Molecules are not static entities. Even at absolute zero temperature atoms in a molecule are actively vibrating. The molecular geometry represented by a static picture on the computer screen or a Dreiding model is therefore only an approximation. The term atom position is usually understood as a position of the atom nucleus, or rather as some kind of average position of the vibrating nucleus. The dimensions of an atom nucleus are negligible compared to average bond lengths, and since its mass is thousands of times larger than the mass of surrounding electrons, the nucleus is the true center of gravity of an atom. The major conceptual difficulty is to decide what is an average position of a nucleus. Nuclear vibrations are anharmonic, and hence, the time average position of a nucleus is not located half way between its extreme positions. Moreover, in molecules containing more than two atoms, nuclei vibrate not only along chemical bonds but also in directions perpendicular to them.Interatomic distances are usually expressed in Ångströms, since distances between chemically bonded atoms are of the order of 1 Å. Also,atomic units are frequently used: 1 a.u. = 1 Bohr = 0.529177249 Å.
The simplest way to specify molecular geometry to the computer is to list cartesian coordinates for each atom. In most cases the right-handed coordinate system is used, whose axes are perpendicular to each other (i.e., orthogonal), as represented in figure below.
Figure: Cartesian coordinate
Cartesian coordinates are usually listed in 3-column format, X, Y, and Z coordinates for each atom. Cartesian coordinates are an efficient representation of molecular geometry for the computer, and have the advantage of including actual spatial orientation of the molecule.
However, they lack the chemical contents for chemists. Chemists prefer to specify and analyze molecular geometry in terms of internal coordinates, i.e., bond lengths and bond angles. The most popular internal coordinates are shown in figure below:
Internal coordinates are efficiently calculated by the computer from cartesian coordinates using the vector operations described above. The bond length , dij, is simply a distance between two bonded atoms i and j, i.e., the length of the vector between atom i and j
The valence angle called also bond angle, teta
ijk, between bonds originating on atom j. The valence angle is always positive and not larger than 180, it is the smaller of the two possible angles.
Figure: Internal coordinates: a) bond length, b) valence angle, c) torsional angle.
The torsional angle, is a dihedral angle, phi
ijkl between two planes passing through atoms i,j,k and j,k,l, respectively. It is an angle between vectors normal (i.e., perpendicular) to these planes, or alternatively, an angle between the lines drawn on these planes perpendicularly to the edge where planes.
Figure: Cartesian coordinate
Cartesian coordinates are usually listed in 3-column format, X, Y, and Z coordinates for each atom. Cartesian coordinates are an efficient representation of molecular geometry for the computer, and have the advantage of including actual spatial orientation of the molecule.
However, they lack the chemical contents for chemists. Chemists prefer to specify and analyze molecular geometry in terms of internal coordinates, i.e., bond lengths and bond angles. The most popular internal coordinates are shown in figure below:
Internal coordinates are efficiently calculated by the computer from cartesian coordinates using the vector operations described above. The bond length , dij, is simply a distance between two bonded atoms i and j, i.e., the length of the vector between atom i and j
The valence angle called also bond angle, tetaijk, between bonds originating on atom j. The valence angle is always positive and not larger than 180, it is the smaller of the two possible angles.
Figure: Internal coordinates: a) bond length, b) valence angle, c) torsional angle.
The torsional angle, is a dihedral angle, phiijkl between two planes passing through atoms i,j,k and j,k,l, respectively. It is an angle between vectors normal (i.e., perpendicular) to these planes, or alternatively, an angle between the lines drawn on these planes perpendicularly to the edge where planes.
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