Source-free exchange-correlation magnetic fields in density functional theory

TL;DR

This paper introduces a source-free formulation of exchange-correlation magnetic fields in density functional theory, improving magnetic moment predictions and spatial field geometries across various materials, including pnictides.

Contribution

It presents a novel source-free approach to exchange-correlation functionals, enhancing accuracy of magnetic properties in DFT calculations.

Findings

Improved total magnetic moments for a wide range of materials.

Good agreement with experimental moments in pnictides.

Significant differences in spatial geometry of ${f B}_{ m xc}$ for pnictides.

Abstract

Spin-dependent exchange-correlation energy functionals in use today depend on the charge density and the magnetization density: $E_{\rm xc}[\rho,{\bf m}]$. However, it is also correct to define the functional in terms of the curl of ${\bf m}$ for physical external fields: $E_{\rm xc}[\rho,\nabla\times{\bf m}]$. The exchange-correlation magnetic field, ${\bf B}_{\rm xc}$, then becomes source-free. We study this variation of the theory by uniquely removing the source term from local and generalized gradient approximations to the functional. By doing so, the total Kohn-Sham moments are improved for a wide range of materials for both functionals. Significantly, the moments for the pnictides are now in good agreement with experiment. We also predict dramatic differences in the spatial geometry of ${\bf B}_{\rm xc}$ for the pnictides. Our source-free method is simple to implement in all existing density functional theory codes.

Figures

Tables

Table 1. Table 1: Magnetic moment (in μ B subscript 𝜇 B \mu_{\rm B} ) per Fe atom for pnictides and per magnetic atom for the rest of the materials. For pnictides the moment is known to be highly sensitive to the structural details and hence the references for experimental structural data are cited in the first column and for magnetic moment in the second column.

Material	Expt	LSDA	LSDASF	GGA	GGASF
LaFeAsOSefat et al. (2008)	0.63Qureshi et al. (2010)	1.60	0.73	1.92	0.59
LaFeAsOgga		1.39	0.7	1.8	0.58
CeFeAsOZhao et al. (2008a)	0.8Zhang et al. (2013)	1.64	0.81	1.95	0.83
PrFeAsOZhao et al. (2008b)	0.5Kimber et al. (2008)	1.53	0.99	1.89	0.85
NdFeAsOQiu et al. (2008)	0.54Tian et al. (2010)	1.24	0.91	1.82	0.93
CaFe2As2Kreyssig et al. (2008)	0.8Goldman et al. (2008)	1.59	0.95	1.86	0.90
SrFe2As2Tegel et al. (2008)	0.94Zhao et al. (2008c)	1.57	0.98	1.84	0.78
BaFe2As2Mittal et al. (2011)	0.87Huang et al. (2008)	1.43	0.87	1.84	0.78
BaFe2As2gga		1.38	0.73	1.67	0.59
FeTeCaglieris et al. (2012)	2.25Li et al. (2009)	2.10	1.73	2.25	1.85
bcc-Fe	2.2	2.15	2.22	2.27	2.16
hcp-Co	1.7	1.63	1.60	1.67	1.61
Ni	0.65	0.64	0.63	0.67	0.65
Ni3Al	0.077	0.17	0.1225	0.1825	0.1725
ZrZn2	0.085	0.21	0.197	0.283	0.257

Table 2. Table 2: Magnetic moment (in μ B subscript 𝜇 B \mu_{\rm B} ) per atom. Calculations are performed using LSDA + U 𝑈 +U , GGA + U 𝑈 +U , LSDA + SF U {}_{\rm SF}+U and GGA + SF U {}_{\rm SF}+U . The values of U 𝑈 U and J 𝐽 J used for these calculations are given in Ref. uj, .

