Getting the Shape of Methyl Internal Rotation Potential Surfaces Right

2770

J. Phys. Chem. 1996, 100, 2770-2783

FEATURE ARTICLE Getting the Shape of Methyl Internal Rotation Potential Surfaces Right Lionel Goodman,* Tapanendu Kundu,† and Jerzy Leszczynski‡ Wright and Rieman Chemistry Laboratories, Rutgers UniVersity, New Brunswick, New Jersey 08903 ReceiVed: April 11, 1995; In Final Form: NoVember 28, 1995X

The effect of the multidimensional character of methyl internal rotation on torsional potential functions in small conjugated methyl molecules is discussed. Partitioned energetics obtained from different ab initio calculation levels are divided into σ and π symmetry components and then further dissected by breakdown of the fully relaxed internal rotation process into separate relaxation steps and their nuclear virials. Important contributions to barrier shape, height, and origin are thus disclosed. The analysis reveals closely balanced antagonistic π and σ contributions as determinants for the barrier height and identifies the dominant contributions to the barrier origin as σ-bonding changes largely resulting from bond lengthening during methyl group rotation. Skeletal hydrogen out-of-plane motion components of the torsional coordinate (e.g. CH2 ethylenic twisting in propene) provide a general mechanism for barrier narrowing in small conjugated methyl molecules. Rydberg jet experiments are discussed as a general means for obtaining accurate torsional frequency information (including ground state) to test internal rotation potential surface models.

I. Introduction The quantum mechanical nature of hindered rotation in molecules was recognized in the early years of quantum theory by Nielson,1 whose work provided the stepping stone for K. S. Pitzer’s two famous papers on internal rotation in ethane2 published in 1937. Since then, many decades of detailed experimental and theoretical studies have attempted to establish the form of internal rotation potential barriers. Much of the recent spelunking into torsional potential barrier heights and shapes has resulted from (1) the ability of high-resolution FTIR, microwave, and jet electronic spectroscopies to obtain precise torsional fundamental and overtone frequencies, leading to accurate torsional potential shape information, and (2) the ability of large scale ab initio computations to accurately predict overall molecular geometries, which include the skeletal flexing that takes place as internal rotation proceeds. The increased clarity and narrow lines available through Rydberg state jet spectroscopy have allowed this spectroscopy to become a powerful tool for probing methyl torsional ground state modes as well as those for the Rydberg states.3-5 The Rydberg experiment reveals the historically missing a2 torsional fundamental and its overtone in acetone, for example.3 In addition, since electron distributions in the Rydberg states approach those for the corresponding radical cations, which are very different from the ground state, new insight into torsional barrier origins can be obtained by comparison of potential energy surfaces hindering internal rotation in the ground state with the Rydberg ones. Early (late 1960s) ideas on the origin of the ethane barrier invoked Pauli-like exchange repulsions between C-H bond orbitals.6 But even for this very well studied molecule, the cause for the barrier is not settled.7,8 Weinhold, for example, proposes † Present address: Physics Department, Indian Institute of Technology, Powai, Bombay 400076, India. ‡ Permanent address: Chemistry Department, Jackson State University, Jackson, MS 39217-0510. X Abstract published in AdVance ACS Abstracts, February 1, 1996.

0022-3654/96/20100-2770$12.00/0

a mechanism involving vicinal interactions between bonding and antibonding bond orbitals,7 and Bader, Wiberg, and colleagues invoke quadrupole polarization of the electron cloud in the C-C bond.8 Small conjugated methyl molecules, capable of internal rotation, such as acetaldehyde, propene, and acetone have had almost as much attention as ethane. For these systems Hehre, Pople, and Devaquet9 (HPD) proposed that it is π interactions that dominate barrier energetics. The simple π fragment ideas by HPDsexpanded by Radom and coworkers10sseem to account for many of the key experimental observations on small conjugated methyl molecules. They have been extended to radical cations, anions, and triplet states11 by taking into account the different orbital occupancies from those in the ground state of the neutral molecule. Alternate barrier origins for individual molecules have been offered. Acetaldehyde has been singled out because of its large CdO bond dipole (causing polarization effects on the methyl C-H bonds12) and because of its oxygen lone pair electrons (leading to a possible weak covalent bond between the carbonyl oxygen and an inplane methyl hydrogen13). But these theories are troubled. Calculated barriers in propene14 and acetaldehyde11 ions do not support a Pauli repulsion origin. Energetic partitionings15 and methyl group geometries16 question the dominance of π interactions. A dipole polarization mechanism predicts that the acetaldehyde barrier, driven by the large CdO dipole, should be increased over that in propene,12 contrary to what is observed, and no bond critical point is found between the oxygen and hydrogen nuclei.15,16 The most primitive approach to internal rotation in a singlerotor molecule like acetaldehyde is one-dimensional, i.e., rigid rotation comprising only the torsional coordinate (see section II). Insight into the physics of internal rotation remained confused because although barrier heights calculated by this model frequently give a rough but reasonable answer (but see section VI), the model fails to correctly predict barrier shapes. This deficiency is because of the multidimensional nature of internal rotation. In propene, for example, the flattened shape © 1996 American Chemical Society

Feature Article and narrow barrier cannot be understood without concomitant coupling to normal modes.17 The same is true for acetaldehyde.18 Thus, our present picture of internal rotation is that of a highly impure vibration comprising large amplitude methyl torsion coupled to out-of-plane normal vibrations, principally involving skeletal hydrogen motions. In any case, we must come to terms with the complex nature of internal rotation. To do this, we must understand partitioned energetics for the multidimensional process and their implications. This article is devoted to an exploration of this view of methyl internal rotation in small conjugated organic molecules. It will concentrate on three key systems: acetaldehyde, propene, and to a lesser extent, acetone. We pay attention to both the shape and height of the internal rotation barrier and to the cooled jet experiments which provide the data base. We have not, in this article, attempted to give a comprehensive review of the extensive internal rotation literature. The reader is referred to Lister, MacDonald, and Owen’s monograph,19 as an excellent basic reference, and to both Payne and Allen’s20 and Veillard’s21 reviews of ab initio barrier calculations through the mid 1970s. Many insights into energy partitioning accompanying internal rotation in these reviews, written 20 years ago, stand today. Internal rotation dynamics is found in the recent fine review by Spangler and Pratt.22 II. What Is Internal Rotation? Considerable confusion has resulted from lack of a unique description for internal rotation. From an energetic viewpoint an internal rotation potential function, V(τ)sτ is the torsional coordinatescan be thought of as the one-dimensional (bidimensional for two rotors) representation of the molecular potential surface passing through the minimum energy pathway with respect to all other degrees of freedom. It is clear that such a description sows its own seeds of destruction, since this adiabatic fully relaxed path for internal rotation meanders around potential hills in nontorsional coordinates and thus cannot be truly one-(or bi-)dimensional! The historical way out of this dillemna has been to describe internal rotation by a direct (but not minimum energy) rigid-rotation path where only the methyl rotation angle is allowed to vary. In this one-dimensional process all skeletal internal degrees of freedom are frozen so that methyl torsion has local mode properties; that is, it is free of coupling to normal vibrations. But this description is faulty. From a quantum mechanical viewpoint the ground state energy must satisfy the variational principle. With few exceptions this translates to unequal methyl C-H bond lengths and C-C-H bond angles for the equilibrium conformation. These unequal bond lengths and angles do not, in general, allow rigid rotation to generate a 3-fold symmetric potential, and therefore it is flawed as a physically reasonable model. The remedy has been equalized methyl bond lengths and hydrogen atoms separated by 120° dihedral angles. We will term rotation of such a C3V symmetry methyl top primitiVe rigid-rotation. Although 3-fold symmetric potentials emanate from such a description, the variational principle is not obeyed. Even fully relaxed rotation is not unambiguous. In this case the torsional angle is not fully defined. Although the “reference” hydrogen rotation angle is precisely fixed, the other two methyl hydrogens do not necessarily undergo the same angles of rotation. To get around this ambiguity in the torsional angle, the internal rotation angle, τ, is frequently defined as the average of the three hydrogen atom rotations from the equilibrium geometry.23 A more precise approach to the internal rotation problem expressing the potential function in terms of three angles loses the appealing simplicity of a single torsional angle.

J. Phys. Chem., Vol. 100, No. 8, 1996 2771 A key question then is how to compare theory and experiment. Barriers are well-defined quantities. As discussed above, they are the difference between stationary points on a full multidimensional potential energy surface. Fitting experimental transitions by an effective one-dimensional potential is a dangerous procedure, the danger of which is illustrated in section V. The outcome cannot correspond exactly to a multidimensional potential calculated by electronic structure methods. Onedimensional torsional potentials of all kinds are not observable quantities, either directly or indirectly. They are merely useful paths connecting the stationary points. III. Nuclear Virials for Internal Rotation The generalized virial theorem24 is

T ) -E + ∑RχR‚FR

(1)

where the nuclear virial ∑RχR‚FR is the contribution of nucleus R to the virial of the forces acting on the electrons, χR is the position vector of R, and FR ) -∇RE is the net force acting on it. The importance of the generalized virial theorem is that it provides an Archimedal principle for discussing internal rotation by identifying strain effects. When the forces acting on the nuclei vanish (as they must at the variational minimum energy equilibrium conformation (E), TE ) -EE. It is useful to break the internal rotation process into two conceptual steps: E w R w FR. The first step, E w R, is rigid rotation; the second goes to the globally minimized fully relaxed barrier top (FR). Here the nuclear virial also vanishes, i.e., TFR ) -EFR. For the nonequilibrium rigid-rotation conformation (R), the forces do not vanish and

TR ) -ER + ∑RχR‚FREwR

(2)

Consequently, the nuclear virial of the rigidly rotated state ΣRχR‚FREwR is given by

