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docs/source/modules/ao_two_e_erf_ints.rst

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+.. _module_ao_two_e_erf_ints: 
+ 
+.. program:: ao_two_e_erf_ints 
+ 
+.. default-role:: option 
+ 
+======================
+ao_two_e_erf_ints
+======================
+
+Here, all two-electron integrals (:math:`erf(\mu r_{12})/r_{12}`) are computed.
+As they have 4 indices and many are zero, they are stored in a map, as defined
+in :file:`utils/map_module.f90`.
+
+The main parameter of this module is :option:`ao_two_e_erf_ints mu_erf` which is the range-separation parameter.
+
+To fetch an |AO| integral, use the
+`get_ao_two_e_integral_erf(i,j,k,l,ao_integrals_erf_map)` function.
+
+
+The conventions are:
+* For |AO| integrals : (ij|kl) = (11|22) = <ik|jl> = <12|12>
+
+
+
+ 
+ 
+ 
+EZFIO parameters 
+---------------- 
+ 
+.. option:: io_ao_two_e_integrals_erf
+ 
+    Read/Write |AO| integrals with the long range interaction from/to disk [ Write | Read | None ]
+ 
+    Default: None
+ 
+.. option:: mu_erf
+ 
+    cutting of the interaction in the range separated model
+ 
+    Default: 0.5
+ 
+ 
+Providers 
+--------- 
+ 
+.. c:var:: ao_integrals_erf_cache
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: ao_integrals_erf_cache	(0:64*64*64*64)
+
+
+    Cache of |AO| integrals for fast access
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_cache_min`
+       * :c:data:`ao_integrals_erf_map`
+       * :c:data:`ao_two_e_integrals_erf_in_map`
+
+
+ 
+.. c:var:: ao_integrals_erf_cache_max
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        integer	:: ao_integrals_erf_cache_min	
+        integer	:: ao_integrals_erf_cache_max	
+
+
+    Min and max values of the AOs for which the integrals are in the cache
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_num`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_cache`
+
+ 
+.. c:var:: ao_integrals_erf_cache_min
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        integer	:: ao_integrals_erf_cache_min	
+        integer	:: ao_integrals_erf_cache_max	
+
+
+    Min and max values of the AOs for which the integrals are in the cache
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_num`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_cache`
+
+ 
+.. c:var:: ao_integrals_erf_map
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        type(map_type)	:: ao_integrals_erf_map	
+
+
+    |AO| integrals
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_num`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_cache`
+       * :c:data:`ao_two_e_integrals_erf_in_map`
+       * :c:data:`mo_two_e_int_erf_jj_from_ao`
+
+ 
+.. c:var:: ao_two_e_integral_erf_schwartz
+
+
+    File : :file:`ao_two_e_erf_ints/providers_ao_erf.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: ao_two_e_integral_erf_schwartz	(ao_num,ao_num)
+
+
+    Needed to compute Schwartz inequalities
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_coef_normalized_ordered_transp`
+       * :c:data:`ao_expo_ordered_transp`
+       * :c:data:`ao_nucl`
+       * :c:data:`ao_num`
+       * :c:data:`ao_power`
+       * :c:data:`ao_prim_num`
+       * :c:data:`n_pt_max_integrals`
+       * :c:data:`nucl_coord`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`mo_two_e_int_erf_jj_from_ao`
+
+ 
+.. c:var:: ao_two_e_integrals_erf_in_map
+
+
+    File : :file:`ao_two_e_erf_ints/providers_ao_erf.irp.f`
+
+    .. code:: fortran
+
+        logical	:: ao_two_e_integrals_erf_in_map	
+
+
+    Map of Atomic integrals
+       i(r1) j(r2) 1/r12 k(r1) l(r2)
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_coef_normalized_ordered_transp`
+       * :c:data:`ao_expo_ordered_transp`
+       * :c:data:`ao_integrals_erf_map`
+       * :c:data:`ao_nucl`
+       * :c:data:`ao_num`
+       * :c:data:`ao_power`
+       * :c:data:`ao_prim_num`
+       * :c:data:`ezfio_filename`
+       * :c:data:`io_ao_two_e_integrals_erf`
+       * :c:data:`n_pt_max_integrals`
+       * :c:data:`nproc`
+       * :c:data:`nucl_coord`
+       * :c:data:`read_ao_two_e_integrals_erf`
+       * :c:data:`zmq_context`
+       * :c:data:`zmq_socket_pull_tcp_address`
+       * :c:data:`zmq_state`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_cache`
+       * :c:data:`mo_two_e_int_erf_jj_from_ao`
