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https://github.com/LCPQ/quantum_package
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1041 lines
22 KiB
ReStructuredText
1041 lines
22 KiB
ReStructuredText
.. _ao_one_e_ints:
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.. program:: ao_one_e_ints
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.. default-role:: option
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==================
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ao_one_e_integrals
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==================
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All the one-electron integrals in the |AO| basis are here.
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The most important providers for usual quantum-chemistry calculation are:
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* `ao_kinetic_integral` which are the kinetic operator integrals on the |AO| basis (see :file:`kin_ao_ints.irp.f`)
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* `ao_nucl_elec_integral` which are the nuclear-elctron operator integrals on the |AO| basis (see :file:`pot_ao_ints.irp.f`)
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* `ao_one_e_integrals` which are the the h_core operator integrals on the |AO| basis (see :file:`ao_mono_ints.irp.f`)
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Note that you can find other interesting integrals related to the position operator in :file:`spread_dipole_ao.irp.f`.
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EZFIO parameters
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----------------
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.. option:: ao_integrals_e_n
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Nucleus-electron integrals in |AO| basis set
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.. option:: io_ao_integrals_e_n
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Read/Write |AO| nucleus-electron attraction integrals from/to disk [ Write | Read | None ]
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Default: None
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.. option:: ao_integrals_kinetic
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Kinetic energy integrals in |AO| basis set
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.. option:: io_ao_integrals_kinetic
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Read/Write |AO| kinetic integrals from/to disk [ Write | Read | None ]
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Default: None
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.. option:: ao_integrals_pseudo
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Pseudopotential integrals in |AO| basis set
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.. option:: io_ao_integrals_pseudo
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Read/Write |AO| pseudopotential integrals from/to disk [ Write | Read | None ]
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Default: None
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.. option:: ao_integrals_overlap
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Overlap integrals in |AO| basis set
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.. option:: io_ao_integrals_overlap
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Read/Write |AO| overlap integrals from/to disk [ Write | Read | None ]
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Default: None
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.. option:: ao_one_e_integrals
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Combined integrals in |AO| basis set
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.. option:: io_ao_one_e_integrals
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Read/Write |AO| one-electron integrals from/to disk [ Write | Read | None ]
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Default: None
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Providers
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---------
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.. c:var:: ao_cart_to_sphe_coef
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.. code:: text
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double precision, allocatable :: ao_cart_to_sphe_coef (ao_num,ao_num)
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integer :: ao_cart_to_sphe_num
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File: :file:`ao_ortho_canonical.irp.f`
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Coefficients to go from cartesian to spherical coordinates in the current basis set
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.. c:var:: ao_cart_to_sphe_inv
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.. code:: text
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double precision, allocatable :: ao_cart_to_sphe_inv (ao_cart_to_sphe_num,ao_num)
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File: :file:`ao_ortho_canonical.irp.f`
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Inverse of :c:data:`ao_cart_to_sphe_coef`
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.. c:var:: ao_cart_to_sphe_num
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.. code:: text
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double precision, allocatable :: ao_cart_to_sphe_coef (ao_num,ao_num)
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integer :: ao_cart_to_sphe_num
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File: :file:`ao_ortho_canonical.irp.f`
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Coefficients to go from cartesian to spherical coordinates in the current basis set
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.. c:var:: ao_cart_to_sphe_overlap
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.. code:: text
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double precision, allocatable :: ao_cart_to_sphe_overlap (ao_cart_to_sphe_num,ao_cart_to_sphe_num)
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File: :file:`ao_ortho_canonical.irp.f`
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|AO| overlap matrix in the spherical basis set
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.. c:var:: ao_deriv2_x
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.. code:: text
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double precision, allocatable :: ao_deriv2_x (ao_num,ao_num)
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double precision, allocatable :: ao_deriv2_y (ao_num,ao_num)
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double precision, allocatable :: ao_deriv2_z (ao_num,ao_num)
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File: :file:`kin_ao_ints.irp.f`
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Second derivative matrix elements in the |AO| basis.
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:math:`{\tt ao\_deriv2\_x} = \langle \chi_i(x,y,z) | \frac{\partial^2}{\partial x^2} |\chi_j (x,y,z) \rangle`
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.. c:var:: ao_deriv2_y
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.. code:: text
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double precision, allocatable :: ao_deriv2_x (ao_num,ao_num)
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double precision, allocatable :: ao_deriv2_y (ao_num,ao_num)
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double precision, allocatable :: ao_deriv2_z (ao_num,ao_num)
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File: :file:`kin_ao_ints.irp.f`
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Second derivative matrix elements in the |AO| basis.
