- Email: [email protected]
Centrifugal distortion effects in the microwave spectra of 1,3-dioxane, tetrahydropyran and piperidine
JOURNAL
OF MOLECULAR
SPECTROSCOPY
60, 312-323
(1976)
Centrifugal Distortion Effects in the of 1,3-Dioxane, Tetrahydropyranl R. s. Department
LOWE
AND
of C/tern&&y, Qzreen’s University,
R.
Microwave Spectra and Piperidine
EEWLEY
k’in,eslon, Ontario,
Can&a,
k’7L 3N6
Applying a bootstrap approach to transition assignment, a first-order centrifugal distortion analysis has been performed on 1,3-dioxane, tetrahydropyran, and piperidine (axial NH conformer). Q-branch transition frequencies for J up to 32 have been combined with previously published or remeasured frequencies for low J, R-, and Q-branch transitions to yield values for the quartic centrifugal distortion parameters. Comparison is made between the results obtained using two forms of the quartic Hamiltonian. INTRODUCTION
Although the theory of centrifugal distortion effects in molecular rotational spectra has been well covered in the literature (I, 2,3), there have been relatively few applications of this theory to heavier molecules, particularly saturated ring compounds (4,5,6). It was therefore thought of interest to obtain a quantitative account of the centrifugal distortion in each of the three oblate six-membered heterocycles : 1,3-dioxane, tetrahydropyran, and the axial NH conformer of piperidine (Fig. 1). Previous studies (7,8,9) yielded rigid rotor rotational constants and found all three molecules to be present exclusively in the chair form, piperidine as both axial and equatorial NH conformers. For reasons given below we have not included equatorial piperidine in the present study. For each of 1,3-diosane and tetrahydropyran (7,s) the effects of nonrigidity were noticed but not included in the earlier analysis. By doing so here for all three molecules we hoped to extend our understanding of the centrifugal distortion effects in this class of compounds. For each molecule the present analysis uses the rigid rotor fit for low J transitions as a starting point. In the case of axial piperidine the published planar moments (9) were helpful in locating the low J transitions. EXPERIMENTAL
With the exception of two R-branch and eight Q-branch transitions measured by Curl and co-workers (10) on a Hewlett-Packard 8460A spectrometer, all the lines used in this analysis were observed using a conventional 100 kHz modulation Hughes-Wilson spectrometer operating between 8 and 40 GHz. All frequency measurements were made at room temperature using oscilloscope display and are considered to be accurate to better than 0.2 MHz. The 1,3-dioxane sample was from K and K Laboratories, the tetrahydropyan from Aldrich Chemical Company and the piperidine from B.D.H. 1The results for 1,3-dioxane and tetrahydropyran were presented Structure and Spectroscopy, Columbus, Ohio, as paper TFlO. 312 Copyright @ 1976 by .4cademic Press, Inc. All rights of reproduction in any form reserved.
at the 30th Symposium
on Molecular
CENTRIFUGAL
EFFECTS,
1,341OXANE x
6-MEMBERED
TETRAHYOROPYRAN
= 0.82852
RINGS
313
AXIAL PIPERIOINE
x = 0.82779
x
= 0.89887
FIG. 1. The molecules examined in this study.
Laboratory Chemicals. All were used without further purification. All computations were performed in double precision arithmetic on the IBM 360/.50 and Burroughs 6700 computers at Queen’s University. ASSIGNMENT
AND ANALYSIS
Essentially all of the transitions listed in Tables I, II, and III were assigned on the basis of goodness of fit except where additional confirmation was possible from Stark effects. Q-branch asymmetry doublets constituted the majority of the high J transitions found between 8 and 40 GHz. They were of b and c type for 1,3-dioxane, a and c type for tetrahydropyran and a type for axial piperidine. C type Q-branch lines could not be located for axial piperidine. This is in agreement with the observations of Buckley, Costain, and Parkin (9). Aside from a small number of allowed but very weak high J CR~,_-2transitions, transitions of high J from other subbranches than those observed were out of the range of our instrument. For equatorial piperidine, because of the small value of the a-component of the dipole moment (0.195 D compared to 1.53 D for tetrahydropyran) only a few tentative assignments of a type Q-branch lines could be made. Also, although identifiable c type transitions should be present, as for axial piperidine, none of these could be found. It was therefore not possible to make a centrifugal distortion analysis for equatorial piperidine. The data were analyzed using two computer programs. One was developed by Dr. W. H. Kirchhoff at the National Bureau of Standards, Washington, D. C. (2) and the other by Professor H. M$llendal at the University of Oslo (II). A. Analysis
with CDANAL
The Kirchhoff program,
CDANAL,
uses as its basis the reduced Hamiltonian
H red
=
H,+
Ha
(1)
H, = A”P,2 + B”P+ + C”Pc2,
(2)
where Hd = t
c
a=a,b,c
~“aaaaPm4 + TIP14 +
r2P24 +
r2P24.
(3)
In Eq. (2) the A”, B”, and C” are the effective rotational constants. These are related to the Kivelson-Wilson rotational constants (1.2) by the following equations (14) A” = A’ _ 127’ bbcc,
(4)
314
LOWE AND
KEWLET
B”
= B’ _
.i 27 ’ a
(5)
C”
= C’ -
+7raabb.
(6)
The quartic distortion constants T”~~~~, 71, 72, and 73 in Eq. (3) are similarly the Kivelson-Wilson coefficients by
related to
I, 7 a(1oa= ‘T’orcrua (a = a, 4 c), 71
=
T'aabl, +
72
=
(A’/.%‘bbcc
~3
=
[S/(B’
T’bbcc
+ +
(7)
~‘ccaa, @‘/S)T’aacc
(8) +
- A’)]daa6b + [S/(/l’
(9)
(C’/S)daabb,
- C’)]d,,ce + [S/(C’
-
B’)ld~ce
(10)
where S = A’+
B’+
C’.
(11)
Finally, Pp, P2*, P34 appearing in Eq. (3) are linear combinations of A’, B’, C’ and the Pa2 (a = a, b, c). As pointed out by Watson (I), there exist only five determinable coefficients in the quartic Hamiltonian. A sixth coefficient, 73, has been added by Kirchhoff to facilitate TABLE I Observed Transitions of 1,3-Dioxane
Transition J(K_jsKl) 0
1 1 I
; 2 ; : 2 2 : 4 z 6 : 7 7 8 a 8" 8 : 9 1: 10 10 1'; 11 11 11 11
'ohs
+ J’(KIlsKi)
0, 0) + l.l)* 0.1)l,0)+ O,l)l,l)+ 1,0)1,2)* 2, 1) *
1 2 2 2 2 2 2 3 3
:* A] : 0: 2) + l,l)* 2, 0) * 0, 5) + 1,5)-r 1,4)* 0. 4) + 0, 6) + 1, 6)+ 2. 4) + 0, 5) + 1,6)* 2. 6) * 2, 6) * 3. 6) + 3, 5) + 4, 4) + 5. 4) + 4. 6) * 5, 5) * 5, 4) * 6, 4) * 5, 5) * 6, 5) + 6, 4) * 3, 8) + 5. 6) * 6. 6) + 6, 5) * 7. 5) * 7. 4) * 9, 4) *
; 3 3 3 5 5 5 5 6 6 6 6 7 7 8 a a a 8 9 9 9 9 10 10 10 11 11 11 11 11 11 11
(MHz 1
1, 0) 2)
0,
is :I 1: 1)
2, 1) 2, 0) 0. 31 1. 21 2. 2)
3, 1, 2. 3. 1, 2, 0. 1. 1. 2, 3, 1. 2. 3, 3, 4. 4. 5, 6. 5. 6. 6, 7, 6, 7. 7, 4. 6. 7. 7, a, 8, 9.
1) 2) 11 0) 4) 4) 5) 5) 5) 51 3) 6) 5) 5) 5) 5) 4) 3) 3) 5) 4) 3) 3) 4) 4) 3) 7) 5) 5) 4) 4) 3) 3)
"ohs - vcalc
P4 Term
(MHZ1
Hz
0.04 0.18 -0.11 0.04 -0.01 0.09 -0.12 -cl.09 0.14 0.07 0.13 -0.11 -0.14 0.00 o.la 0.01 -0.06 -0.07 0.06 0.05 0.15 0.08 0.02 -0.02 n.07 -0.10 -0.17 -0.17 -0.13 0.22 -0.26 0.05 -0.04 0.21 -0.24 -0.04 0.09 -0.24 -0.16 0.14 -0.22 0.15 -0.04
-0.01 -0.02 -0.02 -0.03 -0.03 -0.03 -0.04 -0.04 -0.07 -0.07 -0.11 -0.08 -0.12 -0.12 -0.16 -0.16 -0.13 -0.13 -0.25 -0.25 -0.36 -0.21 -0.52 -0.52 -0.83 -0.83 -0.82 -0.72 -Cl.76 -1.18 -1.12 -0.93 -1.00 -1.42 -1.44 -1.19 -2.08 -1.99 -1.99 -1.76 -1.81 -1.47 -1.57
t(bv)
1 0.3 1.2 -0.8 0.3 -0.1 0.6 -0.8 -0.6 1.0 0.5 0.9 -0.8 -1.0 0.0 1.2 0.1 -0.5 -0.6 0.4 0.4 1.1 0.9 0.1 -0.1 n.5 -n.7 -1.2 -1.2 -Il.9 1.5 -1.8 0.4 -0.2 1.4 -1.7 -0.3 0.6 -1.7 -1.1 1.0 -1.5 1.0 -n.3
Sxlrce
CENTRIFUGAL
EFFECTS, 6-MEMBERED
315
RINGS
TABLE I, Continued
Transition
"obs
12 12
( 3, ( 6.
9) + 12 ( 4, 8) 6) + 12 7, 5)
13 13 14 14 14 14 14 15 15 15 16 16 17 17 17 18 18 18 18
( ( 8, + ( a: 5) .+13 ( 9, 5) * 13 ( 5, 9) + 14 ( 8, 6) * 14 ( 9, (10. (11, ( 9, (11, (12, (10. (12. (11. (13. (14, (12, (13. (14. (15,
5) 5) 4) 6) 5) 4) 6) 5) 6) 5) 4) 6) 6) 5) 4)
;; I;? El ;; 1;;: :] 19 19 20 20 20 21 21 21 22 22 22 23 23 23 24
(171 (18. (14, (15, (16, (15. (17, (20, (15. (16, (18, (17, (18, (19, (18.
