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The determination of Re for the HCl molecule
JOURNAL
OF MOLECULAR
SPECTROSCOPY
The Determination
39, 90-93 (1971)
of R, for the HCI Molecule P. R. BUNKER
Division
of Physics, National Research Council of Canada, Ottawa, Canada
In this paper the value of the isotopically independent quantity R, is obtained for the HCl molecule from the published microwave and infrared results. This is the equilibrium internuclear distance in the Born-Oppenheimer approximation and its value is found to be 1.27460 f 0.00005~.
In the Born-Oppenheimer approximation the equilibrium internuclear distance of a molecule is unaffected by isotopic substitution, since the potential energy curve is independent of nuclear mass. This equilibrium internuclear distance is called R, and we also define B, = h/(8a2pR2c) and Beatomic = h/(8?r2~atomicR,2c), where p is the nuclear reduced mass and patomicis the at’omic reduced mass. From the experimentally derived vibration-rotation energy levels it is possible bo determine the LLeffectiveB, value”, called Yol , which is the coefficient of J(J + 1) in the expression for the energy level (1). If the values of Yol for several isotopes of a molecule are available it is possible (2) to use them to obt,ain the isotopically independent quantities pB, and R, . Further quantities of interest are Rf, which is the isotopically dependent equilibrium internuclear distance in the adiabatic approximation (see Eq. (7~) of (2)), and YOla? which is obt’ained from YOl by subtracting the nonadiabatic contribution (see Eq. (9) or (2)). To determine the nonadiabatic contribution to YOU, and, therefore, to determine Yoi”, it is necessary to use the rot,at,ional g-value. This has been determined by Kaiser (5) for the H35C1molecule. In this paper the value of R, for the HCl molecule is obtained from the published microwave and infrared results (4-7). The method used for determini?g R, is that discussed in (2) and applied there to determine R, (1.12523 f 0.00005 A) for the CO molecule (8). In this method we determine from the data the value of Yol for each isotope and make a graphical plot of EL*~“~~’ Y,n versus 1-l. The Yol axis gives the value of pB, (see Eq. (12) int’ercept of this plot on the ~~~~~~~ of (2)) from which R, can be directly obtained using R, = t(16.8575 f 0.0015)/(clB,)1”2,
(1)
where R, is in A, B, is in cm-‘, and p is in u (unified atomic mass units based on 1/12th the mass of an atom of the 12Cnuclide). The slope of t’his plot gives the 90
R, FOR HCl
91
value of the expression S = /L
15 + 14%
(2) +$h+>B.g, P
where the first term within the curly braces (involving the force constants oi) is the “Dunham correction” to B, (l), the second term is the adiabatic correction, and the last term, involving the rotational g-value gJ , is the nonadiabatic correction. The expression given in Eq. (2) above is that contained within the curly braces in Eq. (12) of (a) and that equation can be written PatomicYol= pB, + S/P.
(3)
The value of YOlfor each isotope is obtained t’o the required accuracy (4 parts in a million) from the equat’ion Y01 = (V - 4Yoz - YU - 0.5 Yz1)/2,
(4)
where Y is the J = 1 * 0 microwave frequency and the values of the three Dunham coefficients on the right side are taken from the infrared results for H35C1 (7); for the other four isotopes for which v has been accurately measured the values of these three Dunham coefficients can be calculated from those of H35C1by using the following approximate isotope relations: (5) These relations are slightly in error and the effect of the approximations is to add a term of negligible magnitude (involving the ui coefficients) to the expression given in Eq. (2) for the slope S of the graph of I.LatomicY~lversus p--l, and to contribute to the (undetected) curvature of the graph. The values of Yol obtained in this way, and other pertinent results, are listed in Table I. The relative uncertainty in the final values of ~atomiCYol is f0.00002 u cm-1 and the absolute uncert’ainty is f0.00004 u cm-l. The graph of ~atomioY,,l versus P--’ is a straight line to within the relative uneertainty and from the intercept (by a linear least-squares analysis) we determine that pB, = 10.37629 f
0.00006 u cm-l.
(6a)
Thus R, = 1.27460 f 0.00005 a,
(fib)
BPrniC (H35C1) = 10.59245 + 0.00006 cm-l.