Material	Expt	LSDA	GGA	LSDASF	GGASF
PrFeAsO	Fe: 0.5	1.40	1.9	0.65	0.63
	Pr: 0.87	0.30	0.30	0.81	0.83
NdFeAsO	Fe: 0.54	1.42	1.84	0.50	0.61
	Nd: 0.9	2.44	1.25	0.80	0.89

Table 3. Table 3: Magnetic moment (in μ B subscript 𝜇 B \mu_{\rm B} ) per Fe atom for pnictides and per magnetic atom for the rest of the materials. LSDA s /GGA s moments are calculated by enhancing the LSDA, GGA magnetization density by 1.12 and 1.14 respectively. LSDA/GGA no source results are obtained by making LSDA/GGA source free (i.e. without any scaling)

Material	Expt	LSDAs	LSDA	GGAs	GGA
			no source		no source
LaFeAsO	0.63	2.11	0.07	2.42	0.06
CeFeAsO	0.8	2.12	0.02	2.37	0.01
PrFeAsO	0.5	2.30	0.15	2.40	0.12
NdFeAsO	0.54	2.20	0.32	2.43	0.30
CaFe2As2	0.8	2.20	0.07	2.46	0.06
SrFe2As2	0.94	2.24	0.06	2.50	0.04
BaFe2As2	0.87	2.40	0.06	2.51	0.05
FeTe	2.25	2.60	0.88	2.67	0.95
bcc-Fe	2.2	2.62	1.91	2.66	1.90
hcp-Co	1.7	1.82	1.28	1.87	1.35
Ni	0.65	0.71	0.53	0.69	0.57
Ni3Al	0.077	0.26	0.01	0.29	0.06
ZrZn2	0.085	0.27	0.17	0.36	0.20

Equations

$$\hat{H}=\sum _{i=1}-\frac{1}{2}{\nabla }_i^2+V_{\mathrm{ext}}({\mathbf{r}}_i)+\vec{\sigma }\cdot {\mathbf{B}}_{\mathrm{ext}}({\mathbf{r}}_i)+\frac{1}{2}\sum _{j\ne i}\frac{1}{\left|{\mathbf{r}}_i-{\mathbf{r}}_j\right|},$$

$${\nabla }^2\varphi (\mathbf{r})=-4\pi \nabla \cdot {\mathbf{B}}_{\mathrm{xc}}(\mathbf{r})$$

$${\tilde{\mathbf{B}}}_{\mathrm{xc}}(\mathbf{r})\equiv {\mathbf{B}}_{\mathrm{xc}}(\mathbf{r})+\frac{1}{4\pi }\nabla \varphi (\mathbf{r})$$

$$E[V_{\mathrm{ext}},B_{\mathrm{ext}}]=\underset{|\Psi ⟩}{\min }⟨\Psi |\hat{H}|\Psi ⟩,$$

$$E[V_{\mathrm{ext}},B_{\mathrm{ext}}]$$

$$=\underset{(\rho ,\mathbf{m})}{\min }\left\{\int d^{3}r{V}_{\mathrm{ext}}(\mathbf{r})\rho (\mathbf{r})+\int d^{3}r{\mathbf{B}}_{\mathrm{ext}}(\mathbf{r})\cdot \mathbf{m}(\mathbf{r})+F[\rho ,\mathbf{m}]\right\},$$

$$F[\rho ,\mathbf{m}]\equiv \underset{|\Psi ⟩\to (\rho ,\mathbf{m})}{\min }⟨\Psi |\hat{T}+{\hat{V}}_{\mathrm{ee}}|\Psi ⟩$$

$$T_s[\rho ,\mathbf{m}]\equiv \underset{|\Psi ⟩\to (\rho ,\mathbf{m})}{\min }⟨\Psi |\hat{T}|\Psi ⟩$$

$$\int d^{3}r\left(\nabla \times {\mathbf{A}}_{\mathrm{ext}}(\mathbf{r})\right)\cdot \mathbf{m}(\mathbf{r})=\int d^{3}r{\mathbf{A}}_{\mathrm{ext}}(\mathbf{r})\cdot \left(\nabla \times \mathbf{m}(\mathbf{r})\right).$$

$$\tilde{F}[\rho ,\nabla \times \mathbf{m}]\equiv \underset{|\Psi ⟩\to (\rho ,\nabla \times \mathbf{m})}{\min }⟨\Psi |\hat{T}+{\hat{V}}_{\mathrm{ee}}|\Psi ⟩,$$

