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Zeeman effect measurements in SO2: An internal magnetic field calibrant
JOURNAL
OF
MOLECULAR
Zeeman
DOUGLAS Department
SPECTROSCOPY
83, 283-287 (1980)
Effect Measurements in SO,: An Internal Magnetic Field Calibrantl J. MILLER, of Chemistry.
ROBERT DELEON, Universit?
ofRochester.
AND
J. S. MUENTER
Rochester,
New
Yd
14627
It is proposed that the Zeeman effect of the 2,,*+ I,, transition in sulfur dioxide be used as a magnetic field calibration standard. The recent accurate beam maser measurements on this transition by Ellenbroek and Dymanus have been confirmed by electric resonance measurements in the I,, level. The Zeeman splittings most useful for calibration purposes depend linearly on the magnetic field. For perpendicular selection rules the Zeeman coefficient is 0.5282(3) MHz/kG while parallel transitions have a 0.3722(3) MHz/kc Zeeman coefficient. The sign of the rotational magnetic moment of SO, has been determined to be negative. INTRODUCTION
The use of an internal calibration standard offers many advantages for most measurement techniques. In microwave spectroscopy, the Stark effect of carbonyl sulfide has long served as a standard for calibrating electric field strengths in electric dipole moment measurements (1). In recent years a very large number of Zeeman effect measurements have been made in a number of different laboratories (2-8). Frequently, the accuracy of these experiments is less than the precision because the effective field over the volume of the sample cannot be determined to sufficient accuracy. Perhaps the greatest advantage of relative measurements arises when data from more than one laboratory are combined (9, 10). The requirements for a microwave spectroscopy Zeeman effect standard are readily stated. Necessary criteria include: (1) a large Zeeman coefficient, preferably first order, (2) a relatively intense and simple spectrum, (3) transitions in a convenient frequency region, (4) a stable and readily available gas-phase compound, and finally (5) a molecule which has known Zeeman properties determined to high absolute accuracy. A Zeeman splitting with coefficients of kO.25 kHz/kG would provide a 5 MHz separation at 10 kG. Relative frequency measurement precisions of 5 kHz would yield 0.1% calibration accuracy, satisfying requirement (1). Problems introduced by nuclear magnetic moments make requirement (2) difficult to satisfy. The large nuclear Zeeman effects are usually only weakly coupled to molecular rotation and are not useful in satisfying (1). However, nuclear moments introduce spectral complexities associated with shielding, shielding anisotropy, and hyperhne structure (1 I ). The elimination of nuclear moments and requirement (4) effectively reduce the selection to com’ This work supported by NSF Grant CHE77-12527 283
0022-2852/80/100283-05$02.00/O Copyright
0 1980 by Academic Press. Inc.
All rights of reproduction
in any form reserved
MILLER,
284
DELEON,
AND MUENTER
pounds containing carbon, oxygen, and sulfur. The obvious candidates are OCS, CO, and SOz. OCS fails criterion (1) and CO fails criterion (3). SOz, and in particular the 2,,2-+ I,, transition at 12 257 MHz, satisfies criteria (1) through (4) very well. The recent molecular beam maser Zeeman effect study of Ellenbroek and Dymanus (12) fulfills the last requirement. It is also desirable that every aspect of the Zeeman effect of a standard be understood and that the accuracy of the standard can be independently confirmed. To this end we have measured the sign of the rotational magnetic moment of SOP and have made accurate measurements of the Zeeman effect in the 1,1 level of SO, using molecular beam electric resonance spectroscopy. We originally proposed this standard two years ago (13) and during the intervening time no alternate proposals have been presented. In view of this, we propose the SOP Zeeman standard and present the additional details on the SO, Zeeman effect. MEASUREMENTS
AND DISCUSSION
The M = 0 + M = + 1 and M = 0 + M = - 1 radio frequency transitions were observed in parallel electric and magnetic fields. The Stark effect has been previously measured to very high precision (14) and the Zeeman contribution to these two transitions can be readily determined. The experimental details and calibration procedures have been discussed elsewhere (8). The results of these measurements, which depend on the sum of the aa and cc components of the rotational magnetic moment and the aa and cc susceptibility anisotropies, are given in Table I. These data depend on four of the five independent Zeeman TABLE
1
Zeeman Measurements on the III State of SO, observed Frequenciesa M=O + M=+l M=O -t).,=-1 Zeeman Splittings
" =
v1 v2
= 1965.00 KHz = 1954.84 KHz
Properties
gaa + gee = (V1tv2)/uNH = 0.6934(3) Axaa
KHz
b
M=+l - M=O M=O - M=-1 Calculated
1349.2(2)
" = 2570.7(2) KHz
+ %c
= (v2-v1)1.6H
2
[ref. 12: 0.6930(4)1
= -0.308(15) KHz/kg' = -1.23(6) x 10-6 erg/g2 mole [ -1.20(31 x 1O-6 erg/g2mole:ref
a.
