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.. _designing_basis_set_normalization:

#################################
Designing Basis Set Normalization
#################################

This section contains notes on how normalization is represented in Chemist's
basis set component.

.. warning::

   References were added by AI and have not been human verified. Please
   report errors.

**********************
What is Normalization?
**********************

An atomic orbital (AO), :math:`\mu`, is said to be normalized if:

.. math::

   \langle \mu | \mu \rangle = 1.

:ref:`ao_basis_set_background` builds a Cartesian AO in three layers: primitive
Gaussians :math:`\chi_p` combine into a contracted Gaussian :math:`G`, which is
then paired with a Cartesian polynomial to give :math:`\mu_{ijk}`. Each layer
introduces something which is not normalized, and correspondingly each layer
contributes one factor to the overall constant. Writing those factors
separately,

.. math::

   \mu_{ijk}(\vec{r}) = \underbrace{N^{AO}_{ijk}}_{\text{AO}}\;
     \underbrace{N^{G}(d, \zeta; \ell)}_{\text{contracted Gaussian}}\;
     x^i y^j z^k \sum_p d_p\,
     \underbrace{N^{\chi}(\zeta_p; \ell)}_{\text{primitive}}\,
     \chi_p(\vec{r}; \zeta_p),

where :math:`\ell = i + j + k`. The three factors are developed below
:cite:`gill1994molecular,fermann2020fundamentals`.

Primitive normalization
=======================

The innermost factor scales a single primitive so that, paired with the
polynomial, it has unit norm:

.. math::

   N^{\chi}(\zeta_p; \ell) =
     \sqrt{\frac{2^{\ell}\left(2\zeta_p\right)^{\ell + 3/2}}
                {\sqrt{\pi^3}\,(2\ell-1)!!}}
   \;=\;
     \left(\frac{2\zeta_p}{\pi}\right)^{3/4}
     \sqrt{\frac{(4\zeta_p)^{\ell}}{(2\ell-1)!!}}.

Note this factor sits *inside* the sum, because it differs from primitive to
primitive. It is therefore not something which can be pulled out in front of
the contraction; folding it in to the contraction coefficients changes the
weights of the primitives. The coefficients are said to be "weight
normalized."

Contracted Gaussian normalization
=================================

Normalizing each primitive does not normalize their sum, because distinct
primitives are not orthogonal. The contracted Gaussian therefore needs its own
factor,

.. math::

   N^{G}(d, \zeta; \ell) =
     \left[\sum_{pq} d_p d_q\, S_{pq}\right]^{-1/2},
   \qquad
   S_{pq} = \left(\frac{2\sqrt{\zeta_p \zeta_q}}
                       {\zeta_p + \zeta_q}\right)^{\ell + 3/2},

where :math:`S_{pq}` is the overlap of the two *already normalized* primitives.
Because they are normalized, :math:`S_{pp} = 1`, so this factor measures
nothing but the non-orthogonality between distinct primitives. It is a single
scalar in front of the sum, and it depends on the whole contraction at once.

Cartesian AO normalization
==========================

The two factors so far depend on :math:`\ell` but not on how the angular
momentum is distributed among the axes. That distribution matters, and the
remaining factor accounts for it:

.. math::

   N^{AO}_{ijk} = \sqrt{\frac{(2\ell-1)!!}
                        {(2i-1)!!\,(2j-1)!!\,(2k-1)!!}}.

By construction :math:`N^{AO}_{ijk} = 1` when all the angular momentum sits on
one axis, *i.e.*, for :math:`z^\ell` and its permutations, and it is larger
otherwise. For a :math:`d` shell it is :math:`\sqrt{3}` for :math:`d_{xy}`
against :math:`1` for :math:`d_{zz}`, which is not a difference one can
neglect.

This is the only one of the three factors which distinguishes the components of
a shell, and it is the one which is usually dropped. Since the components of a
shell are typically stored implicitly, there is nowhere natural to put a
per-component constant, so the common practice is to stop after
:math:`N^{\chi}` and :math:`N^{G}` and let the single-axis component stand in
for the whole shell. Doing so leaves every other component short by a factor of
:math:`N^{AO}_{ijk}`.

