357
J. Phys. Chem. 1993,97, 357-362
Induced Dipole Moments and Intermolecular Force Fields of Rare Gas-CO2 Complexes Studied by Fourier-Transform Microwave Spectroscopy Miooru Iida, Yasuhiro Ohsbima, and Yasuki Endo' Department of Pure and Applied Sciences, The College of Arts and Sciences, The University of Tokyo, Komaba. Meguro-ku, Tokyo 153, Japan Received: August 25, 1992; In Final Form: October 20, 1992
Pure rotational spectra and their Stark effects of X-CO2 type complexes (X: Ne, Kr, and Xe) have been studied by pulsed-nozzle Fourier-transform microwave spectroscopy. Structures, intermolecular force fields, and electric dipole moments in the ground state have been determined, which have been discussed in connection with the previously reported results of Ar-CO2 and Hg-CO2. When the polarizability of the attached atom is small, the dipole moment has been well reproduced by a two-center expansion model for the induction interaction, where the atom is polarized by the electric field arising from the quadrupole moment of COz. However, this simple model deviates systematically as the atoms possess larger polarizabilities, implying a finite size effect of the attached atom. A model calculation to describe intermolecular potential functions has been performed to show that the vdW force constants of X-CO2 have been mainly controlled by the repulsion and dispersion interactions.
Intradpctioa
ChargedistributionsoftheconstituentsofvanderWaals (vdW) complexes are rearranged through intermolecular interactions and thus become slightly but definitely different from those in the isolated molecules. These charge rearrangements inherently cause the induction of the dipole moment, whose magnitude is on the order of 10-1-10-2D. We can regard this induced dipole moment as a probe to investigate the charge redistribution by complex formation, which will be one of the important experimental sources for detailed understandings of the intermolecular interactions. Indeed, accumulated experimental data, mainly contributed from high-resolution spectroscopic studies, have stimulated the development of theoretical models for (semi)quantitative predictions of the induced dipole The induced dipole moment has also a great practical importance, since it allows us to observe pure rotational transitions of vdW complexes with nonpolar constituents. Otherwise, microwave spectroscopy, with high potential for determining structures and vdW force fields, cannot be applied to this category of complexes. Rare g a s 4 0 2 complexes have subgtantial importance as a prototype system for the study of the intermolecular interaction between nonpolar subunits. First, spectroscopic identification has been camedout for A r e 0 2 by Steed, Dixon, and Klemperer: who have confirmed the T-shaped configuration of the complex by the molecular-beam electric resonance method. The dipole moment of this complex has also been determined precisely from a measurement of the Stark effects on the rotational transition^.^ Thisstudy has been followed by observationsof vibration-rotation spectra for Ne-, Ar-, Kr-, and Xe-COz in the vicinity of the C 4 antisymmetric stretching (vt) band of the COz moietyS and of microwave and near-infrared spectra for Ne-,Ar-, and Kr-CO2.6q7 Recently, the ug fundamental band and its combination band with the vdW bending have also been reported for Ar-C02.8.9 An anisotropic nature of the intermolecular interaction has bean explored as well by molecular-beam scattering measurements.lOJ1 Along with these experimental studies, a number of theoretical works have dealt with the construction of the potential energy surface on the basis of ab initio calculations or empirical mOdels.l2-l6 We have recently observed a rotational spectrum of Hg-COZ,l7which is an analogous system to the rare g a s 4 0 2 complexes. It will be of much interest to manifest the general trend in structures, vdW force fields, and dipole moments 0022-3654/58/2097-0357$04.00/0
