Ab initio full charge-density study of the atomic volume of -phase Fr, Ra, Ac, Th, Pa, U, Np, and Pu

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1 PHYSICA EVIEW B VOUME 55, NUMBE 8 15 FEBUAY 1997-II Ab initio fu charge-density study of the atomic voume of -phase Fr, a, Ac, Th, Pa, U, Np, and Pu. Vitos and J. Koár esearch Institute for Soid State Physics, P. O. Box 49, H-1525 Budapest, Hungary H.. Skriver Center for Atomic-scae Materias Physics and Physics Department, Technica University of Denmark, DK-28 yngby, Denmark eceived 2 November 1995 We have used a fu charge-density technique based on the inear muffin-tin orbitas method in firstprincipes cacuations of the atomic voumes of the ight actinides incuding Fr, a, and Ac in their owtemperature crystaographic phases. The good agreement between the theoretica and experimenta vaues aong the series support the picture of itinerant 5 f eectronic states in Th to Pu. The increased deviation between theory and experiment found in Np and Pu may be an indication of correation effects not incuded in the oca density approximation. S I. INTODUCTION To date, it is generay accepted that the ight actinide metas form a transition series with a substantia f bonding. Part of the support for such an itinerant picture of the 5 f eectrons is suppied by the fact that modern one-eectron theory yieds atomic voumes for these eements which agree quite we with the experimenta vaues. 1,2 Furthermore, since f bonding is expected to give rise to compex crysta structures the appearance of tetragona, orthorhombic, and monocinic structures in the actinide series ends additiona support to the itinerant 5 f picture. In fact, even the fcc structure of Th is caused by the 5 f eectrons, 3 and therefore a the ight actinides from Th to Pu are considered to form a 5 f transition series. In spite of their success, the vaidity of the origina inear muffin-tin orbitas MTO cacuations 1 may be questioned because the authors used the atomic sphere approximation ASA, negected spin-orbit interaction, and treated ony the cose-packed fcc phase. It is therefore not surprising that the equiibrium voumes of the ight actinides have been recacuated severa times, graduay reducing the number of approximations invoved. 4 9 However, athough the cacuationa schemes have progressed consideraby it is ony recenty that one has been abe to cacuate the atomic voumes of U, Np, and Pu in their phases. 1 To cacuate the equiibrium voumes of the phases of the actinide metas with up to 16 atoms per primitive ce one needs not ony efficient computer codes but aso to go beyond the ASA. One approach is to use a fu-potentia MTO method as is done by Wis and co-workers. 6,8,9 Another is to retain some of the efficiency of the ASA by using a sphericay symmetric potentia to cacuate the kinetic energy and to generate the nonsphericay symmetric charge density that is subsequenty used in the evauation of the remainder of the energy functiona. In the present work we have adopted the atter fu charge-density FCD approach, which was originay impemented in order to treat attice distortions as we as open crysta structures such as those found in the actinide series. The FCD method has been appied successfuy in the cacuation of the surface energies of the 4d metas 11 and the ight actinides, 12 as we as the atomic voumes and eastic constants of the 4d metas. 13 The cacuated atomic voumes of the ight actinides found in the iterature show considerabe scatter. For instance, the oca-density atomic radius for fcc Th varies from 3.79 bohr in the origina MTO-ASA cacuation 1 to 3.52 bohr in the recent fu-potentia cacuation. 9 The question is therefore whether one can trust the ASA resut, which deviates by ony 1% from the experimenta vaue of 3.76 bohr, 14 or whether the presumaby more accurate fu-potentia resut, which deviates by as much as 6%, means that the ocadensity approximation is inadequate for the ight actinides. The fu-potentia cacuations 9 aso incude resuts obtained by the generaized gradient correction to the oca-density approximation and this improves the agreement with experiment for Th to 3%. However, in view of the success of the fu-potentia cacuations for the 4d series 15 where the gradient correction approximation for Zr, which has approximatey the same number of d eectron as Th, eads to compete agreement with experiment, deviations of 6% and 3% for Th indicate either inadequate numerica approximations or a faiure of the oca density and the generaized gradient correction approximations. In this situation it is important to evauate the atomic radii of