+       * :c:data:`mo_two_e_integrals_erf_in_map`
+
+ 
+.. c:function:: general_primitive_integral_erf:
+
+
+    File : :file:`ao_two_e_erf_ints/two_e_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        double precision function general_primitive_integral_erf(dim,            &
+      P_new,P_center,fact_p,p,p_inv,iorder_p,                        &
+      Q_new,Q_center,fact_q,q,q_inv,iorder_q)
+
+
+    Computes the integral <pq|rs> where p,q,r,s are Gaussian primitives
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`mu_erf`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`add_poly_multiply`
+       * :c:func:`give_polynom_mult_center_x`
+       * :c:func:`multiply_poly`
+
+ 
+ 
+Subroutines / functions 
+----------------------- 
+ 
+.. c:function:: ao_two_e_integral_erf:
+
+
+    File : :file:`ao_two_e_erf_ints/two_e_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        double precision function ao_two_e_integral_erf(i,j,k,l)
+
+
+    integral of the AO basis <ik|jl> or (ij|kl)
+       i(r1) j(r1) 1/r12 k(r2) l(r2)
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`n_pt_max_integrals`
+       * :c:data:`ao_coef_normalized_ordered_transp`
+       * :c:data:`ao_power`
+       * :c:data:`ao_expo_ordered_transp`
+       * :c:data:`ao_prim_num`
+       * :c:data:`ao_nucl`
+       * :c:data:`nucl_coord`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`give_explicit_poly_and_gaussian`
+
+ 
+.. c:function:: ao_two_e_integral_schwartz_accel_erf:
+
+
+    File : :file:`ao_two_e_erf_ints/two_e_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        double precision function ao_two_e_integral_schwartz_accel_erf(i,j,k,l)
+
+
+    integral of the AO basis <ik|jl> or (ij|kl)
+       i(r1) j(r1) 1/r12 k(r2) l(r2)
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`n_pt_max_integrals`
+       * :c:data:`ao_integrals_threshold`
+       * :c:data:`ao_coef_normalized_ordered_transp`
+       * :c:data:`ao_power`
+       * :c:data:`ao_expo_ordered_transp`
+       * :c:data:`ao_prim_num`
+       * :c:data:`ao_nucl`
+       * :c:data:`nucl_coord`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`give_explicit_poly_and_gaussian`
+
+ 
+.. c:function:: ao_two_e_integrals_erf_in_map_collector:
+
+
+    File : :file:`ao_two_e_erf_ints/integrals_erf_in_map_slave.irp.f`
+
+    .. code:: fortran
+
+        subroutine ao_two_e_integrals_erf_in_map_collector(zmq_socket_pull)
+
+
+    Collects results from the AO integral calculation
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_map`
+       * :c:data:`ao_num`
+
+    Called by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_two_e_integrals_erf_in_map`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`end_zmq_to_qp_run_socket`
+       * :c:func:`insert_into_ao_integrals_erf_map`
+
+ 
+.. c:function:: ao_two_e_integrals_erf_in_map_slave:
+
+
+    File : :file:`ao_two_e_erf_ints/integrals_erf_in_map_slave.irp.f`
+
+    .. code:: fortran
+
+        subroutine ao_two_e_integrals_erf_in_map_slave(thread,iproc)
+
+
+    Computes a buffer of integrals
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_num`
+
+    Called by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`ao_two_e_integrals_erf_in_map_slave_inproc`
+       * :c:func:`ao_two_e_integrals_erf_in_map_slave_tcp`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`compute_ao_integrals_erf_jl`
+       * :c:func:`end_zmq_push_socket`
+       * :c:func:`end_zmq_to_qp_run_socket`
+       * :c:func:`push_integrals`
+
+ 
+.. c:function:: ao_two_e_integrals_erf_in_map_slave_inproc:
+
+
+    File : :file:`ao_two_e_erf_ints/integrals_erf_in_map_slave.irp.f`
+
+    .. code:: fortran
+
+        subroutine ao_two_e_integrals_erf_in_map_slave_inproc(i)
+
+
+    Computes a buffer of integrals. i is the ID of the current thread.
+
+    Called by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_two_e_integrals_erf_in_map`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`ao_two_e_integrals_erf_in_map_slave`
+
+ 
+.. c:function:: ao_two_e_integrals_erf_in_map_slave_tcp:
+
+
+    File : :file:`ao_two_e_erf_ints/integrals_erf_in_map_slave.irp.f`
+
+    .. code:: fortran
+
+        subroutine ao_two_e_integrals_erf_in_map_slave_tcp(i)
+
+
+    Computes a buffer of integrals. i is the ID of the current thread.
