The electric dipole moment and the static electric polarizability tensor of 1,1-dideuterocyclopropane: A microwave Fourier transform study

(1990)

JOURNAL OF MOLECULAR SPECTROSCOPY 139,236-240

NOTES The Electric Dipole Moment and the Static Electric Polarizability Tensor of 1,l -Dideuterocyclopropane: A Microwave Fourier Transform Study 1, I-Dideuterocyclopropane, whose pure rotational submillimeter-wave spectrum has been investigated recently by Endo, Chang, and Hirota (I), owes its small electric dipole moment to the slight asymmetry introduced by deuterium substitution into its average structure. In the following we report the results of an experimental determination of this dipole moment by microwave Fourier transform spectroscopy in static electric fields of up to almost 18 000 V/cm. The spectrometer used here, including its specially designed 3-m X-band Stark cell, has been developed by Dreizler and coworkers and was described in detail in Ref. (2). A recent review on microwave Fourier transform spectroscopy, a spectroscopic technique which is ideally suited to investigation of the spectra of almost nonpolar molecules, has been given by Dreizler (3). The sample ( 97.4% D) was purchased from MSD Isotopes (Merck, Canada). The transient decays were recorded at sample pressures close to 5 mTorr and at sample temperatures close to -60°C. Prior to the Stark-effect measurements three low-J transitions were searched for and recorded with our standard microwave Fourier transform spectrometers in order to further improve the vibronic ground state rotational constants and the centrifugal correction constants. The 211 + 212 ’ transition was found at 14 882.803(20) MHz, the 321 + 322 at 11 718.422(2) MHz, and the 431 + 432 at 8 100.516(2) MHz, respectively. A decay-fit routine developed by Haekel and Mader (4) was used for the final determination of these transition frequencies from the observed transient emission signals. The experimental uncertainties in the quoted frequencies correspond to 1 standard deviation. They are noise limited. Typical relaxation times, T2, were around 2 psec. This corresponds to half-halfividths of 80 kHz in cw spectroscopy. Deuterium quadrupole hyperfine splitting and spin-rotation splitting were not resolved. Deuterium hfs-splittings are estimated to be less than 50 kHz. This estimate is based on the assumption of a quadrupole coupling tensor with cylindrical symmetry around the C-D-bond and with a coupling tensor element of 200 kHz in the bond direction. Our three low-J transition frequencies and the high-/ frequencies reported in Ref. (I) were then used to improve the rotational constants and quartic centrifugal correction constants. The centrifugal distortion Hamiltonian corresponding to Watson’s A-reduction (5) was used for this centrifugal analysis. Its expression (in frequency units) is Eif h = A? * + B& + cj2e - D,P - D JKj2j2 c - D Kc .P

- 2sJ2(j;:

-

j;) - a,{j:(j:

- j;) + (j: - j;)jf}.

(I)

In Eq. ( 1) j., & and j, are the operators corresponding to the components of the rotational angular momentum (measured in units of h ) with respect to the principal inertia axes. A, B, and Care the rotational constants, and DJ, D,K, etc. are the quartic centrifugal correction constants. The results of this centrifugal analysis are given in Table I. For the Stark-effect measurements the Stark spectrometer was calibrated using the Stark shifts of the J = 1 -) 0 rotational transition of OCS as calibration standard. The dipole moment of OCS is known with high precision (p = 0.715 19( 3) D, Ref. (6)). From this calibration our electric field strengths have an uncertainty of 0.8%. Our measured Stark splittings for the two X-band transitions are given in Table II. Measurements of the Stark effect of the Ku-band 2 + 2 transition, although of great interest for a better determination of the polarizability tensor (see below), were not possible because a precision Ku-band Stark cell is not yet available. ’ The notation JK.K,, where K, and KCare the K quantum numbers of the limiting symmetric tops, is used to designate the rotational states. 0022-2852190 $3.00 Copyright0 1990 by Academic Press, Inc. Ail rightsof reproductionin any form reserved.

