A fairly old idea for describing the (potentially delocalized) excited states
of molecular crystals, or more generally noncovalent assemblies or aggregates
of (weakly) electronically-coupled chromophores, is the so called
*Frenkel-Davydov Exciton model*. In such a model, a collective excitation
of the entire aggregate is expressed as a linear combination of excitations
that are localized on molecular sites. The $I$th excited state,
$|{\mathrm{\Xi}}_{I}\u27e9$, is thus written

$$|{\mathrm{\Xi}}_{I}\u27e9=\sum _{n}^{\mathrm{sites}}\sum _{i}^{\mathrm{states}}{K}_{n}^{Ii}|{\mathrm{\Psi}}_{n}^{i}\u27e9\prod _{m\ne n}|{\mathrm{\Psi}}_{m}\u27e9,$$ | (12.64) |

where $|{\mathrm{\Psi}}_{n}^{i}\u27e9$ is the $i$th excited state of the $n$th molecular fragment and $|{\mathrm{\Psi}}_{m}\u27e9$ is the ground-state wave function of the $m$th fragment. Eigenstates and energies are found by constructing and diagonalizing the electronic Hamiltonian matrix in this direct product, “exciton-site basis”.

In the *ab initio* Frenkel-Davydov exciton model (AIFDEM) developed by
Morrison and Herbert,^{Morrison:2014, Morrison:2015} the ground-state wave
functions in Eq. (12.64) are single Slater determinants
(obtained from SCF calculations on isolated fragments), and the fragment
excited-state wave functions are linear combinations of singly-excited
determinants:

$$|{\mathrm{\Psi}}_{A}^{\ast}\u27e9=\sum _{ia}{C}^{ia}|{\mathrm{\Phi}}_{A}^{ia}\u27e9.$$ | (12.65) |

The AIFDEM approach computes elements of the exact Hamiltonian,

$$\u27e8{\mathrm{\Psi}}_{A}^{\ast}{\mathrm{\Psi}}_{B}{\mathrm{\Psi}}_{C}\mathrm{\dots}|\widehat{H}|{\mathrm{\Psi}}_{A}{\mathrm{\Psi}}_{B}^{\ast}{\mathrm{\Psi}}_{C}\mathrm{\dots}\u27e9=\sum _{ia\sigma}\sum _{kb\tau}{C}_{\sigma}^{ia}{C}_{\tau}^{kb}\u27e8{\mathrm{\Phi}}_{A}^{ia}{\mathrm{\Phi}}_{B}{\mathrm{\Phi}}_{C}\mathrm{\dots}|\widehat{H}|{\mathrm{\Phi}}_{A}{\mathrm{\Phi}}_{B}^{kb}{\mathrm{\Phi}}_{C}\mathrm{\dots}\u27e9.$$ | (12.66) |

In particular, no dipole-coupling approximation is made (as is often invoked in simple exciton models). Such an approximation may be valid for well-separated chromophores but likely less so for tightly-packed chromophores in a molecular crystal. Overlap matrices

$$\u27e8{\mathrm{\Psi}}_{A}^{\ast}{\mathrm{\Psi}}_{B}{\mathrm{\Psi}}_{C}\mathrm{\dots}|{\mathrm{\Psi}}_{A}{\mathrm{\Psi}}_{B}^{\ast}{\mathrm{\Psi}}_{C}\mathrm{\dots}\u27e9=\sum _{ia\sigma}\sum _{kb\tau}{C}_{\sigma}^{ia}{C}_{\tau}^{kb}\u27e8{\mathrm{\Phi}}_{A}^{ia}{\mathrm{\Phi}}_{B}{\mathrm{\Phi}}_{C}\mathrm{\dots}|{\mathrm{\Phi}}_{A}{\mathrm{\Phi}}_{B}^{kb}{\mathrm{\Phi}}_{C}\mathrm{\dots}\u27e9$$ | (12.67) |

are also required because molecular orbitals located on different fragments are not orthogonal to one another. In order to reduce the number of terms in Eqs. (12.66) and (12.67), the fragment excited states are transformed into the natural transition orbital (NTO) basis (see Section 7.14.2) and then the corresponding orbitals transformation (Section 10.15.2.2) is used to compute matrix elements between non-orthogonal Slater determinants. The size of the exciton-site basis is sufficiently small such that eigenvectors and energies of the exciton Hamiltonian can be printed and saved to scratch files. Transition dipole moments between the ground state and the first ten excited states of the exciton Hamiltonian are also computed.

The cost to compute each matrix element scales with the size of the supersystem
(somewhere between quadratic and quartic with monomer size), since all
fragments must be included in the direct products. To reduce this scaling, a
physically-motivated charge embedding scheme was introduced^{Morrison:2015}
that only treats the excited fragments, and neighbors within a user-specified
distance threshold, with full QM calculation, while the other ground state
fragment interactions are approximated by atomic point charges. In general,
inclusion of neighboring fragments in the QM part of the matrix element
evaluation does not seem to significantly improve the accuracy and diminishes
the cost savings of the charge-embedding procedure. Therefore, the minimal
“0 Å” threshold, where only the excited fragments are described at a QM
level, can be considered optimal. Charge embedding with the minimal threshold
affords an algorithm that scales as ${N}_{\text{F}}^{2}\times \mathcal{O}({n}_{\text{pair}}^{2-4})$, where ${N}_{\text{F}}$ is the number of
fragments and ${n}_{\text{pair}}$ is the size of a pair of fragments.

