[MMTK] NMA and mean square fluctuations
Konrad Hinsen
hinsen at cnrs-orleans.fr
Thu Aug 14 11:18:36 UTC 2008
On 14.08.2008, at 10:49, Matias Saavedra wrote:
> Which is the theoretical basis for the calculation of atomic mean
> square fluctuations with normal modes? In particular the relation
> of mean square fluctuations with boltzmann constant, temperature
> and modes frequencies (eigenvalues). The typical relation in order
> to calculate B-factors. I cannot find a reference wich explain this
> issue with detail.
The B factors are the diagonal elements of the position
autocorrelation matrix. This can be written as a thermal average,
i.e. an integral over the Boltzmann distribution times terms of the
form <(r_i-<r_i>)(r_j-<r_j>)>. For a harmonic potential, this
Boltzmann distribution is a Gaussian distribution which makes it
possible to calculate the integral exactly. The result is that the
position autocorrelation matrix is kT times the inverse of the force
constant matrix.
> Why the calculation of the fluctuations suposes a division of the
> squared eigenvectors by the squared mode frequencies ?
That is just the spectral representation of the inverse force
constant matrix. Any symmetric matrix can be written as
M = sum_i[e_i u_i u_i],
where e_i are its eigenvectors and u_i are its eigenvectors. The
eigenvectors of the (mass weighted) force constant matrix are the
squared frequencies.
> And finally, why the criterion for the applicability of classical
> mechanics is hv<<KbT ? and how this affects the interpretation of NMA?
The separation of energy levels in a quantum harmonic oscillation is
hv. Classical mechanics yields a continuum of energy levels, which
becomes a good approximation when the energy (kT) is high enough to
be made up of many quantum excitations (hv).
Konrad.
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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: hinsen at cnrs-orleans.fr
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