Abstract
We provide a mathematical justification of a spectral approximation scheme known as spectral binning for the Kohn–Sham spin density-functional theory in the presence of an external (nonuniform) magnetic field and a collinear exchange-correlation energy term. We use an extended density-only formulation for modeling the magnetic system. No current densities enter the description in this formulation, but the particle density is split into different spin components. By restricting the exchange-correlation energy functional to be of a collinear LSDA form, we prove a series of results which enable us to mathematically justify the spectral binning scheme using the method of Gamma-convergence, in conjunction with auxiliary steps involving recasting the electrostatic potentials, justifying the spectral approximation by making a spectral decomposition of the Hamiltonian and “linearizing" the latter Hamiltonian.
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1 Introduction
Among the various methods proposed to simplify the numerical simulation of solutions of the non-relativistic Schrödinger equation, the significance of density-functional theory (DFT) in condensed-matter physics and quantum chemistry cannot be overstated [2, 12, 15, 33]. Its theoretical foundations are based upon the seminal work of Hohenberg and Kohn (HK) [13] wherein the HK Theorem was established, stating that the particle density, \(\rho \), determines the external potential up to a constant, and an HK variational principle presented. This led to replacement of the many-body Schrödinger problem with a minimization problem involving a universal density functional. Further advances were made in Lieb’s work [20] which developed a more mathematically tractable formulation based on convex analysis. According to the HK theorem, \(\rho \) can be treated as the fundamental unknown in a many-electron system without magnetic field. Since \(\rho \) only depends on three spatial coordinates, numerical simulations of large molecules and even solids became possible when Kohn and Sham [17] presented an implementation of DFT based upon the idea of replacing the physical system of interacting electrons with a fictitious system (the Kohn–Sham system) of non-interacting electrons, having the same ground state electron density as the physical system. In the Kohn–Sham system, an electron moves in an effective potential consisting of three terms: the external potential induced by the attraction between the electron and the nucleus, the Hartree potential which describes the repulsion between electron and electron, and the exchange-correlation (abbreviated xc) potential which is generated by certain non-classical Coulomb interactions. The latter is not known analytically and, therefore, approximate methods are needed. As part of their work, Kohn and Sham introduced a local density approximation (LDA) for the xc potential. Since then, many other approximations have been proposed such as the generalized gradient approximation (GGA), and the meta-generalized gradient approximation (meta-GGA) (see, e.g., [24]). By solving numerically the Kohn–Sham equation, we can determine the ground state electron density of the original physical system and, via the ground state, other interesting quantities.
Understanding magnetic properties is absolutely crucial in modern material science (atomistic spin-dynamics, complex oxides, saturation magnetization etc.). A magnetic field has two effects on a system of electrons: (i) it tends to align their spins, and (ii) it alters their translational motion. The first effect appears when one adds a term of the form \(-e\hbar m^{-1} \varvec{s} \cdot {\varvec{B}}\) to the Hamiltonian, while the second, diamagnetic effect arises from the usual kinetic energy \((2m)^{-1} | {{\textbf{p}}}|^{2}\) being replaced by \((2m)^{-1} | {{\textbf{p}}}-(e/c) {{\mathcal {A}}}|^{2}\). Here \({{\textbf{p}}}\) is the momentum operator, \({{\mathcal {A}}}\) is the vector potential, \({\varvec{B}}=\nabla \times {{\mathcal {A}}}\) is the magnetic field associated with \({{\mathcal {A}}}\), and \(\varvec{s}\) is the angular momentum vector. Within the numerical practice, one approach is to apply a perturbation method to compute the variations of the characteristic parameters of, say, a molecule, with respect to the outside perturbation. It is interesting to go beyond and consider the full minimization problem of the perturbed energy. In Hartree-Fock theory one typically only takes into account the effect (ii), whereas in non-relativistic DFT it is common to include the spin-dependent term and to ignore (ii) and to study the minimization of the resulting nonlinear functional, which depends upon two densities, one for spin “up” electrons and the other for spin “down” electrons. Each density satisfies a normalisation constraint which can be interpreted as the total number of spin “up” or “down” electrons. The resulting spin-polarized density-functional theory (SDFT) was first developed by von Barth and Hedin in a very general setting [32] (see also [14, 21]). Magnetism of itinerant electrons in solid state materials can be described well by SDFT. This powerful tool underpins quantitative theoretical determination of spin magnetic moments, and it enables one to understand the mechanisms, which lead to the occurrence of magnetism in solid state materials [16, 36, 37].
Most SDFT applications use a restricted version, where local magnetization is constrained along a fixed direction, so-called collinear spin-polarized DFT. In systems with a collinear magnetization (and in which spin-orbit coupling is neglected), spin-up and spin-down states are orthogonal, and hence the density matrix is diagonal. This results in two Kohn–Sham equations, with one Kohn–Sham equation for the majority (\(\uparrow \)) spin channel, and one for the minority (\(\downarrow \)) spin channel. The majority and minority spin channels are connected only through the spin dependence of the exchange-correlation part of the effective one-particle potential. Since the two KS equations can be solved independently, the computational effort for a collinear calculation seems to be just twice the effort for a non-magnetic calculation. However, most magnetic calculations are computationally considerably more demanding since the quantities in question (magnetic moments, energy differences between various magnetic configurations) require much higher accuracy than what is needed for nonmagnetic systems. Systems that can be described by SDFT are all kinds of magnetic materials that assume a collinear magnetic order, e.g. ferromagnetic, antiferromagnetic or ferrimagnetic states [3, 8].
We study a new hybrid spin-density Kohn–Sham DFT in the presence of an external (nonuniform) magnetic field \({\varvec{B}}\) for which the energy minimization problem is given by
where (the reader finds the definitions of \({\text {Tr}}\,( \cdot )\), \({\text {tr}}\,(\cdot )\) and \({{\mathcal {J}}}(\cdot )\) in Sect. 2) \({\varvec{\gamma }}\) is the \(2 \times 2\) spin density matrix, V is the electron-nuclei potential,
Moreover, the collinear, local-spin density approximation (LSDA) introduced by von Barth and Hedin [32] consists in writing
where \(\rho ^{\alpha \alpha }\), \(\alpha =\uparrow ,\downarrow \), are the two eigenvalues of the \(2 \times 2\) density matrix, and \(E_{\textrm{xc}}^\textrm{LDA}\) is the standard exchange-correlation functional in the non-polarized case, which we can write as [17]
The conditions on the function h is given in Assumption 2.1. We justify a spectral approximation scheme, known as spectral binning, developed by Wang et al. [35] for the standard KSLDA model, for the collinear Local Spin Density Approximation (LSDA) Kohn–Sham DFT in the presence of an external magnetic field, even allowing a nonuniform magnetic field. In order to transform the problem into one that can be solved numerically, we discretize the system. In the case of the spatial components, the kth level of discretization will be achieved by using the first k basis functions of the space. For the spectral components on the other hand, we will be using a different type of discretization known as spectral binning. The idea is that at the kth level of discretization, we make a step function with k intervals, and each interval will count the number of eigenvalues in that interval, or “bin”, which is where the name comes from. Since the number of eigenvalues is the same, we ensure that our discretized function remains in the minimization space. To give a mathematical justification of our hybrid, Kohn–Sham collinear LSDA DFT (1.1), we will show that as we increase the levels of iterations, the minimizers of the discretized equations will have Gamma-convergence [6]. This is desirable because the idea of \(\Gamma \)-convergence is that “the limit of the minimizers is the minimizer of the limit”. In other words, suppose we have a sequence of functionals, and each functional has it own minimizing function. If the functionals have \(\Gamma \)-convergence, what we say is that the functionals converge to a limit functional in the sense that minimizing functions will converge to a limit function, and furthermore this limit function will minimize the limit functional. In future, one can numerically search for the rates of convergence. We show convergence with the \(\Gamma \)-convergence method: we first show lower semi-continuity, and then show the existence of a recovery sequence. However, all of the above precludes that the non-discretized minimizing function can be given a spectral approximation. Therefore we must first prove this. If the Hamiltonian under consideration was linear, from spectral theory we would know that a spectral decomposition for the minimizer exists, and therefore we can build a spectral binning function. Unfortunately the Hamiltonian we use is not linear with respect to \({\varvec{\gamma }}\). But what we can do is “linearize” the system with respect to \({\varvec{\gamma }}\) by introducing dummy variables (which we either minimize or maximize over). The idea is that whatever is on the inside of the minimization/maximization will be linear with respect to \({\varvec{\gamma }}\). If we can furthermore show that we can bring the \({\varvec{\gamma }}\) minimization inside the dummy variable variations, we will have a new minimization problem on the inside which will be linear with respect to \({\varvec{\gamma }}\), and therefore open to being approximated by a spectral approximation. The approximation scheme and methodology is described in full in Sect. 3 after a summary of DFT in magnetic fields leading to the model studied herein. The specifics of the \(\Gamma \)-convergence method summarized above is described in Sect. 4, wherein we also state our main result, Theorem 4.1. All proofs are provided in Sect. 5.
Wang et al. [35] shows that the solution of the Kohn–Sham equations within the KSLDA theory (non-magnetic case) can be approximated by its spectral representation, and as one adds more terms to the spectral representation, it \(\Gamma \)-converges to the true solution. This is very useful when one does numerical calculations, such as in [25] where the authors apply this method to studying crystaline structures. A key idea in [35] is that, by reformulating the Kohn–Sham equations into a linear form, one can show that any ground state solution is in fact a Borel function, thus restricting the space of admissible minimizing functions. The first few spectral binning benchmark problem examples exhibit excellent accuracy and convergence characteristics [34]. For the numerical scheme developed for the hybrid model herein, it remains to be seen how well it captures magnetic properties, where the direction of the local magnetization is not constrained to a particular axis but can vary over the space [22] (this type of behaviour is known and has been observed in, for example, bulk \(\gamma \)-Fe, geometrically spin-frustrated lattices as jarosites, halogen salts of erbium, and surfaces; see [4] and references therein), or spin-orbit coupling is included [26].
2 Density functional theory in magnetic fields
2.1 Quantum mechanics
In accordance with the Schrödinger-Pauli equation, the electronic Hamiltonian for a molecular system consisting of N electrons and K nuclei (N and K are fixed throughout this paper) is given by
where \({{\mathcal {A}}}: {{\mathbb {R}}}^{3} \rightarrow {{\mathbb {R}}}^{3}\) is the magnetic vector potential (recall that \({\text {curl}}\,{{\mathcal {A}}}={\varvec{B}}\) is the magnetic field), \({\varvec{\sigma }}_{n}\) contains the Pauli matrices acting on the \(n^\textrm{th}\) spin, viz.
and \(V({\varvec{r}}) = - \sum _{k=1}^{K} Z_{k} / | {\varvec{r}} -{\varvec{R}}_{k} |\), where \({\varvec{R}}_{k}\) denotes the location of the kth nucleus. The N-particle Hilbert space is \(\varvec{{{\mathcal {L}}}}^{2} = \varvec{{{\mathcal {L}}}}_{N}^{2} = \bigotimes _{n=1}^{N} L^{2}({{\mathbb {R}}}^{3} \times \mathbb {C}^{2}) = \bigotimes _{n=1}^{N} L^{2}({{\mathbb {R}}}^{3}; \mathbb {C}^{2})\) and the antisymmetric analogue is \(\varvec{{{\mathcal {L}}}}_\textrm{a}^{2}:= \varvec{{{\mathcal {L}}}}_{\textrm{a},N}^{2}:= \bigwedge _{n=1}^{N} L^{2}({{\mathbb {R}}}^{3}; \mathbb {C}^{2})\). Likewise, for Sobolev spaces \(\varvec{{{\mathcal {H}}}}^{t}:=\varvec{{{\mathcal {H}}}}_{N}^{t}:= H^{t}({{\mathbb {R}}}^{3N} \times (\mathbb {C}^{2})^{N})\) and its antisymmetric analogue \(\varvec{{{\mathcal {H}}}}_\textrm{a}^{t}:= \varvec{{{\mathcal {H}}}}_{\textrm{a}, N}^{t}:= \varvec{{{\mathcal {H}}}}^{t} \cap \varvec{{{\mathcal {L}}}}_\textrm{a}^{2}\). We will only need \(t=1,2\) below. The operator \(H(V,{{\mathcal {A}}})\) is linear and its form domain is \(\varvec{{{\mathcal {H}}}}_\textrm{a}^{1}\) and any \(\Psi \in \varvec{{{\mathcal {H}}}}_\textrm{a}^{1}\) has \(2^N\) components and it satisfies
The one-particle Hilbert space is \({{\mathcal {H}}}=L^2({{\mathbb {R}}}^3;\mathbb {C}^2):=\left\{ \, {\varvec{\phi }}=(\phi ^{\uparrow },\phi ^{\downarrow }) \in (L^2({{\mathbb {R}}}^3))^{2} \,: \,\right. \left. \Vert {\varvec{\phi }} \Vert _{{{\mathcal {H}}}}<\infty \,\right\} \) with inner product
The set of N-particle density matrices describing the pure states is
and the set of N-particle density matrices describing the mixed states, denoted by \({{\mathcal {M}}}_N\), is the convex hull \(\textrm{CH} ({{\mathcal {P}}}_{N})\) of \({{\mathcal {P}}}_N\). Our main object of interest is the ground state energy defined as
or, equivalently, \(E_\textrm{QM}(V,{{\mathcal {A}}})=\min _{\Gamma \in {{\mathcal {P}}}_N} {\text {Tr}}\,\big (H(V, {{\mathcal {A}}})\Gamma \big )\). Since \(H(V, {{\mathcal {A}}})\) is linear and since \({{\mathcal {M}}}_N\) is the convex hull of \({{\mathcal {P}}}_N\), we have \(E_\textrm{QM}(V, {{\mathcal {A}}})=\min _{\Gamma \in {{\mathcal {M}}}_N} {\text {Tr}}\,(H(V, {{\mathcal {A}}})\Gamma )\). A calculation yields
where the (total) electronic density is given by
\({\varvec{j}}_{\rho }\) is the paramagnetic current [9, Sect. 7.4] and \({\varvec{m}}_{\Gamma }\) is the spin density [32].
