# Re: amber-developers: Truncated Octahedron box volume (fwd)

From: Thomas Cheatham <tec3.utah.edu>
Date: Mon, 23 Jul 2007 20:30:19 -0600 (Mountain Daylight Time)

> Thanks for your response. But I could not understand some parts of your
> answer. Let me explain how I look at this issue: The volume of a truncated
> octahedron is defined as
>
> V = 11.3137085 * a^3
>
> in http://en.wikipedia.org/wiki/Truncated_octahedron

Wikepedia is a source of information, however "truncated octahedron" can
be defined in multiple ways. For example, look at the definition used in
Allen & Tildesley which essentially lopes off corners of a cubic box
compared to that implemented in AMBER (as a triclinic unit cell).

> In amber, the box information is defined with 6 values;
>
> 51.9046302 51.9046302 51.9046302 109.4712190 109.4712190 109.4712190
>
> and as far as I understood, each of the first 3 values are the half of a
> cube's edge length (l/2; assuming we have a cube with edge length of 'l').
> From these 6 parameters, someone needs to find what 'a' is.

No, the actual lengths of the periodic unit cell are there, not 1/2 the
box length. If orthorhombic (i.e. rectangular with 90 degree angles),
these are the lengths of each side of the box.

> I do not understand what 109.4712190 is. In your explanation, u define a

This is the angle between the sides; as an example, use ptraj to image
your truncated octahedron without specifying the "familiar" keyword. It
will look like a slanted rectangle. The angles are all 109.47... between
each side.

> parameter 'a' which is not clear to me. It is not the edge length of a
> truncated octahedron defined in the above website.

Do not trust the above mentioned website :-)

You can look through the ptraj or sander or PMEMD code and see:

V = x * y * z * sqrt(1.0 - cos(alpha)**2 - cos(beta)**2 - cos(gamma)**2 +
2.0 * cos(alpha) * cos(beta) * cos(gamma))

x, y, z are the box lengths: x = y = z = a in your example. alpha, beta,
gamma are the box angles which equal 109.47... for a truncated
octahedron.

An easier shape (since the angles are integral and hence do not suffer
from finite precision) which we haven't exploited in AMBER is the rhombic
dodecahedron (12-sided, equal sides) which has x=y=z and alpha=60,
beta=90, gamma=60). This is the shape that is closest to a sphere, yet
space filling.

We discuss this in Current Protocols in Nucleic Acid Chemistry (articles
by Cheatham, Kollman and Brooks) from ~2000-2003. See the attached PDF
for a summary.

-- tec3

Received on Wed Jul 25 2007 - 06:07:26 PDT
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