QuantumPackage/docs/source/modules/ao_one_e_ints.rst

.. _module_ao_one_e_ints: 
 
.. program:: ao_one_e_ints 
 
.. default-role:: option 
 
==================
ao_one_e_integrals
==================

All the one-electron integrals in the |AO| basis are here.

The most important providers for usual quantum-chemistry calculation are:

* `ao_kinetic_integral` which are the kinetic operator integrals on the |AO| basis (see :file:`kin_ao_ints.irp.f`)
* `ao_nucl_elec_integral` which are the nuclear-elctron operator integrals on the |AO| basis (see :file:`pot_ao_ints.irp.f`)
* `ao_one_e_integrals` which are the the h_core operator integrals on the |AO| basis (see :file:`ao_mono_ints.irp.f`)


Note that you can find other interesting integrals related to the position operator in :file:`spread_dipole_ao.irp.f`.
 
 
 
EZFIO parameters 
---------------- 
 
.. option:: ao_integrals_e_n
 
    Nucleus-electron integrals in |AO| basis set
 
 
.. option:: io_ao_integrals_e_n
 
    Read/Write |AO| nucleus-electron attraction integrals from/to disk [ Write | Read | None ]
 
    Default: None
 
.. option:: ao_integrals_kinetic
 
    Kinetic energy integrals in |AO| basis set
 
 
.. option:: io_ao_integrals_kinetic
 
    Read/Write |AO| kinetic integrals from/to disk [ Write | Read | None ]
 
    Default: None
 
.. option:: ao_integrals_pseudo
 
    Pseudopotential integrals in |AO| basis set
 
 
.. option:: io_ao_integrals_pseudo
 
    Read/Write |AO| pseudopotential integrals from/to disk [ Write | Read | None ]
 
    Default: None
 
.. option:: ao_integrals_overlap
 
    Overlap integrals in |AO| basis set
 
 
.. option:: io_ao_integrals_overlap
 
    Read/Write |AO| overlap integrals from/to disk [ Write | Read | None ]
 
    Default: None
 
.. option:: ao_one_e_integrals
 
    Combined integrals in |AO| basis set
 
 
.. option:: io_ao_one_e_integrals
 
    Read/Write |AO| one-electron integrals from/to disk [ Write | Read | None ]
 
    Default: None
 
 
Providers 
--------- 
 
.. c:var:: ao_cart_to_sphe_coef


    File : :file:`ao_one_e_ints/ao_ortho_canonical.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_cart_to_sphe_coef	(ao_num,ao_num)
        integer	:: ao_cart_to_sphe_num	


    Coefficients to go from cartesian to spherical coordinates in the current
    basis set

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_l`
       * :c:data:`ao_num`
       * :c:data:`ao_power`
       * :c:data:`cart_to_sphe_1`
       * :c:data:`cart_to_sphe_2`
       * :c:data:`cart_to_sphe_3`
       * :c:data:`cart_to_sphe_4`
       * :c:data:`cart_to_sphe_5`
       * :c:data:`cart_to_sphe_6`
       * :c:data:`cart_to_sphe_7`
       * :c:data:`cart_to_sphe_8`
       * :c:data:`cart_to_sphe_9`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_cart_to_sphe_inv`
       * :c:data:`ao_cart_to_sphe_overlap`
       * :c:data:`ao_ortho_canonical_coef`

 
.. c:var:: ao_cart_to_sphe_inv


    File : :file:`ao_one_e_ints/ao_ortho_canonical.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_cart_to_sphe_inv	(ao_cart_to_sphe_num,ao_num)


    Inverse of :c:data:`ao_cart_to_sphe_coef`

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_cart_to_sphe_coef`
       * :c:data:`ao_num`


 
.. c:var:: ao_cart_to_sphe_num


    File : :file:`ao_one_e_ints/ao_ortho_canonical.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_cart_to_sphe_coef	(ao_num,ao_num)
        integer	:: ao_cart_to_sphe_num	


    Coefficients to go from cartesian to spherical coordinates in the current
    basis set

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_l`
       * :c:data:`ao_num`
       * :c:data:`ao_power`
       * :c:data:`cart_to_sphe_1`
       * :c:data:`cart_to_sphe_2`
       * :c:data:`cart_to_sphe_3`
       * :c:data:`cart_to_sphe_4`
       * :c:data:`cart_to_sphe_5`
       * :c:data:`cart_to_sphe_6`
       * :c:data:`cart_to_sphe_7`
       * :c:data:`cart_to_sphe_8`
       * :c:data:`cart_to_sphe_9`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_cart_to_sphe_inv`
       * :c:data:`ao_cart_to_sphe_overlap`
       * :c:data:`ao_ortho_canonical_coef`

