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(May 16, 2021)

The Frenkel-Davydov Exciton model is an old idea for describing the (potentially delocalized) excited states of molecular crystals, or more generally noncovalent assemblies or aggregates of weakly electronically-coupled chromophores, as linear combinations of fragment-localized excited states. The $I$th collective excited state, $|{\mathrm{\Xi}}_{I}\u27e9$, is thus written

$$|{\mathrm{\Xi}}_{I}\u27e9=\sum _{n}^{\mathrm{sites}}\sum _{i}^{\mathrm{states}}{K}_{n}^{Ii}|{\mathrm{\Psi}}_{n}^{i}\u27e9\prod _{m\ne n}|{\mathrm{\Psi}}_{m}\u27e9,$$ | (12.67) |

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,
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the ground-state wave
functions in Eq. (12.67) 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:

$$|{\mathrm{\Psi}}_{A}^{\ast}\u27e9=\sum _{ia}{C}^{ia}|{\mathrm{\Phi}}_{A}^{ia}\u27e9.$$ | (12.68) |

The AIFDEM approach computes elements of the exact Hamiltonian,

$$\u27e8{\mathrm{\Psi}}_{A}^{\ast}{\mathrm{\Psi}}_{B}{\mathrm{\Psi}}_{C}\mathrm{\dots}|\widehat{H}|{\mathrm{\Psi}}_{A}{\mathrm{\Psi}}_{B}^{\ast}{\mathrm{\Psi}}_{C}\mathrm{\dots}\u27e9=\sum _{ia\sigma}\sum _{kb\tau}{C}_{\sigma}^{ia}{C}_{\tau}^{kb}\u27e8{\mathrm{\Phi}}_{A}^{ia}{\mathrm{\Phi}}_{B}{\mathrm{\Phi}}_{C}\mathrm{\dots}|\widehat{H}|{\mathrm{\Phi}}_{A}{\mathrm{\Phi}}_{B}^{kb}{\mathrm{\Phi}}_{C}\mathrm{\dots}\u27e9.$$ | (12.69) |

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

$$\u27e8{\mathrm{\Psi}}_{A}^{\ast}{\mathrm{\Psi}}_{B}{\mathrm{\Psi}}_{C}\mathrm{\dots}|{\mathrm{\Psi}}_{A}{\mathrm{\Psi}}_{B}^{\ast}{\mathrm{\Psi}}_{C}\mathrm{\dots}\u27e9=\sum _{ia\sigma}\sum _{kb\tau}{C}_{\sigma}^{ia}{C}_{\tau}^{kb}\u27e8{\mathrm{\Phi}}_{A}^{ia}{\mathrm{\Phi}}_{B}{\mathrm{\Phi}}_{C}\mathrm{\dots}|{\mathrm{\Phi}}_{A}{\mathrm{\Phi}}_{B}^{kb}{\mathrm{\Phi}}_{C}\mathrm{\dots}\u27e9$$ | (12.70) |

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. (12.69) and (12.70), 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
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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. Two additional types of states can be included
in the AIFDEM basis. These may be of the charge transfer-type,where the states are 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.
“Multi-exciton” states, of the type ${|}^{1}({\mathrm{\Phi}}_{A}^{\text{T}}{\mathrm{\Phi}}_{B}^{\text{T}}){\mathrm{\Phi}}_{C}\mathrm{\dots}\u27e9$,
are also easily included in the basis. These states couple triplet wave functions on two different monomers
to an overall singlet, and have been used to study the singlet fission process
in tetracene and pentacene.
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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. (12.69) and (12.70) 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
$x$.
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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

$${\mathbf{h}}^{IJ}={\mathbf{K}}_{I}^{\u2020}{\mathbf{H}}^{[x]}{\mathbf{K}}_{J}.$$ | (12.71) |

The Holstein ($A=B$) and Peierls ($A\ne B$) coupling constants ${g}_{AB\theta}$, expressed in dimensionless
normal mode coordinates, are
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$${g}_{AB\theta}={\left(2{\mu}_{\theta}{\omega}_{\theta}\right)}^{-1/2}\sum _{x}{\stackrel{~}{H}}_{AB}^{[x]}{L}_{x\theta}$$ | (12.72) |

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).
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A basic AIFDEM calculation is requested by setting AIFDEM =
TRUE in the *$rem* section, with additional job-control variables as described below.

AIFDEM

Perform an AIFDEM calculation.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

FALSE
Do not perform an AIFDEM calculation.
TRUE
Perform an AIFDEM calculation.

RECOMMENDATION:

False

AIFDEM_NTOTHRESH

Controls the number of NTOs that are retained in the
exciton-site basis states.

TYPE:

INTEGER

DEFAULT:

99

OPTIONS:

$n$
Threshold percentage of the norm of fragment NTO amplitudes.

RECOMMENDATION:

A threshold of $85\%$ gives a good trade-off of computational time and
accuracy for organic molecules.

AIFDEM_EMBED_RANGE

Specifies the size of the QM region for charge embedding

TYPE:

INTEGER

DEFAULT:

FULL_QM

OPTIONS:

FULL_QM
No charge embedding.
0
Treat only excited fragments with QM.
$n$
Range (in Å) from excited fragments within which to
treat other fragments with QM.