36418.85 23413.36 18854.71 19066.75 13351.00 23342.23 18637.6'0 19020.85 36381.16 23247.19 18336.72 18907.74 19961.05
(
12 7. 5) * l2 ;: 13 17.t Y;%461 5) * 12
3) 2) 6) 6) 5) 6) 5) 1) E) 6) 5) 6) 6) 5) 6)
+ + * + + * + + + + * + + + * + * * + + + * * + * + * + + * + * * +
14 14 14 15 15 15 16 16 17 17 17 18 18 18 18 19 19 19 19 19 19 20 20 20 21 21 21 22 22 22 23 23 23 24
( is :i 8. 3) I 9: ( 5)
( 9, 4) (10, 4) 6, 8) 5) (10, 4) (11, 4) (12. 2) (10. 5) (12, 4) (13, 2) (11, 5) (13, 4) (12. 5) (14, 4) (15, 2) (13, 5) (14. 5) (15, 4) (16. 2) (14, 5) (15. 5) (16. 4) (17. 3) (18, 2) (19, 1) (15, 5) (16, 5) (17, 4) (16, 5) (18, 4) (21, 0) (16, 7) (17, 5) (19, 4) (18. 5) (19, 5) (20, 4) (19. 5)
( ( 9,
a R. Kewley, ref. (8);
'ohs - Vcalc
(!YHr )
J(K_,,K,) + J'(K:,.Ki)
:6:E 19493:09 22952.74 19006.66 22728.27 19085.85 19131.08 22430.53 22947.46 19231.48 19279.69 22040.31 22870.11 19456.84
This work;
(MHz 1
0.03 -O.OA n.25 0.07 -0.22 0.24 0.09 -0.06 0.01 0.10 0.03 -0.08 0.04 -0.19 0.14 -0.02 0.15 0.07 -0.02 -0.O8 -n.ln -0.14 -0.25 0.16 0.04 -n.o5 0.03 0.22
-2.63 -2.45 -2.14 -2.22 -1.60 -2.96 -2.56 -2.67 -4.24 -3.51 -3.02 -3.17 -3.93 -4.10 -3.70 -4.33 -4.73 -4.26 -5.41 -4.84 -4.88 -6.15 -6.35 -5.43 -4.4s -6.96 -7.18 -6.01
t(Av)
0.2 -0.5 1.7 0.2 -1.5 1.6 0.7 -0.4 0.1 0.7 0.2 -0.6 0.3 -1.3 1.0 -0.2 1.0 0.5 -0.2 -0.6 -0.7 -1.0 -1.7 1.1 0.3 -0.4 E -1:9 1.3 -0.9 _'1.: -0:7 1.8 -0.2 1.1 -1.1 -1.1 0.6 -0.1 2.2 0.9 -0.1 -1.3 0.9 -0.8
20915.83 20192.93 25596.29 31465.79 20171.74 20720.08 19330.62 22921.29 21358.09 18437.00 23853.31 17556.25 24367.99 b
P4 Telm
(MHZ)
'
Source
c b : a : b E : b b b b b b : b I! ! b : : b b : E b b c b b b b b b bb b
R.F. Curl et al.. ref. (1p).
comparison of the angular momentum operators between the Watson and KivelsonWilson Hamiltonians. 73 is not, however, a determinable coefficient, but is evaluated once the remaining eight parameters have been determined. The eigenvalue problem is solved by direct diagonalization of the semirigid rotor Hamiltonian in the P rigid rotor basis to give corrected values of the rotational constants and a first-order contribution due to the P4 terms, in the form of the 7 coefficients. A refinement of these constants was obtained by a least-squares fit to the observed frequencies in the following manner. Assignment of the higher J transitions followed the bootstrap approach outlined by Kirchhoff (2). Briefly, the rigid rotor A, B, and C were used to predict the positions of higher _7lines. Those that were expected to lie within 5 MHz of the rigid rotor position were then measured. Using the augmented set of observed frequencies in the leastsquares fit yielded new rotational constants and a first estimate of the 7 coefficients. These new constants were used to predict the positions of still higher J transitions. The analysis was considered completed when a sufficiency condition had been reached,
LOWE AND KE\VLE:Y
316
TABLE II Observed Transitions of Tetrahydmnyran
Transition
'ohs
J(K_,.K,) _ J'(K(T,Ki) 1 1
fl,l)* I,,)l,O)* O.l)1.1) * c, 2) * 1,2)-r l.l)* ?,l)1,2)+ 2, 1) + 2, n) + 1,2)* 2, 2) * 3,1)+ 1,5)+ n, 4) + 1,4)+ 1,6)n, 5) *
2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 5 5 5 6 6
a, :I * 1:aj * 1,8)2,8)+ 4, 6) + 1: 4, 7) * 11 6. 6) + 11 Q, 2) + 6, 7) * ;; (I,2) + 8, F) + 1: 0, 3) * 14 S,lO) t 14 1. 3) + 15 7, 9) * 9, 7) * 1: 2, 3) + 9, 7) * :z 3, 3) + 17 1, 7) + 2, 6) * Y3 1, 7) * 18 2, 7) + 14 1, 8) + 19 2. 8) + 2, 7) * 3. 7) * E 2n 3. 7) * 4, 7) * 20 20 5, 6) * 20 6, 4) + 21 3, 8) + 21 4, 8) + 21 5, 7) + 21 6, R) * 22 5, 8) + 22 5, 7) + 6. 6) + 22 7, 6) * 2': 0, 2) + 24 7, 7) + 25 8, 7) + 25 9, 7) * 9, 7) * ;: 0, 9) + 30 '1, 9) * 32 '3.10) +
9 ' R P 9 9 10 11 11 12 12 13 13 14 14 15 15 15 16 16 17 17 18 18 19 11 19 19 2n 20 20 20 21 21 21 21 22 22 22 22 22 24 25 25 26 29 30 32
1' 1 2 2 2 2 2 2 2 3 3 7 5 4 4 6 5 7
R 9 0
0, 2)
1. 1, 1, 2, n, 1. 1, 2, 2. 3. 3, 1, 2, 3,
2) 1) 1) 1) 3) 3) 2) 2) 2) 1) 0) 3) 3) 2)
0, 1, 1, 0, 2, 2, 1, 3, 2, 4, 4, 6, 1, 6, 2, 8, 2, 5, 3, 7, 9, 4.
5) 5) 5) 6) 5) 7) 7) 7) 7) 5) 6) 5) I! 6) 11 5) 2) 9) 2) 8) 6) 2)
:* 1: 2. 3, 2, 3, 2. 4. 3. 5. 4. 5, 8,
:I 6) 5) 6) 6) 7) 7) 6) 6) 6) 6) 5) 3)
:* 5: 6, 5, 7, a. 7. 1, 9, 0,
:I 6) 5) 7) 6) 5) 5) 2) 6) 6)
;, 2: 3. 3,
:I 8) 8) 9)
1, 4)
P4 Ten
Yobs - Ucal‘
(MHz) 12465.20 12299.00 16086.40 18158.8r) 18515.80 17594.70 17583.40 21764.4n 21289.10 27505.6C 27780.40 77553.60 26785.50 26729.30 3nl91.90 17821.80 27993.77 27993.77 21784.40 33196.24 21775.20 29709.39 29709.39 29701.68 79701.68 21741.40 25711.n6 21676.95 17385.03 25662.04 19839.76 71559.7n 15840.92 37591.66 17715.54 33583.ln 25525.52 19871.98 25466.56 22281.73 25368.88 20964.92 25303.74 25260.35 21349.311 21345.16 25205.47 25123.83 25098.41 24950.37 19731.99 24214.71 29171.7n 29153.44 24727.81 19045.92 29028.98 24878.43 21496.82 18187.11 22021.41 24717.87 24702.43 22952.40 24765.10 32344.55 32169.18 36144.23
(WZ)
(MHz)
-0.11 0.1n -Il.01 0.11 0.24 -0.20 0.02 -cl.14 O.lI, -0.M -0.16 0.24 -fl,n1 n.n1 n.no -0.20 -o.ni o.ni -0.32 0.02 -n.23 0.13 n.13 0.11 0.11 0.13 n.nz O.lC -0.05 0.04 il.07 -n.l6 n.14 -n.lf -0.07 -0.04 -0.18 -0.02 0.08 -n.m -0.04 0.11 n.17 -n.n5 -0.18 0.15 0.14 0.16 O.O? 0.01 -0.10 0.05 0.15 0.00 -0.01 0.11 0.08 -n.o4 -0.17 -0.19 -0.02 0.09 0.00 0.10 -0.03 -0.n2 -0.01 -0.04
-n.n2 -0.01 -n.o3 -0.03 -n.n3 -fl.O4 -n.o4 -n.o,6 -0.06 -0.07 -0.10 -n.11 -0.11 -0.11 -Il.16 -0.12 -n.13 -n.i3 -0.18 -0.21 -Cl.42 -n.34 -n.34 -cl.?6 -11.76 -1.01 -1.35 -1.75 0.14 -2.30 n.17 -2.67 1.07 -3.52 1.72 -4.44 -4.07 2.47 -4.73 3.25 -5.55 -5.58 -6.22 -6.42 -J.Jd -7.77 -7.02 -7.41 -7.83 -8.57 -10.20 -8.64 -9.82 -9.95 -9.97 -12.59 -11.21 -9.33 -4.26 -15.43 -2.30 -10.15 -9.98 -20.22 -9.14 -22.63 -24.28 -32.11
Source
t(av)
-0.9 n.8 -0.6 n.9 1.9 -1.6 0.1 -1.1 0.8 -0.5 -1.3 1.9 -0.1 0.1 -0.1 -1.6 -0.1 0.1 -2.5 n.2 -1.8 1.0 1.0 n.9 0.9 1.0 0.2 n.8 -n.5 0.3 0.7 -1.3 1.2 -1.2 -C.6 -0.7 -1.4 -0.1 0.7 -0.7 -0.7
a a a a a a a a a a a a d
a a a b b a c a b b b b
::: -0.4 -1.4 1.2 1.1 Q.1 0.1 fl.1 -0.8 0.6 1.2 0.0 -0.0 n.9 0.6 -0.3 -1.4 -1.5 -0.5 0.7 0.0 0.9 -0.3 -0.2 -0.1 -0.5
: b b b b b b c b t h b b b b h b b b b b b h b b b b b b b b b b b b b b b b c c c
a
V.M. Rao and R. I:ewley,ref. (I); b
that is, when the mum observed J, Assignment of generated by the
This work;
'
R.F. Curl et al.. ref. (10) -
uncertainties in the calculated transition frequencies, up to the masibecame comparable with the measurement error. lines was facilitated through the use of the least-squares statistics program CDANAL. The most useful statistical indicators are the
CENTRIFUGAL
EFFECTS,
6-MEMBERED
317
RINGS
TABLE III Observed Transitions of Axial Piperidine
Transition J(L_,J,) ;.:I+
.
z;ij+ 2. 0) 0, 3) 1,3)1.2)* 2. 2! 0, 5) 0, 6)
*3
3
* + *
i 3 4 4 4 4 5 6
:*:j: 1:6)+ 2.6)+ 2, 5) + 2, 6)+ 3,6)+ 3, 5) + 3, 6) + 4. 6)+
; 7 7 7 8 B B 9 9
+ *
5; ij *
9
4. 5, 5, 5. 7; 7, 8, 6, 7, 7, 8, 8, 9, 7. 8, 8, 9, 0, 9, 9. 1.
