---
abstract: 'Correlation effects are important for making predictions in the $\delta $ phase of Pu. Using a realistic treatment of the intra–atomic Coulomb correlations we address the long-standing problem of computing ground state properties. The equilibrium volume is obtained in good agreement with experiment when taking into account Hubbard $U$ of the order 4 eV. For this $U,$ the calculation predicts a 5f$^{5}$ atomic–like configuration with L=5, S=5/2, and J=5/2 and shows a nearly complete compensation between spin and orbital magnetic moments.'
author:
- 'S. Y. Savrasov$^{\ast }$ and G. Kotliar$^{+}$'
- '$^{\ast }$Max-Planck-Institut für Festkörperforschung, Heisenbergstr. 1,  70569 Stuttgart, Germany.'
- |
    $^{+}$Department of Physics and Astronomy and Center for Condensed Matter Theory\
    Rutgers University, Piscataway, NJ 08854–8019
date: July 1999
title: 'Ground State Theory of $\delta $–Pu'
---

Metallic plutonium is a key material in the energy industry and understanding its physical properties is of fundamental and technological interest [@Pu]. Despite intensive investigations [@PuBook], its extremely rich phase diagram with six crystal structures as well as its unique magnetic properties are not well understood. It is therefore of great interest to study the ground state of Pu by modern theoretical methods using first principles electronic structure calculations, which take into account the possible strong correlation among the f electrons.

Density functional theory [@DFT] in its local density or generalized gradient approximations (LDA or GGA) is a well-established tool for dealing with such problems. This theory does an excellent job of predicting ground-state properties of an enormous class of materials. However, when applied to Pu [@Pucalc1; @Pucalc2], it runs into serious problems. Calculations of the high-temperature fcc $\delta$ phase have given an equilibrium atomic volume up to 35% lower than experiment [@Pucalc1]. This is the largest discrepancy ever known in density functional based calculations and points to a fundamental failure of existing approximations to the exchange-correlation energy functional.

Many physical properties of this phase are puzzling: large values of the linear term in the specific heat coefficient and of the electrical resistivity are reminiscent of the physical properties of strongly-correlated heavy-fermion systems. On the other hand, the magnetic susceptibility is small and weakly temperature dependent [@fournier]. Moreover, early LDA calculations [@Pucalc2] predicted $\delta $–Pu to be magnetic with a total moment of 2.1 Bohr magnetons in disagreement with experiments.

The reason for these difficulties has been understood for a long time: Pu is located on the border between light actinides with itinerant 5f–electrons and the heavy actinides with localized 5f electrons[@Johannson] . Near this localization-delocalization boundary, the large intra-atomic Coulomb interaction as well as the itineracy of the f electrons have to be considered on the same footing, and it is expected that correlations must be responsible for the anomalous properties. The parameter governing the importance of correlations in electronic structure calculations is the ratio between effective Hubbard interaction $U$ and the bandwidth $W.$ When the distance between atoms is small, the correlation effects may be not important, since the hybridization, and consequently the bandwidth become large. The low-temperature $\alpha $ phase of Pu has an atomic volume which is 25% smaller than the volume of $\delta $ phase. To the extent that the complicated monoclinic structure of the $\alpha $ phase can be modelled by the simplified fcc lattice, it becomes clear that the LDA or GGA calculations which ignore the large effective $U$ converge to the low volume $\alpha $ phase (for which $U/W < 1$). When volume is increased, this ratio is turned around, and LDA loses its predictive power. This results in the long-standing problem of accurate prediction of the volume of $\delta $–Pu.

In the present work it will be shown that a proper treatment of Coulomb correlations allows us to compute the equilibrium atomic volume of $\delta $–Pu in good agreement with experiment. Moreover, our calculations suggest that there is a nearly complete compensation between the spin and the orbital contributions to the total magnetic moment which is consistent with experiment. Thus the strong correlation effects in $\delta $–Pu are not manifest in the static magnetic properties.

