J. Phys. Chem. 1983, 87,3847-3857
for T H F than for alkyl chlorides.18 The proposed mechanism ascribes an important role to the cation, and this appears to agree with the experimental facts. Indeed, the physical and, particularly, the chemical properties of radical anions are strongly cation dependent. A very impressive example could be the function of the cation as an electron carrier in the so-called "atom-transfer" process.lg Equally impressive is the cation-dependent reactivity of naphthalene radical anion toward 5-hexenyl fluoride.20 The cited examples are from studies on dilute solutions of radical anions. So, in order to ascertain that our conclusions are applicable to reactions of radical anions at high concentrations, as well, we made a brief kinetic study of the reaction between alkali-metal fluorenone ketyl anions and benzyl chloride, under a set of conditions. The results are given graphically in Figure 3. Obviously, there is a dramatic cation effect. The rate of the decay of paramagnetism in mixtures of fluorenone ketyl anions and benzyl chloride increases as the counterion changes from Li+ to Na+ to K+. With the latter the reaction is too rapid to be followed by our technique.21 (18) Gas-phase 'ligand affinities" for alkyl halides have begun to become available: Staley, R. H.; Beauchamp, J. L. J. Am. Chem. SOC.1975, 97, 5920; Woodin, R. L.; Beauchamp, J. L. Zbid. 1978, 100, 501. (19) Ward, R. L.; Weissman, S. I. J. Am. Chem. SOC. 1957, 78, 2086. See also: (a) ref 15, p 31. (b) Hirota, H. In ref 11, p 75. (20) Reference 1, p 528.
3847
Experimental Section Carbon-13 nuclear magnetic resonance shifts were measured with a Varian FT-80 NMR spectrometer. Due to the great stability of the instrument no external marker was used. Benzene-d, was sealed in thin-walled capillaries and used to obtain a lock. Solvent shifts were referred to the resonance positions of the T H F carbon bands in neat solvent. Carbon-13 shifts of 1-chloropropane and 1chlorobutane were referred to the relevant carbon resonance in a 30% (v/v) solution of the respective alkyl chloride. Kinetic runs were made by observing the decay of the shift of the a-proton band of T H F at 35 "C, in mixtures being initially 0.70 m in fluorenone ketyl anion and 1.44 m in benzyl chloride, with respect to time. T H F was purified as described previously.22 Registry No. THF,109-99-9; (naphthalene)Li, 7308-67-0; (naphthalene)Na, 3481-12-7; (naphthalene)K, 4216-48-2; (anthracene)Li, 34509-60-9; (anthracene)Na, 12261-48-2; (anthracene)K, 34475-54-2; (benzophenone)Li, 16592-10-2; (benzophenone)K, 4834-86-0;(fluorenone)Li,34474-11-8;(fluorenone)Na, 34474-12-9; (fluorenone)K, 34474-13-0; (p-biphenylyl phenyl ketone)Li, 84658-11-7;(p-biphenylylphenyl ketone)Na, 8465812-8; (p-biphenylylphenyl ketone)K, 31530-50-4. (21) Screttas, C. G. J. Chem. SOC.,Perkin Trans. 2, 1974, 145. (22) Screttas, C. G.; Micha-Screttas, M. J.Org. Chem. 1978,43, 1064.
Ab Initio Prediction of Structures, Force Constants, and Vibrational Frequencies. The Saturated Three-Membered Rings Cyclopropane, Ethylene Oxide, and Ethylene Imine Andrew Komornlckl," Francolse Pauzat, Polyatomics Research Institute, Mountain View, California 94043
and Yves Elllnger Groupe de Chimie Theorique, LEDSS, 53X-F3804 1 Grenobie, France (Received: February 8, 1983: I n Final Form: April 18, 1983)
A comparative ab initio investigation of the geometries, force constants, and vibrational frequencies is presented for a series of saturated three-membered-ring compounds. The calculations have been performed with minimal, split-valence, and split-valence-polarized basis sets. For the spectral results we have found that the minimal basis set results are only of marginal value since they often predict force constants to be too large by over 40%. The errors in the off-diagonal force constants further cause the order of the predicted modes to be grossly in error. At the split-valenceand split-valence-polarized levels we have found that the calculations predict harmonic frequencies to be too high by about 10-15%. On the basis of our calculations we propose several revisions in the currently accepted force field for both cyclopropane and ethylene oxide. Our calculations suggest what we believe to be the most complete force field for ethylene imine. This is especially important since previous experimental refinements have neglected a number of interaction force constants which we show to be important in the correct experimental assignment of the spectra. A scaling of the diagonal elements of the force constant matrix produces reasonably good agreement with experiment. We have also shown how an analysis of the normal modes provides a very useful aid in the interpretation and assignment of experimental spectra, especially in cases where a number of internal coordinates contribute to a molecular vibration. A comparison of the normal modes for cyclopropane, ethylene oxide, and ethylene imine points out the similarity among these compounds and provides a foundation for a functional-groupapproach is interpreting their infrared spectra. For completeness we also include energies and dipole moments evaluated at the SCF level with a near Hartree-Fock basis.
I. Introduction The notion of a functional group is one of the most commonly used concepts in chemical spectroscopy. This concept is largely based on the fact that often force cont Mailing address: 1101 San Antonio Road, Suite 420, Mountain View, CA 94043.
stants and vibrational frequencies which are characteristic of a given group can be transferred from one molecule to another. Thus, the i n f ~ ~ spectrum ed of a molecule is often considered to be a fingerprint, and forms the basis for the identification of many organic species. Although frequencies are the only observable quantities, they can be related to the motions of the nuclei along the normal co-
0022-365418312087-3847$01.50/00 1983 American Chemical Society
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The Journal of Physical Chemistry, Vol. 87, No. 20, 1983
ordinates of a system if the harmonic approximation is known to be valid. The vibrational assignment of frequencies to specific types of displacements is often difficult and in many cases still open to question. By definition, a particular normal mode is a combination of several internal coordinates such as bending, twisting, and rocking. Only in the presence of symmetry constraints do these internal coordinate motions acquire a unique identity. The experimental assignment of observed frequencies to particular normal modes has been resolved in some cases by a comparison of the vibrational modes in an isoelectronic series of related molecules which are also of decreasing symmetry. Thus, assignments are tentatively made for the higher symmetry molecule, where the separation of the various kinds of motions is more clearly defined, while the spectral assignment of the lower symmetry species is made on the basis of consistency and similarity of the normal modes. We present here a theoretical investigation of the structures, force constants, vibrational frequencies, and normal modes of an isoelectronic series which includes the three-membered rings of cyclopropane (C,H,), ethylene oxide (C2H40),and ethylene imine (C2H4NH). Our purpose in undertaking this investigation involves several aspects. First, we would like to compare the calculated force fields and the associated vibrational spectra with current experimental data. By performing a series of calculations with more extensive and hopefully more reliable basis sets we would hope that such a study will demonstrate the level of reliability attainable at this level of theory. The only remaining questions involve the effects of electron correlation and the perturbations due to anharmonic corrections. Second, our aim is to demonstrate how the chemical similarities of these three molecules are manifest in their force fields. A comparison of the normal modes for each of the three molecules should show how symmetry and chemical environment allows us to correlate the nuclear motion in each of these molecules. It is our hope that such investigations will lay the foundation of the systematic theoretical approach for the subsequent vibrational analysis of more complicated systems. The following section outlines the theoretical methods used in this investigation. Section I11 presents a discussion of the calculated structures while section IV presents our calculated force fields, compares them with the currently available experimental data, and in some cases suggests a reassignment of the experimental spectra. Section V draws a corrleation among the normal modes for the three molecules while the final section presents our conclusions. 11. Theoretical Methods
The calculations reported here have all been performed a t the SCF level. The emergence of gradient methods in recent years has allowed the use of some very efficient methods for the location of equilibrium structures and the evaluation of harmonic force con~tants.l-~ In these methods both the energy as well as the gradient are evaluated simultaneously. The geometry optimization was performed within the respective symmetry of each of the (1) A. Komornicki, K. Ishida, K. Morokuma, R. Ditchfield, and M. Conrad, Chem. Phys. Lett., 45, 595 (1977); J. W. McIver, Jr., and A. Komornicki, ibid., 10, 303 (1971). (2) J. W. McIver, Jr., and A. Komornicki, J. Am. Chem. SOC.,94, 2625 (1972); A. Komornicki and J. W. McIver, Jr., ibid., 95, 4512 (1973); 96, 5798 (1974). (3) P. Pulay, “Methods of Electronic Structure Theory”, H. F. Schaefer 111, Ed., Plenum Press, New York, 1977. (4) A. Komornicki, J. D. Goddard, and H. F. Schaefer, J . Am. Chem. SOC.,102, 1763 (1980); A Komornicki, C. E. Dykstra, M. A. Vincent, and L. Radom, ibid.. 103, 1652 (1981).
