J. Phys. Chem. 1984, 88, 586-593
586
or some other heterocycles similar to oxazole used as a ligand. Although there are some disagreements in the vibrational assignment of Pt-N stretching, in pyridine complexes it is generally assigned to the 300-250-cm-' r e g i ~ n . ~ Durig ' . ~ ~ and co-workersZ1 were unable to decide if the 260-cm-' band in cis and the 284-cm-' band in trans pyridine complexes is Pt-N stretching or deformation. Pfeffer et aL3' provide data only for the trans complex but their assignment is more definite: a strong 282-cm-' band is the asymmetric u(Pt-N); medium 256- and 234-cm-' bands are ip G(Pt-N) and oop a(Pt-N), respectively. Intensity considerations are not clear either. In Pfeffer's assignment the strongest band is assigned to Pt-N stretching, not bending. In ammine complexes the situation can be different. N a k a m o t ~ ,defines ~ most metal-nitrogen stretching modes as weak bands. In dihalodiammine complexes one also finds that bending vibrations give stronger bands.35 In assigning the respective vibrations of cisltrans[PtClz(oxa),] we used the intensity considerations of Pfeffer, assigning strong band(s) to stretching motions. Another consideration is that in general one would expect the stretching motion to have somewhat higher frequency that bending motion (particularly oop). The Pt-N and Pt-C1 vibrational assignments for complexes involving ligands similar to oxazole, such as substituted thiazoles36 are also in good agreement with pyridine complexes. In the infrared spectrum of trans-[PtCl,(oxa),] one expects an asymmetric Pt-N stretch and two Pt-N bending modes. The most intense band at 295 cm-' is assigned to Pt-N stretch, while less intense bands at lower frequencies of 276 and 260 cm-' are assigned to ip and oop deformations, respectively. Generally, one would expect that oop deformation would have lower frequency than ip deformation. In the Raman spectrum of this complex only three modes are allowed. Symmetric Pt-C1 stretch was already assigned. The band at 238 cm-' is easily assigned to symmetric Pt-N stretch on the basis of its intensity and frequency. Although the symmetric stretch will in most cases have lower frequency than the asymmetric motion, we are somewhat surprised by the considerable difference in the vibrational frequencies of these, as assigned (295 and 238 cm-I). The third intense band in the Raman spectra at 136 cm-' is assigned to the ip C1-Pt-N deformation. For ~ i s - [ P t C l ~ ( o x atwo ) ~ ] Pt-N stretches and two Pt-N deformation modes are expected in both infrared and Raman. They (35) Perry, C. H.; Athans, D. P.; Young, E. F.; Durig, J. R.; Mitchell, B. R. Spectrochim. Acta, Part A 1967, 23, 1137. (36) Weaver, J. A.; Hambright, P.; Talbert, P. T.; Kang, E.; Thorpe, A.
N. Inorg. Chem. 1970, 9, 268.
are as follows: asymmetric u(Pt-N) (bJ, symmetric v(Pt-N) (al), ip G(Pt-N) (al), and ip G(C1-Pt-N) (bl). In the infrared there are four bands at 281 (m), 260 (w), 250 (m), and 220 (w) cm-I. Two higher frequency bands are assigned to stretching modes and two lower frequency bands to the bending modes. This is definitely a difficult set of bands to assign. We could have assigned the more intense pair, 281 and 250 crn-', to stretching motions, and the less intense pair, 260 and 220 crn-', to bending motions. Possibly, a normal-coordinate analysis would help to resolve ambiguities in these and similar complexes. Raman bands were assigned by frequency comparison with infrared bands. The Pt-C1 bending vibrations are assigned in the literature with good confidence. In these square-planar complexes, metal-halogen bands are relatively isolated. Since there is little coupling with the rest of the molecule, both stretching and bending vibrations of metal-halogen bonds are good group frequencies, little affected by other ligands. (That is not the case for metal-nitrogen bonds, where there can be. considerablecoupling with the rest of the ligand molecule.) There are two Pt-Cl bending modes for each isomer: trans ip G(Pt-Cl) (b2,,) and oop a(Pt-C1) (bl,,); cis ip G(Pt-C1) (al) and oop a(C1-Pt-N) (b,). For similar pyridine and ammine complexes they are found below 180 cm21.21335 For both cis and trans isomers of [PtCl,(~xa)~] these frequencies were beyond the range of our infrared spectrophotometer. In trans- [PtCl,(oxa),] they are not allowed in the Raman spectrum. In cis-[PtCl,(oxa),] there are two Raman bands of medium intensity at 156 and 102 cm-', which are assigned to R-Cl deformations. They are in good agreement with the infrared data for cis-[PtCl,(py),] at 163 and 108 crn-'. The 102-cm-' Raman band in the oxazole complex is in the typical lattice vibration range. We do not have solution spectra, but, becuase of its intensity and good agreement with the infrared vibration in pyridine complexes, we consider it a genuine intramolecular vibration rather than lattice vibration. In the Raman spectra of both oxazole complexes there are a few weak low-frequency bands. Without any analysis we group them together as lattice modes. Two of them at higher frequency (192 and 264 crn-') may not be lattice modes. Acknowledgment. We thank Dr. J. Robert Mooney, Sohio R and D, for useful discussions regarding 'HN M R spectra. J.G.G. especially acknowledgesProfessor Bryce Crawford for many years of encouragement, support, and warm friendship. H e has always set a shining example of excellence in spectroscopy and "joie de vivre"-an enviable combination! Registry No. cis- [PtC12(oxa)2],88 106-62-1; trans- [PtCl,(oxa),], 88195-54-4.
Charge Redistributlon in the Molecular Vibrations of Acetylene, Ethylene, Ethane, Methane, Silane, and the Ammonium Ion. Signs of the M-H Bond Moments Kenneth B. Wiberg* and John J. Wendoloski The Department of Chemistry, Yale University, New Haven, Connecticut 06511, and The Central Research Department, E.I. duPont de Nemours and Co.. Wilmington, Delaware 19801 (Received: June 1, 1983)
The signs of the M-H bond dipoles in the title compounds have been determined by integration of 6-31G** wave functions, and indicate Mt-H- for methane, silane, ethylene, and ethane, and M--Ht for ammonium ion and acetylene. The origin of the dipole moment derivatives for out-of-plane bending modes is considered, and it is attributed to the formation of a bent bond. The signs of the dipole moment derivatives for the stretching modes agree with those expected from the signs of the bond dipoles. The vibrational frequencies calculated with the 6-3 1G** basis set stand in almost constant ratio to the observed (anharmonic) frequencies, but not to the estimated harmonic frequencies. The dipole moment derivatives are calculated with an average error of about 20%.
