The Energy Profile for Rotation about the C-C Bond in Substituted

In the Laboratory

The Energy Profile for Rotation about the C–C Bond in Substituted Ethanes A Multi-Part Experimental and Computational Project for the Physical Chemistry Laboratory Luther E. Erickson and Kevin F. Morris Department of Chemistry, Grinnell College, Grinnell, IA 50112

The potential energy profile for rotation about the C–C bond of ethane or any substituted ethane, as shown in Figure 1 for n-butane, has been the focus of extensive experimental and theoretical investigation for many years (1– 4). An understanding of the factors that determine the potential energy profile is essential to calculate relative energies—and hence preferred geometries—of the many different conformations available to larger molecules. Such understanding is the basis of the molecular modeling techniques that have become an essential tool for research in chemistry and molecular biology. For several years we have linked a pair of traditional physical chemistry lab experiments, a dipole moment determination and the analysis of a high-resolution NMR spectrum in terms of coupling constants and chemical shifts of protons on adjacent carbons. Each experiment provides information about the relative populations of the low-energy rotamers of a substituted ethane, thus permitting an assessment of the energy difference between these rotamers. The effect of solvent properties on the energy difference was typically included in the investigation. Recently we have expanded the scope of the project to include examination of the infrared spectrum of the same substituted ethanes and theoretical calculations of the entire energy profile, including both molecular mechanics and quantum mechanical methods. In addition to introducing these important experimental and computational techniques, the several experiments together constitute a significant research project in which a combination of experimental and theoretical techniques is brought to bear on a single problem. Three to four lab periods are required to complete all aspects of the project. One or more parts of the project— each requiring approximately one 3-hour lab period—can be incorporated into courses with a different weighting of topics. We usually allocate an additional period for completing calculations and writing a formal report. We have sometimes also invited other students and staff to a poster presentation to report results and compare data for the different compounds that the class examined.

900

Experimental Techniques for Determining Rotational Energy Differences Experimental approaches for establishing features of the potential energy profile can be classified into those that are able to detect the individual low-energy rotamers and those that yield an average property, which can be used to estimate the relative populations of low-energy rotamers. When the energy differences between rotamers are small and the energy barrier to the interconversion of rotamers is low, a macroscopic sample will contain an equilibrium mixture of rotamers and the relative populations of the distinct low-energy rotamers will vary with temperature according to a Boltzmann distribution. Whether the separate rotamers can be detected or not, the equilibrium ratio of fractional populations, f j /f i , of

20

15 Rotamer Energy/ (kJ/mol)

Overview and Scope of Project

CH3

CH3

CH3

CH3

CH3

10

CH3

5

0

-5

0

60

120 180 240 Dihedral Angle

300

360

Figure 1. Energy profile for internal rotation of n - butane.

Journal of Chemical Education • Vol. 75 No. 7 July 1998 • JChemEd.chem.wisc.edu

In the Laboratory

any pair of low energy rotamers is given by eq 1. f j /f i = e᎑∆E /RT

(1)

where ∆E is Ej – Ei , the potential energy difference between low-energy rotamers j and i (“gauche” and “anti” or trans in Fig. 1). So if an experimental method is fast enough to detect and determine relative amounts of the separate rotamers, the ratio of populations (or concentrations) of the rotamers yields the energy difference between the minima as ∆E = ᎑RT ln(f j /f i )

(2)

These discussions are often presented in terms of the standard free energy difference between rotamers and the ratio of populations of that pair of rotamers; that is, ∆G° = ᎑RT ln K, where K = f j /f i for the rotamer conversion i → j. On the assumption that the three rotamers have essentially equal entropies and that ∆H = ∆U = ∆E for this intramolecular process, ∆E is identical to ∆G°. Note that, in using eq 1, we need to be careful to employ fractional populations of a pair of rotamers and not, for example fgauche /ftrans , since there are two equalenergy gauche rotamers. Whereas the energy differences between low-energy rotamers can be determined from the relative populations of the rotamers, the rate of interconversion of two rotamers is determined by the energy barrier that must be overcome (∆E*