for these related complexes of the X-CO2 type, where X = Ne, Ar, Kr, Xe, and Hg. In the present study, we report pure rotational spectra and the Stark effects of the Ne-,Kr-, and Xe-CO2 complexes observed by using a pulsed-nozzle Fourier-transform microwave(FTMW) spectrometer. The determined rotational and centrifugal-distortion constants for Kr- and Xe-COz allow us to evaluate the structures and the harmonic vdW force fields. In connection with the previous results of Ar- and Hg-COZ, the dipole moments for the other three complexes determined in the present study are discussed on the basis of a simple induction interaction model,
Experimental !&-dim Rotational spectra of the rare g a s 4 0 2 complexes were observed by using a Fourier-transform microwave spectrometer, which has been reported elsewhere.'* For the measurement of the Ne-COz complex, a mixture of 0.5% C02 diluted in neon was prepared. For Kr-CO2 and Xe-COz, a mixture of 0.25% CO2 and 5% krypton in argon and 0.25% COZ and 2.5% xenon in argon were used as sample gases, respectively. The gases were pulsed through a pulsed nozzle with an 0.8-mm4.d. orifice into an evacuated chamber with a stagnation pressure of about 1.5 atm except for Ne-CO2, whose signal was strongest with a relatively high pressure of 3.5 atm. For measurements of the Stark effects, we mounted two electrodes in the Fabry-Perot cavity. In order to minimize the disturbance to the microwave field and the reduction of the pumping speed, each electrode was made of copper wires of 0.3mm 0.d. with an interval of 1.5 cm stretched between frames, which are made of two 30-cm X 5-cm plastic plates separated 30 cm by AI rods. Static voltage up to +10 kV can be applied to one of the electrodesand up to -1 0 kV to the other. The distance between the two electrodes (about 20 cm) was calibrated by the Stark effect on the J = 1 - 0 transition of 0C3%,whose dipole moment has been determined precisely.19 The calibration was carried out at every experiment, and we found that the distance showed no deviation exceedingthe experimentalaccuracy of 0.4% during several months. The static electric field applied was set parallel to the electric field vector of the microwave, and thus only M = 0 transitions were observed. The line width of the observed transition at 12 GHz was 30 kHz (fwhm) without applying the static electric field, which originated mainly from the Doppler effects. When the transition
-
rQ 1993 American Chemical Society
Iida et al.
358 The Journal of Physical Chemistry, Vol. 97, No. 2, 1993 TABLE I: Observed T m i t i o W of "Kr-CO2 (MHz) transition J' ' . K Kc' J" K." Kc" o b freq obs - calc
-. transition J'
K.'
Kc'
' 2 9 X ~ 0 23 4
0 0 0 0 0 2 2 2 2 2 0 0 0 0 0 2 2 2 2 2 0 0 0
3 4 5 6 7 1 3 2 4 3 3 4 5 6 7 1 3 2 4 3 3 4 5
~
2 3
4 5 6 3 3 4 4 5
5 6 6 a
0 0 0 0 0 2 2 2 2 2 2 2 2
2 3 4 5 6 2 1 3 2 4 3 5 4
1 2 3
4 5 2 2 3 3 4 4 5 5
0 0 0 0 0 2 2 2 2 2 2 2 2
1 2 3 4 5 1 0 2 1 3
2 4 3
5071.148 7602.718' 10129.501' 12649,917' 15 162.429 7604.664 7610.358 10137.669 10151.884 12669.056 12697.442 15 198.422 15247.990
0.002 0.001 0.003 0.002 -0.004 -0.005 -0.001 -0.001
-0.Ooo 0.002 -0.002 0.004 -0.001
Reference 6.
was shifted by 1 MHz by applying an electric field, the line width increased to 40 kHz due to an inhomogeneity of the electric field. Theobserved shifts for strong lines were always reproduced within 3 kHz for every independent measurement, but the reproducibility declined to 5 k H z or more when the signal-to-noise ratio of the transition was smaller. We checked that all the observed Stark shifts were proportional to the squares of the applied electric field within the experimental accuracy of the present measurement.