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`ao_two_e_integrals_erf_in_map_slave`
+
+ 
+.. c:function:: clear_ao_erf_map:
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        subroutine clear_ao_erf_map
+
+
+    Frees the memory of the |AO| map
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_map`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`map_deinit`
+
+ 
+.. c:function:: compute_ao_integrals_erf_jl:
+
+
+    File : :file:`ao_two_e_erf_ints/two_e_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        subroutine compute_ao_integrals_erf_jl(j,l,n_integrals,buffer_i,buffer_value)
+
+
+    Parallel client for AO integrals
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_overlap_abs`
+       * :c:data:`ao_num`
+       * :c:data:`ao_integrals_threshold`
+       * :c:data:`ao_two_e_integral_erf_schwartz`
+
+    Called by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`ao_two_e_integrals_erf_in_map_slave`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`two_e_integrals_index`
+
+ 
+.. c:function:: compute_ao_two_e_integrals_erf:
+
+
+    File : :file:`ao_two_e_erf_ints/two_e_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        subroutine compute_ao_two_e_integrals_erf(j,k,l,sze,buffer_value)
+
+
+    Compute AO 1/r12 integrals for all i and fixed j,k,l
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_overlap_abs`
+       * :c:data:`ao_num`
+       * :c:data:`ao_two_e_integral_erf_schwartz`
+
+    Called by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`mo_two_e_int_erf_jj_from_ao`
+
+ 
+.. c:function:: dump_ao_integrals_erf:
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        subroutine dump_ao_integrals_erf(filename)
+
+
+    Save to disk the |AO| erf integrals
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_map`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`ezfio_set_work_empty`
+
+ 
+.. c:function:: eri_erf:
+
+
+    File : :file:`ao_two_e_erf_ints/two_e_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        double precision function ERI_erf(alpha,beta,delta,gama,a_x,b_x,c_x,d_x,a_y,b_y,c_y,d_y,a_z,b_z,c_z,d_z)
+
+
+    Atomic primtive two-electron integral between the 4 primitives :
+    
+    * primitive 1 : $x_1^{a_x} y_1^{a_y} z_1^{a_z} \exp(-\alpha * r1^2)$
+    * primitive 2 : $x_1^{b_x} y_1^{b_y} z_1^{b_z} \exp(- \beta * r1^2)$
+    * primitive 3 : $x_2^{c_x} y_2^{c_y} z_2^{c_z} \exp(-\delta * r2^2)$
+    * primitive 4 : $x_2^{d_x} y_2^{d_y} z_2^{d_z} \exp(-\gamma * r2^2)$
+    
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`mu_erf`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`integrale_new_erf`
+
+ 
+.. c:function:: get_ao_erf_map_size:
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        function get_ao_erf_map_size()
+
+
+    Returns the number of elements in the |AO| map
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_map`
+
+ 
+.. c:function:: get_ao_two_e_integral_erf:
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        double precision function get_ao_two_e_integral_erf(i,j,k,l,map) result(result)
+
+
+    Gets one |AO| two-electron integral from the |AO| map
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_cache_min`
+       * :c:data:`ao_overlap_abs`
+       * :c:data:`ao_integrals_threshold`
+       * :c:data:`ao_integrals_erf_cache`
+       * :c:data:`ao_two_e_integral_erf_schwartz`
+       * :c:data:`ao_two_e_integrals_erf_in_map`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`map_get`
+       * :c:func:`two_e_integrals_index`
+
+ 
+.. c:function:: get_ao_two_e_integrals_erf:
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        subroutine get_ao_two_e_integrals_erf(j,k,l,sze,out_val)
+
+
+    Gets multiple |AO| two-electron integral from the |AO| map .
+    All i are retrieved for j,k,l fixed.
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_map`
+       * :c:data:`ao_overlap_abs`
+       * :c:data:`ao_integrals_threshold`
+       * :c:data:`ao_two_e_integrals_erf_in_map`
+
+    Called by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`add_integrals_to_map_erf`
+
+ 
+.. c:function:: get_ao_two_e_integrals_erf_non_zero:
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        subroutine get_ao_two_e_integrals_erf_non_zero(j,k,l,sze,out_val,out_val_index,non_zero_int)
+
+
+    Gets multiple |AO| two-electron integrals from the |AO| map .
+    All non-zero i are retrieved for j,k,l fixed.