Spherical shells do not have this problem because the Cartesian-to-spherical
transformation coefficients
:cite:`schlegel1995transformation` have :math:`N^{AO}_{ijk}` built into them,
and so they assume they are being handed Cartesian AOs that are only
normalized up to  :math:`N^{G}`.

***********************************
Why does Chemist need to own this?
***********************************

Chemist's philosophy is to be self-describing. Since the parameters will enter
chemist without a description, Chemist must be the one that establishes the
convention and records it.


.. _normalization_considerations:

****************************
Normalization Considerations
****************************

Topics in this section were considered in designing how normalization is
represented and ultimately were addressed by the design.

.. _n_raw_parameters_preserved:

Raw parameters are preserved
   Normalizing by rewriting the stored coefficients destroys what the user
   supplied.

   - The parameters a user provided should be recoverable unchanged.
   - Normalization constants should be computed from those parameters on
     demand rather than folded into them at construction.
   - Folding normalization in is still useful for handing data to an external
     library, but that should produce a new object rather than mutating the
     original.

.. _angular_momentum_in_primitive:

Angular momentum in primitive
   The primitive normalization factor depends on :math:`\ell` therefore the
   primitive must store it.

.. _n_single_source:

One source of truth
   Consumers should not each re-derive the normalization mathematics.

   - Each factor should be implemented once.
   - Both :math:`N^{\chi} N^{G}`, which integral libraries expect, and the full
     product including :math:`N^{AO}_{ijk}`, which real-space evaluation requires,
     should come from that one implementation, so that the two cannot drift
     apart.

.. _n_component_vs_shell:

Consumers stop at different layers
   Following from :ref:`n_single_source`, the two consumers of normalization
   want different numbers.

   - Code evaluating an AO at a point in space needs the full product,
     :math:`N^{AO}_{ijk}` included.
   - Code interfacing with integral libraries needs the product stopped after
     :math:`N^{G}`, since those libraries apply one scalar per shell and their
     Cartesian-to-spherical coefficients already carry :math:`N^{AO}_{ijk}`.
   - Both must be available, and it must be unambiguous which is which.

Out of Scope
============

Topics in this section were considered, but do not play a role in the current
design.

General contractions
   The design assumes segmented contractions, in which each contracted
   Gaussian has its own primitives. Normalization of general contractions,
   where several contractions share a primitive set, is deferred along with
   general contractions themselves.

Effective core potentials
   ECPs carry their own conventions and are not represented.

Non-Gaussian basis functions
   The normalization constants above are specific to Gaussians. Slater
   orbitals and numerical orbitals would need their own.

Orthonormalization
   AOs are non-orthonormal by default and most code assumes this. We're not
   going to worry about orthonormalizing the basis set.

*******
Summary
*******

:ref:`n_raw_parameters_preserved`
   Constants are computed on demand from the stored parameters. Converting
   between conventions produces a new object rather than mutating an existing
   one.

:ref:`angular_momentum_in_primitive`
   The primitive normalization factor depends on :math:`\ell` therefore the
   primitive must store it.

:ref:`n_single_source`
   The formulas are implemented once as free functions over the parameters
   they depend on, and every class-level accessor delegates to them.

:ref:`n_component_vs_shell`
   ``AOShell`` reports :math:`N^{\chi} N^{G}`; ``AO`` completes the product
   with :math:`N^{AO}_{ijk}`. The two consumers want the same product truncated at
   different points, not two different calculations.

***************
Further Reading
***************

The closed-form normalization constants, and the integral machinery they
belong to, are derived in Gill's review :cite:`gill1994molecular` and, in a
more recent and freely available treatment, by Fermann and Valeev
:cite:`fermann2020fundamentals`. Helgaker, Jørgensen, and Olsen
:cite:`helgaker2000molecular` cover the same ground at textbook length,
including the solid harmonics. For the Cartesian-to-spherical transformation
specifically, and the normalization convention embedded in its coefficients,
see Schlegel and Frisch :cite:`schlegel1995transformation`.

.. bibliography::
   :filter: docname in docnames