E = 391.179 v/cm, H = 7415.7(15)g.
b. Experimental
frequencies with the Stark effect removed.
'elec = 1.63261) was used; this value contains the polarizability contribution to the Stark effect.
12
I
285
ZEEMAN EFFECT IN SO, TABLE II Average Angular Momentum Expressions” General:
= +[J(J+l)
t E(K) - (K+l)aE(K)/aKl
= aE(K)/aK
= +[J(Jtl)
- E(K)
+
(K-l)aE(~)/a~~
CJ b2> = 0
c
2 >=l
*02 CJ s*> = 2 +
(K-3) (K*+3) -l/2
'Jb *> = 2 - 2K(K2t3) -l'*
a.
General
= 2 +
expressions
= 0 and for
ref.
15.
-4
= 2.95523
(rt3) (K2+3)-l'*
from
202, E(K)
= 6.59 x 10
For
= 21K-(K2t3)1'21.
= 3.04412
111, E(K)
= aE(r)/aK
Y = -0.941586
was
used.
parameters of SO, and are in complete agreement with Ellenbroek and Dymanus. The Zeeman effect is given by (15) W(J,T,M)
= -
Jy;y)(1
gd.G))
1
+
J(J + 1) - 3W (25 - 1)(25 + 3)
x C Axi, i
. I
The summations are over the three principal axes. gti and (JT) are the ith components of the rotational g value and the average of the square of the angular momentum about the ith axis, respectively. For the l,, and 20, states, the (J:) can be written in closed form as shown in Table II (IS). Axii = xii - X and is the ith susceptibility anisotropy. Avogadro’s number and Planck’s constant convert Ax from erg/G2.mole to Hz/G2. Using 762.253 Hz/G for the nuclear magneton, Table II, and Ellenbroek and Dymanus’ Zeeman properties,2 the Zeeman effect for the 202+ l,, transition can be written:3 A W = (264.102M, - 78.0128MJH - [3.007 x lo-” (2 - 3M:) + 3.441 x 10e5 (2 - M3]H2. In 2 Results from Ref. (12) are: g,, = -0.6043(3), gb, = -0.11634(12), AX”. = 1.86(4), AxDb= 1.20(3), Axrr = -3.06(5) erg/G2. z For computational convenience, extra significant figures are included.
g,, = -0.08865(10),
286
MILLER, DELEON, AND MUENTER
FIG. 1. Electrodes for dc and rotating rf electric fields. The rf voltages applied to the narrow strips were the same for both electrodes. All segments of one electrode were dc ground potential, while all segments of the other electrode were at +V dc.
