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Analysis of the inversion splittings in methylhydrazine
JOURNAL
OF MOLECULAR
SPECTROSCOPY
138,497-504 (1989)
Analysis of the inversion Splittings in Methylhydrazine NOBUKIMI OHASHI L)epartment of Physics, Faculty of Science, Kanazawa University, Kanazawa, Ishikawa 920. Japan
NORIO MURASE Department of Chemistry, Faculty of Science, The University of Tokyo, Bunkyo-ku Tokyo 113, Japan
KAORU YAMANOUCHI Department of Pure and Applied Sciences, College of Arts and Sciences, The University of Tokyo, Meguro-ku, Tokyo 153. Japan
MASAAKI SUGIE, HARUTOSHI TAKEO, AND CHI MATSUMURA National Chemical Laboratory for Industry, Tsukuba, Ibaraki 305, Japan
AND
Kozo KUCHITSU Department of Chemistry, Nagaoka University of Technology. Nagaoka Niigata 940-21, Japan
Microwavespectraldataon the inversionsplittingsin methylhydrazinehave been analyzedby the use of a generalizedinternal-axismethoddevelopedby Hougen.Inversiontunneling parameters reproducing the observed splittings well have been determined through a least-squares analysis for the inner and outer conformations. Basedon the results obtained from the analysis, a simple tunneling-path model is tested. 0 1989 Academic press, I~C.
INTRODUCTION
Methylhydrazine ( CH3NHNH2) has large-amplitude motions (the N-N torsional vibration, the N-H inversion motion, and the CH3 torsional vibration (l-3)), because of which it is an interesting molecule from spectroscopic and theoretical viewpoints. The existence of the inner and outer conformations has been established by Lattimer and Harmony (I ) by microwave spectroscopy. In a recent microwave study (4), the inversion splittings were found to depend remarkably on the Jand Kquantum numbers for the inner and outer conformations. The “inversion” in methylhydrazine is characterized as the N-H wagging motion accompanied by the torsional motions around the N-N and N-C axes. These three types of large-amplitude motions that occur simultaneously generate vibrational angular momenta along three different directions, being far off the principal axes. For this reason, it is difficult to obtain a simple energy level expression because various 497
0022-2852189
$3.00
Copyright Q 1989 by Academic Press, Inc. All rights of reproduction in any form mewed.
498
OHASHI
ET AL.
vibration-rotation couplings must be taken into account, An effective way to solve this problem is to use a generalized internal-axis method developed by Hougen (5) originally for his analysis of the spectra of the water dimer. The present study aims to make a satisfactory least-squares fit of the microwave spectral data given in Ref. (4) by the use of the formalism developed by Hougen (5). HAMILTONIAN
MATRIX
ELEMENTS
By use of the theory developed in Ref. (5)) we give the Hamiltonian matrix elements for analyzing the microwave spectral data of methylhydrazine. We assume that the imino inversion tunneling occurs only along one tunneling path, which makes it possible to use Hougen’s formalism (5) with a single path parameter. We treat the present problem in two frameworks and in an internal-axis system in which the internal angular momenta are canceled out along the tunneling path. We use a set of effective Hamiltonian operators which is composed of rotational and vibrational terms in the form
He,=Hr+Hv,
(1)
and we designate the inversion-rotation states in the two frameworks as 11; J, K) and 12; J, K). The non-tunneling elements of the Hamiltonian matrix, ( 1; J, K’ 1H, 11; are then set equal to the usual Hamiltonian matrix J,K”)(=(2; J,K’IH,l2;J,K”)), elements for an asymmetric rotor. If the tunneling process that occurs through the rotational operator H, is ignored, the tunneling matrix elements are given by (6) (1; J, K’I Hl2; J, K”) = (2; J, K”IH(1; J, K’)* = (1; J,K’IH,j2;
J, K”)
(2) where D$:Kt is the rotation matrix and X, 8, and CJare the Eulerian-type angles introduced as the “axis-switching” angles characteristic of Hougen’s generalized intemalaxis method (5). The quantity h12 is represented as
where I,!J~” and 3/z”are the vibrational wavefunctions for frameworks 1 and 2, respectively. As seen in Eq. (2), the modifications for the tunneling matrix elements arise from the presence of the Den factors and, generally, they depend on any of X, 8, and 4. In the present two-framework problem, any eigenvalue of the Hamiltonian matrix depends on X + $ instead of X and 4 independently, as proved in the Appendix. For this reason, we represent the tunneling matrix elements as (1; J, ICI Hl2; J, K”) = (2; J, K”( H( 1; J, K’)* = exp[i(K’+
K”)(x
+
$)/2]d$&
(0) -h12
(4)
499
INVERSION SPLITTINGS IN CH,NHNH2
in place of the expression given in Eq. (2)) where dg(:!l is defined as 0, 6)K”Kl = exp[iK”x].d~~K’(e).exp[iK’~].