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:math:`{\tt ao\_deriv2\_x} = \langle \chi_i(x,y,z) | \frac{\partial^2}{\partial x^2} |\chi_j (x,y,z) \rangle`
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.. c:var:: ao_deriv2_z
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.. code:: text
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double precision, allocatable :: ao_deriv2_x (ao_num,ao_num)
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double precision, allocatable :: ao_deriv2_y (ao_num,ao_num)
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double precision, allocatable :: ao_deriv2_z (ao_num,ao_num)
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File: :file:`kin_ao_ints.irp.f`
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Second derivative matrix elements in the |AO| basis.
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:math:`{\tt ao\_deriv2\_x} = \langle \chi_i(x,y,z) | \frac{\partial^2}{\partial x^2} |\chi_j (x,y,z) \rangle`
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.. c:var:: ao_deriv_1_x
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.. code:: text
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double precision, allocatable :: ao_deriv_1_x (ao_num,ao_num)
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double precision, allocatable :: ao_deriv_1_y (ao_num,ao_num)
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double precision, allocatable :: ao_deriv_1_z (ao_num,ao_num)
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File: :file:`spread_dipole_ao.irp.f`
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* array of the integrals of AO_i * d/dx AO_j
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* array of the integrals of AO_i * d/dy AO_j
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* array of the integrals of AO_i * d/dz AO_j
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.. c:var:: ao_deriv_1_y
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.. code:: text
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double precision, allocatable :: ao_deriv_1_x (ao_num,ao_num)
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double precision, allocatable :: ao_deriv_1_y (ao_num,ao_num)
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double precision, allocatable :: ao_deriv_1_z (ao_num,ao_num)
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File: :file:`spread_dipole_ao.irp.f`
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* array of the integrals of AO_i * d/dx AO_j
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* array of the integrals of AO_i * d/dy AO_j
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* array of the integrals of AO_i * d/dz AO_j
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.. c:var:: ao_deriv_1_z
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.. code:: text
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double precision, allocatable :: ao_deriv_1_x (ao_num,ao_num)
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double precision, allocatable :: ao_deriv_1_y (ao_num,ao_num)
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double precision, allocatable :: ao_deriv_1_z (ao_num,ao_num)
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File: :file:`spread_dipole_ao.irp.f`
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* array of the integrals of AO_i * d/dx AO_j
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* array of the integrals of AO_i * d/dy AO_j
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* array of the integrals of AO_i * d/dz AO_j
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.. c:var:: ao_dipole_x
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.. code:: text
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double precision, allocatable :: ao_dipole_x (ao_num,ao_num)
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double precision, allocatable :: ao_dipole_y (ao_num,ao_num)
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double precision, allocatable :: ao_dipole_z (ao_num,ao_num)
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File: :file:`spread_dipole_ao.irp.f`
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* array of the integrals of AO_i * x AO_j
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* array of the integrals of AO_i * y AO_j
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* array of the integrals of AO_i * z AO_j
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.. c:var:: ao_dipole_y
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.. code:: text
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double precision, allocatable :: ao_dipole_x (ao_num,ao_num)
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double precision, allocatable :: ao_dipole_y (ao_num,ao_num)
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double precision, allocatable :: ao_dipole_z (ao_num,ao_num)
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File: :file:`spread_dipole_ao.irp.f`
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* array of the integrals of AO_i * x AO_j
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* array of the integrals of AO_i * y AO_j
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* array of the integrals of AO_i * z AO_j
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.. c:var:: ao_dipole_z
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.. code:: text
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double precision, allocatable :: ao_dipole_x (ao_num,ao_num)
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double precision, allocatable :: ao_dipole_y (ao_num,ao_num)
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double precision, allocatable :: ao_dipole_z (ao_num,ao_num)
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File: :file:`spread_dipole_ao.irp.f`
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* array of the integrals of AO_i * x AO_j
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* array of the integrals of AO_i * y AO_j
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* array of the integrals of AO_i * z AO_j
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.. c:var:: ao_integrals_n_e
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.. code:: text
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double precision, allocatable :: ao_integrals_n_e (ao_num,ao_num)
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File: :file:`pot_ao_ints.irp.f`
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Nucleus-electron interaction, in the |AO| basis set.