= 15.1 kJ mol᎑1 in Fig. 1). The Eyring equation for the rate constant, k1 = (kBT/h) e᎑∆G*/RT (where kB and h are Boltzmann and Plank’s constants), can be used to estimate the interconversion rate from the Gibbs free energy of activation, ∆G*, which for this intramolecular process can be taken as essentially ∆E *. The large effect of ∆G* on the rate of interconversion is shown by the data in Table 1 for the calculated firstorder rate constant, k1, and the mean lifetime, τ = 1/k1, at 298 K as a function of ∆G*. The time scale for some widely used spectroscopic and diffraction techniques that might be used to investigate this energy profile is given in Table 2 (2). X-ray diffraction methods are limited to crystalline solids, and crystal packing forces typically yield a single rotamer, which may not even be the rotamer that is preferred for the same molecule in the gas phase or in solution, so information about the energy profile for isolated or dissolved molecules must be obtained by other methods. Electron diffraction techniques are fast enough to detect individual rotamers, but are limited to relatively small molecules in the gas phase. UV–visible light spectroscopy is also fast enough to distinguish rotamers, but the small differences between spectra of rotamers and the broad absorption bands for dissolved species usually rule out this method. IR and Raman spectroscopy, which are also fast enough to detect and determine concentrations of separate rotamers, were often employed in some of the earliest studies of internal rotation in substituted ethanes (4 ). Microwave spectroscopy can also be employed to detect separate rotamers for rapidly interconverting rotamers for gas phase molecules with reasonable vapor pressures. By contrast, dielectric properties (and electric dipole moments calculated from dielectric constants) yield weighted average values determined by the fractional population of the contributing rotamers. NMR spectroscopy is unique among the spectroscopic methods in permitting determination of both rotamer populations and energy barriers to rotation (5). The time scale for NMR spectroscopy is relatively long, so weighted average spectra are usually observed, especially at room temperature and higher. At lower temperatures, where rates of rotamer interconversion are much slower (Table 2), separate rotamers can often be detected and their concentrations determined. By examining spectra as a function of temperature, both energy differences between low-energy rotamers and energy barriers for interconversion of rotamers can be obtained in favorable cases (6 ). A brief sketch of each of the approaches to the problem that are involved in this project follows. All three experimental approaches permit a determination of the relative population of low-energy rotamers from some experimentally determined parameter(s). The energy difference between low-energy rotamers can then be compared with values calculated from theory. In practice, the theoretical calculations also provide additional insights that can be used to inform and refine the interpretation of the experimental data. The whole project is then a realistic mix of experiment and theory in the best tradition of physical chemistry courses.

Proton NMR Spectroscopy Proton NMR spectroscopy allows an estimate of the rotamer population distribution and associated energy differences from average 3-bond vicinal coupling constants between protons on adjacent carbons, 3JHH (7). These coupling

JChemEd.chem.wisc.edu • Vol. 75 No. 7 July 1998 • Journal of Chemical Education

901

In the Laboratory

constants can be determined from direct measurement of line splittings in some cases. For more complex spectra, some analysis of the coupling pattern—most conveniently by computer simulation to match observed spectra1—is required to determine the coupling constants from the observed frequencies of a complex multiplet. Interpretation of the data requires a knowledge of the way in which coupling constants depend on the dihedral angle, φ. Both empirical and theoretical models agree that 3JHH for H–C–C–H fragments is approximately proportional to cos2φ (8). For symmetric molecules of general formula CH2X–CH 2X or CHX2–CHX 2, coupling constants can be obtained by analyzing the fine structure of the 13Csatellites of the 1% of molecules that contain the isotopically asymmetric molecule, 13CH2X–12 CH2X or 12 CHX2–13CHX2, since the equivalent protons of 12C–12C species give rise to a single peak in the NMR spectrum (9). For unsymmetrically substituted ethanes of general formula CH2X–CH2Y or CHX2–CHY2, J values can be extracted from the proton spectrum itself.