(6~)
and, for example,
BUNKER
92 -
T Molecule
GJW
-
yoz
TABLE Ia
-
-
- Yll .-
H*%l Ds6C1 DWl TWl TWl
625.91924 323.29577 322.34969 222.14378 221.19540
0.0065319 0.0001407 0.0001398 o.OOoO662 0.0000656
atomic
0.30718 0.11328 0.11278 0.06434 0.06393
ye,
“(ucm-l)
YOl 10.59342 5.44880 5.43276 3.737206 3.721181
/iI
(u-‘)
_-
-
10.37724 10.37677 10.37675 10.37662 10.37664
-
1.02138049 ID.525238912 ID.523693197
l3.360224349
I3.358678628 -
* All Dunham coefficients are in cm-l.
The uncertainty in pBe is three times the standard deviat,ion obtained from the least-squares analysis. The slope of the graph is 0.0009 f 0.0001 u2 cm-l, i.e., kstomicYIJ1= 10.37629 + 0.0009 /.&-I
(7)
in u cm-‘. In (S) the value of gJ for H35C1 was determined to be 0.47 f 0.03 nuclear magnetons, and, therefore, me(~Be)(~gJ)/mp = 0.0026 f 0.0002 u2 cm-‘. Thus from Eq. (10) of (2) we deduce that Yor= = B, - 0.0017 /P
(3)
in cm-‘, where the expression for the coefficient of p-2 is 2 I Be3
P
2w,z
1-L
3 +'$kI }_ -
15 + 14Ul - 9az + 15a3 - 23ala2 +
Zl(al:! + al? 2
-0.0017
1
(9)
f 0.0003 u” cm-‘.
We can use the results to determine Rea if we assume that the values of al , and a3 determined in Ref. (7) are not greatly affected by the breaddown of the Born-Oppenheimer approximation. From Eqs. (8) and (9) we can write a2 ,
1 =B e+4B,ak 1= h (Jb2
15 + 14a1 - 9u2 + 15~~ - 23a1a2 +
2l(a12 + al”) 2
(10)
%r2~(R,“)%’
which follows from the definition of Rea. Using the values of the ai given in Ref (7) and the results in Eqs. (6a) and (8) we deduce from Eq. (lo), by approximately expanding the square root, that Ree = { 16.8575/[10.37629 - (0.0017 - O.O0013)ji-‘1)u2 z 1.27460 (1 + 0.060077 /A-‘)
(11)
R, FOR HCI
93
from which 2La(H35C1) -
Rea(D35C1) = 0.00005 8
(12a)
R,“(D3G1)
Rea(T3S1)
(lab)
and
-
= 0.000016 A%
lcl = -3.1
(13)
In the paper by Kaiser (3) Reais called R, .Kaiser obtained the results given in Eq. (12). In summary, from the experimental results of (4-7) we can deduce the result given in Eq. (7) from which the results in Eq. (6) follow. Using the value of g, determined in (S) we can deduce the results in Eqs. (8) and (9). The determination of Reaand ICI (see Eqs. (11) and (13)) depends on accepting the values of al , a2 and a3 given in Ref. (7). The precise effect of the breakdown of the BornOppenheimer approximaOion on the values of the ai needs to be determined but, it has been estimated to be very small (-0.1%) by Tipping and Herman (9). RECEIVED:
January
13, 1971 REFERENCES
1. 2. 5. 4. 6. 6. 7. 8. 9.
J. L. DUNHAM, Phys. Rev. 41,721 (1932). P. R. BUNKER, J. Mol. Speetrosc. 36,306 (1970). E. W. K-USER, J. Chem. Phys. 63,1686 (1970). G. JONESANDW. GORDY, Phys. Rev. 136, Al229 (1964). M. COWAN ANDW. GORDY, Phys. Rev. 111,209 (1958). C. A. BURRUS, W. GORDY, B. BENJAMIN,.~NDR. LIVINGSTON,Phys. Rev. 97, 1661 (1955). D. H. R-INK, B. S. RAO, ANDT. A. WIGGINS, J. Mol. Speckosc. 17,122 (1965). P.R. BUNKER, J. Mol. Spectrosc., 37, 197 (1971). R. TIPPINGAND R. M. HERMAN, J. Chem. Phys. 44.3112 (1966).