To incorporate the effects of correlations we use the LDA + U approach of Anisimov and coworkers [@LDA+U]. This approach recognizes that the failure of LDA is related to the fact that it omits the Hubbard like interaction among electrons in the same shell, irrespectively of their spin orientation. A new orbital–dependent correction to the LDA functional was introduced to describe this effect. In its most recent, rotationally invariant representation, the correction to the LDA functional has the following form [@LDA+Urecent]: $$\Delta E[n]=\frac{1}{2}\sum_{\{\gamma \}}(U_{\gamma _{1}\gamma _{2}\gamma
_{3}\gamma _{4}}-U_{\gamma _{1}\gamma _{2}\gamma _{4}\gamma _{3}})n_{\gamma
_{1}\gamma _{2}}^{c}n_{\gamma _{3}\gamma _{4}}^{c}-E_{dc}  \label{e2}$$ where $n_{\gamma _{1}\gamma _{2}}^{c}$ is the occupancy matrix for the correlated orbital (d or f), and $\gamma $ stands for the combined spin, ($s$), and azimuthal quantum number,($m$), indexes. The electron–electron correlation matrix $U_{\gamma _{1}\gamma _{2}\gamma _{3}\gamma
_{4}}=\left\langle m_{1}m_{3}\left| v_{C}\right| m_{2}m_{4}\right\rangle
\delta _{s_{1}s_{2}}\delta _{s_{3}s_{4}}$ can be expressed via Slater integrals $F^{(i)} $, $ i=0,2,4,6$ in the standard manner [@LDA+Urecent]. The term $E_{dc}$ accounts for the double counting effects. This scheme, known as the ”LDA+U method”, gives substantial improvements over the LDA in many cases[@LDA+Ureview]. The value of the $U$ matrix is an input which can be obtained from a constrained LDA calculations [@CLDA], or just taken from the experiment. The philosophy of this approach is that the delocalized s p d electrons are well described by the LDA while the energetics of the more localized f electrons require the explicit introduction of the Hubbard U. In the spirit of this method, in this work we will treat the s p d electrons by the generalized gradient approximation (GGA) [@GGA] which is believed to be more accurate that the LDA.

Our implementation of the GGA+U functional is based on the localized–orbital representation provided by the linear–muffin–tin–orbital (LMTO) method for electronic structure calculations [@OKA]. It is important to include spin–orbit coupling effects which are not negligible for 5f electrons of Pu. Our calculations include non-spherical terms of the charge density and potential both within the atomic spheres and in the interstitial region [@Sav]. All low-lying semi-core states are treated together with the valence states in a common Hamiltonian matrix in order to avoid unnecessary uncertainties. These calculations are spin polarized and assume the existence of long–range magnetic order. For simplicity, the magnetic order is taken to be ferromagnetic [@Ferro].

We now report our results on the calculated equilibrium volume. To analyze the importance of the correlation effects, our calculations have been performed for several different values of $U$ varying from 0 to 4 eV. For $U$=4 eV we use standard choice of Slater integrals: $F^{(2)}$=10 eV, $F^{(4)}$=7 eV, and $F^{(6)}$=5 eV [@Pu]. For other U’s we have scaled these values proportionally. For each set of $F$’s a full self–consistent cycle minimizing the LDA/GGA+U functionals has been performed for a number of atomic volumes. We calculated the total energy $E$ as a function of both $V$ and $U$. For fixed $U,$ the theoretical equilibrium, $V_{calc},$ is given by the minimum of $E(V)$. Fig. 1 shows the dependence of the calculated–to–experimental equilibrium volume ratio $V_{calc}/V_{exp}$ as a function of the input $U.$ It is clearly seen that the $U$=0 result (LDA) predicts an equilibrium volume which is 38% off the experimental result and the use of GGA gives only slightly improved result ($V_{calc}/V_{exp}$=0.66). On the other hand, switching on a very large repulsion between 5f electrons obviously leads to an overestimate of the inter-atomic distances. An optimal $U$ deduced from this analysis is found to be close to 4 eV when using the GGA expressions for the exchange and correlation.

This estimate of the intra-atomic correlation energy is in excellent agreement with the published conventional data [@PuU]: The value of $U$ deduced from the total energy differences was found to be 4.5 eV. Atomic spectral data give similar value close to 4 eV. Thus, it is demonstrated how significant it is to properly treat Coulomb correlations in predicting the equilibrium properties of this actinide.