The intensities of the infrared bands of polyatomic molecules have been of interest to a number of research groups in the past 0022-3654/84/2088-0586$01.50/0
few years.l-1° Many attempts have been made to develop models that would allow one to understand the differences in intensity 0 1984 American Chemical Society
Signs of M-H Bond Moments
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 587 TABLE I: Optimized Geometries and Energiesa
Figure 1. Electron densities for the hydrogen molecule. The solid lines represent the electron densities for two noninteracting hydrogen atoms, and the dashed line represents the hydrogen molecule.
between different vibrational modes, and that would allow one to predict the intensities of these bands. We have attempted to understand both the static and dynamic dipoles in these molecules via the use of quantum mechanical calculations,”J2 and we have presented evidence for the sign and magnitude of the bond dipoles in hydrocarbons, and for the apparent difference in sign for the C-H bond dipoles derived from the calculations and those obtained from the out-of-plane bending modes. We should now like to examine the problem in more detail and to determine the charge fluxes2 which accompany molecular vibrations.
Hydrogen The hydrogen molecule provides one key to an understanding of the electron population to be assigned to an atom in a molecule. Consider a microscopic observer examining a hydrogen atom (Figure 1). It will “see” 0.5 electron in the region between the hydrogen nucleus and the observer and will conclude that the hydrogen atom has an electron population of 1.O. If this observer examines a hydrogen molecule, it will “see” a somewhat different situation. If the wave function is taken as $ = (4a + &)/(2 + 2 ~ ) ’ for / ~ each electron where the 4’s are 1s wave functions, the electron density distribution will be given by (+a2 + $2 + 24a&)/(1 is). The solid line in Figure 1 represents and 42. For the molecule, it is modified by the overlap population, 24&,, which increases the electron density between the nuclei, and by the normalization factor, 1/(1 + s), which will reduce the electron density in the region outside of the nuclear coordinates. The result will be the dashed line in the figure. The observer will now “see” less than 0.5 electron in the region between the nucleus and the observer. It is clear that two times the electron population from the nucleus to infinity represents a minimum value for the population ~~~~~~~~~
~
coinpd
re calcd
CH, NH,’ SiH, C,H, C,H, C,H,
1.0835 1.01 17 1.4683 1.0569 1.0764 1.0858
E, hartree -40.201 -56.545 -291.229 -76.821 -70.038 -79.238
f,,calcd r e obsd
707 455 900 837 842 235
6.00b 7.60‘ 3.38d 7.0Se 6.21f 5.8Sg
f,,obsd
1.086: 1.035’ * 1.481” 1.060‘ 1.085‘ 1.096m
5.70h (5.5 1)” 3.03” 6.3SP 5.6S4 5.38‘
a Distances are given in A , force constants in nidyn/A for the totally symmetric C-H stretching vibration, and are based o n five-point fits to data with iO.01 and i 0 . 0 2 A changes in bond lengths. The observed force constants are based o n thc estimated harmonic frequencies except for NH,’. All data were obtained with the 6-31G** basis set. 1 hartree = 627.5 kcal/mol. AE = -4.924811rc~t ~ 2 . 7 5 5 9 A r c ~ ’(units: hartrecs, angstroms). AE=-7.1412&”’ t 3.4918Ar”’. A E = - 2 . 1 9 7 5 A r s i ~ ~t . A E = - 3 . 0 1 6 8 A r c ~ ~+ 1 . 6 1 9 1 A r c ~ ~ ; r =c c 1.1861. 1.5515?H2 A E = -e5 . 2 8 2 5 A r c ~ ’ + 2 . 8 5 3 0 A r c ~ ~ ; Y c=c1.3165, LCCH= 121.74“. A E = - 7 . 3 4 2 4 A r c ~ ~t 4 . 0 3 1 5 A ? c ~ ~ ; r c c = 1 . 5 2 6 7 , i C C H = 1 1 1 . 2 2 1 . hGray,D.L.;Robiette,A.G.Mol. Phys. 1979, 37, 1901. Yamaguchi, Y . ; Schaefer, H. I?., I11 J. Chem. Phys. 1980, 73, 2310. Lovejoy, R. W.; Olson, W.B. J. Chem. Phys. 1972, 22, 643 (r,,). Lafferty, W. J.; Thibault, R. 1. J. Mol. Spectrosc. 1964, 14, 79. Duncan, J . L,; Wright, I. J.; Van Lerbcrghc, D. Ibid. 1972, 42, 463. Shaw, D. E.; Lepard, D. W.; Welsh, H. L. J. Chem. Phys. 1965, 42, 3736. Harvey, K. B.; McQuaker, N. R. Ibid. 1971, 55, 4396. This value is based on anharmonic frequencies observed in the solid Ball, D. F.; McKean, D. C.; Spectrochim. Acta 1962, phase. 18, 101 9. P Strey. G.; Mills, I. M. J. Mol. Spectrosc. 1976, 59, 103. Duncan. J. L.; McKean, D. C.; Mallinyon, P. D. Ibid. 1973, 45,221. Kondo, S.; Saeki, S. Spectrochim. Acta Part A 1973, 29, 735.