Result9 Among the r a r e g a d 0 2 complexes, Ar-CO2 has been studied most extensively by microwave spectroscopy. Rotational transitions with KO = 0, 2, and 4 have been observed, leading to a precise determination of the molecular structure and the vdW force constant^.^.^ On the contrary, data for other complexes are yet insufficient for detailed discussions on the structure and the intermolecular force field; transitions with only K,, = 0 have been observed for Ne-CO2 and Kr-CO216 and no rotational spectrum has been reported for Xe-CO2 so far. Therefore, we first observed rotational spectra of Kr-CO2 and Xe-CO2. Unfortunately, we were not able to make an additional observation on Ne-COz because of the limited frequency coverage of our FTMW spectrometer. All the observed transitions were of a-type with K. = 0 or 2, which were consistent with the T-shaped structure with C b symmetry. For Kr-CO2, rotational transitions of only the main isotopic ~peciesof*~Kr (57%naturalabundance)wereexamined, transition frequencies observed are listed in Table I. For mXe-C02,spectra for four main isotopic species, m = 129 (26%), 131 (21%), 132 (27%). and 134 (10%) were observed. Observed frequencies for the m = 129, 132,and 134 species are listed in Table 11. For 131Xe-C02,whose Xe nucleus has a spin of 3/2, the lines were split into two components due to the nuclear quadrupole coupling interaction. The frequencies of the hyperfine components are listed in Table 111. Since all the 0bSe.rved transitions are of AK,, = 0 type and thus insufficient to determine the A rotational constant and all the centrifugal-distortion constants, we have adopted the same procedure developed in the analysis of Hg-CO2.I7 In this analysis, two vdW force constants,f, and f i r for the vdW stretching and bending modes, respectively, are used as parameters to be determined, instead of the A constant and the centrifugaldistortion constants. Five Watson's A constants20were calculated from the rconstants of Kivelsonand Wilson,2I which are expressed by using the harmonic force constants includingf, and fb; the constants for the intramolecular modes were taken from those for the free C02 molecule.22 The A constant was derived from the B and C constants along with the inertial defect, which was also calculated from the harmonic force field, as derived by O h and M~rino.~'The calculated A and A constants as well as the
5 6 7 3 4 4 5 5 '32XC-C02 3 4 5 6 7 3 4 4 5 5 134Xe-C02 3 4 5
J" K," 2 3 4 5 6 2 3 3 4 4 2 3 4 5 6 2 3 3 4 4 2 3 4
0 0 0 0 0 2 2 2 2 2 0 0 0 0 0 2 2 2 2 2 0 0 0
KC
obfreq
ob-celc
2 3 4 5 6 0 2 1 3 2 2 3 4 5 6 0 2 1
6104.700 8136.446 10165.494 12191.179 14212.849 6107.053 8138.771 8144.546 10172.026 10183.585 6071.071 8091.674 10109.632 12124.296 14135.036 6073.351 8093.911 8099.565 10115.986 10127.275 6049.461 8062.900 10073.736
4.001 O.OO0 0.001 O.OO0 4.OO0 4.002 0.005 4,002 -0.002 0.001 O.OO0 0.001 O.OO0 -0.003 0.001 -0.001 -0,002 0.003 0.001 -0.001 0.001 -0.001 O.OO0
1
2 2 3 4
TABLE IIk Observed Tnnsitiolls of 131X402 (MHZ) transition J' Ka' &' F' J" K ." K," F" obfr~q 0 b - d ~ 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 3 3 3 4 4 4 4 5 5
5 5 6 6 6 6
3/2
'12 '12 '12
'12
'12
9/2 "/2 '12 '12 "/2
13/2 '12
ll/z 13/2
15/2
2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5
'12 '12 '12 3/2 '11 '11 9/2 '12 '12 '12 11/2
'12
'12 11/2 13/2
6081.955 6081.955 6082.141 6082.141 8 106.296 8 106.296 8 106.386 8 106.386 10127.915 10127.915 10 127.977 10 127.977 12 146.215 12 146.215 12 146.258 12 146.258
0.003 0.003 0.001 0.002 -0.001 -0.001 0.001 0.001 -0.007 -0.007 0.002 0.002 -0.003 -0.003 0.005 0.005
B and C constants were used to express the ordinary Watson's asymmetric top Hamiltonian in the A-reduced representation.20 Theobserved transition frequencies were fitted to the Hamiltonian in a nonlinear least-squares procedure to determine the Band C rotational constants and thefi andfb force constants. The determined constants for 84Kr-C02, 129Xbc02,and 1 3 2 X ~ are 0 2listed in Table IV. For the "IXeand I'Xe isotopic species, A and all the A constants were fued to the interpolated or extrapolated values. For *31Xbc02,whose spectrum shows hyperfine splittings, the nuclear quadrupole coupling interaction term24was also included in the Hamiltonian and the constant was determined, while the contribution of the xt,i,-xcc term was not included. The results for these less abundant species are listed in Table V. At the next stage of the study, we made measurements on the Stark effects to determine the permanent dipole moments of the complexes. We first examined the Stark effects on the Zor101 transition of Ar-CO2 at 7317.288 MHz, whose dipole moment has been already determined precisely,' for a check of our experimental setup. The observed Stark shifts for the M = 0and 1 components were least-squares fitted by using the secondsrder perturbation method.24 The Stark coefficients thus determined are listed in Table VI. The dipole moments calculated from the two components are consistent within one standard deviation of the fit. The averaged value, 0.067 f 0.003 D, where the error
