Density-functional theory for internal magnetic fields

A density-functional theory is developed based on the Maxwell-Schrödinger equation with an internal magnetic field in addition to the external electromagnetic potentials. The basic variables of this theory are the electron density and the total magnetic field, which can equivalently be represented as a physical current density. Hence, the theory can be regarded as a physical current density-functional theory and an alternative to the paramagnetic current density-functional theory due to Vignale and Rasolt. The energy functional has strong enough convexity properties to allow a formulation that generalizes Lieb's convex analysis formulation of standard density-functional theory. Several variational principles as well as a Hohenberg-Kohn-like mapping between potentials and ground-state densities follow from the underlying convex structure. Moreover, the energy functional can be regarded as the result of a standard approximation technique (Moreau-Yosida regularization) applied to the conventional Schrödinger ground-state energy, which imposes limits on the maximum curvature of the energy (with respect to the magnetic field) and enables construction of a (Fréchet) differentiable universal density functional.

Figure 1

Graphical representation of partial Legendre-Fenchel transforms, represented by double arrows, between functionals in the convex formulation of paramagnetic CDFT. Single arrows represent the change of variables and additive shift by the diamagnetic energy that relate the cycle of partial transforms to the standard energy functional and the Grayce-Harris functional. The designations “concave-gen.” and “convex-gen.” are used for functionals that are generic in their second argument, i.e., they do not have any apparent convexity properties.

Figure 2

Graphical representation of the relations between functionals in the convex formulation of MDFT. Double arrows represent Legendre-Fenchel transformations, while single arrows represent MY regularization and additive shifts of the functionals.

Figure 3

Functionals related to the model energy for an atom in the weak-field regime. MY regularization of the nonconvex $E(v,b)$ (black dotted curve) results in the paraconcave functional $E_{\mathrm{M}}(v,b)$ (blue dash-dot curve), which is nondifferentiable at $b=0$. Subtraction of the magnetic-field energy yields the concave functional ${\overline{E}}_{\mathrm{M}}(v,b)$ (solid blue curve). Addition of the magnetic-field energy to $E(v,b)$ yields the functional $e_{\mathrm{M}}(v,b)$ (red dashed curve), which remains nonconvex. Finally, a double convex conjugation yields the convex envelope ${\overline{e}}_{\mathrm{M}}(v,b)$ (solid red curve), which is constant on the interval between the minima of $e_{\mathrm{M}}(v,b)$.

Figure 4

On the left, the concave functional ${\overline{E}}_{\mathrm{M}}(v,B)$ is displayed together with all the supergradients at two selected points (marked by circles). There is an infinite family of supergradients (dashed lines) at the nondifferentiable point $B=0$. At the differentiable point $B=0.1$, the supergradient (dash-dotted line) is unique. On the right, the convex functional ${\overline{e}}_{\mathrm{M}}(v,{\beta }^{\prime })$ is displayed. The minimum is attained on a finite interval (black figure). The unique subgradient at ${\beta }^{\prime }=0.1$ is also displayed (dash-dotted line). The correspondence between the infinite family of subdifferentials at the maximum of ${\overline{E}}_{\mathrm{M}}$ and the unique subgradient at the minimum of ${\overline{e}}_{\mathrm{M}}$ is an example of the Hohenberg-Kohn-like mapping that exists in the formalism. Another example is the correspondence between the supergradient of ${\overline{E}}_{\mathrm{M}}(v,0.1)$ and the subgradient of ${\overline{e}}_{\mathrm{M}}(v,0.1)$.

Figure 5

Functionals related to the two-state model. MY regularization of the nonconvex $E(v,b)$ (black dotted curve) results in the paraconcave functional $E_{\mathrm{M}}(v,b)$ (blue dash-dotted curve), which is nondifferentiable at $b=0$. Subtraction of the magnetic-field energy yields the concave functional ${\overline{E}}_{\mathrm{M}}(v,b)$. Addition of the magnetic-field energy to $E(v,b)$ yields the functional $e_{\mathrm{M}}(v,b)$ (red solid curve), which remains nonconvex. Finally, a double convex conjugation yields the convex envelope ${\overline{e}}_{\mathrm{M}}(v,b)$ (red dotted curve), which is constant on the interval between the minima of $e_{\mathrm{M}}(v,b)$.

Figure 6

MY regularizations $M_{\nu }E(v,B)$ of the two-state model energy $E(v,B)$ (solid black curve). Approximations obtained with three different values $\nu =0.005,0.05,0.2$ of the regularization parameter are shown (blue dash-dotted curves). Smaller values yield closer approximations and $\nu =0.005$ is sufficiently conservative to preserve the finite, negative curvature around $B=0$.