D’?X,
(5)
DATA ANALYSIS
A least-squares analysis has been carried out on the microwave spectral data given in Ref. (4). For the inner and outer conformations the least-squares fits were made by adjusting hi2, X + $, and 0, along with the rotational constants and centrifugal distortion constants AJ, A,,,, AK, 6J, and &. For the inner conformation, which shows more drastic J and K dependences of the inversion splittings, it was necessary to replace h12 by h12 + h&(J + 1) in order to achieve a reasonable standard deviation. The inversion tunneling parameters thus obtained are given in Table I along with redetermined rotational and centrifugal distortion constants. The rotational constants redetermined in the present fits agree with those in Ref. (4) within quoted errors. The centrifugal distortion constants, A,, etc., obtained in the present analyses reproduce the centrifugal distortion constants denoted by the letter d (with various subscripts) in Ref. (4) within quoted errors except for a small discrepancy in dJK for the inner conformation, and in dKand dwK for the outer conformation. The results of the fittings are listed in Tables II and III. DISCUSSION
The inversion splittings, 12h12( , for J = K = 0 are found to be 152.53 (2.50 = 0.11) MHz and 21.622 (2.5~ = 0.042) MHz for the inner and outer conformations, respectively. These data can be used to determine the potential function for the inversion motion. According to the concept of the tunneling path developed by Hougen (5), the potential function for the inversion motion can be expressed in terms of a path variable,
TABLE I Inversion Tunneling Parameters and Molecular Constants for the Inner and Outer Conformations of Methylhydrazine
hl2
/MHz
-76.266(57)
hl2j/MHZ x
+
-10.810(45)
-0.00179(75)
cb/l-.%d
2.9618(23)
2.8670(39)
e
/rad
0.02104(80)
0.0275(15)
A
/MHZ
36
704.413(95)
38
B
/MHZ
9
689.794(25)
9
591.871(34)
C
/MHZ
8
531.930(24)
8
494.412(30)
AJ
/MHz
AJK
/MHz
AK
/MHz
0.300
O(77)
0.343
3(W)
SJ
/MHz
0.001
7161(15)
0.001
819(16)
6K
/MHz
2(10)
0.002
9127)
apply
to
Numbers last
digits
0.009 -0.030
-0.002
in
parentheses of
the
091.57(13)
03(21)
0.008
90(87)
denote parameters.
-0.032
2.55
and
65(26) 7(12)
the
500
OHASHI ET AL. TABLE II Inversion Splittings (MHz) for the Inner Conformation of Methylhydrazine UPPER
J
LOWER
Ka Kc
J
Ka Kc
OBSa
CALC
o-c
1
0
l-
0
0
0
0.00
2
0
2-
1
0
1
0.00
2
1
l-
1
1
0
0.00
0.05
-0.05
1
-0.02
2
1
17
3
214
1
-0.03 0.05
0.00
0.02
17
3
15
1.59
1.31
0.28
1 -
0.03 -0.05
1
1
l-
0
0
0
2.56
2.49
0.07
3
0
3-
2
1
2
2.42
2.31
0.11
4
0
4-
3
1
3
2.30
2.22
0.08
6
1
5-
5
2
4
7.07
6.63
5
3
2-
6
2
5
9.92
10.34
-0.42
5
3
3-
6
2
4
9.82
10.27
-0.45
9
2
8-
8
3
5
15.74
15.53
0.21
1
1
o-
1
0
1
2.55
2.50
0.05
2
1
l-
2
0
2
2.57
2.50
0.07
3
1
2-
3
0
3
2.62
2.54
0.08 0.04
0.24
4
1
3-
4
0
4
2.63
2.59
5
1
4-
5
0
5
2.70
2.66
1
1
l-
1
0
1
302.42
302.54
2
1
2-
2
0
2
302.59
302.51
3
1
3-
3
0
3
302.34
302.40
-0.06 -0.05
0.04 -0.12 0.08
4
1
4-
4
0
4
302.19
302.24
5
1
5-
5
0
5
302.13
302.04
0.09
6
1
6-
6
0
6
301.82
301.78
0.04
7
1
7-
7
0
7
301.86
301.47
8
1
8-
8
0
8
301.10
301.12
16
2
15
-
16
1
15
281.91
281.90
18
2
17
-
18
1
17
278.73
278.76
-0.03
19
2
18
-
19
1
18
277.06
277.10
-0.04
23
3
21
-
23
2
21
252.50
253.49
-0.99
26
3
24
-
26
2
24
245.74
245.56
27
3
25
-
27
2
25
242.96
242.76
0.20
28
3
26
-
28
2
26
240.09
239.94
0.15
32
4
29
-
32
3
29
210.70
210.52
0.18
BRef.
4.
