**Virial theorem** The virial theorem relates the time average kinetic <T> and time average potential <V> energy of a system governed by a potential that depends on pairwise interactions.

In a stable system consisting of N particles, bound by potential forces, the virial theorem states:

2<T> = -

_{k}∑

^{N}<

**F**_{k}.

**r**_{k}>

where

**F**_{k} represents the force on the kth particle, which is located at position

**r**_{k}.

The word "virial" derives from vis, the Latin word for "force" or "energy".

The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in statistical mechanics; this average total kinetic energy is related to the temperature of the system by the equipartition theorem.

However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in thermal equilibrium.

If the force between any two particles of the system results from a potential energy V(r) = αr

^{n} that is proportional to some power n of the inter-particle distance r, the virial theorem adopts a simple form:

2<T> = n<V

_{tot}>

Here V

_{tot} is the total potential energy of the system and not of an pair of particles.

Interpretation:

If all coordinates in the system are multiplied by a factor C then the potential energy will be increased by a factor Cn.

**Edited: **Why is this so?

Ref: Elementary quantum chemistry - Pilar - p51)

__Source:__ http://en.wikipedia.org/wiki/Virial_theorem