2.2 Spin density functional theory (SDFT)
As explained in the introduction, a magnetic field has two effects on a system of electrons and we usually study them separately under the following names:
-
Spin density functional theory (SDTF) for spin effects: \({{\mathcal {A}}}=0\) and \({\varvec{B}}\ne 0\).
-
Current DFT (CDFT) for orbital effects: \({{\mathcal {A}}}\ne 0\) and \({\varvec{B}}=0\).
In this paper we study the former which arises from the Pauli operator
acting in \({\varvec{{{\mathcal {L}}}}}_\textrm{a}^{2}\); hence no current densities enter but the particle density will be split into different spin components. (For CDFT and other density-variables in DFT with magnetic fields, we refer to [18, 23, 30]). We are thus interested in the SDFT (\({{\mathcal {A}}}=0\), \({\varvec{B}} \ne 0\)) energy expressed by
For \(\Gamma \in {{\mathcal {M}}}_{N}\) with Schwarz kernel \(\Gamma ( {{\textbf{r}}}_{1}, s_{1}, \ldots , {{\textbf{r}}}_{N}, s_{N}; {{\textbf{r}}}^{\prime }, s_{1}^{\prime }, \ldots , {{\textbf{r}}}_{N}^{\prime }, s_{N}^{\prime } )\) we introduce the matrix of spin-polarized electronic densities (the spin density matrix)
where, for \(\alpha , \beta \in \{ \uparrow ,\downarrow \}\), we let
Then the spin angular momentum density \({\varvec{m}}_{\Gamma }\), is given by \( {\varvec{m}}_{\Gamma } = ( \rho _{\Gamma }^{\uparrow \downarrow }+ \rho _{\Gamma }^{\downarrow \uparrow }, i \rho _{\Gamma }^{\downarrow \uparrow }- i\rho _{\Gamma }^{\uparrow \downarrow }, \rho _{\Gamma }^{\uparrow \uparrow }- \rho _{\Gamma }^{\downarrow \downarrow } )^\textrm{T}\). With \({\varvec{B}}=({\varvec{B}}_x,{\varvec{B}}_y,{\varvec{B}}_z)\), we have
We denote the \(2 \times 2\) matrix trace by \({\text {tr}}\,_{\mathbb {C}^2}\). Since \({\varvec{m}}_{\Gamma } = {\text {tr}}\,_{\mathbb {C}^2} \left( \varvec{\sigma } \cdot {\varvec{R}}_{\Gamma } \right) \), and \(\rho _{\Gamma } = \rho _{\Gamma }^{\uparrow \uparrow } + \rho _{\Gamma }^{\downarrow \downarrow }\) is the total electronic density, we have that \({\text {tr}}\,_{\mathbb {C}^2} \left( {\varvec{U}} {\varvec{R}}_{\Gamma } \right) = V \rho _{\Gamma } - \mu {\varvec{B}} \cdot {\varvec{m}}_{\Gamma }\), where \({\varvec{U}}\) is given in (1.2). Then we can recast Eq. (2.4) as \({\text {Tr}}\,\big (H(V,{\varvec{B}})\Gamma \big )={\text {Tr}}\,(H (0,{\varvec{0}}) \Gamma )+\int _{{{\mathbb {R}}}^{3}} {\text {tr}}\,_{\mathbb {C}^2}\left[ {\varvec{U}} {\varvec{R}}_{\Gamma }\right] \, d{{\textbf{r}}}\). When one puts \({\varvec{B}} = \varvec{0}\), one obtains the usual potential energy density V multiplied by the electron density \(\rho _{\Gamma }\), which appears in spin-unpolarized DFT. Similar to standard DFT we apply the constrained search method (see the second equality below), attributed to Levy [19], Valone [31], and Lieb [20], yielding
with \( F({\varvec{R}}_{\Gamma }): = \inf _{\Gamma \in {{\mathcal {M}}}_N,\Gamma \rightarrow {\varvec{R}}_{\Gamma } } {\text {Tr}}\,[H(0,0)\Gamma ]\) and the admissible set of spin density matrices given by \({{\mathcal {J}}}_N ({{\mathcal {M}}}_N):=\left\{ \, {\varvec{R}}_{\Gamma }\;: \; \Gamma \in {{\mathcal {M}}}_N\,\right\} \). Note that (2.5) is the minimum of a functional which depends only on the \(2 \times 2\) spin-density matrix \({\varvec{R}}_{\Gamma }\); the latter explains the name SDFT. In comparison with the linear problem (2.2), which suffers from the curse of dimensionality,Footnote 1 the minimization problem (2.5) is formulated on a space of lower dimension (which is good) but, on the other hand, the problem has become nonlinear. We know that \({{\mathcal {J}}}_N({{\mathcal {M}}}_N)\) consists of all \({\varvec{R}}_{\Gamma } \in {{\mathcal {M}}}_{2\times 2}\big (L^1({{\mathbb {R}}}^3)\big )\) for which \(\varvec{R}_{\Gamma }\) is Hermitian, nonnegative, \(\int _{{{\mathbb {R}}}^3} {\text {tr}}\,_{\mathbb {C}^2}[ {\varvec{R}}_{\Gamma } ] \, d{{\textbf{r}}}=N\), and \(\sqrt{{\varvec{R}}_{\Gamma }} \in {{\mathcal {M}}}_{2\times 2}\) with \(\sqrt{\cdot }\) meant in the Hermitian sense [10]. One of the challenges of DFT is that the functional F cannot be easily evaluated. Kohn and Sham [17] made the problem practically accessible by approximating F by studying a system of non-interacting electrons. Following their approach, we introduce, for \(\Gamma \in {{\mathcal {M}}}_N\), the one-particle density matrix
where, for \(\alpha ,\beta \in \left\{ \uparrow ,\downarrow \right\} ^2\),
Then \({\varvec{R}}_{\Gamma }({{\textbf{r}}})={\varvec{\gamma }}_{\Gamma }({{\textbf{r}}},{{\textbf{r}}})\) and the set of mixed state one-particle density matrices is \({\varvec{{{\mathcal {C}}}}}_N:= \{ {\varvec{\gamma }}_{\Gamma }: \quad \Gamma \in {{\mathcal {M}}}_N \}\), and, identifying the kernel \({\varvec{\gamma }}({{\textbf{r}}}, {{\textbf{r}}}')\) with the corresponding operator of \({\mathcal {S}}(L^2({{\mathbb {R}}}^3; \mathbb {C}^2))\), the space of self-adjoint, bounded operators on \(L^2({{\mathbb {R}}}^3; \mathbb {C}^2)\), Coleman [5] proved that
where \({\text {Tr}}\,( - \Delta {\varvec{\gamma }} ):= {\text {Tr}}\,( - \Delta \gamma ^{\uparrow \uparrow } ) + {\text {Tr}}\,( - \Delta \gamma ^{\downarrow \downarrow } )\). In a similar way, we can define, for \(\lambda > 0\),
By invoking the spectral theory for compact self-adjoint operators, we can write the components \(\gamma ^{\alpha \beta }\) of any \({\varvec{\gamma }} \in {\varvec{{{\mathcal {C}}}}}_{\lambda }\) in the form
Since \({\varvec{\gamma }}_{\Gamma } ( {{\textbf{r}}}, {{\textbf{r}}}) = {\varvec{R}}_{\Gamma }( {{\textbf{r}}})\), we write \( {\varvec{R}}_{{\varvec{\gamma }}} ({{\textbf{r}}}):= {\varvec{\gamma }}({{\textbf{r}}}, {{\textbf{r}}})\) for \({\varvec{\gamma }} \in {\varvec{{{\mathcal {C}}}}}_{N}\). We finally introduce \({\varvec{{{\mathcal {D}}}}}_\lambda := \left\{ {\varvec{R}} \in {{\mathcal {M}}}_{2 \times 2}(L^1({{\mathbb {R}}}^3)): \quad \exists {\varvec{\gamma }} \in {\varvec{{{\mathcal {C}}}}}_\lambda \text{ s.t. } {\varvec{R}} = {\varvec{R}}_{{\varvec{\gamma }}} \right\} \).
2.2.1 Extended Kohn–Sham approach
Following Kohn and Sham [17] we decompose \(F({\varvec{R}})\) as \(F({\varvec{R}})=T_\textrm{KS}({\varvec{R}})+{{\mathcal {J}}}(\rho _{{\varvec{R}}})+{{\mathcal {E}}}_\textrm{xc}({\varvec{R}})\) for all \({\varvec{R}} \in {\varvec{{{\mathcal {D}}}}}_{\lambda }\), where \(T_\textrm{KS}({\varvec{R}})\) is the Kohn–Sham kinetic energy of a non-interacting system given by
The second term \(J(\rho _{{\varvec{R}}})\) is the direct Coulomb energy given by
and the third term \({{\mathcal {E}}}_\textrm{xc}({\varvec{R}})\) is the exchange-correlation energy given by \({{\mathcal {E}}}_\textrm{xc}({\varvec{R}}):=F({\varvec{R}})- T_\textrm{KS}({\varvec{R}}) - J({\varvec{R}})\) for all \( {\varvec{R}} \in {\varvec{{{\mathcal {D}}}}}_{\lambda }\).
2.2.2 Local spin density approximation
Next we address the question on how to choose \({{\mathcal {E}}}_\textrm{xc}({\varvec{R}})\). The local-spin density approximation (in short, LSDA) introduced by von Barth and Hedin [32] consists in the approximation
where \({{\mathcal {E}}}_{\textrm{xc}}^\textrm{LSDA}(\rho ^+, \rho ^-)\) is given in (1.3), \(\rho ^{+}\) and \(\rho ^-\) are the two eigenvalues of the \(2 \times 2\) matrix \({\varvec{R}}\), and \({{\mathcal {E}}}_{\textrm{xc}}^\textrm{LDA}\) is the standard LDA of the exchange-correlation functional in the non-polarized case [17], which takes the form (1.4), where the conditions imposed on the function h is summarized in Assumption 2.1 [35].
Assumption 2.1
-
(P1)
Smoothness condition. The function \(h: {{\mathbb {R}}}^+ \cup \{0\} \rightarrow {{\mathbb {R}}}\) satisfies \(h \in C^1 ({{\mathbb {R}}}^+ \cup \{0\})\).
-
(P2)
Curvative condition. The function h is concave in \({{\mathbb {R}}}^+\).
-
(P3)
Zero density condition: \(h(0)=0\).
-
(P4)
Non-positivity condition: \(h(t) \le 0\) for all \(t \in {{\mathbb {R}}}^+\).
-
(P5)
Decay condition: For \(t> 0\) the function h satisfies \(h^{\prime } (t) \le 0\).
-
(P6)
Growth condition: For \(t>0\), \(C_1 |t|^{4/3} + C_2 \le |h(t)| \le C_3 |t|^{4/3} + C_4\) for some real constants \(C_1> 0, \ C_2 \le 0, \ C_3 > 0\) and \(C_4 \ge 0\).
We can extend h, by reflection, to a function from \({{\mathbb {R}}}\) to \({{\mathbb {R}}}\), setting \(h(t):=h(-t)\) for \(t<0\). This extended function, again denoted by h, is continuous in \({{\mathbb {R}}}\) due to the property (P3). Since h(t) is continuous in \({{\mathbb {R}}}\) and since \(|h(t)| \le C_3 |t|^{4/3} + C_4\), from the upper bound in (P6), Fatou’s Lemma implies that \({{\mathcal {E}}}_\mathrm{{xc}}(\rho _{\gamma })\) is continuous in \(L^{4/3}({{\mathbb {R}}}^3)\). Essentially all LDA xc terms used in practice are covered by Assumption 2.1 (see, e.g., [24]).
Taking into account the approximations in (2.9) and (1.4) in the minimization problem in (2.5) while switching to the one-particle density matrix formulation, we arrive at the minimization problem
where
which takes into account an external field \({\varvec{B}} = ({\varvec{B}}_x, {\varvec{B}}_y, {\varvec{B}}_z)\); see (1.2). In the present work we replace the exchange-correlation term by its collinear version given by \({{\mathcal {E}}}_\textrm{xc}^{ cLSDA} (\rho _{{\varvec{\gamma }}}^{\uparrow \uparrow }, \rho _{{\varvec{\gamma }}}^{\downarrow \downarrow }):= {{\mathcal {E}}}_\textrm{xc}^\textrm{LSDA} (\rho _{{\varvec{\gamma }}}^{\uparrow \uparrow }\), \(\rho _{{\varvec{\gamma }}}^{\downarrow \downarrow })\). For the resulting functional \({{\mathcal {E}}}^\textrm{hy} ({\varvec{\gamma }}):= {{\mathcal {E}}}({\varvec{\gamma }})\), see (2.11) where the last term \({{\mathcal {E}}}_{\textrm{xc}}^\textrm{LSDA}(\rho _{{\varvec{\gamma }}^+}, \rho _{{\varvec{\gamma }}^-})\) is replaced by \({{\mathcal {E}}}_\textrm{xc}^{ cLSDA} (\rho _{{\varvec{\gamma }}}^{\uparrow \uparrow }, \rho _{{\varvec{\gamma }}}^{\downarrow \downarrow })\), we justify the spectral binning approximation scheme.
