Thermodynamic Estimators

Since the VDM, except in the simplest cases, is not exact various estimators for the same quantity will differ. For example the variational principle introduced in section II consists essentially in globally minimizing the squared difference between $\partial\rho/\partial\beta$ and ${\mathcal{H}}\rho$, either of which can be used in estimating the energy. As mentioned above the energy estimator Eq. 3.49 and its kinetic and potential energy pieces do not automatically satisfy the virial theorem for Coulomb systems at low density. As an alternative to Eq. 3.49, one can use the thermodynamic estimators,

$$\left< E \right> = - \left< \frac{\partial}{\partial \beta} \ln \rho \right>,$$ (168)

$$\left< K \right> = - \frac{\lambda}{\beta} \left< \frac{\partial }{\partial \lambda} \ln \rho \right>,$$ (169)

$$\left< V \right> = - \frac{e^2}{\beta} \left< \frac{\partial}{\partial e^2} \ln \rho \right>$$ (170)

for the total, kinetic and potential energy where $\left<\,\dots\,\right>$ denote thermal averages calculate from Eq. 2.4. These estimators satisfy

$$\left< E \right>=\left< K \right>+\left< V \right>$$ (171)

by the following argument. Any function $f=f(\beta \lambda,\beta e^2)$ satisfies

$$\beta \frac{\partial f}{\partial \beta} = \lambda \frac{\pa... ...{\partial \lambda} + e^2 \frac{\partial f}{\partial e^2}\quad.$$ (172)

From Eq. 3.19 it follows that all parameters $q_i=q_i({\bf R'};\beta,\lambda,e^2)$ have this property and therefore so does the variational density matrix.

In the zero temperature limit, the thermodynamic estimators satisfy the virial theorem, which is also satisfied by any exact and any variational Rayleigh-Ritz ground state. From the zero temperature limit of the VDM given by Eq. 3.46 and the $1/\beta$ factor in Eqs. 3.51 and 3.52, it is seen that the symmetry error $\delta({\bf R'})$ is unimportant in this limit. It should be noted that calculating the derivatives for $\left< K \right>$ and $\left< V \right>$ increases the numerical work. The pressure is estimated from

$$3\, \left< P \right> {V^{\!\!\!\!\!\!\:^\diamond}}= 2 \left< K \right> + \left< V \right>\;,$$ (173)

where ${V^{\!\!\!\!\!\!\:^\diamond}}$ is the volume of the simulation cell.