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An ab initio calculation of the rotation and internal-rotation energy levels of the ethyl radical
2 Janaury 1998
Chemical Physics Letters 282 Ž1998. 49–53
An ab initio calculation of the rotation and internal-rotation energy levels of the ethyl radical Allan L.L. East, P.R. Bunker Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ontario, Canada K1A OR6 Received 29 September 1997
Abstract By ab initio calculation at ten values of the internal-rotation angle we have determined the minimum energy path for internal-rotation in the ethyl radical C 2 H 5. Bond-length and bond-angle relaxation are allowed via full optimization, using unrestricted second-order Møller–Plesset perturbation theory, for each choice of torsional dihedral angle. We find that the only significant relaxation is a wagging Žor inversion. of the CH 2 group through "98 about a planar C–CH 2 geometry as the internal-rotation proceeds. The torsional potential can be accurately fitted to a V6 function and the barrier to internal-rotation is calculated to be 26 cmy1. Using our general rotation–contortion Hamiltonian and computer program we have calculated the rotation–torsion energy levels of the molecule using this minimum-energy path. Brief comparison is made to recent experimental measurements of the rotational fine structure of the n 9 s 1 fundamental. q 1998 Elsevier Science B.V.
1. Introduction In connection with our work on the CHq 5 molecular ion w1x we have developed a general rotation– contortion Hamiltonian and computer program that can be used to calculate the rotation–contortion energy levels of any molecule that has one large amplitude Žcontortional. degree of freedom w2x. The most significant aspect of this Hamiltonian is its ability to incorporate a parametrized geometrical path of any desired flexibility, hence avoiding any approximations or perturbation treatment for the large amplitude mode. In the present paper we use the program to calculate the rotation and internal-rotation energy levels of the ethyl radical C 2 H 5 . We use ab initio calculations to determine the optimized structure of the molecule, and the potential energy, along the minimum energy path for internal-rotation.
Our work has been inspired by the observation and analysis of a high resolution spectrum of a fundamental band of the ethyl radical in the gas phase by Sears et al. w3x, having a band origin at 528.1 cmy1 . This fundamental Ž n 9 . corresponds to the out-of-plane or umbrella inversion motion at the CH 2 radical center Žthe CH 2 wagging motion w4,5x.. In this spectrum there is clear evidence for almost free internal-rotation, and from the analysis a torsional barrier height of about 20 cmy1 was determined in Ref. w3x. The approximate model Hamiltonian used in Ref. w3x was unable to precisely fit the data, and we hope that our calculations will lead to a precise simulation of this spectrum. Previous ab initio calculations at correlated levels of theory w6,7x have been concerned with determining the equilibrium structure, harmonic frequencies, and internal-rotation barrier height for use in the
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 1 7 6 - 7
50
A.L.L. East, P.R. Bunkerr Chemical Physics Letters 282 (1998) 49–53
study of hyperfine structure w6x and rate constants for the H q C 2 H 4 | C 2 H 5 reaction w7x. There has been no previous calculation of the rotation and internalrotation energy levels of the ethyl radical based on an ab initio potential surface.
2. The internal-rotation minimum energy path Our ab initio calculations were performed at the unrestricted second-order Møller–Plesset ŽUMP2. level of theory w8x with the cc-pVTZ basis set w9x, using the GAUSSIAN94 program package w10x. Initially, geometry optimizations and harmonic frequencies were determined for the internal-rotation minimum and maximum conformations Žsee Fig. 1.. The minimum energy conformation is traditionally called ‘‘staggered’’ Žsee Pacansky and Dupuis w11x. but has unfortunately been called ‘‘eclipsed’’ by some authors w3,6x; at this conformation the bisector of the CH 2 group is in the same plane as one of the CH bonds of the CH 3 group and it is tilted towards this CH bond out of collinearity with the CC bond by 9.28 Žthe out-of-plane wag angle.. The saddle point between the successive internal-rotation minima has the plane of the CH 2 group in the same plane as one
Fig. 1. The structure of the ethyl radical at equilibrium, and at the saddle point for the internal-rotation, including the molecule fixed Ž xyz . axis system.