’
‘’
”
TABLE 11: Electron Populations at Hydrogens b
compd
P,,,a
Pcorr
CH, SiH, NH,’ C2H, C,H, C,H,
0.51641 0.61626 0.30974 0.52295 0.50066 0.43784
1.0652 1.2712 0.6389 1.0787 1.0327 0.9031
PHC
PMd
1.0648 1.7208 0.4442 1.0822 1.0442 0.8832
0.8816 1.2161 0.5862 0.8892 0.8717 0.7666
Integrated clcctron density from the hydrogen nucleus to -. P,,,, = 2.0627 P,,,. Integrated electron density from the partitioning surface to -. Mulliken population analysis. a
b
~~
(1) Reviews: Person, W. B.; Steele, I). Mol. Spectrosc. 1974, 2, 357. Steele, D. Zbid. 1978, 5 , 106. Gussoni, M. Adu. Infrared Raman Spectrosc. 1980, 6, 61. (2) Decius, J. C. J . Mol. Spectrosc. 1975, 57, 348. Decius, J. C.; Mast, G. B. Zbid. 1978, 70, 294. Mast, G. B.; Decius, J. C. Zbid, 1980, 79, 158. (3) Bode, J. H. G.; Smit, W. M. A. J . Phys. Chem. 1980,84, 198. Bode, J. H. G.; Smit, W. M. A,; Visser, T.; Verkruijsse, H. D. J. Chem. Phys. 1980, 72,6560. Smit, W. M. A.; van Dam, T. Zbid. 1980, 72, 3658. Smit, W. M. A. Zbid. 1979, 70, 5336. Smith, W. M. A.; van Dam, J. J . Mol. Struct. 1982, 88, 273. (4) Gussoni, M.; Abbate, S.; Zerbi, G. J . Chem. Phys. 1979, 71, 3429. Gussoni, M.; Abbate, S.; Dragoni, B.; Zerbi, G. J . Mol. Struct. 1980, 61, 355. Gussoni, M.; Abbate, S.; Sanvito, R.; Zerbi, G. Zbid. 1981, 75, 177. Gussoni, M.; Abate, S.; Zerbi, G. Zbid. 1982, 87, 87. (5) Person, W. B.; Newton, J. H. J. Chem. Phys. 1974,61, 1040. Newton, J. H.; Person, W. B. Zbid. 1976,64, 3036. Neto, B. d. B.; Bruns, R. E. Zbid. 1978,68, 545 1. (6) Kim, K.; McDowell, R. S.; King, W. T. J . Chem. Phys. 1980, 73,36. Kim, K.; King, W. T. Zbid. 1980, 73, 5591. Kim, K.; King, W. T. J . Mol. Struct. 1979, 57, 201. Kim, K.; W. T. J . Chem. Phys. 1979, 71, 1967. (7) Scanlon, K.; Laux, L.; Overend, J. Appl. Spectrosc. 1979, 33, 346. Youngquist, M. J.; Crawford, B., Jr.; Overend, J. J . Phys. Chem. 1979, 83, 2638. (8) Meyer, W.; Pulay, P. J . Chem. Phys. 1972, 56, 2109. Pulay, P.; Fogarasi, G.; Pang, F.; Boggs, J. E. J . Am. Chem. SOC.1979, 101, 2550. (9) Galabov, B.; Orville-Thomas, W. J. J . Chem. SOC.,Faraday Trans. 2 1982, 78, 417. Galabov, B. J . Chem. Phys. 1981, 74, 1599, 4744. (10) Gribov, L. A.; El’yashevich, M. A. Zh. Prikl. Spectrosk. 1978, 29, 1024. Gribov, L. A.; Dement’ev, V. A.; Todorovskii, A. T. Zbid. 1978, 28, 295. Gribov, L. A.; Prokofieva, N. I. J . Mol. Struct. 1982, 84, 39. (11) Wiberg, K. B.; Wendoloski, J. J. J . Am. Chem. SOC.1976, 98, 5465. Ibid. 1978, 100, 723. (12) Wiberg, K. B.; Wendoloski, J. J. J. Comput. Chem. 1981,2,53. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, 6561.
to be assigned to the hydrogen. In order to determine how much it is decreased as a result of bond formation, we have calculated this population using the 6-31G** basis set,I3 which will be used for all calculations reported herein. The population from the nucleus to infinity (designated as was found to be 0.4848. Thus, the scaling factor in going from this value to the total population for each atom is 2.0627.
The Sign and Magnitude of the C-H, N-H, and Si-H Bond Dipoles The same scheme may be applied to the bonds between hydrogen and other atoms, provided the second atom does not contribute significantly to the wave function at the remote side of the hydrogen nucleus. Such is the case for C and N,14and probably also for Si. In order to be consistent throughout, all of the following discussion will use the theoretical equilibrium geometriesI5 as the starting point (Table I). The electron populations have been calculated from the hydrogen nucleus to infinity (PlI2) (Table II).I6 A minimum value for the population which (13) Hariharan, P. C.; Pople, J. A. Chem. Phys. Lett. 1972, 16, 217. (14) Bader, R. F. W.; Messer, R. R. Can.J. Chem. 1974; 52,2260. Bader, R. F. W.; Beddall, P. M. J . Am. Chem. SOC.1973, 95, 305. (1 5) Schwendeman has suggested the use of experimental geometries (Schwendeman, R. A. J . Chem. Phys. 1966, 44, 1966) in calculating vibrational parameters. However, we prefer a consistent approach in which the geometry for which the gradient is zero is used as the reference for calculating the second derivatives of the energy with respect to the coordinates.