Komornicki et al.
three molecules. Convergence in the geometry optimization was deemed satisfactory if the largest component of the gradient was less than 4 X lo4 hartree/bohr. The force constants were evaluated by the finite difference techniques described previously.2 The Cartesian coordinates of each molecule were displaced. Thus, each column of the matrix of second derivatives was then calculated as the difference between two displaced Cartesian gradient vectors. The precision of the calculated force constants was measured by the extent of asymmetry in the calculated force constants as well as by the magnitude of the translational and rotational eigenvalues. In all cases the asymmetry of the calculated force constant matrix was less than 2X au, and the largest of the zero-frequency eigenvalues was only 10-20 cm-’ which assured an adequate separation between the rotational and vibrational motions. All of the calculations were performed with the GRADSCF program ~ y s t e m . ~ For the geometry and force constant calculations we have employed three basis sets of increasing quality. The first basis set was then minimal STO-3G basis used extensively by Pople et al. in calculations of equilibrium geometries and force constants.6 The second was the more flexible 4-31G split-valence basis,’ which has been used quite successfully by Blom and co-workers in their systematic studies of equilibrium structures and force constants for hydrocarbon^.^^^ The third basis set employed was the split-valence-po1arizedlo set often referred to as 6-31G**. This basis includes a split-valence sp space plus a set of d-type polarization functions on each of the heavy atoms, and a set of p-type polarization functions on each of the hydrogens. The exponents used for the d functions were a(C) = 0.8, a(N) = 0.8, and a(0) = 0.9, while for the p functions on hydrogen we used a value of a ( H ) = 1.1. Finally, we have employed a triple-{-polarized basis for the final energies and dipole moments. This basis was constructed from the Huzinaga (lls6p/5s) primitive set as contracted by Dunning’l to a [5s3p/3s] basis. This sp space was augmented by a two-term set of d-type polarization functions with an effective slater exponent of 2.1 for the oxygen and nitrogen atoms, and 2.0 for the carbon atoms. The polarization functions on the hydrogen atoms consisted of a single set of p-type functions with a Gaussian exponent of 1.1. This final basis should provide an estimate for near Hartree-Fock energies and dipole moments for these molecules. For each of the molecules considered, we have evaluated the matrix of second derivatives in a Cartesian coordinate representation. This was done at each of the equilibrium geometries predicted by the respective basis sets. As usual the frequencies and the normal modes were determined by diagonalizing the mass-weighted force constant matrix.2J2 In order to extract the internal valence force constants we evaluated the appropriate B matrix3$’*for (5) GRADSCF is an ab initio gradient program system designed and written by A. Komornicki at Polyatomics Research, and supported on grants and contracts through NASA Ames Research Center. (6) W. J. Hehre, R. F. Stewart, and J. A. Pople, J . Chem. Phys., 51, 2657 (1969); M. D. Newton, W. A. Lathan, W. J. Hehre, and J. A. Pople, ibid., 52, 4064 (1970); L. Radom, W. A. Lathan, W. J. Hehre, and J. A. Pople, J . Am. Chem. Soc., 93, 5339 (1971). (7) R. Ditchfield, W. J. Hehre, and J. A. Pople, J . Chem. Phys., 54, 724 (1971). ( 8 ) C. E. Blom, P. J. Slingerland, and C. Altona, Mol. Phys., 31, 1359 (1976). (9) C. E. Blom and C. Altona, Mol. Phys., 31, 1377 (1976). (10) P. C. Hariharan and J. A. Pople, Mol. Phys., 27, 209 (1974). (11) T. H. Dunning, Jr., J . Chem. Phys., 55, 716 (1971). (12) E. B. Wilson, J. C. Decius, and R. C. Cross, “Molecular Vibrations”, McGraw-Hill, New York, 1955; S. Califano, ‘Vibrational States“. Wiley, London, 1976.
Saturated Three-Membered Rings
The Journal of Physical Chemistry, Vol. 87, No. 20, 1983 3849
TABLE I: Comparison of Calculated and Experimental Geometry of Cyclopropane ( D 3 h ) STO-3G
a
4-31G
6-31G**
exptl" H5
R(C-C) R(C-H)
1.502 1.081
Bondsb 1.502 1.072
1.497 1.076
1.512 1.083
a(H-C-H)
113.8
AngleC 113.7
113.7
114.0
Reference 1 5 .
In angstroms.
-
i'
I
In degrees.
TABLE 11: Comparison of Calculated and Experimental Geometries of Ethylene Oxide ( C 2 u ) STO-3G
a
4-31G
6-31G**
exptla
R(C,-0,) R(C,-C,) R(C2-H,)
1.433 1.482 1.088
Bondsb 1.459 1.461 1.069
1.399 1.452 1.078
1.436 1.472 1.082
o(C-0-C) a(H-C-H) P,(H-C-0) P,(H-C-C) Reference 1 6 .
62.3 114.4 116.6 119.5
AnglesC 60.1 115.6 114.5 119.8
62.5 115.2 115.4 119.8
61.7 116.7
In angstroms.
In degrees.
I
H8 Flgure 1. Coordinate system description and atom numbering used for cyclopropane. The valence angles a describe the CH, deformation while the four angles are used to construct the motions of twist, rock, and wag.
H5
TABLE I11 : Comparison of Calculated and Experimental Geometries of Ethylene lmine (C,) 6-31G**
exptl"
Bondsb 1.485 1.491 1.482 1.466 1.084 1.072 1.084 1.069 1.041 1.000
1.471 1.446 1.078 1.076 0.999
1.481 1.475 1.084 1.083 1.016
AnglesC 114.2 114.4 118.8 118.7 118.5 117.8 115.6 115.2 119.4 120.2 69.8 59.5
114.5 118.7 117.9 115.2 120.2 64.3
115.7
STO-3G
4-31G
' In angstroms.
117.8 119.4 61.0
In degrees.
H4 Figure 2. Coordinate system description and atom numbering used for ethylene oxide. The valence angles a describe the CH, deformation while the four angles are used to construct the motions of twist, rock, and wag.