of the CH bonds in the CH 3 group, and the bisector of the CH 2 group is tilted away from that CH bond out of collinearity with the CC bond by 0.58 Žthe in-plane rock angle.; we call this conformation the ‘‘eclipsed’’ conformation. In 1991, Suter and Ha w6x published UMP2 eclipsed and staggered structures and harmonic frequencies of C 2 H 5 obtained with the smaller 6-31GŽd, p. basis set. Our new results are but little different. The cc-pVTZ bond lengths are 0.1% smaller, and all of the harmonic frequencies are 1–2% smaller except for the umbrella mode, which matches the 6-31GŽd, p. result Ž470 cmy1 . at the minimum conformation, and is 50 cmy1 higher than the 6-31GŽd, p. result Ž390 cmy1 . at the maximum conformation. This is of some relevance to the separability of the internal-rotation mode from the umbrella mode, as the difference in harmonic umbrella frequency between eclipsed and staggered conformations is now only 29 cmy1 Ž470 cmy1 at the minimum and 441 cmy1 at the internal-rotation maximum., reflecting a smaller amount of coupling than would previously have been thought. Hase and coworkers w7x have more recently determined geometries and harmonic frequencies for the minimum at six different levels of ab initio theory. Again their results are similar to ours; the only significant improvement is a reduction of the out-ofplane wag angle by 1.08 at equilibrium with improved Žpost-MP2. correlation. For the determination of the minimum energy path ŽMEP. for internal rotation, we performed full geometry optimizations at the UMP2rcc-pVTZ level of theory at eight more values of the dihedral angle H 7 C 2 C 1 H 3 Ždenoted t 73 , measured in a right-handed sense about C 2 ™ C 1; see Fig. 1.. The torsional angle t is defined as the average of all six dihedral angles between the CH bonds of the CH 3 and CH 2 rotors. At each calculated point the values of the dihedral angle t 73 , of the angle t , and of the resulting potential energy V, are given in Table 1. We can fit this potential using a simple V6 function of t , and the values of this fitted function are also given in Table 1. This fitted function has an internal-rotation barrier height of 36 cmy1 , which we can correct with the zero-point vibrational energies ŽZPVE. of the other 3 N y 7 s 14 vibrational modes Žtaken as half the sum of the UMP2rcc-pVTZ harmonic frequen-
A.L.L. East, P.R. Bunkerr Chemical Physics Letters 282 (1998) 49–53 Table 1 Calculated values of the H 7 C 2 C 1 H 3 dihedral angle t 73 , internalrotation parameter t and the electronic energy Velec at the ten optimized points, and the fitted value of Velec obtained using a V6 cosine potentiala
t 73
t
Velec
fitted Velec
0.00 5.00 b 10.00 b 15.00 b 20.00 b 25.00 b 30.00 b 34.86 35.00 b 40.00 b
0.00 c 4.16 8.31 12.52 16.77 21.11 25.57 30.00 c 30.25 34.97
36.4 34.7 30.1 23.0 15.1 7.5 2.0 0.0 0.0 2.6
36.5 34.8 30.0 23.0 14.9 7.4 1.9 0.0 0.0 2.4
a b c
minor effects being methyl torsional defects Žy0.72 sin 2t in t 73 . and CH 2 rocking Ž0.58 cos 3t in a C 1 C 2 H t .. This accords with the theoretical discussion of the problem given by Sørensen w12x. The structure relaxation can be rationalized in terms of the need to minimize non-bonded H–H interactions. There is practically no methyl tilting Ž0.03 cos 2t in a C 2 C 1 H 3 ., in accord with the results of the fitting to the data w3x.