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_map`
+       * :c:data:`ao_overlap_abs`
+       * :c:data:`ao_integrals_threshold`
+       * :c:data:`ao_two_e_integral_erf_schwartz`
+       * :c:data:`ao_two_e_integrals_erf_in_map`
+
+    Called by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`mo_two_e_int_erf_jj_from_ao`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`map_get`
+       * :c:func:`two_e_integrals_index`
+
+ 
+.. c:function:: insert_into_ao_integrals_erf_map:
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        subroutine insert_into_ao_integrals_erf_map(n_integrals,buffer_i, buffer_values)
+
+
+    Create new entry into |AO| map
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_map`
+
+    Called by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`ao_two_e_integrals_erf_in_map_collector`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`map_append`
+
+ 
+.. c:function:: integrale_new_erf:
+
+
+    File : :file:`ao_two_e_erf_ints/two_e_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        subroutine integrale_new_erf(I_f,a_x,b_x,c_x,d_x,a_y,b_y,c_y,d_y,a_z,b_z,c_z,d_z,p,q,n_pt)
+
+
+    Calculate the integral of the polynomial :
+    
+    $I_x1(a_x+b_x, c_x+d_x,p,q) \, I_x1(a_y+b_y, c_y+d_y,p,q) \, I_x1(a_z+b_z, c_z+d_z,p,q)$
+    
+    between $( 0 ; 1)$
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`mu_erf`
+       * :c:data:`n_pt_max_integrals`
+       * :c:data:`gauleg_t2`
+
+    Called by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`eri_erf`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`i_x1_new`
+
+ 
+.. c:function:: load_ao_integrals_erf:
+
+
+    File : :file:`ao_two_e_erf_ints/map_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        integer function load_ao_integrals_erf(filename)
+
+
+    Read from disk the |AO| erf integrals
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_map`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`cache_map_reallocate`
+       * :c:func:`map_deinit`
+       * :c:func:`map_sort`
+
+ 
+.. c:function:: save_erf_two_e_integrals_ao:
+
+
+    File : :file:`ao_two_e_erf_ints/routines_save_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        subroutine save_erf_two_e_integrals_ao
+
+
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_map`
+       * :c:data:`ezfio_filename`
+       * :c:data:`ao_two_e_integrals_erf_in_map`
+
+    Called by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`routine`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`ezfio_set_ao_two_e_erf_ints_io_ao_two_e_integrals_erf`
+       * :c:func:`ezfio_set_work_empty`
+       * :c:func:`map_save_to_disk`
+
+ 
+.. c:function:: save_erf_two_e_ints_ao_into_ints_ao:
+
+
+    File : :file:`ao_two_e_erf_ints/routines_save_integrals_erf.irp.f`
+
+    .. code:: fortran
+
+        subroutine save_erf_two_e_ints_ao_into_ints_ao
+
+
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`ao_integrals_erf_map`
+       * :c:data:`ezfio_filename`
+       * :c:data:`ao_two_e_integrals_erf_in_map`
+
+    Calls:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:func:`ezfio_set_ao_two_e_ints_io_ao_two_e_integrals`
+       * :c:func:`ezfio_set_work_empty`
+       * :c:func:`map_save_to_disk`
+

docs/source/modules/aux_quantities.rst

@@ -0,0 +1,52 @@
+.. _module_aux_quantities: 
+ 
+.. program:: aux_quantities 
+ 
+.. default-role:: option 
+ 
+==============
+aux_quantities
+==============
+
+
+This module contains some global variables (such as densities and energies)
+which are stored in the EZFIO folder in a different place than determinants.
+This is used in practice to store density matrices which can be obtained from
+any methods, as long as they are stored in the same MO basis which is used for
+the calculations. In |RSDFT| calculations, this can be done to perform damping
+on the density in order to speed up convergence.
+
+The main providers of that module are:
+
+* `data_one_e_dm_alpha_mo` and `data_one_e_dm_beta_mo` which are the
+  one-body alpha and beta densities which are necessary read from the EZFIO
+  folder.
+
+
+Thanks to these providers you can use any density matrix that does not
+necessary corresponds to that of the current wave function.
+ 
+ 
+ 
+EZFIO parameters 
+---------------- 
+ 
+.. option:: data_energy_var
+ 
+    Variational energy computed with the wave function
+ 
+ 
+.. option:: data_energy_proj
+ 
+    Projected energy computed with the wave function
+ 
+ 
+.. option:: data_one_e_dm_alpha_mo
+ 
+    Alpha one body density matrix on the |MO| basis computed with the wave function
+ 
+ 
+.. option:: data_one_e_dm_beta_mo
+ 
+    Beta one body density matrix on the |MO| basis computed with the wave function
+ 

docs/source/modules/becke_numerical_grid.rst

@@ -0,0 +1,833 @@
+.. _module_becke_numerical_grid: 
+ 
+.. program:: becke_numerical_grid 
+ 
+.. default-role:: option 
+ 
+====================
+becke_numerical_grid
+====================
+
+This module contains all quantities needed to build Becke's grid used in general for DFT integration. Note that it can be used for whatever integration in R^3 as long as the functions to be integrated are mostly concentrated near the atomic regions.
+
+This grid is built as the reunion of a spherical grid around each atom. Each spherical grid contains
+a certain number of radial and angular points. No pruning is done on the angular part of the grid.
+
+The main keyword for that module is:
+
+* :option:`becke_numerical_grid grid_type_sgn` which controls the precision of the grid according the standard **SG-n** grids. This keyword controls the two providers `n_points_integration_angular` `n_points_radial_grid`.
