This document proposes a quantum model for describing charge transfer excitations and ionization processes using fluctuating charge densities. It summarizes issues with existing empirical charge models and introduces a quantum model that represents atoms as open systems coupled to a mean field bath. The model reproduces exact quantum behavior in non-interacting limits and can describe delocalized charges. It is shown to accurately model properties of ionization and charge transfer excitations using a single set of parameters for systems like LiF, benzene dimers, and cations. Future work will focus on more rigorous treatment of bath coupling and extensions to polyatomic systems.

1. Empirical potentials for charge transfer excitations
Jiahao Chen and Troy Van Voorhis, MIT Chemistry
Classical models for charge fluctuations Results: ground and CT excitation in LiF
Fluctuating-charge models can describe polarization and charge transfer effects in force fields.
Atomic energy and chemical potential variations with charge
The atomic Hamiltonian matrix elements can be populated from ab initio calculations or even from
tabulated values (Davidson, Hagstrom, Chakravorty, Umar and Fischer, 1991).
fluorine
weakly
noninteracting interacting
1. Have an empirical function for the energy of each isolated atom as a function of charge.
2. The atomic energies are added and coupled with Coulomb interactions. F
3. The total energy is minimized to obtain the charge distribution. (Electronegativity equilibration)
molecular geometry charge distribution
Problems with existing models Li
1. Unphysically symmetric donor→acceptor and acceptor→donor charge transfer excitations.
Energy
quadratic model
The quadratic approximation for the energy means
that both D→A and A→D excitations are symmetrically strongly interacting
DA
- +
distributed around the equilibrium charge distribution. As expected by construction, we correctly reproduce both weak and strong interaction limits in
each subsystem, especially the derivative discontinuity in the chemical potential in the weak limit.
Can fix this with higher order polynomial for the atomic
D+A- energy, but more parameters will be needed and the Ionic-covalent transitions in the S0 (1 1Σ+) and S1 (2 1Σ+) states of LiF
working equations become nonlinear. We fitted data from ab initio MCSCF calculations (Werner and Mayer, 1981) to a simple empirical
D A physical system
potential containing our model and a simple exponential wall.
δ+ δ-
charge transfer
2. Incorrect transition between strongly and weakly interacting limits
a) atom b) diatomic Fit parameters: s0 = 0.252 , Rs = 3.466 Å, Aex = 3232.94 eV, Rex = 0.174 Å
E E
weakly interacting weakly interacting
asymptotes
strongly q strongly ∆q
interacting interacting
Existing fluctuating-charge models employ the electronegativity equilibration assumption,
which implicitly assumes that all atoms are strongly interacting with each other regardless
of separation. This leads to problems with, e.g. incorrect size consistency of polarizabilities.
The agreement with the data is good, especially in the asymptotic limit. In particular, the position
We know from exact quantum mechanics (Perdew, Parr, Levy and Balduz, 1982) of the avoided crossing is correctly predicted (6.9 Å in model vs. 7.0 Å in data).
that a noninteracting quantum system has a piecewise linear variation of energy with
electron population (or equivalently, charge). This leads to the “derivative discontinuity”. Results: benzene dimer and cation
None of the available empirical charge models reproduce the weakly interacting limit correctly, Ab initio data on the benzene dimer and its cation (Pieniazek, Krylov and Bradforth, 2007) can be
although it is possible to mimic the noninteracting limit with explicit geometric dependence in fit simultaneously with our model, together with a Born-Mayer model for all other interactions.
the atomic energies (Chen and Martínez, 2007; 2008) or using discrete topological restrictions
on charge transfer (Chelli and Procacci, 1999).
3. Reference states Fit parameters: s0 = 0.284 , Rs = 3.156 Å, Aex = 2895.703 eV, Rex = 0.426 Å, C6 = 1229.351 Å6.eV
The parameterized atomic energy implicitly assumes a particular choice of atomic state. (Valone
and Atlas, 2006) Which atomic state is the correct choice? E.g. should a potential for sodium atoms
in solution be parameterized for neutral sodium or gas-phase sodium cation, or neither?
A quantum model for charge fluctuations
cation + 0.3 a.u.
cation
Quantum model for isolated atoms
Unlike atoms in molecules, the charge on isolated atoms is well-defined. A basis of charge
eigenstates can be defined as eigenstates of the charge operator neutral
neutral
where = atomic number - electron population
This basis allows explicit matrix representations of the Hamiltonian and observables, e.g.
The dimer cation surface is particularly interesting, as there is a regime where the charge is
completely delocalized across both benzene monomers. This is difficult to model in classical
models with single reference states. The equilibrium geometry shows spontaneous symmetry
breaking; however, this is not unique to our model: similar artifacts can occur in HF and DFT.
Also, we find excellent representation of properties of the ionization process relative to reference
To make an explicit connection between energy and charge, introduce atomic chemical potential μ ab initio data with quantitative accuracy.
which is the Legendre-conjugate quantity of charge. Then we can optimize the wavefunction by
variational optimization, which reduces to finding the lowest eigenvalue and eigenvector of
We can then use the wavefunction to find the corresponding charge
Note: this procedure obeys piecewise linearity as required in the exact noninteracting limit.
Quantum model for open atoms Summary & Outlook
We can treat a full canonical system with interacting atoms or fragments as a grand canonical Here is a quantum model for charge transfer processes on both ground and excited states.
statistical ensemble of open noninteracting atoms.
Our model reproduces the exact quantum mechanical behavior of noninteracting systems.
Using an empirical relation between bath coupling and molecular geometry, we decouple the full
interacting system into noninteracting, open subsystems.
Simple empirical potentials can be developed that can describe charge transfer excitations and
Our key approximation is to assume that open atoms can be described using the above formalism ionization process using a single set of parameters, both at the atomistic and fragment levels.
but with an additional Hamiltonian term coupling to an external “mean field” bath of electrons In particular, it can handle systems with delocalized charges which cannot be adequately
represented using classical fluctuating charge models.
We further assume the empirical forms:
Future work will focus on more rigorous studies of how the bath coupling varies with molecular
geometries, which will allow extensions to polyatomics (multiple components) with formal
which resembles the Wolfsberg-Helmholtz semiempirical coupling between s-type Gaussian justification from mean field theoretic arguments.
basis functions.
Procedure: equalize electronegativities in the presence of an external mean field Acknowledgments
Sponsored by the MIT Center for Excitonics, an Energy Frontier Research Center funded by the
subject to charge conservation US DOE, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0001088.