Specifying the geometry of molecules.
1) Cartesian co-ordinates.
N atoms in a molecule, 3N coordinates, (x, y, z,) per atom. However this is more
information than is needed for the purpose of molecule structure calculations.
H
H
H
H
All three molecules
have the same
molecular
geometry.
O
O
translation
translation
and rotation
O
H
H
The molecular geometry is specified by 3N – 6 coordinates. (Or 3N – 5 if the molecule is
linear.)
2) Internal Coordinates.
Build the molecule up by specifying bond lengths, bond angles, and dihedral angles.
water
atom
atom type
bonded to
bond length
angle with
angle
number
atom
atom
1
H
2
O
1
1.01
3
H
2
1.01
1
110.0
only 3 coordinates needed to specify the molecular geometry
hydrogen peroxide
(can leave out atom number as that is simply the number of the line in the table)
atom bonded to
bond angle with angle dihedral dihedral
type
atom
length
atom
wrt
angle
H
*
*
*
O
1
1.01
*
*
O
2
1.20
1
112.0
*
H
3
1.01
2
112.0
1
90.0
notice the 6 ‘missing’ coordinates indicated by *
H4
O2
H1
O3
methane
H
C
H
H
H
1
2
2
2
*
1.01
1.20
1.01
1.01
1
1
1
*
*
109.4
109.4
109.4
3
3
*
*
*
120.0
-120.0
(or 240.0)
A Z matrix. A very common method of specifying the geometry of the molecule.
A calculational method presented with a molecule in this format:
1) Places atom 1 at the origin (three coordinates ‘fixed’, x1=0, y1=0, z1=0)
2) Places atom 2 on the z axis (two coordinates ‘fixed’, x2=0, y2=0, z2=bond length2)
3) Places atom 3 in the yz plane (one coordinate fixed,
x3=0, y3=bondlength3 sin(angle3), z3=z2+bondlength3 cos(angle3) )
4) Calculates the Cartesian coordinates of the other atoms in this coordinate system
Many calculations then reposition the origin to be at the centre of mass and the axes to be
along symmetry axes (if any).
Many modern calculations have a graphical ‘front-end’ programme. Draw the molecule
on the screen and the Z matrix (or Cartesian coordinates) are worked out by the
programme from the diagram.