0.39 -0.02 0.01
0.18
is a suitably chosen function of the variables representing N-H wagging, N-N torsion, and CHs torsion. When a suitable variable for the tunneling path is unknown, the N-H wagging variable may be taken as an approximate path variable as in Ref. (4) for determining the barrier heights for the N-H inversion. By the use of the inversion splittings 152.5 and 21.62 MHz determined in the present study and the doubleminimum potential reported in Ref. (4), the barrier heights are obtained to be 1958.7(2) and 2474( 2) cm-’ for the inner and the outer conformations, respectively. The inversion splittings obtained above are close to the ab initio values 2290 and 2493 cm-r estimated by use of the 4-3 lG( N* ) basis set (4). As described in Refs. (5) and (6)) the parameters X, 8, and 4 are defined as the Eulerian-type angles for the “backward rotation” or “axis-switching,” which must be applied to the axis system taken in framework 2 to cancel the internal angular momentum generated by the inversion tunneling from framework 1 to 2. These parameters
which
501
INVERSION SPLITTINGS IN CHjNHNH* TABLE III Inversion Splittings (MHz) for the Outer Conformation of Methylhydrazine UPPER
J
b
LOWER
Kc
J
b
Kc
OBSa
CALC
o-c
4
0
4-
3
1
3
0.00
0.74
9
1
9-
8
2
6
2.20
2.36
-0.16
10
2
a-
9
3
7
3.44
3.70
-0.26
-0.74
2
1
1-
2
0
2
0.81
0.82
-0.01
7
1
6-
7
0
7
0.83
0.88
-0.05 -0.01
1
1
o-
0
0
0
42.41
42.42
4
0
4-
3
1
2
42.29
42.28
7
1
7-
6
2
5
39.04
39.00
a
1
8-
7
2
6
38.91
38.92
9
1
9-
8
2
7
38.81
38.81
10
2
0-
9
3
6
32.58
32.65
0.01 0.04 -0.01 0.00 -0.07
11
2
10
-
10
3
8
32.31
32.37
-0.06
13
3
10
-
12
4
8
23.75
23.79
-0.04
13
3
11
-
12
4
9
23.60
23.67
-0.07
2
1
2-
2
0
2
42.36
42.40
-0.04
3
1
3-
3
0
3
42.33
42.36
-0.03
4
1
4-
4
0
4
42.30
42.31
-0.01
5
1
5-
5
0
5
42.26
42.24
0.02
6
1
6-
6
0
6
42.17
42.15
0.02
7
1
7-
7
0
7
42.05
42.05
0.00
8
1
a-
8
0
8
41.99
41.92
0.07
9
1
9-
9
0
9
41.88
41.78
12
2
11
-
12
1
11
37.48
37.61
0.10 -0.13
13
2
12
-
13
1
12
37.29
37.29
0.00
14
2
13
-
14
1
13
36.95
36.93
0.02
17
2
16
-
17
1
16
35.74
35.64
0.10
20
2
19
-
20
1
19
34.27
34.12
0.15
29
3
27
-
29
2
27
23.69
23.70
-0.01
30
3
28
-
30
2
28
22.88
22.91
-0.03
BRef.
4
may be evaluated by assuming a simple model for the inversion tunneling path and by solving numerically the coupled differential equations (49) in Ref. (5)) but they are determined in the present fitting procedure by trial and error. The tunneling path adopted in the present evaluation is described by introducing a path parameter 7 into the reference coordinate system. In the intermediate coordinate system from which the reference coordinate system is derived, the path parameter representing imino wagging, r, is defined as ai = S-*(0,7,O)a?,
i=H,,
ai = S-’ (0 = k17, 0, O)aP,
i = Hz and H3,
ai = C, (7 = LNNC)S-’
(c = k27, 0, O)a?,
where a: denotes the initial position (7), C’Jr) = S1(7r/2, y, -r/2), that the C-N-N-H and N-N-C-H atom adjacent to the methyl group
i= H4,H5andH6,
(6)
(8)
of the ith atom, S-’ is the direction cosine matrix and kl and kz are the constants used to represent dihedral angles change in proportion to T. The H is denoted as H, , and those in the amino group
502
OHASHI ET AL.
as H2 and H3. The definition of 7 in the present model, as depicted in Fig. 1, differs from that of p in Ref. (4), but 19( =kir) corresponds exactly to that defined in Ref. (4). The equilibrium values, 7, and pe, for these imino wagging variables are interrelated as sin(7,) = sin(p,)*cos(LNZN,H,
- 7r/2)/cos(cz),
(9)
where cos(a! - a/2)
= cos(LNzN,H,
- ?r/2).cos(pe).
