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. We know from exact quantum mechanics (Perdew, Parr, Levy and Balduz, 1982) that a noninteracting quantum system has a piecewise linear variation of energy with electron 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 in the atomic energies (Chen and Martínez, 2007; 2008) or using discrete topological restrictions on charge transfer (Chelli and Procacci, 1999). Empirical potentials for charge transfer excitations Jiahao Chen and Troy Van Voorhis, MIT Chemistry Classical models for charge fluctuations Acknowledgments Sponsored by the MIT Center for Excitonics, an Energy Frontier Research Center funded by the US 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 strongly interacting ∆q E strongly interacting b) diatomic a) atom weakly interacting weakly interacting physical system quadratic model Energy charge transfer D - A + D δ+ A δ- 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 means that both D→A and A→D excitations are symmetrically distributed around the equilibrium charge distribution. Can fix this with higher order polynomial for the atomic energy, but more parameters will be needed and the working equations become nonlinear. weakly interacting noninteracting strongly interacting F Li fluorine Atomic energy and chemical potential variations with charge Ionic-covalent transitions in the S 0 (1 1 Σ + ) and S 1 (2 1 Σ + ) states of LiF The agreement with the data is good, especially in the asymptotic limit. In particular, the position of 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 charge eigenstates can be defined as eigenstates of the charge operator where = atomic number - electron population This 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 μ 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. 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 full interacting system into noninteracting, open subsystems. Simple empirical potentials can be developed that can describe charge transfer excitations and ionization 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 molecular geometries, which will allow extensions to polyatomics (multiple components) with formal justification from mean field theoretic arguments. 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 ab initio data with quantitative accuracy. 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. The atomic Hamiltonian matrix elements can be populated from ab initio calculations or even from tabulated 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 be fit simultaneously with our model, together with a Born-Mayer model for all other interactions. Fit parameters: s 0 = 0.284 , R s = 3.156 Å, A ex = 2895.703 eV, R ex = 0.426 Å, C 6 = 1229.351 Å 6. eV We fitted data from ab initio MCSCF calculations (Werner and Mayer, 1981) to a simple empirical potential containing our model and a simple exponential wall. Fit parameters: s 0 = 0.252 , R s = 3.466 Å, A ex = 3232.94 eV, R ex = 0.174 Å asymptotes We can treat a full canonical system with interacting atoms or fragments as a grand canonical statistical ensemble of open noninteracting atoms. Our key approximation is to assume that open atoms can be described using the above formalism but 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 Gaussian basis functions. Procedure: equalize electronegativities in the presence of an external mean field 3. Reference states 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? subject to charge conservation