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Helmholtz (Free) Energy

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  • Helmholtz energy function (Hermann Ludwig Ferdinand von Helmholtz) \(A\) (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

    \[\left.dA\right.=TdS -pdV -TdS-SdT\]

    thus one arrives at


    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.

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