the ight actinides by means of a one-eectron method that is different from the previousy appied methods and thereby obtain a second opinion, as it were. The FCD method is one such method that has been thoroughy tested, e.g., in cacuations of the atomic radii of the 4d metas. It is found that the oca-density FCD radii 13 of the 4d metas from Y to Pd on the average are 1% arger than the corresponding ocadensity fu-potentia vaues 15 and the deviation is traced to the fact that the kinetic energy in the FCD approach is cacuated within the ASA. Since the remainder of the energy functiona is evauated correcty, aso in crysta structures that are not cose packed, we expect the FCD approch to yied atomic radii that are within 1% of the true oca-density radii for the ight actinides /97/558/49476/$ The American Physica Society

2 4948. VITOS, J. KOÁ, AND H.. SKIVE 55 II. FU CHAGE-DENSITY METHOD In the fu charge-density method 11 the kinetic energy of the oca-density functiona is cacuated competey within the ASA, except for the incusion of the so-caed combined correction, 16,17 by means of a sphericay symmetrized charge density, whie the Couomb and the exchange and correation contributions are cacuated by means of the compete, nonsphericay symmetric charge density within nonoverapping, space-fiing Wigner-Seitz ces. In the first appication 11,12 the charge density was expressed in terms of an MTO-ASA one-center expansion corrected in the outer corners of the Wigner-Seitz WS ce by a simpe poynomium to maintain normaization over the ce. Here, we improve the accuracy of the one-center expansion by the use of the tai-canceation theorem to correct the charge density in the overap regions. In the foowing we give a brief outine of the method with specia emphasis on the construction of the charge density. A. Energy functiona The FCD energy functiona is based on a sphericay symmetric atomic-sphere potentia. Within the oca-density approximation we define where T ASA E FCD nr T ASA Enr, 1 E F N d 1 n,asa rv eff rd 3 r 2 4 S is the kinetic energy per site evauated in the ASA and Enr v ext rnrdr 2 1 nrnr drdr rr xc nrnrdr. In these equations E F is the Fermi eve, N () the siteprojected one-eectron-state density, S the atomic Wigner- Seitz radius, n,asa (r) the component of the eectronic charge density at the site, V eff (r) the effective one-eectron potentia corresponding to the ASA energy functiona, v ext (r) the potentia from the nucei, n(r) the tota eectronic charge density that is constructed from the output of a sefconsistent ASA cacuation and refects the proper symmetry of the attice, and xc stands for the exchange-correation energy density. B. Charge density It was shown by Andersen et a. 18,19 that even for open structures such as the diamond structure one may obtain good charge densities by means of an MTO-ASA potentia. In their approach, however, the output charge density is given in a muticenter form which requires doube attice summations and is therefore ess suitabe in tota energy cacuations. Our aim is to rewrite the output ASA charge density in a one-center form 3 n r n r Y rˆ, 4 where is shorthand notation for (,m), Y is a rea harmonic, and r is measured from the origin of the ce at, i.e., r r. This expression is simpe to evauate and we suited for the integration in the Wigner-Seitz ce at. The muffin-tin MT orbitas may be defined 19 as E,,r Y rˆ E,r P E,j,r for r S n,r for r S, where (E,r ) is soution of the radia Schrödinger equation, 2 EV is the kinetic energy in the interstitia region, j and n are, respectivey, reguar and irreguar soutions at the origin of the radia wave equation for the constant potentia V the muffin-tin zero, and P (E,) is the potentia function. If the secuar equation is satisfied and the MT spheres do not overap, the wave function is given within one MT sphere either by the muticenter or by the one-center form E,r 5 E,r E,,r u 6 E,,r u if r S. 7 In the case of overapping spheres, there is, in the region of overap, an uncanceed remainder which is the superposition of the functions f E,,r E,,r n,r f E,,r Y rˆ r coming from the neighboring sites. 19 Here, f E,,r E,r P E,j,r n,r, 9 and (r ) is a step function, which is 1 inside and zero outside the MT sphere at. Thus, in the case of overapping spheres centered at the wave function at can be written as 19 E,r E,,ru f E,,r u if rs. 8 1 Here and in the foowing we use the notation r for r. In the conventiona MTO-ASA method with energyindependent MTOs and with 2 the function f (r ) has the form