H7
I
each molecule which in turn was used to transform the force constants from a Cartesian to an internal coordinate representation. This transformation serves to rigorously project out any translations and rotations from the vibrational modes and yields six true zero-frequency modes whose magnitude is less than 2 X cm-'. To facilitate a comparison with the experimentally derived force fields and the experimental vibrational assignments we transformed the force constants into a set of symmetry-adapted coordinates. Each of the angular coordinates has also been scaled by 1 A in order to give all force constants in units of mdyn/A. A comparison of both the force constants and the normal modes in this symmetry-adapted internal coordinate representation facilitates the assignment of the spectrum and allows us to compare the various types of motions among the three molecules.
Flgure 3. Coordinate system description and atom numbering used for ethylene imine. The motions of the CH, groups are described in the same manner as for cyclopropane and ethylene oxide. The angle y denotes the out-of-plane NH angle.
111. Molecular Geometries The structure of three-membered rings has attracted the attention of both theory and experiment. The unusual bonding in these systems manifests itself by geometrical parameters which show substantial deviation from the values observed in nonstrained systems. We have summarized the results of our optimized geometries in Tables 1-111, where we have compared the results from the three different basis sets with the currently accepted experimental values. Figures 1-3 depict the structures and in-
dicate the coordinates as well as the atom numbering used to define the geometry of each of these molecules. A summary of the final energies and dipole moments, for each basis set and geometry, is presented in Table IV. The final energies and dipole moments are evaluated with the triple- f-polarized basis at the polarized split-valence geometries. Each of these three compounds has been the object of a previous theoretical structural investigation. Thus, cyc l o p r ~ p a n e , ethylene ~J~ oxide,13and ethylene imine14 have
I
3850
The Journal of Physical Chemistty, Vol. 87, No. 20, 1983
TABLE 1V : Summary of Energies and Dipole Moments ( K 1 for C,H,, C,H40, and C,H,NH energy, a u STO-3G 4-31G E-31G** [5~3pldi 3SlPl
-115.666 -116.883 -117.069 -117.098
163 858 053 054
-150.928 -152.626 -152.874 -152.923
504 758 167 933
w,
C,H, STO-3G 4-31'2 6-31G** [5s3pld/3slp] exptl a
Reference 1 6 .
0.0 0.0 0.0 0.0 0.0
1.463 3.008 2.274 2.128
1.880'' Reference 18.
TABLE V : Definition of Internal Coordinates for C,H, - - (Dqh)
R I2
Qi
-131.399 -132.823 -133.048 -133.087
471 902 909 666
Qz
13
9 3
23
Qs
r24 r2,
Q Qi
i-37
Q,
r1ir
Qii
~'~'(8-1-9) ~~(4-2-51 ~r~(6-3-7)
9 4
'36
(1
Q" 3 ;"
D
C2H,0
Komornicki et al.
C,H4NH 1.824 2.250 1.919 1.822 1.890b
been optimized a t either the minimal or a split-valencepolarized level. It is well-known that polarization functions dramatically alter the stabilities of strained ring systems relative to their open-chain analogues. It is also known that polarization functions are required to provide a correct description of angle bends. Since we are primarily interested in the prediction of the vibrational spectra, our approach necessitates the location of equilibrium geometries a t each level of theory. Our calculations are intended as an ab initio prediction of the vibrational spectra at the predicted equilibrium geometry. In this way we may gain confidence in the predictive power of these methods for molecules whose structures are not well characterized experimentally. Although the geometrical structures are not the primary focus of this investigation, we would like to briefly evaluate our structural predictions. Experimental structures are available for cy~lopropane,'~ ethylene oxide,16and ethylene imine.",I8 If we proceed through this series from cyclopropane to ethylene imine and to ethylene oxide, we find that theory correctly predicts a shortening of the ring bond distances. We also observe that the C-X bond is shorter than the adjacent C-C bond. Both of these trends are consistent with the increase in electronegativity of the heteroatom (X). If we next examine the C-H bonds, we find very little variation in the three molecules. In ethylene imine we do find two nonequivalent C-H bonds as expected from the symmetry of this species. Our calculations also reproduce the increase in the valence angle betwen geminal C-H bonds. We also find that the CH2groups are slightly twisted with respect to the ring plane in ethylene imine, whereas the high symmetry of both cyclopropane and ethylene oxide mandates that the plane of the CH, groups be orthogonal to the ring plane. In ethylene imine we find a slight deviation from this ideal angle of 90". Finally, we would like to point out that each of our calculations at the polarized basis level reduces the ring size as well as the C-H bonds. This effect is most pronounced in the C-0 and the C-N bonds. This is consistent with the now known behavior of SCF wave functions in that inclusion of polarization functions tends to shorten bonds (13) W. A. Lathan, L. Radom, P. C. Hariharan, W. J. Hehre, and J. A. Pople, Top. Curr. Chem., 40, 1 (1973). (14) J. Catalan, A. Macias, 0. Mo., and M. Yanez,Mol. Phys., 34, 1429 5) R. J. Butcher and W. J. Jones, J. Mol. Spectrosc. 47, 64 (1973). 6) G. L. Cunningham, Jr., A. W. Boyd, R. J. Myers, W. D. Gwinn, W. I. Le Van, J . Chem. Phys., 19, 676 (1951). 7) B. Bak and S. Skaarup, J . Mol. Struct., 10, 385 (1971). 8) R. D. Johnson, R. J. Myers, and W. D. Gwinn, J . Chem. Phys., 425 (1953).