 
.. c:var:: ao_cart_to_sphe_overlap


    File : :file:`ao_one_e_ints/ao_ortho_canonical.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_cart_to_sphe_overlap	(ao_cart_to_sphe_num,ao_cart_to_sphe_num)


    |AO| overlap matrix in the spherical basis set

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_cart_to_sphe_coef`
       * :c:data:`ao_num`
       * :c:data:`ao_overlap`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_ortho_canonical_coef`

 
.. c:var:: ao_deriv2_x


    File : :file:`ao_one_e_ints/kin_ao_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_deriv2_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv2_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv2_z	(ao_num,ao_num)


    Second derivative matrix elements in the |AO| basis.
    
    .. math::
    
      {\tt ao\_deriv2\_x} =
      \langle \chi_i(x,y,z) | \frac{\partial^2}{\partial x^2} |\chi_j (x,y,z) \rangle
    

    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:`nucl_coord`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_kinetic_integrals`

 
.. c:var:: ao_deriv2_y


    File : :file:`ao_one_e_ints/kin_ao_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_deriv2_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv2_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv2_z	(ao_num,ao_num)


    Second derivative matrix elements in the |AO| basis.
    
    .. math::
    
      {\tt ao\_deriv2\_x} =
      \langle \chi_i(x,y,z) | \frac{\partial^2}{\partial x^2} |\chi_j (x,y,z) \rangle
    

    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:`nucl_coord`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_kinetic_integrals`

 
.. c:var:: ao_deriv2_z


    File : :file:`ao_one_e_ints/kin_ao_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_deriv2_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv2_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv2_z	(ao_num,ao_num)


    Second derivative matrix elements in the |AO| basis.
    
    .. math::
    
      {\tt ao\_deriv2\_x} =
      \langle \chi_i(x,y,z) | \frac{\partial^2}{\partial x^2} |\chi_j (x,y,z) \rangle
    

    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:`nucl_coord`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_kinetic_integrals`

 
.. c:var:: ao_deriv_1_x


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_deriv_1_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv_1_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv_1_z	(ao_num,ao_num)


    * array of the integrals of AO_i * d/dx  AO_j
    
    * array of the integrals of AO_i * d/dy  AO_j
    
    * array of the integrals of AO_i * d/dz  AO_j

    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:`nucl_coord`


 
.. c:var:: ao_deriv_1_y


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_deriv_1_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv_1_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv_1_z	(ao_num,ao_num)


    * array of the integrals of AO_i * d/dx  AO_j
    
    * array of the integrals of AO_i * d/dy  AO_j
    
    * array of the integrals of AO_i * d/dz  AO_j

    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:`nucl_coord`


 
.. c:var:: ao_deriv_1_z


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_deriv_1_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv_1_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_deriv_1_z	(ao_num,ao_num)


    * array of the integrals of AO_i * d/dx  AO_j
    
    * array of the integrals of AO_i * d/dy  AO_j
    
    * array of the integrals of AO_i * d/dz  AO_j

    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:`nucl_coord`


 
.. c:var:: ao_dipole_x


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_dipole_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_dipole_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_dipole_z	(ao_num,ao_num)


    * array of the integrals of AO_i * x AO_j
    
    * array of the integrals of AO_i * y AO_j
    
    * array of the integrals of AO_i * z AO_j

    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:`nucl_coord`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`mo_dipole_x`

 
.. c:var:: ao_dipole_y


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_dipole_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_dipole_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_dipole_z	(ao_num,ao_num)


    * array of the integrals of AO_i * x AO_j
    
    * array of the integrals of AO_i * y AO_j
    
    * array of the integrals of AO_i * z AO_j

    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:`nucl_coord`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`mo_dipole_x`

 
.. c:var:: ao_dipole_z


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_dipole_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_dipole_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_dipole_z	(ao_num,ao_num)


    * array of the integrals of AO_i * x AO_j
    
    * array of the integrals of AO_i * y AO_j
    
    * array of the integrals of AO_i * z AO_j

    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:`nucl_coord`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`mo_dipole_x`

 
.. c:var:: ao_integrals_n_e


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_integrals_n_e	(ao_num,ao_num)


    Nucleus-electron interaction, in the |AO| basis set.
    
