Slater-Jastrow wave function

This short paper summarizes the local energy calculation problem of the calculation of the Slater-Jastrow wave function. This wave function has number of desirable properties for many-body quantum Monte-Carlo calculations of electrons in the presence of ions.

The Slater-Jastrow wave function is a product of Slater determinants and the Jastrow correlation factor:

$$\Psi_T(\{{\mathbf r}_i\}) = \det(A^{\text{up}}) \det(A_{\text{down}}) \exp\left(\sum_{i<j} U_{ij} \right)$$ (1)

Here, the $A^{\text{up}}$ and $A^{\text{down}}$ are defined as the Slater matrices of the single particle up and down orbitals, respectively. That is

$$A = \left[ \begin{array}{ccccl} \phi_1({\mathbf r}_1) & \phi... ...\vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right]$$ (2)

Where $\phi_k$ are molecular orbitals centered at c_k:

$$\phi_k({\mathbf r}) = \exp \left(\frac{-({\mathbf r}- {\mathb... ...mega_k^2 + \nu_k \vert {\mathbf r}- {\mathbf c}_k\vert}\right)$$ (3)

The Jastrow correlation factor U_ij terms are defined in the following manner:

$$U_{ij} = \frac{a_{ij} r_{ij}}{1 + b_{ij} r_{ij}},$$ (4)

where $r_{ij}\equiv \vert{\mathbf r}_i - {\mathbf r}_j\vert$ and

$$a_{ij} = \left\{ \begin{array}{cl} e^2/8D & \text{if $ij$\space a... ...\\ e^2/2D & \text{if $ij$\space are electron-nuclear pairs} \end{array}\right.$$ (5)

This trial wave function (1) has a number of desirable properties:

1.

The corerct cusp conditions for both like and unlike electron spins.

2.

The coorect cusp behavior as the electron-nuclear separation becomes small.

3.

The variational parameters in (1) have a simple physical interpretation at large separations.

(a)

$\beta$ can be related to the polarizability of a molecule

(b)

$\nu^\ast$ the maximum value of $\nu$ is equal to $1/\sqrt{2I}$where I is the first ionization potential.