r,o
Ql2
Q;;
PiO(9-1-2) P 11(g-1-3) P 8-1-3
QZ3
Q,,
twist
I', = P, - P l 0 + P I , - 812 r, = 8 , - 4, + P , - P,
rock
A , = 8,
r3=Pr-P6f
37-13,
-
- P i , - 81, d,, 1 1 2 = P I - 8, - i33 + 8, '13 = 0,- P , - 0, + P ? SI, = P , + P,, - 81, -91,
wag
P 2 - 8 3 - 0 4
n.2=31
n3=P,+P,-5,-o, TABLE VI: Definition of Internal Coordinates for C 2 H 4 0 ('2") Qi
R 12
Qz
I3
R 23
Q iz
r24 f-2 5
Q I3 Q 14
Qb
r36
Qis
r37 al(4-2-5) 01 ,(6-3-7)
Q I6 Qi i
Q, Q,
PAS-2-11 P3( 5 - 2 4 )
Qii
Q, Q, Q, Q7
~ ~ ( 4 - 2 1- 1
Q io
P4(4-2-3) O s ( 7-3-1) P,( 6-3-1) p1(6-3-2) 8 &( 7-3- 2)
I', = P i - B , I', = d , - P , A I= PI - P, .I2= 8, - P,
twist
rock
+ 0 3 - P4 + PI -Pa - 13, + 0, -
+ P,
8,
n , = 4 ,+P,-P,-P, n . 2 = 8 , + 0, - 8,
wag
-
8 5
TABLE V I l : Definition of Internal Coordinates for C2H4NH(C,)
Q, Q,
R 12 R 13
9,
R?3
Q,
r24
Q, Q, Q7
rz 5 ru r3-
Q, Q, Q io
Tlh
0,(4-2-5) 01,(6-3-7) twist
~~(4-2-11 8,(5-2-1) P3(5-2-3) P4(4-2-3) 8,(6-3-1) 8,(7-3-1) 8,(7-3-2) P8(6-3-2) ~~(8-1-2) y 2 (8-1-3
Qii
Qi? Qi,
Qi, Qi,
Q 1, Q 17 Qi, Qi,
Q zc
+ d,
- d, 0,-d A l = P l - P 2 - P 3 + 8, '12= 3, - 8 , - 0, + 6% n.,=d,-P,-8,-J4 a,= 3 , 8, - 4, - 6, I', = 8 ,
-
8,
1.2=P,-?b+
rock
wag
-
relative to both the double-[ and experiment, and only inclusion of electron correlation serves to bring the calculated geometrical parameters into closer agreement with e~periment.'~ IV. Force Constants and Spectra The chemical and structural similarity of these molecules suggests that we examine the various types of internal coordinates which combine to form the normal modes. Due to their similarity, we would like to maintain a com(19) C. E Dykstra and H. F. Schaefer in 'The Chemistry of Ketenes and Allenes", S. Patai, Ed., Wiley-Interscience, 1978, Chapter 1
Saturated Three-Membered Rings
The Journal of Physical Chemistry, Vol. 87, No. 20, 1983 3851
TABLE VIII: Definition of D l h Symmetry Coordinates for C 3 H 6 ( D a h ) description
A,’ SI = Q, S, = Q8
S3
=
+ 9, + 9,
Q, + Q, + Q, + 9, + + Qio
Q, +
Qii +
Qiz
A,’
s,= n , - a,+ n,
A,”
S, =
r , - r, + r,
A,” S6= Q, - Q, Qa
+ Q, - Q, +
- 9 9
S 7 = A l + A 2 + A 3
TABLE IX: Definition of C,, Symmetry Coordinates for C2H40 ( c Z U ) description
C-C ring breathing in-phase CH, symmetric stretch CH, scissors
A, S I= S, = S, = S,=
CH, wag
J32
CH, twist in-phase CH, asymmetric stretch CH, rock
E’ C-C skeletal stretch out-of-phase CH, symmetric stretch CH, scissors CH, wag out-of-plane CH, asymmetric stretch CH, rock CH, twist
mon set of coordinates for all three molecules in order to facilitate comparison, to the extent that this is possible. The ring skeletons are described by the C-C or C-X stretching coordinates, where X is the heteroatom. The motions of the CH2 groups can be broken down into four distinct coordinates. These are the CH2 scissor bending and the CH2 twist, rock, and wag motions. Finally, there are the C-H and N-H stretching motions. For each of these molecules we have defined a set of internal coordinates which are listed in Tables V-VII. The atom numbering is shown in Figures 1-3. In all cases the bond stretching coordinates are denoted by R, the CH, bending coordinates by a,while the p coordinates define the H-C-C or H-C-X angles. Since the sum of the four 0 angles constitutes a redundancy, we have removed this redundancy by constructing only the three twist, rock, and wag coordinates for each CH2 group from the four p coordinates. A comparison of these three molecules is facilitated by recognizing that symmetry correlates the vibrational motions as we proceed from one molecule to another. Each nonredundant set of internal coordinates can be combined into two blocks of coordinates which will transform as the A’ and the A” irreducible representations within C, symmetry. Within C,, symmetry the A’ block will split further into the A, and B, blocks, while the A” block splits into the B1 and the A, irreducible representations. Finally, within the DShpoint group we can map the A, vibrational modes onto the A,’ and E’ vibrational modes, while the B1 modes map onto the A i and E’ modes. In a similar fashion the B2 and A, modes will map onto the A” and E” irreducible representations. Ethylene imine has 18 vibrational modes which within C, symmetry are divided into 10A’ and 8A“. Ethylene oxide belongs to the point group C2”and its 15 vibrational modes transform as 5A1, 3A2, 4B1, and 3Bz. Finally, cyclopropane can be described by a set of 21 nonredundant coordinates which combine into 14 fundamental vibrations. These vibrations transform as 3A,‘, 1A2/, lAl”, 2A2/1,4Ef, and 3E”. We have used the internal coordinates defined in Tables V-VI1 to construct a set of symmetry-adapted coordinates which are listed in Tables VIII-X. We first determined the B-matrix elements for the internal coordinates and then took the appropriate linear combinations.
Q , + 9,. Q,
+ Q, + Q, + Q, s,= n, + n, S , = Q, - Q, + Q, - Q, s,= r , + r , Q,
Q, + Q,
s,= A , f
A,
B,
s,=Qi-Q2
SI, = Q, + Q, Si, = Q, - Q,
- Q 6 - Q,
s,,= n, - n 2
A2
S I ,= Q,
-
Q,
s,,= r , - r ,
-
Q6 + Q,
S I , =A , - A ,
symmetric C-0 stretch C-C stretch symmetric CH stretch CH, bend CH, wag CH stretch CH, twist CH, rock asymmetric C-0 stretch CH stretch asymmetric CH, bend CH, wag CH stretch CH, twist CH, rock
TABLE X : Definition of Symmetry Coordinates for C,H,NH (C,)
.
description
A’ block S , = Q , + Q, S , = Q,
S , = Q 4 + Q, S , = Q, + 9, S , = Q, S , = 9, + Q,,
s,= r , + r, s,=A , + A , s,= n, + n,
Si, = Qi, + Q20 A” block S I ,= Q , - Q2 SI, = Q4 - Q, Si,
=
Qs - 9
7
S I , = Q, - Q,, SI, = r , - r 2 SI, = A , - A , S I , = n, - n, Si, = Qi, - Q 2 0
symmetric ring stretch C-C stretch symmetric C-H stretch (syn) symmetric C-H stretch (syn) N-H stretch CH, symmetric bend scissors symmetric CH, twist symmetric CH, rock symmetric CH, wag N-H bend asymmetric ring deformation asymmetric C-H stretch asymmetric C-H stretch CH, out-of-phase bend asymmetric twist asymmetric rock asymmetric wag N-H wag
Each Cartesian force constant matrix was transformed to this symmetry-adapted internal coordinate representation in order to facilitate our comparisons and analysis. The resulting vibrational spectra are presented in Tables XIXIII. We have included in the assignment the types of vibrations which best describe the motions of the atoms. As is readily seen, in some cases these motions are unique and easily interpreted, while in others they become complicated linear combinations of several types of atomic motions. In order to avoid ambiguity we have chosen the descriptions on the basis of the largest contributions to the internal coordinate displacement vectors in our most reliable set of calculations. A cursory examination of the vibrational spectra reveals that the minimal basis set frequencies are often too large by 20-30 % , while the split-valence and split-valence-Polarized values exceed the experimental frequencies by 10-15%. Our comparison suffers from the fact that the experimental values are the observed frequencies, while our calculated frequencies are pure harmonic values. Futhermore, it has now been established that ab initio SCF theory predicts diagonal force constants which are too large by about 10-20%, while the off-diagonal values are usually in much better agreement with experimentally derived value^.^^*^^ In order to facilitate comparison we chose to scale the diagonal force constants in the internal symmetry
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Komornicki et al.