6) 6) 5) 61 6j 5) 5) 7) 7) 6) 6) 5) 5) 7) 7) 6) 6) 5) 7) 6) 5)
+ + + + + + + + + + + + + + + + + + + + +
10 10 10 11 12 12 12 13 13 13 13 13 13 14 14 14 14 14 15 15 15
0, 0, 1, 1, 2, 1, 1, 2, 2, 3, 2, 3, 3, 4, 3. 4: 4; 5, 5, 6, 6, 7. 8. 8, 8, 9,
7) 6) 6) 5) 5) 7) 6) 6) 5) 5) 8) 6) 5) 5) 61 6i
+ + + + + + + + + + + + + + + +
16 16 16 16 16 17 17 17 17 17 18 1B 18 18 19 10 20 20 21 21 22 24 24 24 25 25
6j + 6) 6) 6) 6) 7) 7) 6) 7) 7)
'ohs
* J’W,>K;)
+ + + + + + + + +
'ohs - "talc
(MHz)
1, 1. 2, 2. 0, 1. 1. 2; 2, 2. 1. 3. 3. 2, 4. 4. 3, 5. 5; 4. 5. 6, 5, 7. 7; 7, 9. 8, 8, 7. 9.
3) 2) 2) 1) 4) 4) 31 3j 4) 5) 5) 4) 5) 5) 4) 5) 5) 4) 5j 5) 4) 51 5) 41 5) 5) 4) 4) 6) 6) 5)
17120.72 2lfl70.58 20793.20 24461.54 22193.37 22193.37 26020.92 26002.8I 17174.39 20991.87 20991.87 17170.92 20988.57 20988.57 17165.32 20984.07 20984.07 17156.98 20977.53 20977.53 17143.65 20968.89 20968.89 17131.00 20957.37 2CQ42.09 17089.68 17065.03 24769.16 24769.1h 20924.fl8
0, 9, 9, 8, 0, 9. 0, 9, 1, 1, 0, 2, 1. 3. 2. 1, 3.
4) 4) 6) 6) 5) 5) 4) 6) 5) 4) 6) 5) 5) 4) 4) 6) 3)
17063.41 17017.42 24754.94 24754.44 20930.86 20897.48 18952.74 24737.38 20873.09 16865.07 24716.23 20839.83 20827.54 16978.32 16747.80 20690.59 20801.16 20778.93 16957.82 16592.32 20757.21 20718.28 16949.77 16388.05 20708.47 20642.46 20656.00 20547.31 2OfiOl.3Q 20427.61 20547.64 24360.94 24311.91 20459.70 24289.41 24210.59
R;
5j 20922.29
:* :i 3: 4) 4. 5) 3, 5) 5, 4) 4, 4) 5. 5) 4. 5) 6, 5) 5, 5) 7, 5) '6, 5) '8, 5) 19, 6) 18, 6) ?I, 5) !O, 6) 19, 6)
P4 Term
(MHz)
(MHZ)
-0.12 -0.26 0.21 0.17 -0.11 -0.02 -0.17 0.29 -0.08 0.17 0.17 0.03 -0.03 -C.O3 0.01 0.01 0.02 -0.19 -(I.12 -0.09 0.19 -0.06 0.05 -0.05 -0.07 (1.15 Q.15 -0.13
-0.5R -0.60 -0.60 -0.63 -1.31 -1.31 -1.34 -1.34 -0.09 -0.13 -0.13 -0.23 -0.34 -0.34 -0.41 -0.58 -0.58 -0.60 -0.85 -0.85 -0.83 -1.15 -1.15 -1.08 -1.49 -1.86 -1.65 -1.71 -2.42 -2.42 -2.26 -2.26 -1.w -2.09 -2.93 -2.93 -2.70 -2.71 -2.54 -3.48 -3.17 -3.07 -4.07 -3.69 -3.75 -3.01 -3.72 -4.71 -4.24 -4.35 -3.32 -4.54 -4.82 -5.c3 -3.57 -5.56 -5.43 -5.81 -6.Ok -6.73 -6.68 -7.81 -7.28 -10.57 -1l.W -8.20 -11.59 -12.47
-0.08 -0.05 n.14 -0.13 0.13 -0.04 -0.09 -F.flZ 0.00 -0.19 0.08 -0.05 0.16
-0.11 0.11 0.08 -0.08 C.06 -0.04 0.15 0.01 -0.11 -0.03 0.05 -o.no -0.10 0.00 -0.05 o.n2 -0.05 0.n9 -0.02 -0.03 0.03 -0.08 0.03 -0.02 -0.05 0.04 0.07
t(Av)
-1.1 -2.3 1.8 1.5 -0.9 -0.2 -1.5 2.6 -0.7 1.5 1.5 0.3 -0.3 -0.3 0.1 0.1 0.2 -1.7 -1.1 -0.8 1.6 -c.5 0.4 -0.4 -0.6 1.3 1.3 -1.1 -0.8 -0.5 _;.: 1:2 -0.4 -0.9 -0.2 0.0 -1.6 0.7 -n.5 1.4 -1.0 1.0 0.7 -0.7 0.6 -0.4 1.3 0.1 -1.0 -0.3 0.5 0.0 -0.9 0.0 -0.7 C.2 -0.5 -E -0.3 0.3 -0.7 i).2 -0.2 -0.6 n.4 0.9
Source
b b : : E b F? b : : b b b b b : b b b : b b b : b b b b b b" : b b b E b b : b b b b b b b b b b b b b b b b b
b This work
standardized which
residuals, t(Av),
AY( = vobs -
which
measure
vealc) differs from an expected t(Av)
the number
of standard
deviations
by
value of zero.
= Av/u(Av).
(12)
TABLE IV Rotational and Centrifugal Distortion Constants (in MHz) of i.3-nioxane. Tetrahydmpyran and Axial Piperidine obtained from Analysis using Kirchhoff's CDANAL Program
Parameter"
1.3-0ioxane
Tetrahydropyran
Axial Pip&dine
A"
4999.925 t 0.011
4673.506 ? 0.009
4494.446 f 0.027
8"
4807.618 r 0.011
4495.071 f n.ons
4395.428 f 0.025
C"
2757.114 t 0.011
26n1.282 f n.oo9
2535.771 f 0.025
(-7.203 t 2.829) x 1O-3
(-5.2W ? 2.48.9)Y 1O-3
-3 (-59.71 * lll.223x 10
(-1.944 t 0.943) x 1ne3
(-1.441 t 0.829) x 1n-3
(-19.63 f 3.40) x inm3
-1 (4.0 t 0.6) x in
(3.0 2 0.5) x 1n-'
-1 (75.23 f 3.94) x 10
T1 12 T3
(-4.28 5 n.'?4)x lO-3
(-3.99 f fl.83)x 1n-3
(-21.60 t 3.40) Y l!Y3'
(-4.42 t n.94) x inm3
-3 (-4.22 t 0.83) x In
-3 (-22.12 f 3.40) x lo
-4 (-7.6 - 9.6) x 10
(-8.49 f 8.44) x ln-4
(-192.6 t 34) x 10-4
'aaaa %hbh 'Cccc
Nunber of lines fitted h Standard deviation of fit (MHz)
90
0.145
68
0.126
68
0.114 --
' UnCWtainty is one standard cww of the parameter quoted. The nlnber of siqnificant firlUresqiven is that necessary to reproduce the observed frequencies. If2 assuminq unit weights
As Kirchhoff has shown (Z), this parameter is much more sensitive to misassignments than the standard deviation of V&s- veaie, a(Av). The final fits are shown in Tables I, II, and III and the resulting coefficients in Table IV. Improved uncertainties in A”, B”, and C” were obtained for the three molecules. Relatively high standard deviations were observed for all of the 7 coefficients. Similar large errors in the T’Swere encountered by Laurie and co-workers (14) in their study of cyclopropene. In all three cases the coefficient 73 was found to be several orders of magnitude larger than the other coefficients. This appears to be a general phenomenon in the analysis of nonplanar and planar molecules using the Kirchhoff program. Since its value is, as Kirchhoff states, (I) arbitrary to within an order of magnitude, and since the contribution of &‘s*) to the frequency was found to be a factor of lo2 smaller than the contributions from the other terms in the energy expression, the discrepancy in magnitudes is not a cause for concern. Also, in the present work, the quartic coefficients were found to be highly correlated. For 1,3-dioxane and tetrahydropyran removal of the most poorly determined coefficient, 7”cccc, from the fit, led to a considerable improvement in the uncertainties of the remainder of the coefficients, while retaining a good fit to the observed frequencies. However, such an arbitrary removal does not seem justified at the present time. Computations for 1,3-dioxane and tetrahydropyran indicate that the observation of the few very weak cR-i,~ transitions of low K-1 is necessary to improve the uncertainty in the value of Pcccc. To test whether the high uncertainties in the quartic coefficients were due to the presence of a sextic contribution in the reduced Hamiltonian, various combinations
CENTRIFUGAL
EFFECTS, 6-MEMBERED
RINGS
319
TABLE V Rotational and Centrifugal Distortion Constants (In Mllzexcept for dWJ and dHK which are dimensionless) of 1.3-flioxane,Tetrahydropyran and Axial Piperidine obtained from Analysis using Mdllendal's MB07 Program
Tetrahydropyran
1.3~rlioxane
ParaneteP
n.nin
Axial Piperldine
i
4999.927 f
4673.502 f 0.009
4494.433 f 0.026
i
4807.613 f 0.010
4495.074 t 0.009
4395.421 f 0.024
F
2757.111 t 0.011
2601.281 f 0.009
2535.759 f.0.024
-3 (1.977 f 0.301) x 10
(2.516 ? 0.227) x lO-3
(11.23 ? 1.60) x lO-3
(-2.088 t O.Q28) x 10“
-1 (1.712 ? 0.034) x 10
(2.515 - 0.665) x 10-l
-2 (9.118 2 0.123) x 10
-2 (-7.413 f 0.145) x 10
(-10.89 f 2.84) x 1O-2
(-1.839 t 0.358) x lO-7
(-3.242 f 0.201) x lO-7
(-13.04 t 3.11) x 10-7
-5 (4.221 t 0.057) x 10
(-3.787 f 0.073) x lfl-'
(-5.748 f 1.492) x lO-5
dJ dJK dK dblJ d?dK
Number of lines fitted
90
68
68
Standard deviation of fit (MHz)
n.145
0.126
0.114
a IUncertaintyis one standard error of the parameter quoted. is that "wessary to reproduce the observed frequrncier.
The number of significant figures
of the sextic coefficients were added to the Kirchhoff basis. In all cases attempted, this led to larger uncertainties in all the coefficients and a poorer fit of the calculated and observed frequencies. This is not an unexpected result for molecules with six heavy atoms and J,,, of 32 (see Ref. (3)). B. Analysis with MB07 The same sets of transitions next were analyzed using the program MB07, written by M$llendal. This employs the following form of Watson’s reduced Hamiltonian (I), In Hred = H,+ Hd: H, = irPa2 + BPb2 f Hd = -dJ(P2)2
cP2,
- d~j@P?
(13) - &P,4 - d~.rH,P2 - &&(H,P,2
The A, i?, and C? in Eq. (13) are again effective rotational Kivelson-Wilson constants by (13) B = A’+
16&i,
B = B’ -
16Rs(A’ - C’)/(B’
c = C’ + 16R&4’ where Re is one of Nielsen’s constants
constants,
+ P,2H,). related
(14) to the
(15) - C’),
B’)/‘(B’ - C’),
(161
(17)
and is given by
R%= A(--27’myy + r’zmz+ dyyyy). In the MB07 program, the eigenvalues of H, are first calculated, then the differences between the observed and calculated frequencies are fitted to the small d coefficients
TABLE Comparison of
!lolecule
of d Coefficients
the Kirchhoff
Coefficlenta'b
1,3-ilioxane
Fit with
VI
Calculated those
from
nbtained
Calculated,
from
tne i Coefficients
in the "1411endal Fit
i's
nbserved.