this expression M, and M2 are the magnetic quantum numbers for the l,, and 2,,2 states, respectively, and H is the magnetic field in gauss. For calibration purposes the most convenient and accurate procedure is to measure the largest Zeeman splitting. For the 2,,2+ I,, transition in Son, these frequency differences have no susceptibility dependence. Thus the frequency differences depend linearly on the magnetic field, which is an added convenience. When perpendicular selection rules apply the highest- and lowest-frequency components are the M2 = 0 + M, = + 1 and M, = 0 + Ml = - 1 transitions, respectively. The separation of these two components is given by pN(gna + g,,)H, and the coefficient is 0.5282(3) MHz/kG. The highest- and lowest-frequency components for parallel selection rules are the M2 = + 1 + M, = + 1 and M2 = -1 to M, = -1 transitions, respectively. This Zeeman splitting is given and the coefficient is by /..~~(0.999781g.u - 0.985075g,, - O.O14705g,,)H 0.3722(3) MHz/kG. The uncertainties quoted here are conservatively arrived at by assuming the worst case combination of the individual uncertainties quoted in Ref. (12). The only remaining unknown detail of the Zeeman effect of SO, is the absolute sign of the rotational magnetic moment. While this sign is not necessary for the use of SO, as a calibrant, a simple experiment can provide this information. For the sake of completeness, this measurement was made. The sign ambiguity arises because, using plane polarized radiation, AM = + 1 transitions cannot be distinguished from AM = - 1 transitions. Circularly polarized radiation is required for this distinction (16, 17). For the electric resonance experiments done here, circularly polarized radiation was generated perpendicular to the magnetic field. The Stark and radiation E fields were generated using two identical electrodes of the form shown in Fig. 1. The radio frequency voltage was applied to only the narrow strips. The rf voltage to the horizontal strip on each electrode was phase shifted4 290” relative to the voltage applied to each vertical strip. This produces a small region, at the intersection of the strips, of radiation which is approximately 75% circularly polarized. This arrangement was evaluated with carbonyl sulfide, where the absolute g value is known to be negative (16). The J = 1, M = 0 + M = - 1 transition was observed to be five times more intense than the J = 1, M = 0 + M = + 1 transition with a +90” phase shift. The relative intensity pattern reversed when a -90” phase shift was used. Identical behavior 4The phase shift was obtained using a Merrimac Lab. Model QH-7-4.9 quadrature hybrid.
ZEEMAN
287
EFFECT IN SO?
was observed for the l,, SO, Zeeman doublet. Thus, SO, and OCS have rotational magnetic moments of the same sign, both negative. CONCLUSION
The 2,, + l,, transition of sulfur dioxide fulfills all the requirements for a magnetic field calibration standard. In addition, no other molecule meets the necessary criteria. The Zeeman properties of this transition in SO, have been measured to an accuracy exceeding 0.1% by Ellenbroek and Dymanus and these results are confirmed here. The Zeeman coefficients are such that calibration accuracy of 0.1% can be obtained with conventional microwave frequency measurements for magnetic fields of the order of 10 kG or larger. Finally, the absolute sign of the rotational magnetic moment of SO, was observed to be negative. RECEIVED:
August
27,
1979 REFERENCES
I. (a) R. G. SHULMAN AND C. H. TOWNES. Ph.vs. Rev. 77, 500-506 (1950); (b) S. A. MARSHALL 4hn J. WEBER. Phys. Rev. 105, 1502-1506 (1957): (c) J. S. MUENTER. J. Chem. Ph?s. 48, 4544-4547 (1968); (d) F. H. DELEEIJWAND A. DYMANUS, Chem. Phxs. Lett. 7, 288-292 (1970). 2. (a) W. H. FLYGARE, W. HDTTNER, R. L. SHOEMAKER, AND P. D. FOSTER, J. Chem. Phys. 50. 1714- 1719 (1969); (b) W. H. FLYGARE AND R. C. BENSON,Mol. Phys. 20. 225-250 (1970): (c) J. MCGURK, C. L. NORRIS,H. L. TICELAAR. AND W. H. FLYGARE. J. Chrm. Ph~s. 58, 3118-3120 (1973). 3. (a) P. K. BHATTACHARYYA AND B. P. DAILEY, J. Chem. Phys. 51, 3051-3057 (1969): (b) P. K. BHATTACHARYYA,H. TAFT, N. SMITH, AND B. P. DAILEY,RCI.. Sci. In.sfrum.46,608-612( 1975). 