Dipole Moments and Force Fields of Rare Gas-COz
The Journal of Physical Chemistry, Vol. 97, No. 2, 1993 359
TABLE Iv: Spectroscopic Constants of 84Kr-C02, 12'xc-C02,
md l32XtCO2*
A (MHz)~
B (MHz) C (MHz) fi (mdyn/A) fb ( m d y n 4 AJ (kHz)b AJK( k H d b AK (kHz)b b j (kHz)' 8~ (kHz)b u (kHz)'
5
8'Kr-C02
129Xe-C02
132Xe-C02
11893.34 (18) 1339.5987 (20) 1196.8041 (16) 0.020074 (13) 0.01 14562 (57) 7.4181 (48) 156.89 (96) -160.32 (96) 0.82225 (53) 98.87 (53) 3.0
11879.05 (78) 1063.9145 (38) 971.9041 (37) 0.021483 (33) 0.012512 (12) 4.0892 (62) 95.31 (89) -95.40 (90) 0.36590 (53) 59.20 (49) 2.5
1 1880.40 (65)
1057.7867 (31) 966.8030 (30) 0.021460 (27) 0.012556 (10) 4.0500 (50) 93.97 (71) -94.03 (72) 0.36038 (43) 58.43 (40) 2.0
Transition frequenciesare fitted by using E , C,J, andfbas parameters. The numtcrs in parentheses represent one standard deviation in the last significant digit. Calculated values by using the determined B, C,J, andfb constants. Errors are also calculated from the variancecovariance matrix for the constants determined in the fits. Standard deviation of the least-squares fits.
TABLE V. S
'~xe-co:C&P ((' B +'1'2I 2 XM
troscopic Constants of la1Xe-C02and
1014.13685 (34) 45.6616 (59) -3.050 (55)
1008.68699 (18) 45.1599 (44)
a The numbers in parentheses represent one standard deviation in the last significant digit. The A constant and all the centrifugal-distortion constants are fixed.
1
0
5071.18
7
1
.
M=l
2
2
/
7
Y
\
5071.10
5
1
0
0
-\
"t
7
1
1
2
.
3
0
4
6
5
1
6
E* 105~2cm-2 Figure 1. Stark shifts of the two Mcomponents of the 202-101 transition of 84Kr-C02plotted against the squares of applied static electric field. shifts. The analysis has been carried out by neglecting the effects of polarizability anisotropy on the Stark shifts. We will show first the validity of the neglect of this effect for the present experimental conditions as follows. The Hamiltonian arising from the polarizability of the complexes can be approximated as a sum of the contributions from the two constituents, an atom (denoted as X) and COS
TABLE VI. Dipole Moments of Ne-, Ar-, Kr-, and Xe-Co2'
transition
Stark coeff IHz/(V/cm)21
u (D)
(D)
llsve
2"Ne-C02 loi-Ow M 0 0.0132 (14) 0.0244 (13) 0.0244 (13) 40Ar-C02 2otlol M = 0 4.0469 (45) 0.0667 (32) M = 1 0.0397 (39) 0.0682 (30) 0.0675 (34) 84Kr-C02 &rlol M = 0 4.1053 (78) 0.0832 (30) M = 1 0.0843 (54) 0.0826 (26) 0.0829 (28) '29xe-Co2303-202 M = 2 0.0626 (52) 0.1027 (42) "2Xe-C02 303-202 M = 2 0.0632 (48) 0.1030 (34) 0.1029 (38)
The numbers in parentheses represent two standard deviations of the least-squares fits in units of the last significant digit.
represents twice the standard deviation of the fit, agrees well with the previously reported value, 0.0679 & 0.0004 D.4 For the Ne-COz complex, the measurement was carried out on the M = 0 component of the 10~400 transition at 6088.244 MHz. Theobserved shift was only 13k H z even under the electric field of 978.3 V/cm, owing to the small dipole moment. The Stark coefficient and the dipole moment are listed in Table VI. For the Kr-CO2 complex, the M = 0 and 1 components of the & r l o l transition were used for the measurement. The plots of the observed frequencies against the squaresof the applied electric field (E)are shown in Figure 1,which represent the proportionality of the Stark shifts with E2 within the experimental accuracy of 5 Wz. The dipole momentsdetermined from the two components are consistent as listed in Table VI. For the Xe-COz complex, the 303-202 transition was used. The M = 0 and 1 components were not resolved up to 980 V/cm, while the M = 2 component was resolved 2500 V/cm. Thus the dipole moment was determined only from the M = 2 component. The values for the 129Xeand 132Xeisotopic species are consistent with each other as given in Table VI.