3. The rotation internal-rotation energy levels
Angles in degrees, energies in cmy1 . This value frozen during geometry optimization. This value determined by symmetry.
cies., obtaining an effective barrier height of 26 cmy1 . This matches exactly the previous results of Suter and Ha with the smaller basis set w6x, although their electronic barrier height and ZPVE correction were almost double ours in size. The t-dependent values of the various other internal coordinates were accurately fitted by the follow˚ ing analytical functions Žwhere bond lengths are in A and bond angles are in degrees.: rC 1 C 2 s 1.4864 q 0.0001 cos 6t ,
Ž 1.
rC 1 H 7 s 1.0768 q 0.0006 cos 3t y 0.0002 cos 6t ,
The internal-coordinate functions of Eqs. Ž1. – Ž6. were first transformed analytically to Cartesian coordinate functions, employing molecule-fixed xyz-axes. These axes were chosen such that the origin was at the centre of mass, the z-axis parallel to the C–C bond axis, and the xz-plane parallel to a line passing through H 6 and H 7 Žsee Fig. 1.. From the Cartesian coordinate functions we can determine the elements of the 4 = 4 extended moment of inertia matrix at any given value of the torsional coordinate t Žsee Ref. w2x.. Inverting this t-dependent matrix gives the t-dependent m matrix elements that we need for the kinetic energy part of the exact rotation–contortion Hamiltonian: Hˆ s Hˆ rot q Hˆ rot ,t q Hˆ t ,
Ž 7.
where
Ž 2.
xyz xyz
Hˆ rot s Ž 1r2 . Ý
rC 1 H 3 s 1.0906 y 0.0041 cos 2t q 0.0003 cos 4t ,
Ý mab Jˆa Jˆb
a
Ž 3. a C 1 C 2 H 7 s 120.89 q 0.58 cos 3t q 0.14 cos 6t ,
Ž 4.
a C 2 C 1 H 3 s 111.63 q 0.03 cos 2t q 0.06 cos 4t ,
Ž 5.
Ž 8.
b
xyz
Hˆ rot ,t s Ž 1r2 . Ý Jˆt , mat Jˆa a xyz
q Ž 1r2 . Ý mat Jˆa Jˆt q Jˆt Jˆa ,
and
t 73 s t y 0.72 sin 2t q 5.45 sin3t .
51
a
Ž 6.
Other internal coordinates are obtained by symmetry, eg. rC 1 H 4Žt . s rC 1 H 3Žt q 2 pr3.. These equations define the minimum energy path we use in the next section. The t-dependence of the bond lengths is almost negligible. The most significant relaxation effect is the CH 2 wag Ž5.45 sin 3t in t 73 ., with
ž
/
Ž 9.
and Hˆ t s Ž 1r2 . mtt Jˆt2 q Ž 1r2 . Jˆt , mtt Jˆt q Ž 1r2 . < m < 1r4 Jˆt , mtt < m
Ž 10 .
A.L.L. East, P.R. Bunkerr Chemical Physics Letters 282 (1998) 49–53
52
In Eqs. Ž8. to Ž10. Jˆx , Jˆy and Jˆz are the components of the total angular momentum operator along molecule-fixed axes, Jˆt is the torsional momentum Table 2 The rotation–torsion energy levels Žin cmy1 . and zero-point energy ŽZPVE. obtained for J s 0, 1, and 2 J
K
Ki
Symmetry
2
2
3
A1 Y A2 Y A2 Y A1 X E X E Y E Y E Y A1 Y A2 X A1 X A2 X A2 X A1 Y E Y E X E X E Y A2 Y A Y A1 Y A2 X E Y E X A2 X A2 X A1 X A1 X A2 Y E Y E X E X E Y A1 Y A2 Y A1 Y A2 X E Y E X A2 Y A2 Y A1 X E Y E X A1 ZPVE
2 1 0 2
1
3
2 1 0 2
1
0
3
1
2 1 0 3
2 1 0 1
0
0
3
0
2 1 0 3 2 1 0
Y
Energy 282.892 57.894 282.892 57.894 173.470 22.776 91.293 16.645 39.522 39.522 200.549 87.508 200.568 87.490 108.168 33.813 46.365 8.998 13.268 13.102 141.732 128.749 62.372 18.910 4.404 197.618 84.566 197.625 84.560 105.232 30.877 43.428 6.061 10.277 10.221 138.796 125.812 59.435 15.972 1.468 137.328 124.344 57.967 14.504 0.000 12.973
Table 3 The wavenumbers Žin cmy1 . for selected n 9 s1 transitions in C2 H5 J
Ki
K upper
K lower
ga
Observedb
Calculated
1 2 3 4 5 2 3 4 5 4 5 4 4 5 5
0 0 0 0 0 1 1 1 1 y2 y2 3 3 3 3
1 1 1 1 1 2 2 2 2 y3 y3 4 4 4 4
0 0 0 0 0 1 1 1 1 y2 y2 3 3 3 3
0 0 0 0 0
536.829 536.783 536.702 536.593 536.460 535.350 535.354 535.333 535.309 534.005 533.993 532.819 532.819 532.808 532.808
536.853 536.798 536.716 536.607 536.473 535.747 535.740 535.722 535.687 534.610 534.596 533.487 533.487 533.481 533.481
0 1 0 1
a
This quantum number is for transitions involving K i s 0, 3, 6, 9, etc. b Ref. w3x.