∑RχR‚FREwR ) ∆ER + ∆TR

(3)

where ∆ER ) ER - EE and ∆TR ) TR - TE. We will show (sections VII and VIII) that for this process ∑RχR‚FREwR is positive (repulsive), and consequently the molecule is left in a strained metastable state. Another useful relationship arises from the vanishing of the nuclear virial at the globally minimized fully relaxed barrier top. Then the decrease in the nuclear virial for the second step, R w FR, is ∑RχR‚FRRwFR ) (ER - EFR). Since ER ) TR + VR and EFR ) TFR + VFR,

∑RχR‚FRRwFR ) (∆TR - ∆TFR) - (∆VFR - ∆VR)

(4)

where ∆VFR refers to VFR - VE, etc. This equation has an important consequence: kinetic and potential energies cancel in the R w FR nuclear virial if the kinetic energy change in going from rigid to fully relaxed descriptions of internal rotation is greater than the nuclear Virial. Its significance is that for this important case there is interchange of kinetic and potential energies between the first and second steps. IV. Methyl Torsional Potentials The tradiational approach is to expand the potential function hindering internal rotation, V(τ), in a Fourier series in the internal rotation angle, τ:

V(τ) ) (1/2)∑Vn(1 - cos nτ)

(5)

2772 J. Phys. Chem., Vol. 100, No. 8, 1996 Only cosine terms arise in the Fourier expansion for systems in which the torsional axis for methyl group rotation lies in the molecular symmetry plane, i.e., if the potential function is symmetrical in the sense V(τ) ) V(τ + π). If the potential function has 3-fold symmetry, as it must for a methyl group which reproduces itself through cyclic permutation of the hydrogens, the only nonzero coefficients in eq 5 are V3, V6, V9, etc. Thus, for propene

V(τ) ) (1/2)V3(1 - cos 3τ) + (1/2)V6(1 - cos 6τ) + (1/2)V9(1 - cos 9τ) + ... (6) (Note that for rigid rotation, the potential function does not fulfill the 3-fold symmetry condition (see section II) and a greater number of coefficients are necessary to describe it.) Higher coefficients than V9 have generally been found to be very small. Only the barrier shape is affected by the V6 term since cos 6(τ+π) ) cos 6τ. The odd terms affect both barrier shape and height, V3 being by far the most important height determinant. The analogus expression for two equivalent methyl tops, in terms of two rotation angles, τ1 and τ2, is

V(τ1,τ2) ) 1/2V3(cos 3τ1 + cos 3τ2) + 1

/2V33(cos 3τ1 cos 3τ2) + 1/2V′33(sin 3τ1 sin 3τ2) + /2V6(cos 6τ1 + cos 6τ2) + ... (7)

1

In this case the barrier height is 2V3. The quantity V3 - V33 represents the potential energy increase by rotating a single methyl group by 60°, and the V′33 term is shape determining. Potential function (5) and hence (6) derive from a pure internal rotation Hamiltonian, H ) Fp2 + V(τ), where p is the momentum associated with methyl top rotation and F, the kinetic energy coefficient,25 relates to methyl rotor moments of inertia. This Hamiltonian does not explicitly take into account interactions between the torsions and other vibrations and thus leads to separation of variables into two categories: normal modes and torsion. In the case of methyl internal rotation the missing interactions can be attributed to skeletal flexing and methyl hydrogen folding occurring during rotation. Thus, eq 5 is only exact for primitive rigid rotation, which freezes all of these motions, in effect uncoupling methyl torsion from the remaining vibrations. The consequence is that the torsional angle dependence of the kinetic energy is frequently ignored in deriving a V(τ) that reproduces an experimental spectrum. In effect, for methyl torsion, T is usually assumed appropriate to a C3V rigid rotor. The ease of obtaining reliable conformer geometries using ab initio wave functions26sconstructed from even moderate basis setssstimulated a number of endeavors to include flexing of the molecular structure in the torsional potential. Early attempts, particularly by Bell,27 Wiberg,16,28 and Radom,10 and more recent studies,29 were focused on barrier heights. These studies concluded that, in general, the height is not greatly affected by the change in skeletal or methyl group geometry that attends methyl torsion (there are exceptions however, e.g., acetone; see next section). The next step was complete torsional potential ab initio calculations which included these motions, first made by Ozkabak, Philis, and Goodman for methyl torsion on acetone30 and by Head-Gordon and Pople for nitro compounds.31 These calculations relaxed the restriction that internal rotations can be represented by pure methyl torsional motions by fitting fully optimized conformer energies to a Fourier expansion (i.e. eq 7) in a single torsional coordinate. The resulting fully relaxed internal rotation potential function expresses the effect of skeletal flexing parametrically since the

Goodman et al. Vn coefficients are now functions of the entire set of molecule internal coordinates and thereby adiabatically mixes methyl internal rotation with other normal vibrations. In contrast, generation of a sizeable number of difficult-to-calculate matrix elements is required by explicit inclusion of internal rotationnormal vibrational mode interaction terms in the Hamiltonian. The advantages of the single Fourier expansion in terms of full geometry optimization are that all of the atomic motions consequent to methyl torsion are revealed, with ease of computation of the potential energy function and energy levels. The consequences of the resulting multidimensional character of the torsional process are that the shape and in some cases the height of the potential function hindering internal rotation, and hence the energy levels associated with methyl torsion, are often drastically altered from the simplified rigid-rotation model. Torsional fundamental frequencies (i.e., energy levels lying deep in the potential well) have been accurately predicted for a range of barriers in a number of molecules using fully relaxed potential functions obtained by this approach,15,18,32-39 particularly by Moule, Smeyers and colleagues, Bell, and Ozkabak and Goodman. The quantitatively meaningful calculated torsional frequencies indicate that the one-dimensional equation (5) is still successful despite the introduction of a parametric dependence of Vn on other degrees of freedom and that the predicted torsional potential terms are physically meaningful. Nino, Munoz-Caro, and Moule have gone beyond the single Fourier expansion in τ by using double (and even triple) expansions which include other large amplitude motions (such as hydrogen wag).18b,39b,c These double expansion potentials predict frequencies even more precisely than the appealingly simple single expansion. V. Probing Methyl Torsions through Rydberg State Spectroscopy Splitting of microwave transitions (caused by interaction between overall rotation and internal rotation of the light methyl group) represents the classical approach for determining internal rotation barriers (and to an extent the form of the potential) in small methyl molecules.25a,40 A lot of attention has also been paid to the torsional transition region in infrared spectroscopy. Because torsional rotation generates only small changes in polarizability, Raman spectroscopy has not played as major a role. In this section we describe a powerful supersonic Rydberg state jet experiment for probing torsional transitions, conceived to get around lack of activity of many torsional vibrations in infrared and Raman spectra.3 The scheme of the experiment is given in Figure 1. A Rydberg state of appropriate symmetry is pumped (by L1). The Rydberg state population is monitored by counting ions formed by absorption of additional photons as the probe laser, L2, scans the ground state torsional vibrational region. The experiment’s power is its unique symmetrybreaking ability obtained by selecting an appropriate Rydberg state. In favorable cases, where the torsional energies are small enough to achieve direct excitation from ground state torsional levels of the jet-cooled molecules, the probe laser will not be needed. We illustrate the power of this approach to the torsional frequency problem with the a2 “antigearing” torsion fundamental and overtone in acetone. Acetone, like other C2V symmetry dimethyl molecules, has two torsional modes: a2, in which the methyl groups rotate in-phase; and b1, which is counter-rotating (“gearing”). Only the b1 vibration is infrared allowed; in acetone it is weakly active. Although both vibrations are Raman allowed, the small polarizability changes associated with methyl rotations and their low frequencies (i.e. proximities to the

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Figure 1. Schematic representation of Rydberg state torsion probing experiment. L1 is a fixed frequency laser, pumping the Rydberg state, and through absorption of an additional photon, generating positive ions. L2 is a tunable probe laser, producing ion current dips when resonant with ground state torsional energy levels.

Rayleigh frequency) make experimental detection of the fundamental difficult and, in acetone, unachieved. Lack of the a2 frequency, and consequently the in-phase-out-of-phase rotation energy gap, leaves the torsion-torsion interaction and its contribution to the torsional potential undetermined. Philis, Berman, and Goodman solved the missing a2 torsion dilemma by using the 1A2(3prn) Rydberg state.3 A transition between the a2 torsion and the A2 Rydberg origin is symmetry allowed: A2Xa2 ) A1 T A1. The torsional fundamental region of acetone and acetone-d6 A2 Rydberg spectra is shown in Figure 2. In this case the low torsional frequencies (Table 1) allowed direct two-photon excitation. Two-photon excitation was used because of wavelength considerations and because the intensities of totally symmetric two-photon transitions are weakened in circularly polarized light. Non-totally symmetric transitions would be intensified. The polarization behavior shown in Figure 3 definitively establishes the a2 ground state torsional fundamental frequency as 77.8 cm-1 (in acetone-d6, 53.4 cm-1). Also determined from the Rydberg spectra are a2 ground state torsional overtone frequencies: 164.4 and 116.1 cm-1 in acetone-h6 and -d6, respectively.5a Large torsion-torsion interaction is revealed by the 47 cm-1 difference in the gearing and antigearing frequencies (Table 1). The new frequency information for acetone-h6 and acetoned6 (i.e., ν12(a2) and 2ν12, Table 1) and the infrared-allowed (ν17(b1)) frequency, six frequencies in all, allow the torsional potential function (eq 7) to be constructed.41 The four Rydberg jet determined potential constants (Table 1) are very different from those obtained by the earlier microwave42 and infrared43 approaches. In particular, the barrier width, strongly affected by V33 and V33′ terms in the potential function, is considerably narrower and the barrier height is more than 200 cm-1 (i.e. nearly 40%) higher than the previous determinations.30 The four potential constants in eq 7 calculated by the ab initio approach described in section IV are given in Table 2. These have been carried out using the three descriptions of internal rotation that we have discussed in section II at different ab initio levels, and they reveal the powerful effect that skeletal relaxation accompanying methyl torsion has on the barrier shape and height. Only the fully relaxed theoretical constants are in qualitative agreement with the experimental ones.30 The barrier height (2V3) illustrates this conclusion. Structural relaxation increases the Cmethyl-C bond length, Cmethyl-C-Cmethyl angle, and the inner (in-plane) hydrogen separation in the (staggered-

Figure 2. Torsional hot band region of the two-photon jet-cooled multiphoton ionization 1A2 (3prn) Rydberg spectrum of acetone. Inset is for acetone-d6. ν12 designates the a2 symmetry torsional mode in which the methyl groups rotate in-phase. Out-of-phase rotation is designated ν17.

