# [gmx-users] principal moments of ellipsoid

Erik Marklund erikm at xray.bmc.uu.se
Tue May 10 13:56:21 CEST 2011

```Mark Abraham skrev 2011-05-10 12.31:
> On 10/05/2011 8:23 PM, lammps lammps wrote:
>> Hi,
>> I want to study the shape change of a macromolecule which seems like
>> a ellipsoid.  So, I need obtain the three principal moments of the
>> molecules.
>> Is the command of g_gyrate helpful for me? The sum of the three
>> principal moments should be equal to the <Rg^2>.  However, I used the
>> command g_gyrate -f traj.gro -s nvt.tpr -n anly.ndx -o shape.xvg -p
>> obtain the Rg and other three qualities.
>> It seems that the sum of the last three qualities is not equal to the
>> Rg^2. Is there something wrong?
>
> Why should that sum equal Rg^2?
>
> Mark
Shouldn't it be half of that sum?

Rg^2 = 1/N Sum([r_k - <r>]^2),

where r are vectors and masses are set to unity for conveniance. If
decomposed in orthonormal axes x, y, and z, it yealds

Rg^2 = 1/N Sum([x_k-<x>]^2 + [y_k-<y>]^2 + [z_k-<z>]^2).

The radius of gyration around e.g. the x-axis is

Rg_x^2 = !/N Sum([y_k-<y>]^2 + [z_k-<z_k>]^2).

Hence Rg^2 = (Rg_x^2+Rg_y^2+Rg_z^2) / 2.

That matches the numbers I get from my tests.

--
-----------------------------------------------
Erik Marklund, PhD student
Dept. of Cell and Molecular Biology, Uppsala University.
Husargatan 3, Box 596,    75124 Uppsala, Sweden
phone:    +46 18 471 4537        fax: +46 18 511 755
erikm at xray.bmc.uu.se    http://folding.bmc.uu.se/

```