Average Electric Dipole Moment The average electric dipole moment for molecules in solution, like the average NMR coupling constant between vicinal protons, is also sensitive to the energy profile for internal rotation (10). Relating the experimentally determined average dipole moment to the relative populations and energies of individual rotamers also requires an estimate of the dipole moment of the individual low-energy rotamers. These quantities, in turn, can be estimated for a particular geometry by a vector addition of the contribution from the individual bonds, the so-called bond moments. Bond moments can be estimated from dipole moments of simpler molecules in which the dipole moment of interest is dominant. Dipole moments can also be calculated for specific geometries by approximate molecular quantum mechanical methods. Infrared and Raman Spectroscopy Infrared and Raman spectroscopy was employed extensively in early experimental investigations of internal rotation. Though it offers the advantage of being able to “see” the separate low energy rotamers, the identification of specific peaks in a complex spectrum with a specific rotamer is not trivial, and the determination of relative concentrations of rotamers requires some knowledge of the extinction coefficients for different bands. We have used the infrared spectral measurements to provide additional qualitative confirmation of relative energies and to illuminate the quantum mechanical calculations of the vibrational spectra of the molecules being examined. Computational Techniques for Determining Rotational Energy Profiles

Molecular Mechanics Calculations Molecular mechanics calculations have been refined in the last 25 years to permit calculation of reasonably reliable relative energies of different conformations of molecules on the basis of a model that treats the molecule as a collection of balls held together by Hooke’s-law springs (11). The total mechanical energy of the molecule (strain energy) is assumed to be attributable to separate contributions from several kinds of distortions from the most stable conformation. Several 902

Figure 2. Proton NMR spectrum of the upfield (more shielded) 13C satellite pattern of 1,2-dibromoethane in CDCl3. Lower trace is DNMR simulation used to determine J values.

Figure 3. Proton NMR spectrum of the upfield (more shielded) 13C satellite pattern of 1,2-dibromoethane in CDCl3. Lower trace is DNMR simulation used to determine J values.

empirical schemes have been proposed to model the system, but all use computer techniques to explore the multidimensional energy surface as a function of 5–6 parameters that depend on bond lengths, bond angles, torsion angles, and van der Waals interactions between atoms. The MM2 and its more recent successor MM3 models developed by Allinger and co-workers have been widely used for small organic compounds (12). These algorithms have now been incorporated into commercially available, user-friendly software that allows convenient construction and manipulation of the structures.2

Quantum Mechanical Calculations Quantum mechanical calculations for molecules have also become accessible to undergraduate laboratories in the last few years so that both semiempirical and ab initio methods to calculate the total electronic energies of molecules with reasonable precision are available at a modest cost (13). The methods are particularly well suited to examining trends in a series of very similar compounds with the same basic structure. A problem with such calculations is that they yield the total electronic energy of the whole molecule. The small differences in rotamer energies that determine the relative populations of the rotamers are a very small fraction of the total electronic energies. Nevertheless, trends in values for a series of compounds can be determined with some confidence. Solvent effects can be modeled by semiempirical methods. In addition, the quantum calculations yield electric dipole moments for a specified geometry, which can be used in calculating the weighted average dipole moment, the value determined experimentally from dielectric constant measurements. They also can be employed to evaluate symmetry properties, to determine force constants for molecular vibrations, and to simulate the vibrational motion of molecules.


In the Laboratory

Additional Details and Student Results for CH2Br–CH2Br

NMR Typical proton NMR spectra of 10% v/v solutions of 1,2-dibromoethane in two solvents, as recorded with a Bruker AF300 high-field NMR spectrometer operating at 300 MHz, are shown in Figures 2 and 3. (NOTE: Since the satellite pattern depends only on 13 C–1H and 1H–1H coupling constants, essentially identical results are obtained with a 60 MHz instrument.) The portion shown is the “upfield” half of the symmetric 13C-satellite pattern, which is centered at J13CH /2 Hz (75 Hz) upfield from the large single peak of the dominant 12 C–12C species and the other half of the pattern for the protons attached to 12C in the 12C–13C species. Proton–proton coupling constants were determined, using simulation package DNMR,1 by treating the spectrum as half of an AA′BB′ pattern with A and B chemical shifts determined from the centers of the two multiplets. Fractional populations of rotamers (as labeled in Table 3) were calculated from the two vicinal coupling constants by assuming (Karplus equation, J = Jt cos2 ϕ) (8) that the trans coupling constant, Jt , is approximately 4Jg, where Jg is the gauche coupling constant, and that each vicinal coupling constant can be written as a weighted sum over the three rotamers in the form J = f 1 J1 + f 2 J2 + f3 J3

(3)

where f1 = f3 = fraction of rotamer 1. This approximation yields f 1 = (4JAB′ – JAB)/(6JAB′ + 3JAB), Jt = JAB / (1 – 1.50 f 1), and f2 = 1 – 2f1, where JAB is the larger of the two vicinal coupling constants. The energy difference ∆E was then calculated from eq 2. The data for 1,2-dibromoethane in CDCl3 and in DMSO-d6 are summarized in Table 3.