+
+The main providers of that module are:
+
+* `n_points_integration_angular` which is the number of angular integration points. WARNING: it obeys to specific rules so it cannot be any integer number. Some of the possible values are [ 50 | 74 | 170 | 194 | 266 | 302 | 590 | 1202 | 2030 | 5810 ] for instance. See :file:`angular.f` for more details.
+* `n_points_radial_grid` which is the number of radial angular points. This can be any strictly positive integer. Nevertheless, a minimum of 50 is in general necessary.
+* `final_grid_points` which are the (x,y,z) coordinates of the grid points.
+* `final_weight_at_r_vector` which are the weights at each grid point
+
+
+For a simple example of how to use the grid, see :file:`example.irp.f`.
+
+The spherical integration uses Lebedev-Laikov grids, which was used from the code distributed through CCL (http://www.ccl.net/).
+See next section for explanations and citation policies.
+
+.. code-block:: text
+
+       This subroutine is part of a set of subroutines that generate
+       Lebedev grids [1-6] for integration on a sphere. The original
+       C-code [1] was kindly provided by Dr. Dmitri N. Laikov and
+       translated into fortran by Dr. Christoph van Wuellen.
+       This subroutine was translated using a C to fortran77 conversion
+       tool written by Dr. Christoph van Wuellen.
+
+       Users of this code are asked to include reference [1] in their
+       publications, and in the user- and programmers-manuals
+       describing their codes.
+
+       This code was distributed through CCL (http://www.ccl.net/).
+
+       [1] V.I. Lebedev, and D.N. Laikov
+           "A quadrature formula for the sphere of the 131st
+            algebraic order of accuracy"
+           Doklady Mathematics, Vol. 59, No. 3, 1999, pp. 477-481.
+
+       [2] V.I. Lebedev
+           "A quadrature formula for the sphere of 59th algebraic
+            order of accuracy"
+           Russian Acad. Sci. Dokl. Math., Vol. 50, 1995, pp. 283-286.
+
+       [3] V.I. Lebedev, and A.L. Skorokhodov
+           "Quadrature formulas of orders 41, 47, and 53 for the sphere"
+           Russian Acad. Sci. Dokl. Math., Vol. 45, 1992, pp. 587-592.
+
+       [4] V.I. Lebedev
+           "Spherical quadrature formulas exact to orders 25-29"
+           Siberian Mathematical Journal, Vol. 18, 1977, pp. 99-107.
+
+       [5] V.I. Lebedev
+           "Quadratures on a sphere"
+           Computational Mathematics and Mathematical Physics, Vol. 16,
+           1976, pp. 10-24.
+
+       [6] V.I. Lebedev
+           "Values of the nodes and weights of ninth to seventeenth
+            order Gauss-Markov quadrature formulae invariant under the
+            octahedron group with inversion"
+           Computational Mathematics and Mathematical Physics, Vol. 15,
+           1975, pp. 44-51.
+
+
+ 
+ 
+ 
+EZFIO parameters 
+---------------- 
+ 
+.. option:: grid_type_sgn
+ 
+    Type of grid used for the Becke's numerical grid. Can be, by increasing accuracy: [ 0 | 1 | 2 | 3 ]
+ 
+    Default: 2
+ 
+ 
+Providers 
+--------- 
+ 
+.. c:var:: alpha_knowles
+
+
+    File : :file:`becke_numerical_grid/integration_radial.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: alpha_knowles	(100)
+
+
+    Recommended values for the alpha parameters according to the paper of Knowles (JCP, 104, 1996)
+    as a function of the nuclear charge
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+
+ 
+.. c:var:: angular_quadrature_points
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: angular_quadrature_points	(n_points_integration_angular,3)
+        double precision, allocatable	:: weights_angular_points	(n_points_integration_angular)
+
+
+    weights and grid points for the integration on the angular variables on
+    the unit sphere centered on (0,0,0)
+    According to the LEBEDEV scheme
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`n_points_radial_grid`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+
+ 
+.. c:var:: dr_radial_integral
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: grid_points_radial	(n_points_radial_grid)
+        double precision	:: dr_radial_integral	
+
+
+    points in [0,1] to map the radial integral [0,\infty]
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`n_points_radial_grid`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+
+ 
+.. c:var:: final_grid_points
+
+
+    File : :file:`becke_numerical_grid/grid_becke_vector.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: final_grid_points	(3,n_points_final_grid)
+        double precision, allocatable	:: final_weight_at_r_vector	(n_points_final_grid)
+        integer, allocatable	:: index_final_points	(3,n_points_final_grid)
+        integer, allocatable	:: index_final_points_reverse	(n_points_integration_angular,n_points_radial_grid,nucl_num)
+
+
+     final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point
+    
+    final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions
+    
+    index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point
+    
+    index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+       * :c:data:`n_points_final_grid`
+       * :c:data:`n_points_radial_grid`
+       * :c:data:`nucl_num`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`aos_grad_in_r_array`
+       * :c:data:`aos_in_r_array`
+       * :c:data:`aos_lapl_in_r_array`
+       * :c:data:`aos_sr_vc_alpha_lda_w`
+       * :c:data:`aos_sr_vc_alpha_pbe_w`
+       * :c:data:`aos_vc_alpha_lda_w`
+       * :c:data:`aos_vc_alpha_pbe_w`
+       * :c:data:`energy_sr_x_lda`
+       * :c:data:`energy_sr_x_pbe`
+       * :c:data:`energy_x_lda`
+       * :c:data:`energy_x_pbe`
+       * :c:data:`mos_in_r_array`
+       * :c:data:`one_e_dm_alpha_at_r`
+       * :c:data:`one_e_dm_and_grad_alpha_in_r`
+
+ 
+.. c:var:: final_weight_at_r
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: final_weight_at_r	(n_points_integration_angular,n_points_radial_grid,nucl_num)
+
+
+    Total weight on each grid point which takes into account all Lebedev, Voronoi and radial weights.