4. B. BAK. E. HAMER, D. H. SUTTER. AND H. DREIZLER. Z. Naturforsch. A 27, 705-707 (1972). 5. R. HONER~AGERAND R. TISCHER, Z. Nuturforsch. A 29. 1695-16% (1974) and Refs. therein. 6. (a) J. VERHOEVENAND A. DYMANUS. J. Chem. Phys. St. 3222-3233 (1970); (b) S. G. KUKOL.ICH. J. Chem. Phys. 54, 8-11 (1971). 7. (a) F. H. DELEEUW AND A. DYMANUS. Chem. Phys. Letr. 7, 288-292 (1970): (b) F. H. DELEEUW AND A. DYMANUS, J. Mol. Spectrosc. 48, 427-445 (1973). 8. (a, R. E. DAVIS AND J. S. MUENTER, Chem. Phys. Lett. 24, 343-345 (1974): (b) R. E. DAVIS AND J. S. MUENTER, J. Chem. Phys. 61, 2940-2945 (1974). 9. J. M. POCHEN. R. G. STONE, AND W. H. FLYGARE, J. Chem. Phys. 51, 4278-4286 (1%9). 10. K. M. MACK AND J. S. MUENTER, J. Chem. Phys. 66, 5278-5283 (1977). II. B. FABRICANT AND J. S. MUENTER. J. Chem. Phys. 66, 5274-5277 (1977). 12. A. W. ELLENBROEKAND A. DYMANUS, Chem. Phys. Lett. 42, 303-306 (1976). 13. D. J. MILLER AND J. S. MUENTER, Paper RCS. 32nd Symposium on Molecular Spectroscopy, 1977. 14. K. M. MACK, Thesis, University of Rochester, 1978; D. PATEL. D. MARGOLESE,AND T. R. DYKE. J. Chem. Phys. 70, 2740-2747 (1979). 15. B. F. BURKE AND M. W. P. STRANDBERG,Phys. Rev. 90. 303-306 16. J. R. ESHBACH AND M. W. P. STRANDBERG,Phys. Rev. 85, 24-34
(1953). (1952).
17. T. R. LAWRENCE, C. H. ANDERSON, AND N. F. RAMSEY. Phvs. Rev.
130, 1865-1870 (1963).
OF
MOLECULAR
Zeeman
DOUGLAS Department
SPECTROSCOPY
83, 283-287 (1980)
Effect Measurements in SO,: An Internal Magnetic Field Calibrantl J. MILLER, of Chemistry.
ROBERT DELEON, Universit?
ofRochester.
AND
J. S. MUENTER
Rochester,
New
Yd
14627
It is proposed that the Zeeman effect of the 2,,*+ I,, transition in sulfur dioxide be used as a magnetic field calibration standard. The recent accurate beam maser measurements on this transition by Ellenbroek and Dymanus have been confirmed by electric resonance measurements in the I,, level. The Zeeman splittings most useful for calibration purposes depend linearly on the magnetic field. For perpendicular selection rules the Zeeman coefficient is 0.5282(3) MHz/kG while parallel transitions have a 0.3722(3) MHz/kc Zeeman coefficient. The sign of the rotational magnetic moment of SO, has been determined to be negative. INTRODUCTION
The use of an internal calibration standard offers many advantages for most measurement techniques. In microwave spectroscopy, the Stark effect of carbonyl sulfide has long served as a standard for calibrating electric field strengths in electric dipole moment measurements (1). In recent years a very large number of Zeeman effect measurements have been made in a number of different laboratories (2-8). Frequently, the accuracy of these experiments is less than the precision because the effective field over the volume of the sample cannot be determined to sufficient accuracy. Perhaps the greatest advantage of relative measurements arises when data from more than one laboratory are combined (9, 10). The requirements for a microwave spectroscopy Zeeman effect standard are readily stated. Necessary criteria include: (1) a large Zeeman coefficient, preferably first order, (2) a relatively intense and simple spectrum, (3) transitions in a convenient frequency region, (4) a stable and readily available gas-phase compound, and finally (5) a molecule which has known Zeeman properties determined to high absolute accuracy. A Zeeman splitting with coefficients of kO.25 kHz/kG would provide a 5 MHz separation at 10 kG. Relative frequency measurement precisions of 5 kHz would yield 0.1% calibration accuracy, satisfying requirement (1). Problems introduced by nuclear magnetic moments make requirement (2) difficult to satisfy. The large nuclear Zeeman effects are usually only weakly coupled to molecular rotation and are not useful in satisfying (1). However, nuclear moments introduce spectral complexities associated with shielding, shielding anisotropy, and hyperhne structure (1 I ). The elimination of nuclear moments and requirement (4) effectively reduce the selection to com’ This work supported by NSF Grant CHE77-12527 283
0022-2852/80/100283-05$02.00/O Copyright
0 1980 by Academic Press. Inc.
