#+TITLE: Slater Determinant
#+SETUPFILE: ../tools/theme.setup
#+INCLUDE: ../tools/lib.org

The slater deteminant is required for the calculation of the
wavefunction, gradient, and derivatives. These quantities will be used
to calculate the local Energy (\[E_L\]).

ψ(x) = det|ϕ₁(x₁)...ϕᵢ(yᵢ)...ϕₙ(xₙ)|

Concerning the gradient and laplacian, in fact what is actually
calculated is the ratio of the gradient/laplacian and the determinant
of the slater matrix:

∇ψ(x)/ψ(x)

and

∇²ψ(x)/ψ(x)

This avoids the unnecessary multiplication and division of by the
determinant ψ(x).


* Headers                                                          :noexport:
  #+begin_src elisp :noexport :results none
(org-babel-lob-ingest "../tools/lib.org")
  #+end_src


  #+begin_src c :tangle (eval h_private_type)
#ifndef QMCKL_MO_HPT
#define QMCKL_MO_HPT

#include <stdbool.h>
  #+end_src

  #+begin_src c :tangle (eval c_test) :noweb yes
#include "qmckl.h"
#include "assert.h"
#ifdef HAVE_CONFIG_H
#include "config.h"
#endif

#include <stdio.h>
#include <math.h>
#include "chbrclf.h"
#include "qmckl_ao_private_func.h"
#include "qmckl_mo_private_func.h"
#include "qmckl_slater_determinant_private_func.h"

int main() {
    qmckl_context context;
    context = qmckl_context_create();

    qmckl_exit_code rc;
  #+end_src

  #+begin_src c :tangle (eval c)
#ifdef HAVE_CONFIG_H
#include "config.h"
#endif

#ifdef HAVE_STDINT_H
#include <stdint.h>
#elif HAVE_INTTYPES_H
#include <inttypes.h>
#endif

#include <stdlib.h>
#include <string.h>
#include <stdbool.h>
#include <assert.h>

#include "qmckl.h"
#include "qmckl_context_private_type.h"
#include "qmckl_memory_private_type.h"
#include "qmckl_memory_private_func.h"
#include "qmckl_ao_private_type.h"
#include "qmckl_ao_private_func.h"
#include "qmckl_mo_private_type.h"
#include "qmckl_mo_private_func.h"
#include "qmckl_slater_determinant_private_type.h"
#include "qmckl_slater_determinant_private_func.h"
  #+end_src

* Context

  The following arrays are stored in the context:

  |-----------------+-------------------------------+------------------------------------|
  | ~type~          | ~char~                        | α (~'A'~) or β (~'B'~) determinant |
  | ~walk_num~      | ~uint64_t~                    | Number of walkers                  |
  | ~det_num~       | ~uint64_t~                    | Number of determinants per walker  |
  | ~fermi_num~     | ~uint64_t~                    | Number of number of fermions       |
  | ~mo_index_list~ | ~mo_index[walk_num][det_num]~ | Index of MOs for each walker       |

  Computed data:
  
  |-------------------+------------------------------------------+----------------------------------------------------------------------------------------|
  | ~det_matrix_list~ | ~[walk_num][det_num][mo_num][fermi_num]~ | The slater matrix for each determinant of each walker.                                 |
  |-------------------+------------------------------------------+----------------------------------------------------------------------------------------|
  | ~det_vgl~         | ~[5][walk_num][det_num]~                 | Value, gradients, Laplacian of the MOs at electron positions                           |
  | ~det_vgl_date~    | ~uint64_t~                               | Late modification date of Value, gradients, Laplacian of the MOs at electron positions |
  |-------------------+------------------------------------------+----------------------------------------------------------------------------------------|