From Exchange Models to Full XC
===============================

Exchange-only CIDER
-------------------

The CIDER23X and CIDER24X files represent an exchange functional.  A common
use is a surrogate-hybrid construction

.. math::

   E_{\mathrm{xc}} = a E_{\mathrm{x}}^{\mathrm{CIDER}}
   + (1-a)E_{\mathrm{x}}^{\mathrm{PBE}}
   + E_{\mathrm{c}}^{\mathrm{PBE}},

with :math:`a=0.25` for the PBE0-surrogate form used in the corresponding
work.  The mixing fraction and external semilocal terms are part of the
functional definition.  They must be supplied explicitly when an
exchange-only model is initialized. :footcite:p:`CIDER23X,CIDER24X`

CIDER26XC functional form
-------------------------

The CIDER26XC models return the full exchange-correlation energy.  In
the notation of the CIDER26XC work, the energy density contains an additive
PBE baseline and learned exchange and correlation corrections,

.. math::

   e_{\mathrm{xc}}^{\mathrm{CIDER}} =
   \sum_\sigma e_{\mathrm{x}}^{\mathrm{PBE}}[n_\sigma]
   + e_{\mathrm{c}}^{\mathrm{PBE}}[n_\uparrow,n_\downarrow]
   + \sum_\sigma e_{\mathrm{x}}^{\mathrm{LDA}}[n_\sigma]
     f_{\mathrm{x}}(\mathbf{X}^{\mathrm{x}}_\sigma)
   + e_{\mathrm{c}}^{\mathrm{PBE}}
     f_{\mathrm{c}}\!\left(\tfrac{1}{2}\sum_\sigma
     \mathbf{X}^{\mathrm{c}}_\sigma\right).

Exchange is evaluated separately for the two spin channels.  Correlation uses
a spin-coupled PBE multiplier and the spin-averaged CIDER feature vector.  The
exchange descriptors are scale invariant.  Correlation additionally receives
density information to learn its nontrivial behavior under uniform coordinate
scaling; see :doc:`uniform_scaling`.

The YAML model contains its additive PBE contributions.  The corresponding
GPAW construction uses ``xmix=1.0``, ``xkernel=None``, and ``ckernel=None``.
CiderPress validates this composition so that the stored full-XC form is
evaluated as trained.  See :doc:`../usage/production_models` for the complete
call.

CIDER26XC feature vectors
-------------------------

The exchange feature vector :math:`\mathbf{X}^{\mathrm{x}}_\sigma` contains
the scale-invariant semilocal descriptors and the three version-j nonlocal
density features used by the CIDER23X nonlocal meta-GGA models
(:doc:`../features/sl` and :doc:`../features/nldf`).  CIDER26XC represents
the kinetic-energy information with the bounded indicator

.. math:: t = \frac{\tau-\tau_0}{\tau+\tau_0},

where :math:`\tau` is the kinetic-energy density and :math:`\tau_0` its
uniform electron gas value as defined in :doc:`../features/sl`.  The
CIDER23X models use the iso-orbital indicator :math:`\alpha` of
:footcite:t:`Sun2013` for this role.  :math:`\alpha` vanishes for
single-orbital densities, which is convenient for nonempirical functional
design, but :math:`t` proved better behaved numerically within the CIDER26XC
models and is bounded to :math:`[-1,1]`.  Combined with the reduced-gradient
descriptor, the two carry the same iso-orbital information.

The correlation feature vector :math:`\mathbf{X}^{\mathrm{c}}_\sigma` carries
one additional descriptor, the density itself, written :math:`X_6 = n` in the
CIDER26XC work.  In the serialized correlation feature list it follows the
semilocal descriptors and precedes the nonlocal density features.  The
scale-invariant features enforce the exchange uniform-coordinate-scaling
constraint.  The density descriptor lets the correlation model represent its
different uniform-coordinate-scaling behavior.

Before entering the Gaussian-process kernel, every feature is mapped onto a
finite interval by the bounded transforms stored in the serialized model
(implemented in
:mod:`ciderpress.dft.transform_data`; the :math:`t` indicator is handled by
the ``SLTMap`` transform).

Self-consistent training
------------------------

For an exchange-only target, exact exchange can be evaluated on a chosen
density matrix.  For a reaction-energy-trained full-XC functional, the density
is part of the prediction: once the model is placed in an SCF loop, orbital
relaxation changes both the features and the remaining Kohn--Sham energy
contributions.

CIDER26XC iterates the training density to account for orbital relaxation.  At
iteration :math:`k`, descriptors and residual labels are constructed from the
current densities, the model is refit, and self-consistent calculations with
that model generate the next densities.  The process is converged when
fixed-density and self-consistent errors agree at the required resolution.
:footcite:t:`CIDER26XC`

CiderPress provides the descriptor, model, kernel, and mapping primitives used
by this process.  Dataset construction and orchestration are supplied by the
surrounding training workflow; :doc:`../workflows/training` describes the
data expected by CiderPress.

Optional long-range correction
------------------------------

``CIDER26XCCHEMD4`` was trained with a D4 term removed from each target and
adds the same term when the model is evaluated.  D4 depends on the nuclear
geometry and does not enter the electronic SCF potential.  The PySCF interface
includes the correction exactly once in the total energy and adds its
analytical nuclear derivative to molecular gradients.

``CIDER26XCCHEM`` uses its learned electronic XC contribution.
``CIDER26XCCHEMD4`` combines that learned contribution with the D4 term
described above, which supplies an explicit long-range asymptote.

.. footbibliography::