588
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984
should be identified with the hydrogen is obtained by multiplying by two. A more realistic value was obtained by multiplying the calculated population 2.0627, the factor found for the hydrogen molecule. This is given as P,,,, in the table. We shall use the symbol P to indicate an electron population. Another, and more physically precise definition of the electron population associated with an atom arises from the concepts of quantum topology developed by Bader.17-20 A bond may be defined by the bond path which is the path of maximum electron density between two bonded atoms. Normal bonds will have a critical point which represents a minimum in electron density along the bond path and a maximum in electron density perpendicular to the bond path. If one starts a t a critical point and develops a set of gradient paths of electron density with respect to coordinate, normal to the bond path (i.e,, paths by which the electron density decreases most rapidly), they will define a partitioning surface which separates one atom from another. A set of such surfaces for all of the bonds in a molecule will lead a set of regions, one for each atom. Within each region, the virial theorem is obeyed, and each may be considered a quantum mechanically distinct region of space.Ig Integration of the charge density within a region leads to the electron population associated with the atom. The values thus obtained for the populations at hydrogen are given in Table I1 as P H . It can be seen that the populations obtained by using the two methods are in remarkably good agreement for the C-H bonds of methane, ethane, ethylene, and acetylene. In examining other cases, we have found that the two measures of electron population at hydrogen are essentially equivalent for hydrocarbons. In the case of ammonium in, the electrostatic effect of the nitrogen serves to attract electron density toward it, shifting the partitioning surface toward the hydrogens. Thus, whereas the partitioning surface for methane is found 0.425 8, f drogen, with ammonium ion the distance decreases to 0.223 the As a result, the electron population (PH)at hydrogen is markedly reduced. The population derived from P1/*is significantly larger. If one were concerned with the properties of the hydrogen atoms in the ammonium ion, PHwould be the appropriate population. However, in considering infrared intensities, we are more concerned with the electron density in the vicinity of the nuclei. Here, P,,,, is probably a more useful quantity. It may be noted that PHleads to an effective charge on nitrogen of 1.22- e, whereas P,,,, leads to a much smaller value of 0.44- e. In any event, the two populations are significantly smaller than unity. A similar situation is found with silane. Here, the silicon repels the electron density associated with the Si-H bond, moving the partitioning surface nearer to the silicon (0.789 8, from the hydrogen), and increasing the region over which the hydrogen population is integrated. This leads to a larger value for PHthan for Pmr The use of PHleads to a model which approaches having four hydride ions associated with a strongly positively charged silicon (effective charge 2.88+), whereas P, leads a more modest electron population at hydrogen, and a much smaller effective charge at silicon (1.09+). Although the magnitudes are different, the sign of the bond moment is the same for both measures of the electron population. In the cases of CzH6,CzH4,CHI, and SiH4,the sign of the X-H dipole is X+-H-. However, with CzH2and NH4+, the sign of the dipole is reversed, X--H+. A recent application of the Hirschfeldzl definition of electron population associated with an atom gives (16) The boundaries between hydrogens were set by using the partitioning surface described in ref 17. (17) Bader, R. F. W. MTP In?. Rev. Sci., Phys. Chem. Ser. Two 1974, 52, 2260. Bader, R. F. W. Acc. Chem. Res. 1975, 8, 34. (18) Bader, R. F. W.; Anderson, S. G.; Duke, A. J. J . Am. Chem. SOC. 1979, 101, 1389. Srebrenik, S.; Bader, R. F. W.; Nguyen-Dang, T. T. J . Chem. Phys. 1978, 68, 3667. (19) Bader, R. F. W.; Tal, Y.; Anderson, S . G.; Nguyen-Dang, T. T. Isr. - . J . Chem. 1980, 19, 8. (20) Bader, R. F. W. J . Chem. Phys. 1980, 73, 2871. (21) Hirshfeld. F. L. Isr. J . Chem. 1977, 16, 198. Theor. Chim. Acta 1977, 44, 129.
Wiberg and Wendoloski Acetylene
5" bend AS=O.l23
Obs
Ethylene
=0.183
%=
I49
4"oop bend aS=O099 p=0.105
Obs
2:;
105
Figure 2. Effect of incomplete orbital following on the dipole moment derivatives for the symmetric bending mode of acetylene, and the outof-plane bending mode of ethylene.
results which are in agreement with the above.22 As we shall show below this has an important relationship to the signs of the dipole moment derivatives. The Mulliken population^^^ (PM) also are given in Table 11. It can be seen that they differ markedly from the populations derived by the other two methods, and lead to the opposite and incorrect sign for the C H bond dipole. Despite the well-known deficiencies with PM,it is interesting to note that there is a reasonable linear relationship between PHand p,:
P H = 1.995PM - 0.693 Here, the rms error is 0.023, and R = 0.998.
Importance of Incomplete Orbital Following in Bending Modes The dp./dS for the out-of-plane bending modes of ethylene and acetylene have been of particular importance in many estimates of the C-H bond dipoles.23 We have shown that standard criteria (Mulliken population analysisz4and localized molecular orbitals25) indicate that no rehybridization occurs in these modes," making a simple geometrical interpretation appear quite reasonable. This leads to pcHeff= 0.74D(C--Hf) for ethylene and pCHeff= 1.19D(C--H+) for acetylene." A similar analysis of the bending modes in methane led to pCHeff= 0.55D(C--Hf). However, as indicated above, ethylene and acetylene have opposite signs for their C-H bond dipoles (Le., CtH- in ethylene, C-H' in acetylene). Therefore, this simple analysis of the out-of-plane bending modes is incorrect. We have attributed the apparent moments derived from these bending modes to the formation of bent bonds.12 The idea of incomplete orbital following in bending motionsz6is now well established, and both our studies12 and those of Chipman, Paulke, and Kirtman2' have suggested that (22) Allen, L. C., private communication.
(23) Wiberg, K. B.; Wendoloski, J. J. Chem. Phys. Lett. 1977, 45, 180. Akiyarna, M . J . Mol. Spectrosc. 1980,84, 49. Kovner, M. A.; Snegirev, B. N. Opz.Spectrosc. (USSR) 1961, 10, 165. (24) Mulliken, R. S. J . Chem. Phys. 1962, 36, 3428. (25) Edmiston, C.; Reudenberg, K. Reu. Mod. Phys. 1963, 35, 457. (26) Nakatsuji, H. J . Am. Chem. SOC.1974, 96, 24. Figeys, H. P.; Berckrnans, D.; Geerlings, P. J . Mol. S t r u t . 1979, 57, 271.
