Dihedral angle

 ( Stereochemistry ) 

A dihedral angle is the angle between two intersecting planes or . 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 , a dihedral angle represents the angle between two . In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common.

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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, $\backslash varphi$ between them is given by:

$\backslash cos\; \backslash varphi\; =\; \backslash frac\{\backslash left\backslash vert\; a\_1\; a\_2\; +\; b\_1\; b\_2\; +\; c\_1\; c\_2\; \backslash right\backslash vert\}\{\backslash sqrt\{a\_1^2+b\_1^2+c\_1^2\}\backslash sqrt\{a\_2^2+b\_2^2+c\_2^2\}\}$

and satisfies $0\backslash le\; \backslash varphi\; \backslash le\; \backslash pi/2.$ It can easily be observed that the angle is independent of $d\_1$ and $d\_2$.

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

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

where is the dot product of the vectors and 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 can be and should be avoided when considering the dihedral angle of two whose boundaries are the same line. In this case, the half planes can be described by a point of their intersection, and three vectors , and such that , and 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)}, and satisfies In this case, switching the two half-planes gives the same result, and so does replacing $\backslash mathbf\; b\_0$ with $-\backslash mathbf\; b\_0.$ In chemistry (see below), we define a dihedral angle such that replacing $\backslash mathbf\; b\_0$ with $-\backslash mathbf\; b\_0$ changes the sign of the angle, which can be between and .

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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 , , , etc. then bond vectors are defined by =−, =−, and =−, more generally.

Martin Kröger (2025). 9783540262107, Springer. ISBN 9783540262107

This is the case for or 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 , and are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval . This dihedral angle is defined by

$\backslash begin\{align\}$

\cos \varphi&=\frac{ (\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\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))}, \end{align} or, using the function atan2,

$\backslash varphi=\backslash operatorname\{atan2\}(\backslash mathbf\{u\}\_2\; \backslash cdot((\backslash mathbf\{u\}\_1\; \backslash times\; \backslash mathbf\{u\}\_2)\; \backslash times\; (\backslash mathbf\{u\}\_2\; \backslash times\; \backslash mathbf\{u\}\_3)),\; |\backslash mathbf\{u\}\_2|\backslash ,(\backslash mathbf\{u\}\_1\; \backslash times\; \backslash mathbf\{u\}\_2)\; \backslash cdot\; (\backslash mathbf\{u\}\_2\; \backslash times\; \backslash mathbf\{u\}\_3)).$

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)

$\backslash begin\{align\}$

\cos \varphi&=\frac{ (\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\mathbf{u}_2 \times \mathbf{u}_3)}\\ \sin \varphi&=\frac{ |\mathbf{u}_2|\,\mathbf{u}_1 \cdot(\mathbf{u}_2 \times \mathbf{u}_3)}, \end{align} or equivalently,

$\backslash varphi=\backslash operatorname\{atan2\}($

|\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 $\backslash varphi$ 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 $\backslash varphi\; =\; \backslash pi$, $\backslash varphi\; =\; +\backslash pi/3$ and $\backslash varphi\; =\; -\backslash pi/3$, which are called the trans, gauche⁺, and gauche⁻ conformations.

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In stereochemistry

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

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.

Eric Anslyn (2025). 9781891389313, University Science. ISBN 9781891389313

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).

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

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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:

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

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

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Higher dimensions In -dimensional Euclidean space, the dihedral angle between the two defined by the equations $$\backslash mathbf\{n\}\_\backslash mathrm\{A\}\; \backslash cdot\; \backslash mathbf\{x\}\; =\; c\_A$$ $$\backslash mathbf\{n\}\_\backslash mathrm\{B\}\; \backslash cdot\; \backslash mathbf\{x\}\; =\; c\_B$$ for vectors and constants and , is given by $$\backslash cos\; \backslash varphi\; =\; \backslash frac\{\; \backslash left\backslash vert\backslash mathbf\{n\}\_\backslash mathrm\{A\}\; \backslash cdot\; \backslash mathbf\{n\}\_\backslash mathrm\{B\}\backslash right\backslash vert\}$$\,.

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See also

• Atropisomer

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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.

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