(-2.f188t0.025)xl"-1 (~.118tn.l:3)xlO-2
dK
(-1.81w.358)xlC-7
dWJ
dJ
(2.516~n.227)xlO
(-7.413tn.l45)xln-?
dK
(-3.?4zt0.2nlM~-7
dWJ
(-3.787*9.n73)x10-5
dWY
(11.23?1.6~)x1P-1
dJ
(?.515tn.fiG5)xlfl-'
dJK
~-1fl.89Q.84)x10-2
dK
(-13.0413.111x10-7
d#J
(-5.748t1.497)xln-5
dWK
b
All centrifuqal dimensionless).
distortion
-3
(i.71zrn.n34)~i~-'
dJY
a
-5
(4.?2ltc.n57Ixln
%K
Axial Piperidine
Fit
(1.177tn.3111)x1n-3
dJ dJK
Tetrahydropyran
from Mollendal
constants
are
Uncertainties in calculated d coefficients calculated variance lnatrix elements.
in MHz
q~ven
(except
d,dJ and dJK which
hy the square
are
root of the
in Eq. (14) by the least squares method using the III’ representation for oblate molecules. The results given in Table V show improved uncertainties in A, B, and I? as well as acceptable standard errors for the five distortion coefficients, except that in the case of axial piperidine the standard errors are somewhat larger due to the absence of c type lines in the data set. DISCUSSION
As was mentioned in the previous section, the percent uncertainties in the 7 coet& cients from the Kirchhoff CDANAL fit are appreciably larger than those in the d coefficients in the M$llendal MB07 fit, see also Tables IV and V. There was found in general to be much less correlation among the quartic coefficients in the M$llendal analysis than for the corresponding r coefficients in the Kirchhoff analysis, an indication that these linear combinations of the 7’s (i.e. the d’s) are better determined than the 7 coefficients themselves. The high correlation between dJ and dwK and between dJK and dwK, discussed by Marstokk and M#llendal (15), was observed in the present case as well. This was interpreted by them in terms of the difficulty in determining centrifugal distortion constants from microwave data alone. Two further points are worth noting with regard to the consistency of the results obtained. The first is that by using the analytical expressions (1, 13) to convert the
CENTRIFUGAL
EFFECTS,
6-MEMBERED
RINGS
321
observed 7’s of Table IV to the small d’s in Eq. (14) we were able to reproduce very well the d’s evaluated by the M$llendal fit, see Table VI. The conversion from 7’s to d’s produces a noticeable damping effect on the standard errors of the coefficients. Since only dg is even slightly dependent on the most poorly determined r coefficient, II 7 ccc, the result of the conversion seems to indicate that the remaining observed 7% are close to the correct ones. Second, transforming the variance-covariance matrix from the Kirchhoff basis to the Mfillendal basis yielded covariance matrix elements for the quartic terms that were in general considerably smaller than the corresponding elements in the 7 matrix. Details of this procedure are given in Appendix 1. The correlation matrix derived from the transformed variance-covariance matrix agreed very well with the correlation matrix actually obtained for the M$llendal fit. This provides further confirmation of the notion that the d’s as linear combinations of the T’S are much less correlated than the s’s themselves and parallels the findings of Carpenter (16) in his work on another oblate molecule, carbonyl fluoride, COF2. He found that certain linear combinations of the quartic parameters could be found that exhibited less correlation than the parameters themselves, with a resultant lowering of the standard deviations quoted for the determinable combinations. The transitions measured for our three molecules appear to contain sufficient information to evaluate the small d’s of the Watson reduced Hamiltonian Hred with small uncertainties. The same is evidently not the case for the 7’s. Much larger centrifugal distortion frequency shifts than those observed in the present study are required to bring the uncertainty limits of the 7’s more closely in line with those for the present sets of d’s. To test this idea we carried out a reanalysis, with the Kirchhoff program, of the recent results for another oblate ring compound, oxetane, CH~CH~CHZ, (Y=O state). When we restricted the oxetane data set to transitions involving J up to 17,2 we obtained high correlation among the quartic coefficients and uncertainties in these T’S of about
10 percent
(the 7 uncertainties
in our present
study
are typically
about
2.5
percent). In the original analysis of the oxetane spectrum (5) A coefficients with uncertainties of better than 6 percent (except for AJ) were obtained using transitions with J up to 48 and including molecules
in our study
of these characteristics uncertainties
comparable
one sextic parameter,
JZJK. Compared
are more rigid in structure should
to oxetane,
and are appreciably
lead to less centrifugal
stretching
so that
to those found for oxetane in Ref. (5) transitions
the three
heavier.
Both
to obtain with J well
above 48 would have to be observed for our molecules. If such high J transitions were accessible the use of sextic term fitting parameters could be made to obtain well-determined 7 coefficients. These considerations indicate that the problem of obtaining small uncertainties in the 7 values is not one inherent to oblate molecules in general. Also, the oxetane results (5) show that it is possible to obtain small uncertainties in centrifugal distortion constants without having available the frequencies of a variety of types of transitions since in the oxetane case, apart from a few low J R-branch lines, the entire data set consists of a type Q-branch 2By setting J,,,, included in the fit.
transitions.
= 17, transitions with centrifugal distortion shifts greater than 50 MHz were not
322
LOWE
AND KEU’LES
CONCLUSION
By including the effects of centrifugal distortion, improved uncertainties have been obtained in the rotational constants of 1,3-diosane, tetrahydropyran and axial NH piperidine. The results indicate that although two different analyses of these effects were able to reproduce the observed spectra with uncertainties approaching the measurement error, the ability to obtain well-determined quartic fitting coefficients can depend strongly on the form of the quartic Hamiltonian used in the fit, as may be seen by comparing the 7 and d coefficients of 1,3-diosane and tetrahydropyran in Tables IV and V. APPENDIX
1
As one of the checks on the calculations performed in this study, the authors found it useful to transform the matrix of covariances from the Kirchhoff basis to the M$llendal basis and compare the result with the matrix of covariances generated in the M$llendal fit. The variance-covariance matrix a2 (henceforth called 1/Cli’-matrix) is given by u2 = ,,z(jWJ)-l
(Al)
where uL2is the standard error of an observation of unit weight, W is a diagonal weight matrix (an identity matrix in the present analysis) and J is the Jacobian matrix, Jlj = (evl)/(aPj). It can be seen, therefore, that the problem of transforming the I/CT/matrix from one basis to the other is one of converting J,, the Jacobian matrix in the Kirchhoff basis to Jd, the Jacobian matrix in the M$llendal basis. Applying the chain rule of differentiation one can write
642) where F = C~=I” vi, with N equal to the number of transitions can be rewritten in the following form, by summing over j :
fitted. Equation
Jd = JP’
(A2)
(A3)
where A’ is the desired transformation matrix, Aij’ = (&,)/‘(adj). However, because the d’s and G’s (G = 6,8, c) are written as linear combinations of the 7’s and (Y”‘s ((Y” = A”, B”, C”) and not vice versa, the matrix actually computed is A, Aij = (adi)/(eTjJ. Equation (A3) now has the form
Jd = J,A-‘.
(A4)
This, of course, requires that A be a square, nonsingular matrix. Fortunately, since the d’s and ~5’sare not dependent upon Kirchhoff’s 73, a zero column in A results, transforming it into an 8 X 8 matrix suitable for inversion. 3 Now, from Eq. (Al), the two forms of the I’CV-matrix are defined as
With the help of Eqs.
Kirchhoff
or2 = (T~~~(J”~J,)-~,
(As)
M$llendal
od2 = cld2(JdJdk1.
(-46)
(A4) and (AS) and a little matrix
3At the same time, the ~3 row and column in
manipulation,
u,~ must also be deleted.
ud2 may be
CENTRIFUGAL
written
EFFECTS,
in the following convenient
6-MEMBERED
RINGS
323
form : (-47)
The actual computation can be performed with the original matrices A and u,~ since the 73 dependence is removed by the matrix multiplication. Finally, the coefficients of the correlation matrix C can be evaluated from the VCVmatrix 02 by the relation Cjk = (~")jk/[(~">jj(~">k~]~ (A@ while the standard
error in parameter
Pj is given by Uj
=
[(U2)jj]’
(A91
ACKNOWLEDGMENTS We should like to thank Dr. W. H. Kirchhoff gram CDANAL and Mr. A. J. Thakkar and Mr. program to the Queen’s University computer. One of us (R.S.L.) acknowledges the receipt Queen’s University School of Graduate Studies. from the National Research Council of Canada.
for providing a copy of the centrifugal distortion proH. B. Schlegel for their aid in adapting the CDANAL of an R. Samuel McLaughlin Scholarship from the This work was supported by operating grant A2046
RECEIVED: July 28, 1975 REFERENCES 1. J. K. G. WATSON,J. Chem. Phys. 46, 1935 (1967). 2. W. H. KIRCHHOFF,J. Mol. Spectrosc. 41, 333 (1972). 3. W. H. KIRCHHOFF,in “Critical Evaluation of Chemical and Physical Structural Information” (D. R. Lide and M. A. Paul, Eds.), pp. 312-322, National Academy of Sciences, Washington, D. C., 1974. 4. L. M. BOGGIA, R. R. FILGUEIRA,0. M. SORARRAIN,AM) D. DAMIANI, .&it. Naturjorsch. 29a, 95 (1974). 5. R. A. CRESWELLAND I. M. MILLS, J. Mol. Spectrosc. 52, 392 (1974). 6. R. PEARSON,JR., A. CHAPLIN,V. LAURIE, ANDJ. SCHWARTZ,J. Chem. Phys. 62, 2949 (1975). 7. V. M. RAO AND R. KEWLEY, Can. J. Chem. 47, 1289 (1969). 8. R. KEWLEY, Can. J. Chem. 50, 1690 (1972). 9. P. J. BUCKLEY,C. C. COSTAIN,ANDJ. E. PARKIN, Chenz. Comm. 668 (1968). 10. R. F. CURL, T. IKEDA, R. S. WILLIAMS,S. LEAVELL,ANDL. H. SCHARPEN,J. Amer. Chem. Sot. 95, 6182 (1973). If. K.-M. MARSTOKKANDH. M$LLENDAL,J. Mol. Struct. 4, 470 (1969); 5, 205 (1970). 12. D. KIVELSONAND E. B. WILSON,J. Chem. Phys. 20, 1575 (1952). 13. W. GORDYANDR. L. COOK, “Microwave Molecular Spectra” (W. West, Ed.), Vol. 9, Part 2, Chap. 8, Interscience, New York, 1970. 14. V. M. STIGLIANI,V. W. LAURIE, ANDJ. C. LI, J. Chem. Phys. 62, 1890 (1975). 15. K.-M. MARSTOKKAM) H. M~LLENDAL,J. Mol. Stract. 8, 234 (1971). 16. J. H. CARPENTER,J. Mol. Spectrosc. 50, 182 (1974).