3 55 Ab initio FU CHAGE-DENSITY STUDY OF THE f r E,r E,r E C r 1 S 2 /S1 E C, 11 where (E,r ) is the orthogona energy derivative of the normaized function (E,r ), E,C,, and are the usua MTO potentia parameters, whie S is the average Wigner-Seitz radius. If the radia function f (r ) may be approximated by a inear combination of Besse functions f r i i C K k i r, 12 we use the spherica Hanke functions of first kind with k i 2 2, we may use the expansion theorem to express the functions centered at the neighboring sites with the coordinates measured from the origin f r Y rˆ f ry rˆ. 13 For the (r ) function this expansion may be carried out anayticay, it is shown in the Appendix. After summing up the square of the wave functions for the occupied states and after some manipuation we obtain the foowing expresssion for the partia component of the charge density in Eq. 4: n r G rm. 14 Here, the function G (r) may be expressed in terms of (E,r), (E,r), f (r), and the expansion coefficients of the function (r ). These expressions and the definition of the energy moments, m are given in the Appendix. A few exampes of the appication of the above formua for cacuating charge density contours are given in ef. 21. C. Couomb energy To cacuate the Couomb energy in Eq. 3 we devide the soid into nonoverapping, space fiing ces around each attice site. The tota eectrostatic energy of the system may be divided into contributions E beonging to the ce at, which are compounded of intrace and interce terms. For simpicity, we ony give expressions for the contribution corresponding to the ce centered at the origin, i.e., E E intra n E inter Q, 15 where we indicate that the interce energy contribution depends on the mutipoe moments defined as Q 41/2 21 WS ce at r S n r Y rˆdr. 16 The intrace energy may be determined in the usua way, e.g., by numerica soution of the -dependent Poisson equation inside the bounding spheres, whie the interce interaction energy beonging to the ce at the origin may be written in the foowing form: 22,23 E inter Q 1 1 2S 21 b S Y bˆ, 421!! Q 21!!21!! C,, S, b Q, 17 where S, () are the conventiona MTO structure constants and C, a rea-harmonic Gaunt coefficient. In Eq. 17 runs over the attice vectors. For neighboring ces with overapping bounding spheres with the radii S C and S C ) the vector b must be arger than zero to ensure the convergence of the summations. An optima chioce for b is b S C S C. 18 A resonabe choice for the parameter is described in ef. 23. D. Computationa detais In the cacuations we used the scaar-reativistic, secondorder MTO Hamitonian within the frozen-core approximation and incuded the combined correction The vaence eectrons were treated sef-consistenty within the ocadensity approximation DA with the Perdew-Zunger parametrization 24 of the resuts of Ceperey and Ader. 25 We incuded the 6p semicore states first pane together with the 5 f,6d, and 7s states second pane as band states. In the first pane we down-foded 26,27 the s, d, and f states and in the second ony the p states. This procedure accounts correcty for the important weak hybrization in the occupied parts of the bandstructure and reduces the rank of the eigenvaue probem to that of the number of active orbitas, i.e., three for the ower pane and 13 for the upper pane. The k-point samping was performed on a uniform grid in the irreducibe wedge of the Briouin zones of the structures using 225, 675, 392, 75, and 48 k points for Th, Pa, U, Np, and Pu, respectivey. III. ATOMIC VOUME OF THE ACTINIDES The resuts of the fu charge-density cacuations for the eary actinides incuding Fr, a, and Ac in their owtemperature crystaographic phases are presented in Fig. 1 and Tabe I. It is seen that the agreement between the cacuated equiibrium atomic Wigner-Seits radii and those obtained experimentay 14 is quite satisfactory over the whoe series. Because the cacuations treat the 5 f eectrons as itinerant and not as ocaized, coreike states, in which case one finds extremey high voumes, 5 one may concude that the 5 f eectrons in the eary actinides, i.e., before Am, are bonding eectrons and that oca-density theory gives a good de-