All rights of reproduction
in any form reserved
MILLER,
284
DELEON,
AND MUENTER
pounds containing carbon, oxygen, and sulfur. The obvious candidates are OCS, CO, and SOz. OCS fails criterion (1) and CO fails criterion (3). SOz, and in particular the 2,,2-+ I,, transition at 12 257 MHz, satisfies criteria (1) through (4) very well. The recent molecular beam maser Zeeman effect study of Ellenbroek and Dymanus (12) fulfills the last requirement. It is also desirable that every aspect of the Zeeman effect of a standard be understood and that the accuracy of the standard can be independently confirmed. To this end we have measured the sign of the rotational magnetic moment of SOP and have made accurate measurements of the Zeeman effect in the 1,1 level of SO, using molecular beam electric resonance spectroscopy. We originally proposed this standard two years ago (13) and during the intervening time no alternate proposals have been presented. In view of this, we propose the SOP Zeeman standard and present the additional details on the SO, Zeeman effect. MEASUREMENTS
AND DISCUSSION
The M = 0 + M = + 1 and M = 0 + M = - 1 radio frequency transitions were observed in parallel electric and magnetic fields. The Stark effect has been previously measured to very high precision (14) and the Zeeman contribution to these two transitions can be readily determined. The experimental details and calibration procedures have been discussed elsewhere (8). The results of these measurements, which depend on the sum of the aa and cc components of the rotational magnetic moment and the aa and cc susceptibility anisotropies, are given in Table I. These data depend on four of the five independent Zeeman TABLE
1
Zeeman Measurements on the III State of SO, observed Frequenciesa M=O + M=+l M=O -t).,=-1 Zeeman Splittings
" =
v1 v2
= 1965.00 KHz = 1954.84 KHz
Properties
gaa + gee = (V1tv2)/uNH = 0.6934(3) Axaa
KHz
b
M=+l - M=O M=O - M=-1 Calculated
1349.2(2)
" = 2570.7(2) KHz
+ %c
= (v2-v1)1.6H
2
[ref. 12: 0.6930(4)1
= -0.308(15) KHz/kg' = -1.23(6) x 10-6 erg/g2 mole [ -1.20(31 x 1O-6 erg/g2mole:ref
a.
E = 391.179 v/cm, H = 7415.7(15)g.
b. Experimental
frequencies with the Stark effect removed.
'elec = 1.63261) was used; this value contains the polarizability contribution to the Stark effect.
12
I
285
ZEEMAN EFFECT IN SO, TABLE II Average Angular Momentum Expressions” General:
t E(K) - (K+l)aE(K)/aKl
= +[J(Jtl)
- E(K)
+
(K-l)aE(~)/a~~
CJ b2> = 0
c
2 >=l
*02 CJ s*> = 2 +
(K-3) (K*+3) -l/2
'Jb *> = 2 - 2K(K2t3) -l'*
a.
General
= 2 +
expressions
= 0 and for
ref.
15.
-4
= 2.95523
(rt3) (K2+3)-l'*
from
202, E(K)
= 6.59 x 10
For
= 21K-(K2t3)1'21.