(10)
The three coupled differential equations given by (49) of Ref. (5) were constructed by using the structural parameters and the equilibrium angles of inversion and torsion (pe and M,) determined by a combined analysis of electron diffraction and ab initio calculation (8), and the equations were then integrated numerically by use of the Runge-Kutta-Gill method subroutine DRKG. Based on considerations in the intermediate-axis system of the final orientation reached by the imino inversion motion compared to the standard orientation of the reference frameworks, a C,(z) corrective rotation (6) seems to be necessary, and we will use such a corrective rotation for comparison purposes here. The actual situation in this molecule, which has no symmetry operations in its equilibrium configuration, may be somewhat more complicated and merits further study. After this correction of the direction cosine matrix (6) from S’( X, 0, 4) to S’( X, 8, 4) C,(z), X, 8, and 4 values are obtained as listed in Table IV. Since the eigenvalues of the Hamiltonian matrix are unchanged under the transformation from X + 4 to X + d + 27r (see Appendix), the X + 4 + 27r values are also given in Table IV for comparison with the x + 4 values in Table I derived from the least-squares analyses. The angles obtained from the numerical integrations agree
Y
cii,
H,
FIG. 1. Definition of the tunneling path parameter 7 representing the imino inversion motion for the inner conformation.
503
INVERSION SPLITTINGS IN CHjNHNH> TABLE IV Angles as Determined from Numerical Integration of the Coupled Differential Equations”
X/rad B/rad &/t-ad
Inner
Outer
-1.462 0.165 -1.775
-1.417 0.165 -1.819
3.046
3.047
X+&+Zx/rad
a The form of the coupleddifferentialequationsis as for Eqs.( 49) of Ref. (5 1.
qualitatively with those from the least-squares fit of the microwave spectral data set. ,4ccordingly, the inversion motion can be interpreted by a tunneling path like that represented in Eqs. (6)-( 8). APPENDIX It is possible for the present two-framework, one-path system to regard the Hamiltonian matrix elements given in Eq. (2 ) as operationally equivalent to those obtained by taking a usual symmetric-top wavefunction multiplied by a vibrational-framework function as the basis set function and using a Hamiltonian operator of the form
H = Ho + Hinv,
(Al)
Hi,-+= v.exp(i~~=).exp(-iBJ,,)-exp(I’xJ,),
(A2)
where Ho is an asymmetric rotor Hamiltonian operator and V represents a vibrational operator such that (1 lVjl>
= (2/V/2)
=o
(A3)
and (1 /Vj2)= By applying a unitary transformation
(2jVj1)=/2,2.
(A4)
S expressed as
S = exp[-i(X
- $)J,/2],
(A5)
to the Hamiltonian operator given in (A 1) above, we have a new Hamiltonian operator H”e” = S-‘HS = Hf;” + HP=“’ mv , HFiY = V.exp[i(X
+ b)J,].exp(-iU,).exp[i(x
(A(j)
+ 4)Jz].
(A7)
Consequently, we find that eigenvalues depend on X + C#J (not x and C$independently) since it is possible to take Ho such that Hrw is equal to the usual asymmetric rotor Hamiltonian, e.g., of Watson’s A-reduction type.
504
OHASHI ET AL.
From a similar discussion with the use of appropriate unitary transformation(s), it is shown that eigenvalues are unchanged under x + 4 + x + $J + 2n and/or e-,--0. ACKNOWLEDGMENTS The authors thank Dr. Jon T. Hougen of the National Institute of Standards and Technology for many helpful discussions and for kindly reading the manuscript. The authors also thank Dr. Laurent H. Coudert for kindly reading the manuscript and valuable discussions. Note added in proof: In the present paper we determined the value of x + 4 in the least squares fit. On the other hand, X - 4 is shown in Ref. (6) to take the value + * or 0 for various motions in the water dimer. The value of x - #Jfor the tunneling motion in the present molecule, which has lower molecular symmetry, should be determined after further detailed consideration. RECEIVED
August 7, 1989 REFERENCES
R. P. LA?TIMERAND M. D. HARMONY,J. Chem. Phys. 53,4575-4583 (1970). R. P. LATTIMERAND M. D. HARMONY,J. Amer. Chem. Sot. 94,35 I-356 ( 1972). J. R. DURIG,W. C. HARRIS,AND D. W. WERTZ, J. Chem. Phys. 50, 1449-1461 ( 1969). N. MURASE,K. YAMANOUCHI,M. SUGIE,H. TAKEO,C. MATSUMURA,Y. HAMADA,M. TSUBOI,AND K. KUCHITSU,J. Mol. Struct. 194,301-316 (1988). 5. J. T. HOUGEN,J. Mol. Spectrosc. 114,395-426 (1985). 6. L. H. COUDERTANDJ. T. HOUGEN,J. Mol. Spectrosc. 130, 86-l 19 ( 1988). 7. E. B. WILSON,JR., J. C. DECIUS,ANDP. C. CROSS,“Molecular Vibrations,” McGraw-Hill, New York,
1. 2. 3. 4.
1955. 8. N. MURASE,K. YAMANOUCHI,T. EGAWA,AND K. KUCHITSU,to be published.
