Local Energy

We wish to compute the local energy of the wave function, defined by

$$E_{\text{local}} \equiv \frac{\hat{H} \Psi_T }{\Psi_T},$$ (6)

where

$$\hat{H} \equiv \frac{\hbar^2}{2m}\nabla^2 + V$$ (7)

The calculation of the potential energy V is straightforward. Therefore, we will focus here the application of the Laplacian operator to the Jastrow wave function. Given the form of the trial wave function, it will prove convenient to define

 

$${\mathcal L}_T \ln (\Psi_T)$$ =

$$\ln (\det(A^{\text{up}})) + \ln (\det(A_{\text{down}})) + \sum_{i<j} U_{ij}$$ = (8)

We now attempt to calculation the action of $\nabla^2$ on $\Psi_T$ in terms of ${\mathcal L}_T$.

$$\nabla^2 \Psi_T({\mathbf R}) \nabla^2\exp({\mathcal L}_T({\mathbf R}))$$ =

$$\nabla \cdot \nabla (\exp({\mathcal L}_T({\mathbf R})))$$ =

$$\nabla \cdot \left[ \exp({\mathcal L}_T({\mathbf R})) \nabla {\mathcal L}_T({\mathbf R})\right]$$ =

$$\nabla^2 {\mathcal L}_T({\mathbf R}) \exp({\mathcal L}_T({\mathbf... ...+ (\nabla {\mathcal L}_T({\mathbf R})) \nabla \exp({\mathcal L}_T({\mathbf R}))$$ =

$$\left[ \nabla^2 {\mathcal L}_T({\mathbf R}) + (\nabla {\mathcal L}_T({\mathbf R}))^2 \right] \psi_T( {\mathbf R})$$ = (9)

This leaves a particularly simple form for the local energy.

$$E_{\text{local}}({\mathbf R}) = \frac{- \hbar^2}{2m} \left[ \... ... \nabla {\mathcal L}_T({\mathbf R}))^2\right] + V({\mathbf R})$$ (10)

Now, we are left with the task of computing $\nabla {\mathcal L}_T$ and $\nabla^2 {\mathcal L}_T$.

The gradient and Laplacian are linear operators, using (8), we can find action of these operators on each of the terms in the sum.