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 589
Signs of M-H Bond Moments TABLE 111: Calculated and Observed Frequencies and Intensities for Tetrahedral Species
CH,n,b
CD,
i
calcd
obsd
vih
1 2 3 4 1 2 3 4
3165 1690 3275 1475 2239 1196 2427 1113
3026 1583 3157 1367 2140 1120 2336 1034
2915 1534 3019 1306 2108 1092 2259 996
N H , + C , d 1 3559
adaQ
uj(obsd)/
Wig
coinpd
wi(calcd) calcd obsd ratio
0.921 0.908 0.922 0.885 0.901 0.913 0.931 0.895
frequencies.
coinpd CH,
0.98 0.73 0.75 0.49 0.51 1.04 NH,' 0.67 0.48 0.72 0.39 0.39 1.00
0.855 0.928 0.845 2.33 3.3 1.4 0.864 2.16 1.3 0.6 ND,' 0.880 0.937 0.860 1.71 0.878 1.35 SiH,'Sf 0.919 0.914 0.930 2.01 1.59 0.79 0.886 2.47 1.80 0.73 SiD, 0.940 0.914 0.938 1.58 1.21 0.77 0.900 1.79 1.30 0.76 a Gray, D. L.; Robiette, A. G. Mol. Phys. 1979, 37, 1901. Bode, J . H. G.; Smit, W. M. A. J. Phys. Chem. 1980, 84, 198. I:erriso, C. C.; Hornig, D. F. J. Chem. Phys. 1960, 32, 1240. Frcdrickson. L. K.; Decius, J. C. Ibid. 1977, 66, 2297. Mast, G. B.; Decius, J. C. J. Mol. Specfrosc. 1980, 79, 158. Ball, D. I;.; McKcan, 1). C. Spectrochim. Acta 1962, 18, 1019. Lcvin. I. W.; King, W. T. J. Chem. Phys. 1962, 37, 1375. Harmonic 2 1852 3 3703 4 1620 1 2517 2 1310 3 2734 4 1215 1 2379 2 1066 3 2357 4 1032 1 1683 2 754 3 1703 4 754
TABLE IV: Change in Electron Population at Hydrogen During Vibrations
3043 1720 3130 1400 2217 1227 2350 1067 2377 2187 975 974 2319 2191 945 914 1681 1582 (689) (689) 1665 1597 699 681
i 1 2 3 iH,) 3 (H,) 4 (H,) 4 (H,)
1 2 3 (HI)
3 (H,) 4 (H,) 4 (H,) SiH,
1 2 3 (HI) 3 (H,) 4 (HI) 4 (H3)
aPH/as?
aP,,,,/asib
-0.388 -0.008 -0.229 0.284 -0.113 0.096
-0.338 -0.002 -0.233 0.253 -0.142 0.138
0.023 0.000 0.1 I5 -0.108 -0.036 0.020
-0.006 0.002 -0.006 0.020 -0.058 0.057
-0.176
0.071
-0.131 0.000 -0.113 0.124 -0.021 0.018
-0.190 0.000
-0.285 -0.001 -0.189 0.198 -0.070 0.068
-0.091 0.001 -0.037 0.036 -0.013 0.008
-0.161 0.155 -0.027 0.022
0.000
-0.197 0.217 -0.071
aPM/aSic
-.
PH is the population troin the partitioning surface to P,,,, is 2.0627 times the population from the hydrogen nucleus PM is the Mulliken population. to w. TABLE V: Symmetry Coordinates for MH, Species
Observed anharmonic frequencies.
relatively little orbital following occurs with C-H bonds. The effect of incomplete orbital following may be visualized as follows. Let us consider the C-H bond in acetylene to be made up of electron distributions assigned to the nuclei, and an overlap population at the center of the bond. Suppose we take the latter as 20% of the total bond population (i.e., 0.4 e) and subtract half of this from each of the carbon and hydrogen populations calculated above. This gives the distribution shown in Figure 2. The observed dipole moment derivative3g6requires that when the C=C-H bonds are bent 5" (AS = 0.123 rad), the dipole moment is 0.183 D, leading to &/as = 1.49 D/rad. In the model, however, complete orbital following (in this case represented by proportional motion of the overlap population), would lead to p = 0.109 D and & / I S = 0.88 D/rad using the point dipole approximation. Clearly, the value is much too small. If the overlap population did not move, p = 0.286 D and dp/dS = 2.32 D/rad, which is too large. If the overlap population moved 50% as much as required for complete orbital following, the correct dp/dS is found. A similar analysis of the out-of-plane bending mode in ethylene also is shown in Figure 2. Here, complete orbital following would lead to a dipole moment derivative with an incorrect sign, and no orbital following gives a value which is only slightly to large. A small amount of orbital following (10%) leads to the correct dp/dS. A smaller degree of orbital following would be expected for ethylene than for acetylene since only the p component of the bond has directional character, and the C-H bond of ethylene has greater p character (67%) than the C-H bond of acetylene (50%). We do not represent the above as a precise physical picture of the process, but only wish to illustrate the importance of incomplete orbital following, and to show how it may lead to an apparent sign of the C-H bond dipole which is opposite to that present in the (27) Chipman, D. M.; Palke, W. E.; Kirtman, B. J . Am. Chem. SOC.1980, 102,3377.
equilibrium molecule. With this introduction, we wish to make a detailed analysis of the vibrational modes of bonds to hydrogen. Dipole Moment Derivatives for Methane, Silane, and Ammonium Ion The three species have two stretching and two bending modes, and only one of each type is infrared active. The force constants were obtained with respect to Cartesian coordinates by the "force" method developed by Pulay.2* Here, the gradients of the energy with respect to the coordinates are obtained analytically for structures in which the symmetrically distinct atoms are moved 0.01 bohr along each of the Cartesian axes in turn, and the second derivatives are obtained numerically. The dipole moment derivatives with respect to the Cartesian displacements (polar tens o r ~ are ~ ~obtained ) at the same time. The vibrational problem was solved in Cartesian coordinates30 and the dipole moment derivatives for the normal coordinates were calculated giving the values listed in Table 111. The calculated frequencies are about 10% too large. In examining a number of hydrocarbon^,^^ we have found a ratio of observed to calculated C-H stretching modes to be 0.92 f 0.02, and for the other modes, the ratio is 0.88 f 0.03. Similar results have been obtained by other investigator^.^^ The major deviation (28) Pulay, P. Mol. Phys. 1969, 17, 197. 1970, 18, 473. 1971, 21, 329. Pulay, P. In "The Force Concept in Chemistry"; Deb, B. Ed., Van Nostrand, New York, 1981. The force constants and dipole moment derivatives for methane, ethane, ethylene, and acetylene have been calculated by using a variety of basis sets (see citations in above reference, and ref 3). The values are somewhat basis set dependent, and therefore they have been recalculated with a common basis set which includes polarization functions at both carbon and hydrogen. (29) Biarge, J. F.; Herranz, J.; Morcillo, J. An. Quim. 1961, A57, 81. Person, W. B.; Newton, J. H. J . Chem. Phys. 1974, 61, 1040. Newton, J. H.; Person, W. D. Ibid. 1976, 64, 3036. Rogers, J. D.; Hilman, J . J. Ibid. 1982, 77, 3615. (30) Gwinn, W. D. J . Chem. Phys. 1971, 55, 477. (31) Wiberg, K. B.; Dempsey, R.; Walters, V., to be published.