3 Approximation scheme and methodology
In the spin DFT setting, we will be minimizing over density matrices, which we denote by \({\varvec{\gamma }}\). Since we will replace \({{\mathbb {R}}}^{3}\) by an open, bounded and simply-connected set \(\Omega \subset {{\mathbb {R}}}^{3}\) (a domain), we first set the scene for density matrices defined on \(\Omega \). Let \({{\mathcal {S}}}({{\mathcal {H}}})\) denote the set of self-adjoint, bounded operators on a Hilbert space \({{\mathcal {H}}}\). Using \(\mathbb {C}^2\) to denote the \(2 \times 2\) matrix structure we write
For \(j \in {{\mathbb {N}}}\), we write an orthonormal basis of \(L^2(\Omega ; \mathbb {C}^2)\) as \(({\varvec{\xi }}_j)_{j=1}^\infty \), where \({\varvec{\xi }}_j({\varvec{r}}) = ( \xi _j^\uparrow ({\varvec{r}}), \xi _j^{\downarrow }({\varvec{r}}) )^\textrm{T}\) \(\in L^2(\Omega ; \mathbb {C}^2)\). With the bra-ket notation, and using the basis \(({\varvec{\xi }}_j)\), we may write \({\varvec{\gamma }}\) using \((\alpha _j)\) corresponding to the basis as \({\varvec{\gamma }}({{\textbf{r}}},{{\textbf{r}}}^\prime ) = \sum _{j=1}^\infty \alpha _j \, | {{\varvec{\xi }}_j({{\textbf{r}}})} \rangle \langle {{\varvec{\xi }}_j({{\textbf{r}}}^\prime )}| \,\). We define
where \({\varvec{{{\mathcal {C}}}}}\) has the corresponding norm \(\left\Vert {\varvec{\gamma }}\right\Vert _{{\varvec{{{\mathcal {C}}}}}}:= {\text {Tr}}\,(|{\varvec{\gamma }}|) + {\text {Tr}}\,(|\nabla |{\varvec{\gamma }}|\nabla |)\).
Recalling that \(0 \le {\varvec{\gamma }}\le 1\) is equivalent to saying that, for the basis expansion of \({\varvec{\gamma }}\) we have \(0 \le \alpha _i \le 1\) for all i, we will define a subspace \({\varvec{{{\mathcal {C}}}}}_N \subset {\varvec{{{\mathcal {C}}}}}\) by
The only difference between the latter space and the space \({\varvec{{{\mathcal {C}}}}}_{N}\) found in (2.6) is that we have replaced \({{\mathbb {R}}}^{3}\) by \(\Omega \) in the definitions in (3.1) and (3.2). Minimizing over \({\varvec{{{\mathcal {C}}}}}_N\) with respect to the \({\varvec{{{\mathcal {C}}}}}\) norm will still give a solution in \({\varvec{{{\mathcal {C}}}}}_N\) [11]. It is clear that we can define the \(2 \times 2\) spin density and the total electronic density in the usual way. We note here that using the basis vector representation, we have that \(\rho _{{\varvec{\gamma }}}({{\textbf{r}}})=\rho _{{\varvec{\gamma }}}({{\textbf{r}}})^{\uparrow \uparrow }+\rho _{{\varvec{\gamma }}}^{\downarrow \downarrow }({{\textbf{r}}})=\sum _{j=1}^\infty \alpha _j |{\varvec{\xi }}_j({{\textbf{r}}})|^2\).
3.1 Reformulation of problem on a bounded domain
It is convenient to introduce the notation
Our first aim is to reformulate the three terms \({{\mathcal {J}}}\), \(E_\textrm{ext}\) and \(E_\textrm{ZZ}\) appearing in \({{\mathcal {E}}}(\cdot )\), viz.
We will explicitly assume, given K fixed, that there exists an \(\epsilon >0\) depending on K only so that for all \(i \ne j\), we have
This essentially follows from the physical condition that the ith and jth positive nuclei at coordinates \({{\textbf{R}}}_i\) and \({{\textbf{R}}}_j\) repel each other. (We note however that this condition does not automatically translate to an infinite collection of nuclei). Writing \({\underline{{{\textbf{R}}}}} =({{\textbf{R}}}_1, \ldots , {{\textbf{R}}}_K)\), we introduce a mollifier function \(f_{{{\textbf{R}}}_i}({{\textbf{r}}})\) and the “potential” function
Then, using Lemma 5.1 and Proposition 5.2 we find that we can write the electrostatic terms as
By \(C_\textrm{self}\) we mean a constant that is independent of \(\rho _{{\varvec{\gamma }}}\) and \(\phi \). It is worth noting that in [35] no justification is given for its analogue of (3.7). Although in [27, p. 259] a very brief outline is given, there is no formal statement or proof of an analogue to Proposition 5.2. We would like to stress that Proposition 5.2 and Lemma 5.1 and their proofs are new. In particular, the repulsion assumption (3.5) is not found elsewhere.
Let us define the \(2 \times 2\) matrix-valued, linear operator \({\mathfrak {B}}\) which acts on \({{\mathcal {S}}}(L^2(\Omega ); \mathbb {C}^2)\) by
where \(\delta \) is the Dirac delta distribution. Then for \({\varvec{\gamma }}\in {\varvec{{{\mathcal {C}}}}}_N\) we have that
Our strategy for reformulating the exchange-correlation energy is to construct its dual functional, taking advantage of the concavity property (P2) of the xc term \({{\mathcal {E}}}_\textrm{xc}\) (we suppress the superscript for simplicity) in Assumption 2.1. Let
From the properties (P2) and (P6) of the exchange-correlation function h, the functional \({\mathfrak {G}}_\mathrm{{xc}}\) is convex and continuous in \(L^{4/3} (\Omega )\). As usual the Legendre-Fenchel dual \({\mathfrak {G}}_\mathrm{{xc}}^{\star }\) of \({\mathfrak {G}}_\mathrm{{xc}}\) is defined as
where \({\mathfrak {G}}_\mathrm{{xc}}^{\star } (u)\) is expressed explicitly in (5.17) via the Legendre transform of \(-h(t)\) with h being the function which enters in the xc energy, see (1.4). In Lemma 5.3 we show that
As a direct consequence of (3.12) we can rewrite the \({{\mathcal {E}}}_\mathrm{{xc}}\) functional as
In our case, we rewrite \({{\mathcal {E}}}_\textrm{xc}^\textrm{cLSDA}\) using duality as above, see Sect. 5.2, and our minimization problem thus becomes
3.2 Spectral theory
We introduce an unbounded local operator \(\Phi ({{\textbf{r}}},{{\textbf{r}}}')=\phi ({{\textbf{r}}}) \ \delta ({{\textbf{r}}},{{\textbf{r}}}')\) for \(\phi \in W_0^{1,2} (\Omega )\) as in (3.7) and with \(\delta \) representing the Dirac delta distribution so that
It is convenient to rephrase the ground state energy as
where the Lagrangian \(L \,: \, L^4 (\Omega ) \times L^4 (\Omega ) \times W_0^{1,2} (\Omega ) \times L^2(\Omega )^3 \times {\varvec{{{\mathcal {C}}}}}_N \rightarrow {{\mathbb {R}}}\) is given by
with the \(2 \times 2\) Hamiltonian \({\varvec{H}}(u,v,\phi ,{\varvec{B}}) = - \dfrac{1}{2} \Delta \otimes \varvec{I}_{2} + \Phi - U -V - {\mathfrak {B}}\) with \(\varvec{I}_{2}\) being the \(2 \times 2\) matrix-valued identity operator and \(\Phi \), U, V, and \({\mathfrak {B}}\) are defined above. Since we can exchange the order of the infimum, the ground-state energy also equals
Next we identify sufficient properties of \(L(u,v,\cdot ,{\varvec{B}},\cdot )\) that allow us to exchange the order of the infimum over \({\varvec{\gamma }}\in {\varvec{{{\mathcal {C}}}}}_N\) and the supremum over \(\phi \in W_0^{1,2} (\Omega )\). In particular, we note that \(L(u,v,\phi ,{\varvec{B}},{\varvec{\gamma }})\) is linear with respect to \({\varvec{\gamma }}\). This explains the first equality in
and it is well-known that we can swap the places of infimum which explains the second equality. In Sect. 5.3 we establish Theorem 5.6 which allows us to interchange \(\inf _{{\varvec{\gamma }}}\) and \(\sup _\phi \) above. Proposition 2.2 in Ekeland [7, Chapter 6] underpins the theorem. This results in
Hence minimization with respect to \({\varvec{\gamma }}\) can be executed while keeping \(\phi \), u and v constant. This makes it possible to use spectral approximation. Moreover, the Lagrangian L depends linearly on \({\varvec{\gamma }}\) and the latter only enters in the trace of \({\varvec{H}}(u,v,\phi ,{\varvec{B}}){\varvec{\gamma }}\). This motivates the definition of the band energy as \(E_\mathrm{{band}} (u,v,\phi ,{\varvec{B}}):= {\text {Tr}}\,({\varvec{H}}(u,v,\phi ,{\varvec{B}}){{\varvec{\gamma }}})\). Utilizing spectral theory, one can form a basis of \(L^2(\Omega ; \mathbb {C}^2)\) using a countable family consisting of the orthonormal eigenvectors \({\varvec{\xi }}_k\) of \({\varvec{H}}(u,v,\phi ,{\varvec{B}})\). We recall that they can be written as \({\varvec{\xi }}_k = ( \xi _k^\uparrow , \xi _k^\downarrow )^\textrm{T}\). In Theorem 5.7, we prove that, for every \(u,v\in L^4 (\Omega )\), \({\varvec{B}} \in L^2(\Omega )\) and every \(\phi \in W_0^{1,2} (\Omega )\) a minimizer of the band energy \(E_\mathrm{{band}} (u,v,\phi ,{\varvec{B}},\cdot )\) in \({\varvec{{{\mathcal {C}}}}}_N\) commutes with the Hamiltonian \({\varvec{H}}(u,v,\phi ,{\varvec{B}})\). In particular, a minimizer \({\varvec{\gamma }} \in {\varvec{{{\mathcal {C}}}}}_{N}\) of \(E_\mathrm{{band}} (u,v,\phi ,{\varvec{B}},\cdot )\) can be written as a spectral function, i.e., \({\text {argmin}}\,_{{\varvec{\gamma }}\in {\varvec{{{\mathcal {C}}}}}_{N}} E_\mathrm{{band}} (u,v,\phi ,{\varvec{B}},{\varvec{\gamma }}) = g( {\varvec{H}}(u,v,\phi ,{\varvec{B}})\), wherein g is the Borel function given in (5.37). Since the minimizer is of the form of a Borel function, it suffices to minimize over Borel functions only. In other words, we will minimize over the space
and the minimization problem becomes
3.3 Spatial discretization
At this stage it is necessary to introduce a spatial discretization of the function spaces \(W_{0}^{1,2}(\Omega )\), \(L^{4}(\Omega )\) and \(L^{2}(\Omega )^3\) by restriction to finite-dimensional subspaces. Specifically, for \(j \in {{\mathbb {N}}}\), let \(W_0^{1,2} (\Omega )_j\) be from a family of finite-dimensional exhausting subspaces of \(W_0^{1,2} (\Omega )\) spanned by the basis \(\{e_1, \ldots , e_j\}\). This can be for example a subspace that corresponds to a finite element discretization. We let \(L^4 (\Omega )_j\) be from a family of finite-dimensional exhausting subspaces of \(L^4 (\Omega )\) spanned by the basis \(\{ d_1, \ldots , d_j \}\). These can be for example piecewise constant functions. We also have \(\{{\mathfrak {d}}_1, \ldots , {\mathfrak {d}}_j \}\) as the basis of \((L^2(\Omega )_j)^{3}\) for the analogous discretization of \(L^2(\Omega )^3\). The restriction of the electrostatic field to \(W_0^{1,2} (\Omega )_j\) is of the form
Accordingly, the restriction of the nuclear charge distribution is
and the restricted dual density potential is
In (3.20)-(3.22) the coefficients \(\{ \varphi _a \}_{a=1}^{j}\), \(\{ {\mathfrak {f}}_a^{\varvec{{{\textbf{R}}}}} \}_{a=1}^{j}\), \(\{ {\mathfrak {u}}_a \}_{a=1}^j\) and \(\{ {\mathfrak {v}}_a \}_{a=1}^{j}\) are sets of real numbers. For the magnetic field term, we define the discretized magnetic field as \({\varvec{B}}_j:= ( {\varvec{B}}_{j,x}, {\varvec{B}}_{j,y}, {\varvec{B}}_{j,z} )^\textrm{T}\), with \({\varvec{B}}_{j,\nu }({{\textbf{r}}}) = \sum _{a=1}^j {{\mathcal {B}}}_{\nu ,a} {\mathfrak {d}}_a({{\textbf{r}}})\), \(\nu = x, y, z\). For the notation, we note that \({\mathfrak {B}}\) and \(\mathfrak {B^j}\), which will be defined in (3.24), are operators. Next, \({\varvec{B}}\) and \({\varvec{B}}_{j}\) are vectors, \({\varvec{B}}_{x}\) and \({\varvec{B}}_{j,x}\) are vector components, and \(\lbrace {{\mathcal {B}}}_{x,a} \rbrace _{a=1}^j\) are real coefficients. Moreover, we define the discrete counterparts to \({\varvec{\gamma }}\), \(\rho _{{\varvec{\gamma }}}\), \(\rho _{{\varvec{\gamma }}}^{\uparrow \uparrow }\) and \(\rho _{{\varvec{\gamma }}}^{\downarrow \downarrow }\); we refer to Sect. 5.5 for details. Next we introduce
The discrete Lagrangians \(L^j\), obtained by restriction of the functional L in (3.15) to \(L^4 ( \Omega )_j \times L^4 ( \Omega )_j \times W_0^{1,2} ( \Omega )_j \times L^2(\Omega )_{j}^{3} \times {\varvec{{{\mathcal {C}}}}}_N^{j}\) follow as
Here \({\varvec{H}}^j(u,v,\phi ,{\varvec{B}})\) denotes the restriction of the operators \({\varvec{H}}^j(u,v,\phi ,{\varvec{B}})\) to the finite-dimensional subspaces \(L^4(\Omega )_j\), \(W_0^{1,2} (\Omega )_j\) and \(L^2(\Omega )_{j}^{3}\), and we define
where
We also define the discrete magnetic operator term as
Formally, A and \({{\mathcal {M}}}\) also depend on j as they are restrictions of operators to the span of \(\{\xi _1,\ldots , \xi _j\}\). We omit this dependence here for simplicity of notation. The discrete band energy is given by
In the following we write \({\varvec{H}}^j \equiv {\varvec{H}}^j (u,v,\phi , {\varvec{B}})\) for simplicity. Similarly to (3.23) we need to introduce the family of discrete constraint sets
Likewise, the corresponding discrete energies \({{\textsf {E}}}_j\) follow as
Theorem 5.6 (see Sect. 5.3) justifies the interchange of \(\inf _{{\varvec{\gamma }}}\) and \(\sup _\phi \) above. In Sect. 5.5 our choices of \(\Phi _{a_{1}, a_{2}}^{j}\), \({\mathfrak {B}}^{j}\) etc., are provided.
