Rotamer

In chemistry, rotamers are chemical species that differ from one another primarily due to rotations about one single bond. Various arrangements of atoms in a molecule that differ by rotation about single bonds can also be referred to as conformations. Conformations, which represent local minima on the potential energy surface, are called conformers. Conformers can differ from one another due to rotation of multiple bonds; rotamers are a subset of conformers. Conformers/rotamers usually differ little in their energies, so they are almost never separable in a practical sense. Rotations about single bonds are subject to small energy barriers. When the time scale for interconversion is long enough for isolation of individual rotamers, the species are termed atropisomers. The ring-flip of substituted cyclohexanes constitutes a common form of conformers.
The study of the energetics of bond rotation is referred to as conformational analysis. In some cases, conformational analysis can be used to predict and explain product selectivity, mechanisms, and rates of reactions. Conformational analysis also plays an important role in rational, structure-based drug design.

Types

Rotating their carbon–carbon bonds, the molecules ethane and propane have three local energy minima. They are structurally and energetically equivalent, and are called the staggered conformers. For each molecule, the three substituents emanating from each carbon–carbon bond are staggered, with each H–C–C–H dihedral angle equal to 60°. The three eclipsed conformations, in which the dihedral angles are zero, are transition states connecting two equivalent energy minima, the staggered conformers.
The butane molecule is the simplest molecule for which single bond rotations result in two types of nonequivalent structures, known as the anti- and gauche-conformers.
For example, butane has three conformers relating to its two methyl groups: two gauche conformers, which have the methyls ±60° apart and are enantiomeric, and an anti conformer, where the four carbon centres are coplanar and the substituents are 180° apart. The energy separation between gauche and anti is 0.9 kcal/mol associated with the strain energy of the gauche conformer. The anti conformer is, therefore, the most stable. The three eclipsed conformations with dihedral angles of 0°, 120°, and 240° are transition states between conformers. Note that the two eclipsed conformations have distinct energies: at 0° the two methyl groups are eclipsed, resulting in higher energy than at 120°, where the methyl groups are eclipsed with hydrogens.

Mathematical analysis

A rough approximate function can illustrate the main features of the conformational analysis for unbranched linear alkanes with rotation around a central C–C bond. The members of this series have the general formula C_2nH_4n+2 with the index n = 1, 2, 3, etc. It can be assumed that the angle strain is negligible in alkanes since the bond angles are all near the tetrahedral ideal. The energy profile is thus periodic with periodicity due to the threefold symmetry of sp³-hybridized carbon atoms. This suggests a sinusoidal potential energy function, typically modelled using a Fourier series truncated to the dominant terms:
Here:

is the dihedral angle in degrees,
are coefficients representing the amplitude of the th harmonic, corresponding to various energy barriers due to torsional influences and asymmetry in steric interactions.
The factor of and the form ensure energy minima at staggered conformations and energy maxima at eclipsed conformations.

For alkanes, the dominant term is usually, reflecting the threefold rotational symmetry. Higher terms may be included for precision where steric effects vary. The primary contribution comes from torsional strain due to alkyl groups eclipsing, captured by the term. Steric interactions rise with the size of substituents, taken into account by the term. The number of carbon atoms clearly influences the size of substituents on the central C–C bond. In general, for unbranched linear alkanes with even-numbered chains, there will be two C_n-1H''_2n-1'' group substituents.
A parameterization using energy values derived from rotational spectroscopy data and theoretical calculations gives the following simplified equation:
Here is given in kcal/mol and. This function largely neglects angle strain and long-range interactions for the members of the series.
While simple molecules can be described by these types of conformations, more complex molecules require the use of the Klyne–Prelog system to describe the conformers.
More specific examples of conformations are detailed elsewhere:

Ring conformation
* Cyclohexane conformations, including with chair and boat conformations among others.
* Cycloalkane conformations, including medium rings and macrocycles
* Carbohydrate conformation, which includes cyclohexane conformations as well as other details.
Allylic strain – energetics related to rotation about the single bond between an sp² carbon and an sp³ carbon.
Atropisomerism – due to restricted rotation about a bond.
Folding, including the secondary and tertiary structure of biopolymers.
Akamptisomerism – due to restricted inversion of a bond angle.
Equilibrium of conformers

Conformers generally exist in a dynamic equilibrium
Three isotherms are given in the diagram depicting the equilibrium distribution of two conformers at various temperatures. At a free energy difference of 0 kcal/mol, this analysis gives an equilibrium constant of 1, meaning that two conformers exist in a 1:1 ratio. The two have equal free energy; neither is more stable, so neither predominates compared to the other. A negative difference in free energy means that a conformer interconverts to a thermodynamically more stable conformation, thus the equilibrium constant will always be greater than 1. For example, the ΔG° for the transformation of butane from the gauche conformer to the anti conformer is −0.47 kcal/mol at 298 K. This gives an equilibrium constant is about 2.2 in favor of the anti conformer, or a 31:69 mixture of gauche:''anti'' conformers at equilibrium. Conversely, a positive difference in free energy means the conformer already is the more stable one, so the interconversion is an unfavorable equilibrium.

Population distribution of conformers

The fractional population distribution of various conformers follows a Boltzmann distribution:
The left hand side is the proportion of conformer i in an equilibrating mixture of M conformers in thermodynamic equilibrium. On the right side, E_k is the energy of conformer k, R is the molar ideal gas constant or 1.987 cal/), and T is the absolute temperature. The denominator of the right side is the partition function.

Factors contributing to the free energy of conformers

The effects of electrostatic and steric interactions of the substituents as well as orbital interactions such as hyperconjugation are responsible for the relative stability of conformers and their transition states. The contributions of these factors vary depending on the nature of the substituents and may either contribute positively or negatively to the energy barrier. Computational studies of small molecules such as ethane suggest that electrostatic effects make the greatest contribution to the energy barrier; however, the barrier is traditionally attributed primarily to steric interactions.
In the case of cyclic systems, the steric effect and contribution to the free energy can be approximated by A values, which measure the energy difference when a substituent on cyclohexane in the axial as compared to the equatorial position. In large rings, there are many accessible low-energy conformations which correspond to the strain-free diamond lattice.

Observation of conformers

The short timescale of interconversion precludes the separation of conformer in most cases. Atropisomers are conformational isomers which can be separated due to restricted rotation. The equilibrium between conformational isomers can be observed using a variety of spectroscopic techniques.
Protein folding also generates conformers which can be observed. The Karplus equation relates the dihedral angle of vicinal protons to their J-coupling constants as measured by NMR. The equation aids in the elucidation of protein folding as well as the conformations of other rigid aliphatic molecules. Protein side chains exhibit rotamers, whose distribution is determined by their steric interaction with different conformations of the backbone. This effect is evident from statistical analysis of the conformations of protein side chains in the Backbone-dependent rotamer library.

Spectroscopy

Conformational dynamics can be monitored by variable temperature NMR spectroscopy. The technique applies to barriers of 8–14 kcal/mol, and species exhibiting such dynamics are often called "fluxional". For example, in cyclohexane derivatives, the two chair conformers interconvert rapidly at room temperature. The ring-flip proceeds at a rates of approximately 10⁵ ring-flips/sec, with an overall energy barrier of 10 kcal/mol. This barrier precludes separation at ambient temperatures. However, at low temperatures below the coalescence point one can directly monitor the equilibrium by NMR spectroscopy and by dynamic, temperature dependent NMR spectroscopy the barrier interconversion.
Besides NMR spectroscopy, IR spectroscopy is used to measure conformer ratios. For the axial and equatorial conformer of bromocyclohexane, ν_CBr differs by almost 50 cm⁻¹.