TABLE 1: Acetone Experimental Ground State Torsional Frequencies and Potential Constants (cm-1) Frequencies torsion

acetone-h6

acetone-d6

ν12′′a

77.8 164.4 124.5

53.4 116.1 96.0

2ν12′′a ν17′′b

Gearing-Antigearing Fundamental Frequency Differences ∆ ≡ ν17 - ν12 ) 46.7 ( 0.8 cm-1 acetone-h6 acetone-d6 ∆ ) 42.6 ( 0.6 cm-1 Potential Constants microwavec infraredd Rydberg jet + IRe a

V3

V33g

V33′g

272 279 381

-12 144

-108 -166

V6

barrierf

0

544 558 763

b

From Rydberg state jet spectra (refs 3 and 5a). From infrared spectra (ref 43). c Reference 42. d Reference 43. e Reference 30. f Barrier height ) 2V3. g Important determinants for barrier shape.

staggered) top-of-the-barrier conformation, and the resulting decrease in H-H repulsion leads to a nearly 300 cm-1 barrier reduction. The fully relaxed values approximate the 760 cm-1 height obtained from the measured frequencies; the other descriptions give much larger (900-1100 cm-1) barriers.3 This feature remains unaffected by basis set extension and correlation (through MP4) inclusion. The large difference between acetone torsional potential surfaces predicted by rigid-rotation and fully relaxed internal rotation descriptions leads to large differences in predicted frequencies for the two fundamentals at any ab initio calculation level that we have attempted. Only the fully relaxed funda-

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Goodman et al. TABLE 2: Acetone ab Initio Calculated Internal Rotation Potential Constants and Frequencies (cm-1) Potential Constantsa experimentc rigid rotation HF/DZd HF/6-31G(d,p)e MP2/6-31G(d,p)e fully relaxed HF/DZ HF/6-31G(d,p) MP2/6-31G(d,p)

V′33

V6

barrierb

144

-166

0

763

485 540.5 543

203 199 192

-208 -205 -197

-52.5 -55 -56

970 1081 1086

393 382 397

154.5 135 128

-181 -171 -161

1.5 -0.5 -0.5

796 764 794

V3

V33

381

Frequenciesf ν12(a2)

ν17(b2)

acetone-h6 experimentg 77.8 ( 0.7 124.5 ( 0.1 rigid rotation HF/DZ 105.7 ( 0.5 157.0 ( 1.2 HF/6-31G(d,p) 118.2 ( 0.3 168.4 ( 0.6 MP2/6-31G(d,p) 121.1 ( 0.2 169.1 ( 0.5 fully relaxed HF/DZ 75.8 ( 0.7 127.7 ( 1.8 HF/6-31G(d,p) 78.7 ( 0.7 128.3 ( 1.7 MP2/6-31G(d,p) 84.0 ( 0.6 130.6 ( 1.3

Figure 3. Effect of cooling and polarization conditions definitively establishing a2 fundamental assignments to -77 and +70 cm-1 bands in acetone 1A2 Rydberg spectrum shown in Figure 2: (a) linearly polarized light and 2.3 atm Ar, (b) linearly polarized light and increased vibrational temperature (caused by removing the argon); and (c) circularly polarized light and 0 atm Ar.

mental frequencies reasonably simulate experiment (Table 2). In contrast, rigid rotation at the same calculation level predicts much too high frequencies. Table 2 shows that predicted firstovertone frequencies for the a2 torsions by the fully relaxed potential are also in good agreement with experiment. As is the case for the fundamentals, rigid-rotation calculations for the acetone overtones are in gross disparity. The important conclusion is that the acetone experimental torsional potential function agrees only with the fully relaxed description of internal rotation. The well-simulated acetone torsional fundamental and overtone frequencies by an effective 3-fold one-dimensional potential, which implicitly contains vibrational averaging over other modes, throws some illumination on the kinetic energy problem since no τ dependence of T is assumed. Good simulation of fundamental (and in many cases overtone) frequencies turns out to be a general conclusion of torsional frequency calculations on small methyl molecules by this approach.15,18,32-39 These results suggest that the form of T(τ) is relatively unimportant for prediction of the low-lying frequencies to within several wavenumbers and that the torsional angle dependence of T can, as a first approximation, be ignored (see section X). VI. Ab Initio Calculation of Internal Rotation Potential Surfaces Our interpretive analysis is based on ab initio calculations designed to focus on interaction type. These partition overall energy terms contributing to the internal rotation process into differences in attractive electron-nuclear (∆Vne), repulsive

2ν12 acetone-h6

acetone-d6

164.4 ( 0.7 116.1 ( 0.7 206.7 ( 4.2 156.8 ( 0.2 231.0 ( 2.8 174.2 ( 0.1 235.8 ( 2.5 178.1 ( 0.1 164.9 ( 3.5 114.8 ( 0.2 167.7 ( 3.4 118.1 ( 0.2 176.1 ( 3.0 125.1 ( 0.2

a Internal rotation potential constants in eq 7. b Barrier height, )2V , 3 is the change in potential energy obtained by rotating both methyl groups through 60° from their equilibrium eclipsed-eclipsed to staggered-staggered positions (see Figure 1 in ref 30). c From Rydberg jet + IR frequency data (Table 1). d Huzinaga-Dunning double-ζ basis set (ref 62). e Pople double valence basis set (ref 26). f Predicted frequencies are averages of torsional sublevel energies (using completely ab initio calculated F constants) with half-maximum splittings as their band-half-widths (indicated if g0.1 cm-1). g a2 and b2 mode observed frequencies are from refs 5a and 43, respectively.

nuclear-nuclear (∆Vnn), and electron-electron (∆Vee) interactions and kinetic (∆T) energies. Allen’s proposal (made over two decades ago) that energy partitioning would elucidate rotational barrier sources44 (even though such decomposition is not unique) did not lead to the expected increased understanding of barrier mechanisms. We have gone an additional step by partitioning the electrostatic attraction and kinetic energy terms into their symmetry components, separating kinetic and core energy differences (∆Vne + ∆T) into σ and π contributions.14,15 Because symmetry terms are frequently antagonistic, the addition of symmetry separation allows considerable insight into barrier origins. This understanding is then strengthened by linking symmetry and natural bond orbital analyses. The attractive and repulsive contributions to the total potential energy change are ∆Va ≡ ∆Vne and ∆Vr ≡ ∆Vee + ∆Vnn, respectively. It is also useful to define the electronic potential ∆Ve ≡ ∆Vee + ∆Vne. These energy calculations were made using restricted Hartree-Fock (HF) and frozen core secondorder Møller-Plesset perturbation theory (MP2) levels applying a series of basis sets ranging from 6-31G(d,p) to 6-311++G(3df,2p). The rationale for the former calculation is that the Hartree-Fock methodswhich even with modest basis sets predicts a reasonable barriersshould correctly describe the physics of the barrier. More quantitative aspects are given by the extensive polarization function basis sets with the MP2 electron correlation corrections. Unless otherwise indicated, we report in this article the MP2/6-311G(3df,2p) (valence triple-ζ augmented by d and f polarization functions on heavy atoms, and p functions on hydrogens) results, with geometry optimization carried out at the same level. Basis set studies indicate that barrier height convergence requires basis sets at least at the triple-ζ level with additional polarization functions.45 Since

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MP2 calculations are not variational, perturbation theory partitioned energies are not on the same footing as the variational HF ones. Because it is important in energy-partitioning calculations that the virial theorem is satisfied, we always check MP2 partitioned energies, obtained for at least two basis sets, against HF ones for sign consistency. Details are found in refs 14 and 15. Torsional potential functions were determined by fitting the three- or four-term potentials of eqs 6 and 7 to total methyl rotational conformer ab initio energies obtained by rotating the methyl group(s) successively every 15° up to 180°, as described in section IV and in our papers on the individual molecules.14-18a,23,30,32 Methyl conformer energies obtained with full geometry bond lengths and internal angles all frozen at their geometry in the equilibrium conformer were utilized for rigid rotation. Primitive rigid rotation has an additional C3 methyl group symmetry constraint achieved by equalizing methyl bond lengths and setting internal dihedral angles equal to 120°. Since rigid rotation does not generate 3-fold potentials (see section II), some error is generated for rigid-rotation potential functions by use of three- or four-term eqs 6 or 7. This error does not appear in the partitioned energetics reported in the tables because these are derived from full 180° rotation. VII. Propene We now address the question, what skeletal motions accompany internal rotation? We do this by analyzing the methyl torsional potential for propene in some detail. Propene serves as the simplest model system for ethylenic molecules containing methyl groups. Advantage can be taken of propene’s small size, allowing insight into basis set, electron correlation contribution, and skeletal-flexing effects on the shape and height of the potential function hindering methyl torsion. The equilibrium geometry is the Cs symmetry eclipsed conformation shown in Figure 4. This geometry defines τ ) 0. The top of the barrier corresponds to the staggered conformation, τ ) 180°, which also possesses Cs symmetry. The most significant structural change between the twoconformations is lengthening of the Cmethyl-C bond (by nearly 0.01 Å). There are also small changes in methyl angles: C-Cmethyl-Hip and Hop1-C-Hop2 expands; Hip-C-Hop contracts. (a) Barrier Shape. The power of the nuclear virial to interpret torsional motion is shown in Figures 5 and 6. Examination of the rigid rotation virial, ∑RχR‚FREwR (solid curve in Figure 5), shows that it increases until about 35° and then decreases slightly as rotation proceeds to the staggered conformation.46 It is helpful to recall that the nuclear virial is calculated for frozen parameters and therefore is an expression of the missing relaxations. Consideration of nuclear virials for stepwise inclusion of these missing relaxations further elucidates ∑RχR‚FREwR. The first step achieves a partially relaxed conformation (PR) by allowing all in-plane flexings to occur, but the out-of-plane ones are still omitted. The τ dependence of ∑RχR‚FREwPR is shown in Figure 6a. The nuclear virial for this process passes through a maximum near 30° and vanishes at 60°. Another partially relaxed conformation can be achieved by allowing all skeletal out-of-plane relaxations (but now freezing the in-plane ones). ∑RχR‚FREwPR for this process (dashed curve in Figure 5) monotonically increases on going to the staggered conformation. Thus, the nonmonotonic behavior of the rigid-rotation nuclear virial is the combined result of two effects: (1) the missing skeletal out-of-plane flexings which can occur only for intermediate torsional angles and (2) missing in-plane skeletal flexing arising from changes in the

Figure 4. MP2/6-311G(3df,2p) optimized geometries for propene equilibrium eclipsed (E) and top-of-barrier staggered (S) conformers.