OF MOLECULAR
SPECTROSCOPY
60, 312-323
(1976)
Centrifugal Distortion Effects in the of 1,3-Dioxane, Tetrahydropyranl R. s. Department
LOWE
AND
of C/tern&&y, Qzreen’s University,
R.
Microwave Spectra and Piperidine
EEWLEY
k’in,eslon, Ontario,
Can&a,
k’7L 3N6
Applying a bootstrap approach to transition assignment, a first-order centrifugal distortion analysis has been performed on 1,3-dioxane, tetrahydropyran, and piperidine (axial NH conformer). Q-branch transition frequencies for J up to 32 have been combined with previously published or remeasured frequencies for low J, R-, and Q-branch transitions to yield values for the quartic centrifugal distortion parameters. Comparison is made between the results obtained using two forms of the quartic Hamiltonian. INTRODUCTION
Although the theory of centrifugal distortion effects in molecular rotational spectra has been well covered in the literature (I, 2,3), there have been relatively few applications of this theory to heavier molecules, particularly saturated ring compounds (4,5,6). It was therefore thought of interest to obtain a quantitative account of the centrifugal distortion in each of the three oblate six-membered heterocycles : 1,3-dioxane, tetrahydropyran, and the axial NH conformer of piperidine (Fig. 1). Previous studies (7,8,9) yielded rigid rotor rotational constants and found all three molecules to be present exclusively in the chair form, piperidine as both axial and equatorial NH conformers. For reasons given below we have not included equatorial piperidine in the present study. For each of 1,3-diosane and tetrahydropyran (7,s) the effects of nonrigidity were noticed but not included in the earlier analysis. By doing so here for all three molecules we hoped to extend our understanding of the centrifugal distortion effects in this class of compounds. For each molecule the present analysis uses the rigid rotor fit for low J transitions as a starting point. In the case of axial piperidine the published planar moments (9) were helpful in locating the low J transitions. EXPERIMENTAL
With the exception of two R-branch and eight Q-branch transitions measured by Curl and co-workers (10) on a Hewlett-Packard 8460A spectrometer, all the lines used in this analysis were observed using a conventional 100 kHz modulation Hughes-Wilson spectrometer operating between 8 and 40 GHz. All frequency measurements were made at room temperature using oscilloscope display and are considered to be accurate to better than 0.2 MHz. The 1,3-dioxane sample was from K and K Laboratories, the tetrahydropyan from Aldrich Chemical Company and the piperidine from B.D.H. 1The results for 1,3-dioxane and tetrahydropyran were presented Structure and Spectroscopy, Columbus, Ohio, as paper TFlO. 312 Copyright @ 1976 by .4cademic Press, Inc. All rights of reproduction in any form reserved.
at the 30th Symposium
on Molecular
CENTRIFUGAL
EFFECTS,
1,341OXANE x
6-MEMBERED
TETRAHYOROPYRAN
= 0.82852
RINGS
313
AXIAL PIPERIOINE
x = 0.82779
x
= 0.89887
FIG. 1. The molecules examined in this study.
Laboratory Chemicals. All were used without further purification. All computations were performed in double precision arithmetic on the IBM 360/.50 and Burroughs 6700 computers at Queen’s University. ASSIGNMENT
AND ANALYSIS
Essentially all of the transitions listed in Tables I, II, and III were assigned on the basis of goodness of fit except where additional confirmation was possible from Stark effects. Q-branch asymmetry doublets constituted the majority of the high J transitions found between 8 and 40 GHz. They were of b and c type for 1,3-dioxane, a and c type for tetrahydropyran and a type for axial piperidine. C type Q-branch lines could not be located for axial piperidine. This is in agreement with the observations of Buckley, Costain, and Parkin (9). Aside from a small number of allowed but very weak high J CR~,_-2transitions, transitions of high J from other subbranches than those observed were out of the range of our instrument. For equatorial piperidine, because of the small value of the a-component of the dipole moment (0.195 D compared to 1.53 D for tetrahydropyran) only a few tentative assignments of a type Q-branch lines could be made. Also, although identifiable c type transitions should be present, as for axial piperidine, none of these could be found. It was therefore not possible to make a centrifugal distortion analysis for equatorial piperidine. The data were analyzed using two computer programs. One was developed by Dr. W. H. Kirchhoff at the National Bureau of Standards, Washington, D. C. (2) and the other by Professor H. M$llendal at the University of Oslo (II). A. Analysis
with CDANAL
The Kirchhoff program,
CDANAL,
uses as its basis the reduced Hamiltonian
H red
=
H,+
Ha
(1)
H, = A”P,2 + B”P+ + C”Pc2,
(2)
where Hd = t
c
a=a,b,c
~“aaaaPm4 + TIP14 +
r2P24 +
r2P24.
(3)
In Eq. (2) the A”, B”, and C” are the effective rotational constants. These are related to the Kivelson-Wilson rotational constants (1.2) by the following equations (14) A” = A’ _ 127’ bbcc,
(4)
314
LOWE AND
KEWLET
B”
= B’ _
.i 27 ’ a
(5)
C”
= C’ -
+7raabb.
(6)
The quartic distortion constants T”~~~~, 71, 72, and 73 in Eq. (3) are similarly the Kivelson-Wilson coefficients by
related to
I, 7 a(1oa= ‘T’orcrua (a = a, 4 c), 71
=
T'aabl, +
72
=
(A’/.%‘bbcc
~3
=
[S/(B’
T’bbcc
+ +
(7)
~‘ccaa, @‘/S)T’aacc
(8) +
- A’)]daa6b + [S/(/l’
(9)
(C’/S)daabb,
- C’)]d,,ce + [S/(C’
-
B’)ld~ce
(10)
where S = A’+
B’+
C’.
(11)
Finally, Pp, P2*, P34 appearing in Eq. (3) are linear combinations of A’, B’, C’ and the Pa2 (a = a, b, c). As pointed out by Watson (I), there exist only five determinable coefficients in the quartic Hamiltonian. A sixth coefficient, 73, has been added by Kirchhoff to facilitate TABLE I Observed Transitions of 1,3-Dioxane
Transition J(K_jsKl) 0
1 1 I
; 2 ; : 2 2 : 4 z 6 : 7 7 8 a 8" 8 : 9 1: 10 10 1'; 11 11 11 11
'ohs
+ J’(KIlsKi)
0, 0) + l.l)* 0.1)l,0)+ O,l)l,l)+ 1,0)1,2)* 2, 1) *
1 2 2 2 2 2 2 3 3
:* A] : 0: 2) + l,l)* 2, 0) * 0, 5) + 1,5)-r 1,4)* 0. 4) + 0, 6) + 1, 6)+ 2. 4) + 0, 5) + 1,6)* 2. 6) * 2, 6) * 3. 6) + 3, 5) + 4, 4) + 5. 4) + 4. 6) * 5, 5) * 5, 4) * 6, 4) * 5, 5) * 6, 5) + 6, 4) * 3, 8) + 5. 6) * 6. 6) + 6, 5) * 7. 5) * 7. 4) * 9, 4) *
; 3 3 3 5 5 5 5 6 6 6 6 7 7 8 a a a 8 9 9 9 9 10 10 10 11 11 11 11 11 11 11
(MHz 1
1, 0) 2)
0,
is :I 1: 1)
2, 1) 2, 0) 0. 31 1. 21 2. 2)
3, 1, 2. 3. 1, 2, 0. 1. 1. 2, 3, 1. 2. 3, 3, 4. 4. 5, 6. 5. 6. 6, 7, 6, 7. 7, 4. 6. 7. 7, a, 8, 9.
1) 2) 11 0) 4) 4) 5) 5) 5) 51 3) 6) 5) 5) 5) 5) 4) 3) 3) 5) 4) 3) 3) 4) 4) 3) 7) 5) 5) 4) 4) 3) 3)
"ohs - vcalc
P4 Term
(MHZ1
Hz
0.04 0.18 -0.11 0.04 -0.01 0.09 -0.12 -cl.09 0.14 0.07 0.13 -0.11 -0.14 0.00 o.la 0.01 -0.06 -0.07 0.06 0.05 0.15 0.08 0.02 -0.02 n.07 -0.10 -0.17 -0.17 -0.13 0.22 -0.26 0.05 -0.04 0.21 -0.24 -0.04 0.09 -0.24 -0.16 0.14 -0.22 0.15 -0.04
-0.01 -0.02 -0.02 -0.03 -0.03 -0.03 -0.04 -0.04 -0.07 -0.07 -0.11 -0.08 -0.12 -0.12 -0.16 -0.16 -0.13 -0.13 -0.25 -0.25 -0.36 -0.21 -0.52 -0.52 -0.83 -0.83 -0.82 -0.72 -Cl.76 -1.18 -1.12 -0.93 -1.00 -1.42 -1.44 -1.19 -2.08 -1.99 -1.99 -1.76 -1.81 -1.47 -1.57
t(bv)
1 0.3 1.2 -0.8 0.3 -0.1 0.6 -0.8 -0.6 1.0 0.5 0.9 -0.8 -1.0 0.0 1.2 0.1 -0.5 -0.6 0.4 0.4 1.1 0.9 0.1 -0.1 n.5 -n.7 -1.2 -1.2 -Il.9 1.5 -1.8 0.4 -0.2 1.4 -1.7 -0.3 0.6 -1.7 -1.1 1.0 -1.5 1.0 -n.3
Sxlrce
CENTRIFUGAL
EFFECTS, 6-MEMBERED
315
RINGS
TABLE I, Continued
Transition
"obs
12 12
( 3, ( 6.
9) + 12 ( 4, 8) 6) + 12 7, 5)
13 13 14 14 14 14 14 15 15 15 16 16 17 17 17 18 18 18 18
( ( 8, + ( a: 5) .+13 ( 9, 5) * 13 ( 5, 9) + 14 ( 8, 6) * 14 ( 9, (10. (11, ( 9, (11, (12, (10. (12. (11. (13. (14, (12, (13. (14. (15,
5) 5) 4) 6) 5) 4) 6) 5) 6) 5) 4) 6) 6) 5) 4)
;; I;? El ;; 1;;: :] 19 19 20 20 20 21 21 21 22 22 22 23 23 23 24
(171 (18. (14, (15, (16, (15. (17, (20, (15. (16, (18, (17, (18, (19, (18.