operator, and < m < is the determinant of the m matrix. For the potential function we take V Ž t . s Ž V6r2 . Ž 1 q cos 6t . ; V6 s 26 cmy1 .
Ž 11 .
This Hamiltonian is solved by a combination of numerical integration and matrix diagonalization as described in Ref. w2x. In making this calculation of the rotation and internal-rotation energies we found that satisfactory convergence in the matrix diagonalization stage could be achieved with a basis set that included internal-rotation functions up to those having K i s 12, where K i is the internal rotation quantum number. The energies, and symmetries in the G12 molecular symmetry group, are given in Table 2. We also converted our Table 2 results to n 9 s 1 rovibrational transition wavenumbers by calculating the appropriate D K s 1 differences and adding the experimental band origin value of 528.1 cmy1 , and in Table 3 compare some of these results to those assigned by Sears et al. w3x.
A.L.L. East, P.R. Bunkerr Chemical Physics Letters 282 (1998) 49–53
53
4. Discussion
References
It is clear that the internal-rotation motion in the ethyl radical is almost free with a barrier of about 25 cmy1 , and that a rigid coaxial C 3v –C 2v internal-rotation model neglects primarily only the out-of-plane CH 2 wagging relaxation. We do not achieve a close fitting of the data with this purely ab initio calculation. The next stage is to attempt a least squares fitting of the data, in which we adjust the minimum energy path and internal-rotation potential parameters, both in the ground state and in the wagging fundamental. Centrifugal distortion will almost certainly also have to be included. Work along these lines will be initiated, in collaboration with the authors of Ref. w3x, in the hope of precisely understanding their spectrum and what is required to reproduce it.
w1x A.L.L. East, M. Kolbuszewski, P.R. Bunker, J. Phys. Chem. Ž1997., in press. w2x A.L.L. East, P.R. Bunker, J. Mol. Spectrosc. 183 Ž1997. 157. w3x T.J. Sears, P.M. Johnson, P. Jin, S. Oatis, J. Chem. Phys. 101 Ž1996. 781. w4x J.M. Hollas, Modern Spectroscopy, Wiley, Chichester, UK, 1987. w5x This mode was referred to as the CH 2 rocking motion in Refs. w3,12x. w6x H.U. Suter, T.-K. Ha, Chem. Phys. 154 Ž1991. 227. w7x W.L. Hase, H.B. Schlegel, V. Balbyshev, M. Page, J. Phys. Chem. 100 Ž1996. 5354. w8x C. Møller, M.S. Plesset, Phys. Rev. 46 Ž1934. 618. w9x T.H. Dunning Jr., J. Chem. Phys. 90 Ž1989. 1007. w10x M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. DeFrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez, J.A. Pople, GAUSSIAN 94, Gaussian Inc., Pittsburgh PA, 1995. w11x J. Pacansky, M. Dupuis, J. Chem. Phys. 68 Ž1978. 4276. w12x G.O. Sørensen, J. Chem. Phys. 105 Ž1996. 3942.
Acknowledgements We thank Dr. T.J. Sears and Dr. G.O. Sørensen for their encouragement and advice.