Figure 5. Nuclear virials for propene methyl internal rotation: (solid curve) rigid rotation; (dashed curve) all out-of-plane skeletal motions allowed to occur, but in-plane skeletal relaxations frozen.

molecular skeleton on going from the equilibrium eclipsed conformation to the top-of-barrier staggered one. To determine which of the missing out-of-plane motions are important determinants of nuclear virial (1) we examine separate motions of the skeletal hydrogens, Ha, Hb, and Hc (Figure 4). These are shown in Figure 6b. The τ dependence of the Hc wag resembles that of the full out-of-plane nuclear virial (1), suggesting that the large amplitude flexings of the skeletal hydrogens represent a significant portion of the “methyl torsion” coordinate. Examination of ethylenic CH2 hydrogen atom out-

2776 J. Phys. Chem., Vol. 100, No. 8, 1996

Goodman et al. TABLE 3: Flexing Model Dependence of MP2/ 6-311G(3df,2p) ab Initio Calculated Potential Constants for Methyl Torsion in Propene (cm-1) method a

experiment model Ib model IIc model IIId model IVe model Vf model VIg

V3

V6

V9

693.7 718.4 718.6 719.3 718.1 745.1 733.1

-14.0 -15.0 8.1 -2.8 12.0 -3.4 -0.6

0.2 -0.1 -0.8 0.4 1.4 0.0

a Reference 48. b All skeletal and method group hydrogen relaxations independently allowed. c Relaxations in model I calculation restricted by clamping Ha and Hb (i.e., CH2 hydrogens) out-of-plane motions. Hc out-of-plane motion remains free. d Relaxations in model I restricted by clamping Hc (hydrogen adjacent to methyl group) out-of-plane motion. Ha and Hb out-of-plane motions remain free. e All skeletal hydrogen out-of-plane motions clamped. f Rigid rotation. g Primitive rigid rotation with a C3V symmetry methyl group having identical C-H lengths and 120° HCH angles.

Figure 6. (a) Nuclear virial for partially relaxed propene methyl internal rotation. All in-plane skeletal relaxations occur, but out-ofplane skeletal hydrogen motions frozen. (b) Dependence of ethylenic hydrogen twisting angle (solid curve) and ethylenic Hc out-of-plane wagging angle (dashed curve) on the methyl torsional angle. The twist angle is calculated from the average of fully relaxed methyl internal rotation Ha and Hb displacements. See Figure 4 for hydrogen atom designations.

of-plane displacements (za and zb) obtained from the optimized conformer geometries as the methyl group rotates not only confirms this conclusion but also shows that the two CH2 hydrogens move in concert (but not equivalently) in opposite directions. Projection of these displacements onto ethylenic twisting and CH2 wagging coordinates, Sφ ) (za - zb)/x2 and Sω ) (za + zb)/x2, respectively, shows that the coefficient of Sφ (Rφ) is much greater than the coefficient of Sω (Rω) for any angle, 60° > τ > 0°.17 Thus, the CH2 out-of-plane displacements are predominantly twisting of the ethylenic CH2 group. The maximum ethylenic twisting angle (dashed curve in Figure 6b) approaches 3°, at τ ) 30°. The maximum out-of-plane displacement of Hc is ∼4°. We conclude that methyl torsion in propene initiates both ethylenic twisting and out-of-plane wagging of the hydrogen adjacent to the methyl group. We will show that these motions, intrinsic parts of the torsional coordinate, cannot be neglected in understanding the shape of the internal rotation potential surface. In spectroscopic language, methyl torsion is coupled to normal modes involving CH2 twist and Hc wag. The former mode is assigned to infrared transitions observed at 990 cm-1 for the A′′ sublevel and 1188 cm-1 (for E), the wag, at 575 cm-1.47 These modes, particularly twist, lie well above the 188 cm-1 methyl torsion frequency48 and indicate that substantial energy separation between large amplitude and normal vibrations is no guarantee for lack of coupling. Spectroscopic conse-

quences of the coupling between torsion and skeletal hydrogen out-of-plane normal modes such as isotopic, intensity, splitting, and intramolecular vibrational relaxation effects are envisioned. A further use of stepwise partitioning in terms of molecular relaxations is to gain insight into the mechanisms controlling the width of the internal rotation barrier. We do this by comparing ab initio potentials calculated using six models incorporating progressively increasing restrictions for flexing motion during the methyl rotation.17 These are as follows. I. The first model is fully relaxed internal rotation involving complete geometry optimization including independent optimization of all skeletal hydrogen out-of-plane wagging angles. This model has no constraints. II. The optimization in I is restricted by clamping the CH2 hydrogen out-of-plane motion. All Hc internal degrees of freedom are allowed. III. The optimization in I is restricted by clamping the skeletal hydrogen adjacent to the methyl group (Hc) out-of-plane motion. All Ha and Hb degrees of freedom are allowed. IV. The optimization in I is restricted by clamping all skeletal hydrogen atom out-of-plane motions. V. This model consists of rigid rotation. All bond distances and bond angles are frozen at equilibrium geometry. VI. This model consists of primitive rigid rotation, as in V, but methyl group dihedral angles and C-H bond distances are set equal to 120° and the average length computed by model I, respectively. Table 3 compares the experimental internal rotation potential constants to those calculated by these six flexing models. All of the experimentally derived potential constants for internal rotation in propene, whether originating from microwave, infrared, or Raman data, lead to a substantial negative V6 term. The generally accepted value is -14.0 cm-1.48 All models which include in-plane flexing (i.e., models I-IV) give essentially the same value for the barrier height determining V3 term. Primitive rigid rotation and rigid-rotation models (V and VI) predict higher barriers, parallel to the acetone discussion in section V. However, the shape-determining V6 term strongly depends on the kind of flexing. In-plane skeletal flexing alone (model IV) yields V6 > 0. If, in addition to the in-plane optimization, out-of-plane motion of Hc is introduced (model II), V6 is still >0. Model III introduces, in addition to the inplane optimizations of model IV, out-of-plane motions of the CH2 hydrogens. The outcome is different from models II and IV. The sign of V6 is reversed from that of model IV, but the magnitude of V6 is too small. Finally model I takes into account

Feature Article

J. Phys. Chem., Vol. 100, No. 8, 1996 2777 TABLE 5: Propene Partitioned Methyl Torsion Energy Differences (cm-1) energy differencea

Figure 7. Effect of phase relationship between V3 and V6 functions in eq 6 on internal rotation barrier width: (dashed curve) in-phase; (solid curve) out-of-phase (as for propene and acetaldehyde).

TABLE 4: Electron Correlation Dependence of Propene Internal Rotation Barrier and Torsional Frequencies (cm-1)a first overtonee methodb experimentf HF CISD MP2 MP3 MP4(SDQ) MP4(SDTQ)

barrier

fundamentalc,d

A

E

693.7 774.5 727.3 718.4 695.4 692.3 689.1

188.1 201.1

357.7 383.9

359.3 385.5

192.5 189.3 189.1 188.5

366.4 360.0 359.0 357.8

368.4 362.3 361.3 360.2

a All theoretical barriers and frequencies are calculated using 6-311G(3df,2p) fully relaxed potential constants. b Potential constants are calculated at MP2 optimized geometries. c Calculated frequencies utilize the fully relaxed ab initio MP2/6-311G(3df,2p) F constant ) 7.1731. d Average of A and E components split by e0.1 cm-1. e 2(A) r 0(A) and 2(E) r 0(E) transitions. f Reference 48.

independent wagging distortions of all three skeletal hydrogens. The result is that the magnitude of V6 is large, 15.0 cm-1, and its sign is opposite that of V3 (the excellent agreement with the experimental value should be noted). The V6 sign conclusions hold at all levels of HF and electron-correlated calculations that we have attempted (see sections VI and VII(b)).17 There are clear conclusions. Coupling of methyl torsion to totally symmetric skeletal stretching vibrations influences the barrier energy. The shape of the propene methyl internal rotation potential function can only be understood by incorporating skeletal hydrogen out-of-plane motions in the large amplitude methyl torsion. The dominant motions are wagging of the hydrogen adjacent to the methyl carbon and ethylenic twist. Figure 7 illustrates the narrowing of the methyl torsional potential barrier that they cause. Increased splittings and crowded together torsional energy levels near the bottom of the widened potential well are consequences. As seen through our discussion of repulsive nuclear virials, the barrier narrowing has its origin in the strain relief provided by out-of-plane C-H bond relaxations accompanying breaking of the equilibrium Cs symmetry. Because this is ultimately an electronic effect derived solely from electrostatic interactions, the out-of-phase relationship between the V3 and V6 functions in eq 5 should occur for many conjugated methyl molecules. Previous explanations have involved the variation in methyl top moment, of inertia with torsional angle.49 (b) Barrier Energetics. We start with the electron correlation dependence of the ab initio calculated potential barrier height (V3) given in Table 4.17 It is clear from Table 4 that electron correlation effects on the barrier height are large and must be considered in any attempt to accurately predict the internal rotation potential function. An MP4(SDTQ) correction