Disc~ioa
Permraeat Dipole Moment8 Induced by Complex Formation. In the present study, we have determined the electric dipole moments of the rare gas-C02complexesfrom the observed Stark
where a z z is the Z Z component of the polarizability tensor and Ez is the static electric field along the space-fixed Z-axis. The first term in the right-hand side is constant and produces no shift to rotational transitions, because the polarizability of a free atom is isotropic. Therefore, contribution of the term in q 1 on the Stark shifts can be estimated by only using the polarizability anisotropy of the free C02 molecule. From the reported polarizability anisotropy, 2.03 A3,25 the coefficients for all the observed transitions are calculated to be less than 10-3 Hz/(V/ cm)z,which are well within the experimental errors of the present measurements. The consistencies between the different M components of Ar-COZ and Kr-CO2 also show that the contribution of the polarizability anisotropy can be neglected within the present experimental accuracy. The electric dipole moments of the Ne-, Kr-, and Xe-CO2 complexes determined in the present study are summarized in Table VII. In Table VI1 are also listed the results for the Arand Hg-COZcomplexes. As the two constituentsof the complexes are both nonpolar, dipole moments of the complexes are entirely induced by complex formation. Among the various kinds of intermolecular interactions, the induction effect is expected to account for the major part of the induced dipole moment. Even without the dipole moment, the COZmolecule possesses higherorder electric multipole moments, which make a nonuniform electric field around COz. The electric field distorts the charge distribution of the attached atom with a finite polarizability, yielding to the induced dipole moment. Here we adopt a simple model in which the CO2 molecule is regarded as a point quadrupole moment at its center of mass.The electric field at the position of the atom, X,is then expressed as E = %P
(cos e)
R4 where R is the distance between C and X and 8 is the angle between R and the C02 molecular axis. The observed distances arc large enough to ensure the validity of q 2; the contribution of the multipole moments higher than quadrupole is calculated to be less than I%, if the values from an ab initio calculationz6are
360 The Journal of Physical Chemistry, Vol. 97, No.2, 1993
Iida et al.
TABLE W: C~mpuisOnof tbe M o k u l u Con~trntsof X-CO2 ( X Ne, Ar, Isr, XC, lad Hg)' 2QNe-Co2 aAr-C02 %r-C02 132xC-c02 ax
(A')b
0.395 36.0 (17) 3.2904 0.01207d 0.0080 (4) +0.0176 4.0091
C,@) (au)C R (A) P (mdyn/A) p k (total). J F (YP).
5-
(dlsp).
fp* (ind). fbh 0ndyn.A) fb(rep). fb(dhp)'
-0.oO04 0.00317d +0.0039 -0.001 1
h d (ind)'
+O.oO04
(cm-I)
'"1 "a(Y V,
1.64 114.5 (76) 3.5048 0.01738 0.0155 (10) +0.0339 -0.0176 -0.OOO9 0.009428 +0.0108 -0.0023 +o.m9 37.5 29.4 0.147 7.13 0.0679 0.0669
38.6 17.9 0.178 9.60 0.0244 0.0199
vb
A f L (degY
2.48 162 (14) 3.629 0.02007 0.0166 (14) +0.0363 -0.0188
-0.o009 0.01 146 +0.013 1 -0.0026 +0.0010 34.4 31.6 0.130 6.71 0.0829 0.0883
4.04 282 (28) 3.818 0.02146 0,0191 (18) +0.0418 -0.0218 -0.OOO9 0.01256 +0.0148 -0.0034 +0.0011 33.2 32.8 0.124 6.52 0.1029 0.1178
"Hg-CO2 5.1 (232) 3.695 0.02072 (0.02072) (+0.0457) (-0.0233) -0.0016 0.01205 (+0.0136) (-0.0034) +o.oo 18 31.2 32.1 0.122 6.59 0.1070 0.1695
(D) (D)* a Data for Ne402 are taken from ref 5; Ar-CO2 from refs 4 and 6; Hg-COZ from ref 17. Polarizability of the attached atom, X. From ref 34. Isotropic dispersion coefficient. From ref 33. Calculated by using the centrifugal-distortionconstana from ref 5. Caiculated vaiue. See text for details. f Root-mean-square amplitudes of the vdW coordinates. #ut4
h k
O
.