Chemical Physics Letters 282 Ž1998. 49–53
An ab initio calculation of the rotation and internal-rotation energy levels of the ethyl radical Allan L.L. East, P.R. Bunker Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ontario, Canada K1A OR6 Received 29 September 1997
Abstract By ab initio calculation at ten values of the internal-rotation angle we have determined the minimum energy path for internal-rotation in the ethyl radical C 2 H 5. Bond-length and bond-angle relaxation are allowed via full optimization, using unrestricted second-order Møller–Plesset perturbation theory, for each choice of torsional dihedral angle. We find that the only significant relaxation is a wagging Žor inversion. of the CH 2 group through "98 about a planar C–CH 2 geometry as the internal-rotation proceeds. The torsional potential can be accurately fitted to a V6 function and the barrier to internal-rotation is calculated to be 26 cmy1. Using our general rotation–contortion Hamiltonian and computer program we have calculated the rotation–torsion energy levels of the molecule using this minimum-energy path. Brief comparison is made to recent experimental measurements of the rotational fine structure of the n 9 s 1 fundamental. q 1998 Elsevier Science B.V.
1. Introduction In connection with our work on the CHq 5 molecular ion w1x we have developed a general rotation– contortion Hamiltonian and computer program that can be used to calculate the rotation–contortion energy levels of any molecule that has one large amplitude Žcontortional. degree of freedom w2x. The most significant aspect of this Hamiltonian is its ability to incorporate a parametrized geometrical path of any desired flexibility, hence avoiding any approximations or perturbation treatment for the large amplitude mode. In the present paper we use the program to calculate the rotation and internal-rotation energy levels of the ethyl radical C 2 H 5 . We use ab initio calculations to determine the optimized structure of the molecule, and the potential energy, along the minimum energy path for internal-rotation.
Our work has been inspired by the observation and analysis of a high resolution spectrum of a fundamental band of the ethyl radical in the gas phase by Sears et al. w3x, having a band origin at 528.1 cmy1 . This fundamental Ž n 9 . corresponds to the out-of-plane or umbrella inversion motion at the CH 2 radical center Žthe CH 2 wagging motion w4,5x.. In this spectrum there is clear evidence for almost free internal-rotation, and from the analysis a torsional barrier height of about 20 cmy1 was determined in Ref. w3x. The approximate model Hamiltonian used in Ref. w3x was unable to precisely fit the data, and we hope that our calculations will lead to a precise simulation of this spectrum. Previous ab initio calculations at correlated levels of theory w6,7x have been concerned with determining the equilibrium structure, harmonic frequencies, and internal-rotation barrier height for use in the
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 1 7 6 - 7
50
A.L.L. East, P.R. Bunkerr Chemical Physics Letters 282 (1998) 49–53
study of hyperfine structure w6x and rate constants for the H q C 2 H 4 | C 2 H 5 reaction w7x. There has been no previous calculation of the rotation and internalrotation energy levels of the ethyl radical based on an ab initio potential surface.
2. The internal-rotation minimum energy path Our ab initio calculations were performed at the unrestricted second-order Møller–Plesset ŽUMP2. level of theory w8x with the cc-pVTZ basis set w9x, using the GAUSSIAN94 program package w10x. Initially, geometry optimizations and harmonic frequencies were determined for the internal-rotation minimum and maximum conformations Žsee Fig. 1.. The minimum energy conformation is traditionally called ‘‘staggered’’ Žsee Pacansky and Dupuis w11x. but has unfortunately been called ‘‘eclipsed’’ by some authors w3,6x; at this conformation the bisector of the CH 2 group is in the same plane as one of the CH bonds of the CH 3 group and it is tilted towards this CH bond out of collinearity with the CC bond by 9.28 Žthe out-of-plane wag angle.. The saddle point between the successive internal-rotation minima has the plane of the CH 2 group in the same plane as one
Fig. 1. The structure of the ethyl radical at equilibrium, and at the saddle point for the internal-rotation, including the molecule fixed Ž xyz . axis system.