    :math:`\langle \chi_i | -\sum_A \frac{1}{|r-R_A|} | \chi_j \rangle`

    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_charge`
       * :c:data:`nucl_coord`
       * :c:data:`nucl_num`
       * :c:data:`read_ao_integrals_e_n`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_one_e_integrals`
       * :c:data:`ao_ortho_canonical_nucl_elec_integrals`
       * :c:data:`ao_ortho_lowdin_nucl_elec_integrals`
       * :c:data:`mo_integrals_n_e`

 
.. c:var:: ao_integrals_n_e_per_atom


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_integrals_n_e_per_atom	(ao_num,ao_num,nucl_num)


    Nucleus-electron interaction in the |AO| basis set, per atom A.
    
    :math:`\langle \chi_i | -\frac{1}{|r-R_A|} | \chi_j \rangle`

    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`
       * :c:data:`nucl_num`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`mo_integrals_n_e_per_atom`

 
.. c:var:: ao_kinetic_integrals


    File : :file:`ao_one_e_ints/kin_ao_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_kinetic_integrals	(ao_num,ao_num)


    Kinetic energy integrals in the |AO| basis.
    
    :math:`\langle \chi_i |\hat{T}| \chi_j \rangle` 
    

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_deriv2_x`
       * :c:data:`ao_num`
       * :c:data:`read_ao_integrals_kinetic`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_one_e_integrals`
       * :c:data:`mo_kinetic_integrals`

 
.. c:var:: ao_one_e_integrals


    File : :file:`ao_one_e_ints/ao_one_e_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_one_e_integrals	(ao_num,ao_num)
        double precision, allocatable	:: ao_one_e_integrals_diag	(ao_num)


    One-electron Hamiltonian in the |AO| basis.

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_integrals_n_e`
       * :c:data:`ao_kinetic_integrals`
       * :c:data:`ao_num`
       * :c:data:`ao_pseudo_integrals`
       * :c:data:`do_pseudo`
       * :c:data:`read_ao_one_e_integrals`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`fock_matrix_ao_alpha`
       * :c:data:`hf_energy`
       * :c:data:`scf_energy`

 
.. c:var:: ao_one_e_integrals_diag


    File : :file:`ao_one_e_ints/ao_one_e_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_one_e_integrals	(ao_num,ao_num)
        double precision, allocatable	:: ao_one_e_integrals_diag	(ao_num)


    One-electron Hamiltonian in the |AO| basis.

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_integrals_n_e`
       * :c:data:`ao_kinetic_integrals`
       * :c:data:`ao_num`
       * :c:data:`ao_pseudo_integrals`
       * :c:data:`do_pseudo`
       * :c:data:`read_ao_one_e_integrals`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`fock_matrix_ao_alpha`
       * :c:data:`hf_energy`
       * :c:data:`scf_energy`

 
.. c:var:: ao_ortho_canonical_coef


    File : :file:`ao_one_e_ints/ao_ortho_canonical.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_ortho_canonical_coef	(ao_num,ao_num)
        integer	:: ao_ortho_canonical_num	


    matrix of the coefficients of the mos generated by the
    orthonormalization by the S^{-1/2} canonical transformation of the aos
    ao_ortho_canonical_coef(i,j) = coefficient of the ith ao on the jth ao_ortho_canonical orbital

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_cart_to_sphe_coef`
       * :c:data:`ao_cart_to_sphe_overlap`
       * :c:data:`ao_cartesian`
       * :c:data:`ao_num`
       * :c:data:`ao_overlap`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_ortho_canonical_coef_inv`
       * :c:data:`ao_ortho_canonical_nucl_elec_integrals`
       * :c:data:`ao_ortho_canonical_overlap`
       * :c:data:`mo_coef`
       * :c:data:`mo_num`

 
.. c:var:: ao_ortho_canonical_coef_inv


    File : :file:`ao_one_e_ints/ao_ortho_canonical.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_ortho_canonical_coef_inv	(ao_num,ao_num)


    ao_ortho_canonical_coef^(-1)

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_num`
       * :c:data:`ao_ortho_canonical_coef`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`mo_coef_in_ao_ortho_basis`

 
.. c:var:: ao_ortho_canonical_num


    File : :file:`ao_one_e_ints/ao_ortho_canonical.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_ortho_canonical_coef	(ao_num,ao_num)
        integer	:: ao_ortho_canonical_num	


    matrix of the coefficients of the mos generated by the
    orthonormalization by the S^{-1/2} canonical transformation of the aos
    ao_ortho_canonical_coef(i,j) = coefficient of the ith ao on the jth ao_ortho_canonical orbital