4 495. VITOS, J. KOÁ, AND H.. SKIVE 55 FIG. 1. Fu charge-density FCD resuts for the equiibrium Wigner-Seitz radii of the first eight eements in the seventh row of the periodic tabe compared with experimenta vaues 14. The cacuations have been performed in the DA for the crystaographic phases indicated in Pearson notation at the top of the figure. scription of the ground state of these metas. In the comparison between theory and experiments one finds a systematic overestimate of the binding, i.e., owering of the cacuated voume reative to the experimenta voume, as one approaches Pu. One aso observes that this trend is unaffected by the appearance of the distorted phases in Pa Pu. The deviation is therefore most easiy interpreted as due to correation effects not incuded in the presenty used oca-density approximation. However, before one reaches such a concusion one must ascertain that the deviation may not be caused by approximations, numerica or otherwise, in the technique used to sove the effective one-eectron Kohn- Sham equations, i.e., that the cacuated atomic voumes represent a cose approximation to the true oca-density vaues. The ony major approximation that remains in our impementation of the fu charge-density method is the negect of spin-orbit couping. The effect of this approximation has been addressed in MTO-ASA cacuations, 4,5 Tabe I, and in fu-potentia cacuations. 6 Brooks 4 found an increase in atomic voume between Np and Pu and concuded that part of the deviation in Pu coud be accounted for by spin-orbit couping. A simiar concusion may be reached from the cacuations by Söderind et a. 5 In contrast, the work of Wis and Eriksson, 6 based on fuy reativistic cacuations, concuded that spin-orbit couping did not ead to an increase in atomic voume between Np and Pu and therefore coud not account for the deviation between theory and experiments in these metas. ecenty, Wis and co-workers 6,8,9 have cacuated the atomic voumes of Th, Pa, U, and Np in their phases by means of the fu-potentia MTO method. As may be seen in Fig. 2 and Tabe I, where we compare the fu potentia and fu charge-density resuts, the agreement with the present atomic voumes is not good. In fact, because the fu-potentia method invoves fewer approximations than the present fu charge-density technique, it woud appear that the 6% difference in cacuated atomic radius 18% in voume for Th disquaifies the present vaues for the whoe series as the true oca-density resuts. Obviousy the cose agreement between theory and experiment obtained in the TABE I. Atomic Wigner-Seitz radii in bohr for the ight actinides in their ow-temperature crystaographic phases, upper pane, and in the fcc structure, ower pane. The cacuated vaues were a obtained within the oca-density approximation. In the upper pane we ist fu charge-density and fu-potentia resuts and in the ower pane ASA resuts. Fr a Ac Th Pa U Np Pu Structure ci2 ci2 cf4 cf4 ti2 oc4 op8 mp16 Expt. a Present Wis b Eriksson c Söderind d Structure cf4 cf4 cf4 cf4 cf4 cf4 cf4 cf4 Skriver e Söderind f Brooks g Söderind h a eference 14. b Tabe I, ef. 6. c Figure 1, ef. 8 and Fig. 11, ef. 6. d oca-density resuts, ef. 9. e Pure ASA resuts, ef. 1. f ASA pus combined correction, ef. 5. g ASA pus spin orbit, ef. 4. h ASA pus combined correction and spin orbit, ef. 5.

5 55 Ab initio FU CHAGE-DENSITY STUDY OF THE the eaier actinides, Fr U, where spin-orbit couping does not affect the voumes in a significant way. Furthermore, the fact that the cacuated voume is a decreasing function of the atomic number simiar to the fcc resuts means that in the DA the crysta structures are not responsibe for the increasing difference between theory and experiment as one approaches Pu. Based on earier MTO-ASA cacuations it seems that spin-orbit couping may expain haf of the discrepancy in Pu and that therefore the remainder may be attributed to correation effects not incuded in the ocadensity approximation. However, such a concusion must await cacuations incuding not ony the correct crysta structures for Np and Pu but aso spin-orbit couping. FIG. 2. Comparison between cacuated Wigner-Seitz radii for the ight actinides in their crystaographis phases. Open circes from ef. 6, open diamonds from ef. 9, open trianges from ef. 8, and fied squares from the present FCD cacuations. Open circes with soid ine are experimenta vaues from ef. 14. present work is not a measure of the accuracy of the physica and numerica approximations of the FCD approach, but if numerica accuracy can be estabished it is a measure of the vaidity of the oca-density approximation. The four sets of MTO-ASA cacuations presented in Tabe I assumed the fcc structure, 1 incuded spin-orbit couping, 4 combined-correction, 5 or combined-correction and spin-orbit couping. 