* Debug the hessian

*Program to check the hessian matrix*

The program compares the result of the first and last code for the
hessian. First of all the 4D hessian and after the 2D hessian.

Provided:
| mo_num                     | integer | number of MOs                        |
| optimization_method        | string  | Method for the orbital optimization: |
|                            |         | - 'full' -> full hessian             |
|                            |         | - 'diag' -> diagonal hessian         |
| dim_list_act_orb           | integer | number of active MOs                 |
| list_act(dim_list_act_orb) | integer | list of the actives MOs              |
|                            |         |                                      |

Internal:
| m                                 | integer          | number of MOs in the list          |
|                                   |                  | (active MOs)                       |
| n                                 | integer          | number of orbitals pairs (p,q) p<q |
|                                   |                  | n = m*(m-1)/2                      |
| H(n,n)                            | double precision | Original hessian matrix (2D)       |
| H2(n,n)                           | double precision | Hessian matrix (2D)                |
| h_f(mo_num,mo_num,mo_num,mo_num)  | double precision | Original hessian matrix (4D)       |
| h_f2(mo_num,mo_num,mo_num,mo_num) | double precision | Hessian matrix (4D)                |
| i,j,p,q,k                         | integer          | indexes                            |
| threshold                         | double precision | threshold for the errors           |
| max_error                         | double precision | maximal error in the 4D hessian    |
| max_error_H                       | double precision | maximal error in the 2D hessian    |
| nb_error                          | integer          | number of errors in the 4D hessian |
| nb_error_H                        | integer          | number of errors in the 2D hessian |

#+BEGIN_SRC f90 :comments org :tangle debug_hessian_list_opt.irp.f
program debug_hessian_list_opt

  implicit none

  ! Variables

  double precision, allocatable :: H(:,:),H2(:,:), h_f(:,:,:,:), h_f2(:,:,:,:)
  integer                       :: n,m
  integer                       :: i,j,k,l
  double precision              :: max_error, max_error_H
  integer                       :: nb_error, nb_error_H
  double precision              :: threshold
  
  m = dim_list_act_orb !mo_num

  ! Definition of n  
  n = m*(m-1)/2

  PROVIDE mo_two_e_integrals_in_map ! Vérifier pour suppression

  ! Hessian 
  if (optimization_method == 'full') then
    print*,'Use the full hessian matrix'
    allocate(H(n,n),H2(n,n))  
    allocate(h_f(m,m,m,m),h_f2(m,m,m,m))

    call hessian_list_opt(n,m,list_act,H,h_f)
    call first_hessian_list_opt(n,m,list_act,H2,h_f2)
    !call hessian_opt(n,H2,h_f2)

    ! Difference
    h_f = h_f - h_f2
    H = H - H2
    max_error = 0d0
    nb_error = 0    
    threshold = 1d-12

    do l = 1, m
      do k= 1, m
        do j = 1, m
          do i = 1, m
            if (ABS(h_f(i,j,k,l)) > threshold) then
              print*,h_f(i,j,k,l)
              nb_error = nb_error + 1
              if (ABS(h_f(i,j,k,l)) > ABS(max_error)) then
                max_error = h_f(i,j,k,l)
              endif
            endif
          enddo
        enddo
      enddo
    enddo

    max_error_H = 0d0
    nb_error_H = 0

    do j = 1, n
      do i = 1, n
        if (ABS(H(i,j)) > threshold) then
          print*, H(i,j)
          nb_error_H = nb_error_H + 1

          if (ABS(H(i,j)) > ABS(max_error_H)) then
            max_error_H = H(i,j)
          endif

        endif
      enddo
    enddo 

    ! Deallocation
    deallocate(H, H2, h_f, h_f2)

  else

    print*, 'Use the diagonal hessian matrix'
    allocate(H(n,1),H2(n,1))
    call diag_hessian_list_opt(n,m,list_act,H)
    call first_diag_hessian_list_opt(n,m,list_act,H2)
    
    H = H - H2
  
    max_error_H = 0d0
    nb_error_H = 0
 
    do i = 1, n
      if (ABS(H(i,1)) > threshold) then
        print*, H(i,1)
        nb_error_H = nb_error_H + 1
 
        if (ABS(H(i,1)) > ABS(max_error_H)) then
          max_error_H = H(i,1)
        endif
 
      endif
    enddo
   
 endif
  
  print*,''
  if (optimization_method == 'full') then
    print*,'Check of the full hessian'
    print*,'Threshold:', threshold
    print*,'Nb error:', nb_error
    print*,'Max error:', max_error
    print*,''
  else
    print*,'Check of the diagonal hessian'
  endif
   
  print*,'Nb error_H:', nb_error_H
  print*,'Max error_H:', max_error_H
 
end program
#+END_SRC