of the CH bonds in the CH 3 group, and the bisector of the CH 2 group is tilted away from that CH bond out of collinearity with the CC bond by 0.58 Žthe in-plane rock angle.; we call this conformation the ‘‘eclipsed’’ conformation. In 1991, Suter and Ha w6x published UMP2 eclipsed and staggered structures and harmonic frequencies of C 2 H 5 obtained with the smaller 6-31GŽd, p. basis set. Our new results are but little different. The cc-pVTZ bond lengths are 0.1% smaller, and all of the harmonic frequencies are 1–2% smaller except for the umbrella mode, which matches the 6-31GŽd, p. result Ž470 cmy1 . at the minimum conformation, and is 50 cmy1 higher than the 6-31GŽd, p. result Ž390 cmy1 . at the maximum conformation. This is of some relevance to the separability of the internal-rotation mode from the umbrella mode, as the difference in harmonic umbrella frequency between eclipsed and staggered conformations is now only 29 cmy1 Ž470 cmy1 at the minimum and 441 cmy1 at the internal-rotation maximum., reflecting a smaller amount of coupling than would previously have been thought. Hase and coworkers w7x have more recently determined geometries and harmonic frequencies for the minimum at six different levels of ab initio theory. Again their results are similar to ours; the only significant improvement is a reduction of the out-ofplane wag angle by 1.08 at equilibrium with improved Žpost-MP2. correlation. For the determination of the minimum energy path ŽMEP. for internal rotation, we performed full geometry optimizations at the UMP2rcc-pVTZ level of theory at eight more values of the dihedral angle H 7 C 2 C 1 H 3 Ždenoted t 73 , measured in a right-handed sense about C 2 ™ C 1; see Fig. 1.. The torsional angle t is defined as the average of all six dihedral angles between the CH bonds of the CH 3 and CH 2 rotors. At each calculated point the values of the dihedral angle t 73 , of the angle t , and of the resulting potential energy V, are given in Table 1. We can fit this potential using a simple V6 function of t , and the values of this fitted function are also given in Table 1. This fitted function has an internal-rotation barrier height of 36 cmy1 , which we can correct with the zero-point vibrational energies ŽZPVE. of the other 3 N y 7 s 14 vibrational modes Žtaken as half the sum of the UMP2rcc-pVTZ harmonic frequen-
A.L.L. East, P.R. Bunkerr Chemical Physics Letters 282 (1998) 49–53 Table 1 Calculated values of the H 7 C 2 C 1 H 3 dihedral angle t 73 , internalrotation parameter t and the electronic energy Velec at the ten optimized points, and the fitted value of Velec obtained using a V6 cosine potentiala
t 73
t
Velec
fitted Velec
0.00 5.00 b 10.00 b 15.00 b 20.00 b 25.00 b 30.00 b 34.86 35.00 b 40.00 b
0.00 c 4.16 8.31 12.52 16.77 21.11 25.57 30.00 c 30.25 34.97
36.4 34.7 30.1 23.0 15.1 7.5 2.0 0.0 0.0 2.6
36.5 34.8 30.0 23.0 14.9 7.4 1.9 0.0 0.0 2.4
a b c
minor effects being methyl torsional defects Žy0.72 sin 2t in t 73 . and CH 2 rocking Ž0.58 cos 3t in a C 1 C 2 H t .. This accords with the theoretical discussion of the problem given by Sørensen w12x. The structure relaxation can be rationalized in terms of the need to minimize non-bonded H–H interactions. There is practically no methyl tilting Ž0.03 cos 2t in a C 2 C 1 H 3 ., in accord with the results of the fitting to the data w3x.