The exciton-site basis can be expanded to include higher-lying fragment excited
states which affords the wave function increased variational flexibility, and
can significantly improve the accuracy for polar systems and delocalized
excited states. The number of fragment excited states included in the basis is
specified by the CIS_N_ROOTS keyword, which must be $\ge 1$. A cost
effective means of including polarization effects is to use the XPol method to
compute fragment ground state, and the nature of the atomic point charges is
therefore controlled by keywords specified in the *$xpol* input section, as
described in Section 12.12. Fragment excited states are then
computed using the XPol-polarized MOs. Charge transfer-type states of the form
$|{\mathrm{\Phi}}_{A}^{+}{\mathrm{\Phi}}_{B}^{-}{\mathrm{\Phi}}_{C}\mathrm{\dots}\u27e9$, where ${\mathrm{\Phi}}_{A}^{\pm}$
are cationic or anionic determinants from unrestricted SCF calculations on the
isolated fragments, can also be included in the basis.

The exciton-site basis states are spin-adapted to form proper ${\widehat{S}}^{2}$ eigenstates. Their multiplicity determines that of the target excited state and this must be specified by setting CIS_SINGLETS or CIS_TRIPLETS to TRUE. The number of terms included in Eqs. (12.66) and (12.67) can be rationally truncated at some fraction of the norm of the fragment NTO amplitudes, in order to reduce cost at the expense of accuracy, although the approximation is controllable by means of the truncation threshold. Computation time scales approximately quadratically with the number of terms and a threshold of about 85% has been found to maintain acceptable accuracy for organic molecules with reasonable cost. The fragment orbitals and excited states may be computed with any SCF and single-excitation theory, including DFT and TDDFT, however the coupling matrix elements are always computed with a CIS-like Hamiltonian with no DFT exchange-correlation. Both OpenMP and MPI parallel implementations are available; the former distributes two-electron integral computation across cores in a node as in a traditional excited-state calculation, the latter can distribute matrix element evaluations across hundreds of cores with minimal overhead.

There are many chemical processes of interest where motion of nuclei induces
electronic transitions, in a breakdown of the Born-Oppenheimer approximation.
In order to investigate such processes it is useful to calculate some quantity
that codifies the coupling of adiabatic electronic states dues to nuclear
motion. In molecular electronic structure theory this quantity is the
nonadiabatic coupling (or “derivative coupling”) vector $\u27e8{\mathrm{\Psi}}_{I}|(\partial /\partial x)|{\mathrm{\Psi}}_{J}\u27e9$, which describes how the nuclear
position derivative $\partial /\partial x$ couples adiabatic (Born-Oppenheimer)
electronic states ${\mathrm{\Psi}}_{I}$ and ${\mathrm{\Psi}}_{J}$. In solid-state physics these ideas
are typically not discussed in terms of the Born-Oppenheimer approximation but
rather in terms of the so-called “Holstein” and “Peierls” exciton/phonon coupling constants (see below). Within the framework of the AIFDEM,
each of these quantities can be computed from a common intermediate
${\mathbf{H}}^{[x]}$, which is the derivative of the AIFDEM Hamiltonian matrix
with respect to some nuclear coordinate
$x$.^{Morrison:2017a, Morrison:2017b} Diagonal matrix elements ${H}_{AA}^{[x]}\equiv \partial {H}_{AA}/\partial x$ describe how the exciton-site energies are
modulated by nuclear motion and off-diagonal matrix elements ${H}_{AB}^{[x]}\equiv \partial {H}_{AB}/\partial x$ describe how nuclear motion modifies the
energy-transfer couplings.

Once ${\mathbf{H}}^{[x]}$ has been constructed, then the nonadiabatic coupling vector between eigenstates ${\mathrm{\Psi}}_{I}$ and ${\mathrm{\Psi}}_{J}$ of the exciton Hamiltonian is simply

$${\mathbf{h}}^{IJ}={\mathbf{K}}_{I}^{\u2020}{\mathbf{H}}^{[x]}{\mathbf{K}}_{J}.$$ | (12.68) |

The Holstein ($A=B$) and Peierls ($A\ne B$) coupling constants ${g}_{AB\theta}$, expressed in dimensionless
normal mode coordinates, are^{Morrison:2017b}

$${g}_{AB\theta}={\left(2{\mu}_{\theta}{\omega}_{\theta}\right)}^{-1/2}\sum _{x}{\stackrel{~}{H}}_{AB}^{[x]}{L}_{x\theta}$$ | (12.69) |

where the matrix $\mathbf{L}$ is the transformation between Cartesians
coordinates and normal (phonon) modes. The columns of $\mathbf{L}$ contain the
normalized Cartesian displacements of normal mode $\theta $, whose frequency and
effective mass are ${\omega}_{\theta}$ and ${\mu}_{\theta}$, respectively. The tilde
on, ${\stackrel{~}{H}}_{AB}^{[x]}$ indicates that the matrix element derivatives
have been orthogonalized (including the derivative of the orthogonalization
transformation).^{Morrison:2017b}