$$\begin{array}{rl}{\tilde{\mathbf{B}}}_{\mathrm{xc}}(\mathbf{r})& \equiv {\frac{\delta {\tilde{E}}_{\mathrm{xc}}[\rho ,\nabla \times \mathbf{m}]}{\delta \mathbf{m}(\mathbf{r})}|}_{\rho }\\ & =\int d^3{r}^{\prime }\frac{\delta \left(\nabla \times \mathbf{m}({\mathbf{r}}^{\prime })\right)}{\delta \mathbf{m}(\mathbf{r})}{\frac{\delta {\tilde{E}}_{\mathrm{xc}}[\rho ,\nabla \times \mathbf{m}]}{\delta \left(\nabla \times \mathbf{m}({\mathbf{r}}^{\prime })\right)}|}_{\rho }\\ & =\nabla \times {\mathbf{A}}_{\mathrm{xc}}(\mathbf{r}),\end{array}$$

$$\frac{{\delta }^2{\tilde{E}}_{\mathrm{xc}}}{\delta \mathbf{m}(\mathbf{r})\delta \mathbf{m}({\mathbf{r}}^{\prime })}$$

$$=f_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\prime })+\int d^3{r}^{\prime \prime }M(\mathbf{r},{\mathbf{r}}^{\prime \prime })f_{\mathrm{xc}}({\mathbf{r}}^{\prime \prime },{\mathbf{r}}^{\prime }),$$

$$M(\mathbf{r},{\mathbf{r}}^{\prime \prime })={\nabla }_r\otimes {\nabla }_r\frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime \prime }|}$$

Full text

Density functional theory (DFT)Hohenberg and Kohn (1964); Kohn and Sham (1965) has proven enormously successful for calculating the electronic structure of both molecules and solids. Lattice structures, phonon spectra and many other properties are now routinely calculated. Magnetism presents more of a mixed picture. Simple magnets, such as elemental solids (Fe, Co and Ni), are well-described by the local spin density approximation (LSDA)Perdew and Wang (1992) or the generalized gradient approximations (GGA)Perdew et al. (1997), at least as far as total moments are concerned. However, both LSDA and GGA perform poorly for the iron pnictide and related materialsDai (2015); Mazin (2010); Kordyuk (2012); Dai et al. (2012) for which they greatly overestimate the moments by factors of two or more (see Fig. 1). This has been an impediment to investigating the microscopic magnetic structureYildirim (2008); Singh and Du (2008); Ma et al. (2008); Sharma et al. (2009); Ortenzi et al. (2015), related response functionsEssenberger et al. (2012) and superconductivity Mazin et al. (2008a); Lischner et al. (2015); Essenberger et al. (2016) of these materials with density functional methods.

Most approximate spin-dependent exchange-correlation energy functionals, $E_{\rm xc}[\rho,{\bf m}]$, use the density and magnetization as their argumentsvon Barth and Hedin (1972); Sandratskii and Guletskii (1986); Kubler et al. (1988); Gidopoulos (2007). This form is dictated by the many-body Hamiltonian used originally in the context of spin DFT (SDFT) by von Barth and Hedinvon Barth and Hedin (1972):

[TABLE]

where $V_{\rm ext}$ and ${\bf B}_{\rm ext}$ are the external scalar potential and magnetic fields, respectively; and the sum runs to the number of particles. The external magnetic field was assumed to be an unconstrained vector field in the original formulation of SDFT. Physical magnetic fields are not unconstrained but rather the curl of a vector potential, i.e. ${\bf B}_{\rm ext}=\nabla\times{\bf A}_{\rm ext}$. With this constraint it is possible to demonstrate (see Eschrig et al. (1985); Capelle and Gross (1997) and Appendix) that the exchange-correlation functional can be chosen to depend on the spin current $\nabla\times{\bf m}({\bf r})$ instead of ${\bf m}({\bf r})$: $\widetilde{E}_{\rm xc}[\rho,\nabla\times{\bf m}]$.