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:math:`\langle \chi_i | -\sum_A \frac{1}{|r-R_A|} | \chi_j \rangle`
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.. c:var:: ao_integrals_n_e_per_atom
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.. code:: text
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double precision, allocatable :: ao_integrals_n_e_per_atom (ao_num,ao_num,nucl_num)
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File: :file:`pot_ao_ints.irp.f`
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Nucleus-electron interaction in the |AO| basis set, per atom A.
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:math:`\langle \chi_i | -\frac{1}{|r-R_A|} | \chi_j \rangle`
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.. c:var:: ao_kinetic_integrals
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.. code:: text
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double precision, allocatable :: ao_kinetic_integrals (ao_num,ao_num)
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File: :file:`kin_ao_ints.irp.f`
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Kinetic energy integrals in the |AO| basis.
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:math:`\langle \chi_i |\hat{T}| \chi_j \rangle`
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.. c:var:: ao_one_e_integrals
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.. code:: text
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double precision, allocatable :: ao_one_e_integrals (ao_num,ao_num)
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double precision, allocatable :: ao_one_e_integrals_diag (ao_num)
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File: :file:`ao_one_e_ints.irp.f`
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One-electron Hamiltonian in the |AO| basis.
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.. c:var:: ao_one_e_integrals_diag
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.. code:: text
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double precision, allocatable :: ao_one_e_integrals (ao_num,ao_num)
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double precision, allocatable :: ao_one_e_integrals_diag (ao_num)
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File: :file:`ao_one_e_ints.irp.f`
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One-electron Hamiltonian in the |AO| basis.
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.. c:var:: ao_ortho_canonical_coef
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.. code:: text
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double precision, allocatable :: ao_ortho_canonical_coef (ao_num,ao_num)
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integer :: ao_ortho_canonical_num
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File: :file:`ao_ortho_canonical.irp.f`
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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
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.. c:var:: ao_ortho_canonical_coef_inv
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.. code:: text
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double precision, allocatable :: ao_ortho_canonical_coef_inv (ao_num,ao_num)
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File: :file:`ao_ortho_canonical.irp.f`
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ao_ortho_canonical_coef^(-1)
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.. c:var:: ao_ortho_canonical_num
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.. code:: text
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double precision, allocatable :: ao_ortho_canonical_coef (ao_num,ao_num)
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integer :: ao_ortho_canonical_num
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File: :file:`ao_ortho_canonical.irp.f`
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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
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.. c:var:: ao_ortho_canonical_overlap
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.. code:: text
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double precision, allocatable :: ao_ortho_canonical_overlap (ao_ortho_canonical_num,ao_ortho_canonical_num)
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File: :file:`ao_ortho_canonical.irp.f`
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overlap matrix of the ao_ortho_canonical. Expected to be the Identity
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.. c:var:: ao_overlap
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.. code:: text
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double precision, allocatable :: ao_overlap (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_x (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_y (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_z (ao_num,ao_num)
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File: :file:`ao_overlap.irp.f`
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Overlap between atomic basis functions:
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:math:`\int \chi_i(r) \chi_j(r) dr`
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.. c:var:: ao_overlap_abs
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.. code:: text
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double precision, allocatable :: ao_overlap_abs (ao_num,ao_num)
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File: :file:`ao_overlap.irp.f`
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Overlap between absolute values of atomic basis functions:
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:math:`\int |\chi_i(r)| |\chi_j(r)| dr`
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.. c:var:: ao_overlap_x
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.. code:: text
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double precision, allocatable :: ao_overlap (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_x (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_y (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_z (ao_num,ao_num)
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File: :file:`ao_overlap.irp.f`
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Overlap between atomic basis functions:
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:math:`\int \chi_i(r) \chi_j(r) dr`
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.. c:var:: ao_overlap_y
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.. code:: text
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double precision, allocatable :: ao_overlap (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_x (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_y (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_z (ao_num,ao_num)
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File: :file:`ao_overlap.irp.f`
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Overlap between atomic basis functions:
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:math:`\int \chi_i(r) \chi_j(r) dr`
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.. c:var:: ao_overlap_z
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.. code:: text
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double precision, allocatable :: ao_overlap (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_x (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_y (ao_num,ao_num)
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double precision, allocatable :: ao_overlap_z (ao_num,ao_num)
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File: :file:`ao_overlap.irp.f`
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Overlap between atomic basis functions:
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:math:`\int \chi_i(r) \chi_j(r) dr`
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.. c:var:: ao_pseudo_integrals
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.. code:: text
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double precision, allocatable :: ao_pseudo_integrals (ao_num,ao_num)
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File: :file:`pot_ao_pseudo_ints.irp.f`
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Pseudo-potential integrals in the |AO| basis set.