∆E ∆T A′(σ) A′′(π) ∆T(σ+π) ∆V ) ∆Vr + ∆Va ∆Vnn ∆Vee ∆Vne A′(σ) A′′(π) ∆Va ) ∆Vne(σ+π) ∆Vr ) ∆Vee + ∆Vnn

fully relaxed

step Ib

step IIc

step IIId

719

743

-14

-10

-4056 3402 -654 1373 -30 877 -26 785

-1229 3464 2235 -1492 -1392 2064

-3620 -235 -3855 3842 -32 108 -31 194

793 172 965 -977 2623 2345

90 014 -30 980 59 034 -57 662

37 187 -39 350 -2164 672

57 074 10 068 67 143 -63 302

-4247 -1697 -5944 4968

a Difference [MP2/6-311G(3df,2p)] between staggered (180°) and equilibrium eclipsed (0°) conformers (Figure 4). Positive energy differences are destabilizing ones (barrier forming); negative differences are stabilizing (barrier reducing). b Step I represents rigid rotation where the molecule is fixed at its eclipsed geometry except that the methyl group rotates to its staggered position. c Step II. In addition to step I the Cmethyl-Cadj bond is lengthened to its value in the fully relaxed conformation (Figure 4). Energy differences are between this step and the rigid rotation step. d Step III. Other flexings to realize the fully relaxed staggered geometry. Energy differences are between the fully relaxed and step II metastable conformers.

indicates the importance of the correlation effect, as seen from the nearly 90 cm-1 flattening of the potential barrier compared to a HF level calculation. Even a modest MP2 correction decreases the barrier by nearly 60 cm-1. When extensive polarization functions and high-order correlation corrections are included,14 i.e., at the MP4(SDTQ)/6-311G(3df,2p) level, the height is calculated at 689 cm-1, only 5 cm-1 below the experimentally established barrier.48,50 To predict torsional frequencies, the kinetic energy constant, F, is required. In the following discussion a purely ab initio calculated average F from nine MP2/6-311G(3df,3p) fully optimized conformer geometries was employed.17 We start with the fully relaxed model I prediction of the propene fundamental frequency using the MP4(SDTQ)/6-311G(3df,2p) potential function (Table 4). Comparison to the gas phase frequency48 shows impressive agreement, i.e., to within 1 cm-1! The litmus test for the importance of the skeletal hydrogen out-of-plane displacements to the potential shape is that freezing of the Various hydrogen out-of-plane flexings carried out indiVidually or in concert causes large (>10 cm-1) disparity between the predicted and experimental fundamental frequencies. Parallel behavior is found for the overtone transition observed in the Raman spectrum of gaseous propene at 358 cm-1.48 The overtone frequency components are impressively predicted at 357.8 and 360.2 cm-1 (!) by the fully relaxed MP4(SDTQ) potential function, compared to much poorer prediction at the HF level (Table 4). Large disparities are found between the Raman frequency and predictions where skeletal hydrogen outof-plane displacements are missing (model IV) at all basis sets and correlation levels.17 These disparities are even larger than for the fundamental, and as with the fundamental, introduction of either ethylenic CH2 or Hc wagging motion alone produces only minor improvements. We now consider the effects of in-plane skeletal flexings, shown in Table 3 and by the discussion in section VII(a) to influence V3 and consequently the barrier height. Our focus is on the barrier energetics (Table 5), divided into individual attractive and repulsive terms with symmetry partitioning, where applicable. For fully relaxed rotation the kinetic energy sum, (∆E + ∆T) ) 65 cm-1 (Table 5), is ∆E. Concomitant with this reversal of the nuclear virial from repulsive to attractive, there is a change of sign of the potential energy term, ∆V, which goes from negative to positive. In accord with eq 4, kinetic and potential energies have been exchanged! What is happening is that the repulsive nuclear virial for the metastable rotamer generated by step I is distributed into potential energy terms by the Cmethyl-Cadj bond relaxation of step II.14 This change comes from an increase in ∆Vne for step II, which overwhelms decreases in the repulsive terms, ∆Vee and ∆Vnn. Thus, the energetics of steps I + II are qualitatiVely the same as for the fully relaxed process. The magnitudes are different

Goodman et al. because of the negative nuclear virial residue for the still incompletely relaxed internal rotation process defined by steps I + II. Both σ and π electron contributions to the kinetic energy decrease for step II. But the σ decrease is much greater than for π, so much so that it overwhelms the overall π contribution for steps I + II. It is because of the σ term that the overall steps I + II kinetic energy change becomes negative. Parallel effects are found for the ∆Vne(π) and ∆Vne(σ) terms. Both are increased, but the step II increment in ∆Vne(σ) is so much greater than for ∆Vne(π) that ∆Vne(total) is controlled by ∆Vne(σ). The change in sign of ∆Vee caused by step II needs emphasis. This term is now antibarrier, implying that four-electron interactions after Cmethyl-Cadj relaxation no longer are barrier forming. Thus, it is relaxation of the Cmethyl-Cadj bond that leads to barrier energetics dominated by σ electron interactions.14 The remaining skeletal flexings in step III modulate the large negative nuclear virial incurred in step II; that is, both σ and π components of ∆T increase. Although the individual energy and total energy corrections are relatively small (-10 cm-1 for the latter), the effect of this step is important. It stabilizes the metastable molecule possessing the stretched Cmethyl-Cadj bond of step II. But the physics of the origin of the barrier is not changed from step II. The outcome is that although the propene rigid rotation barrier energy is not very different from the fully relaxed one, conclusions about the physics of barrier formation in propene drawn from a one-dimensional (rigid frame) view of internal rotation can be misleading. Both repulsive potential and kinetic energies are barrier forming for rigid rotation, but they have opposite senses for the fully relaxed process. It is the barrierforming attractive potential that dominates for the overall fully relaxed process. And only for this latter process is the barrier formed by an increase in the total potential energy. This indepth analysis shows that the electron attribution of energetics of the barrier-forming term also depends on the rotational process. When multidimensional, σ-bonding changes dominate; when one-dimensional, π. The extreme sensitivity of the energy decomposition to geometric relaxation casts doubt on the value of antagonistic ∆V and ∆T components to reveal driving forces for the energy change. An illustration of this dilemma was provided by Ruedenberg’s analysis of H2+ bond energetics carried out 30 years ago.53 Here ∆E ) E(H2+) - E(H+H+) is negative because ∆V < 0 in spite of ∆T > 0 (just the reverse of internal rotation energetics). Nonetheless, Ruedenberg’s detailed analysis showed that the bond formation is a consequence of behavior of the kinetic energy.53b In a parallel fashion relaxed barrier energies conceal the strain effects, which arise from steric and bond-weakening interactions. To dissect the interplay of σ and π barrier-forming terms with geometric relaxation effects, we carry out two additional calculations. The first (section VIII) considers the energy difference between eclipsed and staggered geometries for the propene2+ π cation (both ionizing electrons are from the frontier a′′ ethylenic π orbital). Although no experimental barrier is available for the propene2+ π cation, this species, by changing the balance between σ and π electron interactions, provides insight into the complex competition between these interactions in forming the internal rotation barrier. The second (section IX) considers energy decomposition for methyl torsion in acetaldehyde. In this molecule we combine symmetry partitioning with dissection in terms of natural bond orbitals (NBO). Weinhold7,54,55 has shown the power of NBO analysis in revealing barrier sources in terms of chemically appealing bond

Feature Article

J. Phys. Chem., Vol. 100, No. 8, 1996 2779 TABLE 6: Propene2+ π Cation Partitioned Methyl Torsion Energy Differences (cm-1) energy differencea ∆E ∆T A′(σ) A′′(π) ∆T(σ+π) ∆V ) ∆Vr + ∆Va ∆Vnn ∆Vee ∆Vne A′(σ) A′′(π) ∆Va ) ∆Vne(σ+π) ∆Vr ) ∆Vee + ∆Vnn

fully relaxed

step I (rigid rotation)

1241

1547

2771 -4242 -1471 2712 36 111 8746

5984 -5493 491 1056 107 649 63 753

-47 204 5059 -42 145 44 857

-163 209 -7138 -170 347 171 402

a Difference [MP2/6-311G(3df,2p)] between staggered (180°) and equilibrium eclipsed (0°) conformers (Figure 5).

localizations. NBO energy changes are also more realistic than the large symmetry-decomposed energy terms. Because acetaldehyde possesses only six bonds and both σ and π lone pairs, it is particularly appropriate for this kind of dissection.56

of the π electrons have been removed in the ion. It follows that repulsions stemming from π electrons (comprising π-π and π-σ interactions) act to decrease ∆Vee, since the electron repulsion term is antibarrier in the neutral molecule. If only π interactions are considered, as in the HPD π fragment model, the interactions that contribute to stabilization of the neutral propene eclipsed conformation are either reversed, not present, or unchanged in the +2 π cation, with consequent predicted lowering of the barrier. There is another important change from neutral propene. The barrier is much more sensitiVe to geometric relaxation in the cation. Referring to Table 6, the fully relaxed barrier is 25% (306 cm-1) lower than calculated for the step I rigid frame internal rotation process, compared to the much smaller 3% (12 cm-1) decrease found for the neutral molecule. This difference illustrates the complex and frequently confusing interplay between geometric relaxation and σ and π interactions in the barrier origin mechanism for conjugated methyl molecules.