3.4 Spectral binning
We now will give the discretization for the minimizers following Wang et al. [35]. As the original \({\varvec{H}}\) was a self-adjoint \(2 \times 2\) matrix operator, so is \({\varvec{H}}^j\). Hence, we can make a spectral discretization of it, in the form \({\varvec{H}}^j = \int _{{\text {spec}}\,({\varvec{H}}^j)} \lambda \, d {\varvec{P}}^j(\lambda )\), where \({\varvec{P}}^j\) is the resolution of the identity, and \({\text {spec}}\,({\varvec{H}}^j)\) is the spectrum. For \({\varvec{\gamma }}_j\) there also exist bounded Borel functions \(g^j \,: \, {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) so that \( {\varvec{\gamma }}_j= \int _{{\text {spec}}\,({\varvec{H}}^j)} g^j(\lambda ) \, d {\varvec{P}}^j(\lambda ) =: g^j({\varvec{H}}^j)\). Then
with the spectral measure
We let \(\mathscr {B}\) denote the space of real-valued, bounded Borel functions defined on \({{\mathbb {R}}}\). If, for every a, (3.29)–(3.30) and (3.31) are known, the determination of \({\varvec{\gamma }}^{j}\) for fixed \((\phi ,u,v,{\varvec{B}})\) amounts to minimizing
Minimizing over all Borel functions is not numerically feasible and therefore we need to restrict to a finite-dimensional subspace \(\mathscr {B}_{k}\) of \(\mathscr {B}\). Let \(\lbrace s_q\rbrace _{q=1}^k\) be a basis of \(\mathscr {B}_{k}\) such that the minimization problem above is replaced by
With this in mind, we introduce the subset
With this definition, the discrete band energy will become
Similarly the number of electrons will be
Therefore, with \(c= ( c_1, \dots , c_k ) \in {{\mathbb {R}}}^k\) the minimization becomes
We proceed to specifying the basis \(\lbrace s_q \rbrace _{q=1}^{k}\) of \(\mathscr {B}_{k}\). Let \({{\mathcal {I}}}=[\lambda _{l}, \lambda _{r}] \subset {{\mathbb {R}}}\) be a fixed interval and recall that \(\lambda _\textrm{min}^{j}\), resp. \(\lambda _\textrm{max}^{j}\), is the smallest, resp. largest, eigenvalue of the operator \({\varvec{H}}^{j}\). By introducing the ‘bins’ \(\{ t_{q} \}_{q=0}^{k}\) chosen such that
we can partition the interval \({{\mathcal {I}}}\) into k subintervals with endpoints \(\lbrace t_q \rbrace _{q=0}^k\). With such a partition, we define the piece-wise constant functions
and we choose \(\mathscr {B}_{k}\) to be the span of \(\{ s_{t_{q}} \}_{q=1}^{k}\); note that \(\overline{{\text {span}}\,_{q=1, \ldots , k} s_{t_{q}} } = L^{2}({{\mathcal {I}}})\). For a density operator \({\varvec{\gamma }}_{k}^{j} \in {{\mathcal {C}}}_{N,k}^{{\varvec{H}}^{j}}\) we thus obtain
via the functional calculus with respect to the basis \(\{ s_{t_{q}} \}_{q=1}^{k}\). The band energy corresponding to this \({\varvec{\gamma }}_{k}^{j}\) is given by
Similarly the number of electrons will be
Note that \(n_{q}^{k,j}\) can be understood as the number of eigenvalues in the qth ‘bin’ \((t_{q-1}, t_{q})\) which explains the terminology “Spectral binning" introduced in [35]. Similarly, \(w_{q}^{k,j}\) is interpreted as the band energy in each bin. In this way, we minimize
If we are solving for this numerically, we need to find the values of \(w_{q}^{k,j}\) and \(n_{q}^{k,j}\). Sylvester’s Theorem [28] enables us to compute \(n_{q}^{k,j}\) but we are unable to do the same to find \(w_{q}^{k,j}\). However, we will use the \(n_{q}^{k,j}\) to make an approximation. We note that for an interval \((t_{q-1},t_q)\), its centre of mass \(m_q^{k,j}\) will be given by
However, a reasonable approximation for the centre of mass is the centre of interval, i.e.,
In particular, \(w^{k,j}_q = m_q^{k,j} n^{k,j}_q \approx m_q n^{k,j}_q\). Hence, we will define the approximate trace as
where in comparison to \({\text {Tr}}\,( {\varvec{H}}^j {\varvec{\gamma }}^{j} )\) we have replaced \(w^{k,j}_q\) with \(m_q n^{k,j}_q\). The discrete spectral band energy is therefore given by
4 Main result and strategy of its proof
Our main theorem states that in the limit of the number of spatial discretizations \(j \rightarrow \infty \), and consequently in the limit of the number of spectral discretizations \(k_j \rightarrow 0\), the family of ground state energies of the spatially and spectrally discrete energy functional converges to the full ground state energy given by (assuming it is attained)
where \({{\mathcal {E}}}^\textrm{hy}(\cdot )\) is given in (1.1). Before we state the theorem we describe the strategy for proving it and, in the process, we introduce the mathematical objects used in its formulation.
4.1 Strategy
We prove convergence with respect to spectral and spatial discretization using the method of \(\Gamma \)-convergence. In order to apply the latter method, we rewrite the extended energy functional in (3.15) by introducing the indicator function \(I_{M}\) associated with a set M as
Then, resuming our discussion about (3.17) above,
where (4.3) is obtained by rearranging the terms according to their dependence on u, v, \(\phi \) and \({\varvec{\gamma }}\). Note that the minimization over \({\varvec{{{\mathcal {C}}}}}_N\) is replaced by the minimization over \({\varvec{{{\mathcal {C}}}}}_N^{{\varvec{H}}(u,v,\phi ,{\varvec{B}})}\). This ensures the existence of a spectral function and is justified by (3.19) and (5.38) from the proof of Theorem 5.6. The overall idea is to show convergence with respect to both spectral and spatial discretization using three nested \(\Gamma \)-convergence proofs.
4.1.1 Limit functionals
In order to clarify the steps of our proof, we introduce the following functionals
defined by
Resuming from (4.3) we thus consider the minimization problem
4.1.2 Functionals involving spectral and spatial approximations
Let j refer to the dimension of the finite element spaces \(W_0^{1,2} (\Omega )_j\) and \(L^4(\Omega )_j\), as in Sect. 3.3, and let \(k_j\) refer to the number of subdivisions in the binning approximation, as in Sect. 3.4; recall that \(k_j \rightarrow \infty \) as \(j \rightarrow \infty \). Next we introduce the approximated constrained set of density operators
with piecewise constant functions \(\left\{ s_{t_q} (\lambda )\ \right\} _{q=1}^{k_j}\) defined in (3.35) and the discrete band energies \(E_\mathrm{{band}}^{j,k_{j}} (u,v,\phi , {\varvec{B}}, \cdot ) \,: \, {\varvec{{{\mathcal {C}}}}} \rightarrow {{\mathbb {R}}}\)
where \({\widetilde{{\text {Tr}}\,}}_{k_j} (\cdot )\) is the approximation of the trace operator described in (3.39). Analogously to (4.4)– (4.7) we introduce, for \(j \in {{\mathbb {N}}}\), functionals
defined by
and
This is in conjunction with the family of energies
It is worth to point out here that the convergence of the discrete magnetic field \({\varvec{B}}_j\) to the true magnetic field \({\varvec{B}}\) takes place in the indicator function \(I_{{\varvec{{{\mathcal {C}}}}}_{N, k_j}^{{\varvec{H}}^j}}\), which is why there is no \({\varvec{B}}_j\) dependence in \(S_1^{j,k_j}\), \(S_2^{j,k_j}\) and \(S_3^{j,k_j}\). We justify inserting \(-I_{W_0^{1,2}(\Omega )_j}(\phi )\), in addition to \(I_{L^4(\Omega )_j}(u)\) and \(I_{L^4(\Omega )_j}(v)\), since we are finding the minimizer over these restricted spaces.
Since \(n^{k,j}_q\) is the number of eigenvalues in the qth interval, we have that
and, therefore,
Collecting (3.33) and (3.40) for \({{\varvec{\gamma }}}_{j,k_{j}} \in {\varvec{{{\mathcal {C}}}}}_{N, k_j}^{{\varvec{H}}^j (u_j,v_j, \phi _j,{\varvec{B}}_j)}\), that is, the spectral approximation of \({{\varvec{\gamma }}}_j \in {\varvec{{{\mathcal {C}}}}}_N^{{\varvec{H}}^j (u,v,\phi ,{\varvec{B}})}\), the approximate trace operator depending on \(k_j \in {{\mathbb {N}}}\) is given by
with the centers of mass \(m_q =\dfrac{t_q + t_{q-1}}{2}\) for \(\ 1 \le q \le k_j\), as defined in (3.38).
We can finally formulate the main theorem.
Theorem 4.1
Let Assumption 2.1 hold, and let the interval \([\lambda _{l}, \lambda _{r}]\) be partitioned by \(\{ t_{q} \}_{q=0}^{k}\); see (3.34). Then, in the limit of the number of spatial discretization \(j \rightarrow \infty \), and consequently in the limit of the number of spectral discretizations \(k_j \rightarrow 0\), the family of ground state energies of the spatially and spectrally discrete energy functional converges to the full ground state energy in (4.1), that is,
In particular, the theorem implies that
5 Proofs
5.1 Recasting of electrostatic energies
To prove the equality
we need the following lemma in which we transform \(E_\mathrm{{ZZ}}\) in (3.3) to an integral form. Here the \({{\textbf{R}}}_i\), \(Z_i\) are spatial coordinates and nuclear charge, respectively, of the ith nuclei. Bear in mind also the definition of \({\mathfrak {f}}({{\textbf{r}}},{\underline{{{\textbf{R}}}}})\) in (3.6).
Lemma 5.1
Let \(f_{{\varvec{R}}_i}({{\textbf{r}}})\) be a regularised and bounded function with support on a small ball around \({{\textbf{R}}}_i\), so that \(\int _{\Omega } f_{{\varvec{R}}_i}({{\textbf{r}}}) \, d{{\textbf{r}}}=1\). Let \(\epsilon >0\) be defined as in (3.5), and in particular let us choose the \(f_{{\varvec{R}}_i}\) so that for all \(1 \le i \le K\), \({\text {diam}}\,(\textrm{supp}\,f_{{\varvec{R}}_i} ) < \epsilon /3\). Then
where \(C_\mathrm{{self}}\) depends only on dimension, minimum distance between the nuclear coordinates \({{\textbf{R}}}_i\), maximum nuclear charge, and choice of the functions \(f_{{\varvec{R}}_i}\).
We use the previous lemma in the proof of the following proposition.
Proposition 5.2
Let \({{\mathcal {J}}}\), \(E_\mathrm{{ext}}\), \(E_\mathrm{{ZZ}}\) be defined as above. Then
where \(C_\mathrm{{self}}\) depends only on dimension, minimum distance between the nuclear coordinates \({{\textbf{R}}}_i\), maximum nuclear charge, and choice of the functions \(f_{{\varvec{R}}_i}\).
We first give a proof of Lemma 5.1.
Proof of Lemma 5.1
Assume \({{\textbf{r}}}\in \textrm{supp}\,f_{{\varvec{R}}_i}\) and \({{\textbf{r}}}^\prime \in \textrm{supp}\,f_{{\varvec{R}}_j}\), for \(i \ne j\). Then,
By our choice of the supports of \(f_{{\varvec{R}}_i}\) for \(1 \le i \le M\), we know that \(| {{\textbf{R}}}_i-{{\textbf{R}}}_j|> \epsilon \) and \(| {{\textbf{r}}}_i-{{\textbf{r}}}_j|> \epsilon /3\). In addition, both \(| {{\textbf{r}}}-{{\textbf{R}}}_i|\) and \(| {{\textbf{r}}}- {{\textbf{R}}}_j|\) are bounded above by \(\epsilon /3\). Hence,
and \(| {{\textbf{R}}}_i-{{\textbf{R}}}_j| - |( {{\textbf{r}}}-{{\textbf{R}}}_i)+( {{\textbf{R}}}_j-{{\textbf{r}}}^\prime ) +({{\textbf{R}}}_i-{{\textbf{R}}}_j)| < 2\epsilon / 3\). Thus, we have
where \(C_0\) is a constant that only depends on the minimum distance between the nuclei points of \({{\textbf{R}}}\). Thus for \(i \ne j\),
where \(|g_{i,j}| < C_0.\) Hence,
In the final line, the first term is \(E_\mathrm{{ZZ}}\). The second term is a constant \(C_1\), where
The constant \(C_1\) depends on the minimum distance between nuclei and the maximum charge of the system. For the third term, by linear transformations of \({{\textbf{R}}}_i\) and \(f_{{\varvec{R}}_i}\) to the origin, it suffices to show that we can bound the integral
For a fixed \({{\textbf{r}}}^\prime \), we have
where the constant \(C_2\) depends only on the choice of the function f and the dimension (in this case 3), and is finite due to smoothness of f. Hence it follows that
with \(C_3\), as \(C_1\), depending on the function f and the maximum charge of the \(Z_i\), but in addition depending on the dimension (3 in this case). Writing \(C_\mathrm{{self}} = C_1 + C_3\), we finally arrive at (5.1). \(\square \)
We now can give a proof of Proposition 5.2.
