# Helmholtz Energy

Helmholtz energy function (Hermann Ludwig Ferdinand von Helmholtz) $$A$$ (for arbeit):

$\left.A\right.=U-TS$

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

$\left.dA\right.=dU-TdS-SdT$

From the second law of thermodynamics one obtains

$\left.dA\right.=TdS -pdV -TdS-SdT$

thus one arrives at

$\left.dA\right.=-pdV-SdT.$

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

$dA=\left(\dfrac{\partial A}{\partial T}\right)_V dT + \left(\dfrac{\partial A}{\partial V}\right)_T dV$

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

$\left.A\right.=-k_B T \ln Q_{NVT}$

where $$k_B$$ is the Boltzmann constant, T is the temperature, and $$Q_{NVT}$$ is the canonical ensemble partition 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]):

$\dfrac{A-A_{ {\mathrm{classical}} }}{N} = \dfrac{\hbar^2}{24m(k_BT)^2} \langle F^2 \rangle$

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