Existing fluctuating-charge models employ the electronegativity equilibration assumption,which implicitly assumes that all atoms are strongly interacting with each other regardlessof separation. This leads to problems with, e.g. incorrect size consistency of polarizabilities.

We know from exact quantum mechanics (Perdew, Parr, Levy and Balduz, 1982)that a noninteracting quantum system has a piecewise linear variation of energy withelectron population (or equivalently, charge). This leads to the “derivative discontinuity”.

None of the available empirical charge models reproduce the weakly interacting limit correctly,although it is possible to mimic the noninteracting limit with explicit geometric dependence inthe atomic energies (Chen and Martínez, 2007; 2008) or using discrete topological restrictionson charge transfer (Chelli and Procacci, 1999).

Empirical potentials for charge transfer excitationsJiahao Chen and Troy Van Voorhis, MIT Chemistry

Classical models for charge fluctuations

AcknowledgmentsSponsored by the MIT Center for Excitonics, an Energy Frontier Research Center funded by theUS DOE, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0001088.

2. Incorrect transition between strongly and weakly interacting limits

A quantum model for charge fluctuations

Problems with existing models

Results: ground and CT excitation in LiF

Results: benzene dimer and cation

Summary & Outlook

Fluctuating-charge models can describe polarization and charge transfer effects in force fields.

q

E

stronglyinteracting

∆q

E

stronglyinteracting

b) diatomica) atom

weakly interacting weakly interacting

physical system

quadratic model

Ener

gy

charge transfer

D-A+

Dδ+Aδ-

D+A-

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.3. The total energy is minimized to obtain the charge distribution. (Electronegativity equilibration)

molecular geometry charge distribution

1. Unphysically symmetric donor→acceptor and acceptor→donor charge transfer excitations.

The quadratic approximation for the energy meansthat both D→A and A→D excitations are symmetricallydistributed around the equilibrium charge distribution.

Can fix this with higher order polynomial for the atomicenergy, but more parameters will be needed and theworking equations become nonlinear.

weaklyinteractingnoninteracting

strongly interacting

F

Li

fluorine

Atomic energy and chemical potential variations with charge

Ionic-covalent transitions in the S0 (1 1Σ+) and S1 (2 1Σ+) states of LiF

The agreement with the data is good, especially in the asymptotic limit. In particular, the positionof the avoided crossing is correctly predicted (6.9 Å in model vs. 7.0 Å in data).

Quantum model for isolated atoms

Quantum model for open atoms

Unlike atoms in molecules, the charge on isolated atoms is well-defined. A basis of chargeeigenstates can be defined as eigenstates of the charge operator

where = atomic number - electron populationThis basis allows explicit matrix representations of the Hamiltonian and observables, e.g.

To make an explicit connection between energy and charge, introduce atomic chemical potential μwhich is the Legendre-conjugate quantity of charge. Then we can optimize the wavefunction byvariational 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.

Here is a quantum model for charge transfer processes on both ground and excited states.

Our model reproduces the exact quantum mechanical behavior of noninteracting systems.Using an empirical relation between bath coupling and molecular geometry, we decouple the fullinteracting system into noninteracting, open subsystems.

Simple empirical potentials can be developed that can describe charge transfer excitations andionization process using a single set of parameters, both at the atomistic and fragment levels.In particular, it can handle systems with delocalized charges which cannot be adequately represented using classical fluctuating charge models.

Future work will focus on more rigorous studies of how the bath coupling varies with moleculargeometries, which will allow extensions to polyatomics (multiple components) with formaljustification from mean field theoretic arguments.

The dimer cation surface is particularly interesting, as there is a regime where the charge iscompletely delocalized across both benzene monomers. This is difficult to model in classicalmodels with single reference states. The equilibrium geometry shows spontaneous symmetrybreaking; 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 referenceab initio data with quantitative accuracy.

As expected by construction, we correctly reproduce both weak and strong interaction limits ineach subsystem, especially the derivative discontinuity in the chemical potential in the weak limit.

The atomic Hamiltonian matrix elements can be populated from ab initio calculations or even fromtabulated values (Davidson, Hagstrom, Chakravorty, Umar and Fischer, 1991).

cation + 0.3 a.u.

cation

neutral neutral

Ab initio data on the benzene dimer and its cation (Pieniazek, Krylov and Bradforth, 2007) can befit simultaneously with our model, together with a Born-Mayer model for all other interactions.

Fit parameters: s0 = 0.284 , Rs = 3.156 Å, Aex = 2895.703 eV, Rex = 0.426 Å, C6 = 1229.351 Å6.eV

We fitted data from ab initio MCSCF calculations (Werner and Mayer, 1981) to a simple empiricalpotential containing our model and a simple exponential wall.

Fit parameters: s0 = 0.252 , Rs = 3.466 Å, Aex = 3232.94 eV, Rex = 0.174 Å

asymptotes

We can treat a full canonical system with interacting atoms or fragments as a grand canonicalstatistical ensemble of open noninteracting atoms.

Our key approximation is to assume that open atoms can be described using the above formalismbut with an additional Hamiltonian term coupling to an external “mean field” bath of electrons

We further assume the empirical forms:

which resembles the Wolfsberg-Helmholtz semiempirical coupling between s-type Gaussianbasis functions.Procedure: equalize electronegativities in the presence of an external mean field

3. Reference statesThe parameterized atomic energy implicitly assumes a particular choice of atomic state. (Valoneand Atlas, 2006) Which atomic state is the correct choice? E.g. should a potential for sodium atomsin solution be parameterized for neutral sodium or gas-phase sodium cation, or neither?

subject to charge conservation