Proof of Proposition 5.2
We consider the Poisson equation (in dimension \(d=3\)),
By considering the Green’s function, we find that the unique solution of (5.6) is given by
We have, by comparing to the end of the proof of Lemma 5.1, that
Putting this into the Green’s function we arrive at
Hence,
Applying (5.8), we have
The constant \( C =2 C_1 \int \rho _{{\varvec{\gamma }}} d{{\textbf{r}}}= 2C_1 N\) is independent of \(R_{{\varvec{\gamma }}}\), and in the final line we used the result from Lemma 5.1. Going back to the Poisson equation, we rearrange (5.6) to arrive at
Taking the integrals of both sides and using (5.10), we get
We will now show that the unique solution \(\Phi \) of the Poisson equation will also give the maximal value of the following maximization expression, that is to say,
We define
Since we are taking a supremum over the \(\phi \), it suffices to show that for all \(\phi \in W_0^{1,2}\) we have \(I(\Phi ) \ge I(\phi )\). Taking the integral over the zero function, we have
We use the basic inequality \(ab \le \frac{1}{2} (a^2+b^2)\) to get
This proves \(I(\Phi ) \ge I(\phi )\), and hence the desired result follows. \(\square \)
5.2 Recasting of XC energy
As a consequence of (3.11), \({\mathfrak {G}}_\mathrm{{xc}}^{\star }\) is introduced as the pointwise supremum of a family of continuous affine functions. By (P2), \({\mathfrak {G}}_\mathrm{{xc}}\) is convex. By (P2) and (P6), \({\mathfrak {G}}_\mathrm{{ xc}}\) is continuous in \(L^{4/3}(\Omega )\). Important properties of \({\mathfrak {G}}_\mathrm{{xc}}^{\star }\) are summarized in Lemma 5.3 below, wherein \(\langle {u},{v} \rangle \) denotes the dual product of \(u \in L^4 (\Omega ), \ v \in L^{4/3} (\Omega )\). It is defined by
Lemma 5.3
The functional \({\mathfrak {G}}_\mathrm{{xc}}^{\star }\) defined in (3.11) is convex and lower semi-continuous on \(L^4(\Omega )\) and satisfies
The function \(h^{\star } (x) \,: \, {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) is the Legendre transform of \(-h(t)\) and fulfills the growth conditions
for real constants \(C_5>0, \ C_6 >0, \ C_7\) and \(C_8\). For \({\mathfrak {G}}_\mathrm{{xc}} (\rho _{{\varvec{\gamma }}})=-{{\mathcal {E}}}_\mathrm{{xc}} (\rho _{{\varvec{\gamma }}})\) it holds that
Proof
From [7], we know that a dual function formulated as in (3.11) is both convex and lower semi-continuous, (5.17) also follows from [7, Proposition 1.2 Chapter IV]. The growth conditions (5.18) follow from applying (P6) to the Legendre transform of h(t). Since \({\mathfrak {G}}_\mathrm{{xc}}\) is also convex and lower semi-continuous, we have from [7, Proposition 4.1 Chapter I] that \({\mathfrak {G}}_\mathrm{{xc}}^{\star \star } = {\mathfrak {G}}_\mathrm{{xc}}\). Hence equality (5.19) follows. \(\square \)
As in [35], we will use the dual function method, twice in fact, for two different terms, namely \(E_\mathrm{{xc}}^\textrm{LDA}(2\rho ^{\uparrow \uparrow })\) and \(E_\mathrm{{xc}}^\textrm{LDA} (2\rho ^{\downarrow \downarrow })\). This means that we can write
Therefore, we define the operators U and V so that
Then we have that
The operators U and V are linear with respect to \({\varvec{\gamma }}\).
5.3 Interchange of infimum and supremum
First we identify sufficient properties of \(L(u,v, \cdot ,{\varvec{B}},\cdot )\) that allow us to exchange the order of the infimum over \({\varvec{\gamma }}\in {\varvec{{{\mathcal {C}}}}}_N\) and the supremum over \(\phi \in W_0^{1,2} (\Omega )\).
Lemma 5.4
For all \(u,v \in L^4(\Omega )\), \(\phi \in W_0^{1,2}(\Omega )\), the functional \(L(u,v,\phi ,{\varvec{B}},\cdot )\) is convex and lower semi-continuous with respect to \({\varvec{\gamma }}\). In addition, for any fixed \(u,v \in L^4(\Omega )\), \({\varvec{B}} \in L^2(\Omega )\), and \(\phi \in W_0^{1,2}(\Omega )\),
Proof
All of the terms involving \({\varvec{\gamma }}\) in \(L(u,v,\phi ,{\varvec{B}},{\varvec{\gamma }})\) are linear, and so the convexity is immediate from that. Note in particular that because the exchange correlation energy was a function of \(\rho _{{\varvec{\gamma }}}^{\uparrow \uparrow } \) and \( \rho _{{\varvec{\gamma }}}^{\downarrow \downarrow }\), the resulting inner products which come up are linear with respect to \({\varvec{\gamma }}\). For semi-continuity, we note that \(\frac{1}{2}{\text {Tr}}\,\left( -\Delta {\varvec{\gamma }}\right) \) is lower semi-continuous in \({\varvec{{{\mathcal {C}}}}}_N\). Indeed, \(\frac{1}{2}{\text {Tr}}\,\left( -\Delta {\varvec{\gamma }}\right) \) is lower semi-continuous if, for all \({\varvec{\gamma }}_0 \in {\varvec{{{\mathcal {C}}}}}_N\), and for all real \(\epsilon >0\) we can find a neighbourhood \({\varvec{{{\mathcal {N}}}}}_{0}\) containing \({\varvec{\gamma }}_0\) so that for all \({\varvec{\gamma }}\in {\varvec{{{\mathcal {N}}}}}_x\), we have that \(\frac{1}{2}{\text {Tr}}\,\left( -\Delta {\varvec{\gamma }}\right) \le \frac{1}{2}{\text {Tr}}\,\left( -\Delta {\varvec{\gamma }}_0 \right) +\epsilon \).
For a given \({\varvec{\gamma }}_0\) and \(\epsilon >0\), consider the \({\varvec{{{\mathcal {C}}}}}\)-ball centered on \({\varvec{\gamma }}_0\) with radius \(\epsilon /2\). Therefore, for all \({\varvec{\gamma }}\) in this ball, \(\left\Vert {\varvec{\gamma }}_0-{\varvec{\gamma }}\right\Vert _{{\varvec{{{\mathcal {C}}}}}}< \epsilon /2\). With \((\alpha _{0,k})\) denoting the basis expansion of \({\varvec{\gamma }}_0\) and \((\alpha _k)\) denoting the basis expansion of \({\varvec{\gamma }}\), we thus have
Therefore, it follows that
whence lower semi-continuity holds. Moving onto the density terms, we have
where we write \( {\text {Tr}}\,(- \Delta {{\varvec{\gamma }}}^{\mu \mu }):= \sum _{k=1}^\infty \alpha _k |\nabla \xi _k^{\mu }|^2\); slightly abusing notation. Invoking the Gagliardo-Nirenberg-Sobolev embedding theorem, we also have \(\left\Vert \rho _{{\varvec{\gamma }}}^{\mu \mu }\right\Vert _{L^3} \le C {\text {Tr}}\,(- \Delta {{\varvec{\gamma }}}^{\mu \mu })\). Since we are working on a bounded domain \(\Omega \), we get
where the constant C changes from left to the right above. Hence, we have that if \({{\varvec{\gamma }}}_i \rightarrow {{\varvec{\gamma }}}\) in \({\varvec{{{\mathcal {C}}}}}\) as \(i \rightarrow \infty \), then \(\rho _{{{\varvec{\gamma }}}_i}^{\mu \mu } \rightarrow \rho _{{\varvec{\gamma }}}^{\mu \mu }\) in \(L^{2}(\Omega )\). Hence by applying Hölder’s inequality with \(p=q=2\) we have continuity, and therefore lower semi-continuity, for the terms
as well as for the magnetic field terms:
Let us concretely demonstrate the continuity of the terms by considering as an example the term \(\int _\Omega \rho _{{\varvec{\gamma }}} \phi d {{\textbf{r}}}\). Using Hölder’s inequality and inequality (5.23) we arrive at
meaning that \({\varvec{\gamma }}\mapsto \int _\Omega \rho _{{\varvec{\gamma }}} \phi d {{\textbf{r}}}\) is continuous with respect to \(\Vert \cdot \Vert _{{\varvec{{{\mathcal {C}}}}}}\). To show coercivity, we will apply Hölder’s inequality to the terms in L. We write L as
where \(C_{u,v,\phi } = \int _{\Omega } \left( -C_d |\nabla \phi |^2 +{\mathfrak {f}}\phi \right) d {{\textbf{r}}}+C_\mathrm{{self}} + {\mathfrak {G}}_\mathrm{{xc}}^*(u) + {\mathfrak {G}}_\mathrm{{xc}}^*(v)\). By applying the triangle and Hölder (\(p=2=q\)) inequalities to \({\text {Tr}}\,({\mathfrak {B}} {\varvec{\gamma }})\), we have
Next, we note that
Furthermore, by considering their eigenvalue representation, and with \(\textrm{Re}\) and \(\textrm{Im}\) being the real and imaginary parts respectively, we have that
By considering the quadratic expansion, we have that
Therefore it follows that
By inserting inequalities (5.26)-(5.28) into (5.25), we arrive at
Therefore, when we apply Hölder’s inequality with \(p=q=2\) for operators \(\Phi \), U and V to (5.24), and use inequality (5.29), we get
Next, by interpolation, we have that \(\left\Vert \rho _{{\varvec{\gamma }}}\right\Vert _{L^2(\Omega )} \le \left\Vert \rho _{{\varvec{\gamma }}}\right\Vert _{L^{1}(\Omega )}^{1/4} \left\Vert \rho _{{\varvec{\gamma }}}\right\Vert _{L^3(\Omega )}^{3/4}\), and by the Gagliardo-Nirenberg-Sobolev inequality, there exists a constant C independent of \(\rho _{{\varvec{\gamma }}}\) so that \(\left\Vert \rho _{{\varvec{\gamma }}}\right\Vert _{L^3(\Omega )} = \left\Vert \sqrt{\rho _{{\varvec{\gamma }}}}\right\Vert _{L^6(\Omega )}^{2} \le C \left\Vert \nabla \sqrt{\rho _{{\varvec{\gamma }}}}\right\Vert _{L^2(\Omega )}^{2}\). We note that \(\left\Vert \rho _{{\varvec{\gamma }}}\right\Vert _{L^1(\Omega )} = N\), and, therefore,
Hence, we arrive at
where, on the right-hand side,
depends on \(N,u,v,\phi ,{{\textbf{B}}}\), but is independent of \({{\varvec{\gamma }}}\). By writing \({\text {Tr}}\,(-\Delta {{\varvec{\gamma }}})= {\text {Tr}}\,(|\nabla |{{\varvec{\gamma }}}|\nabla |)\) and utilizing \(\frac{1}{2}\left( {\text {Tr}}\,({{\varvec{\gamma }}})-N \right) =0\), we get
We recall the definition of \(\left\Vert \cdot \right\Vert _{{\varvec{{{\mathcal {C}}}}}} = {\text {Tr}}\,(|\cdot |) +{\text {Tr}}\,(|\nabla |\cdot |\nabla |)\). Using the inequality (5.22), and that for f, g positive we have \(\big |\nabla \sqrt{f+g}\big |\le \big |\nabla \sqrt{f}\big | +\big |\nabla \sqrt{g}\big |\), and we thus obtain
As a consequence,
and it follows that
Since
it follows that for fixed u, v and \(\phi \) that \(\lim _{\left\Vert {{\varvec{\gamma }}}\right\Vert _{{\varvec{{{\mathcal {C}}}}}} \rightarrow \infty } L(u,v,\phi ,{\varvec{B}},{{\varvec{\gamma }}}) = +\infty \), as desired. \(\square \)
Next we establish concavity.
Lemma 5.5
For all \(u,v \in L^4(\Omega )\), \({{\varvec{\gamma }}} \in {\varvec{{{\mathcal {C}}}}}_N\), the functional \(L(u,v,\cdot ,{\varvec{B}},{{\varvec{\gamma }}})\) is concave and upper semi-continuous with respect to \(\phi \). In addition, for any fixed \(u,v \in L^4(\Omega )\), \({\varvec{B}} \in L^2(\Omega )\), and \({{\varvec{\gamma }}} \in {\varvec{{{\mathcal {C}}}}}_N\),
Proof
We write \(L(u,v,\phi ,{\varvec{B}},{{\varvec{\gamma }}})\) as
where
Recalling (3.13), \({\text {Tr}}\,(\Phi {{\varvec{\gamma }}})\) and also \(\int _{\Omega } {\mathfrak {f}}({{\textbf{r}}},{\underline{{{\textbf{R}}}}}) \phi ({{\textbf{r}}}) d{{\textbf{r}}}\) are linear with respect to \(\phi \), they are also concave. Since the second term is a negative quadratic, it is also concave with respect to \(\phi \).
Next, we show that \(L(u,v,\phi ,{\varvec{B}},{{\varvec{\gamma }}})\) is upper semi-continuous with respect to \(\phi \). Again, for fixed \({{\varvec{\gamma }}}\) and b, we have that \({\text {Tr}}\,(\Phi {{\varvec{\gamma }}})\) and \(\int _{\Omega } {\mathfrak {f}}({{\textbf{r}}},{\underline{{{\textbf{R}}}}}) \phi ({{\textbf{r}}}) \, d{{\textbf{r}}}\) are continuous with respect to \(\phi \). The quadratic term is also upper semi-continuous, see for example [6, Proposition 2.1].