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.. c:var:: ao_pseudo_integrals_local
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.. code:: text
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double precision, allocatable :: ao_pseudo_integrals_local (ao_num,ao_num)
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File: :file:`pot_ao_pseudo_ints.irp.f`
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Local pseudo-potential
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.. c:var:: ao_pseudo_integrals_non_local
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.. code:: text
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double precision, allocatable :: ao_pseudo_integrals_non_local (ao_num,ao_num)
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File: :file:`pot_ao_pseudo_ints.irp.f`
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Non-local pseudo-potential
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.. c:var:: ao_spread_x
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.. code:: text
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double precision, allocatable :: ao_spread_x (ao_num,ao_num)
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double precision, allocatable :: ao_spread_y (ao_num,ao_num)
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double precision, allocatable :: ao_spread_z (ao_num,ao_num)
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File: :file:`spread_dipole_ao.irp.f`
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* array of the integrals of AO_i * x^2 AO_j
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* array of the integrals of AO_i * y^2 AO_j
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* array of the integrals of AO_i * z^2 AO_j
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.. c:var:: ao_spread_y
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.. code:: text
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double precision, allocatable :: ao_spread_x (ao_num,ao_num)
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double precision, allocatable :: ao_spread_y (ao_num,ao_num)
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double precision, allocatable :: ao_spread_z (ao_num,ao_num)
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File: :file:`spread_dipole_ao.irp.f`
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* array of the integrals of AO_i * x^2 AO_j
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* array of the integrals of AO_i * y^2 AO_j
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* array of the integrals of AO_i * z^2 AO_j
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.. c:var:: ao_spread_z
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.. code:: text
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double precision, allocatable :: ao_spread_x (ao_num,ao_num)
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double precision, allocatable :: ao_spread_y (ao_num,ao_num)
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double precision, allocatable :: ao_spread_z (ao_num,ao_num)
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File: :file:`spread_dipole_ao.irp.f`
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* array of the integrals of AO_i * x^2 AO_j
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* array of the integrals of AO_i * y^2 AO_j
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* array of the integrals of AO_i * z^2 AO_j
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.. c:var:: give_polynomial_mult_center_one_e_erf
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.. code:: text
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subroutine give_polynomial_mult_center_one_e_erf(A_center,B_center,alpha,beta,&
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power_A,power_B,C_center,n_pt_in,d,n_pt_out,mu_in)
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File: :file:`pot_ao_erf_ints.irp.f`
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Returns the explicit polynomial in terms of the :math:`t` variable of the following polynomial:
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:math:`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)` .
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.. c:var:: give_polynomial_mult_center_one_e_erf_opt
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.. code:: text
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subroutine give_polynomial_mult_center_one_e_erf_opt(A_center,B_center,alpha,beta,&
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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)
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File: :file:`pot_ao_erf_ints.irp.f`
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Returns the explicit polynomial in terms of the :math:`t` variable of the following polynomial:
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:math:`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)` .
