pressure, so $b_2^{\rm int}$ must be negative.

The second virial coefficient
=============================

The second virial coefficient (\[b2int\]) is evaluated as $$b_2^{\rm int} =\frac{1}{N} \int dE \left[ \rho_2(E) -\rho_2^0(E)
\right]
e^{-\beta(E-2\Delta)},
\label{b2e}$$ where the level densities $\rho_2(E)$ are given by $$\left\{
\begin{array}{l}
\rho_2(E)=\mathop{\rm Tr}\nolimits_2 \delta(E-{\cal H}) \\
\rho_2^0(E)= \mathop{\rm Tr}\nolimits_2 \delta(E-{\cal H}_0).
\end{array}
\right.
\label{rhos}$$ The level densities are obtained from the formula $${\cal G}_+(E) = \frac{1}{E+i\epsilon-{\cal H}}={\cal P}
\frac{1}{E-{\cal H}}
-i\pi\delta(E-{\cal H}),
\label{Green}$$ which transfers the problem to the determination of the Green’s function, for which the general equation holds $${\cal G}(z) = {\cal G}_0(z)+{\cal G}_0(z) V {\cal G}(z),
\label{Geq}$$ with $z$ a complex number. The density $\rho(E)$ is obtained from ${\cal G}$ as $$\rho(E)=-\frac{1}{\pi} \mathop{\rm Im}\nolimits
\mathop{\rm Tr}\nolimits {\cal G}_+ (E).
\label{rhoG}$$

The states of the unperturbed 2-vacancy Hamiltonian are denoted by two wavenumbers $\vec{k}_1$ and $\vec{k}_2$ with the property $$\begin{aligned}
{\cal H}_0 |\vec{k}_1\,\vec{k}_2\rangle &=&E_0(\vec{k}_1, \vec{k}_2)
|\vec{k}_1\,\vec{k}_2\rangle\nonumber\\
&=&\left(2\Delta+\varepsilon(\vec{k}_1)
+\varepsilon(\vec{k}_2) \right) |\vec{k}_1\,\vec{k}_2\rangle.
\label{states}\end{aligned}$$ The matrix elements of $V$ are obtained from (\[vk\]) $$\langle\vec{k}_1\,\vec{k}_2| V |\vec{k}_1'\,\vec{k}_2'\rangle =
\frac{2U}{N} \sum_{\vec{G}}
\delta_{\vec{k}_1+\vec{k}_2,\vec{k}_1'+\vec{k}_2'+\vec{G}},
\label{vmat}$$ which shows that a representation in center of mass and relative coordinates will be advantageous. Thus we introduce $$\vec{K}=\vec{k}_1+\vec{k}_2,\mbox{~~~~~}\vec{k}=(\vec{k}_1-
\vec{k}_2)/2.
\label{kk}$$ The sum over $\vec{G}$ is eliminated by choosing the 