OF MOLECULAR
SPECTROSCOPY
138,497-504 (1989)
Analysis of the inversion Splittings in Methylhydrazine NOBUKIMI OHASHI L)epartment of Physics, Faculty of Science, Kanazawa University, Kanazawa, Ishikawa 920. Japan
NORIO MURASE Department of Chemistry, Faculty of Science, The University of Tokyo, Bunkyo-ku Tokyo 113, Japan
KAORU YAMANOUCHI Department of Pure and Applied Sciences, College of Arts and Sciences, The University of Tokyo, Meguro-ku, Tokyo 153. Japan
MASAAKI SUGIE, HARUTOSHI TAKEO, AND CHI MATSUMURA National Chemical Laboratory for Industry, Tsukuba, Ibaraki 305, Japan
AND
Kozo KUCHITSU Department of Chemistry, Nagaoka University of Technology. Nagaoka Niigata 940-21, Japan
Microwavespectraldataon the inversionsplittingsin methylhydrazinehave been analyzedby the use of a generalizedinternal-axismethoddevelopedby Hougen.Inversiontunneling parameters reproducing the observed splittings well have been determined through a least-squares analysis for the inner and outer conformations. Basedon the results obtained from the analysis, a simple tunneling-path model is tested. 0 1989 Academic press, I~C.
INTRODUCTION
Methylhydrazine ( CH3NHNH2) has large-amplitude motions (the N-N torsional vibration, the N-H inversion motion, and the CH3 torsional vibration (l-3)), because of which it is an interesting molecule from spectroscopic and theoretical viewpoints. The existence of the inner and outer conformations has been established by Lattimer and Harmony (I ) by microwave spectroscopy. In a recent microwave study (4), the inversion splittings were found to depend remarkably on the Jand Kquantum numbers for the inner and outer conformations. The “inversion” in methylhydrazine is characterized as the N-H wagging motion accompanied by the torsional motions around the N-N and N-C axes. These three types of large-amplitude motions that occur simultaneously generate vibrational angular momenta along three different directions, being far off the principal axes. For this reason, it is difficult to obtain a simple energy level expression because various 497
0022-2852189
$3.00
Copyright Q 1989 by Academic Press, Inc. All rights of reproduction in any form mewed.
498
OHASHI
ET AL.
vibration-rotation couplings must be taken into account, An effective way to solve this problem is to use a generalized internal-axis method developed by Hougen (5) originally for his analysis of the spectra of the water dimer. The present study aims to make a satisfactory least-squares fit of the microwave spectral data given in Ref. (4) by the use of the formalism developed by Hougen (5). HAMILTONIAN
MATRIX
ELEMENTS
By use of the theory developed in Ref. (5)) we give the Hamiltonian matrix elements for analyzing the microwave spectral data of methylhydrazine. We assume that the imino inversion tunneling occurs only along one tunneling path, which makes it possible to use Hougen’s formalism (5) with a single path parameter. We treat the present problem in two frameworks and in an internal-axis system in which the internal angular momenta are canceled out along the tunneling path. We use a set of effective Hamiltonian operators which is composed of rotational and vibrational terms in the form
He,=Hr+Hv,
(1)
and we designate the inversion-rotation states in the two frameworks as 11; J, K) and 12; J, K). The non-tunneling elements of the Hamiltonian matrix, ( 1; J, K’ 1H, 11; are then set equal to the usual Hamiltonian matrix J,K”)(=(2; J,K’IH,l2;J,K”)), elements for an asymmetric rotor. If the tunneling process that occurs through the rotational operator H, is ignored, the tunneling matrix elements are given by (6) (1; J, K’I Hl2; J, K”) = (2; J, K”IH(1; J, K’)* = (1; J,K’IH,j2;
J, K”)
(2) where D$:Kt is the rotation matrix and X, 8, and CJare the Eulerian-type angles introduced as the “axis-switching” angles characteristic of Hougen’s generalized intemalaxis method (5). The quantity h12 is represented as
where I,!J~” and 3/z”are the vibrational wavefunctions for frameworks 1 and 2, respectively. As seen in Eq. (2), the modifications for the tunneling matrix elements arise from the presence of the Den factors and, generally, they depend on any of X, 8, and 4. In the present two-framework problem, any eigenvalue of the Hamiltonian matrix depends on X + $ instead of X and 4 independently, as proved in the Appendix. For this reason, we represent the tunneling matrix elements as (1; J, ICI Hl2; J, K”) = (2; J, K”( H( 1; J, K’)* = exp[i(K’+
K”)(x
+
$)/2]d$&
(0) -h12
(4)
499
INVERSION SPLITTINGS IN CH,NHNH2
in place of the expression given in Eq. (2)) where dg(:!l is defined as 0, 6)K”Kl = exp[iK”x].d~~K’(e).exp[iK’~].
