[MMTK] Normalized distance vectors

Konrad Hinsen research at khinsen.fastmail.net
Fri Aug 16 10:50:38 UTC 2013

Lars Skjærven writes:

 > Sorry for the vague formulation. I definitively don't want to
 > normalize the vectors so that k(r=1). I was referring to the
 > positional internal parts of the second derivatives.  To
 > illustrate, take a 3x3 super-element of H (for atoms i and j)
 > H_ij. Then the elements of the first row of H_ij is calculated as
 > k(r) * (v_x * v_x , v_x * v_y, v_x*v_z), where the vector v is the
 > difference vector between atom i and j: (x_j-x_i , y_j-y_i,
 > z_j-z_i). The questions was whether it is customary to normalize
 > the vector v (but not for use in k(r)!).

I think I see what you are getting at, but I wouldn't call it
"normalization". It's just the result of computing the second
derivatives of an harmonic potential. If you start from

  U_ij = k(r_ij) (dx_i - dx_j)^2,

where dx_i is the (vector) displacement of atom i, then you just
compute the second derivative with respect to dx_i (treating r_ij as a
constant), and you get the right result, which indeed involves the
distance vector divided by its length. One way to see quickly that it
must be divided by its length, or at least *some* length, is
dimensional analysis: the elements of the Hessian have the units of a
force constant, so anything coming after k(r) must be dimensionless.

 > I think we figured out that it should be normalized, and my error
 > was related to the ANM function:

 > function k(r):
 >   if r>15: return 0
 >   else: return (1 / r^2)

If the return value of that function is supposed to be the pair force
constant, then the function is just wrong. Alternatively, if the
function is right, then it's supposed to return something else than
the pair force constant.

There is no "choice of normalization" in the mathematics. There is only
a choice in where to divide by r^2 in the program.

Konrad Hinsen
Centre de Biophysique Moléculaire, CNRS Orléans
Synchrotron Soleil - Division Expériences
Saint Aubin - BP 48
91192 Gif sur Yvette Cedex, France
Tel. +33-1 69 35 97 15
E-Mail: research AT khinsen DOT fastmail DOT net
ORCID: http://orcid.org/0000-0003-0330-9428
Twitter: @khinsen

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