OF MOLECULAR
SPECTROSCOPY
The Determination
39, 90-93 (1971)
of R, for the HCI Molecule P. R. BUNKER
Division
of Physics, National Research Council of Canada, Ottawa, Canada
In this paper the value of the isotopically independent quantity R, is obtained for the HCl molecule from the published microwave and infrared results. This is the equilibrium internuclear distance in the Born-Oppenheimer approximation and its value is found to be 1.27460 f 0.00005~.
In the Born-Oppenheimer approximation the equilibrium internuclear distance of a molecule is unaffected by isotopic substitution, since the potential energy curve is independent of nuclear mass. This equilibrium internuclear distance is called R, and we also define B, = h/(8a2pR2c) and Beatomic = h/(8?r2~atomicR,2c), where p is the nuclear reduced mass and patomicis the at’omic reduced mass. From the experimentally derived vibration-rotation energy levels it is possible bo determine the LLeffectiveB, value”, called Yol , which is the coefficient of J(J + 1) in the expression for the energy level (1). If the values of Yol for several isotopes of a molecule are available it is possible (2) to use them to obt,ain the isotopically independent quantities pB, and R, . Further quantities of interest are Rf, which is the isotopically dependent equilibrium internuclear distance in the adiabatic approximation (see Eq. (7~) of (2)), and YOla? which is obt’ained from YOl by subtracting the nonadiabatic contribution (see Eq. (9) or (2)). To determine the nonadiabatic contribution to YOU, and, therefore, to determine Yoi”, it is necessary to use the rot,at,ional g-value. This has been determined by Kaiser (5) for the H35C1molecule. In this paper the value of R, for the HCl molecule is obtained from the published microwave and infrared results (4-7). The method used for determini?g R, is that discussed in (2) and applied there to determine R, (1.12523 f 0.00005 A) for the CO molecule (8). In this method we determine from the data the value of Yol for each isotope and make a graphical plot of EL*~“~~’ Y,n versus 1-l. The Yol axis gives the value of pB, (see Eq. (12) int’ercept of this plot on the ~~~~~~~ of (2)) from which R, can be directly obtained using R, = t(16.8575 f 0.0015)/(clB,)1”2,
(1)
where R, is in A, B, is in cm-‘, and p is in u (unified atomic mass units based on 1/12th the mass of an atom of the 12Cnuclide). The slope of t’his plot gives the 90
R, FOR HCl
91
value of the expression S = /L
15 + 14%
(2) +$h+>B.g, P
where the first term within the curly braces (involving the force constants oi) is the “Dunham correction” to B, (l), the second term is the adiabatic correction, and the last term, involving the rotational g-value gJ , is the nonadiabatic correction. The expression given in Eq. (2) above is that contained within the curly braces in Eq. (12) of (a) and that equation can be written PatomicYol= pB, + S/P.
(3)
The value of YOlfor each isotope is obtained t’o the required accuracy (4 parts in a million) from the equat’ion Y01 = (V - 4Yoz - YU - 0.5 Yz1)/2,
(4)
where Y is the J = 1 * 0 microwave frequency and the values of the three Dunham coefficients on the right side are taken from the infrared results for H35C1 (7); for the other four isotopes for which v has been accurately measured the values of these three Dunham coefficients can be calculated from those of H35C1by using the following approximate isotope relations: (5) These relations are slightly in error and the effect of the approximations is to add a term of negligible magnitude (involving the ui coefficients) to the expression given in Eq. (2) for the slope S of the graph of I.LatomicY~lversus p--l, and to contribute to the (undetected) curvature of the graph. The values of Yol obtained in this way, and other pertinent results, are listed in Table I. The relative uncertainty in the final values of ~atomiCYol is f0.00002 u cm-1 and the absolute uncert’ainty is f0.00004 u cm-l. The graph of ~atomioY,,l versus P--’ is a straight line to within the relative uneertainty and from the intercept (by a linear least-squares analysis) we determine that pB, = 10.37629 f
0.00006 u cm-l.
(6a)
Thus R, = 1.27460 f 0.00005 a,
(fib)
BPrniC (H35C1) = 10.59245 + 0.00006 cm-l.