590
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984
TABLE VI: Force Constants and Dipole Moment Derivatives for Tetrahedral Species
~
Sign of ,u Mode
CH4
NH;
Wiberg and Wendoloski
SiH4 compd
symmetry coord
force constant" obsd
calcd
5.16 (5.84) 0.56 (0.58) 5.17 (5.38) 0.51 (0.55) 0.22 (0.22) 2.84 0.4 1 2.74 (3.03) 0.51 (0.52) 0.03 (0.04) 5.51 0.63 5.29 0.52 0.08
5.95 0.67 5.78 0.63 0.21 3.36 0.49 3.16 0.63 0.11 7.52 0.60 1.45 0.68 0.14
6/1/6S
obsd
calcd
0.75 0.37
1.01 0.39
1.43" 1.80
1.89 2.4 1
2.8" 1.1
2.3 1 1.13
~~
CH,
s, s2
s 3 s4
(f3q4)
+
+
4
SiH,
S, s 2
s3 s, Figure 3. Signs of the dipoles created by the antisymmetric stretching and bending modes of MH4 species.
is found with the out-of-plane bending modes for alkenes where the ratio is somewhat smaller. If the calculated frequencies are about 10% too large, the corresponding force constants are about 20%too large. It is interesting to note that the dipole moment derivatives are also about 20% too large. Somewhat larger deviations were found with the ammonium ion, but this might be expected since the observed values are derived from solid-phase spectra, and an ionic medium might well perturb the vibrations of the ion. Force constants and dipole moment derivatives are usually presented in terms of symmetry coordinates. The symmetry coordinates for the tetrahedral species are given in Table V. The force constants and dp/dS are compared with the observed (anharmonic) data in Table VI. In order to avoid any possible abiguity concerning the signs of dp/dS, the modes are shown in Figure 3 along with the resulting dipoles. These data present an interesting challenge with regard to interpretation. The sign of dp/dS3 (the stretching mode) is the same for methane and silane, but is reversed with ammonium ion. On the other hand, the signs of dp/dS4 (the bending mode) are the same for methane and ammonium ion, but is reversed for silane. Let us first consider the stretching mode. The dipole moment derivatives in molecules such as these have frequently been discussed in terms of contributions of two types: (a) from static bond dipoles, and (b) from charge fluxesZ (or related quantities') which modify the effects of the static bond dipoles. It has generally been considered that all three species have the same sign of the C-H or N-H bond dipole (Le., H+ in each case) and the signs of dp/dS3 have been ascribed to large charge flux terms., NOWthat it is clear that the hydrogen is the negative end of the bond dipole in methane and silane, and the positive end in the ammonium ion, the charge fluxes must be reconsidered. We prefer to calculate the charge fluxes as the change in electron population at an atom for a given change in the symmetry coordinate (Le., dPH/dS). It is possible that the motion of the nuclei will lead to a shift of the partitioning surface and a change in population which does not correlate with the change in electron density near the nucleus. In order to examine this possibility, we have calculated both the population as defined by the partitioning surface (PH)and the population derived by multiplying that from the hydrogen nucleus to infinity by 2.0627 (Parr). Both sets of values are given in Table IV. It may be noted that a positive value of dPH/dS indicates an increase in electron population and a more negative (or less positive) charge associated with the hydrogen. It can be seen that the change in electron population is essentially zero for the symmetric bending mode (S,) of methane, and is small for the antisymmetric bending mode (S4). It is considerably larger for the stretching modes (SIand S3),and has a sign leading toward electrical neutrality on stretching. However, the values are much too small to reverse the sign of the dipole (32) Pulay, P. Mod. Theor. Chem. 1977, 4, 153. Pople, J. A,; Schlegel, H. B.; Krishanan, R.; DeFrees, D. J.; Binkley, J. S.; Mrisch, M. J.; Whiteside, R.A,; Hout, R. F.; Hehre, W. J. Int. J . Quantum Chem., Quantum Chem. Symp. 1981, 15, 269.
Cf,,,)
NH,'
S, s2
s3 s4
V,,) " The observed anharmonic force constants are given first, and
the estimated harmonic force constants are given in parentheses. Duncan, J. L.; Mills, I . M . Spectrochim. Acta 1964, 20, 523. Bode, J. H. G.;Smit, W. M . A . J. Phys. Chem. 1980, 84, 198. Ball, D. I . ; McKean, D. C. Spectrochim. Acta 1962, 18, 1019. Harvey, K. B.; McQuaker, N. R. J. Chem. Phys. 1971, 55, 4396. The force constants must be considered as approximate since it is known that the vibrational frequencies for the ammonium ion are dependent on both phase and counterion. e These values differ from those in the cited literature because of a difference in the symmetry coordinates which were used,
moment derivatives. For example, with methane, PH is 1.0649 at 1.0835 k,and only drops to 1.0510 at 1.1032 k. Thus, the sign of the dipole moment derivative for the C-H stretching mode must correspond to that of the bond dipole, and this agrees with the experimental observations. A similar conclusion is reached on examining the antisymmetric stretching mode (S,) of the ammonium ion and silane. The reversal of sign found with the ammonium ion results from the change in sign of the N H dipole as compared to CH or SiH. The bending mode (S,) of the ammonium ion gives a large negative dipole moment derivative, corresponding to a positive charge on hydrogen moving in the direction shown in Figure 3. With methane, the bond dipole would predict a small positive dp/dQ, but a small negative value is observed, which corresponds to the bent bond moment described above. Finally, with silane, the electron population at hydrogen is so large that its effect cannot be cancelled by the bent bond moment, and a positive dp/dQ is observed. The conclusions reached above concerning the relationship between the sign of the M-H bond dipole and the sign of dp/dS for the stretching modes is reinforced by the observationsof Kondo and Saeki33with regard to difluoromethane. Here, the sign of the dipole moment derivative was reversed with respect to that for methane. Our calculation of the electron population for the hydrogens of difluoromethane leads to a value of 0.995 e. This is reversed from that found in methane, and should lead to a reversed sign for +/as. Dipole Moment Derivatives for Ethane, Ethylene and Acetylene The force constants and dipole moment derivative (polar tensors) with respect to Cartesian coordinates were calculated as described above. The calculated vibrational frequencies and dp/dQ are given in Table VII. Again, the ratio of the observed (anharmonic) frequencies to the calculated values is essentially constant, with somewhat low ratios found only with the out-ofplane bending modes. The more precise experimental values for the dipole moment derivatives are available for a ~ e t y l e n e ,and ~ . ~ here there is an (33)
Kondo, S.;Nakanaga, T.; Saeki, S. J . Chem. Phys. 1980, 73, 5490.
