# Helmholtz energy function

Helmholtz energy function (Hermann Ludwig Ferdinand von Helmholtz) Definition of ${\displaystyle A}$ (for arbeit):

${\displaystyle A:=U-TS}$

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

${\displaystyle \left.dA\right.=dU-TdS-SdT}$

From the second law of thermodynamics one obtains

${\displaystyle \left.dA\right.=TdS-pdV-TdS-SdT}$

thus one arrives at

${\displaystyle \left.dA\right.=-pdV-SdT}$.

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

${\displaystyle dA=\left({\frac {\partial A}{\partial T}}\right)_{V}dT+\left({\frac {\partial A}{\partial V}}\right)_{T}dV}$

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

${\displaystyle \left.A\right.=-k_{B}T\ln Q_{NVT}}$

where ${\displaystyle k_{B}}$ is the Boltzmann constant, T is the temperature, and ${\displaystyle Q_{NVT}}$ 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]):

${\displaystyle {\frac {A-A_{\mathrm {classical} }}{N}}={\frac {\hbar ^{2}}{24m(k_{B}T)^{2}}}\langle F^{2}\rangle }$

where ${\displaystyle \langle F^{2}\rangle }$ is the mean squared force on any one atom due to all the other atoms.