236

237

NOTES TABLE I

Rotational Constants, Quartic Centrifugal Correction Constants, Electric Dipole Moment, and Molecular ( Static) Electrical Polarizability Anisotropy of 1,I-Dideuterocyclopropane A

/ MHz

18 835.681

6

(60)

B

/ MHz

16 370.279

5

(20)

C

/ MHz

11 409.274

4

(20)

DJ

/ MHz

0.011

246

(04)

DJK

/ MHz

0.005

088

(10)

DK

/ MHz

0.007

046

(40)

bJ

/ MHz

0.003

027

(20)

bK

/ MHz

0.005

556

(80)

P

/D

0.011

5

(10)

al-a,,

/ 10-24

cm3

-0.948

(128)

For the analysis of the low-J Stark splittings we have used the rigid rotor Hamiltonian, supplemented by the potential energy of the “permanent” electric dipole moment, ii. and of the field-induced electric dipole moment, z - B, in the static exterior electric field, E: H Stark= _jj.E’-

@.,.E

(2)

where z is the (static) molecular polarizability tensor. If referred to the molecular principal inertia axes system the Hamiltonian corresponding to this classical energy expression takes on the form &a& = -/&Ezcos(aZ)

- fE:(a,cos*(aZ)

+ a*rcosZ(bZ) + a,cos2(cZ))

= I& + I&

(3)

In Eq. (3 ) cos(gZ) (g = a, b, c) is used for the direction cosines of the angles between the gyrating molecular principal inertia axes and the exterior field axis. Eistarkwas treated as a perturbation of the rigid rotor Hamiltonian. Within the eigenfunction basis of the rigid rotor the permanent dipole moment contributes only within second-order perturbation theory ( 7), while the polarizabihty already leads to first-order contributions. Since the polar&ability contribution has exactly the same mathematical form as the magnetic susceptibility contribution in rotational Zeeman-effect spectroscopy (8), we can use the energy expression derived earlier for the susceptibility contribution without having to go through the mathematical derivation again. We simply have to replace the magnetic field, Hz, by the electric field, Ez, and the magnetic susceptibility tensor elements, Xa, by the electric polarizability tensor elements, crgg.The explicit energy contribution of the polarizability tensor thus takes on the form (4) (Compare Eq. (111.12) of Ref. (8).) In Eq. (4) 01= (a,, + crbb+ a,)/3 is the so-called (static) molecular bulk polarizabihty and ($) (g = a, b, c) are the asymmetric top expectation values of the squares of the angular momentum operators measured in units of h. For low-J states the latter are easily calculated with high precision from the closed analytical expressions for the rigid rotor energy levels as functions of the rotational constants as derivatives with respect to the latter (Ref. ( 8)) p. I25 ) For cyclopropane Eq. (4) can be simplified further if the small difference between (Y, and abb which is introduced by deuterium substitution is neglected. Within this approximation we did set 01,”= 01~~ = n!, and Lyce= al. With this notation Eq. (4) can be rearranged into the form W~~J.I(.K,,M~ = -&Et

+ [u,,,,,

+ bw~~&*lE:(~,

- q)

(5)

238

NOTES

with J(J+ 1) - 3(jZ) u(r,x’ic) = - 3(2J - 1)(2J + 3)

(6)

and b

MxEkc)= (2J-

I(J + 1) - 3(.Q 1)(2J+ 3)J(J+

(7)

I)

A very similar form is obtained for the second order dipole contribution: wd(J,&‘Q

=

b,&&)

+

E,,,,,@&d,

(8)

where the coethcients A(J,&&) and BCJ,x.lG)are calculated as perturbation KONABI was used for these calculations.

sums. Botzkor’s program

kHz 300 _ 250 _ 200 _

150 -

10050 -

0

1

kHz

I 5

I 10

I 15

I 20

I 25

I 30

I 35

I 20

I 25

I 30

35

107

V2/cm2

431 -432

600 i 500 -

400 300 -

(

I 5

I 10

I 15

I 107v2/cm2

FIG. I. Plots of the Stark-splittings of the 3 2 1 + 3 2 2 and the 4 3 1 + I,1-dideuterocyclopropane versus the square of the applied Stark field.

4 3

2 rotational transitions of

239

NOTES TABLE II Stark-satellite Frequencies of the 3 2 I + 3 2 2 and 4 3 1 + 4 3 2 Rotational Transitions of 1,l -Dideuterocyclopropane 3’

I(,’

K,’

E / (V /

->

J”

cm)

M=2

K,”

15160.6

17524.7

*-contr. a-contr. D

11718.483 0.061 0.059 -0.008 0.010

11718.532 0.110 0.131 -0.017 -0.004

11718.614 0.192 0.222 -0.029 -0.001

11718.675 0.253 0.296 -0.040 -0.004

ohs. b p-contr. a-contr. a

11718.431 0.009 0.026 0.000 -0.017

11718.472 0.058 0.058 0.000 0.000

11718.520 0.098 0.099 0.000 ~-0.001

11718.555 0.133 0.132 0.000 -0.007

b

E / (V / cm)

obs . b

p-contr. a-contr. L1 obs. n=3

b

p-contr. a-contr. a obs. M=2

-) 322 11676.6

J' K,* Kc' -) J" Ka" Kc"