All that remains is to show that, for fixed \(u,v,{\varvec{B}}\) and \({{\varvec{\gamma }}}\), as \(\left\Vert \phi \right\Vert _{W_0^{1,2}}\) tends to infinity, \(L(u,v,\phi ,{\varvec{B}},{{\varvec{\gamma }}})\) will tend to negative infinity. By applying Hölder’s inequality to the linear terms with \(p=2=q\), we arrive at
where for the final inequality we applied the Poincaré inequality, which is where the constant \(C_\mathrm{{ poin}}\) comes from. Next, by Young’s inequality,
with \(a=\left\Vert \phi \right\Vert _{W_0^{1,2}(\Omega )} \) and \(b=\left\Vert \rho _{{\varvec{\gamma }}} + {\mathfrak {f}}({{\textbf{r}}},{{\textbf{R}}})\right\Vert _{L^2(\Omega )}\), we have that
Let us choose \(\epsilon \) small enough so that \(C_\mathrm{{poin}}-\epsilon >0\). This means that
Then,
and, therefore,
Hence, \(\lim _{\left\Vert \phi \right\Vert _{W_0^{1,2}(\Omega )} \rightarrow \infty } L(u,v,\phi ,{\varvec{B}},{{\varvec{\gamma }}})=-\infty \). \(\square \)
Having prepared the auxiliary results above, we are now ready to show that it is possible to exchange the orders of the infimum and supremum when computing the energy.
Theorem 5.6
With the assumptions as before,
Proof
For fixed \(u,v \in L^4 (\Omega )\) we want to apply [7, Proposition 2.2] in the functional \((\phi ,{{\varvec{\gamma }}}) \mapsto L(u,v,\phi ,{\varvec{B}},{{\varvec{\gamma }}})\). The convergence in \({{\varvec{\gamma }}}\) is with respect to the \(\text{ weak}^{*}\)-topology in \({\varvec{{{\mathcal {C}}}}}\). Since \(W_0^{1,2} (\Omega )\) and \({\varvec{{{\mathcal {C}}}}}_N\) are convex, non-empty sets, [7, Proposition 2.2] is valid provided the following conditions are met for arbitrary \(u,v \in L^4 (\Omega )\):
(p1) For all \(\phi \in W_0^{1,2} (\Omega )\), \(L(u,v,\phi ,{\varvec{B}},\cdot )\) is convex and lower semi-continuous.
(p2) For all \({{\varvec{\gamma }}} \in {\varvec{{{\mathcal {C}}}}}_N\), \(L(u,v,\cdot ,{\varvec{B}},{{\varvec{\gamma }}})\) is concave and upper semi-continuous.
(p3) There exists a \(\phi _0 \in W_0^{1,2} (\Omega )\) such that
(p4) There exists \({{\varvec{\gamma }}}_0 \in {\varvec{{{\mathcal {C}}}}}_N\) such that \(\displaystyle \lim _{\Vert \phi \Vert _{W^{1,2} (\Omega )} \rightarrow 0} L(u,v,\phi ,{\varvec{B}},{{\varvec{\gamma }}}_0)=-\infty \).
The properties (p1) and (p3) follow from Lemma 5.4, Lemma 5.5 implies the properties (p2) and (p4). Hence [7, Proposition 2.2] applies. This ensures that there exists at least one saddle point \((\phi , {{\varvec{\gamma }}})\) of \(L(u,v,\cdot ,{\varvec{B}},\cdot )\) for every \(u,v \in L^4 (\Omega )\) and that the order of infimum in \({{\varvec{\gamma }}}\) and supremum in \(\phi \) can be exchanged without affecting the ground-state energy. \(\square \)
5.4 Commutativity of band energy
Theorem 5.7
With the assumptions stated above, we have that, for every \(u,v\in L^4 (\Omega )\), \({\varvec{B}} \in L^2(\Omega )\) and every \(\phi \in W_0^{1,2} (\Omega )\) a minimizer of the band energy \(E_\mathrm{{band}} (u,v,\phi ,{\varvec{B}},\cdot )\) in \({\varvec{{{\mathcal {C}}}}}_N\) commutes with the Hamiltonian \({\varvec{H}}(u,v,\phi ,{\varvec{B}})\).
Proof
For fixed u, v, and \(\phi \), the operator \({\varvec{H}}(u,v,\phi ,{\varvec{B}})\) on \(L^2(\Omega ; \mathbb {C}^2)\) is self-adjoint, unbounded, and semi-bounded below. Moreover, since \({\varvec{H}}(u,v,\phi ,{\varvec{B}})\) is bounded below, there exists a smallest eigenvalue, say \(\lambda _1\). Let \(\lambda _k\) and \({\varvec{\xi }}_k\) denote the \(k^{th}\) eigenvalue and eigenvector, respectively, where the eigenvalues are in increasing order. It then follows that
By self-adjointness we have
Since \({\varvec{\xi }}_i\) are eigenvectors of \({\varvec{H}}\), we have
If we wish to minimize the expression \(\inf _{{{\varvec{\gamma }}} \in {\varvec{{{\mathcal {C}}}}}_N} \sum _{i=1}^\infty \lambda _i \left\langle {{\varvec{\gamma }}} {\varvec{\xi }}_i, {\varvec{\xi }}_i \right\rangle \), we remember that \({{\varvec{\gamma }}} \in {\varvec{{{\mathcal {C}}}}}_N\) requires that \({\text {Tr}}\,({{\varvec{\gamma }}}) = N\). Therefore, the infimum over \({{\varvec{\gamma }}}\) corresponds to taking the N eigenvectors that have the correspondingly lowest eigenvalues. Hence,
Therefore, it follows that there exists a Borel function g so that
Since the minimizer is of the form of a Borel function, it suffices to minimize over Borel functions only. In other words, we will minimize over the restricted space
Operators \(\hat{{\varvec{\gamma }}}\) in \(\hat{{\varvec{{{\mathcal {C}}}}}}_N\) have a spectral decomposition, and therefore \(E_\mathrm{{band}} (u,v,\phi ,{\varvec{B}},\hat{{\varvec{\gamma }}})\) commutes with the Hamiltonian \({\varvec{H}}(u,v,\phi ,{\varvec{B}})\). Because we have shown that minimizing over \({\varvec{{{\mathcal {C}}}}}_N\) reduces to minimizing over \(\hat{{\varvec{{{\mathcal {C}}}}}}_N\), it follows that \(E_\mathrm{{band}} (u,v,\phi ,{\varvec{B}},{\varvec{\gamma }})\) will also commute with the Hamiltonian \({\varvec{H}}(u,v,\phi ,{\varvec{B}})\) for \({\varvec{\gamma }} \in {\varvec{{{\mathcal {C}}}}}_N \). \(\square \)
5.5 Spatial discretization
We define the discrete counterparts to \({\varvec{\gamma }}\), \(\rho _{{\varvec{\gamma }}}\), \(\rho _{{\varvec{\gamma }}}^{\uparrow \uparrow }\) and \(\rho _{{\varvec{\gamma }}}^{\downarrow \downarrow }\). First, however, we define the basis vectors \({\varvec{\xi }}_a \in W_0^{1,2}(\Omega ; \mathbb {C}^2)_j\), for \(1 \le a \le j\), by \({\varvec{\xi }}_a = ( \xi _a^\uparrow , \xi _a^\downarrow )^\textrm{T}\). We have
where
and, analogously, for \(\gamma _j^{\downarrow \uparrow }( {{\textbf{r}}}, {{\textbf{r}}}^\prime )\) and \( \gamma _j^{\downarrow \downarrow }( {{\textbf{r}}}, {{\textbf{r}}}^\prime )\). For \(\rho _{{\varvec{\gamma }}}\), we recall that
where the second equality comes from the spectral definition of \({{\varvec{\gamma }}}\). Therefore, we will define
Note that \(\alpha _k\) is not necessarily equal to \(\rho _{j,a_1,a_2}\), since we still want \(\rho _j\) to integrate to the total density, N. We justify our choices of \(\Phi _{a_1,a_2}^j\) and \({\mathfrak {B}}^j\). The others follow in an analogous way. We initially defined \(\Phi \) in order that
where the second equality comes from our definition of \(\rho _{{\varvec{\gamma }}}\). Since we have
it then follows that
where the final equality comes from our definition of \(\Phi _{a_1, a_2}^j.\) Let us now take a look at the other example, \({\mathfrak {B}}^j\). In particular we look at the \({{\textbf{B}}}_z\) component. We recall by definition of the trace that
Therefore, with \( {\varvec{B}}_{j,z}({{\textbf{r}}}) = \sum _{k=1}^j {{\mathcal {B}}}_{z,k} {\mathfrak {d}}_k({{\textbf{r}}})\) and
we arrive at
Repeating a similar process with \({{\textbf{B}}}_x\) and \({{\textbf{B}}}_y\), we obtain
5.6 Proof of main Theorem 4.1
We first establish the convergence of the exact band energies \({\text {Tr}}\,( {\varvec{H}}^j (u_j,v_{j}, \phi _j, {\varvec{B}}_j) {{\varvec{\gamma }}}_j)\). Secondly, we validate the convergence of the approximate trace operators.
5.6.1 Convergence of exact band energies
We first establish \(\Gamma \)-convergence of the exact band energies \({\text {Tr}}\,({\varvec{H}}^j (u_j,v_j, \phi _j, {\varvec{B}}_j) {\varvec{\gamma }}_j)\). For given sequences \((u_j)_{j\in {{\mathbb {N}}}}, (v_j)_{j\in {{\mathbb {N}}}} \subset L^4(\Omega )\), \(({\varvec{B}}_j)_{j \in {{\mathbb {N}}}} \subset L^2(\Omega )\), \((\phi _j)_{j\in {{\mathbb {N}}}} \subset W_0^{1,2} (\Omega )\) with \(u_j \rightarrow u, v_{j} \rightarrow v\) in \(L^4 (\Omega )\), \({\varvec{B}}_j \rightarrow {\varvec{B}}\) in \(L^2(\Omega )^3\), and \(\phi _j \rightarrow \phi \) in \(W_0^{1,2} (\Omega )\) for \(j \rightarrow \infty \) we are going to establish the \(\Gamma \)-convergence result
with respect to the \(\text{ weak}^{\star }\)-topology in \({\varvec{{{\mathcal {C}}}}}\) as \(j \rightarrow \infty \). Recalling the properties of \(\Gamma \)-convergence, we need to verify: (i) The lower semi-continuity of
(ii) The existence of a recovery sequence.
Part (i) is shown in Lemma 5.8, and Part (ii) is shown in Lemma 5.9.
Lemma 5.8
Let \(u_j \rightharpoonup u\) and \(v_j \rightharpoonup v\), both in \(L^4 (\Omega ), \ \phi _j \rightharpoonup \phi \) in \(W_0^{1,2} (\Omega )\) and \({\varvec{B}}_j \rightharpoonup {\varvec{B}}\) in its usual space \(L^2(\Omega )^3\), as \(j \rightarrow \infty \), be four given sequences. Then, for arbitrary \(({{\varvec{\gamma }}}_j)_{j \in {{\mathbb {N}}}} \subset {\varvec{{{\mathcal {C}}}}}\) with \({{\varvec{\gamma }}}_j \rightharpoonup ^\star {{\varvec{\gamma }}}\) in \({\varvec{{{\mathcal {C}}}}}\),
We call the band energies \({\text {Tr}}\,( {\varvec{H}}^j (u_j,v_{j}, \phi _j, {\varvec{B}}_j) {{\varvec{\gamma }}}_j)\) exact because no spectral discretization of the trace operator enters.
Proof
We will consider four disjoint and exhaustive cases: 1. Let \({{\varvec{\gamma }}} \in {\varvec{{{\mathcal {C}}}}}_N^{{\varvec{H}}(u,v,\phi ,{\varvec{B}})}\) and \(({{\varvec{\gamma }}}_j)_{j \in {{\mathbb {N}}}} \subset {\varvec{{{\mathcal {C}}}}}\) be a sequence with \({{\varvec{\gamma }}}_j \overset{\star }{\rightarrow }\ {{\varvec{\gamma }}}\) such that there exists a \(j_1 \in {{\mathbb {N}}}\) such that \({{\varvec{\gamma }}}_j \in {\varvec{{{\mathcal {C}}}}}_{N,k_j}^{{\varvec{H}}^j(u_j,v_{j},\phi _j,{\varvec{B}}_j)}\) for all \(j \ge j_1\). By the lower semi-continuity, we have \(\lim _{j \rightarrow \infty } \inf {\text {Tr}}\,(-\Delta {{\varvec{\gamma }}}_j) \ge {\text {Tr}}\,(-{{\varvec{\gamma }}} )\) and by the compact embedding of \(W_0^{1,2} (\Omega )\) in \(L^2 (\Omega ), \ {{\varvec{\gamma }}}_j \rightarrow {{\varvec{\gamma }}}\) in \({\varvec{{{\mathcal {C}}}}}\) implies that \(\rho _{{{\varvec{\gamma }}}_j}^{\mu \mu } \rightarrow \rho _{{{\varvec{\gamma }}}}^{\mu \mu }\) in \(L^2 (\Omega )\). This yields for \(\Phi _j, \ U_j, V_j\) corresponding to \(\phi _j, u_j, v_j\), defined as in (3.20) and (3.22),
The second equality is true because of the weak convergence of \(( \phi _j)\) and \(( u_j )\), and because
where we used Hölder’s inequality and the \(L^2\)-convergence of \((\rho _{{{\varvec{\gamma }}}_j})_{j}\). The magnetic term also follows easily,
Hence, \(\lim _{j\rightarrow \infty } {\text {Tr}}\,( {\varvec{H}}^j (u_j,v_{j},\phi _j, {\varvec{B}}_j ) {{\varvec{\gamma }}}_j) \ge {\text {Tr}}\,({\varvec{H}}(u, v,\phi ,{\varvec{B}}) {{\varvec{\gamma }}})\). This coincides with (5.41) for the case considered here, completing this case.