3. The rotation internal-rotation energy levels
Angles in degrees, energies in cmy1 . This value frozen during geometry optimization. This value determined by symmetry.
cies., obtaining an effective barrier height of 26 cmy1 . This matches exactly the previous results of Suter and Ha with the smaller basis set w6x, although their electronic barrier height and ZPVE correction were almost double ours in size. The t-dependent values of the various other internal coordinates were accurately fitted by the follow˚ ing analytical functions Žwhere bond lengths are in A and bond angles are in degrees.: rC 1 C 2 s 1.4864 q 0.0001 cos 6t ,
Ž 1.
rC 1 H 7 s 1.0768 q 0.0006 cos 3t y 0.0002 cos 6t ,
The internal-coordinate functions of Eqs. Ž1. – Ž6. were first transformed analytically to Cartesian coordinate functions, employing molecule-fixed xyz-axes. These axes were chosen such that the origin was at the centre of mass, the z-axis parallel to the C–C bond axis, and the xz-plane parallel to a line passing through H 6 and H 7 Žsee Fig. 1.. From the Cartesian coordinate functions we can determine the elements of the 4 = 4 extended moment of inertia matrix at any given value of the torsional coordinate t Žsee Ref. w2x.. Inverting this t-dependent matrix gives the t-dependent m matrix elements that we need for the kinetic energy part of the exact rotation–contortion Hamiltonian: Hˆ s Hˆ rot q Hˆ rot ,t q Hˆ t ,
Ž 7.
where
Ž 2.
xyz xyz
Hˆ rot s Ž 1r2 . Ý
rC 1 H 3 s 1.0906 y 0.0041 cos 2t q 0.0003 cos 4t ,
Ý mab Jˆa Jˆb
a
Ž 3. a C 1 C 2 H 7 s 120.89 q 0.58 cos 3t q 0.14 cos 6t ,
Ž 4.
a C 2 C 1 H 3 s 111.63 q 0.03 cos 2t q 0.06 cos 4t ,
Ž 5.
Ž 8.
b
xyz
Hˆ rot ,t s Ž 1r2 . Ý Jˆt , mat Jˆa a xyz
q Ž 1r2 . Ý mat Jˆa Jˆt q Jˆt Jˆa ,
and
t 73 s t y 0.72 sin 2t q 5.45 sin3t .
51
a
Ž 6.
Other internal coordinates are obtained by symmetry, eg. rC 1 H 4Žt . s rC 1 H 3Žt q 2 pr3.. These equations define the minimum energy path we use in the next section. The t-dependence of the bond lengths is almost negligible. The most significant relaxation effect is the CH 2 wag Ž5.45 sin 3t in t 73 ., with
ž
/
Ž 9.
and Hˆ t s Ž 1r2 . mtt Jˆt2 q Ž 1r2 . Jˆt , mtt Jˆt q Ž 1r2 . < m < 1r4 Jˆt , mtt < m
Ž 10 .
A.L.L. East, P.R. Bunkerr Chemical Physics Letters 282 (1998) 49–53
52
In Eqs. Ž8. to Ž10. Jˆx , Jˆy and Jˆz are the components of the total angular momentum operator along molecule-fixed axes, Jˆt is the torsional momentum Table 2 The rotation–torsion energy levels Žin cmy1 . and zero-point energy ŽZPVE. obtained for J s 0, 1, and 2 J
K
Ki
Symmetry
2
2
3
A1 Y A2 Y A2 Y A1 X E X E Y E Y E Y A1 Y A2 X A1 X A2 X A2 X A1 Y E Y E X E X E Y A2 Y A Y A1 Y A2 X E Y E X A2 X A2 X A1 X A1 X A2 Y E Y E X E X E Y A1 Y A2 Y A1 Y A2 X E Y E X A2 Y A2 Y A1 X E Y E X A1 ZPVE
2 1 0 2
1
3
2 1 0 2
1
0
3
1
2 1 0 3
2 1 0 1
0
0
3
0
2 1 0 3 2 1 0
Y
Energy 282.892 57.894 282.892 57.894 173.470 22.776 91.293 16.645 39.522 39.522 200.549 87.508 200.568 87.490 108.168 33.813 46.365 8.998 13.268 13.102 141.732 128.749 62.372 18.910 4.404 197.618 84.566 197.625 84.560 105.232 30.877 43.428 6.061 10.277 10.221 138.796 125.812 59.435 15.972 1.468 137.328 124.344 57.967 14.504 0.000 12.973
Table 3 The wavenumbers Žin cmy1 . for selected n 9 s1 transitions in C2 H5 J
Ki
K upper
K lower
ga
Observedb
Calculated
1 2 3 4 5 2 3 4 5 4 5 4 4 5 5
0 0 0 0 0 1 1 1 1 y2 y2 3 3 3 3
1 1 1 1 1 2 2 2 2 y3 y3 4 4 4 4
0 0 0 0 0 1 1 1 1 y2 y2 3 3 3 3
0 0 0 0 0
536.829 536.783 536.702 536.593 536.460 535.350 535.354 535.333 535.309 534.005 533.993 532.819 532.819 532.808 532.808
536.853 536.798 536.716 536.607 536.473 535.747 535.740 535.722 535.687 534.610 534.596 533.487 533.487 533.481 533.481
0 1 0 1
a
This quantum number is for transitions involving K i s 0, 3, 6, 9, etc. b Ref. w3x.
