Dihedral angle is defined by points ,
,
, and
(
):
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(7.22) |
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(7.23) |
The first derivatives of with respect to Cartesian coordinates are:
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(7.24) |
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(7.25) |
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(7.26) |
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(7.27) |
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(7.28) |
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(7.29) |
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(7.30) |
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(7.31) |
These equations for the derivatives have a numerical instability when the angle goes to 0. Thus, the following set of equations is used instead [#!Sch93!#]:
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(7.32) |
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(7.33) |
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(7.34) |
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(7.35) |
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(7.36) |
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(7.37) |
The only possible instability in these equations is when the length of
the central bond of the dihedral, , goes to 0. In such a case,
which should not happen, the derivatives are set to 0. The expressions for
an improper dihedral angle, as opposed to a dihedral or dihedral angle,
are the same, except that indices
are permuted to
.
In both cases, covalent bonds
,
, and
are defining
the angle.