We now discuss the calculated GGA+U electronic structure of $\delta $–Pu for the optimal value of $U$=4 eV. Fig. 2 shows the energy bands in the vicinity of the Fermi level. They originate from the extremely wide 6s–band strongly mixed with the 5d–orbitals which are strongly hybridized among themselves. The resulting band complex has a bandwidth of the order of 20 eV. On top of this structure there exist a weakly hybridized set of levels originating from the 5f–orbitals.

In order to understand the physics behind the formation of spin and orbital moment in the f–shell, it is instructive to visualize the orbital characters as ”fat bands”[@Fat]. The one–electron wave function has an expansion $\psi _{{\bf k}j}({\bf r})=\sum A _{lms}^{{\bf k}j} \phi _{lms}({\bf r})$ where $\phi _{lms}({\bf r})$ are the solutions of the radial Schrödinger equation normalized to unity within atomic sphere. The information about partial $lms$ characters of the state with given ${{\bf k}j}$ is contained in the coefficients $|A _{lms}^{{\bf k}j}|^2$. Sum over all $lms$ in the latter quantity gives unity (we neglect by a small contribution from the interstitial region) since one band carries one electron per cell. At the same time, sum over all $j$ in $|A _{lms}^{{\bf k}j}|^2$ is also equal to one since each $lms$ describes one state. Fixing a particular $lms$, we can visualize this partial character on top of the band structure by widening each band $E_{{\bf k}j}$ proportionally to $|A_{lms}^{{\bf k}j}|^2$. A maximum width $\Delta $ which corresponds to $\sum _j |A_{lms}^{{\bf k}j}|^2$=1 should be appropriately chosen. Now, at the absence of hybridization, each band originates from a particular $lms$ state, and therefore there exists only one ”fat band” for given $lms$ which has the maximum width $\Delta $. When hybridization is switched on, there can be many bands which have the particular $lms$ character, they will all be widened as $\Delta |A_{lms}^{{\bf k}j}|^2$, while the sum of individual widths for all bands is now equal to $\Delta $. The width of the band is then proportional to its $lms$ character. This technique [@Fat] gives us an important information on the distribution of atomic levels as well as their hybridization in a solid. For f–electrons of Pu, it is convenient to work in the spherical harmonics representation in which the f–f block of the Hamiltonian is found to be nearly diagonal.

The result of such ”fat bands” analysis for 5f–orbitals is shown on Fig. 2. In order to distinguish the states with different $m$’s and spins we have used different colors. (-3 $\equiv $ red, -2 $\equiv $ green, -1 $\equiv $ blue, 0 $\equiv $ magenta, +1 $\equiv $ cyan, +2 $\equiv $ yellow, +3 $%
\equiv $ gray). Two consequences are seen from this coloured spaghetti: First, spin–up and spin–down bands are all split by the values governed by the effective $U$ and the occupancies of the levels. These are just the well known lower and upper Hubbard subbands. Second, only spin–up states with $m$=-3,-2,-1,0, and +1 are occupied while all other states are empty. This simply implies 5f$^{5}$ like atomic configuration for $\delta $–Pu which is filled according to the Hund rule. Note that spin-orbit coupling is crucial for the existence of such an occupation scheme. In the absence of spin orbit coupling the occupancies of the levels with $\pm m$ are the same which automatically produces zero orbital moment.

Besides providing the experimentally observed volume of the $\delta $–Pu, our calculation suggests a simple picture of the electronic structure of this material and sheds new light on its puzzling physical properties discussed in the introduction.

The ”fat bands” shown in Fig. 2, suggest a physical picture in which the f electrons are in atomic states forming a multiplet of the 5f$^5$ configuration with L=5, S=5/2 spin orbit coupled to J=5/2. Crystal fields can split this multiplet into a doubly degenerate state transforming according to $\Gamma_7$ representation of the cubic group and a quartet transforming according to $\Gamma_8$ representation [@PuBook], but cannot remove the orbital degeneracy completely. In a dynamic picture, the f electrons will fluctuate between the degenerate configurations, until this degeneracy is removed by the Kondo effect with the delocalized electrons in the s-p-d band. Therefore the experimentally observed characteristic heavy fermion behavior in this system, namely, the large high–temperature resistivity and the large linear T coefficient of the specific heat arises naturally in this picture [@PuBook].