= 3.04412
111, E(K)
= aE(r)/aK
Y = -0.941586
was
used.
parameters of SO, and are in complete agreement with Ellenbroek and Dymanus. The Zeeman effect is given by (15) W(J,T,M)
= -
Jy;y)(1
gd.G))
1
+
J(J + 1) - 3W (25 - 1)(25 + 3)
x C Axi, i
. I
The summations are over the three principal axes. gti and (JT) are the ith components of the rotational g value and the average of the square of the angular momentum about the ith axis, respectively. For the l,, and 20, states, the (J:) can be written in closed form as shown in Table II (IS). Axii = xii - X and is the ith susceptibility anisotropy. Avogadro’s number and Planck’s constant convert Ax from erg/G2.mole to Hz/G2. Using 762.253 Hz/G for the nuclear magneton, Table II, and Ellenbroek and Dymanus’ Zeeman properties,2 the Zeeman effect for the 202+ l,, transition can be written:3 A W = (264.102M, - 78.0128MJH - [3.007 x lo-” (2 - 3M:) + 3.441 x 10e5 (2 - M3]H2. In 2 Results from Ref. (12) are: g,, = -0.6043(3), gb, = -0.11634(12), AX”. = 1.86(4), AxDb= 1.20(3), Axrr = -3.06(5) erg/G2. z For computational convenience, extra significant figures are included.
g,, = -0.08865(10),
286
MILLER, DELEON, AND MUENTER
FIG. 1. Electrodes for dc and rotating rf electric fields. The rf voltages applied to the narrow strips were the same for both electrodes. All segments of one electrode were dc ground potential, while all segments of the other electrode were at +V dc.
this expression M, and M2 are the magnetic quantum numbers for the l,, and 2,,2 states, respectively, and H is the magnetic field in gauss. For calibration purposes the most convenient and accurate procedure is to measure the largest Zeeman splitting. For the 2,,2+ I,, transition in Son, these frequency differences have no susceptibility dependence. Thus the frequency differences depend linearly on the magnetic field, which is an added convenience. When perpendicular selection rules apply the highest- and lowest-frequency components are the M2 = 0 + M, = + 1 and M, = 0 + Ml = - 1 transitions, respectively. The separation of these two components is given by pN(gna + g,,)H, and the coefficient is 0.5282(3) MHz/kG. The highest- and lowest-frequency components for parallel selection rules are the M2 = + 1 + M, = + 1 and M2 = -1 to M, = -1 transitions, respectively. This Zeeman splitting is given and the coefficient is by /..~~(0.999781g.u - 0.985075g,, - O.O14705g,,)H 0.3722(3) MHz/kG. The uncertainties quoted here are conservatively arrived at by assuming the worst case combination of the individual uncertainties quoted in Ref. (12). The only remaining unknown detail of the Zeeman effect of SO, is the absolute sign of the rotational magnetic moment. While this sign is not necessary for the use of SO, as a calibrant, a simple experiment can provide this information. For the sake of completeness, this measurement was made. The sign ambiguity arises because, using plane polarized radiation, AM = + 1 transitions cannot be distinguished from AM = - 1 transitions. Circularly polarized radiation is required for this distinction (16, 17). For the electric resonance experiments done here, circularly polarized radiation was generated perpendicular to the magnetic field. The Stark and radiation E fields were generated using two identical electrodes of the form shown in Fig. 1. The radio frequency voltage was applied to only the narrow strips. The rf voltage to the horizontal strip on each electrode was phase shifted4 290” relative to the voltage applied to each vertical strip. This produces a small region, at the intersection of the strips, of radiation which is approximately 75% circularly polarized. This arrangement was evaluated with carbonyl sulfide, where the absolute g value is known to be negative (16). The J = 1, M = 0 + M = - 1 transition was observed to be five times more intense than the J = 1, M = 0 + M = + 1 transition with a +90” phase shift. The relative intensity pattern reversed when a -90” phase shift was used. Identical behavior 4The phase shift was obtained using a Merrimac Lab. Model QH-7-4.9 quadrature hybrid.