0.15
*
O
I
-
n
2 0.100.05
-
0.00 0
0.01
0.02
0.03
axR" I k1 Figure 2. Observbd dipole moments for X-COz plotted against ax/#. The solid line corresponds to the calculated value of 3Qax(&(ms e))/ 2R.
adopted. The dipole moment induced by the electric field is thus p
3Qax
= a x E = -(
R4
P2(cos e))
(3)
where ax is the polarizability of the attached atom. In eq 3, angular dependence, P2(cos e), is averaged over the vdW bending vibration. This vibrational average can be approximated as (Pz(cose)) = P2(cos e,,), where 8," = 90" + A&,,, and AB, is the root-mean-square amplitude for thevdW bendiig. Thecalculated values by using Q = -4.3 X mu2' are listed in Table VII. Those calculated dipole moments are plotted against a x / @ in Figure 2, along with the observed dipole moments. Since e., zs 90" for every X 4 0 2 type complex, calculated values are approximately proportional to a x / P , as seen in Figure 2. For the complexes with smaller ax, e.g., Ne-COz, Ar-CO2, and KrCOZ,agreements between the observed and calculated values are satisfactory. However, the calculated values deviate systematically from the observed ones for the complexes containing the atom with larger polarizability, as seen in Table VI1 and Figure 2.
Evidently, the model represented in eq 3 is too simple to reproduce the observed dipole moments for the complexes with larger UX. First, we will consider the effects other than the induction interaction. Contributionof the charge transfer is ruled
out since the negative values of ccobc - k l c require an electron transfer on the order of 0.1% from C02 to the attached atom. Judging from other vdW complexescontainingKr, Xe,or Hg,m.B electrons would be transferred in an opposite direction, if there were any charge transfer. The distortion from the linearity of the O==C--O angle causes the dipole moment of 0.024 D/deg, calculated from the reported dipole moment function for the q bending m0de.3~ If ~robc- lrcrlc originates entirely from this type of distortion, the COz molecule in Hg-CO2 should be bent by 2.6'. This value is unrealistically large, comparing with the O=C=O zero-point bending amplitude of 5.3" for the ground vibrational state or with thevalue of 0.6O for the distortion of BF3 from the planarity in the more stronglybound BF3-CO complex?' Therefore, the discrepancy should be ascribed predominantly to the insufficient description of the induction interaction by eq 3. Especially, we have to account for a finite size effect of the attached atom, since the electric field due to the C02 moiety is quite nonuniform. For example, the electric field by the C02 quadrupole moment at R = 3.7 A, which is a typical distance between the attached atom and C02 in the complex, becomes larger by 1 1%. if the distance is made shorter by only 0.1 A. Even though the contribution of the higher multipole moments is negligible for the distance between the centers of the attached atom and C02, it becomes considerable in a shorter distance up to around 2 A, where the atom still has a substantial amount of electron density. Further complication is introduced by the fact that the charge distribution inside the atom is also nonuniform, and thus the polarizability of the atom is a function of the position. In this regard, the present experimental results would be a benchmark for more sophisticated analyses, such as the distrib uted-multipole/distributed-polarizabilityanalysis.'-3 It is worth noting that the discrepancy between the observed and the calculated values for Hg-CO2 is extremely large, as much as almost 60%. On the contrary, the discrepancies for all the rare gae-CO2 complexes are smaller within 2096, although Hg and the rare gas atoms have the same *Soelectronic ground state. Further investigations of vdW complexes containing other IIB metal atoms will be of much interest to examine whether the complexes with smaller a x , such as Cd and Zn, also show large discrepancies between the observed and the calculated dipole moments. IntcrmdccuhrForceComtmte. The harmonic forceconstants for the vdW stretching and bending have been determined in the present study for the Kr-COz and X b c 0 2 complexes, which are listed in Table VI1 along with the values for other related complexes. The data for A r e 0 2 and Hg-CO2 are taken form refs 4 and 6 and ref 17, respectively. The constants for N e 4 0 2
Dipole Moments and Force Fields of Rare G a s 4 0 2
The Journal of Physical Chemistry, Vol. 97, No. 2, I993 361 coordinate, S, = AR
i
0.025
4
C J
J J
156C12(9O0) 42c6(9O0)
-
f,
e
. -4 e
E
1
4
R,8
R,lO
(8)
w
W
0
\
c m
.