An immediate consequence of this is that the functional derivative of $\widetilde{E}_{\rm xc}[\rho,\nabla\times{\bf m}]$ with respect to ${\bf m}({\bf r})$ is of the form $\widetilde{\bf B}_{\rm xc}({\bf r})\equiv\delta\widetilde{E}_{\rm xc}/\delta{\bf m}({\bf r})=\nabla\times{\bf A}_{\rm xc}({\bf r})$ which implies $\nabla\cdot{\bf B}_{\rm xc}({\bf r})=0$. In other words, the exchange-correlation magnetic field is source-free.

An essential aspect of this version of SDFT, and one which has been seemingly overlooked in the past, is that the Kohn-Sham magnetization ${\bf m}({\bf r})$ obtained from the exact $\widetilde{\bf B}_{\rm xc}$ is not itself exact, but rather only its curl is. The difference between the two is a curl-free function which is therefore the gradient of some scalar function: $\nabla f({\bf r})$. Thus one loses some information about the magnetization by using the source-free theory but not, as it turns out, the total moment. For finite systems, the total moment obtained from the Kohn-Sham magnetization using $\widetilde{\bf B}_{\rm xc}$ is also exact because the integral of $\nabla f({\bf r})$ over all space is zero. This is not true for periodic boundary conditions, however. In this case, the functional domain has to be augmented with the total moment vector ${\bf M}$, thus $\widetilde{E}_{\rm xc}\equiv\widetilde{E}_{\rm xc}[\rho,\nabla\times{\bf m},{\bf M}]$. This is analogous to the macroscopic polarization required as an extra variable in presence of an external electric field applied to a solidGonze et al. (1995). The variable conjugate to ${\bf M}$ now has to be included in the calculation; this variable is clearly a constant magnetic field and corresponds to an ${\bf A}_{\rm ext}$ which diverges at large distance.

Functionals in common use, such as LSDA and GGA, are not, in general, source-free. This is confirmed simply by computing $\nabla\cdot{\bf B}_{\rm xc}({\bf r})$ for a magnetic material. One may therefore reasonably ask: how can LSDA or GGA be modified so that they do have this property, i.e. how can any approximate ${\bf B}_{\rm xc}$ be made source-free? We appeal to Helmholtz’s theorem which states that any vector field on a domain in $\mathbb{R}^{3}$, which is twice differentiable can be decomposed into a curl-free component and a source-free component. This decomposition is unique for given boundary conditions. Thus let $\phi$ be the solution to Poisson’s equation (in atomic units)

[TABLE]

and define

[TABLE]

then $\nabla\cdot\widetilde{\bf B}_{\rm xc}({\bf r})=0$, i.e. $\widetilde{\bf B}_{\rm xc}$ is source-free. It is important to note that that the scalar part of the potential, $V_{\rm xc}({\bf r})$, is not directly affected by this procedure. This modified functional has certain intrinsic properties: (a) it is still correct for homogeneous electron gas (HEG) because ${\bf B}^{\rm HEG}_{\rm xc}$ is a constant implying that $\nabla\cdot{\bf B}_{\rm xc}({\bf r})=0$ and therefore this modification has no effect, (b) since $\widetilde{\bf B}_{\rm xc}$ is obtained by solving Poisson’s equation, the functional is intrinsically non-local, in other words, the field at ${\bf r}$ depends on the magnetization everywhere, (c) $\widetilde{\bf B}_{\rm xc}$ is necessarily non-collinear, (d) ${\bf m}({\bf r})\times\widetilde{\bf B}_{\rm xc}({\bf r})$ is non-zero and hence will contribute to spin-dynamics even in the absence of the external fieldSharma et al. (2007), (e) $\widetilde{\bf B}_{\rm xc}$ is not, in general, the functional derivative of an energy functional (see Appendix), (f) the procedure is simple to implement in any code since all codes have a Poisson equation solver, and (g) very little computational effort is needed for the modification.