D’?X,
(5)
DATA ANALYSIS
A least-squares analysis has been carried out on the microwave spectral data given in Ref. (4). For the inner and outer conformations the least-squares fits were made by adjusting hi2, X + $, and 0, along with the rotational constants and centrifugal distortion constants AJ, A,,,, AK, 6J, and &. For the inner conformation, which shows more drastic J and K dependences of the inversion splittings, it was necessary to replace h12 by h12 + h&(J + 1) in order to achieve a reasonable standard deviation. The inversion tunneling parameters thus obtained are given in Table I along with redetermined rotational and centrifugal distortion constants. The rotational constants redetermined in the present fits agree with those in Ref. (4) within quoted errors. The centrifugal distortion constants, A,, etc., obtained in the present analyses reproduce the centrifugal distortion constants denoted by the letter d (with various subscripts) in Ref. (4) within quoted errors except for a small discrepancy in dJK for the inner conformation, and in dKand dwK for the outer conformation. The results of the fittings are listed in Tables II and III. DISCUSSION
The inversion splittings, 12h12( , for J = K = 0 are found to be 152.53 (2.50 = 0.11) MHz and 21.622 (2.5~ = 0.042) MHz for the inner and outer conformations, respectively. These data can be used to determine the potential function for the inversion motion. According to the concept of the tunneling path developed by Hougen (5), the potential function for the inversion motion can be expressed in terms of a path variable,
TABLE I Inversion Tunneling Parameters and Molecular Constants for the Inner and Outer Conformations of Methylhydrazine
hl2
/MHz
-76.266(57)
hl2j/MHZ x
+
-10.810(45)
-0.00179(75)
cb/l-.%d
2.9618(23)
2.8670(39)
e
/rad
0.02104(80)
0.0275(15)
A
/MHZ
36
704.413(95)
38
B
/MHZ
9
689.794(25)
9
591.871(34)
C
/MHZ
8
531.930(24)
8
494.412(30)
AJ
/MHz
AJK
/MHz
AK
/MHz
0.300
O(77)
0.343
3(W)
SJ
/MHz
0.001
7161(15)
0.001
819(16)
6K
/MHz
2(10)
0.002
9127)
apply
to
Numbers last
digits
0.009 -0.030
-0.002
in
parentheses of
the
091.57(13)
03(21)
0.008
90(87)
denote parameters.
-0.032
2.55
and
65(26) 7(12)
the
500
OHASHI ET AL. TABLE II Inversion Splittings (MHz) for the Inner Conformation of Methylhydrazine UPPER
J
LOWER
Ka Kc
J
Ka Kc
OBSa
CALC
o-c
1
0
l-
0
0
0
0.00
2
0
2-
1
0
1
0.00
2
1
l-
1
1
0
0.00
0.05
-0.05
1
-0.02
2
1
17
3
214
1
-0.03 0.05
0.00
0.02
17
3
15
1.59
1.31
0.28
1 -
0.03 -0.05
1
1
l-
0
0
0
2.56
2.49
0.07
3
0
3-
2
1
2
2.42
2.31
0.11
4
0
4-
3
1
3
2.30
2.22
0.08
6
1
5-
5
2
4
7.07
6.63
5
3
2-
6
2
5
9.92
10.34
-0.42
5
3
3-
6
2
4
9.82
10.27
-0.45
9
2
8-
8
3
5
15.74
15.53
0.21
1
1
o-
1
0
1
2.55
2.50
0.05
2
1
l-
2
0
2
2.57
2.50
0.07
3
1
2-
3
0
3
2.62
2.54
0.08 0.04
0.24
4
1
3-
4
0
4
2.63
2.59
5
1
4-
5
0
5
2.70
2.66
1
1
l-
1
0
1
302.42
302.54
2
1
2-
2
0
2
302.59
302.51
3
1
3-
3
0
3
302.34
302.40
-0.06 -0.05
0.04 -0.12 0.08
4
1
4-
4
0
4
302.19
302.24
5
1
5-
5
0
5
302.13
302.04
0.09
6
1
6-
6
0
6
301.82
301.78
0.04
7
1
7-
7
0
7
301.86
301.47
8
1
8-
8
0
8
301.10
301.12
16
2
15
-
16
1
15
281.91
281.90
18
2
17
-
18
1
17
278.73
278.76
-0.03
19
2
18
-
19
1
18
277.06
277.10
-0.04
23
3
21
-
23
2
21
252.50
253.49
-0.99
26
3
24
-
26
2
24
245.74
245.56
27
3
25
-
27
2
25
242.96
242.76
0.20
28
3
26
-
28
2
26
240.09
239.94
0.15
32
4
29
-
32
3
29
210.70
210.52
0.18
BRef.
4.