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_cart_to_sphe_coef`
       * :c:data:`ao_cart_to_sphe_overlap`
       * :c:data:`ao_cartesian`
       * :c:data:`ao_num`
       * :c:data:`ao_overlap`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_ortho_canonical_coef_inv`
       * :c:data:`ao_ortho_canonical_nucl_elec_integrals`
       * :c:data:`ao_ortho_canonical_overlap`
       * :c:data:`mo_coef`
       * :c:data:`mo_num`

 
.. c:var:: ao_ortho_canonical_overlap


    File : :file:`ao_one_e_ints/ao_ortho_canonical.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_ortho_canonical_overlap	(ao_ortho_canonical_num,ao_ortho_canonical_num)


    overlap matrix of the ao_ortho_canonical.
    Expected to be the Identity

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_num`
       * :c:data:`ao_ortho_canonical_coef`
       * :c:data:`ao_overlap`


 
.. c:var:: ao_overlap


    File : :file:`ao_one_e_ints/ao_overlap.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_overlap	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_z	(ao_num,ao_num)


    Overlap between atomic basis functions:
    
    :math:`\int \chi_i(r) \chi_j(r) dr`

    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:`nucl_coord`
       * :c:data:`read_ao_integrals_overlap`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_cart_to_sphe_overlap`
       * :c:data:`ao_ortho_canonical_coef`
       * :c:data:`ao_ortho_canonical_overlap`
       * :c:data:`ao_ortho_lowdin_coef`
       * :c:data:`ao_ortho_lowdin_overlap`
       * :c:data:`fps_spf_matrix_ao`
       * :c:data:`mo_overlap`
       * :c:data:`s_half`
       * :c:data:`s_half_inv`
       * :c:data:`s_inv`
       * :c:data:`s_mo_coef`

 
.. c:var:: ao_overlap_abs


    File : :file:`ao_one_e_ints/ao_overlap.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_overlap_abs	(ao_num,ao_num)


    Overlap between absolute values of atomic basis functions:
    
    :math:`\int |\chi_i(r)| |\chi_j(r)| dr`

    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:`nucl_coord`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_two_e_integral_alpha`
       * :c:data:`mo_two_e_int_erf_jj_from_ao`
       * :c:data:`mo_two_e_integral_jj_from_ao`
       * :c:data:`mo_two_e_integrals_vv_from_ao`

 
.. c:var:: ao_overlap_x


    File : :file:`ao_one_e_ints/ao_overlap.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_overlap	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_z	(ao_num,ao_num)


    Overlap between atomic basis functions:
    
    :math:`\int \chi_i(r) \chi_j(r) dr`

    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:`nucl_coord`
       * :c:data:`read_ao_integrals_overlap`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_cart_to_sphe_overlap`
       * :c:data:`ao_ortho_canonical_coef`
       * :c:data:`ao_ortho_canonical_overlap`
       * :c:data:`ao_ortho_lowdin_coef`
       * :c:data:`ao_ortho_lowdin_overlap`
       * :c:data:`fps_spf_matrix_ao`
       * :c:data:`mo_overlap`
       * :c:data:`s_half`
       * :c:data:`s_half_inv`
       * :c:data:`s_inv`
       * :c:data:`s_mo_coef`

 
.. c:var:: ao_overlap_y


    File : :file:`ao_one_e_ints/ao_overlap.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_overlap	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_z	(ao_num,ao_num)


    Overlap between atomic basis functions:
    
    :math:`\int \chi_i(r) \chi_j(r) dr`

    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:`nucl_coord`
       * :c:data:`read_ao_integrals_overlap`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_cart_to_sphe_overlap`
       * :c:data:`ao_ortho_canonical_coef`
       * :c:data:`ao_ortho_canonical_overlap`
       * :c:data:`ao_ortho_lowdin_coef`
       * :c:data:`ao_ortho_lowdin_overlap`
       * :c:data:`fps_spf_matrix_ao`
       * :c:data:`mo_overlap`
       * :c:data:`s_half`
       * :c:data:`s_half_inv`
       * :c:data:`s_inv`
       * :c:data:`s_mo_coef`

 
.. c:var:: ao_overlap_z


    File : :file:`ao_one_e_ints/ao_overlap.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_overlap	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_overlap_z	(ao_num,ao_num)


    Overlap between atomic basis functions:
    
    :math:`\int \chi_i(r) \chi_j(r) dr`

    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:`nucl_coord`
       * :c:data:`read_ao_integrals_overlap`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_cart_to_sphe_overlap`
       * :c:data:`ao_ortho_canonical_coef`
       * :c:data:`ao_ortho_canonical_overlap`
       * :c:data:`ao_ortho_lowdin_coef`
       * :c:data:`ao_ortho_lowdin_overlap`
       * :c:data:`fps_spf_matrix_ao`
       * :c:data:`mo_overlap`
       * :c:data:`s_half`
       * :c:data:`s_half_inv`
       * :c:data:`s_inv`
       * :c:data:`s_mo_coef`

 
.. c:var:: ao_pseudo_integrals


    File : :file:`ao_one_e_ints/pot_ao_pseudo_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_pseudo_integrals	(ao_num,ao_num)


    Pseudo-potential integrals in the |AO| basis set.

