A Born–Oppenheimer approximation for path integrals with an application to electron solvation in polarizable fluids

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Abstract
Path integral techniques are very useful for exploring electron and proton solvation and electron transfer in po- larizable molecular fluids. * The solvated electron or proton is found to sustain energy excitations which are small com- pared to the energy spacing in the molecular electronic energy levels. This means that the dipolar fluctuations in the molecules, the source of molecular polarizability, are rapid compared to the motions of the solvated electron with a concomitant separation in time scales. These sys- tems can thus be treated in the Born-Oppenheimer (BO) approximation. In this paper we introduce what to our knowledge is a new path integral method for treating the above problems which we call the free energy Born- Oppenheimer (FEBO) approximation. In a dynamical system with slow and fast degrees of freedom the Born-Oppenheimer approximation consists of solving for the energy eigenvalues and corresponding en- ergy eigenfunctions keeping the slow degrees of freedom fixed. This yields energy eigenvalues and eigenfunctions which are parametrically dependent on the slow degrees of freedom and are called adiabatic states and energy sur- faces. The slow degrees of freedom move on these adiabatic energy surfaces and one solves the Schriidinger equation for this motion. In the path integral formalism one then has to perform a separate path integral calculation for each of these adiabatic surfaces. This approach does not readily lend itself to Monte Carlo simulations or to analytical the- ories using path integrals. In this paper we derive an alter- native formulation which does not require separate calcu- lations for each potential energy surface. This formulation is shown to be as accurate as the full Born-Oppenheimer approximation. We apply it to the problem of electron sol- vation in polarizable fluids but we expect it to be quite useful in other applications. A useful model for describing polarizable systems is the Drude oscillator model."" The solvation of an excess electron in a fluid of Drude oscillators is due in large part to the many-body polarization energy. The usual approach is to calculate a pseudopotential describing the electron-