36418.85 23413.36 18854.71 19066.75 13351.00 23342.23 18637.6'0 19020.85 36381.16 23247.19 18336.72 18907.74 19961.05
(
12 7. 5) * l2 ;: 13 17.t Y;%461 5) * 12
3) 2) 6) 6) 5) 6) 5) 1) E) 6) 5) 6) 6) 5) 6)
+ + * + + * + + + + * + + + * + * * + + + * * + * + * + + * + * * +
14 14 14 15 15 15 16 16 17 17 17 18 18 18 18 19 19 19 19 19 19 20 20 20 21 21 21 22 22 22 23 23 23 24
( is :i 8. 3) I 9: ( 5)
( 9, 4) (10, 4) 6, 8) 5) (10, 4) (11, 4) (12. 2) (10. 5) (12, 4) (13, 2) (11, 5) (13, 4) (12. 5) (14, 4) (15, 2) (13, 5) (14. 5) (15, 4) (16. 2) (14, 5) (15. 5) (16. 4) (17. 3) (18, 2) (19, 1) (15, 5) (16, 5) (17, 4) (16, 5) (18, 4) (21, 0) (16, 7) (17, 5) (19, 4) (18. 5) (19, 5) (20, 4) (19. 5)
( ( 9,
a R. Kewley, ref. (8);
'ohs - Vcalc
(!YHr )
J(K_,,K,) + J'(K:,.Ki)
:6:E 19493:09 22952.74 19006.66 22728.27 19085.85 19131.08 22430.53 22947.46 19231.48 19279.69 22040.31 22870.11 19456.84
This work;
(MHz 1
0.03 -O.OA n.25 0.07 -0.22 0.24 0.09 -0.06 0.01 0.10 0.03 -0.08 0.04 -0.19 0.14 -0.02 0.15 0.07 -0.02 -0.O8 -n.ln -0.14 -0.25 0.16 0.04 -n.o5 0.03 0.22
-2.63 -2.45 -2.14 -2.22 -1.60 -2.96 -2.56 -2.67 -4.24 -3.51 -3.02 -3.17 -3.93 -4.10 -3.70 -4.33 -4.73 -4.26 -5.41 -4.84 -4.88 -6.15 -6.35 -5.43 -4.4s -6.96 -7.18 -6.01
t(Av)
0.2 -0.5 1.7 0.2 -1.5 1.6 0.7 -0.4 0.1 0.7 0.2 -0.6 0.3 -1.3 1.0 -0.2 1.0 0.5 -0.2 -0.6 -0.7 -1.0 -1.7 1.1 0.3 -0.4 E -1:9 1.3 -0.9 _'1.: -0:7 1.8 -0.2 1.1 -1.1 -1.1 0.6 -0.1 2.2 0.9 -0.1 -1.3 0.9 -0.8
20915.83 20192.93 25596.29 31465.79 20171.74 20720.08 19330.62 22921.29 21358.09 18437.00 23853.31 17556.25 24367.99 b
P4 Telm
(MHZ)
'
Source
c b : a : b E : b b b b b b : b I! ! b : : b b : E b b c b b b b b b bb b
R.F. Curl et al.. ref. (1p).
comparison of the angular momentum operators between the Watson and KivelsonWilson Hamiltonians. 73 is not, however, a determinable coefficient, but is evaluated once the remaining eight parameters have been determined. The eigenvalue problem is solved by direct diagonalization of the semirigid rotor Hamiltonian in the P rigid rotor basis to give corrected values of the rotational constants and a first-order contribution due to the P4 terms, in the form of the 7 coefficients. A refinement of these constants was obtained by a least-squares fit to the observed frequencies in the following manner. Assignment of the higher J transitions followed the bootstrap approach outlined by Kirchhoff (2). Briefly, the rigid rotor A, B, and C were used to predict the positions of higher _7lines. Those that were expected to lie within 5 MHz of the rigid rotor position were then measured. Using the augmented set of observed frequencies in the leastsquares fit yielded new rotational constants and a first estimate of the 7 coefficients. These new constants were used to predict the positions of still higher J transitions. The analysis was considered completed when a sufficiency condition had been reached,
LOWE AND KE\VLE:Y
316
TABLE II Observed Transitions of Tetrahydmnyran
Transition
'ohs
J(K_,.K,) _ J'(K(T,Ki) 1 1
fl,l)* I,,)l,O)* O.l)1.1) * c, 2) * 1,2)-r l.l)* ?,l)1,2)+ 2, 1) + 2, n) + 1,2)* 2, 2) * 3,1)+ 1,5)+ n, 4) + 1,4)+ 1,6)n, 5) *
2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 5 5 5 6 6
a, :I * 1:aj * 1,8)2,8)+ 4, 6) + 1: 4, 7) * 11 6. 6) + 11 Q, 2) + 6, 7) * ;; (I,2) + 8, F) + 1: 0, 3) * 14 S,lO) t 14 1. 3) + 15 7, 9) * 9, 7) * 1: 2, 3) + 9, 7) * :z 3, 3) + 17 1, 7) + 2, 6) * Y3 1, 7) * 18 2, 7) + 14 1, 8) + 19 2. 8) + 2, 7) * 3. 7) * E 2n 3. 7) * 4, 7) * 20 20 5, 6) * 20 6, 4) + 21 3, 8) + 21 4, 8) + 21 5, 7) + 21 6, R) * 22 5, 8) + 22 5, 7) + 6. 6) + 22 7, 6) * 2': 0, 2) + 24 7, 7) + 25 8, 7) + 25 9, 7) * 9, 7) * ;: 0, 9) + 30 '1, 9) * 32 '3.10) +
9 ' R P 9 9 10 11 11 12 12 13 13 14 14 15 15 15 16 16 17 17 18 18 19 11 19 19 2n 20 20 20 21 21 21 21 22 22 22 22 22 24 25 25 26 29 30 32
1' 1 2 2 2 2 2 2 2 3 3 7 5 4 4 6 5 7
R 9 0
0, 2)
1. 1, 1, 2, n, 1. 1, 2, 2. 3. 3, 1, 2, 3,
2) 1) 1) 1) 3) 3) 2) 2) 2) 1) 0) 3) 3) 2)
0, 1, 1, 0, 2, 2, 1, 3, 2, 4, 4, 6, 1, 6, 2, 8, 2, 5, 3, 7, 9, 4.
5) 5) 5) 6) 5) 7) 7) 7) 7) 5) 6) 5) I! 6) 11 5) 2) 9) 2) 8) 6) 2)
:* 1: 2. 3, 2, 3, 2. 4. 3. 5. 4. 5, 8,
:I 6) 5) 6) 6) 7) 7) 6) 6) 6) 6) 5) 3)
:* 5: 6, 5, 7, a. 7. 1, 9, 0,
:I 6) 5) 7) 6) 5) 5) 2) 6) 6)
;, 2: 3. 3,
:I 8) 8) 9)
1, 4)
P4 Ten
Yobs - Ucal‘
(MHz) 12465.20 12299.00 16086.40 18158.8r) 18515.80 17594.70 17583.40 21764.4n 21289.10 27505.6C 27780.40 77553.60 26785.50 26729.30 3nl91.90 17821.80 27993.77 27993.77 21784.40 33196.24 21775.20 29709.39 29709.39 29701.68 79701.68 21741.40 25711.n6 21676.95 17385.03 25662.04 19839.76 71559.7n 15840.92 37591.66 17715.54 33583.ln 25525.52 19871.98 25466.56 22281.73 25368.88 20964.92 25303.74 25260.35 21349.311 21345.16 25205.47 25123.83 25098.41 24950.37 19731.99 24214.71 29171.7n 29153.44 24727.81 19045.92 29028.98 24878.43 21496.82 18187.11 22021.41 24717.87 24702.43 22952.40 24765.10 32344.55 32169.18 36144.23
(WZ)
(MHz)
-0.11 0.1n -Il.01 0.11 0.24 -0.20 0.02 -cl.14 O.lI, -0.M -0.16 0.24 -fl,n1 n.n1 n.no -0.20 -o.ni o.ni -0.32 0.02 -n.23 0.13 n.13 0.11 0.11 0.13 n.nz O.lC -0.05 0.04 il.07 -n.l6 n.14 -n.lf -0.07 -0.04 -0.18 -0.02 0.08 -n.m -0.04 0.11 n.17 -n.n5 -0.18 0.15 0.14 0.16 O.O? 0.01 -0.10 0.05 0.15 0.00 -0.01 0.11 0.08 -n.o4 -0.17 -0.19 -0.02 0.09 0.00 0.10 -0.03 -0.n2 -0.01 -0.04
-n.n2 -0.01 -n.o3 -0.03 -n.n3 -fl.O4 -n.o4 -n.o,6 -0.06 -0.07 -0.10 -n.11 -0.11 -0.11 -Il.16 -0.12 -n.13 -n.i3 -0.18 -0.21 -Cl.42 -n.34 -n.34 -cl.?6 -11.76 -1.01 -1.35 -1.75 0.14 -2.30 n.17 -2.67 1.07 -3.52 1.72 -4.44 -4.07 2.47 -4.73 3.25 -5.55 -5.58 -6.22 -6.42 -J.Jd -7.77 -7.02 -7.41 -7.83 -8.57 -10.20 -8.64 -9.82 -9.95 -9.97 -12.59 -11.21 -9.33 -4.26 -15.43 -2.30 -10.15 -9.98 -20.22 -9.14 -22.63 -24.28 -32.11
Source
t(av)
-0.9 n.8 -0.6 n.9 1.9 -1.6 0.1 -1.1 0.8 -0.5 -1.3 1.9 -0.1 0.1 -0.1 -1.6 -0.1 0.1 -2.5 n.2 -1.8 1.0 1.0 n.9 0.9 1.0 0.2 n.8 -n.5 0.3 0.7 -1.3 1.2 -1.2 -C.6 -0.7 -1.4 -0.1 0.7 -0.7 -0.7
a a a a a a a a a a a a d
a a a b b a c a b b b b
::: -0.4 -1.4 1.2 1.1 Q.1 0.1 fl.1 -0.8 0.6 1.2 0.0 -0.0 n.9 0.6 -0.3 -1.4 -1.5 -0.5 0.7 0.0 0.9 -0.3 -0.2 -0.1 -0.5
: b b b b b b c b t h b b b b h b b b b b b h b b b b b b b b b b b b b b b b c c c
a
V.M. Rao and R. I:ewley,ref. (I); b
that is, when the mum observed J, Assignment of generated by the
This work;
'
R.F. Curl et al.. ref. (10) -
uncertainties in the calculated transition frequencies, up to the masibecame comparable with the measurement error. lines was facilitated through the use of the least-squares statistics program CDANAL. The most useful statistical indicators are the
CENTRIFUGAL
EFFECTS,
6-MEMBERED
317
RINGS
TABLE III Observed Transitions of Axial Piperidine
Transition J(L_,J,) ;.:I+
.
z;ij+ 2. 0) 0, 3) 1,3)1.2)* 2. 2! 0, 5) 0, 6)
*3
3
* + *
i 3 4 4 4 4 5 6
:*:j: 1:6)+ 2.6)+ 2, 5) + 2, 6)+ 3,6)+ 3, 5) + 3, 6) + 4. 6)+
; 7 7 7 8 B B 9 9
+ *
5; ij *
9
4. 5, 5, 5. 7; 7, 8, 6, 7, 7, 8, 8, 9, 7. 8, 8, 9, 0, 9, 9. 1.
