Distances vs. Angles

1
Distances vs. Angles
http//www.math.iastate.edu/wu/math597.html
http//www.math.iastate.edu/wu/Math597HW0000/index
.htm

• Math/BCB/ComS597
• Zhijun Wu
• Department of Mathematics

2
Transformation between Coordinate Systems
Cartesian Coordinates
Inter-Atomic Distances
Dihedral Angles
3
Bond Lengths, Bond Angles, Dihedral Angles
The structure is determined by the bond lengths
and the angle.
a
b
c
dihedral angle
c
N
a
C
N
Ca
b
bond angle
4
Bond Lengths, Bond Angles, Dihedral Angles
dihedral angle
aijkl
nab
nbc
a xj xi b xk xj c xl xk
a
b
c
xi
nab unit normal of a-b plane. nbc unit normal
of b-c plane.
xk
aijk
xl
ajkl
xj
5
Vector Inner and Cross Products
nab
a
b
ab
6
Vector Inner and Cross Products
a
b
c
7
Bond Lengths, Bond Angles, Dihedral Angles
dihedral angle
aijkl
nab
nbc
a
b
c
xi
xk
aijk
xl
ajkl
xj
8
Bond Lengths, Bond Angles, Dihedral Angles
dihedral angle
aijkl
nab
nbc
a
b
c
xi
xk
aijk
xl
ajkl
xj
9
Bond Lengths, Bond Angles, Dihedral Angles
dihedral angle
aijkl
nab
nbc
a
b
c
xi
aikl
xk
aijk
xl
ajkl
xj
10
Bond Lengths, Bond Angles, Dihedral Angles
xi
aikl
a
xk
aijk
b
xl
ajkl
c
xj