0
0.010-
~
Here, the experimentallydetermined R is used as theequilibrium distance, &. The dispersion coefficient, c6(9O0), can be calculated by using the isotropic term, c6(’), determined experimentally from the frequency-dependent polarizabilities,’) and the following relation of the anisotropic term, c6(2), to the previously reported polarizabilities of the C02 moleculezs
E 0.020
6
~
--8 1@ax
0.005
-
(9) W
-000.0 0
2
4
6
ux I ~3 lilgwe 3. Determined vdW force constants for X-COz plotted against ax.
Cloesdcircles, open circles, and closed squaresrepresent the observed
The induction contributions are also readily derived by using the quadrupolemoment of C02 and the polarizabilitiesof the attached atom. The repulsion coefficients,c12(9O0),are evaluated in terms of other known parameters by using the equilibrium condition for the R coordinate
h, calculated h, and observed ji,respectively.
are calculated from the centrifugal-distortion constants reported in ref 5. The harmonic frequencies and the root-mean-square amplitudes for the vdW vibrations have been calculated by using the CF-matrix methodg2and are also listed in Table VII. The valuesof R listed in Table VI1 are calculated from the determined B constants, after the correction for the vibrational effects.” Figure 3, in which the determined vdW force constants are plotted against ax, shows a general trend that the force constants for the vdW stretching and bending become larger for the complexes with larger ax. However, thevalues for the Kr-, Xc-, and Hg-COz complexes are quite similar to each other in spite of their different ax. These force constants, which characterize the shape of the vdW potential energy surface around the equilibrium configuration, are controlled by three different kinds of intermolecular interactions, namely, induction,dispersion,and repulsion. For a semiquantitativeevaluation of these interactions from already known bulk properties, we have performed a simple model calculation as follows. We expressthe intermolecular potential function for the X 4 0 2 complexes as a sum of the above mentioned three terms, Le., arising from the repulsion, dispersion, and induction interactions V(R,@ Vrep + Vdbp + ynd (4) Sinceour interest is mainly on thecharacterization of the potential function very near the equilibrium configuration,each interaction term is represented in the form of a twecenter expansion,although this model must be too simple to describe the global feature of the intermolecular potential. Here the repulsion and dispersion terms are expressed by using a commonly used (12,6)-type Lcnnard-Jones potential
where the angular dependent C,coefficients ( I = 12 or 6) are expanded as
c,(e) = cto)+ C
, ( ~ ) Pe) ~+(...~
(6) For the induction term, only the dominant interaction, the quadrupolbquadrupole-induced dipole type, is considered
The vdW stretching force constant is derived as a second derivative of the potential function with the vdW stretching
Contributions of these three terms to the vdW stretching force constants thus calculated are listed in Table VII. The calculated fi constants are also listed in Table VI1 and plotted in Figure 3. Errors quoted originate entirely from the uncertainties of the observed C6(O)coefficients. Except for Ne-COz, thevalues agree reasonably well with the observed ones, indicatingthe satisfactory description of the intermolecular interactions along the R coordinate around the potential minimum. It is notable that the repulsion interaction makes the dominant positive contributions tofi and the dispersion interaction cancels about half of these contributions, while the induction term is almost negligible. The c6(’) coefficient for Hg-CO2, which has not been reported so far, is estimated and listed in Table VI1from the previously determined
fi.” The vdW bending force constant is calculated as a second derivative of the intermolecular potential with the bending coordinate, Sb = &Ad
where the angular dependences higher than P~(cos0 ) are not considered. As in the case of the vdW stretching force constants, the second (dispersion) and the third (induction) terms can be readily calculated, but it is difficult to evaluatethe fmt (repulsion) term from already known bulk properties. Here, we regard differences between the observed values and the dispersion plus induction contributionsas the repulsion contribution. The results are listed in Table VII. Again, the repulsion interaction makes a dominant contribution and both the dispersion and induction terms are relatively small, indicating the importance of steric effects in the characterization of the anisotropic part of the intermolecular potential. In conclusion,the intermolecular interaction potentials around the equilibrium configuration for the X-CO2 type complexesare mainly controlled by the repulsion and dispersion interactions, while the induction term contributes a minor part. It is in contrast to the permanent dipole moments induced by complex formation for the complexes, which are ascribed predominantly to the induction interaction, as discussed in the previous section.
Acknowledgment. The present study was partly supported by a grant aid of the Ministry of Education, Science, and Culture of Japan (No. 63470007).
