(6)
knowing the value of a, one calculates the value of x,, satisfying KHACHKURUZOV(5) the limiting value of the rotational state is Jlim = @Jo,){
(4) gives x,,
1K&,4 Y. - 3)1/t % - 312);
with Y, = u + x0 = 3(eX0- l)/(e”” - 2). The quantity
equation
(6). (7)
c( is given by
cx = 0.52{[(G,,(0)/D,)(2Y,-3)2/Y,(Y,-3)]2~
11/2-(9/4Y,)+(3/Y;)I}1’3
(8)
and J:, CZJi,(l
--2X).
Hence, for the Morse potential function, the quantities J,, can be evaluated knowing the molecular constants. The HULBURT-HIRSCHFELDER’~) potential function function involving five constants. It is given by Q(X) = (1 -em”)2+Cx3
(9) x0, Y, and therefore is a modified
Jlim, CYand
form of the Morse
eC’*(l +/IX),
(10)
where b = 2+[7-(l/u2){15d2-(8~,,x,/B,)}],‘12~, (’ = 1 -d/u,
(11) (12)
and d = l-cc,w,/6B;.
(13)
It is worth noting that the relation for b given by KHACHKURUZOV~‘) is in error. In order to obtain the value of x0 corresponding to the point of inflexion, the condition expressed by equation (2) is used. The method outlined by KHACHKURUZOV~~) gives
Jlim= @D,l~,)y~ e -““JW3)
le-""-2-(~/2)f2(xo)ll,
(14)
Upper
limits of rotational
states of
1149
H, ,Hi, CH and CH’
where f(x) = x3 e-‘“(1 + bx),
(15)
and fi(x), fi(x), f3( x ) are the successive derivatives off(x). The quantity o! is given by x = 0.52[{G,,(O) eZ”PAl(x)Y
1+3(x)/4&4-
l2/yW3
(16)
where 4i(x) = 2(e”- l)+cf(x),
(17)
$3(x) = 2(e”- 1) + cf3(x).
(18)
and
Thus, one can calculate J, from Ji = Ji,(l -2a).
However, approximate
(19)
values for G,,(O) have to be used from known values of G(0). RESULTS
In the present study, for the systems H,, H:, CH and CH+, the molecular constants given by BOND et al.(‘) were used. Two sets of calculations corresponding to the Morse function and the Hulburt-Hirschfelder function, respectively, were made using the spectral data. The results are given in Table 1 for the various electronic states. The approximate values of G,,(O) are such that a is not much less than unity either for the Morse function or for the Hulburt-Hirschfelder function. As a consequence, one cannot use the approximate relation J, N Jli,(l -a) (20) TABLE 1. UPPERLIMITSOFROTATIONALSTATESOF
System HZ
Electronic state X1x+ B’Z.+p
Cl l-I: AI, a3Z+ d31-I; k31-I, nsrt,
H,,H:,CH
AND CH+*
n
JiE
a”
JM 0
JEll
1.4398 0.8983 1.5266 1.4112
0.3596 0.2329 0.3634 0.3312
0.1671 0.0861 0.1676 0.1247
16 24 15 21
0.3454 0.3359 0.3354 0.3337
20 53 20 25 23 24 24 25
19 26 19 24
1.4532 1.4176 1.4304 1.4301
38 72 38 43 40 42 42 43
23 27 24 26
0.1038 0.1411 0.1272 0.1119
20 21 23
a”
J!f
H:
X*x+g
1.4022
42
0.3352
24
26
0.1065
23
CH
X’l-I A28 B2ZCCZZ-
2.1755 2.4620 5.3857 2.4673
62 53 36 52
0.2922 0.3442 1.2329 0.3478
40 30 32 29
38 36 34 42
0.1222 0.1466 0.3229 0.1631
33 30 23 35
CH+
‘TI
3.3491
31
0.6303
16
23
0.1795
19
* The superscripts M and H for J,i,, a and J, in the above table refer to the calculated function and Hulburt-Hirschfelder function respectively.
values using Morse
1150
S. SUNDARAM
in these studies. It is easily seen that, when G,,(O) is very small compared with D,, J, 1: Jlim as a is negligible. Further, it is obvious that the approximation proposed by KHACHKURUZOV(~) for G,,(O) will not influence the values of Jlim obtained in this investigation for the various electronic states of the different systems. The values of Jlim depend on D,, o,, and x0 only for Morse potential function or the function proposed by HulburtHirschfelder. Acknowledgement-The
author wishes to thank Mrs. A. SHADAC~PAN for assistance
with computations
REFERENCES 1. J. W. BOND, K. M. WATSON and J. A. WELCH, Alomic Theory qf’Gas Dynamic.), Addison-Wesley, Mass. (1965). 2. G. HERZBERG, Spectra qf Diatomic Molecules, 2nd Ed, D. Van Nostrand, New York (1950). 3. H. M. HULBURT and J. 0. HIRSCHFELDER,J. them. Phys. 35, 1901 (1961). 4. G. A. KHACHKURUZOV, Opt. Spectrosc. 21, 91 (1966). 5. G. A. KHACHKURUZOV, Opt. Spectrosc. 22, 1I (1967). 6. S. SUNDARAM, Spectroscopia Molecular 21, I (1972).
Reading,