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QuantumPackage/docs/source/modules/ao_one_e_ints.rst

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2019-03-07 15:12:02 +01:00
.. _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:
$\int_{0}^{\pi} \cos(\theta)^n d\theta$.
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