Coordinates Summary

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Coordinates Summary
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

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Transformation between Coordinate Systems
Cartesian Coordinates
Inter-Atomic Distances
Dihedral Angles
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From Cartesian Coordinates to Distances
4
From Distances to Cartesian Coordinates
5
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
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Bond Lengths, Bond Angles, Dihedral Angles
dihedral angle
aijkl
nab
nbc
a
b
c
xi
aikl
aijk
xk
xl
ajkl
xj
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Atomic Distances vs. Dihedral Angles
xi
aikl
a
xk
aijk
b
xl
ajkl
c
xj
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Cartesian Coordinates vs. Dihedral Angles
dihedral angle
aijkl
nab
nbc
a
b
c
xi
aikl
aijk
xk
xl
xj
ajkl
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Cartesian Coordinates vs. Dihedral Angles
xi
a c - a c cos aikl b c - b
c cos ajkl a b c a b c sin
aijk sin ajkl sin aijkl xl xk c
a
aijk
xj
aikl
ajkl
aijkl
xl
b
c
xk
xi (ui, vi, wi) xj (uj, vj, wj) xk (uk, vk,
wk) xl ?
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Cartesian Coordinates vs. Dihedral Angles
xi
na a / a nb b / b nab a b /
(a b sin aijk) c1 - c cos aikl
na c2 - c cos ajkl nb c3 c sin ajkl
sin aijkl nab c c1 c2 c3 xl xk c
a
aijk
xj
aikl
ajkl
b
xl
xk
c
nab
nb
aijkl
na
a local coordinate system
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Homework Assignment 4 Due 6pm, Fri, October 14th

• Compute the dihedral angles a and ß given the
  coordinates of the atoms in the picture.
• xi (36.671, 74.732, 224.213), xj (36.457,
  75.719, 225.271)
• xk (35.028, 76.253, 225.116), xl (34.044,
  75.462, 225.516)
• xm (32.682,75.923,225.362)

xi
a
xj
ß
xl
xm
xk
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Homework Assignment 4 Due 6pm, Fri, October 14th

• Compute the dihedral angles a and ß given the
  distances between the pairs of atoms in the
  picture.
• dij 1.4626445227, dik 2.4141870267, dil
  3.0218931152, djk 1.5333694923
• djl 2.4389840097, djm 3.7816030992, dkl
  1.3243628656, dkm 2.3818337473
• dlm 1.4461262047

xi
a
xj
ß
xl
xm
xk
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Homework Assignment 4 Due 6pm, Fri, October 14th

• Compute the distances dil and djm given the bond
  lengths, bond angles, and dihedral angles in the
  picture.
• dij 1.4626445227, djk 1.5333694923, dkl
  1.3243628656, dlm 1.4461262047
• p-aijk 1.2679532884, p-ajkl 1.0996644141,
  p-aklm 1.0733153864
• aijkl 1.3198177215, ajklm 3.1219755045

xi
aijk
xj
xl
ajkl
aijkl
aklm
xm
xk
ajklm
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Homework Assignment 4 Due 6pm, Fri, October 14th

• Compute the coordinates for xl and xm given the
  bond lengths, bond angles, and bond angles in the
  picture.
• 
  xi (36.671, 74.732,
  224.213)
• 
  xj (36.457, 75.719,
  225.271)
• 
  xk (35.028, 76.253, 225.116)
• dij 1.4626445227, djk 1.5333694923, dkl
  1.3243628656, dlm 1.4461262047
• p-aijk 1.2679532884, p-ajkl 1.0996644141,
  p-aklm 1.0733153864
• aijkl 1.3198177215, ajklm 3.1219755045

xi
aijk
xj
xl
ajkl
aijkl
aklm
xm
xk
ajklm