The Journal of Physical Chemistry, Vol. 87,No. 20, 1983
TABLE XI:
Comparison of Calculated (Harmonic) and Experimental Frequencies (cm-' ) for C,H, ( D s h )
assign
description
A,' 'I '2
'3
A, '4
STO-3G
exptla
6-31G * *
4-31G
CH stretch CH, d e € ring stretch
3038 1479 1188
3673 1828 1445
3484 1722 1377
3322 1673 1285
3150 1575 1222
3302 1668 1297
3132 1571 1236
CH, wag
1070
1364
1295
1292
1226
1219
1157
CH, twist
1126
1380
1309
1278
1 2 12
1260
1196
CH stretch CH, rock
3102 8 54
3816 964
3620 914
3407 94 0
3232 89 2
3386 916
3212 869
CH stretch CH, def CH, wag ring def
3024 1438 1028 868
3652 1753 1393 1116
3463 1659 1307 1066
3307 1636 1229 94 8
3136 1549 1154 898
3288 1604 1188 958
3119 1517 1114 910
CH stretch CH, twist + rock twist + rock
3082 1188 739
3810 1431 892
3614 1372 830
3383 1347 846
3207 1291 788
3363 1322 799
3189 1270 74 1
A," A:' ' 6
'1
E' 'R ' 9
'
IO
'I1
E"
'
I,
' '
13 14
a
Reference 2 1 and 30.
TABLE XII: Comparison of Calculated (Harmonic) and Experimental Frequencies (cm-l ) for C,H,O (C," ) assign
description
A, V l
V I V 3 V4 '5
A2 ' 6
'7 '2
Bl *9 'IO 'I1
'12
B,
'
13
u I4 "IS
exptla
STO-3G
4-31G
6-31G**
CH stretch CH, d e € ring + CH, def CH, wag ring + CH, wag
3024 1498 1270 1120 877
3623 1832 1525 1421 1097
3433 1722 1451 1335 1049
3334 1686 1372 1260 89 2
3162 1585 1302 1189 844
3278 1694 1425 1303 99 1
3108 1591 1358 1221 94 5
CH stretch CH, rock - twist CH2 twist + rock
3065 1168 1020
3765 1351 1172
3573 1280 1113
3413 1313 1120
3235 1245 1063
3351 1285 1151
3177 1214 1096
CH stretch CH, def CH, wag ring + CH, wag
2978 1470 1159 822
3605 1760 1364 1227
3416 1664 1286 1166
3320 1658 1306 862
3150 1566 1237 80 6
3264 1640 1290 978
3096 1549 1223 914
CH stretch CH, rock + twist CH, rock - twist
3065 1147 808
3773 1307 94 5
3581 1252 884
3431 1236 909
3254 1183 852
3367 1286 885
3194 1233 823
a Reference 36.
TABLE XIlI:
Comparison of Calculated (Harmonic) and Experimental Frequencies (cm-' ) for C,H4NH (C,)
assign
STO-3G
4-31G
6-31G**
description
expt14
NH stretch CH, a-stretch CH, s-stretch CH, def NH bend; CH, rock, twist ring, NH bend CH, wag, twist, rock NH bend; CH, rock, wag ring + CH2 rock CH, rock, twist; NH bend
3346 3079 3015 1483 1210 1095 1089 997 856 772
3929 3797 3646 1810 1530 1484 1382 1156 1084 920
3728 3604 3456 1704 1467 1407 1301 1099 1026 855
3758 3408 3311 1680 1378 1333 1240 1065 929 839
3565 3231 3141 1584 1323 1258 1174 1014 874 767
3790 3369 3282 1675 1397 1367 1250 1083 968 847
3595 3195 3113 1577 1341 1296 1177 1029 9 15 781
CH, CH, CH, CH, CH,
3079 3003 1462 1268 1237 1131 903 81 7
3792 3630 1756 1525 1371 1354 1167 1047
3598 3440 1661 1452 1302 1280 1109 979
3391 3299 1656 1406 1300 1280 1027 930
3214 3130 1567 1338 1234 1215 960 87 3
3354 3271 1630 1393 1265 1248 1010 978
3179 3103 1542 1328 1203 1180 94 2 922
A' '1 '2
'3 4 ' ' 5 '6
'7 "R "Q
uIO
A" b'll
" '
I2
13
vl3 vIs '16
u17 vIX a
a-stretch s-stretch def rock, NH wag twist, rock, wag 'HZ wag CH, rock, twist; NH wag ring + CH, rock
References 40 and 41.
coordinate representation by a factor of 0.9. The resulting frequencies are presented as the second column for each of the calculated spectra. A scaling of the diagonal force
constants, especially for the larger basis sets, brings the calculated frequencies within 1-3% of the experimental values. The remaining errors can be attributed mainly to
Saturated Three-Membered Rings
the neglect of anharmonic terms in the treatment of the vibrational problem. Cyclopropane. The vibrational spectrum of cyclopropane has been extensively studied both theoretically and experimentally. There have appeared several normal-coordinate analyseszsz6 and ab initio force field s t u d i e ~ . Absolute ~ ~ ~ ~ ~band ~ intensity studies have also been r e p ~ r t e d . ~The ~ , ~high ~ symmetry of this molecule allows for the most nonambiguous vibrational assignment of any of the three molecules considered in this series. A comparison of the predicted spectra from the three levels of theory shows the same ordering for the high-frequency modes, namely, the C-H stretching and the CH2bending modes. We find the most serious discrepancies in the assignment of the vibrational spectra in the mid-frequency region of lloCt1300 cm-l where we find the wag, twist, and ring breathing modes. The experimental order is u3 (ring), and u5 (twist), and u4 (wag). The three basis sets predict a different order in this region of the spectrum. The split-valence basis predicts the order to be v4, v3, and us, while the split-valence-polarized basis predicts the experimentally derived order. The minimal basis set order appears to be almost random. For example, this level of theory predicts the symmetric ring breathing mode us to lie slightly above the degenerate rock and twist ( ~ 1 3 ) ,while both larger basis sets predict the ring breathing mode u3 to lie lower in frequency, while the experimental assignment places both of these bands a t the same frequency. These inconsistencies between the three basis sets point out the need for a good wave function to describe the structure of the ring which the split-valence-polarized basis should provide. An understanding of these effects can only be gained by comparing the potential parameters, the force constants, and not the observables themselves, the frequencies. This point has extensively been discussed by Pulay in a number of paper^.^,^^ In Table XIV we compare the three calculated force fields. Our comparison here is in terms of the symmetry-adapted coordinates defined in Table VIII. A cursory examination of these results reveals the reasons for the differences in the three predicted spectra. The diagonal force constants change by as much as 40% as we improve the basis set. However, we do find relatively consistent agreement between the split-valence and split-valence-polarized basis sets. In the Al' block we find that the minimal basis set predicts the sign of the interaction force constant F1,2 (CH stretch/CH, bend) to be negative, while the remaining basis sets predict a positive value. In the A2" block the split-valence-polarized calculations predict the F6,, interaction force constant (CH stretch/CH2 rock) to be about half of the split-valence value. For the degenerate representations the three basis sets predict a much more consistent description of the force constants. (20) S. J. Cyvin, Spectrochim. Acta, 16, 1022 (1960). (21) J. L. Duncan and G. R. Burns, J . Mol. Spectrosc., 30, 253 (1969). (22) R. A. R. Pearce and I. W. Levin, Spectrochim. Acta, Part A , 32, 1135 (1976). (23) L. Nemes, Acta Chim.Sci. Hung., 59, 75 (1969). (24) J. L. Duncan and D. C. McKean, J . Mol. Spectrosc., 27, 117 (1968). (25) T. Hirokawa, M. Hayashi, and H. Murata, J . Sci. Hiroshima Unio., Ser. A , 37, 271 (1973). (26) J. R. Robins, S. J. Daunt, and H. F. Shurvell, J. Raman Spectrosc., 5, 411 (1976). (27) M. Dupuis and J. Pacansky, J . Chem. Phys., 76, 2511 (1982). (28) T. P. Lewis and I. W. Levin, Theor. Chim. Acta, 19, 55 (1970). (29) S. Kondo, T. Nakanaga, and A. Saeki, Spectrochim. Acta, Part A , 35, 181 (1979). (30) I. W . Levin and R. A. R. Pearce, J . Chem. Phys., 69, 2196 (1978). (31) P. Pulay, G. Fogarasi, F. Pang, and J. E. Boggs, J . Am. Chem. Soc., 101, 2550 (1979).