VIII. Propene2+ π Cation

IX. Acetaldehyde

The MP2/6-311G(3df,2p) optimized geometries are given in Figure 8. The cation equilibrium geometry retains the Cs symmetry and eclipsed equilibrium conformation of neutral propene, but the CdC bond is lengthened and the Cmethyl-Cadj bond shortened. The methyl group is also tilted about 9° from the Cmethyl-Cadj bond axis. As for neutral propene, the top-ofbarrier geometry is Cs staggered. Parallel to the geometry changes that take place upon internal rotation in neutral propene the Cmethyl-Cadj bond lengthens (by 0.004 Å) on going to the staggered conformer. In the cation, internal rotation shortens the CdC bond by 0.008 Å, an effect not present in neutral propene. The barrier energy and partitioned energy terms are listed in Table 6. The calculated fully relaxed barrier (1241 cm-1)14 is roughly double that for neutral propene (719 cm-1). ∆T is negative and reasonably satisfies the virial theorem; the positive ∆V term forms the barrier. Its sign is controlled by increased repulsive potential energies (∆Vee and ∆Vnn > 0), and the antibarrier ∆Vne term modulates the magnitude. These results are opposite to that found for neutral propene. The decrease in ∆T arises solely from π electrons, contrasted to a σ electron origin in the neutral molecule. Other reversals from neutral propene are the ∆Vne(σ) term, now antibarrier, and the (now) barrier-forming ∆Vne(π) term. However, the σ contribution dominates, and consequently ∆Vne(total) is antibarrier. It is also clear that the reversal in sign of ∆Vee, from a decrease in the neutral molecule to an increase in the cation, involves the increased role of σ electron interactions, since half

We now consider changes in the barrier potential by altering the nonpolar propene double bond to the polar one in acetaldehyde. The presence of an oxygen lone electron pair in the aldehyde provides another electronic difference. Both molecules have Cs equilibrium symmetry with the in-plane hydrogen atom eclipsing (s-cis) the double bond linkage and top-of-barrier staggered conformations obtained by 180° methyl group rotation. Both are isoelectronic, with identical numbers of π and σ electrons.51 These factors are electronic/structural in nature. However, from the multidimensional view of internal rotation that we are emphasizing, the fewer vibrational degrees of freedom in acetaldehyde provide an additional important difference: altered couplings between methyl torsion and other large amplitude vibrations. Acetaldehyde differs from propene in that it possesses a unique skeletal hydrogen out-of-plane vibration, the aldehyde hydrogen was fundamental at ∼764 cm-1, well separated from transitions involving the methyl torsional mode.57 Parallel structural changes induced by methyl rotation in the two molecules are lengthening of the Cmethyl-C bond (0.007 Å), smaller than in propene, and greater expansion of the CCmethylHip angle (1.5°) (Figure 9). Another change is substantial contraction of the OCCmethyl angle. The most noteworthy difference is that, in contrast to propene, the acetaldehyde methyl group is tilted (0.5° for eclipsed and 1.4° for staggered) from the Cmethyl-Cald bond axis in both conformations. (a) Barrier Shape. We proceed to analyze the τ dependence of the rigid-rotation nuclear virial (shown in Figure 10 by open

Figure 8. MP2/6-311G(3df,2p) optimized geometries for propene2+ π cation equilibrium eclipsed and top-of-barrier staggered conformers.

2780 J. Phys. Chem., Vol. 100, No. 8, 1996

Figure 9. MP2/6-311G(3df,2p) optimized geometries for acetaldehyde equilibrium eclipsed and top-of-barrier staggered conformers.

circles). Unlike in propene, it increases monotonically as rotation proceeds. The 0 curve is for ∑RχR‚FREwPR, where all in-plane flexings are allowed to occur, but the aldehyde hydrogen out-of-plane one is omitted. Like in propene, it passes through a maximum near 30°, vanishing at 60°. The virial for the partially relaxed conformation obtained by freezing all inplane relaxations (∆ curve) shows that allowing aldehyde hydrogen out-of-plane relaxation causes ∑Rχa‚FaEwPR to also monotonically increase, but for intermediate rotation angles its magnitude is decreased from ∑RχR‚FREwR. Thus, the virial analysis shows two kinds of relaxation effects: (1) aldehyde hydrogen out-of-plane flexing, which occurs only for intermediate torsional angles, and (2) in-plane skeletal flexing, arising from changes in the molecular geometry on going to the staggered conformation. These conclusions are parallel to those found for propene. The τ dependence of the aldehyde hydrogen wagging angle, calculated in a manner similar to propene (] curve in the upper part of Figure 10), predicts a maximum wagging angle >2°, occurring when τ is near 30°. Since the turning point for one quantum of the methyl torsion (∼210 cm-1 above the internal rotation potential minimum23) is calculated to be 32°, activation of the methyl torsion fundamental initiates ∼5° oscillations (twice the wagging angle) of the aldehyde hydrogen above and below the skeletal plane. The wagging motion oscillation exceeds 3° even at the turning points for the torsion zero-point level (∼17°). The conclusion obtained from Figure 10 is that methyl torsion is strongly coupled to aldehyde hydrogen outof-plane wag. The simplicity of the acetaldehyde “torsion”s contaminated only by the aldehyde hydrogen wagsallows the

Goodman et al.

Figure 10. (lower) Nuclear virials for acetaldehyde internal rotation: circles, rigid rotation; triangles, all out-of-plane skeletal motions allowed to occur, but in-plane skeletal relaxations frozen; squares, all in-plane skeletal relaxations occur, but out-of-plane aldehyde hydrogen motion frozen. (upper) Dependence of aldehyde hydrogen out-of-plane wagging angle on the methyl torsional angle: diamonds, methyl group allowed to fold; squares, methyl group taken as C3V rigid top.

effect of folding of the methyl top to be delineated. The calculated wag displacement for rotation of a C3V rigid methyl top (but at the same time allowing full skeletal flexing) (0 curve in the upper part of Figure 10) shows that wagging motion is still initiated, although with some diminution in the maximum oscillation angle. We conclude that folding of the methyl group, i.e. local methyl group symmetry, plays only a minor role in the coupling mechanism and that the coupling largely arises from oVerall lowered molecular symmetry when 0° < τ < 60°. The experimental internal rotation barrier in acetaldehyde has been the object of intense scrutiny.4,40 All studies, microwave, infrared, and Rydberg state jet spectroscopies, conclude that the sign of V6 is negative, and the most recent ones conclude that V6 is near -13 cm-1. As in the case of propene, we analyze the barrier shape by comparing ab initio potentials calculated using models incorporating progressively increasing restrictions for flexing motion during the methyl rotation. These are as follows: I. complete geometry optimization along all internal coordinates during the methyl rotation II. partial geometry optimization which shuts off out-of-plane motion of the aldehyde hydrogen III. rigid rotation, shutting off all skeletal and aldehyde hydrogen flexing motions

Feature Article

J. Phys. Chem., Vol. 100, No. 8, 1996 2781

TABLE 7: Skeletal Flexing Dependence of Acetaldehyde Internal Rotation Potential Constants (cm-1)a method

V3

V6

V9

experimentb

407.9 412.2 412.5 500.7 444.7

-12.9 -8.1 3.2 34.2 -0.9

-0.3 -0.5 31.6 -1.0

model Ic model IId model IIIe model IVf

a MP4 SDTQ/6-311G(3df,2p) level at MP2/6-311G(3df,2p) geometry. b Reference 40c. c All skeletal and methyl group hydrogen relaxations independently allowed. d Relaxations in model I calculation restricted by clamping aldehyde hydrogen out-of-plane motion. e Rigid rotation. f Primitive rigid rotation with a C3V symmetry methyl group having identical C-H lengths and 120° HCH angles.

IV. as in III, but regarding the methyl group as a rigid C3V symmetry top with identical hydrogen C-H distances and HCH angles (primitive rigid rotation) Table 7 compares internal rotation potential constants, calculated for these four models, to the experimental constants. The two models which include in-plane flexing (I and II) give essentially the same value for the barrier height determining V3 term. Rigid-rotation and primitive rigid rotation models (III and IV) predict higher barriers, parallel to the results of analogus model calculations for propene. However, the shape-determining V6 term, as for propene, strongly depends on the flexing details. The difference between flexing models which include (model I) or clamp (model II) out-of-plane motion of the aldehyde hydrogen is dramatic. The sign of V6 is negative only when out-of-plane wagging of the aldehyde hydrogen is included. The important conclusion is that the negatiVe sign of V6 can only be understood by requiring coupling of the two large amplitude motions: methyl torsion and out-of-plane aldehyde hydrogen wag.18 (b) Barrier Energetics. The electron correlation dependence of the barrier height determining V3 term is significant; that is, there is a more than 50 cm-1 lowering of the HF barrier.45 At the MP4(SDTQ) level, the height is calculated as 412 cm-1, only 4 cm-1 above 408 cm-1, the most recent experimentally established barrier.40,58 These results are parallel to propene and affirm the general principle that accurate calculation of torsional barrier heights requires correlation corrections. The virial theorem is poorly satisfied at the MP2/6-311G(3df,2p) level, since ∆E + ∆T > 250 cm-1, an appreciable fraction of ∆E itself. However, a variational HF/6-31G(d,p) calculation gives good virial theorem satisfaction, ∆E + ∆T < 7 cm-1 (Table 7) despite the modest basis set. We emphasize that although there are large quantitative differences between HF/6-31G(d,p) and MP/26-311G(3df,2p) partitioned energies, both calculation levels yield the same qualitative behaviors shown in Tables 7 and 8. Table 8 shows the change in attractive, repulsive, and total potential energies. Attractive and repulsive potential changes have opposite sense for fully relaxed rotation, the dominant attractive term forming the barrier.13b,44 However, they reverse senses for rigid rotation, ∆Va now antibarrier and ∆Vee (and consequently ∆Vr) barrier forming. These reversals underline the role that repulsions play in the strain effects illustrated in Figure 10. Comparison of energy partitioning calculations for acetaldehyde in Table 8 to those for propene in Table 5 shows that potential terms for fully relaxed rotation all have the same sense as in propene, including inequalitites and dominance of barrier-forming σ interactions over antibarrier π. The relative sizes of the σ and π term components provide an important difference between the two molecules, however. As in propene, the barrier-forming ∆V term, for the multidimensional fully relaxed rotation process, arises entirely from