2. Let \({{\varvec{\gamma }}} \in {\varvec{{{\mathcal {C}}}}}_N^{{\varvec{H}}(u,v,\phi ,{\varvec{B}}_j)}\) and \(({{\varvec{\gamma }}}_j)_{j \in {{\mathbb {N}}}} \subset {\varvec{{{\mathcal {C}}}}}\) be a sequence such that there exists an index \(j_2 \in {{\mathbb {N}}}\) so that \({{\varvec{\gamma }}}_j \not \in {\varvec{{{\mathcal {C}}}}}_{N, k_j}^{{\varvec{H}}^j (u_j, v_{j}, \phi _j,{\varvec{B}}_j)}\) for all \(j \ge j_2\). Then in this case, (5.41) holds because as \(j \rightarrow \infty \), \({\varvec{\gamma }}_j\) will not be in the space \({\varvec{{{\mathcal {C}}}}}_{N,k_j}^{{\varvec{H}}^j(u_j, v_{j}, \phi _j,{\varvec{B}}_j)}\), and therefore
3. Let \({\varvec{\gamma }}\not \in {\varvec{{{\mathcal {C}}}}}_N\) and \(({{\varvec{\gamma }}}_j)_{j\in {{\mathbb {N}}}} \subset {\varvec{{{\mathcal {C}}}}}\) be a sequence such that there exists an index \(j_3 \in {{\mathbb {N}}}\) so that \({{\varvec{\gamma }}}_j \not \in {\varvec{{{\mathcal {C}}}}}_{N,k_j}^{{\varvec{H}}^j (u_j,v_{j}, \phi _j,{\varvec{B}}_j)}\) for all \(j \ge j_3\). In this case (5.41) is satisfied trivially because both the left-hand side and the right-hand side are equal to infinity, and therefore each other.
4. Finally, we consider the case where \({{\varvec{\gamma }}} \not \in {\varvec{{{\mathcal {C}}}}}_N^{{\varvec{H}}(u,v,\phi ,{\varvec{B}})}\), and show that there does not exist a sequence \({{\varvec{\gamma }}}_j \overset{\star }{\rightharpoonup }\ {{\varvec{\gamma }}}\) and \(j_4 \in {{\mathbb {N}}}\) so that \({{\varvec{\gamma }}}_j \in {\varvec{{{\mathcal {C}}}}}_{N,k_j}^{{\varvec{H}}^j (u_j,v_j,\phi _j, {\varvec{B}}_j)}\) for all \(j \ge j_4\). Let \(({\varvec{\xi }}_i)_{i \in {{\mathbb {N}}}} \subset W_0^{1,2} (\Omega ; \mathbb {C}^2)\) be the eigenvectors of \({\varvec{H}}(u,v,\phi ,{\varvec{B}})\). As in the proof of Theorem 5.6, these \(({\varvec{\xi }}_i)_{i \in {{\mathbb {N}}}}\) form an orthonormal basis of \(L^2(\Omega ;\mathbb {C}^2)\). Similarly, for each \(j \in {{\mathbb {N}}}\) let \(( {\varvec{\xi }}_i^j)_{i \in {{\mathbb {N}}}} \subset W_0^{1,2} (\Omega ;\mathbb {C}^2)\) be the eigenvectors of \({\varvec{H}}^j (u_j, v_j, \phi _j, {\varvec{B}}_j)\). By the spatial discretization of the Hamiltonian, we know that for every \(i \in {{\mathbb {N}}}\), we have
However, since in this case we assume that \({{\varvec{\gamma }}} \not \in {\varvec{{{\mathcal {C}}}}}_N^{{\varvec{H}}(u,v,\phi ,{\varvec{B}})}\), there must exist at least one eigenvector of \({\varvec{H}} (u, v, \phi ,{\varvec{B}})\) that is not an eigenvector of \({{\varvec{\gamma }}}\). Without loss of generality suppose \({\varvec{\xi }}_1\) is this eigenvector. Therefore, when we take the eigenvector representation of \({{\varvec{\gamma }}}\), \({{\varvec{\gamma }}} {\varvec{\xi }}_1 =\sum _{q=1}^{\infty } c_{1q} {\varvec{\xi }}_q\), there must exist some index \(p \in {{\mathbb {N}}}, \ p \ne 1\), with \(c_{1p} \ne 0\). By orthogonality of the eigenvectors, we have that
However, by the weak-star convergence of \({\varvec{\gamma }}_j \rightharpoonup ^\star {\varvec{\gamma }}\), we know that \(\langle {{{\varvec{\gamma }}} {\varvec{\xi }}_1},{ {\varvec{\xi }}_p} \rangle = \lim _{j \rightarrow \infty } \langle {{{\varvec{\gamma }}}_j {\varvec{\xi }}_1},{ {\varvec{\xi }}_p} \rangle .\) For a contradiction, suppose that there exists an index \(j_4 \in {{\mathbb {N}}}\) so that \({{\varvec{\gamma }}}_j \in {\varvec{{{\mathcal {C}}}}}_{N,k_j}^{{\varvec{H}}^j (u_j,v_j,\phi _j, {\varvec{B}}_j )}\) for all \(j \ge j_4\). It then follows that
where we know that the Borel functions \(g_j\) in (5.44) do exist, because \({{\varvec{\gamma }}}_j \in {\varvec{{{\mathcal {C}}}}}_{N,k_j}^{{\varvec{H}}^j (u_j,v_{j},\phi _j,{\varvec{B}}_j)}\) for large j. Therefore, for an arbitrary \(p \ne 1\), we have that
where the final line comes from the orthogonality of the \({\varvec{\xi }}_j\), and in the penultimate line we used equations (5.43) and (5.44). Therefore we arrive at the contraction that \(c_{1,p}\) is both zero and non-zero. Therefore, if \({{\varvec{\gamma }}} \not \in {\varvec{{{\mathcal {C}}}}}_N^{{\varvec{H}}(u,v,\phi ,{\varvec{B}})}\), we cannot have \(({{\varvec{\gamma }}}_j)_{j \in {{\mathbb {N}}}}\) with \({{\varvec{\gamma }}}_j \in {\varvec{{{\mathcal {C}}}}}_{N,k_j}^{{\varvec{H}}^i(u_j, v_j,\phi _j, {\varvec{B}}_j)}\). Therefore, this final case cannot occur, and inequality (5.41) holds trivially. \(\square \)
Lemma 5.9
Let \(u_j \rightarrow u\) and \(v_j \rightarrow v\) in \(L^4 (\Omega )\), \({\varvec{B}}_{j} \rightarrow {\varvec{B}}\) in \(L^2(\Omega )^3\), \(\phi _j \rightarrow \phi \) in \(W_0^{1,2} (\Omega )\) for \(j \rightarrow \infty \) be four given sequences. Then there exists a recovery sequence \(({{\varvec{\gamma }}}_j)_{j \in {{\mathbb {N}}}} \subset {\varvec{{{\mathcal {C}}}}}, \ {{\varvec{\gamma }}}_j \overset{\star }{\rightarrow }\ {{\varvec{\gamma }}}\), such that
and, consequently, in view of Lemma 5.8
with respect to the \(\text{ weak}^{\star }\)-topology in \({\varvec{{{\mathcal {C}}}}}\) in the limit \(j \rightarrow \infty \).
We omit the proof of Lemma 5.9 because, with obvious modifications, it is similar to the proof of [35, Lemma 4.1] (Details are provided in [29]). However, we provide an outline:
Proof
(Outline of the proof of Lemma 5.9) We consider two disjoint cases. The first case, where we assume \({{\varvec{\gamma }}} \not \in {\varvec{{{\mathcal {C}}}}}_N^{{\varvec{H}} (u,v,\phi ,{\varvec{B}})}\) is standard and we omit it here. For the second case, we suppose that \({{\varvec{\gamma }}} \in {\varvec{{{\mathcal {C}}}}}_N^{{\varvec{H}} (u,v,\phi ,{\varvec{B}})}\). Then, \({{\varvec{\gamma }}}=\sum _{i=1}^{\infty } \alpha _i \, | { {\varvec{\xi }}_i} \rangle \langle { {\varvec{\xi }}_i}| \, =\sum _{i=1}^{\infty } \alpha _i {\varvec{\xi }}_i \otimes {\varvec{\xi }}_i\).
As in Lemma 5.8, \(( {\varvec{\xi }}_i)_{i \in {{\mathbb {N}}}}\) are the eigenvectors of \({\varvec{H}} (u,v,\phi ,{\varvec{B}})\). For each \(j \in {{\mathbb {N}}}\), let \(( {\varvec{\xi }}_i^j)_{i \in {{\mathbb {N}}}}\) be the corresponding eigenvectors of \({\varvec{H}}^j (u_j,v_{j}, \phi _j,{\varvec{B}}_j)\). The idea is to construct a recovery sequence in three steps. For the first step, we construct a sequence of finite-rank operators \(({{\varvec{\gamma }}}_j)_{j \in {{\mathbb {N}}}}\) from the eigenvectors of \({\varvec{H}}^j (u_j,v_{j}, \phi _j,{\varvec{B}}_j)\). In the second step, we construct for each \({{\varvec{\gamma }}}_j\) a sequence \(({{\varvec{\gamma }}}_{k,j})_{k \in {{\mathbb {N}}}}\) by the spectral binning approximation. For the third and final step, we choose a diagonal sequence \(({{\varvec{\gamma }}}_{k_j,j})_{j \in {{\mathbb {N}}}}\), which will yield the recovery sequence for this case. \(\square \)
Now that we have the \(\Gamma \)-convergence result, we can show that we have convergence at the first level of approximation. However, also we need the following lemma.
Lemma 5.10
For every \(\phi _j \rightharpoonup \phi \) in \(W_0^{1,2}(\Omega )\), \({\varvec{B}}_j \rightharpoonup {\varvec{B}}\) in \(L^2(\Omega )^3\) and every \(u_j \rightharpoonup u\), \(v_j \rightharpoonup v \) in \(L^4(\Omega )\), equi-coercivity with respect to the weak-\(\star \) topology in \({\varvec{{{\mathcal {C}}}}}\) holds for the family
Proof
We give an outline of the proof here, for the detailed proof we refer to [29]. Similar to the proof of lower semi-continuity in Lemma 5.4, we show that
where
and the constant \(C_{11}\) results from the Gagliardo–Nirenberg–Sobolev inequality. Note that \(C_{12}\) is well defined and bounded, because weakly convergent series are bounded. Since \({\text {Tr}}\,(-\Delta {{\varvec{\gamma }}}) \ge \Vert \nabla \sqrt{\rho _{{\varvec{\gamma }}}}\Vert _{L^2(\Omega )}^2\), the kinetic energy is the dominating \(\gamma \) term in the inequality. Therefore, for any \(t \in {\mathbb {R}}\) the level sets
are bounded, indeed, \(t \ge (1/2) \Vert {{\varvec{\gamma }}}\Vert _{{\varvec{{{\mathcal {C}}}}}} - C_{12} \Vert \nabla \sqrt{\rho _{{\varvec{\gamma }}}}\Vert _{L^2(\Omega )}^{\frac{3}{2}} - (N/2)\), where we used that \(\left\Vert \cdot \right\Vert _{{\varvec{{{\mathcal {C}}}}}} \ge {\text {Tr}}\,(-\Delta \cdot )\). By the results in [6, Prop 7.7] this shows that for every j and \(k_j\), the level sets (5.48) are precompact and hence equi-coercive. \(\square \)
We now show that we have convergence in the first level, that is to say, spatial discretisation.
Lemma 5.11
If \(\phi _j \rightharpoonup \phi \) in \(W_0^{1,2}(\Omega )\), \({\varvec{B}}_j \rightharpoonup {\varvec{B}}\) in \(L^2(\Omega )^3\), and \(u_j \rightharpoonup u, v_j \rightharpoonup v\) in \(L^4(\Omega )\) then
Proof
This comes directly applying [6, Thm 7.8], because we have shown equi-coercivity in Lemma 5.10 and the existence of a recovery sequence in Lemma 5.9. \(\square \)
5.6.2 Convergence of spectral binning operator \(E_\mathrm{{band}}^{j,k_j}\)
Next we establish \(\Gamma \)-convergence of \(E_\mathrm{{band}}^{j,k_j}\) with approximation of the trace operator. In the previous section we showed convergence of the exact trace operator. We will now record the convergence result with the spectral binning operator \(E_\mathrm{{band}}^{j,k_j}\), which was defined in (3.40).
Lemma 5.12
Let \(u_j \rightharpoonup u\) and \(v_j \rightharpoonup v\) in \(L^4(\Omega )\), \({\varvec{B}}_j \rightharpoonup {\varvec{B}}\) in \(L^2(\Omega )^3\), \(\phi _j \rightharpoonup \phi \) in \(W_0^{1,2}(\Omega )\) as \(j \rightarrow \infty \) and \({{\varvec{\gamma }}} \in {\varvec{{{\mathcal {C}}}}}_{N, k_j}^{{\varvec{H}}^j(u_j, v_{j}, \phi _j)}\) for all \(j \in {\mathbb {N}}\). Then
With obvious modifications, its proof is similar to the proof of [35, Lemma 4.5]; the proof is written out in [29].
Having the convergence of \({\widetilde{{\text {Tr}}\,}} (\cdot )\) to \({\text {Tr}}\,(\cdot )\) under control, we easily establish the announced \(\Gamma \)-convergence result. Utilizing Lemma 5.9 and Lemma 5.12, and using the same recovery sequence as the one constructed in Lemma 5.9, the proof is straightforward and thus we omit it here; details are provided in [29].
