Molecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.

Molecular geometry determination

The molecular geometry can be determined by various spectroscopic methods and diffraction methods. IR, microwave and Raman spectroscopy can give information about the molecule geometry from the details of the vibrational and rotational absorbances detected by these techniques. X-ray crystallography, neutron diffraction and electron diffraction can give molecular structure for crystalline solids based on the distance between nuclei and concentration of electron density. Gas electron diffraction can be used for small molecules in the gas phase. NMR and FRET methods can be used to determine complementary information including relative distances,

dihedral angles,

angles, and connectivity. Molecular geometries are best determined at low temperature because at higher temperatures the molecular structure is averaged over more accessible geometries (see next section). Larger molecules often exist in multiple stable geometries (conformational isomerism) that are close in energy on the potential energy surface. Geometries can also be computed by ab initio quantum chemistry methods to high accuracy. The molecular geometry can be different as a solid, in solution, and as a gas.

The position of each atom is determined by the nature of the chemical bonds by which it is connected to its neighboring atoms. The molecular geometry can be described by the positions of these atoms in space, evoking bond lengths of two joined atoms, bond angles of three connected atoms, and torsion angles (dihedral angles) of three consecutive bonds.

The influence of thermal excitation

Since the motions of the atoms in a molecule are determined by quantum mechanics, one must define “motion� in a quantum mechanical way. The overall (external) quantum mechanical motions translation and rotation hardly change the geometry of the molecule. (To some extent rotation influences the geometry via Coriolis forces and centrifugal distortion, but this is negligible for the present discussion.) A third type of motion is vibration, which is the internal motion of the atoms in a molecule. The molecular vibrations are harmonic (at least to good approximation), which means that the atoms oscillate about their equilibrium, even at the absolute zero of temperature. At absolute zero all atoms are in their vibrational ground state and show zero point quantum mechanical motion, that is, the wavefunction of a single vibrational mode is not a sharp peak, but an exponential of finite width. At higher temperatures the vibrational modes may be thermally excited (in a classical interpretation one expresses this by stating that “the molecules will vibrate faster�), but they oscillate still around the recognizable geometry of the molecule.

To get a feeling for the probability that the vibration of molecule may be thermally excited, we inspect the Boltzmann factor \exp\left( -\frac{\Delta E}{kT} \right) , where \Delta E is the excitation energy of the vibrational mode, k the Boltzmann constant and T the absolute temperature. At 298K (25 Â°C), typical values for the Boltzmann factor are: 0.089 for ΔE = 500 cm^−1 ; ΔE = 0.008 for 1000 cm^−1 ; 7 10^−4 for ΔE = 1500 cm^−1. That is, if the excitation energy is 500 cm^−1, then about 9 percent of the molecules are thermally excited at room temperature. The lowest excitation vibrational energy in water is the bending mode (about 1600 cm^−1). Thus, at room temperature less than 0.07 percent of all the molecules of a given amount of water will vibrate faster than at absolute zero.

As stated above, rotation hardly influences the molecular geometry. But, as a quantum mechanical motion, it is thermally excited at relatively (as compared to vibration) low temperatures. From a classical point of view it can be stated that more molecules rotate faster at higher temperatures, i.e., they have larger angular velocity and angular momentum. In quantum mechanically language: more eigenstates of higher angular momentum become thermally populated with rising temperatures. Typical rotational excitation energies are on the order of a few cm^−1.

The results of many spectroscopic experiments are broadened because they involve an averaging over rotational states. It is often difficult to extract geometries from spectra at high temperatures, because the number of rotational states probed in the experimental averaging increases with increasing temperature. Thus, many spectroscopic observations can only be expected to yield reliable molecular geometries at temperatures close to absolute zero, because at higher temperatures too many higher rotational states are thermally populated.

Bonding

Molecules, by definition, are most often held together with covalent bonds involving single, double, and/or triple bonds, where a "bond" is a shared pair of electrons (the other method of

In geometry, a vertex (plural vertices) is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces (typically triangles) in 3D models, where each such point is given as a vector.

Definitions

Of an angle

The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place.

Of a polytope

A vertex is a corner point of a polygon, polyhedron, or other higher dimensional polytope, formed by the intersection of edges, faces or facets of the object: a vertex of a polygon is the point of intersection of two edges, a vertex of a polyhedron is the point of intersection of three or more edges or faces, and a vertex of a d-dimensional polytope is the intersection point of d or more edges, faces or facets.

In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians; otherwise, it is called "concave" or "reflex". More generally, a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, and concave otherwise.

Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope, and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices. However, in graph theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. There is also a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, and if a polygon is approximated by a smooth curve there will be a point of extreme curvature near each polygon vertex. However, a smooth curve approximation to a polygon will also have additional vertices, at the points where its curvature is minimal.

Of a plane tiling

A vertex of a plane tiling or tessellation is a point where three or more tiles meet; generally, but not always, the tiles of a tessellation are polygons and the vertices of the tessellation are also vertices of its tiles. More generally, a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such as simplicial complexes are its zero-dimensional faces.

Principal vertex

A polygon vertex x_i of a simple polygon P is a principal polygon vertex if the diagonal [x_{(i-1)},x_{(i+1)}] intersects the boundary of P only at x_{(i-1)} and x_{(i+1)}. There are two types of principal vertices: ears and mouths.

Ears

A principal vertex x_i of a simple polygon P is called an ear if the diagonal [x_{(i-1)},x_{(i+1)}] that bridges x_i lies entirely in P. (see also convex polygon)

Mouths

A principal vertex x_i of a simple polygon P is called a mouth if the diagonal [x_{(i-1)},x_{(i+1)}] lies outside the boundary of P. (see also concave polygon)

Vertices in computer graphics

In computer graphics, objects are often represented as triangulated polyhedra in which the object vertices are associated not only with three spatial coordinates but also with other graphical information necessary to render the object correctly, such as colors, reflectance properties, textures, and surface normals; these properties are used in rendering by a vertex shader, part of the vertex pipeline.