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_num`
       * :c:data:`ao_pseudo_integrals_local`
       * :c:data:`ao_pseudo_integrals_non_local`
       * :c:data:`do_pseudo`
       * :c:data:`pseudo_klocmax`
       * :c:data:`pseudo_kmax`
       * :c:data:`read_ao_integrals_pseudo`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_one_e_integrals`
       * :c:data:`mo_pseudo_integrals`

 
.. c:var:: ao_pseudo_integrals_local


    File : :file:`ao_one_e_ints/pot_ao_pseudo_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_pseudo_integrals_local	(ao_num,ao_num)


    Local pseudo-potential

    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:`nucl_charge`
       * :c:data:`nucl_coord`
       * :c:data:`nucl_num`
       * :c:data:`pseudo_klocmax`
       * :c:data:`pseudo_v_k_transp`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_pseudo_integrals`

 
.. c:var:: ao_pseudo_integrals_non_local


    File : :file:`ao_one_e_ints/pot_ao_pseudo_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_pseudo_integrals_non_local	(ao_num,ao_num)


    Non-local pseudo-potential

    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:`nucl_charge`
       * :c:data:`nucl_coord`
       * :c:data:`nucl_num`
       * :c:data:`pseudo_kmax`
       * :c:data:`pseudo_lmax`
       * :c:data:`pseudo_v_kl_transp`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_pseudo_integrals`

 
.. c:var:: ao_spread_x


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_spread_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_spread_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_spread_z	(ao_num,ao_num)


    * array of the integrals of AO_i * x^2 AO_j
    
    * array of the integrals of AO_i * y^2 AO_j
    
    * array of the integrals of AO_i * z^2 AO_j

    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:`nucl_coord`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`mo_spread_x`

 
.. c:var:: ao_spread_y


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_spread_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_spread_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_spread_z	(ao_num,ao_num)


    * array of the integrals of AO_i * x^2 AO_j
    
    * array of the integrals of AO_i * y^2 AO_j
    
    * array of the integrals of AO_i * z^2 AO_j

    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:`nucl_coord`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`mo_spread_x`

 
.. c:var:: ao_spread_z


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        double precision, allocatable	:: ao_spread_x	(ao_num,ao_num)
        double precision, allocatable	:: ao_spread_y	(ao_num,ao_num)
        double precision, allocatable	:: ao_spread_z	(ao_num,ao_num)


    * array of the integrals of AO_i * x^2 AO_j
    
    * array of the integrals of AO_i * y^2 AO_j
    
    * array of the integrals of AO_i * z^2 AO_j

    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:`nucl_coord`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`mo_spread_x`

 
.. c:function:: give_polynomial_mult_center_one_e_erf:


    File : :file:`ao_one_e_ints/pot_ao_erf_ints.irp.f`

    .. code:: fortran

        subroutine give_polynomial_mult_center_one_e_erf(A_center,B_center,alpha,beta,&
      power_A,power_B,C_center,n_pt_in,d,n_pt_out,mu_in)


    Returns the explicit polynomial in terms of the $t$ variable of the
    following polynomial:
    
    $I_{x1}(a_x, d_x,p,q) \times I_{x1}(a_y, d_y,p,q) \times I_{x1}(a_z, d_z,p,q)$.

    Calls:

    .. hlist::
       :columns: 3

       * :c:func:`i_x1_pol_mult_one_e`
       * :c:func:`multiply_poly`

 
.. c:function:: give_polynomial_mult_center_one_e_erf_opt:


    File : :file:`ao_one_e_ints/pot_ao_erf_ints.irp.f`

    .. code:: fortran

        subroutine give_polynomial_mult_center_one_e_erf_opt(A_center,B_center,alpha,beta,&
      power_A,power_B,C_center,n_pt_in,d,n_pt_out,mu_in,p,p_inv,p_inv_2,p_new,P_center)


    Returns the explicit polynomial in terms of the $t$ variable of the
    following polynomial:
    
    $I_{x1}(a_x, d_x,p,q) \times I_{x1}(a_y, d_y,p,q) \times I_{x1}(a_z, d_z,p,q)$.