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 591
Signs of M-H Bond Moments TABLE V11: Calculated and Observed Frequencies and Intensities for Ethane, Ethylene, and Acetylene
TABLE V111: Symmetry Coordinates for C, Species
-
-
A,, A,,
E tha ne S, = ( r , + r , r , r , + r , r6)/6‘12 S , = ( r , + r 2 + r , - r 4 - r j -r6)/6’12 S,=(P, + P 5 + P 6 - P l - P ~ - P ~ ) / 6 1 ’ 2
A,
b t livlene S I = i r , + r , + r , + r4)/2”’ S, = R s, = (201, -0,- p2 t 201, - p3 - /3,)/(12),’2
A, B,,
S, = r a S , = i r,
B,, B,, B,,
s, = ( b , - S2)/21/2 S, = ( r , - r 2 + r , - r 4 ) / 2
-___
___ Wi i calcd
coni pd
C,H,,””‘
e\t
ut
1 2 3 A,, 4 A,, 5 6 b, 7 8 9 E, 10 11 12
3167 1566 1051 330 3160 1535 3216 1629 1330 3241 1633 891
3064 1437 995 290 3061 1444 3063 1504 1211 3119 1518 840
2954 1388 995 289 2896 1379 2969 1469 1190 2985 1472 821
0.933 0.886 0.947 0.876 0.917 0.898 0.890 0.902 0.890 0.921 0.901 0.921
1 2 3 4 5 6 7 8 9 10 1I 12
2282 1279 894 234 2268 1172 2388 1164 1063 2403 1167 643
2147 1183 863 205 2171 1116 2277 1059 984 2310 1105 604
2083 1155 843 200 2087 1077 2225 1041 970 2236 1081 594
0.913 0.903 0.943 0.854 0.920 0.919 0.932 0.879 0.913 0.930 0.926 0.924
1 2 3 4 5 6 7 8 9
3306 1843 1489 A, 1151 B,, 3360 1351 B,, 1092 B,, 1100 B,, 3387 I O 906 B,, 11 3282 12 1603
3153 1655 1370 1044 3232 1245 969 969 3234 843 3147 1473
3026 1630 1342 1023 3103 1220 949 949 3105 826 3021 1444
0.915 0.881 0.901 0.889 0.923 0.903 0.869 0.855 0.917 0.912 0.920 0.901
1 2461 1698 1087 814 2508 1098 826 909 2522 651 2372 1187
2330 1539 1000 738 2381 1028 798 731 2418 604 2266 1094
2260 1518 985 726 2310 1011 780 720 2345 584 2220 1078
0.918 0.894 0.906 0.892 0.921 0.920 0.944 0.792 0.930 0.897 0.936 0.908
1 3677 3495 3375 2 2222 2008 1974 3 3564 3415 3282 4 800 624 613 5 877 747 730
0.918 0.888 0.921 0.766 0.832
1 2971 2782 2705 2 1946 1793 1759 3 2619 2508 2439 4 668 529 511 5 644 558 539
0.911 0.904 0.931 0.765 0.837
A,,
c 2 D6
C,H,d,F
A,
c, D4
2 3 4 5 6 7 8 9 10 11 12
C,H,f..h
2, 2, IIg 11,
c 2
D,
uiiobsd)/ wi(calcd) calcd
IadaQ
I
obsd ratio
1.31 1.47 1.12 0.07 (0.42)
1.54 0.37 0.25
1.66 0.55 0.36
1.08 1.49 1.44
B,,
xg 0.93 1.04 1.12 0.07 (0.36)
1.10 0.31 0.18
1.20 1.09 0.45 1.45 0.27 1.50
xu 11, 11,
+
+ +
- r2- r,
+ r4)/2
s, = (0,- 8, - 8, + 0,112 s,= ( b , + 6,)/21’2
SI” = (8, - 8 2 + 03 -0,112 S I ,= ( y I + r , - r , - r 4 ) / 2 SI, = (201, - p, - p, - 201,
+ p , + p,)/cl2)”2
Acetvlciie s, = ( r , + r 2 ) / 2 ’ ” S,=R .y3 = ( r , - r , ) / ~ 2 ” ~ s, = (01, - a,)/2’’2 s, = ( a , t a,)/2“2
6 is the out-of-plane angle a t the methylene group and torsional angle at t h e double bond.
T
is t h e
Sign of ,u 1.53
1.37 0.90
0.96 0.80 0.83 0.06 (0.11) 0.74 0.59 0.80 0.41 0.48 1,17
1.16
0.99 0.85
0.71 0.56 0.79 0.05 (0.03) 0.51 0.44 0.86 0.32 0.35 1.09
1.47
1.32 0.90
4
H-CZC-H c+
1.54
1.46 0.90
1.08
0.96 0.89
1.13
1.06 0.94
Nyquist, I . M.; Mills, 1. M.; Person, W. B.; Crawtord, B., J r . Kondu, S.; Saeki, S . J. C h c m Phys. 1957, 26, 552. Specrrochim. Acta, Part A 1973, 29, 735. Nakagawa, 1.; Shinianouchi, T. J. Mol. Spectrosc. 1971, 39, 255. Duncan, .I.L.; McKean. 0.C.; Mallinson. P. D. ibid. 1973, 45,221.