6) 6) 5) 61 6j 5) 5) 7) 7) 6) 6) 5) 5) 7) 7) 6) 6) 5) 7) 6) 5)
+ + + + + + + + + + + + + + + + + + + + +
10 10 10 11 12 12 12 13 13 13 13 13 13 14 14 14 14 14 15 15 15
0, 0, 1, 1, 2, 1, 1, 2, 2, 3, 2, 3, 3, 4, 3. 4: 4; 5, 5, 6, 6, 7. 8. 8, 8, 9,
7) 6) 6) 5) 5) 7) 6) 6) 5) 5) 8) 6) 5) 5) 61 6i
+ + + + + + + + + + + + + + + +
16 16 16 16 16 17 17 17 17 17 18 1B 18 18 19 10 20 20 21 21 22 24 24 24 25 25
6j + 6) 6) 6) 6) 7) 7) 6) 7) 7)
'ohs
* J’W,>K;)
+ + + + + + + + +
'ohs - "talc
(MHz)
1, 1. 2, 2. 0, 1. 1. 2; 2, 2. 1. 3. 3. 2, 4. 4. 3, 5. 5; 4. 5. 6, 5, 7. 7; 7, 9. 8, 8, 7. 9.
3) 2) 2) 1) 4) 4) 31 3j 4) 5) 5) 4) 5) 5) 4) 5) 5) 4) 5j 5) 4) 51 5) 41 5) 5) 4) 4) 6) 6) 5)
17120.72 2lfl70.58 20793.20 24461.54 22193.37 22193.37 26020.92 26002.8I 17174.39 20991.87 20991.87 17170.92 20988.57 20988.57 17165.32 20984.07 20984.07 17156.98 20977.53 20977.53 17143.65 20968.89 20968.89 17131.00 20957.37 2CQ42.09 17089.68 17065.03 24769.16 24769.1h 20924.fl8
0, 9, 9, 8, 0, 9. 0, 9, 1, 1, 0, 2, 1. 3. 2. 1, 3.
4) 4) 6) 6) 5) 5) 4) 6) 5) 4) 6) 5) 5) 4) 4) 6) 3)
17063.41 17017.42 24754.94 24754.44 20930.86 20897.48 18952.74 24737.38 20873.09 16865.07 24716.23 20839.83 20827.54 16978.32 16747.80 20690.59 20801.16 20778.93 16957.82 16592.32 20757.21 20718.28 16949.77 16388.05 20708.47 20642.46 20656.00 20547.31 2OfiOl.3Q 20427.61 20547.64 24360.94 24311.91 20459.70 24289.41 24210.59
R;
5j 20922.29
:* :i 3: 4) 4. 5) 3, 5) 5, 4) 4, 4) 5. 5) 4. 5) 6, 5) 5, 5) 7, 5) '6, 5) '8, 5) 19, 6) 18, 6) ?I, 5) !O, 6) 19, 6)
P4 Term
(MHz)
(MHZ)
-0.12 -0.26 0.21 0.17 -0.11 -0.02 -0.17 0.29 -0.08 0.17 0.17 0.03 -0.03 -C.O3 0.01 0.01 0.02 -0.19 -(I.12 -0.09 0.19 -0.06 0.05 -0.05 -0.07 (1.15 Q.15 -0.13
-0.5R -0.60 -0.60 -0.63 -1.31 -1.31 -1.34 -1.34 -0.09 -0.13 -0.13 -0.23 -0.34 -0.34 -0.41 -0.58 -0.58 -0.60 -0.85 -0.85 -0.83 -1.15 -1.15 -1.08 -1.49 -1.86 -1.65 -1.71 -2.42 -2.42 -2.26 -2.26 -1.w -2.09 -2.93 -2.93 -2.70 -2.71 -2.54 -3.48 -3.17 -3.07 -4.07 -3.69 -3.75 -3.01 -3.72 -4.71 -4.24 -4.35 -3.32 -4.54 -4.82 -5.c3 -3.57 -5.56 -5.43 -5.81 -6.Ok -6.73 -6.68 -7.81 -7.28 -10.57 -1l.W -8.20 -11.59 -12.47
-0.08 -0.05 n.14 -0.13 0.13 -0.04 -0.09 -F.flZ 0.00 -0.19 0.08 -0.05 0.16
-0.11 0.11 0.08 -0.08 C.06 -0.04 0.15 0.01 -0.11 -0.03 0.05 -o.no -0.10 0.00 -0.05 o.n2 -0.05 0.n9 -0.02 -0.03 0.03 -0.08 0.03 -0.02 -0.05 0.04 0.07
t(Av)
-1.1 -2.3 1.8 1.5 -0.9 -0.2 -1.5 2.6 -0.7 1.5 1.5 0.3 -0.3 -0.3 0.1 0.1 0.2 -1.7 -1.1 -0.8 1.6 -c.5 0.4 -0.4 -0.6 1.3 1.3 -1.1 -0.8 -0.5 _;.: 1:2 -0.4 -0.9 -0.2 0.0 -1.6 0.7 -n.5 1.4 -1.0 1.0 0.7 -0.7 0.6 -0.4 1.3 0.1 -1.0 -0.3 0.5 0.0 -0.9 0.0 -0.7 C.2 -0.5 -E -0.3 0.3 -0.7 i).2 -0.2 -0.6 n.4 0.9
Source
b b : : E b F? b : : b b b b b : b b b : b b b : b b b b b b" : b b b E b b : b b b b b b b b b b b b b b b b b
b This work
standardized which
residuals, t(Av),
AY( = vobs -
which
measure
vealc) differs from an expected t(Av)
the number
of standard
deviations
by
value of zero.
= Av/u(Av).
(12)
TABLE IV Rotational and Centrifugal Distortion Constants (in MHz) of i.3-nioxane. Tetrahydmpyran and Axial Piperidine obtained from Analysis using Kirchhoff's CDANAL Program
Parameter"
1.3-0ioxane
Tetrahydropyran
Axial Pip&dine
A"
4999.925 t 0.011
4673.506 ? 0.009
4494.446 f 0.027
8"
4807.618 r 0.011
4495.071 f n.ons
4395.428 f 0.025
C"
2757.114 t 0.011
26n1.282 f n.oo9
2535.771 f 0.025
(-7.203 t 2.829) x 1O-3
(-5.2W ? 2.48.9)Y 1O-3
-3 (-59.71 * lll.223x 10
(-1.944 t 0.943) x 1ne3
(-1.441 t 0.829) x 1n-3
(-19.63 f 3.40) x inm3
-1 (4.0 t 0.6) x in
(3.0 2 0.5) x 1n-'
-1 (75.23 f 3.94) x 10
T1 12 T3
(-4.28 5 n.'?4)x lO-3
(-3.99 f fl.83)x 1n-3
(-21.60 t 3.40) Y l!Y3'
(-4.42 t n.94) x inm3
-3 (-4.22 t 0.83) x In
-3 (-22.12 f 3.40) x lo
-4 (-7.6 - 9.6) x 10
(-8.49 f 8.44) x ln-4
(-192.6 t 34) x 10-4
'aaaa %hbh 'Cccc
Nunber of lines fitted h Standard deviation of fit (MHz)
90
0.145
68
0.126
68
0.114 --
' UnCWtainty is one standard cww of the parameter quoted. The nlnber of siqnificant firlUresqiven is that necessary to reproduce the observed frequencies. If2 assuminq unit weights
As Kirchhoff has shown (Z), this parameter is much more sensitive to misassignments than the standard deviation of V&s- veaie, a(Av). The final fits are shown in Tables I, II, and III and the resulting coefficients in Table IV. Improved uncertainties in A”, B”, and C” were obtained for the three molecules. Relatively high standard deviations were observed for all of the 7 coefficients. Similar large errors in the T’Swere encountered by Laurie and co-workers (14) in their study of cyclopropene. In all three cases the coefficient 73 was found to be several orders of magnitude larger than the other coefficients. This appears to be a general phenomenon in the analysis of nonplanar and planar molecules using the Kirchhoff program. Since its value is, as Kirchhoff states, (I) arbitrary to within an order of magnitude, and since the contribution of &‘s*) to the frequency was found to be a factor of lo2 smaller than the contributions from the other terms in the energy expression, the discrepancy in magnitudes is not a cause for concern. Also, in the present work, the quartic coefficients were found to be highly correlated. For 1,3-dioxane and tetrahydropyran removal of the most poorly determined coefficient, 7”cccc, from the fit, led to a considerable improvement in the uncertainties of the remainder of the coefficients, while retaining a good fit to the observed frequencies. However, such an arbitrary removal does not seem justified at the present time. Computations for 1,3-dioxane and tetrahydropyran indicate that the observation of the few very weak cR-i,~ transitions of low K-1 is necessary to improve the uncertainty in the value of Pcccc. To test whether the high uncertainties in the quartic coefficients were due to the presence of a sextic contribution in the reduced Hamiltonian, various combinations
CENTRIFUGAL
EFFECTS, 6-MEMBERED
RINGS
319
TABLE V Rotational and Centrifugal Distortion Constants (In Mllzexcept for dWJ and dHK which are dimensionless) of 1.3-flioxane,Tetrahydropyran and Axial Piperidine obtained from Analysis using Mdllendal's MB07 Program
Tetrahydropyran
1.3~rlioxane
ParaneteP
n.nin
Axial Piperldine
i
4999.927 f
4673.502 f 0.009
4494.433 f 0.026
i
4807.613 f 0.010
4495.074 t 0.009
4395.421 f 0.024
F
2757.111 t 0.011
2601.281 f 0.009
2535.759 f.0.024
-3 (1.977 f 0.301) x 10
(2.516 ? 0.227) x lO-3
(11.23 ? 1.60) x lO-3
(-2.088 t O.Q28) x 10“
-1 (1.712 ? 0.034) x 10
(2.515 - 0.665) x 10-l
-2 (9.118 2 0.123) x 10
-2 (-7.413 f 0.145) x 10
(-10.89 f 2.84) x 1O-2
(-1.839 t 0.358) x lO-7
(-3.242 f 0.201) x lO-7
(-13.04 t 3.11) x 10-7
-5 (4.221 t 0.057) x 10
(-3.787 f 0.073) x lfl-'
(-5.748 f 1.492) x lO-5
dJ dJK dK dblJ d?dK
Number of lines fitted
90
68
68
Standard deviation of fit (MHz)
n.145
0.126
0.114
a IUncertaintyis one standard error of the parameter quoted. is that "wessary to reproduce the observed frequrncier.
The number of significant figures
of the sextic coefficients were added to the Kirchhoff basis. In all cases attempted, this led to larger uncertainties in all the coefficients and a poorer fit of the calculated and observed frequencies. This is not an unexpected result for molecules with six heavy atoms and J,,, of 32 (see Ref. (3)). B. Analysis with MB07 The same sets of transitions next were analyzed using the program MB07, written by M$llendal. This employs the following form of Watson’s reduced Hamiltonian (I), In Hred = H,+ Hd: H, = irPa2 + BPb2 f Hd = -dJ(P2)2
cP2,
- d~j@P?
(13) - &P,4 - d~.rH,P2 - &&(H,P,2
The A, i?, and C? in Eq. (13) are again effective rotational Kivelson-Wilson constants by (13) B = A’+
16&i,
B = B’ -
16Rs(A’ - C’)/(B’
c = C’ + 16R&4’ where Re is one of Nielsen’s constants
constants,
+ P,2H,). related
(14) to the
(15) - C’),
B’)/‘(B’ - C’),
(161
(17)
and is given by
R%= A(--27’myy + r’zmz+ dyyyy). In the MB07 program, the eigenvalues of H, are first calculated, then the differences between the observed and calculated frequencies are fitted to the small d coefficients
TABLE Comparison of
!lolecule
of d Coefficients
the Kirchhoff
Coefficlenta'b
1,3-ilioxane
Fit with
VI
Calculated those
from
nbtained
Calculated,
from
tne i Coefficients
in the "1411endal Fit
i's
nbserved.