0.39 -0.02 0.01
0.18
is a suitably chosen function of the variables representing N-H wagging, N-N torsion, and CHs torsion. When a suitable variable for the tunneling path is unknown, the N-H wagging variable may be taken as an approximate path variable as in Ref. (4) for determining the barrier heights for the N-H inversion. By the use of the inversion splittings 152.5 and 21.62 MHz determined in the present study and the doubleminimum potential reported in Ref. (4), the barrier heights are obtained to be 1958.7(2) and 2474( 2) cm-’ for the inner and the outer conformations, respectively. The inversion splittings obtained above are close to the ab initio values 2290 and 2493 cm-r estimated by use of the 4-3 lG( N* ) basis set (4). As described in Refs. (5) and (6)) the parameters X, 8, and 4 are defined as the Eulerian-type angles for the “backward rotation” or “axis-switching,” which must be applied to the axis system taken in framework 2 to cancel the internal angular momentum generated by the inversion tunneling from framework 1 to 2. These parameters
which
501
INVERSION SPLITTINGS IN CHjNHNH* TABLE III Inversion Splittings (MHz) for the Outer Conformation of Methylhydrazine UPPER
J
b
LOWER
Kc
J
b
Kc
OBSa
CALC
o-c
4
0
4-
3
1
3
0.00
0.74
9
1
9-
8
2
6
2.20
2.36
-0.16
10
2
a-
9
3
7
3.44
3.70
-0.26
-0.74
2
1
1-
2
0
2
0.81
0.82
-0.01
7
1
6-
7
0
7
0.83
0.88
-0.05 -0.01
1
1
o-
0
0
0
42.41
42.42
4
0
4-
3
1
2
42.29
42.28
7
1
7-
6
2
5
39.04
39.00
a
1
8-
7
2
6
38.91
38.92
9
1
9-
8
2
7
38.81
38.81
10
2
0-
9
3
6
32.58
32.65
0.01 0.04 -0.01 0.00 -0.07
11
2
10
-
10
3
8
32.31
32.37
-0.06
13
3
10
-
12
4
8
23.75
23.79
-0.04
13
3
11
-
12
4
9
23.60
23.67
-0.07
2
1
2-
2
0
2
42.36
42.40
-0.04
3
1
3-
3
0
3
42.33
42.36
-0.03
4
1
4-
4
0
4
42.30
42.31
-0.01
5
1
5-
5
0
5
42.26
42.24
0.02
6
1
6-
6
0
6
42.17
42.15
0.02
7
1
7-
7
0
7
42.05
42.05
0.00
8
1
a-
8
0
8
41.99
41.92
0.07
9
1
9-
9
0
9
41.88
41.78
12
2
11
-
12
1
11
37.48
37.61
0.10 -0.13
13
2
12
-
13
1
12
37.29
37.29
0.00
14
2
13
-
14
1
13
36.95
36.93
0.02
17
2
16
-
17
1
16
35.74
35.64
0.10
20
2
19
-
20
1
19
34.27
34.12
0.15
29
3
27
-
29
2
27
23.69
23.70
-0.01
30
3
28
-
30
2
28
22.88
22.91
-0.03
BRef.
4
may be evaluated by assuming a simple model for the inversion tunneling path and by solving numerically the coupled differential equations (49) in Ref. (5)) but they are determined in the present fitting procedure by trial and error. The tunneling path adopted in the present evaluation is described by introducing a path parameter 7 into the reference coordinate system. In the intermediate coordinate system from which the reference coordinate system is derived, the path parameter representing imino wagging, r, is defined as ai = S-*(0,7,O)a?,
i=H,,
ai = S-’ (0 = k17, 0, O)aP,
i = Hz and H3,
ai = C, (7 = LNNC)S-’
(c = k27, 0, O)a?,
where a: denotes the initial position (7), C’Jr) = S1(7r/2, y, -r/2), that the C-N-N-H and N-N-C-H atom adjacent to the methyl group
i= H4,H5andH6,
(6)
(8)
of the ith atom, S-’ is the direction cosine matrix and kl and kz are the constants used to represent dihedral angles change in proportion to T. The H is denoted as H, , and those in the amino group
502
OHASHI ET AL.
as H2 and H3. The definition of 7 in the present model, as depicted in Fig. 1, differs from that of p in Ref. (4), but 19( =kir) corresponds exactly to that defined in Ref. (4). The equilibrium values, 7, and pe, for these imino wagging variables are interrelated as sin(7,) = sin(p,)*cos(LNZN,H,
- 7r/2)/cos(cz),
(9)
where cos(a! - a/2)
= cos(LNzN,H,
- ?r/2).cos(pe).
