irpjast/README.org

* IRPJAST

  CHAMP's Jastrow factor computation using the IRPF90 method

  Original equation:
  
 $$
 \sum_{i=2}^{Ne} \sum_{j=1}^i \sum_{pkl} \sum_a^{Nn} c_{apkl}\, r_{ij}^k\, ( R_{ia}^l + R_{ja}^l) ( R_{ia} R_{ja})^m 
 $$

 Expanding, one obtains:
 
 $$
 \sum_{i=2}^{Ne} \sum_{j=1}^i \sum_{pkl} \sum_a^{Nn} c_{apkl} R_{ia}^{p-k-l}\, r_{ij}^k\, R_{ja}^{p-k+l} + c_{apkl} R_{ia}^{p-k+l}\, r_{ij}^k\, R_{ja}^{p-k-l}
 $$

 The equation is symmetric if we exchange $i$ and $j$, so we can rewrite

 $$
 \sum_{i=1}^{Ne} \sum_{j=1}^{Ne} \sum_{pkl} \sum_a^{Nn} c_{apkl} R_{ia}^{p-k-l}\, r_{ij}^k\, R_{ja}^{p-k+l} 
 $$
 
 If we reshape $R_{ja}^p$ as a matrix $R_{j,al}$ of size 
 $N_e \times N_n(N_c+1)$, 
 for every $k$, we can pre-compute the matrix product

 $$
 C_{i,al}^{k} = \sum_j r_{ij}^k\, R_{i,al}
 $$
 which can be computed efficiently with BLAS.
 We can express the total Jastrow as:

 $$
 \sum_{i=1}^{Ne} \sum_{pkl} \sum_a^{Nn}
 c_{apkl} R_{ia}^{p-k-l}\, C_{i,a(p-k+l)}^k
 $$