operator, and < m < is the determinant of the m matrix. For the potential function we take V Ž t . s Ž V6r2 . Ž 1 q cos 6t . ; V6 s 26 cmy1 .
Ž 11 .
This Hamiltonian is solved by a combination of numerical integration and matrix diagonalization as described in Ref. w2x. In making this calculation of the rotation and internal-rotation energies we found that satisfactory convergence in the matrix diagonalization stage could be achieved with a basis set that included internal-rotation functions up to those having K i s 12, where K i is the internal rotation quantum number. The energies, and symmetries in the G12 molecular symmetry group, are given in Table 2. We also converted our Table 2 results to n 9 s 1 rovibrational transition wavenumbers by calculating the appropriate D K s 1 differences and adding the experimental band origin value of 528.1 cmy1 , and in Table 3 compare some of these results to those assigned by Sears et al. w3x.
A.L.L. East, P.R. Bunkerr Chemical Physics Letters 282 (1998) 49–53
53
4. Discussion
References
It is clear that the internal-rotation motion in the ethyl radical is almost free with a barrier of about 25 cmy1 , and that a rigid coaxial C 3v –C 2v internal-rotation model neglects primarily only the out-of-plane CH 2 wagging relaxation. We do not achieve a close fitting of the data with this purely ab initio calculation. The next stage is to attempt a least squares fitting of the data, in which we adjust the minimum energy path and internal-rotation potential parameters, both in the ground state and in the wagging fundamental. Centrifugal distortion will almost certainly also have to be included. Work along these lines will be initiated, in collaboration with the authors of Ref. w3x, in the hope of precisely understanding their spectrum and what is required to reproduce it.
w1x A.L.L. East, M. Kolbuszewski, P.R. Bunker, J. Phys. Chem. Ž1997., in press. w2x A.L.L. East, P.R. Bunker, J. Mol. Spectrosc. 183 Ž1997. 157. w3x T.J. Sears, P.M. Johnson, P. Jin, S. Oatis, J. Chem. Phys. 101 Ž1996. 781. w4x J.M. Hollas, Modern Spectroscopy, Wiley, Chichester, UK, 1987. w5x This mode was referred to as the CH 2 rocking motion in Refs. w3,12x. w6x H.U. Suter, T.-K. Ha, Chem. Phys. 154 Ž1991. 227. w7x W.L. Hase, H.B. Schlegel, V. Balbyshev, M. Page, J. Phys. Chem. 100 Ž1996. 5354. w8x C. Møller, M.S. Plesset, Phys. Rev. 46 Ž1934. 618. w9x T.H. Dunning Jr., J. Chem. Phys. 90 Ž1989. 1007. w10x M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. DeFrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez, J.A. Pople, GAUSSIAN 94, Gaussian Inc., Pittsburgh PA, 1995. w11x J. Pacansky, M. Dupuis, J. Chem. Phys. 68 Ž1978. 4276. w12x G.O. Sørensen, J. Chem. Phys. 105 Ž1996. 3942.
Acknowledgements We thank Dr. T.J. Sears and Dr. G.O. Sørensen for their encouragement and advice.