# Helmholtz energy function

**Helmholtz energy function** (Hermann Ludwig Ferdinand von Helmholtz)
Definition of (for *arbeit*):

where *U* is the internal energy, *T* is the temperature and *S* is the entropy.
*(TS)* is a *conjugate pair*. The differential of this function is

From the second law of thermodynamics one obtains

thus one arrives at

- .

For *A(T,V)* one has the following *total differential*

The following equation provides a link between classical thermodynamics and statistical mechanics:

where is the Boltzmann constant, *T* is the temperature, and is the canonical ensemble partition function.

## Ideal gas

*Main article: Ideal gas Helmholtz energy function*

## Quantum correction

A quantum correction can be calculated by making use of the Wigner-Kirkwood expansion of the partition function, resulting in (Eq. 3.5 in ^{[1]}):

where is the mean squared force on any one atom due to all the other atoms.