RECOMMENDATION:

The minimal threshold of zero Å typically
maintains accuracy while significantly reducing computational time.

AIFDEM_CTSTATES

Include charge-transfer-like cation/anion pair states
in the AIFDEM basis.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE
Include CT states.
FALSE
Do not include CT states.

RECOMMENDATION:

None

AIFDEM_SINGFIS

Include multiexciton states
in the AIFDEM basis.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE
Include multiexciton states.
FALSE
Do not include multiexciton states.

RECOMMENDATION:

None

For calculations on large systems, it may be desirable to break up the calculation of the AIFDEM matrix elements into batches. The user can specify can specify the first and last matrix element for a calculation with the following variables.

AIFDEM_SEGSTART

Indicates the index of the first matrix element to be computed.

TYPE:

INTEGER

DEFAULT:

NONE

OPTIONS:

$n$
First matrix element of chunk to be computed.

RECOMMENDATION:

Needs to be used with AIFDEM_SEGEND

AIFDEM_SEGEND

Indicates the index of the last matrix element to be computed.

TYPE:

INTEGER

DEFAULT:

NONE

OPTIONS:

$n$
Last matrix element of chunk to be computed.

RECOMMENDATION:

Needs to be used with AIFDEM_SEGSTART

When computing AIFDEM matrix elements in batches, one can avoid running fragment scf calculations for each job, by first running a job that runs only the fragment SCF calculations. These are saved in the scratch directory. These may then be read into subsequent calculations. The following variables control this utility.

AIFDEM_FRGM_WRITE

Fragment SCF calculations only.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE
Only fragment scf calculations are carried out, no computation of matrix elements.
FALSE
Regular AIFDEM calculation as specified by other REM variables.

RECOMMENDATION:

None

AIFDEM_FRGM_READ

Skips fragment scf calculations.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE
Skips fragment scf calculations, only computation of matrix elements.
FALSE
Regular AIFDEM calculation as specified by other REM variables.

RECOMMENDATION:

Requires a prior calculation that computes fragment scf data.

For charge-embedded AIFDEM calculations, set XPOL = TRUE in
the *$rem* section and then select the type of embedding charges via the
*$xpol* input section, as described in Section 12.12 and illustrated
in the following example.

$molecule 0 1 --H2O 0 0 1 O 1.74078 1.59716 -1.49814 H 2.22908 2.18316 -2.08914 H 0.88038 2.04726 -1.32684 --H2O 1 0 1 O 1.31998 -1.18934 -1.91734 H 1.49988 -0.22974 -1.89044 H 1.69058 -1.52594 -1.07704 --H2O 2 0 1 O -0.68982 2.59476 -0.72224 H -1.14372 3.37086 -1.07364 H -1.35592 1.84986 -0.78334 --H2O 3 0 1 O -1.27512 -1.77394 -1.69524 H -0.32252 -1.52884 -1.85604 H -1.53992 -2.30454 -2.45644 $end $rem BASIS aug-cc-pvdz EXCHANGE HF CIS_N_ROOTS 3 CIS_TRIPLETS FALSE XPOL TRUE AIFDEM TRUE AIFDEM_EMBED_RANGE 0 AIFDEM_NTOTHRESH 90 NTO_PAIRS 1 $end $xpol embed charges charges CHELPG $end

To compute AIFDEM derivatives ${\mathbf{H}}^{[x]}$ and ${\mathbf{S}}^{[x]}$ of the Hamiltonian and overlap matrices, the user should request a standard AIFDEM job and in addition set CIS_STATE_DERIV = 1. Currently, the AIFDEM derivatives do not support charge embedding so the keyword AIFDEM_EMBED_RANGE must be omitted from these jobs, which precludes the use of XPol wavefunctions for the fragments. Furthermore, only one excited state per fragment is supported so the keyword CIS_N_ROOTS = 1. is required.

The derivatives of the AIFDEM Hamiltonian matrix and overlap matrix are printed in the output file in sets of the three Cartesian coordinates that belong to a single atom. For convenience, the orthogonalized AIFDEM Hamiltonian matrix elements are saved in the scratch directory, $QCSCRATCH/aifdem_deriv. These are organized such that the derivatives for each unique matrix element are stored in individual files in the order of the atomic Cartesian coordinates. These files can facilitate external calculation of exciton/phonon coupling constants.

$molecule 0 1 --frgm 0 0 1 He 0.000 0.000 0.000 He 0.000 0.000 1.400 --frgm 1 0 1 He 0.000 0.000 2.800 He 0.000 0.000 4.200 $end $rem BASIS = cc-pvdz EXCHANGE = hf AIFDEM = true CIS_N_ROOTS = 1 CIS_SINGLETS = true CIS_TRIPLETS = false CIS_STATE_DERIV = 1 NTO_PAIRS = 1 MEM_TOTAL = 1000 MEM_STATIC = 1000 MAX_CIS_CYCLES = 200 MAX_SCF_CYCLES = 200 THRESH = 10 AIFDEM_NTOTHRESH = 100 $end