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.. c:var:: i_x1_pol_mult_one_e
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.. code:: text
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recursive subroutine I_x1_pol_mult_one_e(a,c,R1x,R1xp,R2x,d,nd,n_pt_in)
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File: :file:`pot_ao_ints.irp.f`
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Recursive routine involved in the electron-nucleus potential
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.. c:var:: i_x2_pol_mult_one_e
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.. code:: text
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recursive subroutine I_x2_pol_mult_one_e(c,R1x,R1xp,R2x,d,nd,dim)
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File: :file:`pot_ao_ints.irp.f`
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Recursive routine involved in the electron-nucleus potential
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.. c:var:: pseudo_dz_k_transp
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.. code:: text
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double precision, allocatable :: pseudo_v_k_transp (pseudo_klocmax,nucl_num)
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integer, allocatable :: pseudo_n_k_transp (pseudo_klocmax,nucl_num)
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double precision, allocatable :: pseudo_dz_k_transp (pseudo_klocmax,nucl_num)
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File: :file:`pot_ao_pseudo_ints.irp.f`
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Transposed arrays for pseudopotentials
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.. c:var:: pseudo_dz_kl_transp
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.. code:: text
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double precision, allocatable :: pseudo_v_kl_transp (pseudo_kmax,0:pseudo_lmax,nucl_num)
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integer, allocatable :: pseudo_n_kl_transp (pseudo_kmax,0:pseudo_lmax,nucl_num)
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double precision, allocatable :: pseudo_dz_kl_transp (pseudo_kmax,0:pseudo_lmax,nucl_num)
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File: :file:`pot_ao_pseudo_ints.irp.f`
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Transposed arrays for pseudopotentials
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.. c:var:: pseudo_n_k_transp
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.. code:: text
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double precision, allocatable :: pseudo_v_k_transp (pseudo_klocmax,nucl_num)
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integer, allocatable :: pseudo_n_k_transp (pseudo_klocmax,nucl_num)
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double precision, allocatable :: pseudo_dz_k_transp (pseudo_klocmax,nucl_num)
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File: :file:`pot_ao_pseudo_ints.irp.f`
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Transposed arrays for pseudopotentials
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.. c:var:: pseudo_n_kl_transp
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.. code:: text
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double precision, allocatable :: pseudo_v_kl_transp (pseudo_kmax,0:pseudo_lmax,nucl_num)
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integer, allocatable :: pseudo_n_kl_transp (pseudo_kmax,0:pseudo_lmax,nucl_num)
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double precision, allocatable :: pseudo_dz_kl_transp (pseudo_kmax,0:pseudo_lmax,nucl_num)
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File: :file:`pot_ao_pseudo_ints.irp.f`
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Transposed arrays for pseudopotentials
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.. c:var:: pseudo_v_k_transp
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.. code:: text
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double precision, allocatable :: pseudo_v_k_transp (pseudo_klocmax,nucl_num)
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integer, allocatable :: pseudo_n_k_transp (pseudo_klocmax,nucl_num)
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double precision, allocatable :: pseudo_dz_k_transp (pseudo_klocmax,nucl_num)
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File: :file:`pot_ao_pseudo_ints.irp.f`
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Transposed arrays for pseudopotentials
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.. c:var:: pseudo_v_kl_transp
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.. code:: text
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double precision, allocatable :: pseudo_v_kl_transp (pseudo_kmax,0:pseudo_lmax,nucl_num)
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integer, allocatable :: pseudo_n_kl_transp (pseudo_kmax,0:pseudo_lmax,nucl_num)
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double precision, allocatable :: pseudo_dz_kl_transp (pseudo_kmax,0:pseudo_lmax,nucl_num)
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File: :file:`pot_ao_pseudo_ints.irp.f`
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Transposed arrays for pseudopotentials
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.. c:var:: s_half
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.. code:: text
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double precision, allocatable :: s_half (ao_num,ao_num)
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File: :file:`ao_overlap.irp.f`
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:math:`S^{1/2}`
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.. c:var:: s_half_inv
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.. code:: text
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double precision, allocatable :: s_half_inv (AO_num,AO_num)
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File: :file:`ao_overlap.irp.f`
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:math:`X = S^{-1/2}` obtained by SVD
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.. c:var:: s_inv
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.. code:: text
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double precision, allocatable :: s_inv (ao_num,ao_num)
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File: :file:`ao_overlap.irp.f`
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Inverse of the overlap matrix
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Subroutines / functions
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-----------------------
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.. c:function:: give_all_erf_kl_ao
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.. code:: text
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subroutine give_all_erf_kl_ao(integrals_ao,mu_in,C_center)
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File: :file:`pot_ao_erf_ints.irp.f`
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Subroutine that returns all integrals over :math:`r` of type :math:`\frac{ \erf(\mu * |r-R_C|) }{ |r-R_C| }`
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.. c:function:: give_polynomial_mult_center_one_e
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.. code:: text
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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)
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File: :file:`pot_ao_ints.irp.f`
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Returns the explicit polynomial in terms of the "t" variable of the following
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:math:`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)` .