The Journal of Physical Chemistry, Vol. 87, No. 20, 1983 3853
TABLE XIV: Harmonic Force Constants for C3H, ( D 3 h ) Fij, mdynla STO-3G 4-31G 6-31G**
A, F1,l
F1,2 3.1'
F,2 F2,3
F,3
A2
6.223 -0.130 -0.338 7.535 0.164 0.995
4.632 0.078 -0.373 6.222 0.147 0.877
4.833 0.093 -0.312 6.148 0.151 0.850
1.360
1.201
1.076
2.051
1.727
1.694
7.735 0.034 0.454
6.174 0.081 0.426
6.091 0.044 0.408
7.864 -0.160 0.447 -0.600 7.549 0.104 -0.063 0.984 -0.008 1.607
5.251 -0.185 0.399 -0.702 6.205 0.136 0.016 0.848 -0,010 1.388
5.391 -0.188 0.378 -0.662 6.138 0.142 0.016 0.820 -0.012 1.293
7.709 0.072 -0.079 0.462 -0.191 1.257
6.112 0.132 -0.077 0.427 -0.162 1.050
6.026 0.093 --0.080 0.397 -0.176 1.001
A7:,4
F,5,
erimentallv derived force field has had six interaction force constants constrained to be zero. Our calculations indicate that the magnitude of these force constants is comparable to some of those retained in the experimental analysis, and thus hardly justifies neglect. Of particular interest are the nonbonded CH stretch/C-C stretch interactions, and the bonded CH stretch/CH2 rock interactions, which we predict with opposite signs, and a magnitude of about 0.07 mdyn/A. Further analysis reveals that a number of our predicted interactions are nearly a factor of 2 too large, when compared with the experimentally derived force field, while some such as the C-C stretch/C-C stretch and the C-C stretch/C-H stretch bonded interactions are in excellent agreement. Duncan and Burns have used several relationships in the refinement of their force field which we examine through our calculations. They have shown that the interactions of C-H stretch/CH, deformation in the A' and the E' blocks should be equal, i.e., F2,3= Fg,lo.We find this to be nearly the case, since F2,3 = 0.151 mdyn/A, and F,,,, = 0.142 mdyn/A. The experimentally derived value however is only 0.085 mdyn/A. Likewise they have used the relationship that Fl,2= -2F8,g which involves the CH stretch/ring interactions. We find the sign relationship to hold, but suggest that the proportionality constant should be reversed to yield 0.5 and not 2.0. Our calculations yield Fl,2= 0.093, and FeV9 = -0.188. Another comparison can be made between F6,7 and F12,13 which involves the CH stretch/CH2 rock interaction force constants. Both of these have been constrained to a value of zero in the experimental refinement, yet our calculations predict values of 0.044 and 0.093 mdyn/A. Finally, another discrepancy which we have found involves the CH stretch/ CH2wag nonbonded interactions. Our calculations predict a value which differs from the experimental analysis by over a factor of 7. In light of these differences, and the fact that the Duncan and Burns force field is a result of several constraints, we would like to suggest a reexamination of the currently accepted force field for cyclopropane. Ethylene Oxide. The infrared spectrum of ethylene oxide has been the subject of a number of investigat i o n ~ . ~The ~ - most ~ ~ recent work of Nakanaga36has focused on the examination of the Coriolis interactions among the v5, u12, and ~ 1 modes 5 of C2H40. Using these data, this latest investigation has reinterpreted the spectrum and proposed several new assignments along with a revised force field. This is especially pertinent since a number of previous investigations have assigned the CH2 rock and twist motions to a variety of bands in the observed ~ p e c t r u m . ~Up ~ -until ~ ~ this most recent work, the
0.013 -0.010
-0.144
Units are mdyn/A. Values for t h e internal valence force constants result from symmetrized force field of Duncan and Burns.2'
(32) N. W. Cant and W. J. Armstead, Spectrochim Acta, Part A , 31, 839 (1975). (33) W. J. Potts, Spectrochim. Acta, 21, 511 (1965). (34) R. C. Lord and B. N o h , J. Chem. Phys., 24, 656 (1956). (35) J. E. Bertie and D. A. Othen, Can. J . Chem., 51, 1155 11973). (36) T. Nakanaga, J. Chem. Phys., 73, 5451 (1980). (37) J. M. Freeman and T. Henshall, Can J . Chem., 46, 2135 (1968). (38) N. Yoshimizu, C. Hirose, and S. Maeda, Bull. Chem. SOC.Jpn. 48, 3529 (1975). (39) T. Hirokawa, M. Hayashi, and H. Murata, J . Sci. Hiroshima Uniu., Ser. A , 37, 283 (1973). (40) Deleted in proof. See ref 35. (41) J. Le Brumant, C. R. Hebd. Seances Acad. Sci., Ser. E , 264,1107 (1967). (42) M. Spiekermann, D. Bougeard, and B. Schrader, J . Comput. Chem., 3, 354 (1982).