Dipole Moment The dipole moment of 1,2-dibromoethane was determined from measurements of the dielectric constant, ε, and the refractive index, n, of solutions of the compound in the nonpolar solvent cyclohexane (Table 4). Dielectric constant measurements were made with a 1960 vintage Sargent oscillometer and refractive indices were determined with an Abbe refractometer. Linear plots of the ε and n2 vs mole fraction 1,2-dibromoethane were used to calculate the orientation polarization, P2µo, and the dipole moment, µ (10). To determine the fractional population of trans and gauche rotamers, it is necessary also to know the dipole moment of the gauche rotamers, µg. This quantity was calculated by setting the dipole moment of 1-bromoethane, 2.03 D, equal to the group moment for the CH2Br group, M, and using that value and the geometry of the molecule (assuming tetrahedral angles

and a 60° angle between the projection of the group moments onto the plane normal to the C–C axis) to calculate µg2 = 8M2/3. Since the molar polarization, used to calculate the dipole moment, is proportional to the square of the dipole moment, the weighted average dipole moment is related to µg by eq 4: µ2 = f1 µg2 + f 2 µ t2 + f 3 µg2 = (1 – f t ) µg2 µ t2

(4) 3µ 2/8M2

where f 2 = f t and = 0 (by symmetry), so ft = 1 – and f g = f1 = f 3 = (1– f t)/2. Fractional populations calculated in this way were then used to calculate ∆E from eq 2. Results are summarized in Table 4 for 1,2-dibromoethane in cyclohexane solvent at room temperature, 23 °C.

Infrared Spectral Evidence for Gauche and Trans Rotamers The infrared spectrum of a thin film of pure liquid 1,2dibromoethane, recorded with a Nicolet Model 5SXB FTIR spectrometer, is shown in Figure 4. Several prominent peaks that have been reliably identified with either gauche or trans rotamers are labeled in the spectrum (3, 14). It is clear that

gauche (1419 cm –1)

Figure 4. Infrared spectrum of liquid film of 1,2-dibromoethane showing assignment of distinct peaks for each rotamer.


903

In the Laboratory

Molecular Mechanics The Spartan version of MM2 was employed to calculate the conformational energy at fixed dihedral angles between the Br–C bonds at 30° intervals from 0 to 180°. The results of those calculations, with additional data points at 2° intervals from 60 to 78° to establish the dihedral angle for the low-energy gauche conformation more accurately, are given in Table 5. The energy profile generated by a spline fit to these data is plotted in Figure 5. Note that the gauche conformation has its minimum energy at 68°, and not at the 60° assumed in the analysis of both NMR and dipole data. The calculations, which make no attempt to take into account intermolecular interactions, are for isolated molecules in the gas phase. Quantum Mechanics We have sometimes required students to complete both semiempirical and ab initio quantum calculations, but the time required to generate the whole rotational energy profile by such calculations is prohibitive. We now have them use the minimum energy conformations identified by the molecular mechanics and the high-energy eclipsed conformations as input for ab initio calculations for this molecule with the 3-21G* basis set option available in Spartan. These energy values are then used to calculate ∆E, the energy difference between gauche and trans rotamers, and ∆E *, the activation energy barrier for rotation. The two low-energy conformations are also used as input for an ab initio calculation of the dipole moment and the vibrational frequencies for comparison with their experimentally determined values. The results of these calculations are collected in Table 6, which summarizes all ∆E values obtained in this project and the conditions of the determination and in Table 7, which lists the calculated vibrational frequencies of gauche and trans rotamers. Figure 904

50

MM2 Strain Energy / (kJ/mol)

both rotamers are present at a significant concentration in the room temperature spectrum, but that the trans rotamer is much more important, in spite of the fact that there are two gauche rotamers but only one trans. A quantitative determination of the relative concentration of the rotamers is hampered by a lack of information about their relative absorbancy indices (extinction coefficients). We have limited the assignment to the identification of significant well resolved peaks in an uncluttered part of the spectrum. Provisions for controlling the temperature of the sample would permit determination of ∆E from the change in relative intensities of the peaks as a function of temperature. We have not attempted to examine gas-phase infrared spectra in this project, but the vapor pressure of 1,2-dibromoethane should be great enough to obtain goodquality spectra of the vapor phase of the compound at temperatures not much above room temperature.

