# Helmholtz (Free) 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.