Saturated Three-Membered Rings
The Journal of Physical Chemistry. Vol. 87, No. 20, 1983
3855
assignment of Cant and A r m ~ t e a dwas ~ ~ considered the TABLE XVI: Harmonic Force Constants for C,H,O C,, most reliable. Fii, mdyn/A STO-3G 4-31G 6-31G** An examination of the frequencies in Table XI1 for ethylene oxide reveals some of the same problems which 4.102 7.334 5.658 we encountered in cyclopropane. As in all of our calcu7.990 6.069 6.574 lations, we consider the split-valence-polarized basis to be 7.378 6.291 6.091 1.023 0.910 0.902 the most reliable. In all cases the most consistent as1.771 1.647 1.664 signment is found in the high-frequency region, while the 0.038 -0.555 -0.223 most ambiguous assignment is found in the low- to mid0.093 0.134 0.219 frequency range. In the CH stretching region the calcu-0.133 -0.195 -0.192 lations reproduce well the experimental ordering. Ex0.424 0.579 0.544 perimental data3* indicate that v6 lies slightly above ~ 1 3 , 0.086 -0.046 0.106 -0.365 -0.363 -0.329 while the experimentally refined force field of N a k a ~ ~ a g a ~ ~ -0.373 -0.435 -0.370 reverses this ordering. Our calculations clearly agree with 0.117 0.141 0.136 this revised order. Likewise our calculations reproduce the 0.082 -0.001 0.009 experimental order for v2, v3, and vl0 which involve the CH2 -0.138 -0.193 -0.190 bend and the wag. The motions which include the twist, 8.687 4.524 rock, and wag coordinates present the most serious prob5.683 7.390 6.281 6.086 lems. Our calculations are unable to reproduce the cur1.019 0.891 0.887 rently accepted order for the modes v4, v7, vll, and ~ 1 4 .All 1.545 1.458 1.458 of these involve the CH2 wag, rock, and twist. Our best 0.513 0.290 0.461 calculations still predict v4 to lie slightly higher than v7, -0.576 -0.511 -0.565 while the experimental assignment predicts the reverse 0.544 0.736 0.813 0.132 0.073 0.139 ordering. At the polarized basis set level we find that a 0.031 0.011 0.030 scaling of the diagonal force constants shifts modes v4 and -0.124 -0.194 -0.194 vll below the value found for v4. We see an improved situation for the order of the modes v5, us, v12, and vi5 where 7.555 6.239 6.006 our best calculations do reproduce the experimentally 1.246 1.049 1.089 derived order, while at the split-valence basis level we find 0.544 0.509 0.524 -0.003 -0.056 -0.063 the symmetric ring breathing modes v5 and v12 to lie below 0.077 0.085 0.072 the B2 symmetry mode ~ 1 5 . This we find to be incom0.155 0.133 0.176 patible with the currently accepted experimental assignment, as well as with the more reliable polarized basis set 7.535 6.201 5.969 results. We thus would suggest that the correct ordering 1.708 1.463 1.512 of these bands in this narrow frequency range requires the 0.541 0.504 0.515 -0.044 -0.090 -0.090 use of polarization functions in the theoretical description. 0.097 0.110 0.088 Since all of these modes are within a very narrow frequency 0.005 0.004 0.048 range, only a knowledge of the normal modes and their symmetry can serve to unambiguously support their asference between the twist/rock interactions found for FTB8 signment. The narrow frequency range and the fact that and F14,15. some of these modes may involve large-amplitude vibraExperimentally the most anomalous assignment has tions may require the introduction of anharmonic corinvolved the twist and rock motions identified and vl, us, rections. These modes however cannot couple directly to and ~ 1 4 ~, 1 in 5 the A, and B2 blocks. The original assigneach other, since they are each of a different symmetry. ment by P ~ t t was s ~ used ~ by Cant and A r m ~ t e a d who ,~~ Our spectra are derived from the force constants which also proposed a refined force field. Their force field, the we present in Table XVI. Since in most cases the minimal most complete a t the time, suffered from the same defibasis set results are uniformly 20-30% too high, we will ciencies which we have discussed above for cyclopropane, confine our discussion to just the split-valence and ponamely, the neglect of a number of important interaction larized basis set results. In the A, block we find that these force constants. In particular all of the off-diagonal terms latter two basis sets predict diagonal force constants which in the A2 and B2 blocks were neglected. This approxiare quite close to each other. A striking feature here is the mation is undoubtedly responsible for the uncertain aslow value of F1,l, the symmetric C-0 stretching force signment of the v7 band. Their refined force field suggested constant, which the split-valence basis set predicts to be that the rocks have a higher frequency than the twists, much lower. The interaction force constants show the while the force field of Nakanaga% suggests that the twists largest deviation from one basis set to another. Most are higher in frequency than the rocks. If we use the noticeable are the ring C-0 interactions with the C-C and symmetry correlation described at the begining of this the C-H bonds involving F1,2 and F1,3, respectively. The section, along with the known assignments for cyclosplit-valence basis predicts F1,2to be small and positive, propane, we would also place the twists above the rocks, while the polarized basis predicts this interaction to be in agreement with the most recent experimental assignsignificantly larger and negative. For F1,3, although the ment. As in cyclopropane, we would propose our best force sign remains the same, the larger basis set predicts a value field as a basis for further experimental refinements. For of over twice the split-valence result. We would tend to both v7, us, as well as vI4, ~ 1 5 we , are unable to label these attribute these discrepancies to the inadequacy of the modes as pure rocks or twists, since our force field predicts smaller basis set in correctly describing the bonding endisplacement coordinates which show an almost equal vironment of this strained ring system. In the B1block admixture of both of these types of motions. It would have the most obvious difference again involves the asymmetric been highly desirable and informative to compare our results with the experimentally derived force field used by C-0 stretch. We find the most consistent set of force Nakanaga; however, his coordinates do not lend themselves constants in the B2 and A2 blocks. Of interest is the dif-
3856
The Journal of Physical Chemistry, Vol. 87, No. 20, 1983
easily to such a comparison. Ethylene Imine. The spectrum of ethylene imine has been the least studied of any of the three molecules considered in this study. T o the best of our knowledge no force field, either experimental or theoretical, is currently available. Furthermore, the low symmetry of this species has made the assignment of the bands extremely difficult. In the early 1960s P ~ t t had s ~ recorded ~ the spectra in the vapor phase and his assignment is currently accepted. A few years later Mitchel et al.43had reexamined the spectrum and proposed a revised assignment. As in cyclopropane and ethylene oxide the high-frequency modes are the most unambiguous to assign. These involve the NH stretch, the four CH stretching modes, as well as the two CH2 deformations. This, however, leaves 11 modes to be assigned. The experimental assignment in the above two investigations labels the modes as pure twists, wags, or ring deformations. The low symmetry of this species, by definition, allows all of these motions to contribute to any of the normal modes. In order to assign the 11 lower frequency modes we first decided to look at the 5 modes v8, vg, vl0 and v17, v18. Our calculations reproduce the experimental order at the split-valence level. At the polarized basis set level we do predict a reversal of the order between v g and v18. These two modes lie so close in frequency that a slight change in the interaction force constants in either symmetry block could shift the position of these modes sufficiently to reverse this order. Our calculations do identify mode ~ 1 as 7 primarily a rock, in combination with the NH wag, with a substantial component of the CH2 twist, while v18 is primarily the ring and rock. This is a reversal of the assignment by P ~ t t and s ~ in~ complete disagreement with the assignment proposed by Mitchell et al.43 This latter work places the antisymmetric ring deformation as v14, while we assign v14 to be a combination of the rock and NH wag. The remaining six low-frequency modes involve v5, v6, v7 in the A' block, and the modes ~ 1 4 ~, 1 5 and , v16 in the A" block. Both experiment and theory place these modes in a relatively narrow frequency range of less than 200 cm-'. Furthermore, there is no agreement regarding the order of these modes between our theoretical calculations and the current experimental assignment. Experimentally the highest frequency mode in this set is assigned to ~ 1 4 which , is either the CH2 twist, as assigned by Potts, or the ring deformation, as assigned by Mitchell. We however identify this mode as the rock with NH wag, as mentioned above. A t the polarized basis set level our calculations predict v5 to lie only a few cm-l above ~ 1 4 . The third mode in this set we assign to v6 which we describe as the symmetric ring breathing mode with a strong component of the NH bend. Our calculations suggest that Mitchel et al. assigned the band at 1268 cm-' erroneously to the A" block as the ring mode, while it should be placed as v6 in the A' block. Our two best calculations do predict v16 and v7 to be the lowest frequencies of the six modes considered in this range, while the experimental assignment places v6 and v7 to be the lowest. Although no experimental force field is available, we would like to suggest our calculated force constants for ethylene imine as a suitable starting point of the further experimental refinement. A comparison of the diagonal force constants for the three calculated force fields is presented in Table XVII. As in our discussion of the previous two molecules, we find the most consistent set (43) R. W. Mitchell, J. C. Burr, Jr., and J. A. Merritt, Spectrochim. Acta, Part A , 23, 195 (1967).
Komornicki et ai.