40

30

20

10

-180

-120

-60

0

60

120

180

Dihedral Angle

Figure 5 Energy profile for internal rotation of 1,2-dibromoethane as calculated by molecular mechanics methods using MM2 running under Spartan software.

Br

Br

Br

Br

A u mode:1300 cm -1

Bu mode:1427 cm -1

Br

Br Br

B mode:1262 cm -1

Br

B mode:1444 cm -1

Figure 6. Fundamental vibrational modes of 1,2-dibromoethane as depicted by ab initio quantum mechanical methods with Spartan software.


In the Laboratory

Analysis of Data and Suggestions for Further Work In addition to discussing—in a formal report or in a poster presentation—the results obtained in the several experiments and calculations, students are required to do some further analysis that makes specific predictions about the effect of temperature on some of the properties investigated. This analysis uses the terminology and concept of the partition function from statistical thermodynamics to help to make that important concept more concrete. The partition function for internal rotation is simply Qrot = Σ e(᎑Ei / RT ), where the sum is taken over the three rotamers whose energies are Ei. The fractional population of the ith rotamer is then given by fi = e(᎑Ei / RT )/ Σ e(᎑Ei /RT )

(5)

Thus, any measured property, P, can be expressed as Σf i Pi , where Pi is the value of the property in the ith rotamer whose fractional population is fi. The calculated variation in fractional populations of rotamers, vicinal coupling constants (eq 3), and the simulated NMR spectrum of 1,2-dibromoethane between ᎑ 60 and + 60 °C are shown in Figures 7–9. The point of the calculation is to make some prediction about the sensitivity of the NMR spectrum to temperature changes in order to assess the feasibility of doing a variable-temperature

NMR experiment to extract ∆E by reversing the calculation. Similar calculations could be carried out to predict the effect of temperature on the dipole moment of the molecule. Such calculations suggest a fruitful direction to expand the project: investigate the temperature dependence of the NMR spectrum, the IR spectrum, or the dipole moment to obtain the fractional populations and ∆E by a regression analysis. Semiempirical calculations at the AM1 level, available in Spartan, permit an examination of the effect of solvent on the rotamer energies as well. We have sometimes suggested that students run such calculations for gas phase, hexadecane solvent, and water solvent to explore the effect of solvent polarity on the energy profile, which their experimental data reveal. Energy values obtained by those calculations are included in the data in Table 6. 1.0

0.8

Fraction

6 shows the vibrational modes (which can be animated by Spartan) that contribute to the observed IR spectrum in the 1200–1500 cm ᎑1 range and their calculated frequencies. Though the calculated and observed frequencies differ by several wave numbers and the ordering of the peaks does not match experiment (14 ), the calculation correctly predicts only two allowed IR bands (with u symmetry) in this range for the trans rotamer.

0.6

ft=f2

0.4

fg =f1 =f3 0.2

0.0 -80

-60

-40

-20

0

20

40

60

80

T/ o C Figure 7. Calculated effect of temperature on the fractional population of rotamers of 1,2-dibromoethane based on ∆E = 3.56 kJ mol᎑1.

14

Jij / H z

12

JAB

10

8

JAB'

6

4 -80

-60

-40

-20

0

20

40

60

80

o

T/ C Figure 8. Calculated effect of temperature on weighted average vicinal proton–proton spin coupling constants of rotamers of 1,2dibromoethane based on ∆E = 3.56 kJ mol᎑1.


905

In the Laboratory

Notes 1. DNMR, which is capable of simulating both complex coupling patterns and the effects of rate processes on the spectrum, is available as part of WINDNMR (Reich, H. J. J. Chem. Educ. Software 1995, 3D(2), 17–33; abstracted in J. Chem. Educ. 1995, 72, 1086–1087). 2. We have used Spartan (now available in PC or Macintosh versions for about $300) from Wavefunction, Inc., the CAChe programs from Oxford Molecular, and Alchemy and Sybyl from Tripos. For this project, we have favored having students use Spartan, which includes molecular mechanics, semiempirical and ab initio calculation modules and convenient builder and display modules in the same package.