(6~)
and, for example,
BUNKER
92 -
T Molecule
GJW
-
yoz
TABLE Ia
-
-
- Yll .-
H*%l Ds6C1 DWl TWl TWl
625.91924 323.29577 322.34969 222.14378 221.19540
0.0065319 0.0001407 0.0001398 o.OOoO662 0.0000656
atomic
0.30718 0.11328 0.11278 0.06434 0.06393
ye,
“(ucm-l)
YOl 10.59342 5.44880 5.43276 3.737206 3.721181
/iI
(u-‘)
_-
-
10.37724 10.37677 10.37675 10.37662 10.37664
-
1.02138049 ID.525238912 ID.523693197
l3.360224349
I3.358678628 -
* All Dunham coefficients are in cm-l.
The uncertainty in pBe is three times the standard deviat,ion obtained from the least-squares analysis. The slope of the graph is 0.0009 f 0.0001 u2 cm-l, i.e., kstomicYIJ1= 10.37629 + 0.0009 /.&-I
(7)
in u cm-‘. In (S) the value of gJ for H35C1 was determined to be 0.47 f 0.03 nuclear magnetons, and, therefore, me(~Be)(~gJ)/mp = 0.0026 f 0.0002 u2 cm-‘. Thus from Eq. (10) of (2) we deduce that Yor= = B, - 0.0017 /P
(3)
in cm-‘, where the expression for the coefficient of p-2 is 2 I Be3
P
2w,z
1-L
3 +'$kI }_ -
15 + 14Ul - 9az + 15a3 - 23ala2 +
Zl(al:! + al? 2
-0.0017
1
(9)
f 0.0003 u” cm-‘.
We can use the results to determine Rea if we assume that the values of al , and a3 determined in Ref. (7) are not greatly affected by the breaddown of the Born-Oppenheimer approximation. From Eqs. (8) and (9) we can write a2 ,
1 =B e+4B,ak 1= h (Jb2
15 + 14a1 - 9u2 + 15~~ - 23a1a2 +
2l(a12 + al”) 2
(10)
%r2~(R,“)%’
which follows from the definition of Rea. Using the values of the ai given in Ref (7) and the results in Eqs. (6a) and (8) we deduce from Eq. (lo), by approximately expanding the square root, that Ree = { 16.8575/[10.37629 - (0.0017 - O.O0013)ji-‘1)u2 z 1.27460 (1 + 0.060077 /A-‘)
(11)
R, FOR HCI
93
from which 2La(H35C1) -
Rea(D35C1) = 0.00005 8
(12a)
R,“(D3G1)
Rea(T3S1)
(lab)
and
-
= 0.000016 A%
lcl = -3.1
(13)
In the paper by Kaiser (3) Reais called R, .Kaiser obtained the results given in Eq. (12). In summary, from the experimental results of (4-7) we can deduce the result given in Eq. (7) from which the results in Eq. (6) follow. Using the value of g, determined in (S) we can deduce the results in Eqs. (8) and (9). The determination of Reaand ICI (see Eqs. (11) and (13)) depends on accepting the values of al , a2 and a3 given in Ref. (7). The precise effect of the breakdown of the BornOppenheimer approximaOion on the values of the ai needs to be determined but, it has been estimated to be very small (-0.1%) by Tipping and Herman (9). RECEIVED:
January
13, 1971 REFERENCES
1. 2. 5. 4. 6. 6. 7. 8. 9.
J. L. DUNHAM, Phys. Rev. 41,721 (1932). P. R. BUNKER, J. Mol. Speetrosc. 36,306 (1970). E. W. K-USER, J. Chem. Phys. 63,1686 (1970). G. JONESANDW. GORDY, Phys. Rev. 136, Al229 (1964). M. COWAN ANDW. GORDY, Phys. Rev. 111,209 (1958). C. A. BURRUS, W. GORDY, B. BENJAMIN,.~NDR. LIVINGSTON,Phys. Rev. 97, 1661 (1955). D. H. R-INK, B. S. RAO, ANDT. A. WIGGINS, J. Mol. Speckosc. 17,122 (1965). P.R. BUNKER, J. Mol. Spectrosc., 37, 197 (1971). R. TIPPINGAND R. M. HERMAN, J. Chem. Phys. 44.3112 (1966).