ZEEMAN
287
EFFECT IN SO?
was observed for the l,, SO, Zeeman doublet. Thus, SO, and OCS have rotational magnetic moments of the same sign, both negative. CONCLUSION
The 2,, + l,, transition of sulfur dioxide fulfills all the requirements for a magnetic field calibration standard. In addition, no other molecule meets the necessary criteria. The Zeeman properties of this transition in SO, have been measured to an accuracy exceeding 0.1% by Ellenbroek and Dymanus and these results are confirmed here. The Zeeman coefficients are such that calibration accuracy of 0.1% can be obtained with conventional microwave frequency measurements for magnetic fields of the order of 10 kG or larger. Finally, the absolute sign of the rotational magnetic moment of SO, was observed to be negative. RECEIVED:
August
27,
1979 REFERENCES
I. (a) R. G. SHULMAN AND C. H. TOWNES. Ph.vs. Rev. 77, 500-506 (1950); (b) S. A. MARSHALL 4hn J. WEBER. Phys. Rev. 105, 1502-1506 (1957): (c) J. S. MUENTER. J. Chem. Ph?s. 48, 4544-4547 (1968); (d) F. H. DELEEIJWAND A. DYMANUS, Chem. Phxs. Lett. 7, 288-292 (1970). 2. (a) W. H. FLYGARE, W. HDTTNER, R. L. SHOEMAKER, AND P. D. FOSTER, J. Chem. Phys. 50. 1714- 1719 (1969); (b) W. H. FLYGARE AND R. C. BENSON,Mol. Phys. 20. 225-250 (1970): (c) J. MCGURK, C. L. NORRIS,H. L. TICELAAR. AND W. H. FLYGARE. J. Chrm. Ph~s. 58, 3118-3120 (1973). 3. (a) P. K. BHATTACHARYYA AND B. P. DAILEY, J. Chem. Phys. 51, 3051-3057 (1969): (b) P. K. BHATTACHARYYA,H. TAFT, N. SMITH, AND B. P. DAILEY,RCI.. Sci. In.sfrum.46,608-612( 1975). 4. B. BAK. E. HAMER, D. H. SUTTER. AND H. DREIZLER. Z. Naturforsch. A 27, 705-707 (1972). 5. R. HONER~AGERAND R. TISCHER, Z. Nuturforsch. A 29. 1695-16% (1974) and Refs. therein. 6. (a) J. VERHOEVENAND A. DYMANUS. J. Chem. Phys. St. 3222-3233 (1970); (b) S. G. KUKOL.ICH. J. Chem. Phys. 54, 8-11 (1971). 7. (a) F. H. DELEEUW AND A. DYMANUS. Chem. Phys. Letr. 7, 288-292 (1970): (b) F. H. DELEEUW AND A. DYMANUS, J. Mol. Spectrosc. 48, 427-445 (1973). 8. (a, R. E. DAVIS AND J. S. MUENTER, Chem. Phys. Lett. 24, 343-345 (1974): (b) R. E. DAVIS AND J. S. MUENTER, J. Chem. Phys. 61, 2940-2945 (1974). 9. J. M. POCHEN. R. G. STONE, AND W. H. FLYGARE, J. Chem. Phys. 51, 4278-4286 (1%9). 10. K. M. MACK AND J. S. MUENTER, J. Chem. Phys. 66, 5278-5283 (1977). II. B. FABRICANT AND J. S. MUENTER. J. Chem. Phys. 66, 5274-5277 (1977). 12. A. W. ELLENBROEKAND A. DYMANUS, Chem. Phys. Lett. 42, 303-306 (1976). 13. D. J. MILLER AND J. S. MUENTER, Paper RCS. 32nd Symposium on Molecular Spectroscopy, 1977. 14. K. M. MACK, Thesis, University of Rochester, 1978; D. PATEL. D. MARGOLESE,AND T. R. DYKE. J. Chem. Phys. 70, 2740-2747 (1979). 15. B. F. BURKE AND M. W. P. STRANDBERG,Phys. Rev. 90. 303-306 16. J. R. ESHBACH AND M. W. P. STRANDBERG,Phys. Rev. 85, 24-34
(1953). (1952).
17. T. R. LAWRENCE, C. H. ANDERSON, AND N. F. RAMSEY. Phvs. Rev.
130, 1865-1870 (1963).