362 The Journal of Physical Chemistry, Vol. 97, No. 2, 1993 References and Notes (1) Buckingham,A. D.; Fowler, P. W.; Stone, A. J.In?.Reu. Phys. Chcm. 1986.5, 107. (2) Fowler, P. W.; Stone, A. J. J. Phys. Chcm. 1987,91,509. (3) Le Sueur, C. R.;Stone, A. J.; Fowler, P. W. J . Phys. Chcm. 1987, 91,509. (4) Steed, J. M.;Dixon, T. A.; Klemperer, W. J. Chcm. Phys. 1979.70, 4095. (5) Randall, R. W.; Walsh, M. A.; Howard, B. J. Furaduy Discuss. Chem. Soc. 1988,85,13. (6) Fraser, G.T.;Pine, A. S.;Suenram, R. D. J. Chem. Phys. 1988,88, 6157. (7) Pine, A. S.;Fraser, G. T. J. Chcm. Phys. 1988,89,100. ( 8 ) Sharpe, S.W.; Sheeke, R.;Wittig, C.; Beaudct, R. A. Chcm. Phys. Lctr. 1988,151,267. (9) Sharpe,S. W.;Reifschneider, D.; Wittig,C.; Bsaudet, R. A. J. Chcm. Phys. 1991, 94,233. (10) Keil, M.;Parker, G. A.; Kuppermann, A. Chem. Phys. Lett. 1978, 59,443.Parker, G.A.;Keil, M.;Kuppermann, A. J . Chcm. Phys. 1983,78, 1145. (11) Loesch, H. J. Chcm. Phys. 1976,18,431. (12) Parker, G.A.; Snow, R. L.;Pack, R. T. J. Chcm. Phys. 1976,64, 1668. (13) Preston, R.K.; Pack, R. T. J . Chcm. Phys. 1977,66,2480. (14) Stroud, C. L.;Raff, L. M.J. Chcm. Phys. 1980,72, 5479. (15) Keil, M.; Parker, G. A. J. Chem. Phys. 1985,82,1947. (16) Hough, A. M.;Howard, B. J. J. Chem. Soc., Furuduy Tram. 2 1987, 83, 191.
Iida et al. (17) Iida, M.;Ohshima, Y.;Endo, Y. J. Chem. Phys. 1991,95. 4772. (18) Iida, M.;Ohshima, Y.; Endo, Y . J . Chcm. Phys. 1991,94,6989. (19) Reinartz, J. M.L.J.; Dymanus, A. Chcm. Phys. Lcrt. 1974,24,346. (20) Watson, J. K. G. J . Chcm. Phys. 1%7,46, 1935. (21) Kivehn, D.; Wilson, E. B., Jr. J. Chem. Phys. 1953,21,1229. (22) P a r k u , M.A.; Suzuki, I.; Overend. J. J. Chcm. Phys. 1965,42, 2335. (23) Oka,T.;Morino, Y. J . Mol. Smcrrosc. 1961.6.472. (24 Townes, C. H.; Schawlow, A. L. Microwwc Spktroscopy; Dover: New York, 1975. (25) Alms,G.R.;Burnham, A. K.; Flygare, W. H. J . Chcm. Phys. 1975, 63,3321. (26) Amos, R. D.; Handy, N. C.; Knowlcs, P. J.; Rice, J. E.; Stone, A. J. J. Phys. Chem. 1985,89,2186. (27) Buckingham, A. D.; Disch, R. L.;Dunmur, D. A. J. Am. Chem.Soc. 1968.90,3104. (28) Campbell, E. J.; Buxton,L.W.; Keenan, M.R.; Flygare,W. H. Phys. Rev. A 1981,24,812. (29) Shea, J. A.; Campbell, E. J. J. Chcm. Phys. 1984,81,5326. (30) Suzuki, I. J. Mol. Spectrosc. 1980,80, 12. (31) Janda,K.C.;&mstein,L.S.;Stted,J.M.;Novick,S.E.;Klrmperer, W. J. Am. Chcm. Soc. 1978, 100,8074. (32) Wilson, E.B., Jr.; Decius, J. C.; Cross, P. C. Mofeculur Mbrurlons; McGraw-Hill: New York, 1955. (33) Pack, R. T.J . Chcm. Phys. 1974,61,2091. (34) Radzig, A.A.; Smirov, B. M.Reference Duru on Atoms, Moleculrs, und Ions; Springer: Berlin, 1985.