    Called by:

    .. hlist::
       :columns: 3

       * :c:func:`nai_pol_mult_erf`

    Calls:

    .. hlist::
       :columns: 3

       * :c:func:`i_x1_pol_mult_one_e`
       * :c:func:`multiply_poly`

 
.. c:function:: i_x1_pol_mult_one_e:


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        recursive subroutine I_x1_pol_mult_one_e(a,c,R1x,R1xp,R2x,d,nd,n_pt_in)


    Recursive routine involved in the electron-nucleus potential

    Called by:

    .. hlist::
       :columns: 3

       * :c:func:`give_polynomial_mult_center_one_e`
       * :c:func:`give_polynomial_mult_center_one_e_erf`
       * :c:func:`give_polynomial_mult_center_one_e_erf_opt`
       * :c:func:`i_x1_pol_mult_one_e`
       * :c:func:`i_x2_pol_mult_one_e`

    Calls:

    .. hlist::
       :columns: 3

       * :c:func:`i_x1_pol_mult_one_e`
       * :c:func:`i_x2_pol_mult_one_e`
       * :c:func:`multiply_poly`

 
.. c:function:: i_x2_pol_mult_one_e:


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        recursive subroutine I_x2_pol_mult_one_e(c,R1x,R1xp,R2x,d,nd,dim)


    Recursive routine involved in the electron-nucleus potential

    Called by:

    .. hlist::
       :columns: 3

       * :c:func:`i_x1_pol_mult_one_e`

    Calls:

    .. hlist::
       :columns: 3

       * :c:func:`i_x1_pol_mult_one_e`
       * :c:func:`multiply_poly`

 
.. c:var:: pseudo_dz_k_transp


    File : :file:`ao_one_e_ints/pot_ao_pseudo_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: pseudo_v_k_transp	(pseudo_klocmax,nucl_num)
        integer, allocatable	:: pseudo_n_k_transp	(pseudo_klocmax,nucl_num)
        double precision, allocatable	:: pseudo_dz_k_transp	(pseudo_klocmax,nucl_num)


    Transposed arrays for pseudopotentials

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`nucl_num`
       * :c:data:`pseudo_dz_k`
       * :c:data:`pseudo_klocmax`
       * :c:data:`pseudo_n_k`
       * :c:data:`pseudo_v_k`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_pseudo_integrals_local`

 
.. c:var:: pseudo_dz_kl_transp


    File : :file:`ao_one_e_ints/pot_ao_pseudo_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: pseudo_v_kl_transp	(pseudo_kmax,0:pseudo_lmax,nucl_num)
        integer, allocatable	:: pseudo_n_kl_transp	(pseudo_kmax,0:pseudo_lmax,nucl_num)
        double precision, allocatable	:: pseudo_dz_kl_transp	(pseudo_kmax,0:pseudo_lmax,nucl_num)


    Transposed arrays for pseudopotentials

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`nucl_num`
       * :c:data:`pseudo_dz_kl`
       * :c:data:`pseudo_kmax`
       * :c:data:`pseudo_lmax`
       * :c:data:`pseudo_n_kl`
       * :c:data:`pseudo_v_kl`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_pseudo_integrals_non_local`

 
.. c:var:: pseudo_n_k_transp


    File : :file:`ao_one_e_ints/pot_ao_pseudo_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: pseudo_v_k_transp	(pseudo_klocmax,nucl_num)
        integer, allocatable	:: pseudo_n_k_transp	(pseudo_klocmax,nucl_num)
        double precision, allocatable	:: pseudo_dz_k_transp	(pseudo_klocmax,nucl_num)


    Transposed arrays for pseudopotentials

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`nucl_num`
       * :c:data:`pseudo_dz_k`
       * :c:data:`pseudo_klocmax`
       * :c:data:`pseudo_n_k`
       * :c:data:`pseudo_v_k`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_pseudo_integrals_local`

 
.. c:var:: pseudo_n_kl_transp


    File : :file:`ao_one_e_ints/pot_ao_pseudo_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: pseudo_v_kl_transp	(pseudo_kmax,0:pseudo_lmax,nucl_num)
        integer, allocatable	:: pseudo_n_kl_transp	(pseudo_kmax,0:pseudo_lmax,nucl_num)
        double precision, allocatable	:: pseudo_dz_kl_transp	(pseudo_kmax,0:pseudo_lmax,nucl_num)