We perform an additional modification of the functional which effectively enhances the spin splitting. It comprises of a simple scaling of the input magnetization $E_{\rm xc}[\rho,{\bf m}]\rightarrow E_{\rm xc}[\rho,s{\bf m}]$ and then a further scaling of the resultant magnetic field ${\bf B}_{\rm xc}\rightarrow s{\bf B}_{\rm xc}$ in order to keep the functional variational with respect to ${\bf m}$. This scaling is reminiscent of that performed by Ortenzi el al.Ortenzi et al. (2012). The scaling factor $s$ is chosen empirically and we find that $s=1.12$ and $s=1.14$ are good choices for LSDA and GGA respectively, for a diverse set of materials. We note that this factor, though empirical, is not a material-dependent parameter.

In the present work we apply this procedure to LSDA and GGA: first we enhance the strength of the exchange splitting and then modify the functional in a unique way to become source-free. These new source-free functionals have the effect that: (a) the good moments of LSDA for elemental solids are retained, (b) the large overestimation of pnictide moments is cured, and (c) the GGA now yields better results than LSDA for both classes of materials. These results are illustrated in Fig. 1, where we show percentage errors for both common and pnictide materials, with both LSDA and GGA and their source-free counterparts. Our implementation is publicly available in the Elk Codeelk and so can be applied to many more such materials.

The percentage deviation in the magnetic moment from experiment are presented in Fig. 1(a). We note that the magnetic moments calculated using LSDA or GGA for simple magnets (Fe, Co and Ni) are already in very good agreement with experiment with maximum deviation of 8%. This is in contrast the moments for pnictides which deviate strongly from experiment with a maximum error of 278%. The moments calculated using source-free LSDA and GGA are also presented in Fig. 1(a) and for simple magnets they are of the same quality as that of the unmodified functionals. The fact that the integrated moments are the same does not necessarily imply that the magnetization densities are similar at each point in space. However, we find that for the simple magnets the two densities are fairly close at each point in space. In the case of pnictides, the moments show dramatic improvement. At a first glance it appears that LSDASF/GGASF substantially reduces the Fe moment for all pnictides compared to the corresponding LSDA/GGA value. A closer inspection, however, reveals that this reduction is, as it should be, highly selective in that the moment in SrFe2As2 is reduced by $\sim$30% while on LaFeASO by $\sim$60% compared to the LSDA/GGA results. The maximum deviation is now less than 25% for all materials with the exception of NdFeAsO and PrFeAsO (see Table 1).

For NdFeAsO and PrFeAsO the source-free functionals provide a considerable improvement over unmodified LSDA or GGA, but the percentage deviation from experiment is still relatively large. We find the reason behind this to be the moment on the rare earth atoms. In these materials the moment of the rare-earth atom is known to be strongly coupled to the moment on the Fe atomsTian et al. (2010) and the rare earth moment is not accurately described by the source-free functional alonemwu . In order to treat these we use the well-established methodLiechtenstein et al. (1995) of applying an on-site Coulomb repulsion $U$. It is important to mention that $U$ was applied only to the $f$-states of the rare-earth atom and chosen to reproduce the experimental moment of that atom only. Nevertheless, this substantially improves the moment on the Fe sites (see Table 2). Like experiments we find the moment on rare-earths to be in-plane and oriented perpendicular to the moment on the Fe atoms. The effect of the source-free functional is particularly apparent in these cases since without it the correct moment on either atom cannot be obtained for any choice of $U$.

A material that also requires special attention is LaFeAsO, perhaps the most studied pnictide of all. Reported experimental values of the magnetic moment range from 0.36$\mu_{\rm B}$de la Cruz et al. (2008) to 0.8$\mu_{\rm B}$Li et al. (2010) making it difficult to know to what our theoretical results should be compared. Perhaps the best choice is with the more recent experimental value which lies in between these two extremes, 0.63$\mu_{\rm B}$Qureshi et al. (2010). This experiment was performed at low temperature (2K) which is closest to our theoretical ideal of zero temperature. Another reasonable choice would be 0.8$\mu_{\rm B}$Li et al. (2010) since, like our theoretical work, these experiments are performed on single crystals. In either case, our results with source-free functionals still show a maximum deviation of 25%.