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`alpha_knowles`
+       * :c:data:`angular_quadrature_points`
+       * :c:data:`grid_points_radial`
+       * :c:data:`m_knowles`
+       * :c:data:`n_points_radial_grid`
+       * :c:data:`nucl_charge`
+       * :c:data:`nucl_num`
+       * :c:data:`weight_at_r`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_grid_points`
+       * :c:data:`n_points_final_grid`
+
+ 
+.. c:var:: final_weight_at_r_vector
+
+
+    File : :file:`becke_numerical_grid/grid_becke_vector.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: final_grid_points	(3,n_points_final_grid)
+        double precision, allocatable	:: final_weight_at_r_vector	(n_points_final_grid)
+        integer, allocatable	:: index_final_points	(3,n_points_final_grid)
+        integer, allocatable	:: index_final_points_reverse	(n_points_integration_angular,n_points_radial_grid,nucl_num)
+
+
+     final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point
+    
+    final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions
+    
+    index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point
+    
+    index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+       * :c:data:`n_points_final_grid`
+       * :c:data:`n_points_radial_grid`
+       * :c:data:`nucl_num`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`aos_grad_in_r_array`
+       * :c:data:`aos_in_r_array`
+       * :c:data:`aos_lapl_in_r_array`
+       * :c:data:`aos_sr_vc_alpha_lda_w`
+       * :c:data:`aos_sr_vc_alpha_pbe_w`
+       * :c:data:`aos_vc_alpha_lda_w`
+       * :c:data:`aos_vc_alpha_pbe_w`
+       * :c:data:`energy_sr_x_lda`
+       * :c:data:`energy_sr_x_pbe`
+       * :c:data:`energy_x_lda`
+       * :c:data:`energy_x_pbe`
+       * :c:data:`mos_in_r_array`
+       * :c:data:`one_e_dm_alpha_at_r`
+       * :c:data:`one_e_dm_and_grad_alpha_in_r`
+
+ 
+.. c:var:: grid_points_per_atom
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: grid_points_per_atom	(3,n_points_integration_angular,n_points_radial_grid,nucl_num)
+
+
+    x,y,z coordinates of grid points used for integration in 3d space
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`alpha_knowles`
+       * :c:data:`angular_quadrature_points`
+       * :c:data:`grid_points_radial`
+       * :c:data:`m_knowles`
+       * :c:data:`n_points_radial_grid`
+       * :c:data:`nucl_charge`
+       * :c:data:`nucl_coord`
+       * :c:data:`nucl_num`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_grid_points`
+       * :c:data:`one_e_dm_alpha_in_r`
+       * :c:data:`weight_at_r`
+
+ 
+.. c:var:: grid_points_radial
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: grid_points_radial	(n_points_radial_grid)
+        double precision	:: dr_radial_integral	
+
+
+    points in [0,1] to map the radial integral [0,\infty]
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`n_points_radial_grid`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+
+ 
+.. c:var:: index_final_points
+
+
+    File : :file:`becke_numerical_grid/grid_becke_vector.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: final_grid_points	(3,n_points_final_grid)
+        double precision, allocatable	:: final_weight_at_r_vector	(n_points_final_grid)
+        integer, allocatable	:: index_final_points	(3,n_points_final_grid)
+        integer, allocatable	:: index_final_points_reverse	(n_points_integration_angular,n_points_radial_grid,nucl_num)
+
+
+     final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point
+    
+    final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions
+    
+    index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point
+    
+    index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+       * :c:data:`n_points_final_grid`
+       * :c:data:`n_points_radial_grid`
+       * :c:data:`nucl_num`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`aos_grad_in_r_array`
+       * :c:data:`aos_in_r_array`
+       * :c:data:`aos_lapl_in_r_array`
+       * :c:data:`aos_sr_vc_alpha_lda_w`
+       * :c:data:`aos_sr_vc_alpha_pbe_w`
+       * :c:data:`aos_vc_alpha_lda_w`
+       * :c:data:`aos_vc_alpha_pbe_w`
+       * :c:data:`energy_sr_x_lda`
+       * :c:data:`energy_sr_x_pbe`
+       * :c:data:`energy_x_lda`
+       * :c:data:`energy_x_pbe`
+       * :c:data:`mos_in_r_array`
+       * :c:data:`one_e_dm_alpha_at_r`
+       * :c:data:`one_e_dm_and_grad_alpha_in_r`
+
+ 
+.. c:var:: index_final_points_reverse
+
+
+    File : :file:`becke_numerical_grid/grid_becke_vector.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: final_grid_points	(3,n_points_final_grid)
+        double precision, allocatable	:: final_weight_at_r_vector	(n_points_final_grid)
+        integer, allocatable	:: index_final_points	(3,n_points_final_grid)
+        integer, allocatable	:: index_final_points_reverse	(n_points_integration_angular,n_points_radial_grid,nucl_num)
+