5 These cacuations a made use of the pressure reation 28 rather than the tota energy to obtain equiibrium radii, and this procedure is reativey insensitive to the pressence of the 6p semicore states to be discussed beow. The atomic radii obtained in these cacuations agree with each other to within ess than 1%, which may be considered satisfactory. The ony exception is Pu, where spinorbit couping eads to an increase in atomic radius of 5%. Thus, if one negects Pu, it seems that sma deviations in numerica techniques resuts in a scatter of approximatey 1% in the cacuated atomic radii. Hence, we may consider this number as an ad hoc measure of confidence. Since we cannot assess the accuracy of the numerica impementation of the fu-potentia method 6,8,9 we must judge the accuracy of the FCD cacuations on the basis of resuts for eements other than the actinides. ecenty, we have cacuated the atomic voumes of the 4d metas by the exact procedure used here for the ight actinides, i.e., 4p semicore, two energy panes, and s, p, d, and f orbitas, and compared the resuts with fu-potentia cacuations. We find that the FCD method yieds atomic voumes for the metas Y, Zr, Nb, Mo, Tc, u, h, and Pd, which on the average are ony 1% arger than the corresponding fu-potentia cacuations. 15 Hence, we expect the fu-charge density resuts presented in Figs. 1 and 2 to represent an accurate estimate of the true oca-density atomic voumes of the actinide metas in their phases. IV. CONCUSION Based on the above considerations, it appears that the fu charge-density resuts, shown in Fig. 1, represent a cose approximation to the true oca-density vaues, at east for the ACKNOWEDGMENTS This work was supported in part by Grant No. EB- CIPA-CT of the Commission of the European Communities and esearch Project Nos. OTKA 295 and 1674 of the Hungarian Nationa Scientific esearch Foundation. Center for Atomic-scae Materias Physics is sponsored by the Danish Nationa esearch Foundation. APPENDIX The step function (r ) in Eq. 8 may be expanded in terms of rea harmonics, i.e., r ry rˆ, where the partia components (r) are k r21 1 1/2 2 1 k/2 k 2k 1 r2 2 S 2 2r 2k1 2k1 D m ˆ and D m (ˆ ) are the matrix eements of finite rotations defined, for exampe, in ef. 29. The energy moments in Eq. 14 are given by m j * j,occ. E j E p b E j E q j b j where b denote the eigenvectors and E j the one-eectron energies obtained from the band cacuation. The quantities G (r) in Eq. 14 are defined as G r C r r r C F r C F F r r r C F r,,

6 4952. VITOS, J. KOÁ, AND H.. SKIVE 55 and 1 G r C r r 1 G C F r r, 1 rg r, 11 G r C r r. Here F r C f r r, with f (r) defined from Eq. 13. In these expressions and run over neighbors with overapping ASA spheres. 1 H.. Skriver, O.K. Andersen, and B. Johansson, Phys. ev. ett. 41, M.S.S. Brooks, B. Johansson, and H.. Skriver, in Handbook on the Physics and Chemistry of the Actinides, edited by A.J. Freeman and G.H. ander North-Hoand, Amsterdam, 1984, Vo. 1, p B. Johansson,. Ahuja, O. Eriksson, and J.M. Wis, Phys. ev. ett. to be pubished. 4 M.S.S. Brooks, J. Phys. F 13, P. Söderind,. Nordström,. Yongming, and B. Johansson, Phys. ev. B 42, J.M. Wis and O. Eriksson, Phys. ev. B 45, J. van Ek, P.A. Sterne, and A. Gonis, Phys. ev B 48, 16 28, O. Eriksson, J.M. Wis, P. Söderind, Joost Mesen,. Ahuja, A.M. Boring, and B. Johansson, J. Aoys Comp. 213/214, P. Söderind, O. Eriksson, B. Johansson, and J.M. Wis, Phys. ev B 5, P. Söderind, J.M. Wis, B. Johansson, and O. Eriksson, Phys. ev. B to be pubished. 11. Vitos, J. Koár, and H.. Skriver, Phys. ev. B 49, J. Koár,. Vitos, and H.. Skriver, Phys. ev. B 49, Vitos, J. Koár, and H.. Skriver unpubished. 14 D.A. Young, Phase Diagrams of the Eements University of Caifornia Press, Berkeey, V. Ozoins and M. Köring, Phys. ev. B 48, O.K. Andersen, Phys. ev. B 12, H.. Skriver, The MTO Method Springer-Verag, Berin, O.K. Andersen, Z. Pawowska, and O. Jepsen, Phys. ev. B 34, O.K. Andersen, A.V. Postnikov, and S.Yu. Savrasov, in Appications of Mutipe Scattering Theory in Materias Science, edited by W.H. Buter, P.H. Dederichs, A. Gonis, and.. Weaver Materias esearch Society, Pittsburgh, PA, 1992, pp O.K. Andersen and.g. Wooey, Mo. Phys. 26, Vitos, J. Koár, and H.. Skriver, in Stabiity of Materias, Vo. 355 of NATO ASI Series E: Appied Sciences, edited by A. Gonis, P.E.A. Turchi, and J. Kudrnovsky Kuwer Academic Pubishers, The Netherands, 1995, pp A. Gonis, E.C. Sowa, and P.A. Sterne, Phys. ev. ett. 66, Vitos and J. Koár, Phys. ev. B 51, J. Perdew and A. Zunger, Phys. ev. B 23, D.M. Ceperey and B.J. Ader, Phys. ev. ett. 45, O.K. Andersen, O. Jepsen, and D. Götze, in Highights of Condensed-Matter Theory, edited by F. Bassani, F. Fumi, and M.P. Tosi North-Hoand, New York, W.. ambrecht and O.K. Andersen, Phys. ev. B 34, D.G. Pettifor, Commun. Phys. 1, A.. Edmonds, Anguar Momentum in Quantum Mechanics, edited by Eugene Wigner and obert Hofstadter Princeton University Press, Princeton, 1957.