(-2.f188t0.025)xl"-1 (~.118tn.l:3)xlO-2
dK
(-1.81w.358)xlC-7
dWJ
dJ
(2.516~n.227)xlO
(-7.413tn.l45)xln-?
dK
(-3.?4zt0.2nlM~-7
dWJ
(-3.787*9.n73)x10-5
dWY
(11.23?1.6~)x1P-1
dJ
(?.515tn.fiG5)xlfl-'
dJK
~-1fl.89Q.84)x10-2
dK
(-13.0413.111x10-7
d#J
(-5.748t1.497)xln-5
dWK
b
All centrifuqal dimensionless).
distortion
-3
(i.71zrn.n34)~i~-'
dJY
a
-5
(4.?2ltc.n57Ixln
%K
Axial Piperidine
Fit
(1.177tn.3111)x1n-3
dJ dJK
Tetrahydropyran
from Mollendal
constants
are
Uncertainties in calculated d coefficients calculated variance lnatrix elements.
in MHz
q~ven
(except
d,dJ and dJK which
hy the square
are
root of the
in Eq. (14) by the least squares method using the III’ representation for oblate molecules. The results given in Table V show improved uncertainties in A, B, and I? as well as acceptable standard errors for the five distortion coefficients, except that in the case of axial piperidine the standard errors are somewhat larger due to the absence of c type lines in the data set. DISCUSSION
As was mentioned in the previous section, the percent uncertainties in the 7 coet& cients from the Kirchhoff CDANAL fit are appreciably larger than those in the d coefficients in the M$llendal MB07 fit, see also Tables IV and V. There was found in general to be much less correlation among the quartic coefficients in the M$llendal analysis than for the corresponding r coefficients in the Kirchhoff analysis, an indication that these linear combinations of the 7’s (i.e. the d’s) are better determined than the 7 coefficients themselves. The high correlation between dJ and dwK and between dJK and dwK, discussed by Marstokk and M#llendal (15), was observed in the present case as well. This was interpreted by them in terms of the difficulty in determining centrifugal distortion constants from microwave data alone. Two further points are worth noting with regard to the consistency of the results obtained. The first is that by using the analytical expressions (1, 13) to convert the
CENTRIFUGAL
EFFECTS,
6-MEMBERED
RINGS
321
observed 7’s of Table IV to the small d’s in Eq. (14) we were able to reproduce very well the d’s evaluated by the M$llendal fit, see Table VI. The conversion from 7’s to d’s produces a noticeable damping effect on the standard errors of the coefficients. Since only dg is even slightly dependent on the most poorly determined r coefficient, II 7 ccc, the result of the conversion seems to indicate that the remaining observed 7% are close to the correct ones. Second, transforming the variance-covariance matrix from the Kirchhoff basis to the Mfillendal basis yielded covariance matrix elements for the quartic terms that were in general considerably smaller than the corresponding elements in the 7 matrix. Details of this procedure are given in Appendix 1. The correlation matrix derived from the transformed variance-covariance matrix agreed very well with the correlation matrix actually obtained for the M$llendal fit. This provides further confirmation of the notion that the d’s as linear combinations of the T’S are much less correlated than the s’s themselves and parallels the findings of Carpenter (16) in his work on another oblate molecule, carbonyl fluoride, COF2. He found that certain linear combinations of the quartic parameters could be found that exhibited less correlation than the parameters themselves, with a resultant lowering of the standard deviations quoted for the determinable combinations. The transitions measured for our three molecules appear to contain sufficient information to evaluate the small d’s of the Watson reduced Hamiltonian Hred with small uncertainties. The same is evidently not the case for the 7’s. Much larger centrifugal distortion frequency shifts than those observed in the present study are required to bring the uncertainty limits of the 7’s more closely in line with those for the present sets of d’s. To test this idea we carried out a reanalysis, with the Kirchhoff program, of the recent results for another oblate ring compound, oxetane, CH~CH~CHZ, (Y=O state). When we restricted the oxetane data set to transitions involving J up to 17,2 we obtained high correlation among the quartic coefficients and uncertainties in these T’S of about
10 percent
(the 7 uncertainties
in our present
study
are typically
about
2.5
percent). In the original analysis of the oxetane spectrum (5) A coefficients with uncertainties of better than 6 percent (except for AJ) were obtained using transitions with J up to 48 and including molecules
in our study
of these characteristics uncertainties
comparable
one sextic parameter,
JZJK. Compared
are more rigid in structure should
to oxetane,
and are appreciably
lead to less centrifugal
stretching
so that
to those found for oxetane in Ref. (5) transitions
the three
heavier.
Both
to obtain with J well
above 48 would have to be observed for our molecules. If such high J transitions were accessible the use of sextic term fitting parameters could be made to obtain well-determined 7 coefficients. These considerations indicate that the problem of obtaining small uncertainties in the 7 values is not one inherent to oblate molecules in general. Also, the oxetane results (5) show that it is possible to obtain small uncertainties in centrifugal distortion constants without having available the frequencies of a variety of types of transitions since in the oxetane case, apart from a few low J R-branch lines, the entire data set consists of a type Q-branch 2By setting J,,,, included in the fit.
transitions.
= 17, transitions with centrifugal distortion shifts greater than 50 MHz were not
322
LOWE
AND KEU’LES
CONCLUSION
By including the effects of centrifugal distortion, improved uncertainties have been obtained in the rotational constants of 1,3-diosane, tetrahydropyran and axial NH piperidine. The results indicate that although two different analyses of these effects were able to reproduce the observed spectra with uncertainties approaching the measurement error, the ability to obtain well-determined quartic fitting coefficients can depend strongly on the form of the quartic Hamiltonian used in the fit, as may be seen by comparing the 7 and d coefficients of 1,3-diosane and tetrahydropyran in Tables IV and V. APPENDIX
1
As one of the checks on the calculations performed in this study, the authors found it useful to transform the matrix of covariances from the Kirchhoff basis to the M$llendal basis and compare the result with the matrix of covariances generated in the M$llendal fit. The variance-covariance matrix a2 (henceforth called 1/Cli’-matrix) is given by u2 = ,,z(jWJ)-l
(Al)
where uL2is the standard error of an observation of unit weight, W is a diagonal weight matrix (an identity matrix in the present analysis) and J is the Jacobian matrix, Jlj = (evl)/(aPj). It can be seen, therefore, that the problem of transforming the I/CT/matrix from one basis to the other is one of converting J,, the Jacobian matrix in the Kirchhoff basis to Jd, the Jacobian matrix in the M$llendal basis. Applying the chain rule of differentiation one can write
642) where F = C~=I” vi, with N equal to the number of transitions can be rewritten in the following form, by summing over j :
fitted. Equation
Jd = JP’
(A2)
(A3)
where A’ is the desired transformation matrix, Aij’ = (&,)/‘(adj). However, because the d’s and G’s (G = 6,8, c) are written as linear combinations of the 7’s and (Y”‘s ((Y” = A”, B”, C”) and not vice versa, the matrix actually computed is A, Aij = (adi)/(eTjJ. Equation (A3) now has the form
Jd = J,A-‘.
(A4)
This, of course, requires that A be a square, nonsingular matrix. Fortunately, since the d’s and ~5’sare not dependent upon Kirchhoff’s 73, a zero column in A results, transforming it into an 8 X 8 matrix suitable for inversion. 3 Now, from Eq. (Al), the two forms of the I’CV-matrix are defined as
With the help of Eqs.
Kirchhoff
or2 = (T~~~(J”~J,)-~,
(As)
M$llendal
od2 = cld2(JdJdk1.
(-46)
(A4) and (AS) and a little matrix
3At the same time, the ~3 row and column in
manipulation,
u,~ must also be deleted.
ud2 may be
CENTRIFUGAL
written
EFFECTS,
in the following convenient
6-MEMBERED
RINGS
323
form : (-47)
The actual computation can be performed with the original matrices A and u,~ since the 73 dependence is removed by the matrix multiplication. Finally, the coefficients of the correlation matrix C can be evaluated from the VCVmatrix 02 by the relation Cjk = (~")jk/[(~">jj(~">k~]~ (A@ while the standard
error in parameter
Pj is given by Uj
=
[(U2)jj]’
(A91
ACKNOWLEDGMENTS We should like to thank Dr. W. H. Kirchhoff gram CDANAL and Mr. A. J. Thakkar and Mr. program to the Queen’s University computer. One of us (R.S.L.) acknowledges the receipt Queen’s University School of Graduate Studies. from the National Research Council of Canada.
for providing a copy of the centrifugal distortion proH. B. Schlegel for their aid in adapting the CDANAL of an R. Samuel McLaughlin Scholarship from the This work was supported by operating grant A2046
RECEIVED: July 28, 1975 REFERENCES 1. J. K. G. WATSON,J. Chem. Phys. 46, 1935 (1967). 2. W. H. KIRCHHOFF,J. Mol. Spectrosc. 41, 333 (1972). 3. W. H. KIRCHHOFF,in “Critical Evaluation of Chemical and Physical Structural Information” (D. R. Lide and M. A. Paul, Eds.), pp. 312-322, National Academy of Sciences, Washington, D. C., 1974. 4. L. M. BOGGIA, R. R. FILGUEIRA,0. M. SORARRAIN,AM) D. DAMIANI, .&it. Naturjorsch. 29a, 95 (1974). 5. R. A. CRESWELLAND I. M. MILLS, J. Mol. Spectrosc. 52, 392 (1974). 6. R. PEARSON,JR., A. CHAPLIN,V. LAURIE, ANDJ. SCHWARTZ,J. Chem. Phys. 62, 2949 (1975). 7. V. M. RAO AND R. KEWLEY, Can. J. Chem. 47, 1289 (1969). 8. R. KEWLEY, Can. J. Chem. 50, 1690 (1972). 9. P. J. BUCKLEY,C. C. COSTAIN,ANDJ. E. PARKIN, Chenz. Comm. 668 (1968). 10. R. F. CURL, T. IKEDA, R. S. WILLIAMS,S. LEAVELL,ANDL. H. SCHARPEN,J. Amer. Chem. Sot. 95, 6182 (1973). If. K.-M. MARSTOKKANDH. M$LLENDAL,J. Mol. Struct. 4, 470 (1969); 5, 205 (1970). 12. D. KIVELSONAND E. B. WILSON,J. Chem. Phys. 20, 1575 (1952). 13. W. GORDYANDR. L. COOK, “Microwave Molecular Spectra” (W. West, Ed.), Vol. 9, Part 2, Chap. 8, Interscience, New York, 1970. 14. V. M. STIGLIANI,V. W. LAURIE, ANDJ. C. LI, J. Chem. Phys. 62, 1890 (1975). 15. K.-M. MARSTOKKAM) H. M~LLENDAL,J. Mol. Stract. 8, 234 (1971). 16. J. H. CARPENTER,J. Mol. Spectrosc. 50, 182 (1974).