This heavy fermion behavior however should not appear in the magnetic susceptibility. The GGA + U calculation suggests that the magnetic moment of the low lying configurations of the f electrons is much smaller than the $5 \mu_B $ that one would obtain if we ignore the orbital angular momentum and assumed that the spin is fully polarized. The combination of strong Coulomb interactions and spin–orbit coupling reduce the crystal–field effects and give rise to a large magnetic moment which nearly cancels the spin moment. In an atomic picture, the $5f^5$ configuration with L=5, S=5/2 and J=5/2 has a total moment given by $M_{tot}$ $=\mu _{B}gJ=0.7\mu
_{B}$, with Lande’s g-factor of 0.28. This simple relation breaks down in the presence of crystal fields, but in both the $\Gamma_7$ or the $\Gamma_8$ representation the g factor is further reduced from the atomic estimate.

The GGA + U calculation gives a spin moment $ M_{S}$ 5.1 Bohr magnetons which is slightly increased relative to the 5 Bohr magnetons expected in a pure f$^{5}$ atomic configuration due to the polarization of the band electrons outside the muffin–tin shell. Evaluation of the orbital and total moments is in general a more difficult problem [@Cal]. We have estimated the average of $\langle {\bf k}j\left|
l_{z}\right| {\bf k}j\rangle $ summed over all occupied states $|{\bf k}%
j\rangle .$ This leads to a value $M_{L}$=-3.9 $\mu
_{B.}$ for the orbital moment. The total calculated moment $M_{tot}=M_{S}+M_{L}$ is thus reduced to 1.2 $\mu _{B}$ . It worth noting that an atomic analogue of this estimate, $ M_{tot}$ $=\mu _{B}\left| L-2S\right| $ gives exactly zero for our 5f$^{5}$ ground state[@Another]. A remarkable outcome of the calculation is clearly seen: [*A nearly complete compensation of spin and orbital contributions occurs*]{} for metallic $\delta $–Pu.

In this picture the weakly temperature independent susceptibility which is observed in $\delta $–Pu [@PuBook; @fournier] is the result of a a very large Van Vleck contribution and of a very small magnetic moment which results from the near cancellation of two large orbital and spin moments.

In a recent paper [@Eriksson] Eriksson and coworkers introduced a different approach to the anomalous properties of $\delta$ plutonium. In their calculation a fraction of the f-electrons is treated as core electrons while the rest are treated as delocalized. Using a combination of the constrained LDA calculation with the atomic multiplets data they obtain the correct equilibrium volume when four f-electrons are part of the core, while one f electron is itinerant. The basic difference between the methods, is the different treatment of the f electrons. In this paper all the f electrons on equal footing, and their itineracy is reduced by the Hubbard U relative to the predictions of LDA or GGA calculations. Since our approach and that of Eriksson et. al. lead to different ground state configurations of the localized f electrons ($f^5$ vs $f^4$), further experimental spectroscopic studies of $\delta $–Pu would be of interest.

In conclusion, using a realistic value of the Hubbard $U$=4 eV incorporated into the density functional GGA calculation, we have been able to describe ground state properties of $\delta $–Pu in good agreement with experimental data. This theory correctly predicts the equilibrium volume of the $\delta $ phase and suggest that nearly complete cancellation occurs between spin and orbital moments. The main shortcoming of the present calculation is the assumed long range spin and orbital order. This is the essential limitation of the LDA+U approach (or of any [*static*]{} mean field theory ) in order to capture the effects of correlations this approach it has to impose some form of long–range order. Static mean field theories are unable to capture subtle many–body effects such as the formation of local moments and their subsequent quenching via the Kondo effect. These deficiencies will be removed by ab initio [*dynamical*]{} mean field [@DMFT] calculations for which codes are currently being developed. We believe however, that our main conclusions, i.e. that correlations lead to the correct lattice constant and a reduction of the moment, relative to the LDA results, are robust consequences of the strong correlations presented in this material, and will be reproduced by more accurate treatments of the electron correlations.

