Born–Oppenheimer Interatomic Forces from Simple, Local Kinetic Energy Density Functionals

Abstract

Rapid calculation of Born-Oppenheimer (B-O) forces is essential for driving the so-called quantum region of a multi-scale molecular dynamics simulation. The success of density functional theory (DFT) with modern exchange-correlation approximations makes DFT an appealing choice for this role. But conventional Kohn-Sham DFT, even with various linear-scaling implementations, really is not fast enough to meet the challenge of complicated chemo-mechanical phenomena (e.g. stress-induced cracking in the presence of a solvent). Moreover, those schemes involve approximations that are difficult to check practically or to validate formally. A popular alternative, Car-Parrinello dynamics, does not guarantee motion on the B-O surface. Another approach, orbital-free DFT, is appealing but has proven difficult to implement because of the challenge of constructing reliable orbital-free (OF) approximations to the kinetic energy (KE) functional. To be maximally useful for multi-scale simulations, an OF-KE functional must be local (i.e. one-point). This requirement eliminates the two-point functionals designed to have proper linear-response behavior in the weakly inhomogeneous limit. In the face of these difficulties, we demonstrate that there is a way forward. By requiring only that the approximate functional deliver high-quality forces, by exploiting the "conjointness" hypothesis of Lee, Lee, and Parr, by enforcing a basic positivity constraint, and by parameterizing to a carefully selected, small set of molecules we are able to generate a KE functional that does a good job of describing various H q Si m O n clusters as well as CO (providing encouraging evidence of transferability). In addition to that positive result, we discuss several major negative results. First is definitive proof that the conjointness hypothesis is not correct, but nevertheless is useful. The second is the failure of a considerable variety of published KE functionals of the generalized gradient approximation type. Those functionals yield no minimum on the energy surface and give completely incorrect forces. In all cases, the problem can be traced to incorrect behavior of the functionals near the nuclei. Third, the seemingly obvious strategy of direct numerical fitting of OF-KE functional parameters to reproduce the energy surface of selected molecules is unsuccessful. The functionals that result are completely untransferable.

The energy-differences based exact criterion for testing approximations to the functional for the kinetic energy of non-interacting electrons

“…The exchangecorrelation energy functional is a minor component of the total energy, whereas the kinetic energy is a major one. Whereas even the simplest approximations to the exchange-correlation energy functional applied within the Kohn-Sham framework lead to a reasonable description of many important properties of atoms, molecules, and solids, known approximations to T s [ρ] used to approximate δT s [ρ] δρ in variational orbital-free calculations lead usually to qualitatively wrong results [9][10][11][12]. Although an acceptable approximation to δT s [ρ] δρ is the ultimate goal and an indispensable component of any successful orbital-free computational framework, related quantities are frequently subject of numerical comparisons instead.…”

Comment on “Single-point kinetic energy density functionals: A pointwise kinetic energy density analysis and numerical convergence investigation”

“…P92 20 and E00 21 tested by X-C) and our GGAs (VT84F, PBE2) is that the former do not predict binding in small molecules at all, while the latter do predict semi-quantitatively correct binding in those molecules and simple solids 6,17,18 . This categorical distinction and the broad utility of these GGAs is suggestive of having achieved a universal functional (though certainly not a proof).…”

“…GGA is generalized gradient approximation, in which G([n]; r) ≈ F θ (s). Then, among various informative tests, they explored the convergence behavior (with respect to plane-wave cutoff) of the mildly empirical GGA OFKE functional "PBE2" which came from our group several years ago 6 .…”

Semilocal Approximations for the Kinetic Energy

“…Three approximations (n = 2, 3, and 4) of the above general form were considered in Ref. 35. The coefficients C (n) i and a (n) are given in Table 16.1.…”