+
+     final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point
+    
+    final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions
+    
+    index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point
+    
+    index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+       * :c:data:`n_points_final_grid`
+       * :c:data:`n_points_radial_grid`
+       * :c:data:`nucl_num`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`aos_grad_in_r_array`
+       * :c:data:`aos_in_r_array`
+       * :c:data:`aos_lapl_in_r_array`
+       * :c:data:`aos_sr_vc_alpha_lda_w`
+       * :c:data:`aos_sr_vc_alpha_pbe_w`
+       * :c:data:`aos_vc_alpha_lda_w`
+       * :c:data:`aos_vc_alpha_pbe_w`
+       * :c:data:`energy_sr_x_lda`
+       * :c:data:`energy_sr_x_pbe`
+       * :c:data:`energy_x_lda`
+       * :c:data:`energy_x_pbe`
+       * :c:data:`mos_in_r_array`
+       * :c:data:`one_e_dm_alpha_at_r`
+       * :c:data:`one_e_dm_and_grad_alpha_in_r`
+
+ 
+.. c:var:: m_knowles
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        integer	:: m_knowles	
+
+
+    value of the "m" parameter in the equation (7) of the paper of Knowles (JCP, 104, 1996)
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+
+ 
+.. c:var:: n_points_final_grid
+
+
+    File : :file:`becke_numerical_grid/grid_becke_vector.irp.f`
+
+    .. code:: fortran
+
+        integer	:: n_points_final_grid	
+
+
+    Number of points which are non zero
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`n_points_radial_grid`
+       * :c:data:`nucl_num`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`aos_grad_in_r_array`
+       * :c:data:`aos_in_r_array`
+       * :c:data:`aos_lapl_in_r_array`
+       * :c:data:`aos_sr_vc_alpha_lda_w`
+       * :c:data:`aos_sr_vc_alpha_pbe_w`
+       * :c:data:`aos_vc_alpha_lda_w`
+       * :c:data:`aos_vc_alpha_pbe_w`
+       * :c:data:`energy_sr_x_lda`
+       * :c:data:`energy_sr_x_pbe`
+       * :c:data:`energy_x_lda`
+       * :c:data:`energy_x_pbe`
+       * :c:data:`final_grid_points`
+       * :c:data:`mos_grad_in_r_array`
+       * :c:data:`mos_in_r_array`
+       * :c:data:`mos_lapl_in_r_array`
+       * :c:data:`one_e_dm_alpha_at_r`
+       * :c:data:`one_e_dm_and_grad_alpha_in_r`
+       * :c:data:`potential_sr_c_alpha_ao_lda`
+       * :c:data:`potential_sr_x_alpha_ao_lda`
+       * :c:data:`potential_sr_x_alpha_ao_pbe`
+       * :c:data:`potential_x_alpha_ao_lda`
+       * :c:data:`potential_x_alpha_ao_pbe`
+
+ 
+.. c:var:: n_points_grid_per_atom
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        integer	:: n_points_grid_per_atom	
+
+
+    Number of grid points per atom
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`n_points_radial_grid`
+
+
+ 
+.. c:var:: n_points_integration_angular
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        integer	:: n_points_radial_grid	
+        integer	:: n_points_integration_angular	
+
+
+    n_points_radial_grid = number of radial grid points per atom
+    
+    n_points_integration_angular = number of angular grid points per atom
+    
+    These numbers are automatically set by setting the grid_type_sgn parameter
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`grid_type_sgn`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`angular_quadrature_points`
+       * :c:data:`final_grid_points`
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+       * :c:data:`grid_points_radial`
+       * :c:data:`n_points_final_grid`
+       * :c:data:`n_points_grid_per_atom`
+       * :c:data:`one_e_dm_alpha_in_r`
+       * :c:data:`weight_at_r`
+
+ 
+.. c:var:: n_points_radial_grid
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        integer	:: n_points_radial_grid	
+        integer	:: n_points_integration_angular	
+
+
+    n_points_radial_grid = number of radial grid points per atom
+    
+    n_points_integration_angular = number of angular grid points per atom
+    
+    These numbers are automatically set by setting the grid_type_sgn parameter
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`grid_type_sgn`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`angular_quadrature_points`
+       * :c:data:`final_grid_points`
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+       * :c:data:`grid_points_radial`
+       * :c:data:`n_points_final_grid`
+       * :c:data:`n_points_grid_per_atom`
+       * :c:data:`one_e_dm_alpha_in_r`
+       * :c:data:`weight_at_r`
+
+ 
+.. c:var:: weight_at_r
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: weight_at_r	(n_points_integration_angular,n_points_radial_grid,nucl_num)
+
+
+    Weight function at grid points : w_n(r) according to the equation (22)
+    of Becke original paper (JCP, 88, 1988)
+    
+    The "n" discrete variable represents the nucleis which in this array is
+    represented by the last dimension and the points are labelled by the
+    other dimensions.