TABLE 8: Acetaldehyde Partitioned Internal Rotation Energy Differences (cm-1) energy differencea ∆E ∆T A′(σ) A′′(π) ∆T(σ+π) ∆V ) ∆Vr + ∆Va ∆Vnn ∆Vee ∆Vne A′(σ) A′′(π) ∆Va ) ∆Vne(σ+π) ∆Vr ) ∆Vee + ∆Vnn

fully relaxed

step Ib

step IIc

step IIId

373

415

-6

-36

-6321 5955 -366 739 -4431 -692

-4213 5495 1282 -867 1658 3737

-2500 -171 -2671 2665 -22 126 -21 747

392 631 1023 -1059 16 037 17 318

61 924 -56 062 5862 -5123

45 123 -51 385 -6262 2330

39 533 7005 46 538 -43 873

-22 732 -11 682 -34 414 33 355

a Difference [HF/6-31G(d,p)] between staggered (180°) and equilibrium eclipsed (0°) conformers (Figure 8). Also see footnote a, Table 5. b-d See corresponding footnotes, Table 5.

TABLE 9: Acetaldehyde Barrier Forming Bond Energy Changes (cm-1) energy differenceb bonda

fully relaxed

rigid rotation

C-Cmethyl(σ) C-Hmethyl(op)c LP O(n) C-O(σ) LP O(σ)

2223 1100 709 340 176

50 1163 917 110 329

a Natural bond orbitals calculated at the HF 6-31G(d,p) level (ref 56). b Energy differences between population-weighted natural bond orbital energies for the staggered-equilibrium eclipsed conformers given in Figure 8. Threshold: 100 cm-1. c Total of C-Hmethyl(op) contributions is twice this value.

∆Vne, and its principal contribution arises through relaxation of the Cmethyl-Cadj bond. The magnitude of the bond lengthening provides an indicator of the repulsive nuclear virial. In acetaldehyde it lengthens by 0.007 Å, compared to 0.0095 Å for propene. Our symmetry analysis shows that σ-bonding changes resulting from the lengthening are the major determinants for the barrier energy in propene and acetaldehyde. To further pinpoint the origin of the barrier in terms of chemical bonds, the individual NBO energy changes are presented in Table 9 for both fully relaxed and rigid rotation. The calculations are at the HF 6-31G(d,p) level in order to avoid complexities generated by higher level basis sets. Examination of Table 9 shows that the dominant barrier-forming term for fully relaxed rotation is the increase in C-C σ energy, exceeding the C-Hmethyl(op) term by more than a factor of 2.56 The C-C σ term all but Vanishes for rigid rotation, whereas the hyperconjugatiVe C-H term scarcely changes, rationalizing the sensitivity of the energy decomposition to geometric relaxation. Thus, it is weakening of the C-C σ bond that is the barrier source for relaxed rotation changing to loss of hyperconjugation for rigid rotation. There are also less important interactions between σ orbitals localized in the CdO and C-Hald bonds and C-Hmethyl orbitals causing charge transfer back and forth between these bonds.56 Similar but more pronounced results are found for propene.59 Thus, the barrier origin in propene and acetaldehyde appears to stem from a partial unmaking of the bond between Cmethyl and the adjacent carbon atom. While this conclusion seems to be at variance with the π fragment model, which neglects repulsive and attractive interactions in the σ framework, the role of π electrons is not ignorable: π effects contribute to the strain, as demonstrated by the π electron kinetic energy increase as rigid rotation

2782 J. Phys. Chem., Vol. 100, No. 8, 1996 proceeds. But overall energetics, and hence barrier heights, are controlled by σ interactions. X. Future Directions In the foregoing discussions of internal rotation for three basic conjugated methyl molecules, acetone, propene, and acetaldehyde, the multidimensional nature of internal rotation is seen to be key in understanding physical mechanisms controlling shape and barrier height of the torsional potential surfaces. A principal purpose of this article has been to provide a qualitative physical model. Despite the success of most simulations of torsional potential terms and frequencies, significant quantitative differences from some of the experimental values remain. It seems reasonable to assume that these disparities relate to the absence of any τ dependence of T and that this affects the comparison with experiment. No matter what the level of sophistication of ab initio calculation, some of the empirical potential terms and experimental frequencies always differ from those predicted when no dependence of T on τ is assumed. Recently, Munoz-Caro, Nino, and Moule60 have taken a step in including T-τ variation by including a cos 9τ term in the potential function for acetaldehyde. A small (∼1 cm-1) effect is produced on the barrier. Clearly more studies are needed; it may be possible to describe the experimental spectrum using an operator of the form T(τ) + V(τ), where V(τ) is a one-dimensional cut through the potential surface and T(τ) is a correctly formulated kinetic energy. The ab initio approach to the torsional energy problem described in the previous sections averages energy over all of the vibrational motions. This is a static quantity independent of the nuclear masses and of the vibrational state of the molecule. In this formulation, the nuclei do not vibrate; they are fixed at the potential minimum. While this type of approach allows appreciation of barrier heights and potential shapes, it glosses over specific vibrational effects on the torsional motion. For example, it does not allow one to ascertain the effect of deuteration, designed to change the amplitude of a specific CH out-of-plane mode, on the torsional energy. The quantity E(τ) actually depends on both vibrational quantum numbers and nuclear masses. Even the contribution of the zero-point vibrations, found to be negligible in the static approach, may be affected. Exploration of this kind of dynamical approach to internal rotation potentials should provide a good opportunity for a serious dialogue between theory and experiment, e.g., jet experiments involving selective deuterations. Zero-kinetic-energy (ZEKE) photoelectron spectroscopy has potential for revealing torsional spectroscopic information in radical cations. The technique has been sufficiently developed61 so that barrier tunneling induced splittings as well as accurate torsional frequencies could be measured, allowing shape and height of cation internal rotation barriers to be obtained. Comparison of accurate experimental cationic and ground state torsional potential curves should allow an incisive test of internal rotation potential models. Another main need is to connect the implications of these methyl barrier studies to other classes of rotational barrier problems. Some interesting and important classes, all of which have undergone spectroscopic and/or computational rotational barrier studies, include NO2, CF3, NO, NH2, OH, and phenyl rotors. In this regard extension of the energy and natural bond orbital analyses to aromatic skeletons would be salutary. XI. Concluding Remarks We have discussed several consequences of the multidimensional nature of methyl internal rotation in three basic conjugated

Goodman et al. molecules: propene, acetaldehyde, and acetone. The negative sign of the V6 coefficient in the internal rotation potential functions for acetaldehyde and propene ground states cannot be understood without appreciation of impure torsional motion. This out-of-phase relationship between the V3 and V6 component functions in the torsional potential arises from coupling of torsional and skeletal hydrogen out-of-plane wagging motions. This connection between the sign of V6 and torsional purity expands the usefulness of microwave spectra to obtain information about the canonical nature of the torsional vibration. Without this connection only the form of the potential barrier could be obtained. Another consequence of the multidimensional nature of internal rotation is the major role that strain plays in torsional energetics. In each of the molecules that we discuss, rigid rotation leaves the molecule in a strained metastable conformation, with the barrier appearing to originate from a kinetic energy and repulsive Coulombic potential increase. The latter, part of the “steric hindrance effect”, arises from four-electron interactions (containing Pauli repulsions6) and from nuclear repulsions. In the strained molecule the repulsive nuclear virial contribution to the kinetic energy increase arises from the π electrons. To remove both the repulsive nuclear virial and repulsive potential, the molecule has to relax by lengthening of the Cmethyl-Cadj bond. Other relaxations appear to be relatively ineffective. The greater sensitivity of σ energies to Cmethyl-Cadj bond lengthening overwhelms the π contribution, and in the stretched C-C bond conformation it is σ electron interactions which dominate barrier energetics. These σ effects, larger in propene than in acetaldehyde, have their origin in the partial unmaking of the bond between Cmethyl and the adjacent carbon atom. In this respect there is a parallel between the small conjugated molecules discussed here and ethane.8 There is a temptation to identify the increase in σ electronnuclear attraction energy as the “barrier origin”. Breakdown of the energy difference between eclipsed and staggered geometries in terms of antagonistic ∆V and ∆T components does not, in and of itself, reveal sources. However, natural bond orbital analysis, when combined with symmetry energy partitioning, allows identification of the C-Cmethyl bond weakening as the major barrier-forming term. One further conclusion is clear: because σ and π interactions have different sensitivities to skeletal relaxation, models for the physical origin of barriers which do not take into account the multidimensional nature of internal rotation are suspect and give a false sense of intuition. Acknowledgment. Financial support by the National Science Foundation and Donors of the Petroleum Research Fund, administered by the American Chemical Society, and computational grants by the Pittsburgh Supercomputer Center are gratefully acknowledged. We take this opportunity to especially acknowledge the contributions of members of the Rutgers internal rotation group: Professors John G. Philis, and Surya N. Thakur, Drs. Joel Berman, Ali G. Ozkabak, and Ding Guo, and Ms. Hongbing Gu. References and Notes (1) Nielson, H. H. Phys. ReV. 1932, 40, 445. (2) Pitzer, K. S. J. Chem. Phys. 1937, 5, 466, 473. (3) Philis, J. G.; Berman, J. M.; Goodman, L. Chem. Phys. Lett. 1990, 167, 16. (4) Gu, H.; Kundu, T.; Goodman, L. J. Phys. Chem. 1993, 97, 7194. (5) (a) Kundu, T.; Thakur, S. N.; Goodman, L. J. Chem. Phys. 1992, 97, 5410. (b) Philis, J. G.; Goodman, L. J. Chem. Phys. 1993, 98, 3795. (6) Pitzer, R. M. Acc. Chem. Res. 1983, 207, 16. (7) Brunck, T. K.; Weinhold, F. J. Am. Chem. Soc. 1979, 101, 1700.

