QM/ωFQ: Quantum Mechanics/Frequency-Dependent Fluctuating Charges (and Fluctuating Dipoles)
*******************************************************************************************

The Quantum Mechanics/Frequency-Dependent Fluctuating Charges (QM/ωFQ)
and Frequency-Dependent Charges and Fluctuating Dipoles (QM/ωFQFμ)
methods are multiscale models designed to describe the properties of a
chemical system perturbed by a plasmonic substrate, where the latter
is described with the polarizable ωFQ or ωFQFμ models. [1] [2] [3] [4]
[5] [6] The QM/ωFQ(Fμ) model has been applied, among others, to
Surface-Enhanced Raman Scattering [2] and Surface-Enhanced
Fluorescence [7]. In these models, each atom is endowed with complex,
frequency-dependent electric variables: fluctuating charges in ωFQ,
and fluctuating charges and dipoles in ωFQFμ. Charge transfer between
atoms is governed by a classical Drude-like conduction mechanism,
while short-range quantum tunneling effects are effectively accounted
for through a distance-dependent damping function. The response is
evaluated at a given frequency \omega of the external perturbation.

These methods can be used, for example, to study the spectra of an
organic molecule adsorbed on a metal nanoparticle. The electromagnetic
field probing the spectroscopic response can resonate with the
nanoparticle and originate surface plasmons that greatly enhance the
response of the adsorbed molecule. As in the standard QM/FQ model used
for solvation, in QM/ωFQ the target molecule is described quantum-
mechanically (typically using DFT) while the substrate is described
classically.

The ωFQ charges, and the ωFQFμ charges and dipoles of the atoms in the
substrate are obtained by solving the following complex linear
systems. [1] [2] [3]

   \left[\overline{\mathbf{K}}\,\mathbf{T}^{qq} -
   z_q(\omega)\mathbf{I}_N\right]\mathbf{q}(\omega) =
   -\overline{\mathbf{K}}\mathbf{V}(\omega)

   \left[ \begin{pmatrix} \overline{\mathbf{K}} & \mathbf{0} \\
   \mathbf{0} & \mathbf{I}_{3N} \end{pmatrix} \begin{pmatrix}
   \mathbf{T}^{qq} & \mathbf{T}^{q\mu} \\ \mathbf{T}^{\mu q} &
   \mathbf{T}^{\mu\mu} \end{pmatrix} - \begin{pmatrix}
   z_q(\omega)\mathbf{I}_N & \mathbf{0} \\ \mathbf{0} &
   z_\mu(\omega)\mathbf{I}_{3N} \end{pmatrix} \right] \begin{pmatrix}
   \mathbf{q}(\omega) \\ \boldsymbol{\mu}(\omega) \end{pmatrix} =
   \begin{pmatrix} \overline{\mathbf{K}} & \mathbf{0} \\ \mathbf{0} &
   \mathbf{I}_{3N} \end{pmatrix} \begin{pmatrix} -\mathbf{V}(\omega)
   \\ \mathbf{E}(\omega) \end{pmatrix}

where \mathbf{q}(\omega) and \boldsymbol{\mu}(\omega) are the vectors
of complex, frequency-dependent fluctuating charges and dipoles,
respectively, and \omega is the frequency of the external
perturbation. \mathbf{V}(\omega) and \mathbf{E}(\omega) denote the
potential and electric field, respectively, originating from the
perturbed molecular density.

The matrices \mathbf{T}^{qq}, \mathbf{T}^{q\mu}, and
\mathbf{T}^{\mu\mu} describe charge–charge, charge–dipole, and
dipole–dipole interactions, respectively. Moreover, the matrix
\overline{\mathbf{K}} encodes atomistic charge transport and is
constructed from a Drude-like conduction model including short-range
tunneling corrections. The quantities z_q(\omega) and z_\mu(\omega)
introduce the frequency dependence of the intraband and interband
electronic response.

The coupling of the ωFQ/ωFQFμ models with a QM Hamiltonian can be done
by introducing the QM/ωFQ(Fμ) interaction operator as

   H_\text{QM/ωFQ} = \sum_i^{N} q_i(\omega) V[\rho](r_i),
   H_{\text{QM/ωFQF}\mu} = \sum_i^{N} q_i(\omega) V[\rho](r_i) -
   \sum_i^{N} \mu_i(\omega) E[\rho](r_i),

where N is the number of atoms, q_i and \mu_i are the i-th ωFQFμ
charge and dipole located at position r_i, respectively. V[\rho](r_i)
and E[\rho](r_i) are the electric potential and field generated by the
QM system on the same point.

Since the charges (and dipoles) depend on the QM density, explicit
terms also appear within response equations that are solved to
simulate spectroscopic and excited-state properties of the QM system.
Note that this is also true in the standard QM/FQ method for
solvation, but the difference here is that the interaction terms also
carry an imaginary part and are frequency-dependent.

A screening is included for the interaction between MM atoms and the
QM density to avoid unstable results if numerical integration points
are accidentally close to MM atoms. The screened 1/r_{ij} has the
form:

   \text{ERF: =}  & \frac{erf(ar_{ij})}{r_{ij}} \\ \text{EXP: =}  &
   \frac{(1-exp(-ar_{ij}))^2}{r_{ij}} \\ \text{GAUS: =} &
   \frac{erf(r_{ij}/a)}{r_{ij}}

where a can be changed with the QMSCREENFACTOR keyword.


Input options
=============

The input scheme for a QM/ωFQ(Fμ) calculation is identical to that for
a regular QM/FQ calculation and is explained in the corresponding
manual page.

The number of parameters that should be specified is greater. For this
reason we provide several examples that show how to build a sample
input and that already contain parameters for the most common
substrates.

[1] T.Giovannini et al. *Do We Really Need Quantum Mechanics to
    Describe Plasmonic Properties of Metal Nanostructures?* ACS
    Photonics 9 (2022) 3025

[2] P.Lafiosca et al. *QM/Classical Modeling of Surface Enhanced Raman
    Scattering Based on Atomistic Electromagnetic Models* JCTC 19
    (2023) 3616

[3] S.Sodomaco et al. *Computational Insights into the Adsorption of
    Ligands on Gold Nanosurfaces* JPC A 127 (2023) 10282

[4] T. Giovannini, A. Puglisi, M. Ambrosetti, C. Cappelli,
    *Polarizable QM/MM approach with fluctuating charges and
    fluctuating dipoles: the QM/FQFμ model*, Journal of Chemical
    Theory and Computation, 15, 2233-2245 (2019)

[5] 12. Nicoli et al. *Atomistic Multiscale Modeling of Colloidal
        Plasmonic Nanoparticles* ACS Phys. Chem. Au 4 (2024) 669

[6] S.Sodomaco et al. *Atomistic QM/Classical Modeling of Surface-
    Enhanced Infrared Absorption* JPC C 130 (2026) 1919

[7] P.Grobas Illobre et al. *Multiscale modeling of surface enhanced
    fluorescence* Nanoscale Adv. 6 (2024) 3410