Lemma 5.13
For every \(\phi _j \rightharpoonup \phi \) in \(W_0^{1,2}(\Omega )\) every \(u_j \rightharpoonup u, v_j \rightharpoonup v \) in \(L^4(\Omega )\), \({\varvec{B}}_j \rightharpoonup {\varvec{B}}\) in \(L^2(\Omega )^3\), and any \(({{\varvec{\gamma }}}_{j} )_{j}\) in \({\varvec{{{\mathcal {C}}}}}\) with \( {{\varvec{\gamma }}}_{j} \overset{\star }{ \rightharpoonup }\ {{\varvec{\gamma }}}\), as \(j \rightarrow \infty \), one has
The next result for \({\widetilde{{\text {Tr}}\,}}(\cdot )\) is analogous to Lemma 5.10.
Lemma 5.14
If \(u_j \rightharpoonup u, v_j \rightharpoonup v\) in \(L^4(\Omega )\), \({\varvec{B}}_j \rightharpoonup {\varvec{B}}\) in \(L^2(\Omega )^3\), and \(\phi _j \rightharpoonup \phi \) in \(W_0^{1,2}(\Omega )\), then for every \({{\varvec{\gamma }}} \in {\varvec{{{\mathcal {C}}}}}\), equi-coercivity on \({\varvec{{{\mathcal {C}}}}}\) holds for the family of functionals
We use Lemma 5.10, especially (5.47), Lemma 5.12, in conjunction with standard arguments found in [6] to prove the latter result; the proof is similar to [35, Lemma 4.7]. Details are found in [29].
We now obtain the next level of convergence from Lemma 5.10 and Lemma 5.12:
Lemma 5.15
If \(\phi _j \rightharpoonup \phi \) in \(W_0^{1,2}(\Omega )\), \({\varvec{B}}_j \rightharpoonup {\varvec{B}}\) in \(L^2(\Omega )^3\), \(v_j \rightharpoonup u\) in \(L^4(\Omega )\) and \(u_j \rightharpoonup u\) in \(L^4(\Omega )\), then
5.6.3 Convergence of \(S_3^{j,k_j}\).
Next we show \(\Gamma \)-convergence of \(S_3^{j,k_j}\) from (4.13).
Lemma 5.16
If \(u_j \rightharpoonup u, v_j \rightharpoonup v\) in \(L^4(\Omega )\) then, for \(j \rightarrow \infty \), 
with respect to \(\phi \) in the weak topology of \(W_0^{1,2}(\Omega )\).
Proof
From Lemma 5.15 for every \(u,v \in L^4(\Omega )\) and all \(u_j \rightharpoonup u,\) \(v_j \rightharpoonup v\) in \(L^4(\Omega )\), \({\varvec{B}}_j \rightharpoonup {\varvec{B}}\) in \(L^2(\Omega )^3\), and \(\phi \in W_0^{1,2}(\Omega )\),
which covers the \({{\varvec{\gamma }}}\) term in \(S_3\); the dependency on \({{\varvec{\gamma }}}\) is given explicitly in (4.6). It is straightforward to treat the remaining terms, whence we omit the details; see [29] for details. \(\square \)
We now show that these functionals are equi-coercive.
Lemma 5.17
If \(u_j \rightharpoonup u, v_j \rightharpoonup v\) in \(L^4(\Omega )\) then the family of functionals \( \{ \phi \mapsto -S_3^{j,k_j}(u_j, v_j, \phi ) \}_{j \in {\mathbb {N}}}\) is equi-coersive with respect to the weak topology in \(W_0^{1,2}(\Omega )\).
Proof
By Hölder’s inequality, we have that
where we introduce the term \(\varepsilon _{k_j}\), given by \({\widetilde{{\text {Tr}}\,}}\left( {\varvec{H}}^j(u_j,v_j, \phi ,{\varvec{B}}_j)\tilde{{{\varvec{\gamma }}}_j} \right) = {\text {Tr}}\,{\varvec{H}}^j (u_j,v_j,\phi ,{\varvec{B}}_j) \tilde{{{\varvec{\gamma }}}}_j - \varepsilon _{k_j}\). Due to Lemma 5.15, \(\varepsilon _{k_j} \rightarrow 0\) as \( j \rightarrow \infty \). In (5.52) we also introduced \(\tilde{{{\varvec{\gamma }}}}_j \in {\varvec{{{\mathcal {C}}}}}_{N, k_j}^{{\varvec{H}}^j(u_j,v_j, \phi ,{\varvec{B}}_j)}\) and \(\tilde{{{\varvec{\gamma }}}} \in {\varvec{{{\mathcal {C}}}}}_{N, k}^{{\varvec{H}}(u,v, \phi ,{\varvec{B}})}\) as the minimizers of the left and right sides of (5.50), respectively.
By expanding the \({\text {Tr}}\,( {\varvec{H}} \tilde{{{\varvec{\gamma }}}})\) as above, it follows that
with a constant \(C_{13} > 0\) originating from Poincaré’s inequality, and with further constants \(C_{14}:= \Vert {\mathfrak {f}}({{\textbf{r}}}, \underline{{\varvec{R}}}) \Vert _{L^2(\Omega )} + \sup _{j \in {\mathbb {N}}} \Vert \rho _{{\tilde{{{\varvec{\gamma }}}}}_j}\Vert _{L^2(\Omega )}\),
Inequality (5.53) shows that \(-S_3^{j,k_j}(u_j,v_j, \phi )\) is an equi-coercive functional, and similar to Lemma 5.10, \(-S_3^{j,k_j}(u_j,v_j, \phi )\) is equi-coercive to with respect to the weak topology in \(W_0^{1,2}(\Omega )\). \(\square \)
To conclude this section, we establish convergence for this level of our model by using standard arguments [6] in conjunction with Lemmas 5.16 and 5.17.
Lemma 5.18
If \(u_j \rightharpoonup u\) and \(v_j \rightharpoonup v\) in \(L^4(\Omega )\) then
5.6.4 Convergence of \(S_{i}^{j,k_j}\), \(i=1, 2\)
We proceed to establishing \(\Gamma \)-convergence of the operators \(S_{i}^{j,k_j}\), \(i=1, 2\). We need the following lemma for this purpose.
Lemma 5.19
Proof
From the definition of \({\mathfrak {G}}_\mathrm{{xc}}^*\), the sequence \(\left( {\mathfrak {G}}_\mathrm{{xc}}^*(u_j) \right) \) is bounded above. The reverse Fatou Lemma, together with \(u_j\) being the projections onto the space \(L^4(\Omega )_j\), then yields
Finally, we know that \({\mathfrak {G}}_\mathrm{{xc}}^*\) is lower semi-continuous, and therefore we conclude (5.55). \(\square \)
Establishing \(\Gamma \)-convergence of \(S_{i}^{j,k_j}\) now follows easily if we apply Lemma 5.18, in conjunction with weakly lower semi-continuity of \({\mathfrak {G}}_\mathrm{{xc}}^*(u)\), together with Lemma 5.19.
Lemma 5.20
For \(v_j \rightharpoonup v\), the family \(\{ S_2^{j,k_j}(u,v) \}_{j\in {\mathbb {N}}}\) converges in the \(\Gamma \)-sense, i.e, for \(j \rightarrow \infty \), 
with respect to the weak topology in \(L^4(\Omega )\).
For the next result, we provide details of the proof.
Lemma 5.21
For \(v_j \rightharpoonup v\) in \(L^4(\Omega )\), the family \(\{ S_2^{j,k_j}(u,v) \}_{j \in {\mathbb {N}}}\) is equi-coersive with respect to the weak topology in \(L^4(\Omega )\).
Proof
In view of Lemma 5.17 and the bounds from (5.18) in Lemma 5.3, there exist real constants \(C_{16} > 0\) and \(C_{17}\) such that
The estimate (5.56) implies natural bounds from below on the functional \(S_2^{j,k_j}\),
where \({\tilde{\phi }} = 0\) is a test function in \(W_0^{1,2}(\Omega )\), \(\lambda _{l}\) denotes the lower bound of the binning interval \([\lambda _{l}, \lambda _{r}]\) for \({\varvec{H}}^j(u,v,{\tilde{\phi }},{\varvec{B}})\), and \(\lambda _1^{{\varvec{H}}^j(u,v, {\tilde{\phi }},{\varvec{B}})}\) denotes the lowest eigenvalue of \({\varvec{H}}^j(u,v,{\tilde{\phi }},{\varvec{B}})\). Let \(\lambda _{l} = \lambda _1^{{\varvec{H}}^j(u,v,{\tilde{\phi }},{\varvec{B}})} + C_j\). We know that \(\sup _j |C_j|\) is uniformly bounded, because \(\lambda _{l}\) is only a functional of \({\tilde{\phi }}\), u and v, and independent of spatial discretization. If \({\varvec{\xi }}_1^{{\varvec{H}}^j(u,v, {\tilde{\phi }},{\varvec{B}})}\) denotes the normalized eigenvector of \({\varvec{H}}^j(u,v, {\tilde{\phi }}, {\varvec{B}})\) we can derive a lower bound of \(\lambda _1^{{\varvec{H}}^j(u,v,{\tilde{\phi }},{\varvec{B}})}\) by the ellipticity of the underlying variational problem
Using (5.56)–(5.58), we bound \(T^{j,k_j}(u)\) below by a coersive functional which is independent of j and \(k_j\):
In the limit \(\Vert u\Vert _{L^4(\Omega )} \rightarrow \infty \) the term \(C_{16}\Vert u\Vert _{L^4(\Omega )}^4\) dominates, hence we have \(T^{j,k_j} \rightarrow \infty \). \(\square \)
Lemma 5.22
If \(v_j \rightharpoonup v\) in \(L^4(\Omega )\), then \(\lim _{j \rightarrow \infty } \inf _{v \in L^4(\Omega ) }S_2^{j,k}(u,v_j) = \inf _{v \in L^4(\Omega ) }S_2(u,v)\).
Proof
We apply [6] using Lemmas 5.19 and 5.21. \(\square \)
We now state and prove the analogous results for \(S_1\).
Lemma 5.23
The family of functionals \(\{ S_1^{j,k_j}(v) \}_{j\in {\mathbb {N}}}\) converges in the \(\Gamma \)-sense, i.e, for \(j \rightarrow \infty \), 
with respect to the weak topology in \(L^4(\Omega )\).
Proof
We first consider the \(\liminf \) condition. We recall from (4.4) that \( S_1(v) = C_\mathrm{{self}} +{\mathfrak {G}}_\mathrm{{xc}}^*(v) + \inf _{v} S_2(u,v)\). The \(\liminf \) condition holds for \(\inf _{v} S_2(u,v)\) by Lemma 5.22, and holds for \({\mathfrak {G}}_\mathrm{{xc}}^*(v)\) by the same arguments as in the proof of Lemma 5.21.
The \(\limsup \) condition follows for the same reasons too. \(\square \)
Our second result follows in a similar fashion.
Lemma 5.24
The family of functionals \(\{ S_1^{j,k_j}(v) \}_{j \in {\mathbb {N}}}\) is equi-coersive with respect to the weak topology in \(L^4(\Omega )\).
Proof
The lower bound (5.57) on \(S_2^{j,k_j}\) implies
The only term depending on v inside the \(\inf _u\) is then \( \lambda _1^{{\varvec{H}}^j(u,v, {\tilde{\phi }},{\varvec{B}})}\), and as in (5.58), we have the lower bound \(\lambda _1^{{\varvec{H}}^j(u,v, {\tilde{\phi }},{\varvec{B}})} \ge - \left\Vert v\right\Vert _{L^2(\Omega )}\). However, using (5.56) with v as the argument gives us \( {\mathfrak {G}}_\mathrm{{xc}}^*(v) \ge C_{18} \left\Vert v\right\Vert _{L^4(\Omega )}^{4} - C_{19}|\Omega |\). Therefore, the dominating term in (5.59) will be greater than \( C_{18} \left\Vert v\right\Vert _{L^4(\Omega )}^{4}\), thus giving us equi-coercivity. \(\square \)
5.6.5 Main result
Finally we are ready to prove the main theorem.
Proof
(Proof of Theorem 4.1) We apply [6, Thm 7.8] with Lemmas 5.23 and 5.24. \(\square \)
6 Conclusion
We present a mathematical analysis of DFT for modeling magnetic systems using an extended density-only formulation. No current densities enter the description in this formulation, but the particle density is split into different spin components. By restricting the exchange-correlation energy functional to be of an LSDA form (but further constrained to facilitate the analysis), we prove a series of results which enable us to establish our main result; a spectral (binning) approximation scheme. We prove the main result using the method of \(\Gamma \)-convergence. Auxiliary steps involves recasting the electrostatic potentials and justifying the spectral approximation by making a spectral decomposition of the Hamiltonian. In turn, we justify the spectral decomposition by “linearizing" the Hamiltonian. It is worth to note that, if we could by-pass the linearizing step of the proof, this would open up the possibility of using the general LSDA term as well as other xc functionals, such as GGAs. We leave this challenge for a future work.
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Notes
This depends on the way the integrals are evaluated numerically. For other approaches, e.g., numerical tensor methods (Tensor-train decompostion and MALS) this might be different.
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Acknowledgements
The first author acknowledges financial support from the Dr Perry James Browne Research Centre on Mathematics and its Applications. The second author is grateful to the University of Sussex for the award of a Ph.D. studentship. The authors also thank the anonymous referees for their constructive feedback and in particular for pointing out a few minor errors.
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Melgaard, M., Syrjanen, V.J.J. Spectral approximation scheme for a hybrid, spin-density Kohn–Sham density-functional theory in an external (nonuniform) magnetic field and a collinear exchange-correlation energy. J Math Chem 62, 711–760 (2024). https://doi.org/10.1007/s10910-023-01557-6
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DOI: https://doi.org/10.1007/s10910-023-01557-6
Keywords
- Density functional theory
- Kohn–Sham energy functional
- Magnetic field
- Spin-density
- Spatial and spectral discretizations
- \(\Gamma \)-convergence