Feature Article (8) Bader, R. F. W.; Cheeseman, J. R.; Laidig, K. E.; Wiberg, K. B.; Breneman, C. J. Am. Chem. Soc. 1990, 112, 6530. (9) Hehre, W. J.; Pople, J. A.; Devaquet, A. J. P. J. Am. Chem. Soc. 1976, 98, 664. (10) For example: Radom, L.; Baker, J.; Gill, P. M. W.; Nobes, R. H.; Riggs, N. W. J. Mol. Struct. 1985, 126, 271. Pross, A.; Radom, L.; Riggs, N. V. J. Am. Chem. Soc. 1980, 102, 2253. (11) Dorigo, A. E.; Pratt, D. W.; Houk, K. N. J. Am. Chem. Soc. 1987, 109, 6591. (12) Hadad, C. M.; Foresman, J. B.; Wiberg, K. B. J. Phys. Chem. 1993, 97, 4293. (13) (a) Jorgensen, W. L.; Allen, L. C. J. Am. Chem. Soc. 1971, 93, 567. (b) Munoz-Caro, C.; Nino, A.; Moule, D. C. Theor. Chim. Acta 1994, 88, 299. (14) Kundu, T.; Goodman, L.; Leszczynski, J. J. Chem. Phys. 1995, 103, 1523. (15) Goodman, L.; Kundu, T.; Leszczynski, J. J. Am. Chem. Soc. 1995, 117, 2082. (16) Wiberg, K. B.; Martin, E. J. Am. Chem. Soc. 1985, 107, 5035. (17) Goodman, L.; Leszczynski, J.; Kundu, T. J. Am. Chem. Soc. 1993, 115, 11991. (18) (a) Goodman, L.; Leszczynski, J.; Kundu, T. J. Chem. Phys. 1994, 100, 1274. (b) Nino, A.; Munoz-Caro, C.; Moule, D. C. J. Phys. Chem. 1994, 98, 1519. (19) Lister, D. G.; MacDonald, J. N.; Owen, N. L. Internal Rotation and InVersion: An Introduction To Large Amplitude Motions In Molecules; Academic Press: New York, 1978. (20) Payne, P. W.; Allen, L. C. In Applications of Electronic Structure Theory; Schaefer, H. F., Ed.; Plenum: New York, 1977; pp 29-108. (21) Veillard, A. In Internal Rotation in Molecules; Orville-Thomas, W. J., Ed.; John Wiley and Sons: New York, 1974; pp 385-421. (22) Spangler, L. H.; Pratt, D. W. In Jet Spectroscopy and Molecular Dynamics; Hollas, J. M., Phillips, D., Eds.; Chapman and Hall: London, 1995; pp 369-398. (23) For example: Ozkabak, A. G.; Goodman, L. J. Chem. Phys. 1992, 96, 5958. (24) (a) Slater, J. C. J. Chem. Phys. 1933, 1, 687. (b) Nelander, B. Ibid. 1969, 51, 469. (25) (a) Swalen, J. D.; Costain, C. C. J. Chem. Phys. 1959, 31, 1562. (b) Grant, D. M.; Pugmire, R. J.; Livingston, R. C.; Strong, K. A.; McMurry, H. L.; Brugger, R. M. Ibid. 1970, 52, 4424. (26) (a) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213. (b) Hehre, W. J.; Random, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (27) Crighton, J. S.; Bell, S. J. Mol. Spectrosc. 1985, 112, 285; 1986, 118, 383. (28) Wiberg, K. B.; Murko, M. A. J. Phys. Chem. 1987, 91, 3616. Wiberg, K. B.; Murcko, M. A.; Laidig, K. E.; MacDougall, P. J. Ibid. 1990, 94, 6956. (29) Head-Gordon, M.; Pople, J. A. J. Phys. Chem. 1993, 97, 1147. (30) Ozkabak, A. G.; Philis, J. G.; Goodman, L. J. Am. Chem. Soc. 1990, 112, 7854. (31) Head-Gordon, M.; Pople, J. A. Chem. Phys. Lett. 1990, 173, 585. (32) Ozkabak, A. G.; Goodman, L. Chem. Phys. Lett. 1991, 176, 19. (33) Goodman, L.; Ozkabak, A. G. J. Mol. Struct. (THEOCHEM) 1992, 261, 367. (34) Bell, S. J. Mol. Struct. 1994, 125, 320. (35) Smeyers, Y. G.; Senent, M. L.; Botella, V.; Moule, D. C. J. Chem. Phys. 1993, 98, 2754. (36) Smeyers, Y. G.; Senent, M. L.; Penalver, F. Y. J. J. Mol. Struct. (THEOCHEM) 1993, 287, 117. (37) Senent, M. L.; Moule, D. C.; Smeyers, Y. G. Chem. Phys. Lett. 1994, 221, 512.

J. Phys. Chem., Vol. 100, No. 8, 1996 2783 (38) Munoz-Caro, C.; Nino, A.; Moule, D. C. J. Mol. Struct. (THEOCHEM) 1994, 315, 9. (39) (a) Munoz-Caro, C.; Nino, A.; Moule, D. C. Chem. Phys. 1994, 186, 221. (b) Nino, A.; Munoz-Caro, C.; Moule, D. C. J. Mol. Struct. (THEOCHEM) 1994, 318, 237. (c) Senent, M. L.; Moule, D. C.; Smeyers, Y. G. J. Phys. Chem. 1995, 99, 7970, 8515. (40) (a) Belov, S. P.; Yu Tretyakov, M.; Kleiner, I.; Hougen, J. T. J. Mol. Spectrosc. 1993, 160, 61. (b) Kleiner, I.; Hougen, J. T.; Suenram, R. D.; Lovas, F. J.; Godefroid, M. Ibid. 1991, 148, 38. (c) Kleiner, I.; Hougen, J. T.; Suenram, R. D.; Lovas, F. J. 1992, 153, 578. (41) The connection between potential constants and torsional energy levels is obtained by variationally solving the torsional Schrodinger equation, Hφ ) Eφ. Details about internal rotation Hamiltonian matrices, kinetic energy terms, and convergence to torsional energy levels can be found in many papers, e.g., refs 19, 25, 27, and 35. In transforming frequencies to potential constants for a two-rotor molecule like acetone, the average frequency of the two torsional fundamentals strongly influences the V3 and V33 terms; the splitting between them, the important barrier shape determining V′33 term. It was the historical lack of a measured splitting between the two torsional fundamentals that prevented accurate assessment of the barrier width. (42) Nelson, R.; Pierce, L. J. Mol. Spectrosc. 1965, 16, 344. (43) Groner, P.; Guirgis, G. A.; Durig, J. R. J. Chem. Phys. 1987, 86, 565. (44) (a) Allen, L. C. Chem. Phys. Lett. 1968, 2, 597. (b) Davidson, R. B.; Allen, L. C. J. Chem. Phys. 1971, 54, 2828. (45) Leszczynski, J.; Goodman, L. J. Chem. Phys. 1993, 99, 4867. (46) Note that details of the τ dependence of partially relaxed nuclear virials suffer the inaccuracies mentioned in section II for rigid rotation; however, trends are unaffected. (47) Silvi, B.; Labarbe, P.; Perchard, J. P. Spectrochim. Acta 1973, 29A, 263. (48) Durig, J. R.; Guirgis, G. A.; Bell, S. J. Phys. Chem. 1989, 93, 3487. (49) (a) Ewing, C. S.; Harris, D. O. J. Chem. Phys. 1970, 52, 6268. (b) Lees, R. M. Ibid. 1973, 59, 2690. (50) The harmonic zero-point-energy correction to the 693 cm-1 experimental barrier48 is estimated17 as -8 cm-1, indicating that the propene barrier energy is nearly entirely electronic in nature. (51) Cs symmetry allows the potential terms to be additionally broken down into contributions from a′ and a′′ orbitals. Since the molecular skeleton remains planar and the methyl group valence orbitals can be classified as π and σ types,52 this symmetry partitioning naturally separates π and σ effects. (52) Hoffman, R. Pure Appl. Chem. 1970, 24, 567. (53) (a) Ruedenberg, K. ReV. Mod. Phys. 1962, 34, 326. (b) Feinberg, M. J.; Ruedenberg, K. J. Chem. Phys. 1971, 54, 1495. (c) Feinberg, M. J.; Ruedenberg, K.; Mehler, E. AdV. Quantum Chem. 1970, 5, 27. (54) Wesenberg, G.; Weinhold, F. Int. J. Quantum Chem. 1982, 21, 487. (55) Reed, A. E.; Weinhold, F. Isr. J. Chem. 1991, 31, 277. (56) Guo, D.; Goodman, L. J. Phys. Chem., submitted (57) Hollenstein, H.; Gunthard, H. H. Spectrochim. Acta 1971, 27A, 2027. (58) The harmonic zero-point-energy correction is estimated to range from -8 to -19 cm-1.15,34 (59) Gu, H.; Goodman, L. J. Am. Chem. Soc., submitted. (60) Munoz-Caro, C.; Nino, A.; Moule, D. C. J. Chem. Soc., Faraday Trans. 1995, 91, 399. (61) For example: Blush, J. A.; Chen, P.; Wiedmann, R. T.; White, M. G. J. Chem. Phys. 1993, 98, 3557. (62) Huzinaga, S. J. J. Chem. Phys. 1965, 42, 1293. Dunning, T. H. Ibid. 1970, 53, 2823.

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