Note that the two steps which comprise this method (i.e. scaling and making the functional source-free) must be performed in combination; each applied alone yields unreasonable results (see Table 3 in Appendix).

A means of summarizing the overall quality of results is the root-mean-square percentage error (RMSPE) which one expects to be reduced on improving the functional, by going, for example, from LSDA to the more sophisticated GGA. To the contrary, the value of RMSPE for LSDA is 90.2% and for GGA is 143%, i.e. the quality of results deteriorates on improving the functional by adding gradients. These errors are greatly reduced to 34.3% with LSDASF and 30.6% with GGASF. Furthermore, once the LSDA${}_{\rm SF}+U$ and GGA${}_{\rm SF}+U$ results are considered the RMSPE are 16% for source-free LSDA and 11% for source-free GGA, indicating that removal of the source term results in GGASF performing better than LSDASF.

It is also appropriate to ask under which circumstances do these source-free functionals fail? For the class of materials which are well known to be close to quantum critical points and for which spin-fluctuations dominate the physicsAguayo et al. (2004); Singh and Mazin (2002), i.e. Ni3Al and ZnZr2, the functional fails in that percentage errors compared to experiments, despite being an improvement over LSDA/GGA, are still large. Another aspect where LSDASF and GGASF are deficient is the fact that they are potential functionals and not energy functionals. Thus one cannot compare the total energies of various structural and magnetic configurations. Hence in the present work we have used experimental crystal structures. However, PBE is well known to be excellent for structural optimization in most cases, including magnetic pnictides, with lattice parameters very close to experiments and small deviations in the position of As atom Yin et al. (2008); Yildirim (2009); Colonna et al. (2011); Mazin et al. (2008b); Sharma et al. (2009). Hence one could perform a full optimization using PBE followed by a calculation using LSDASF or GGASF. To test this we have performed such calculations for BaFe2As2 and LaFeAsO and find that the moment obtained using LSDA and GGA are still highly over-estimated (maximum error of 180%), while the moments obtained using LSDASF and GGASF are again within 25% of experiment (see Table I).

It is enlightening to see how ${\bf B}_{\rm xc}$ of LSDA and GGA and their new source-free versions differ spatially. We plot this for the case of BaFe2As2 in Fig. 2. The field lines for LSDA are unphysical in the sense that they begin and end at different points whereas the source-free field lines are always closed. This means that they have to follow more complicated paths in the crystal, a fact evident from Fig. 2.

To summarize: motivated by an exact property of spin current DFT, we removed the source term from the ${\bf B}_{\rm xc}$ of LSDA and GGA. The spin splitting was also enhanced by a simple scaling of the input magnetization and output field. The resulting functionals were found to produce moments which were in better agreement with experiment. This improvement was particularly pronounced for the pnictides where errors were reduced from 100-200% down to 25% or less. We hope that our findings will spur the development of exchange-correlation energy functionals whose resultant magnetic fields are manifestly source-free.

I Acknowledgments

The authors would like to thank Kieron Burke and Carsten Ullrich for useful discussions.

II Appendix

II.1 Showing ${\bf B}_{\rm xc}$ can be chosen source-free

Let

[TABLE]

where $\hat{H}$ is given in Eq. (1) and the minimization is over all $N$-electron states $|\Psi\rangle$, be the total energy as a functional of the external potential and magnetic field. This can be written as a constrained minimization

[TABLE]

where

[TABLE]

is a universal functional of density and magnetization; $\hat{T}$ and $\hat{V}_{\rm ee}$ are the kinetic and electron-electron interaction parts of the Hamiltonian, respectively. Likewise, the non-interacting kinetic energy functional is defined as

[TABLE]

from which is obtained the exchange-correlation energy functional $E_{\rm xc}[\rho,{\bf m}]\equiv F[\rho,{\bf m}]-T_{s}[\rho,{\bf m}]-E_{\rm H}[\rho]$, where $E_{\rm H}$ is the usual Hartree energy.