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`grid_points_per_atom`
+       * :c:data:`n_points_radial_grid`
+       * :c:data:`nucl_coord_transp`
+       * :c:data:`nucl_dist_inv`
+       * :c:data:`nucl_num`
+       * :c:data:`slater_bragg_type_inter_distance_ua`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+
+ 
+.. c:var:: weights_angular_points
+
+
+    File : :file:`becke_numerical_grid/grid_becke.irp.f`
+
+    .. code:: fortran
+
+        double precision, allocatable	:: angular_quadrature_points	(n_points_integration_angular,3)
+        double precision, allocatable	:: weights_angular_points	(n_points_integration_angular)
+
+
+    weights and grid points for the integration on the angular variables on
+    the unit sphere centered on (0,0,0)
+    According to the LEBEDEV scheme
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`n_points_radial_grid`
+
+    Needed by:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`final_weight_at_r`
+       * :c:data:`grid_points_per_atom`
+
+ 
+ 
+Subroutines / functions 
+----------------------- 
+ 
+.. c:function:: cell_function_becke:
+
+
+    File : :file:`becke_numerical_grid/step_function_becke.irp.f`
+
+    .. code:: fortran
+
+        double precision function cell_function_becke(r,atom_number)
+
+
+    atom_number :: atom on which the cell function of Becke (1988, JCP,88(4))
+    r(1:3)                       :: x,y,z coordinantes of the current point
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`nucl_dist_inv`
+       * :c:data:`slater_bragg_type_inter_distance_ua`
+       * :c:data:`nucl_coord_transp`
+       * :c:data:`nucl_num`
+
+ 
+.. c:function:: derivative_knowles_function:
+
+
+    File : :file:`becke_numerical_grid/integration_radial.irp.f`
+
+    .. code:: fortran
+
+        double precision function derivative_knowles_function(alpha,m,x)
+
+
+    Derivative of the function proposed by Knowles (JCP, 104, 1996) for distributing the radial points
+
+ 
+.. c:function:: example_becke_numerical_grid:
+
+
+    File : :file:`becke_numerical_grid/example.irp.f`
+
+    .. code:: fortran
+
+        subroutine example_becke_numerical_grid
+
+
+    subroutine that illustrates the main features available in becke_numerical_grid
+
+    Needs:
+
+    .. hlist::
+       :columns: 3
+
+       * :c:data:`n_points_final_grid`
+       * :c:data:`final_weight_at_r`
+       * :c:data:`n_points_radial_grid`
+       * :c:data:`grid_points_per_atom`
+       * :c:data:`final_grid_points`
+       * :c:data:`nucl_coord`
+       * :c:data:`nucl_num`
+
+ 
+.. c:function:: f_function_becke:
+
+
+    File : :file:`becke_numerical_grid/step_function_becke.irp.f`
+
+    .. code:: fortran
+
+        double precision function f_function_becke(x)
+
+
+
+ 
+.. c:function:: knowles_function:
+
+
+    File : :file:`becke_numerical_grid/integration_radial.irp.f`
+
+    .. code:: fortran
+
+        double precision function knowles_function(alpha,m,x)
+
+
+    Function proposed by Knowles (JCP, 104, 1996) for distributing the radial points :
+    the Log "m" function ( equation (7) in the paper )
+
+ 
+.. c:function:: step_function_becke:
+
+
+    File : :file:`becke_numerical_grid/step_function_becke.irp.f`
+
+    .. code:: fortran
+
+        double precision function step_function_becke(x)
+
+
+    Step function of the Becke paper (1988, JCP,88(4))
+