Going beyond the frozen core approximation: Development of coordinate-dependent pseudopotentials and application to ${\rm Na}_2^+$ Na 2+

Abstract

Mixed quantum/classical (MQC) simulations treat the majority of a system classically and reserve quantum mechanics only for a few degrees of freedom that actively participate in the chemical process(es) of interest. In MQC calculations, the quantum and classical degrees of freedom are coupled together using pseudopotentials. Although most pseudopotentials are developed empirically, there are methods for deriving pseudopotentials using the results of quantum chemistry calculations, which guarantee that the explicitly-treated valence electron wave functions remain orthogonal to the implicitly-treated core electron orbitals. Whether empirical or analytically derived in nature, to date all such pseudopotentials have been subject to the frozen core approximation (FCA) that ignores how changes in the nuclear coordinates alter the core orbitals, which in turn affects the wave function of the valence electrons. In this paper, we present a way to go beyond the FCA by developing pseudopotentials that respond to these changes. In other words, we show how to derive an analytic expression for a pseudopotential that is an explicit function of nuclear coordinates, thus accounting for the polarization effects experienced by atomic cores in different chemical environments. We then use this formalism to develop a coordinate-dependent pseudopotential for the bonding electron of the sodium dimer cation molecule and we show how the analytic representation of this potential can be used in one-electron MQC simulations that provide the accuracy of a fully quantum mechanical Hartree-Fock (HF) calculation at all internuclear separations. We also show that one-electron MQC simulations of Na(2)(+) using our coordinate-dependent pseudopotential provide a significant advantage in accuracy compared to frozen core potentials with no additional computational expense. This is because use of a frozen core potential produces a charge density for the bonding electron of Na(2)(+) that is too localized on the molecule, leading to significant overbinding of the valence electron. This means that FCA calculations are subject to inaccuracies of order ~10% in the calculated bond length and vibrational frequency of the molecule relative to a full HF calculation; these errors are fully corrected by using our coordinate-dependent pseudopotential. Overall, our findings indicate that even for molecules like Na(2)(+), which have a simple electronic structure that might be expected to be well-treated within the FCA, the importance of including the effects of the changing core molecular orbitals on the bonding electrons cannot be overlooked.

To Be or Not to Be in a Cavity: The Hydrated Electron Dilemma

“…This obvious sensitivity to the details of the construction of the pseudopotential make it abundantly clear that a more rigorous, dynamical approach to the construction of a water pseudopotential that responds to a changing environment may be necessary to shed light on the simulation of this system. 23 The other reason that the LGS non-cavity picture of the hydrated electron has not been well received lies in the fact that it initially appears counterintuitive. Because scientists studying solvated electrons have spent decades developing the idea that closed-shell molecules are intrinsically repelled by excess electrons, the idea of an electron that completely overlaps with these molecules seems to go against our instincts.…”

Erratum: “Going beyond the frozen core approximation: Development of coordinate-dependent pseudopotentials and application to ${\rm Na}_2^+ $ Na 2+” [J. Chem. Phys. 138, 054110 (2013)]

“…We then carried out the same procedure in our original work of fitting the numerically calculated ξ (r 1 , r 2 ; R) functions at a number of internuclear distances to an analytic function. 1…”

“…This function contains ten fitting parameters, b(R) − j (R), and Figure 2 shows a fit of ξ (r 1 , r 2 ; R) with the above function at a representative internuclear distance. As done previously, 1 we then interpolated the pseudopotential by fitting the fitting parameters of Eq. (1) to rational polynomials of various degrees in R. As in our original manuscript, the fitting parameters behave smoothly, yielding a continuous, nested expression for our coordinatedependent pseudopotential for Na + 2 .…”

Comment on “Going beyond the frozen core approximation: Development of coordinate-dependent pseudopotentials and application to Na2+” [J. Chem. Phys. 138, 054110 (2013)]

“…Phys. 138, 054110 (2013)] Hermann Stoll, 1,a) Patricio Fuentealba, 2,b) and László von Szentpály 1 The recent paper by Kahros and Schwartz, "Going beyond the frozen core approximation: Development of coordinate-dependent pseudopotentials and application to Na 2 + ," 1 contains important omissions and inconsistencies, therefore we feel obliged to draw attention to the following points:…”

“…However, problems might arise from using finite basis sets for the pointwise determination of the local pseudopotentials, as mentioned by KSch in Ref. 1. Additional problems may be caused by their use of non-norm-conserving PK pseudo-orbitals, whose valence densities are reduced with respect to the all-electron case.…”