Dihedral angle

This article is about the geometry term. For other uses, see Dihedral.

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.

Definitions

A dihedral angle is an angle between two intersecting planes on a third plane perpendicular to the line of intersection.

A torsion angle is a particular example of a dihedral angle, used in stereochemistry to define the geometric relation of two parts of a molecule joined by a chemical bond.

Dihedral angles in stereochemistry

See also: Alkane stereochemistry and Conformational isomerism

Configuration names	syn n-Butane Newman projection	syn n-Butane sawhorse projection

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

In stereochemistry every set of three (not co-linear) atoms of a molecule defines a plane. When two such planes intersect, the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation. 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 cabon atoms. The syn-conformation shown above, with a dihedral angle of 60° is less stable than the anti- configuration 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).

Dihedral angles of proteins

Ramachandran plot of a protein, showing backbone dihedral angles

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 cis isomer is mainly observed in Xaa-Pro peptide bonds (where Xaa is any amino acid).

The sidechain dihedral angles of proteins are denoted as χ₁-χ₅ for each successive bond along that chain. The χ₁ dihedral angle is defined by atoms N-C^α-C^β-C^γ, the χ₂ dihedral angle is defined by atoms C^α-C^β-C^γ-C^δ, and so on.

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 neighbouring backbone and sidechain dihedrals; for example, the gauche⁺ conformation is rarely followed by the gauche⁺ conformation (and vice versa) because of the increased likelihood of atomic collisions.

Calculation of a dihedral angle

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

$a_1 x + b_1 y + c_1 z + d_1 = 0$

$a_2 x + b_2 y + c_2 z + d_2 = 0$

the dihedral angle, $\varphi$ between them is given by:

$\cos \varphi = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}.$

An alternative method is to calculate the angle between the vectors,$\mathbf{n}_{A}$ and $\mathbf{n}_{B}$, which are normal to the planes..

$\cos \varphi = -\frac{ \mathbf{n}_A \cdot \mathbf{n}_B}{|\mathbf{n}_A | |\mathbf{n}_B|}$

where $\mathbf{n}_A \cdot \mathbf{n}_B$ is the dot product of the vectors and $|\mathbf{n}_A | |\mathbf{n}_B|$ is the product of their lengths.

Dihedral angles in polyhedra

See also: Table of polyhedron dihedral angles

Every polyhedron, regular and irregular, convex and concave, has a dihedral angle at every edge.

A dihedral angle (also called the face angle) is the internal angle at which two adjacent faces meet. An angle of zero degrees means the face normal vectors are antiparallel and the faces overlap each other (Implying part of a degenerate polyhedron). An angle of 180 degrees means the faces are parallel (like 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 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 edges AP , BP and CP, the dihedral angle between the faces containing APC and BPC is:

$\cos\varphi = \frac{ \cos (\angle APB) - \cos (\angle APC) \cos (\angle BPC)}{ \sin(\angle APC) \sin(\angle BPC)}$

External links

• The Dihedral Angle in Woodworking at Tips.FM
• Analysis of the 5 Regular Polyhedra gives a step-by-step derivation of these exact values.