TABLE XVII: Comparison of Diagonal Force Constants for C2H,NH
Fit, mdy4.4
STO-3G
4-31G
6-31G**
6.628 7.652 7.544 7.569 8.531 1.005 1.252 0.514 1.676 0.739
4.584 5.442 6.118 6.255 7.816 0.893 1.111 0.486 1.534 0.506
5.263 5.910 6.024 6.098 7.949 0.876 1.081 0.484 1.474 0.538
7.933 7.539 7.565 1.000 1.743 0.510 1.478 2.075
5.089 6.093 6.233 0.876 1.514 0.483 0.377 1.406
5.600 6.001 6.079 0.860 1.514 0.473 1.303 1.420
TABLE XVIII: Comparison of Diagonal Force Constants (mdynl.4 ) for C,q-Symmetry6-31G** description
C3H,
C,H,O
C,H,NH
5.205 5.019 6.107 6.107 0.840 1.001 0.404 1.293
6.574 5.658 6.048 6.048 0.902 1.089 0.524 1.664
5.910 5.263 6.024 6.098 0.876 1.081 0.484 1.474
6.082 6.082 5.390 0.820 1.463 0.398 1.148
6.028 6.028 5.683 0.887 1.512 0.514 1.458
6.001 6.079 5.600 0.860 1.514 0.473
A'
c-c C-X + C-X CH + CH CH + CH
CH, def twist rock wag A' CH-CH CH-CH1
c-x - c-x CH, def twist rock wag
1.303
of force constants between the split-valence and split-valence-polarized calculations. The force constants do show a considerable amount of similarity if we choose to compare them with those calculated for cyclopropane and ethylene oxide. We offer such a comparison of the diagonal force constants in Table XVIII, where we find all of the unique motions to have diagonal force constants which appear to be very transferable from one molecule to another. Here we have transformed the Cartesian force constants of all three molecules to a common C, symmetry coordinate representation. V. Correlation among the Vibrational Modes In the preceding section we have examined the spectra of the three molecules and have compared our derived force fields with those available from experimental data. It is one of the tenets of molecular vibrational spectroscopy that force constants for various functional groups are roughly transferable from one molecule to another. In fact, the similarity of functional groups is the basis of much of chemistry. We may now ask a related question, namely, whether the normal modes are also transferable. A t the beginning of section IV we stated the formal theoretical symmetry relationships which exist among the three molecules. Our approach here is to see if the formal symmetry mappings and the relative order of the modes holds for these three isoelectronic molecules. All three ring systems have six types of coordinate motions in common.
The Journal of Physical Chemistty, Vol. 87,No. 20, 1983 3857
Saturated Three-Membered Rings SYMMETRY CORRELATION OF NORMAL MODES Type
D3h(C3H6)
CZV(C2H40)
CH-stretch
CH -deformation 2
CH2 twist
CH2 rock
CH
2
rock
CH2 wag
Ring
Figure 4. Formal symmetry mappings for the seven types of nuclear motions present in cyclopropane, ethylene oxMe, and ethylene imine.
We label these as the CH stretch, CH2 deformation, CH2 twist, rock, and wag, and finally the ring motions. These six types of motions are schematically depicted in Figure 4, along with their symmetry mapping. The lowest symmetry molecule, ethylene imine, is unique in that the NH group motions can mix with any of the other six coordinates and thus forms a strong perturbation in this molecular system. The first type of motion which we consider involves the CH stretch motions. Due to their high frequency these modes are nearly pure in the sense that over 99% of these modes arise out of the CH stretch internal coordinates. The experimental order for cyclopropane is vg (Ai'), v12 (E"), v1 (Al', and vs (E'). This order is maintained for all three molecules. There are however two small discrepancies which we have already alluded to in our previous discussions. These involve the order of the two highest modes for both ethylene oxide and ethylene imine. If the symmetry correlation were strictly observed, then the modes of A2 symmetry should be higher in C2H40 as should the A" mode in C2H4NH. The deformations of the CH2 group are the next highest in frequency. Here we find in all three molecules that the higher symmetry band lies a t the higher frequency, in contrast to the trend observed in modes which involve the CH stretch. Using symmetry considerations alone, these modes could mix with both the CH2 wag as well as the ring motions in cyclopropane and in ethylene oxide, and with all of the remaining internal coordinates in ethylene imine. Yet these modes remain relatively pure with contributions over 95% from the CH2 deformation internal coordinates. The remaining types of motions do mix in very strongly as we decrease the symmetry of the molecule. If we examine our assigned spectra, we find that the twists and rocks are combined together even for the highest symmetry
species. We do find that the order predicted for cyclopropane ~ 1 (E"), 3 v5 (AT), v7 (Ai'), and ~ 1 (E") 4 does also hold for ethylene oxide where we find v7 (A2), ~ 1 (BJ, 4 vs (A2),and ~ 1 (B2). 5 As our assignment shows, it is much harder to place such unambiguous labels for the modes of ethylene imine. However, we can extend this mapping to ethylene imine and assign as primarily twists and rocks the modes ~ 1 (A"), 5 v5 (A'), v17 (A"), and vl0 (A'). We do stress that even for ethylene oxide the rocks and twists do mix in strongly with each other, each contributing coefficients of 0.5 or greater to the above-mentioned modes, while in ethylene imine these modes are also contaminated and perturbed by the motions of the NH group. A similar situation obtains for the motions which involve the wag and ring motions. In cyclopropane we find relatively little mixing between these two motions, and the order predicted is v3 (Al'), v4 ( A i ) , vl0 (E'), and vll (E'). It is pleasing to note that these modes map onto v3 (Al), vll (BJ, v4 (Al), v5 (Al), and vI2 (B,) within Czusymmetry for ethylene oxide. As we have discussed in the treatment of the individual spectra, it is difficult to distinguish uniquely between wags and ring motions for modes v3, v5, and v12. As with the twists and rocks we are now better able to assign the order of the ethylene imine spectrum. If we assume that the mode at 1268 cm-' has been experimentally incorrectly assigned, then our symmetry correlation arguments predict the wag and ring motions to be 1268 (A'), 1131 (A"), 1095 (A'), 997 (A'), 856 (A'), and 807 (A") cm-'. As with the twists and rocks, the displacement vectors contain large contributions from the NH group bending and wag motions.
VI. Conclusion Ab initio calculations have had widespread success in predicting interaction force constants, and in the refinement of experimental force fields. We have predicted and examined the spectra of three isoelectronic molecules which are also related by symmetry. Our calculations have pointed out the similarities and differences in the spectra. On the basis of a critical examination of the force fields we offer sufficient cause for a reexamination of the currently accepted force fields, especially in cases where the experimental refinement is the result of a constrained variation. No attempt has been made to fit the calculated force constants to the experimentally observed spectrum. We have calculated spectra which result from a scaling (by 0.9) of the diagonal force constants in order to compensate for the lack of correlation in the SCF wave function. Our analysis has also shown that force constants for individual types of motions are roughly transferable. Finally, an examination of the displacement vectors in an internal symmetry coordinate representation has shown that a unique label, such as twist, rock, or wag, is an oversimplified description and that the normal modes are indeed best described as combinations of these types of motions in the absence of symmetry constraints. Acknowledgment. We thank Professor David Dixon for a critical reading of this manuscript. This work was supported by NASA contract NAS2-10559 and by NASA grant NCC2-154. Registry No. Cyclopropane,75-19-4;ethylene oxide, 75-21-8; ethylene imine, 151-56-4.