The authors are indebted to E. Abrahams, O. K. Andersen, O. Gunnarsson, A. I. Lichtenstein, and J. R. Schrieffer for many helpful discussions. The work was supported by the DOE division of basic energy sciences, grant No. DE-FG02-99ER45761.

For a review, see, [*e.g*]{}., [*Plutonium - A General Survey*]{}, edited by K. H. Lieser (Verlag Chemie, 1974)

For a review, see, [*e.g.*]{}, [*The Actinides: Electronic Strucuture and Related Properties*]{}, edited by A. J. Freeman and J. B. Darby, Vols. 1 and 2 (Academic Press, New York, 1974);

For a review, see, [*e.g*]{}., [*Theory of the Inhomogeneous Electron Gas*]{}, edited by S. Lundqvist and S. H. March (Plenum, New York, 1983).

For recent calculations, see, P. Söderlind, O. Eriksson, B. Johansson, and J. M. Wills, Phys. Rev. B [**50**]{}, 7291 (1994); J. C. Boettger, M. D. Jones, R. C. Albers, and D. J. Singh, cond-mat/9811199

I. V. Solovyev, A. I. Liechtenstein, V. A. Gubanov, V. P. Antropov, and O. K. Andersen, Phys. Rev. B [**43**]{}, 14414 (1991);

S. Meot-Raymond and J. Fournier J. Alloys and Compounds [**232**]{} 119 (1996).

B. Joahnsson J. Magn. Magn. Mater 47 (1985) 231;

V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B [**44**]{}, 943 (1991);

A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 (1995);

For a review, see, V. I. Anisimov, F. Aryasetiawan, and A. I. Liechtenstein, J. Phys.: Cond. Matter [**9**]{}, 767 (1997);

P. H. Dederichs, S. Blügel, R. Zeller, and H. Akai, Phys. Rev .Lett. [**53**]{}, 2512 (1984);

J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. [**77**]{}, 3865 (1996).

O. K. Andersen, Phys. Rev. B [**12**]{}, 3060 (1975).

A short description of the method can be found in: S. Y. Savrasov, Phys. Rev. B [**54**]{}, 16470 (1996);

In fact, our own and previous [@Pucalc2] total energy calculations suggest that the antiferromagnetic state has slightly lower energy. A realistic theory should not assume any long–range order at all. Unfortunately, it is very hard to treat short–range order with the existing techniques (through the use of supercells), while alternative methods based on dynamical mean–field theory and Green functions are still in developmental stage.

See, [*e.g*]{}., J. P. Desclaux and A. J. Freeman, in[*Handbook on the Physics and Chemistry of the Actinides*]{}, edited by A. J.. Freeman and G. H. Lander (Elsevier, Amsterdam, 1984), Vol. 1;

O. Jepsen and O. K. Andersen, Z. Phys. B [**97**]{}, 35 (1997);

M. Singh, J. Callaway, and C. S. Wang, Phys. Rev. B[**14**]{}, 1214 (1976);

Note that we have found another metastable solution of the functional \[e2\] corresponding to the occupation of the levels with $%
m=-3,-2,-1,+1,+2.$ The orbital moment of this configuration is about 3 and carries a large magnetic moment. The energy of this configuration is however found to lie 0.5 Ry higher than the energy of the ground state configuration with high orbital moment.

O. Eriksson, J. D. Becker, A. V. Balatsky, and J. M. Wills, J. Alloys and Comp. [**287**]{}, 1 (1999);

A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. [**68**]{}, 13 (1996);

FIGURE CAPTIONS

Fig.1. Calculated theoretical volume (normalized to the experiment) of $\delta $–Pu as a function of the Hubbard $U$ within the LDA+U and GGA+U approaches.

Fig.2. Calculated energy bands of $\delta $–Pu using GGA+U method with $U$=4 eV. Spin and orbital characters of the f-bands are shown with the color: ($m$=-3 $\equiv $ red, -2 $\equiv $ green, -1 $\equiv $ blue, 0 $\equiv $ magenta, +1 $\equiv $ cyan, +2 $\equiv $ yellow, +3 $%
\equiv $ gray). Boxes from the left and from the right show approximate positions of the f levels.