However, if we assume that the external magnetic field is physical, i.e. ${\bf B}_{\rm ext}({\bf r})=\nabla\times{\bf A}_{\rm ext}({\bf r})$, and the magnetization tends to zero at large distance, then the classical energy of the external magnetic field can be written as

[TABLE]

Thus we can define another universal functional $\widetilde{F}[\rho,\nabla\times{\bf m}]$ as

[TABLE]

with analogous functionals $\widetilde{T}[\rho,\nabla\times{\bf m}]$ and $\widetilde{E}_{\rm xc}[\rho,\nabla\times{\bf m}]$. On the space of densities obtained from physical external magnetic fields we have that $E_{\rm xc}[\rho,{\bf m}]=\widetilde{E}_{\rm xc}[\rho,\nabla\times{\bf m}]$. This implies that the total energy obtained from both functionals is also the same for physical densities.

The equality of the functionals does not hold in general for densities obtained from external magnetic fields which have a source term. A consequence of this is that the unconstrained functional derivative of $E_{\rm xc}$ with respect to ${\bf m}({\bf r})$ is different for the two functionals, i.e. ${\bf B}_{\rm xc}({\bf r})\neq\widetilde{\bf B}_{\rm xc}({\bf r})$ in general. The functional derivative of $\widetilde{E}_{\rm xc}$ can be further evaluated as

[TABLE]

proving that $\widetilde{\bf B}_{\rm xc}$ is indeed source-free. We note that this derivation also holds for the case where ${\bf A}_{\rm ext}$ and ${\bf m}$ are lattice-periodic. In this case, the surface term which had to be zero in order to derive Eq. (9), sums to zero over the faces of the periodic box.

II.2 Showing $\widetilde{\bf B}_{\rm xc}$ is not the functional

derivative of an $\widetilde{E}_{\rm xc}$

If $\widetilde{\bf B}_{\rm xc}$ was indeed the functional derivative of a scalar, then it would follow that the second derivative would be symmetric in its arguments. It is simple to demonstrate that

[TABLE]

where

[TABLE]

and $f_{\rm xc}({\bf r},{\bf r}^{\prime})$ is a $3\times 3$ matrix. If $M$ had been a Dirac delta function then $\tilde{f}_{\rm xc}$ would be symmetric in its arguments. This is not the case though and hence $\widetilde{\bf B}_{\rm xc}$ is not the functional derivative of an $\widetilde{E}_{\rm xc}$. However, $M$ is highly localized in $|{\bf r}-{\bf r}^{\prime\prime}|$.

II.3 Computational details

The full potential linearized augmented plane wave (LAPW) method implemented within the Elk codeelk is used in the present work. All calculations are performed in the presence of the spin-orbit coupling. To obtain the Pauli spinor states, the Hamiltonian containing only the scalar potential is diagonalized in the LAPW basis: this is the first-variational step. The scalar states thus obtained are then used as a basis to set up a second-variational Hamiltonian with spinor degrees of freedomSingh and Nordström (2006). This is more efficient than simply using spinor LAPW functions, but care must be taken to ensure that there is a sufficient number of first-variational eigenstates for convergence of the second-variational problem. For example, $394$ states per k-point were used for the pnictides to ensure convergence of the second variational step. We use a k-point set of $20\times 20\times 10$ for pinictides and $20\times 20\times 20$ for the rest of the materials. A smearing width of 0.027eV was used.

II.4 Separate scaling and removal of source term

It is also interesting to investigate separately the effect of making the functional source-free and scaling it. We found that the purely source-free LSDA and GGA functionals lead to highly under estimated moments (see Table 3). This is due to suppression of the $z$-projected moment. The magnetization density obtained by purely source-free LSDA and GGA is also highly non-collinear (i.e. x and y projected moments are as significant as Mz). The scaling alone of the LSDA/GGA has, as expected, rather trivial effect of increasing the moment universally since the whole m(r) is uniformly scaled. One could envisage using the scaling parameter so as to reproduce experimental moment for the materials. This, however, has the disadvantage that the scaling parameter is then highly material-dependent. The combination of the two things (scaling and source-free) leads to good agreement with experiments for a wide set of materials and most importantly the scaling factor depends only on the choice of original functional.