chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...

and

crystallography Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...

, a symmetry element is a point, line, or plane about which

symmetry operation In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a turn rotation of a regular triangle about its center (geometry), center, a refle ...

s can take place. In particular, a symmetry element can be a mirror plane, an

axis of rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

(either proper and improper), or a center of inversion. For an object such as a

molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...

or a

crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...

, a symmetry element corresponds to a set of symmetry operations, which are the

rigid transformation In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformation ...

s employing the symmetry element that leave the object unchanged. The set containing these operations form one of the

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

s of the object. The elements of this symmetry group should not be confused with the "symmetry element" itself. Loosely, a symmetry element is the geometric set of fixed points of a symmetry operation. For example, for rotation about an axis, the points on the axis do not move and in a reflection the points that remain unchanged make up a plane of symmetry.

Identity

The identity symmetry element is found in all objects and is denoted ''E''. It corresponds to an operation of doing nothing to the object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity element. An object having no symmetry elements other than E is called asymmetric. Such an object is necessarily chiral. : R-bromochlorofluoroiodomethane-2D

Mirror planes

Mirror planes are denoted by σ. In a molecule that also has an axis of symmetry, a mirror plane that includes the axis is called a vertical mirror plane and is labeled σ , while one perpendicular to the axis is called a horizontal mirror plane and is labeled σ . A vertical mirror plane that bisects the angle between two C2 axes is called a dihedral mirror plane, σ . XeF4 molecule

Rotational symmetry

Rotational symmetry, also known as radial symmetry, is represented by an axis about which the object rotates in its corresponding symmetry operation. A group of proper rotations is denoted as ''C'', where the degrees of rotation that restore the object is 360/n (''C''= 180º rotation, ''C''= 120º rotation, ''C''= 90º rotation, ''C''= 72º rotation). The ''C'' notation is also used for the related, more abstract,

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

. An

improper rotation In geometry, an improper rotation. (also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion) is an isometry in Euclidean space that is a combination of a Rotation (geometry), rotation about an axis and a reflection ( ...

is the composition of a rotation about an axis, and reflection in a plane perpendicular to that axis. The order in which the rotation and reflection are performed does not matter (that is, these operations commute). Improper rotation is also defined as the composition of a rotation about an axis, and inversion about a point on the axis. These definitions are equivalent because inversion about a point is equivalent to rotation by 180° about any axis, followed by mirroring about a plane perpendicular to that axis. The symmetry elements for improper rotation are the rotation axis, and either the mirror plane, the inversion point, or both. The improper rotation group of order 2''n'' is denoted ''S''.

Inversion

For inversion, denoted i, there must be a point in the center of an object that is the inversion center. Inversion consists of passing each point through the center of inversion and out to the same distance on the other side of the molecule. In the inversion operation for 3D coordinates, the inversion center is the origin (0,0,0). When an object is inverted, the position vector of a point in an object, ⟨x,y,z⟩, is inverted to ⟨-x,-y,-z⟩.

Gallery

File:Sigmav.png, Example of vertical mirror plane. , Example of C symmetry element. File:Ferrocene-from-xtal-3D-balls.png,

Ferrocene Ferrocene is an organometallic chemistry, organometallic compound with the formula . The molecule is a Cyclopentadienyl complex, complex consisting of two Cyclopentadienyl anion, cyclopentadienyl rings sandwiching a central iron atom. It is an o ...

molecule, an object having ''S'' symmetry. Its symmetry elements are: a vertical rotation axis, a horizontal plane, and an inversion point at the center.

References