(10)
The three coupled differential equations given by (49) of Ref. (5) were constructed by using the structural parameters and the equilibrium angles of inversion and torsion (pe and M,) determined by a combined analysis of electron diffraction and ab initio calculation (8), and the equations were then integrated numerically by use of the Runge-Kutta-Gill method subroutine DRKG. Based on considerations in the intermediate-axis system of the final orientation reached by the imino inversion motion compared to the standard orientation of the reference frameworks, a C,(z) corrective rotation (6) seems to be necessary, and we will use such a corrective rotation for comparison purposes here. The actual situation in this molecule, which has no symmetry operations in its equilibrium configuration, may be somewhat more complicated and merits further study. After this correction of the direction cosine matrix (6) from S’( X, 0, 4) to S’( X, 8, 4) C,(z), X, 8, and 4 values are obtained as listed in Table IV. Since the eigenvalues of the Hamiltonian matrix are unchanged under the transformation from X + 4 to X + d + 27r (see Appendix), the X + 4 + 27r values are also given in Table IV for comparison with the x + 4 values in Table I derived from the least-squares analyses. The angles obtained from the numerical integrations agree
Y
cii,
H,
FIG. 1. Definition of the tunneling path parameter 7 representing the imino inversion motion for the inner conformation.
503
INVERSION SPLITTINGS IN CHjNHNH> TABLE IV Angles as Determined from Numerical Integration of the Coupled Differential Equations”
X/rad B/rad &/t-ad
Inner
Outer
-1.462 0.165 -1.775
-1.417 0.165 -1.819
3.046
3.047
X+&+Zx/rad
a The form of the coupleddifferentialequationsis as for Eqs.( 49) of Ref. (5 1.
qualitatively with those from the least-squares fit of the microwave spectral data set. ,4ccordingly, the inversion motion can be interpreted by a tunneling path like that represented in Eqs. (6)-( 8). APPENDIX It is possible for the present two-framework, one-path system to regard the Hamiltonian matrix elements given in Eq. (2 ) as operationally equivalent to those obtained by taking a usual symmetric-top wavefunction multiplied by a vibrational-framework function as the basis set function and using a Hamiltonian operator of the form
H = Ho + Hinv,
(Al)
Hi,-+= v.exp(i~~=).exp(-iBJ,,)-exp(I’xJ,),
(A2)
where Ho is an asymmetric rotor Hamiltonian operator and V represents a vibrational operator such that (1 lVjl>
= (2/V/2)
=o
(A3)
and (1 /Vj2)= By applying a unitary transformation
(2jVj1)=/2,2.
(A4)
S expressed as
S = exp[-i(X
- $)J,/2],
(A5)
to the Hamiltonian operator given in (A 1) above, we have a new Hamiltonian operator H”e” = S-‘HS = Hf;” + HP=“’ mv , HFiY = V.exp[i(X
+ b)J,].exp(-iU,).exp[i(x
(A(j)
+ 4)Jz].
(A7)
Consequently, we find that eigenvalues depend on X + C#J (not x and C$independently) since it is possible to take Ho such that Hrw is equal to the usual asymmetric rotor Hamiltonian, e.g., of Watson’s A-reduction type.
504
OHASHI ET AL.
From a similar discussion with the use of appropriate unitary transformation(s), it is shown that eigenvalues are unchanged under x + 4 + x + $J + 2n and/or e-,--0. ACKNOWLEDGMENTS The authors thank Dr. Jon T. Hougen of the National Institute of Standards and Technology for many helpful discussions and for kindly reading the manuscript. The authors also thank Dr. Laurent H. Coudert for kindly reading the manuscript and valuable discussions. Note added in proof: In the present paper we determined the value of x + 4 in the least squares fit. On the other hand, X - 4 is shown in Ref. (6) to take the value + * or 0 for various motions in the water dimer. The value of x - #Jfor the tunneling motion in the present molecule, which has lower molecular symmetry, should be determined after further detailed consideration. RECEIVED
August 7, 1989 REFERENCES
R. P. LA?TIMERAND M. D. HARMONY,J. Chem. Phys. 53,4575-4583 (1970). R. P. LATTIMERAND M. D. HARMONY,J. Amer. Chem. Sot. 94,35 I-356 ( 1972). J. R. DURIG,W. C. HARRIS,AND D. W. WERTZ, J. Chem. Phys. 50, 1449-1461 ( 1969). N. MURASE,K. YAMANOUCHI,M. SUGIE,H. TAKEO,C. MATSUMURA,Y. HAMADA,M. TSUBOI,AND K. KUCHITSU,J. Mol. Struct. 194,301-316 (1988). 5. J. T. HOUGEN,J. Mol. Spectrosc. 114,395-426 (1985). 6. L. H. COUDERTANDJ. T. HOUGEN,J. Mol. Spectrosc. 130, 86-l 19 ( 1988). 7. E. B. WILSON,JR., J. C. DECIUS,ANDP. C. CROSS,“Molecular Vibrations,” McGraw-Hill, New York,
1. 2. 3. 4.
1955. 8. N. MURASE,K. YAMANOUCHI,T. EGAWA,AND K. KUCHITSU,to be published.