‘
‘’ Golikc. K. C.; Mills, I . M.; Person, W. B.; Crawford. B., Jr. J. Chem. Phys. 1956, 25, 1266. St rey, G.; Mills, I. M. J . Mol. Specrrosc. 1976, 59, 103. Stnit. W. M. A.; Van Straaten, A . J.; V i w r . T. J. Mol. Struct. 1976, 48, 177. Kim. K.; King, W. T. Ihid. 1979, 37, 201. SuLuki. I.; Ovcrend, J . Specrrochim. Acta, Part A 1969, 2.5, 977.
-
t
H-C-C-H
t
t
Figure 4. Signs of the dipoles created by bending and stretching modes of C2species.
excellent agreement between observed and calculated values. The data for ethylene are less precise because of overlapping bands.34 Nevertheless, there is generally good agreement between the calculated and observed values. Finally, with ethane, the problem of overlapping bands is particularly severe,35and the agreement between calculated and observed values is less satisfactory. The symmetry coordinates for these compounds are given in Table VI11 and led to the force constants and &/as summarized in Table IX. Again, the symmetry coordinates and signs of the resultant dipoles are presented diagrammatically in Figure 4. Only the A2,, mode of ethane is considered here, a more detailed (34) Strey, G.; Mills, I. M. J . Mol. Spectrosc. 1976, 59, 103. (35) Golike, R. C.; Mills, I. M.; Person, W. B.; Crawford, B.,Jr. J . Chern. Phys. 1956, 25, 1266.
592
The Journal of Physical Chemistry, Vol. 88. No. 3, 1984
TABLE 1X: Force Constants and Dipole Moment Derivatives
for C, Species force constant
coiiipd
inode *2U
fs,s f6.6
*g
fl,, fi.2 fl,,
fz,, f2J
f,,, Au
f4,4
B1g
f5.S fS.6
BIU
f,,, f,,,
'32,
fa,,
B2u
f,,, f9,lO flO,l"
B3u
fl1,ll fll,,, fl,,,,
Xg
1'1.1 f2,2
ZU
f,,,
I'g
f4,4
'1,
f5,S
obsd
calcd
5.37 0.63 5.64 0.37 -0.06 9.40 -0.3 1 0.49 0.30 5.66 0.40 0.66 0.30 0.22 5.49 -0.18 0.49 5.60 0.09 0.45 6.35 16.34 6.38 0.16 0.34
5.84 0.68 6.15 0.47 -0.10 11.45 -0.3 1 0.5 7 0.33 6.03 0.21 0.74 0.39 0.29 6.07 0.20 0.54 6.12 0.10 0.53 7.05 19.8 6.78 0.16 0.37
Wiberg and Wendoloski TABLE X: Change in Electron Population at Hydrogen for C, Species
IS!.l/SS I -___ calcd
-0.287 -0.251 0.275 -0.054 0.060 -0.403 -0.01 I -0.375 0.485
obsd 1.08
0.28
1.30 0.07 CZH4
-0.01 8
1.05
1.14
0.80
0.91
0.09 0.63
0.08
0.30
0.25
1.26
1.42
1.49
1.56
0.04 7 -0.376 0.354 -0.1 19 0.085 -0.550 -0.262 -0.558 0.643 -0.005 -0.007
C2%
0.74
Kondo, S.; Saeki, S. Spectrochim. A c t a , Part A 1973, 29, 735. Duncan, J. L.; McKean, D. C.; Mallinson, P. D. J. Mol. Spectrosc. 1971, 39, 255. The force constants given above differ from those listed by DMM because they scaled their bending force constants with thc C-H bond length, whereas we used r = 1 .O A , and because we used a different definition of the synimetry coordinates for Strey, G.; Mills, I . M. Ibid. 1976, 59, 103. S,, S,,and S,.
-0.239 -0.202 0.257 -0.059 0.091 -0.309 0.009 -0.385 0.55 1 -0.054 0.032 -0.328 0.267 -0.084 0.103 -0.555 -0.195 -0.557 0.557
(CCH bcnd)
-0.01 0
(g oop bend) (ti oop bend)
-0.01 1
(sym CH str) (antisym CH str)
(antisyin CCH bend) (syni CH str) (oop bend) (CH s t r )
(CH s t r )
(CCH bend) (syni CH str)
(C=C str) (antisym CH str)
A2U
a
analysis for ethane will be presented at a later time in connection with intensity measurements for the ethyl halides.36 As with methane, the signs of the &/as for the stretching and bending modes of ethane and ethylene have the same sign. In addition, whereas ethane and ethylene have the same sign of ap/aS for their stretching modes, the sign is reversed with acetylene. Our electron population analysis indicates that ethane and ethylene have the same sign for their C-H bond dipole, and that it is reversed with acetylene. Thus, again, the signs of the dipole moment derivatives for the stretching modes corresponds to the signs of the C-H bond moments. The dipole moment derivatives for the bending modes are dominated by the bent bond moment as described above, and as a result the C-H stretching and bending modes of ethane and ethylene have the same signs. The changes in electron population for some symmetry coordinates of the three hydrocarbons are summarized in Table X. The results are rather similar to those found with methane. The C-H stretching modes have fairly large values of BP/BS corresponding to a decrease in population as the bond is stretched, and the sign is the same for all three hydrocarbons. The largest value is found with acetylene, and it is this case in which dissociation to C-H+ is most facile. Again, the magnitudes of the changes are too small to affect the sign of the dipole moment derivatives. The bending modes generally gave small values of dP/BS. Here, the in-plane modes, which should lead to a change in hybridization, resulted in an increase in electron population at hydrogen for deformations which lead to an increase in p character in the bond, and a decrease in population when the s character increases. This is in accord with one's expectation since the carbon s orbital is more electronegative than the p orbital. It may be significant to note the difference between the modes in which the symmetry of the C-C bond is preserved and those in which it is lost. In the former case, no C-C dipole will be (36) Wiberg, K. B.; Walters, V.; Rossi, A., to be published.
B3u
Hb
+0.102 C=c -0.059H/ /
nu
/)l-O.O30 +0.076