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.. c:function:: int_gaus_pol
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.. code:: text
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double precision function int_gaus_pol(alpha,n)
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File: :file:`pot_ao_ints.irp.f`
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Computes the integral:
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:math:`\int_{-\infty}^{\infty} x^n \exp(-\alpha x^2) dx` .
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.. c:function:: nai_pol_mult
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.. code:: text
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double precision function NAI_pol_mult(A_center,B_center,power_A,power_B,alpha,beta,C_center,n_pt_in)
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File: :file:`pot_ao_ints.irp.f`
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Computes the electron-nucleus attraction with two primitves.
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:math:`\langle g_i | \frac{1}{|r-R_c|} | g_j \rangle`
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.. c:function:: nai_pol_mult_erf
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.. code:: text
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double precision function NAI_pol_mult_erf(A_center,B_center,power_A,power_B,alpha,beta,C_center,n_pt_in,mu_in)
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File: :file:`pot_ao_erf_ints.irp.f`
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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|}` .
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.. c:function:: nai_pol_mult_erf_ao
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.. code:: text
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double precision function NAI_pol_mult_erf_ao(i_ao,j_ao,mu_in,C_center)
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File: :file:`pot_ao_erf_ints.irp.f`
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Computes the following integral : :math:`\int_{-\infty}^{infty} dr \chi_i(r) \chi_j(r) \frac{\erf(\mu |r-R_C|)}{|r-R_C|}` .
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.. c:function:: overlap_bourrin_deriv_x
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.. code:: text
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subroutine overlap_bourrin_deriv_x(i_component,A_center,B_center,alpha,beta,power_A,power_B,dx,lower_exp_val,overlap_x,nx)
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File: :file:`spread_dipole_ao.irp.f`
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.. c:function:: overlap_bourrin_dipole
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.. code:: text
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subroutine overlap_bourrin_dipole(A_center,B_center,alpha,beta,power_A,power_B,overlap_x,lower_exp_val,dx,nx)
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File: :file:`spread_dipole_ao.irp.f`
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.. c:function:: overlap_bourrin_spread
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.. code:: text
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subroutine overlap_bourrin_spread(A_center,B_center,alpha,beta,power_A,power_B,overlap_x,lower_exp_val,dx,nx)
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File: :file:`spread_dipole_ao.irp.f`
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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
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.. c:function:: overlap_bourrin_x
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.. code:: text
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subroutine overlap_bourrin_x(A_center,B_center,alpha,beta,power_A,power_B,overlap_x,lower_exp_val,dx,nx)
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File: :file:`spread_dipole_ao.irp.f`
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.. c:function:: v_e_n
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.. code:: text
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double precision function V_e_n(a_x,a_y,a_z,b_x,b_y,b_z,alpha,beta)
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File: :file:`pot_ao_ints.irp.f`
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Primitve nuclear attraction between the two primitves centered on the same atom.
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:math:`p_1 = x^{a_x} y^{a_y} z^{a_z} \exp(-\alpha r^2)`
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:math:`p_2 = x^{b_x} y^{b_y} z^{b_z} \exp(-\beta r^2)`
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.. c:function:: v_phi
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.. code:: text
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double precision function V_phi(n,m)
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File: :file:`pot_ao_ints.irp.f`
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Computes the angular :math:`\phi` part of the nuclear attraction integral:
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:math:`\int_{0}^{2 \pi} \cos(\phi)^n \sin(\phi)^m d\phi` .
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.. c:function:: v_r
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.. code:: text
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double precision function V_r(n,alpha)
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File: :file:`pot_ao_ints.irp.f`
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Computes the radial part of the nuclear attraction integral:
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:math:`\int_{0}^{\infty} r^n \exp(-\alpha r^2) dr`
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.. c:function:: v_theta
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.. code:: text
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double precision function V_theta(n,m)
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File: :file:`pot_ao_ints.irp.f`
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Computes the angular :math:`\theta` part of the nuclear attraction integral:
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:math:`\int_{0}^{\pi} \cos(\theta)^n \sin(\theta)^m d\theta`
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.. c:function:: wallis
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.. code:: text
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double precision function Wallis(n)
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File: :file:`pot_ao_ints.irp.f`
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Wallis integral:
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:math:`\int_{0}^{\pi} \cos(\theta)^n d\theta` .
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