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 the presence of a plasmonic substrate, where the latter is described using the polarizbale ωFQ or ωFQFμ models. [1] [2] [3] [4] [5] [6] The QM/ωFQ(Fμ) model has been applied, among others, to the calculation of 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 ω of the external perturbation.

Thus, 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, which originate surface plasmons that can greatly enhance the response of the molecule adsorbed on it. As in the standard QM/FQ model used for modeling solvation, in QM/ωFQ the target molecule is descrbied 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]

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

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

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

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

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 in case numerical integration points are accidentally close to MM atoms. The screened \(1/r_{ij}\) has the form:

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

where \(a\) can be changed with the QMSCREENFACTOR keyword.

Input options¶

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

The number of parameters that should be specified is greater, and for this reason we have setup several examples showing how to practically build a sample input and that already contain parameters for the most common substrates.