In order to investigate the local photochemical behaviour of molecular crystals, it is practical to use a multiscale method which partitions the system in an excited and a ground state region.
First, a molecule (or several) is selected from the unit cell to become the :term:`model system<Model system>`. It is placed in the middle of a large supercell from which a surrounding shell of molecules is selected, producing a cluster (the :term:`real system<Real system>`). A molecule is selected if any of its atoms fall within a chosen distance of the centroid of the model system.
With this partitioned cluster, we use the :term:`ONIOM` energy expression:cite:`Dapprich1999` to recover a multiscale energy:
E_{ONIOM} = E_{mh} + E_{rl} - E_{ml}
For straightforward :term:`mechanical embedding<Mechanical embedding>`, the terms E_{mh}, E_{rl} and E_{ml} are simply the energy of the :term:`model system<Model system>` at a high level of theory, the :term:`real system<Real system>` at a low level of theory, and the :term:`model system<Model system>` at a low level of theory.
However this scheme only includes intersystem interactions at the low (ground state) level of theory. To include intersystem Coulombic interactions in the excited state, we use point charges. This is called :term:`electrostatic embedding<Electrostatic embedding>` in the general :term:`ONIOM` literature and the :term:`Embedded Cluster (EC) model<EC>` in this implementation.
The :term:`mh` calculation is embedded in point charges located at the atomic sites of the surrounding cluster molecules. The value of these point charges is not uniquely defined and a discussion is offered in a different :ref:`section<pop>`. In order to avoid the double counting of electrostatic interactions, point charges must also be included in the :term:`ml` term.
Embedded cluster models have successfully been used by Ciofini's group in characterising excited states in molecular crystals.:cite:`Presti2014,Presti2016,Presti2016a`
The :term:`EC` method described above represents short range interactions to a reasonable extent. However the long range interactions, which are predominantly electrostatic, are completely omitted. In fact they cannot be approximated by increasing the size of the real system because the Madelung sum is conditionally convergent:cite:`Kittel1986`. To remedy this, we use an Ewald embedding scheme:cite:`Klintenberg2000,Derenzo2000` in :term:`mh` where a large array of point charges at lattice positions is generated and then fitted to match the Ewald potential. The combination of point charge Ewald embedding and :term:`EC` is the :term:`Ewald Embedded Cluster (EEC) method<EEC>`.
This method requires some justification. First of all, the long range electrostatic charges of the crystal are not cancelled in the :term:`ml` term. If we wished, we could embed :term:`rl` and :term:`ml` in Ewald fitted point charges. However when we perform geometry optimisation, the surrounding cluster is fixed in place. Therefore the additional computation of the Ewald point charges in the ground state Hamiltonians would only add a correcting constant term.
Another first-glance objection is that the :term:`mh` charges from the surrounding molecules which were included in the :term:`EC` model have potentially been modified to match the Ewald potential, thus rendering the cancellation of ground state interactions by the embedding of :term:`ml` inexact. However by definition the Ewald potential contains the totality of the Coulombic potential of the crystal, both short and long ranged. Furthermore a spherical region of the Ewald point charge array of a chosen radius can be chosen to remain of fixed charge, providing a 'buffer zone' from any highly deviated charges which might break the point charge approximation.
A major omission from the :term:`EC` and :term:`EEC` models is the electrostatic response to the excitation of the :term:`model system<Model system>` by the surrounding cluster. To recover mutual polarisation effects, we employ an extreme model where the entire crystal is excited at an electrostatic equilibrium. The model system is embedded in Ewald point charges as in :term:`EEC` at the optimised ground state position. A population analysis is carried out on the model system whose charges are then redistributed in the embedding supercell and gain fitted to the Ewald potential. This loop is repeated until self-consistency.
The method is adapted from the work of Wilbraham et al.:cite:`Wilbraham2016a,Presti2017` Self-consistent Ewald embedding schemes were previously used in the determination of NMR parameters:cite:`Weber2010`
This scheme is termed the Self-Consistent Ewald Embedded Cluster (:term:`SC-EEC`-S1). It accurately represents short range electrostatic interactions from a mutually polarising delocalised excitation. Alternatively, the self consistent loop can be performed in the ground state which would give similar results to :term:`EEC` (:term:`SC-EEC`-S0).