Empirical potentials for charge transfer excitations Jiahao Chen and Troy Van Voorhis, MIT ChemistryClassical 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 distributionProblems with existing models Li1. 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 transfer2. 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 interactingExisting 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. The agreement with the data is good, especially in the asymptotic limit. In particular, the positionWe 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 withelectron population (or equivalently, charge). This leads to the “derivative discontinuity”. Results: benzene dimer and cationNone 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 bealthough 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 restrictionson 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.eVThe 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?A quantum model for charge fluctuations cation + 0.3 a.u. cationQuantum 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 andOur 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 formalwhich 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.