Dihedral angle

This article is about the geometry concept. For the aeronautics concept, see Dihedral (aeronautics). For other uses, see Dihedral.

Angle between two half-planes (α, β, pale blue) in a third plane (red) perpendicular to line of intersection.

A dihedral angle is the angle between two intersecting planes or half-planes. It is a plane angle formed on a third plane, perpendicular to the line of intersection between the two planes or the common edge between the two half-planes. In higher dimensions, a dihedral angle represents the angle between two hyperplanes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common.

Mathematical background

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:

$\cos \varphi ={\frac {\left\vert a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}\right\vert }{{\sqrt {a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}{\sqrt {a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}}$

and satisfies $0\leq \varphi \leq \pi /2.$ It can easily be observed that the angle is independent of $d_{1}$ and $d_{2}$.

Alternatively, if n_A and n_B are normal vector to the planes, one has

$\cos \varphi ={\frac {\left\vert \mathbf {n} _{\mathrm {A} }\cdot \mathbf {n} _{\mathrm {B} }\right\vert }{|\mathbf {n} _{\mathrm {A} }||\mathbf {n} _{\mathrm {B} }|}}$

where n_A · n_B is the dot product of the vectors and |n_A| |n_B| is the product of their lengths.

The absolute value is required in above formulas, as the planes are not changed when changing all coefficient signs in one equation, or replacing one normal vector by its opposite.

However the absolute values can be and should be avoided when considering the dihedral angle of two half planes whose boundaries are the same line. In this case, the half planes can be described by a point P of their intersection, and three vectors b₀, b₁ and b₂ such that P + b₀, P + b₁ and P + b₂ belong respectively to the intersection line, the first half plane, and the second half plane. The dihedral angle of these two half planes is defined by

$\cos \varphi ={\frac {(\mathbf {b} _{0}\times \mathbf {b} _{1})\cdot (\mathbf {b} _{0}\times \mathbf {b} _{2})}{|\mathbf {b} _{0}\times \mathbf {b} _{1}||\mathbf {b} _{0}\times \mathbf {b} _{2}|}}$,

and satisfies $0\leq \varphi <\pi .$ In this case, switching the two half-planes gives the same result, and so does replacing $\mathbf {b} _{0}$ with $-\mathbf {b} _{0}.$ In chemistry (see below), we define a dihedral angle such that replacing $\mathbf {b} _{0}$ with $-\mathbf {b} _{0}$ changes the sign of the angle, which can be between −π and π.

In polymer physics

In some scientific areas such as polymer physics, one may consider a chain of points and links between consecutive points. If the points are sequentially numbered and located at positions r₁, r₂, r₃, etc. then bond vectors are defined by u₁=r₂−r₁, u₂=r₃−r₂, and u_i=r_i+1−r_i, more generally. This is the case for kinematic chains or amino acids in a protein structure. In these cases, one is often interested in the half-planes defined by three consecutive points, and the dihedral angle between two consecutive such half-planes. If u₁, u₂ and u₃ are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval (−π, π]. This dihedral angle is defined by

${\begin{aligned}\cos \varphi &={\frac {(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}}\\\sin \varphi &={\frac {\mathbf {u} _{2}\cdot ((\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3}))}{|\mathbf {u} _{2}|\,|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}},\end{aligned}}$

or, using the function atan2,

$\varphi =\operatorname {atan2} (\mathbf {u} _{2}\cdot ((\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3})),|\mathbf {u} _{2}|\,(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})).$

This dihedral angle does not depend on the orientation of the chain (order in which the point are considered) — reversing this ordering consists of replacing each vector by its opposite vector, and exchanging the indices 1 and 3. Both operations do not change the cosine, but change the sign of the sine. Thus, together, they do not change the angle.

A simpler formula for the same dihedral angle is the following (the proof is given below)

${\begin{aligned}\cos \varphi &={\frac {(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}}\\\sin \varphi &={\frac {|\mathbf {u} _{2}|\,\mathbf {u} _{1}\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}},\end{aligned}}$

or equivalently,

$\varphi =\operatorname {atan2} (|\mathbf {u} _{2}|\,\mathbf {u} _{1}\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3}),(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})).$

This can be deduced from previous formulas by using the vector quadruple product formula, and the fact that a scalar triple product is zero if it contains twice the same vector:

$(\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3})=[(\mathbf {u} _{2}\times \mathbf {u} _{3})\cdot \mathbf {u} _{1}]\mathbf {u} _{2}-[(\mathbf {u} _{2}\times \mathbf {u} _{3})\cdot \mathbf {u} _{2}]\mathbf {u} _{1}=[(\mathbf {u} _{2}\times \mathbf {u} _{3})\cdot \mathbf {u} _{1}]\mathbf {u} _{2}$

Given the definition of the cross product, this means that $\varphi$ is the angle in the clockwise direction of the fourth atom compared to the first atom, while looking down the axis from the second atom to the third. Special cases (one may say the usual cases) are $\varphi =\pi$, $\varphi =+\pi /3$ and $\varphi =-\pi /3$, which are called the trans, gauche⁺, and gauche⁻ conformations.

In stereochemistry

Main article: Torsion angle

Configuration names according to dihedral angle	syn n-Butane in the gauche− conformation (−60°) Newman projection	syn n-Butane sawhorse projection

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

A torsion angle, found in stereochemistry, is a particular example of a dihedral angle describing the geometric relation of two parts of a molecule joined by a chemical bond. Every set of three non-colinear atoms of a molecule defines a half-plane. As explained above, when two such half-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. 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).

Geometry

See also: Table of polyhedron dihedral angles

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 a polyhedron that is isotoxal and/or isohedral has the same value. This includes the 5 Platonic solids, the 13 Catalan solids, the 4 Kepler–Poinsot polyhedra, the 2 convex quasiregular polyhedra, and the 2 infinite families of bipyramids and trapezohedra.

Law of cosines for dihedral angle

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:

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

This can be deduced from the spherical law of cosines, but can also be found by other means.

Higher dimensions

In m-dimensional Euclidean space, the dihedral angle ⁠$\varphi$⁠ between the two hyperplanes defined by the equations $$\mathbf {n} _{\mathrm {A} }\cdot \mathbf {x} =c_{A}$$ $$\mathbf {n} _{\mathrm {B} }\cdot \mathbf {x} =c_{B}$$ for vectors n_A, n_B, x ∈ R^m and constants c_A and c_B, is given by $$\cos \varphi ={\frac {\left\vert \mathbf {n} _{\mathrm {A} }\cdot \mathbf {n} _{\mathrm {B} }\right\vert }{|\mathbf {n} _{\mathrm {A} }||\mathbf {n} _{\mathrm {B} }|}}\,.$$

See also

• Atropisomer

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.