Molecular symmetry in the context of "Nitrogen dioxide"

Dimethyl sulfoxide (DMSO) is an organosulfur compound with the formula (CH₃)₂S=O. This colorless liquid is the sulfoxide most widely used commercially. It is an important polar aprotic solvent that dissolves both polar and nonpolar compounds and is miscible in a wide range of organic solvents as well as water. It has a relatively high boiling point. DMSO is metabolised to compounds that leave a garlic-like taste in the mouth after DMSO is absorbed by skin.

In terms of chemical structure, the molecule has idealized C_s symmetry. It has a trigonal pyramidal molecular geometry consistent with other three-coordinate S(IV) compounds, with a nonbonded electron pair on the approximately tetrahedral sulfur atom.

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 about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each a special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation.It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have improper rotation symmetry.

It is important to note the distinction between rotary reflection and rotary inversion symmetry operations and their associated symmetry elements. Rotary reflections are generally used to describe the symmetry of individual molecules and are defined as a 360°/n rotation about an n-fold rotation axis followed by a reflection over a mirror plane perpendicular to the n-fold rotation axis. Rotoinversions are generally used to describe the symmetry of crystals and are defined as a 360°/n rotation about an n-fold rotation axis followed by an inversion through the origin. Although rotary reflection operations have a rotoinversion analogue and vice versa, rotoreflections and rotoinversions of the same order need not be identical. For example, a 6-fold rotoinversion axis and its associated with symmetry operations are distinct from those resulting from a 6-fold reflection axis.

In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.

Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1).