Machine learning the derivative discontinuity of density-functional theory

Machine learning is a powerful tool to design accurate, highly non-local, exchange-correlation functionals for density functional theory. So far, most of those machine learned functionals are trained for systems with an integer number of particles. As such, they are unable to reproduce some crucial and fundamental aspects, such as the explicit dependency of the functionals on the particle number or the infamous derivative discontinuity at integer particle numbers. Here we propose a solution to these problems by training a neural network as the universal functional of density-functional theory that (i) depends explicitly on the number of particles with a piece-wise linearity between the integer numbers and (ii) reproduces the derivative discontinuity of the exchange-correlation energy. This is achieved by using an ensemble formalism, a training set containing fractional densities, and an explicitly discontinuous formulation.

CYFFD Parameterization Method for Cylindrical Components of Aircrafts

This paper proposes a parameterization method using cylindrical coordinates based free-form deformation(CYFFD) technique by introducing a coordinate transformation method and a virtual lattice method. The method is suitable for axisymmetric and non-axisymmetric cylindrical applications. CYFFD is able to deform radially and circumferentially and to maintain first order and curvature continuity across frame border. First, the coordinate transformation step helps capture geometrical characteristics of cylindrical objects to conduct radial and circumferential deformation. Due to the need of delicate shape design, FFD lattice need be set up closely around cylinder-like objects and this will cause the boundary of FFD frame to intersect with the objects, which lead to derivative discontinuity at the intersection. The virtual lattice method is introduced to reuse some control points as virtual ones so that first order and curvature continuity can be preserved. A cylinder deformation example compares the capability of CYFFD with that of conventional FFD for radial and circumferential deformation and keeping derivative continuity. An airplane nose example shows the possibility to use CYFFD and NFFD together for complex shape. A nacelle deformation example and fitting example show that CYFFD is valuable for non-axisymmetric cylindrical objects with complex shapes. The optimization example on cylinder nose shape indicates that CYFFD can give good optimization results and it is valuable for parameterizing cylinder-like objects.

Scattering in quantum mechanics under quaternionic Dirac delta potential

In this paper, the Schrödinger equation for quaternionic quantum mechanics with a Dirac delta potential has been investigated. The derivative discontinuity condition for the quaternionic wave function has been derived and the boundary conditions for the quaternionic wave function have been applied. Probability current densities for different regions of the problem have been determined along with reflection and transmission coefficients.

The derivative discontinuity of the exchange–correlation functional

Manifestations of the derivative discontinuity of the energy in density functional theory are demonstrated in simple systems in chemistry and physics.