Literature Cited

Figure 9. Simulation of effect of temperature on the proton NMR spectrum of the upfield (more shielded) 13C satellite pattern of 1,2dibromoethane in CDCl3 based on J values shown in Table 6.

Not all substituted ethanes are as well behaved as 1,2dibromoethane, but almost any similar small molecule could be used for parts of the project. The similar 1,2-dichloroethane has a smaller ∆E, so the NMR spectrum is less spread out and harder to decipher by simulation. The 1,2-dicyanoethane, suggested for dipole moment measurements (10), works fine. The ether 1,2-dimethoxyethane has more degrees of freedom, so the vibrational mode calculations take a very long time. The same is true of ethylene glycol (which is also too viscous for convenient handling) and succinic acid. Asymmetrically substituted molecules like 1-bromo-2-chloroethane avoid the problem of deciphering spectra from the lowintensity 13C satellites. Trisubstituted molecules such as 1bromo-2-chloropropane have three different energy rotamers, so the analysis presented here must be modified to take that into account. The Journal of Physical Chemistry regularly reports theoretical calculations and experimental investigations on similar molecules, including solvent effects, so the current literature could be consulted for additional suggestions (15–18). Acknowledgments Financial support was provided by the National Science Foundation (ILI grant USE-925169 for equipment for a computational chemistry laboratory and a Chemistry Research Equipment grant in support of purchase of NMR spectrometer) and the Camille and Henry Dreyfus Foundation (Scholar and Postdoctoral Fellow grants to LEE and KFM, respectively).

906

1. Eliel, E. L.; Wilen, S. H.; Mander, L. N. Stereochemistry of Organic Compounds; Wiley: New York, 1994; Chapter 10. 2. Nasipuri, D. Stereochemistry of Organic Compounds: Principles and Applications; Wiley: New York, 1991; Chapter 9. 3. Orville-Thomas, W. J. Internal Rotation in Molecules; Wiley: New York, 1974. 4. Mizushuma, S. Structure of Molecules and Internal Rotation; Academic: New York, 1954. 5. Harris, R. K. Nuclear Magnetic Resonance Spectroscopy; Longman: Essex, UK, 1986. 6. Brunelle, J. A.; Letendre, L. J.; Weltin, E. E.; Brown, J. H.; Bushweller, C. H. J. Phys. Chem. 1992, 96, 9225. 7. Bovey, F. A. NMR Spectroscopy; Academic: New York, 1988. 8. Karplus, M. J. Am. Chem. Soc. 1963, 85, 2870. 9. Jackman, L. M.; Sternhell, S. Applications of Nuclear Magnetic Resonance Spectroscopy in Organic Chemistry, 2nd ed.; Pergamon: New York, 1969; p 140. 10. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry, 5th ed.; McGraw-Hill: New York, 1989; experiment 31 based on Braun, C. L.; Stockmayer, W. H.; Orwall, R. A. J. Chem. Educ. 1970, 47, 287. 11. Boyd, D. B.; Lipkowitz, K. B. J. Chem. Educ. 1982, 59, 269. 12. Allinger, N; Burkhart, H. Molecular Mechanics; American Chemical Society: Washington, DC, 1982. 13. Hehre, W. J.; Radom, L.; Schleyer, P. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. Hehre, W. J.; Burke, L. D.; Shusterman, A. J.; Pietro, W. J. Experiments in Computational Organic Chemistry; Wavefunction, Inc.: Irvine, CA, 1993. 14. Powling, J.; Bernstein, H. J. J. Am. Chem. Soc. 1951, 73, 1815. 15. Dixon, D. A.; Matsuzawa, N.; Walker, S. C. J. Phys. Chem. 1992, 96, 10740. 16. Brown, J. H.; Bushweller, C. H. J. Phys. Chem. 1994, 98, 11411. 17. Wiberg, K. B.; Keith, T. A.; Frisch, M. J.; Murcho, M. J. Phys. Chem. 1995, 99, 9072. 18. Kazorian, S. G.; Poliakoff, M. J. Phys. Chem. 1995, 99, 8624.


The Energy Profile for Rotation about the C-C Bond in Substituted

Recommend Documents