Improved stress-energy tensor derivation from Green's function

In chapter 4 of the book Quantum fields in curved space by Birrell and Davies, the authors provide a derivation of the Casimir energy for a massless scalar field (in 4-D) using the method of images and Green’s function. The new improved stress-energy tensor which is traceless has been used to calculate the vacuum expectation value. The conformally invariant traceless tensor is given by the following expression (Eq. 4.35 of the book)

$$ T_{munu} = frac{2}{3}, partial_muphi partial_nuphi,- frac{1}{6},eta_{munu}eta^{rhosigma}partial_rhophi,partial_sigmaphi,-frac{1}{3}phipartial_mupartial_nuphi,+ eta_{munu}phiBoxphi.$$

This for the energy density component reduces to

$$ T_{00} = frac{1}{2} (partial_tphi)^2 + frac{1}{6} (partial_iphi)^2 – frac{1}{4}phi partial_t^2phi – frac{1}{12}phipartial_i^2phi.$$
where the sum over the spatial index $i=1,2,3$ is implied.

In all the previous calculations the book has used the Hadamard’s Green’s function $D^{(1)}$ to calculate the canonical stress-energy tensor which is defined as

$$ D^{(1)}(x,x’) = langle 0| {phi(x),phi(x’)}|0rangle.$$

However, the book does not show how to find the improved stress-energy tensor from the Green’s function but uses $D^{(1)}$ to calculate the Casimir energy. I am stuck with the relation between the improved stress-energy tensor and the Green’s function. The first two terms of $T_{00}$ can be found easily by doing the following

$$ lim_{x’,x” rightarrow x} left(frac{1}{4}partial_{t^{”}}partial_{t^{‘}} + frac{1}{12}partial_{x_i^{”}}partial_{x_i^{‘}}right) D^{(1)}(x’,x”)$$

where $x=(t,x_i)$. But I am stuck with how to get the next two terms in the expression of $T_{00}$. Any guidance is highly appreciated. Thanks.