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Angle between two planes (α, β, green) in a third plane (pink) which cuts the line of intersection at right angles

A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimension, a dihedral angle represents the angle between two hyperplanes.^[1]

Mathematical background

When the two intersecting planes are described in terms of Cartesian coordinates by the two equations

a1x+b1y+c1z+d1=0{displaystyle a_{1}x+b_{1}y+c_{1}z+d_{1}=0}

a2x+b2y+c2z+d2=0{displaystyle a_{2}x+b_{2}y+c_{2}z+d_{2}=0}

the dihedral angle, φ{displaystyle varphi } between them is given by:

cos⁡φ=|a1a2+b1b2+c1c2|a12+b12+c12a22+b22+c22.{displaystyle cos varphi ={frac {leftvert a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}rightvert }{{sqrt {a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}{sqrt {a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}}.}

An alternative method is to calculate the angle between the vectors, n_A and n_B, which are normal to the planes.

cos⁡φ=|nA⋅nB||nA||nB|{displaystyle cos varphi ={frac {leftvert mathbf {n} _{mathrm {A} }cdot mathbf {n} _{mathrm {B} }rightvert }{|mathbf {n} _{mathrm {A} }||mathbf {n} _{mathrm {B} }|}}}

where n_A · n_B is the dot product of the vectors and |n_A| |n_B| is the product of their lengths.^[2]

Any plane can also be described by two non-collinear vectors lying in that plane; taking their cross product yields a normal vector to the plane. Thus, a dihedral angle can be defined by three vectors, b₁, b₂ and b₃, forming two pairs of non-collinear vectors.^[3]

φ=atan2⁡(([b1×b2]×[b2×b3])⋅b2|b2|,[b1×b2]⋅[b2×b3]).{displaystyle varphi =operatorname {atan2} left({big (}[mathbf {b} _{1}times mathbf {b} _{2}]times [mathbf {b} _{2}times mathbf {b} _{3}]{big )}cdot {frac {mathbf {b} _{2}}{|mathbf {b} _{2}|}},[mathbf {b} _{1}times mathbf {b} _{2}]cdot [mathbf {b} _{2}times mathbf {b} _{3}]right).}

Stereochemistry

Configuration names according to dihedral angle	syn n-Butane Newman projection	syn n-Butane sawhorse projection

Free energy diagram of n-butane as a function of dihedral angle.

In chemistry, a torsion angle is defined as a particular example of a dihedral angle, describing the geometric relation of two parts of a molecule joined by a chemical bond.^[4][5] Every set of three not-colinear atoms of a molecule defines a plane. When two such planes intersect (i.e., a set of four consecutively-bonded atoms), the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation.^[6]Stereochemical arrangements corresponding to angles between 0° and ±90° are called syn (s), those corresponding to angles between ±90° and 180° anti (a). Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called clinal (c) and those between 0° and ±30° or ±150° and 180° are called periplanar (p).

The two types of terms can be combined so as to define four ranges of angle; 0° to ±30° synperiplanar (sp); 30° to 90° and −30° to −90° synclinal (sc); 90° to 150° and −90° to −150° anticlinal (ac); ±150° to 180° antiperiplanar (ap). The synperiplanar conformation is also known as the syn- or cis-conformation; antiperiplanar as anti or trans; and synclinal as gauche or skew.

For example, with n-butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The syn-conformation shown above, with a dihedral angle of 60° is less stable than the anti-conformation with a dihedral angle of 180°.

For macromolecular usage the symbols T, C, G⁺, G⁻, A⁺ and A⁻ are recommended (ap, sp, +sc, −sc, +ac and −ac respectively).

Proteins

Depiction of a protein, showing backbone dihedral angles

A Ramachandran plot (also known as a Ramachandran diagram or a [φ,ψ] plot), originally developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan,^[7] is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein structure. The figure at right illustrates the definition of the φ and ψ backbone dihedral angles^[8] (called φ and φ′ by Ramachandran).

In a protein chain three dihedral angles are defined as φ (phi), ψ (psi) and ω (omega), as shown in the diagram. The planarity of the peptide bond usually restricts ω to be 180° (the typical trans case) or 0° (the rare cis case). The distance between the C^α atoms in the trans and cis isomers is approximately 3.8 and 2.9 Å, respectively. The vast majority of the peptide bonds in proteins are trans, though the peptide bond to the nitrogen of proline has an increased prevalence of cis compared to other amino-acid pairs.^[9]

The sidechain dihedral angles tend to cluster near 180°, 60°, and −60°, which are called the trans, gauche⁺, and gauche⁻ conformations. The stability of certain sidechain dihedral angles is affected by the values φ and ψ.^[10] For instance, there are direct steric interactions between the Cγ of the side chain in the gauche⁺ rotamer and the backbone nitrogen of the next residue when ψ is near -60°.^[11]

Converting from dihedral angles to Cartesian coordinates in chains

It is common to represent polymers backbones, notably proteins, in internal coordinates; that is, a list of consecutive dihedral angles and bond lengths. However, some types of computational chemistry instead use cartesian coordinates. In computational structure optimization, some programs need to flip back and forth between these representations during their iterations. This task can dominate the calculation time. For processes with many iterations or with long chains, it can also introduce cumulative numerical inaccuracy. While all conversion algorithms produce mathematically identical results, they differ in speed and numerical accuracy.^{[12][non-primary source needed]}

Geometry

Every polyhedron has a dihedral angle at every edge describing the relationship of the two faces that share that edge. This dihedral angle, also called the face angle, is measured as the internal angle with respect to the polyhedron. An angle of 0° means the face normal vectors are antiparallel and the faces overlap each other, which implies that it is part of a degenerate polyhedron. An angle of 180° means the faces are parallel, as in a tiling. An angle greater than 180° exists on concave portions of a polyhedron.

Every dihedral angle in an edge-transitive polyhedron has the same value. This includes the 5 Platonic solids, the 13 Catalan solids, the 4 Kepler–Poinsot polyhedra, the two quasiregular solids, and two quasiregular dual solids.

Given 3 faces of a polyhedron which meet at a common vertex P and have edges AP, BP and CP, the cosine of the dihedral angle between the faces containing APC and BPC is:^[13]

cos⁡φ=cos⁡(∠APB)−cos⁡(∠APC)cos⁡(∠BPC)sin⁡(∠APC)sin⁡(∠BPC){displaystyle cos varphi ={frac {cos(angle mathrm {APB} )-cos(angle mathrm {APC} )cos(angle mathrm {BPC} )}{sin(angle mathrm {APC} )sin(angle mathrm {BPC} )}}}

See also

• Atropisomer

References

External links

• The Dihedral Angle in Woodworking at Tips.FM


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  Analysis of the 5 Regular Polyhedra gives a step-by-step derivation of these exact values.