    Transposed arrays for pseudopotentials

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`nucl_num`
       * :c:data:`pseudo_dz_kl`
       * :c:data:`pseudo_kmax`
       * :c:data:`pseudo_lmax`
       * :c:data:`pseudo_n_kl`
       * :c:data:`pseudo_v_kl`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_pseudo_integrals_non_local`

 
.. c:var:: pseudo_v_k_transp


    File : :file:`ao_one_e_ints/pot_ao_pseudo_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: pseudo_v_k_transp	(pseudo_klocmax,nucl_num)
        integer, allocatable	:: pseudo_n_k_transp	(pseudo_klocmax,nucl_num)
        double precision, allocatable	:: pseudo_dz_k_transp	(pseudo_klocmax,nucl_num)


    Transposed arrays for pseudopotentials

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`nucl_num`
       * :c:data:`pseudo_dz_k`
       * :c:data:`pseudo_klocmax`
       * :c:data:`pseudo_n_k`
       * :c:data:`pseudo_v_k`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_pseudo_integrals_local`

 
.. c:var:: pseudo_v_kl_transp


    File : :file:`ao_one_e_ints/pot_ao_pseudo_ints.irp.f`

    .. code:: fortran

        double precision, allocatable	:: pseudo_v_kl_transp	(pseudo_kmax,0:pseudo_lmax,nucl_num)
        integer, allocatable	:: pseudo_n_kl_transp	(pseudo_kmax,0:pseudo_lmax,nucl_num)
        double precision, allocatable	:: pseudo_dz_kl_transp	(pseudo_kmax,0:pseudo_lmax,nucl_num)


    Transposed arrays for pseudopotentials

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`nucl_num`
       * :c:data:`pseudo_dz_kl`
       * :c:data:`pseudo_kmax`
       * :c:data:`pseudo_lmax`
       * :c:data:`pseudo_n_kl`
       * :c:data:`pseudo_v_kl`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_pseudo_integrals_non_local`

 
.. c:var:: s_half


    File : :file:`ao_one_e_ints/ao_overlap.irp.f`

    .. code:: fortran

        double precision, allocatable	:: s_half	(ao_num,ao_num)


    :math:`S^{1/2}`

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_num`
       * :c:data:`ao_overlap`


 
.. c:var:: s_half_inv


    File : :file:`ao_one_e_ints/ao_overlap.irp.f`

    .. code:: fortran

        double precision, allocatable	:: s_half_inv	(AO_num,AO_num)


    :math:`X = S^{-1/2}` obtained by SVD

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_num`
       * :c:data:`ao_overlap`

    Needed by:

    .. hlist::
       :columns: 3

       * :c:data:`eigenvalues_fock_matrix_ao`

 
.. c:var:: s_inv


    File : :file:`ao_one_e_ints/ao_overlap.irp.f`

    .. code:: fortran

        double precision, allocatable	:: s_inv	(ao_num,ao_num)


    Inverse of the overlap matrix

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_num`
       * :c:data:`ao_overlap`


 
 
Subroutines / functions 
----------------------- 
 
.. c:function:: give_all_erf_kl_ao:


    File : :file:`ao_one_e_ints/pot_ao_erf_ints.irp.f`

    .. code:: fortran

        subroutine give_all_erf_kl_ao(integrals_ao,mu_in,C_center)


    Subroutine that returns all integrals over $r$ of type
    $\frac{ \erf(\mu * | r - R_C | ) }{ | r - R_C | }$

    Needs:

    .. hlist::
       :columns: 3

       * :c:data:`ao_num`

 
.. c:function:: give_polynomial_mult_center_one_e:


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        subroutine give_polynomial_mult_center_one_e(A_center,B_center,alpha,beta,power_A,power_B,C_center,n_pt_in,d,n_pt_out)


    Returns the explicit polynomial in terms of the "t" variable of the following
    
    $I_{x1}(a_x, d_x,p,q) \times I_{x1}(a_y, d_y,p,q) \times I_{x1}(a_z, d_z,p,q)$.

    Called by:

    .. hlist::
       :columns: 3

       * :c:func:`nai_pol_mult`

    Calls:

    .. hlist::
       :columns: 3

       * :c:func:`i_x1_pol_mult_one_e`
       * :c:func:`multiply_poly`

 
.. c:function:: int_gaus_pol:


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        double precision function int_gaus_pol(alpha,n)


    Computes the integral:
    
    $\int_{-\infty}^{\infty} x^n \exp(-\alpha x^2) dx$.

 
.. c:function:: nai_pol_mult:


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        double precision function NAI_pol_mult(A_center,B_center,power_A,power_B,alpha,beta,C_center,n_pt_in)


    Computes the electron-nucleus attraction with two primitves.
    
