Widom insertion method

The Widom Insertion Method is a statistical thermodynamic approach to the calculation of material and mixture properties. It is named for Benjamin Widom, who derived it in 1963. In general, there are two theoretical approaches to determining the statistical mechanical properties of materials. The first is the direct calculation of the overall Partition Function of the system, which directly yields the system free energy. The second approach, known as the Widom Insertion method, instead derives from calculations centering on one molecule. The Widom Insertion method directly yields the chemical potential of one component rather than the system free energy. This approach is most widely applied in molecular computer simulations but has also been applied in the development of analytical statistical mechanical models.

As originally formulated by Benjamin Widom in 1963, the approach can be summarized by the equation:

${\displaystyle \mathbf {B} _{i}={\frac {\rho _{i}}{a_{i}}}=\left\langle \exp \left(-{\frac {\psi _{i}}{k_{B}T}}\right)\right\rangle }$

where ${\displaystyle \mathbf {B} _{i}}$ is called the 'insertion parameter', ${\displaystyle \rho _{i}}$ is the number density of species ${\displaystyle i}$, ${\displaystyle a_{i}}$ is the activity of species ${\displaystyle i}$, ${\displaystyle k_{B}}$ is the Boltzmann constant, and ${\displaystyle T}$ is temperature, and ${\displaystyle \psi }$ is the interaction energy of an inserted particle with all other particles in the system. The average is over all possible insertions. This can be understood conceptually as fixing the location of all molecules in the system and then inserting a particle of species ${\displaystyle i}$ at all locations through the system, averaging over a Boltzmann factor in its interaction energy over all of those locations.

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