M=4

= 3 21

7800.0

ohs. n=3

K,”

b

p-contr. amcontr. (J

=431-,432 11672.8

15156.7

17489.7

17497.5

8100.622 0.108 0.123 --0.005 -0.010

8100.774 0.258 0.275 -0.010 0.007

8100.974 0.458 0.464 -0.017 0.011

8101.113 0.597 0.617 -0.023 0.002

8101.111 0.595 0.618 -0.023 0.002

8100.570 0.054

8100.780 0.264 0.261 -0.004 0.007

8100.858 0.342 0.348 -0.006 0.000

8100.857 0.341

-0.001 0.014

8100.659 0.147 0.155 -0.003 0.005

8100.542 0.026 0.031 0.001 -0.006

8100.584 0.068 0.069 0.003 -0.004

8100.651 0.133 0.117 0.005 0.011

8100.670 0.152 0.155 0.007 ~0.010

8100.669 0.151 0.155 0.007 0.011

7796.1

0.069

0.349

0.005 0.001

Note. 6 is the observed splitting with respect to the zero-field frequency. p is the splitting calculated by second-order perturbation theory from the optimized dipole moment given in Table I. 01is the splitting calculated by first-order theory from the polarizability anisotropy given in Table 1. CT= 6 - c, - CYis the difference between the observed and the calculated Stark splitting.

From Eqs. (5) and (8) it follows that the Stark shifts. if plotted against E:, should follow straight lines. which indeed is the case (compare Fig. I ). Furthermore each observed Stark shift corresponds to an equation which is linear in r: and ((Y~-CX,,),with coefficients which can be calculated with high precision from the quantum numbers and from the measured rotational constants. Thus the normal equations which correspond to the observed Stark shifts were set up and were solved for ~3 and ( aL- a,,) in the standard least-squares procedure. Our result is given at the bottom of Table I. The correlation coefficient between ~2 and (a, - CY,,) is 0.606. With rf and ((Y~-cY,,)now known, the dipole and polarization contributions to the Stark shifts can be calculated separately. The result is given in Table II. Admittedly the polarization contribution is comparatively small in the two X-band transitions which were accessible to experiment. This too explains the comparatively large uncertainty in ( CX-ol,,). On the other hand, in the Stark-shifts of the 2 1 I +

240

NOTES

2 1 2 transition, the polarizabihty anisotropy contributes up to 80 kHz at fields close to 17 kV/cm. This indicates that especially for molecules with small dipole moments precise static polarizability anisotropies can be determined by experiments as described here. Experimental knowledge of these anisotropies would be of interest for instance in the theory of intermolecular interactions at close range. ACKNOWLEDGMENTS We thank Professor H. Dreizler for providing the spectrometer and for critically reading the manuscript. Financial support by Deutsche Forschungsgemeinschaft under Grant Su 41/ 14 is gratefully acknowledged. REFERENCES I. 2. 3. 4. 5. 6. 7. 8.

Y. ENDO, M. C. CHANG, AND E. HIROTA, J. Mol. Spectrosc. 126,63-71 (1987). E. FLIEGEAND H. DREIZLER,Z. Naturforsch. A 42, 72-78 ( 1987). H. DREIZLER,Mol. Phys. 59, 1-28 ( 1986). J. HAEKELAND H. MKDER, Z. Naturforsch. A 43,203-306 ( 1988). W. GORDY AND R. L. COOK, “Microwave Molecular Spectra,” 3rd ed., p. 331, Wiley, New York 1984. J. M. L. REINARTZAND A. DYMANUS, Chem. Phys. Lett. 24,346-351 ( 1974). S. GOLDEN AND E. B. WILSON, JR., J. Chem. Phys. 16,669-685 ( 1948). D. H. SUTTER AND W. H. FLYGARE,Top. Curr. Chem. 63,89-196 ( 1976). 0. BOTTCHER N. HEINEKING D. H. SUTTER

Abteilung fir Chemische Physik in Institut ftir Physikalische Chemie Christian Albrechts Universitiit zu Kiel Ludewig Meyn Str. D-2300 Kiel, West Germany Received August 4, 1989