    :math:`\langle g_i | \frac{1}{|r-R_c|} | g_j \rangle`

    Calls:

    .. hlist::
       :columns: 3

       * :c:func:`give_polynomial_mult_center_one_e`

 
.. c:function:: nai_pol_mult_erf:


    File : :file:`ao_one_e_ints/pot_ao_erf_ints.irp.f`

    .. code:: fortran

        double precision function NAI_pol_mult_erf(A_center,B_center,power_A,power_B,alpha,beta,C_center,n_pt_in,mu_in)


    Computes the following integral :
    
    .. math::
    
      \int dr (x-A_x)^a (x-B_x)^b \exp(-\alpha (x-A_x)^2 - \beta (x-B_x)^2 )
      \frac{\erf(\mu | r - R_C | )}{ | r - R_C | }$.
    

    Calls:

    .. hlist::
       :columns: 3

       * :c:func:`give_polynomial_mult_center_one_e_erf_opt`

 
.. c:function:: nai_pol_mult_erf_ao:


    File : :file:`ao_one_e_ints/pot_ao_erf_ints.irp.f`

    .. code:: fortran

        double precision function NAI_pol_mult_erf_ao(i_ao,j_ao,mu_in,C_center)


    Computes the following integral :
    $\int_{-\infty}^{infty} dr \chi_i(r) \chi_j(r) \frac{\erf(\mu | r - R_C | )}{ | r - R_C | }$.

    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`

 
.. c:function:: overlap_bourrin_deriv_x:


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        subroutine overlap_bourrin_deriv_x(i_component,A_center,B_center,alpha,beta,power_A,power_B,dx,lower_exp_val,overlap_x,nx)



    Called by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_deriv_1_x`

    Calls:

    .. hlist::
       :columns: 3

       * :c:func:`overlap_bourrin_x`

 
.. c:function:: overlap_bourrin_dipole:


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        subroutine overlap_bourrin_dipole(A_center,B_center,alpha,beta,power_A,power_B,overlap_x,lower_exp_val,dx,nx)



    Called by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_dipole_x`

 
.. c:function:: overlap_bourrin_spread:


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        subroutine overlap_bourrin_spread(A_center,B_center,alpha,beta,power_A,power_B,overlap_x,lower_exp_val,dx,nx)


    Computes the following integral :
     int [-infty ; +infty] of [(x-A_center)^(power_A) * (x-B_center)^power_B * exp(-alpha(x-A_center)^2) * exp(-beta(x-B_center)^2) * x ]
     needed for the dipole and those things

    Called by:

    .. hlist::
       :columns: 3

       * :c:data:`ao_spread_x`

 
.. c:function:: overlap_bourrin_x:


    File : :file:`ao_one_e_ints/spread_dipole_ao.irp.f`

    .. code:: fortran

        subroutine overlap_bourrin_x(A_center,B_center,alpha,beta,power_A,power_B,overlap_x,lower_exp_val,dx,nx)



    Called by:

    .. hlist::
       :columns: 3

       * :c:func:`overlap_bourrin_deriv_x`

 
.. c:function:: v_e_n:


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        double precision function V_e_n(a_x,a_y,a_z,b_x,b_y,b_z,alpha,beta)


    Primitve nuclear attraction between the two primitves centered on the same atom.
    
    $p_1 = x^{a_x} y^{a_y} z^{a_z} \exp(-\alpha r^2)$
    
    $p_2 = x^{b_x} y^{b_y} z^{b_z} \exp(-\beta  r^2)$

 
.. c:function:: v_phi:


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        double precision function V_phi(n,m)


    Computes the angular $\phi$ part of the nuclear attraction integral:
    
    $\int_{0}^{2 \pi} \cos(\phi)^n \sin(\phi)^m d\phi$.

 
.. c:function:: v_r:


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        double precision function V_r(n,alpha)


    Computes the radial part of the nuclear attraction integral:
    
    $\int_{0}^{\infty} r^n  \exp(-\alpha  r^2)  dr$
    

 
.. c:function:: v_theta:


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        double precision function V_theta(n,m)


    Computes the angular $\theta$ part of the nuclear attraction integral:
    
    $\int_{0}^{\pi} \cos(\theta)^n \sin(\theta)^m d\theta$

 
.. c:function:: wallis:


    File : :file:`ao_one_e_ints/pot_ao_ints.irp.f`

    .. code:: fortran

        double precision function Wallis(n)


    Wallis integral:
    
    $\int_{0}^{\pi} \cos(\theta)^n d\theta$.