PHYSICAL REVIEW B VOLUME 60, NUMBER 6 1 AUGUST 1999-II Noncollinear magnetic ordering in small chromium clusters C. Kohl and G. F. Bertsch Institute for Nuclear Theory­Department of Physics, University of Washington, Seattle, Washington 98195 Received 16 February 1999 We investigate noncollinear effects in antiferromagnetically coupled clusters using the general, rotationally invariant form of local spin-density theory. The coupling to the electronic degrees of freedom is treated with relativistic nonlocal pseudopotentials and the ionic structure is optimized by Monte Carlo techniques. We find that small chromium clusters (N 13) strongly favor noncollinear configurations of their local magnetic moments due to frustration. This effect is associated with a significantly lower total magnetization of the noncollinear ground states, ameliorating the disagreement between Stern-Gerlach measurements and previous collinear calculations for Cr12 and Cr13 . Our results further suggest that the trend to noncollinear configurations might be a feature common to most antiferromagnetic clusters. S0163-1829 99 10929-9 I. INTRODUCTION maximal spin alignment in the atom (3d54s1): all valence electrons have parallel spins, i.e., the total magnetization is Magnetic properties of transition-metal clusters have be- S 3. This leads to a variety of unique effects including an come the subject of intensive research, both from the unusually short dimer bonding length or a repulsion for FE experimental1­4 and theoretical point of view.5­8 One of the coupling at small distances.21 The other reason why we most interesting and challenging aspects of that field is the chose chromium is the possibility to compare with the thor- subtle interplay between geometric structure and magnetic ough ab initio study of Cr clusters by Cheng and Wang.22 ordering that has mostly been investigated for ferromagnetic These authors recently employed the conventional LSDA for 3d clusters and 4d clusters. Finite-size effects and a reduced Nat 15, finding all clusters to be antiferromagnetically dimensionality often lead to a significantly different mag- coupled. Their structures can serve as a benchmark to check netic behavior from the bulk with clusters having enhanced our computations concerning the restriction to collinear atomic moments,9,10 larger anisotropy energies,11 or an al- configurations-an aspect that is crucial to extract the impor- tered temperature dependence of the magnetization.12 tance of noncollinear effects. Almost all theory to date employed the local spin-density In Sec. II we outline the fundamentals of our theoretical approximation LSDA with the assumption that the spin- approach and briefly motivate the structure optimization. density matrix is diagonal in some particular frame. In that This involves the rotationally invariant LSDA to find the special case, the spins are automatically collinear along a electronic ground state and a relativistic, nonlocal pseudopo- fixed quantization axis. The only generalized spin-density tential for their interaction with the ions. In Sec. III we dis- calculation for clusters that treats the electron spin as a vec- cuss some numerical tests and present our results. We find tor observable and a function of position has recently been noncollinear spin configurations for all investigated clusters. performed by Car and co-workers.13 They have shown that We demonstrate the influence of noncollinearity on the ge- noncollinear configurations exist in Fe3 and Fe5, although ometry and on the total magnetic moment and discuss how the effect on structure and energetics of these ferromagnetic this reduces the discrepancy concerning the magnetization of FE clusters is not very pronounced. On the other hand, an some chromium clusters between the experiment by Bloom- unconstrained orientation of the quantization axis is known field and co-workers23 and some previous theoretical to play a key role in describing various nonferromagnetic results.22,24 systems like the phase of bulk iron,14 disordered systems,15 or ultrathin Fe films with a partial antiferromagnetic AF II. THEORETICAL AND NUMERICAL BACKGROUND coupling.16 Furthermore, the work on Fe/Cr Ref. 17 and Ag/Cr multilayers18 demonstrated how the competition be- The density-functional theory in its most general form, as tween AF ordering and frustration of the Cr moments leads developed by Barth and Hedin,25 allows the orientation of to noncollinear arrangements in the form of a spin-density each spin to vary with position. The wave functions are de- wave. Although the importance of frustration in AF systems scribed by complex two-component spinors ( , ), seems to be evident and was discussed in detail for embed- where and denote the spin indices, and the degrees of ded Cr clusters by Pastor and co-workers,19 and more gener- freedom are the elements of the single-particle spin-density ally by Manninen and co-workers,20 the possibility of non- matrix collinear effects has not yet been considered for AF clusters. In this paper, we present a general local spin-density cal- * r . 1 culation for clusters of AF materials. Besides the motivation r i, r i i, given above, we have decided to explore noncollinear effects in chromium clusters for two reasons: First, chromium is Assuming this matrix to be diagonal, the usual local spin- particularly challenging amongst the 3d elements due to its density functionals are parameterized in terms of (r ) 0163-1829/99/60 6 /4205 7 /$15.00 PRB 60 4205 ©1999 The American Physical Society 4206 C. KOHL AND G. F. BERTSCH PRB 60 : atoms has a negligible effect on the orientation of their mag- (r ) and (r ) : (r ) only. In that special case, the spins are necessarily collinear along the chosen quantization netic moment, although its magnitude becomes somewhat axis and the exchange-correlation potential is obtained via bigger. Due to the varying integration radius, however, mag- Vxc nitudes of local magnetic moments from different clusters Exc , / . However, rotational invariance re- quires that the true variables are the eigenvalues n cannot be compared easily. (r ) and n As usual, we only treat the valence electrons explicitly, (r ) of the spin-density matrix (r ). We can thus apply standard local spin-density functionals we chose the formula taking care of the ionic core with a pseudopotential approxi- of Perdew and Wang26 by evaluating the potential in a lo- mation. We use the relativistic pseudopotential from cally diagonal frame. The transformation is carried out fol- Goedecker and co-workers,28 which contains a local part lowing the work of Ku¨bler et al.27 who used the spin-1/2 plus a sum of separable Gaussians, optimized to represent a rotation matrix transferable nonlocal pseudopotential on a coordinate mesh. The multiplication of the wave functions with the nonlocal r r part can be limited to a small region around the ions as the cos 2 e(i/2) (r ) sin 2 e( i/2) (r ) radialprojectorsfalloffratherquickly.However,20integra- U r 2 tions within the covalent radius of each atom need to be r r performed to correctly account for nonlocal effects in chro- sin 2 e(i/2) (r ) cos 2 e( i/2) (r ) mium. The energetics at small ionic separations inside the clusters further requires us to include the 3s and 3p semi- to locally diagonalize the spin-density matrix: core electrons into the variational space. Our pseudopotential also includes spin-orbit terms that fix the orientation of the U total magnetization M to the ionic structure, thus giving rise r r U* r n r . 3 to magnetic anisotropy. The implementation of the L *S op- By working in this representation we express Exc/ erator is not too costly because we have to deal with a com- by Exc/ n plex spinor structure anyway. Spin-orbit effects enable us to plus the introduction of local spin rotation angles (r ) and (r ) that are the local azimuthal and polar angles test the validity of the usually applied atomic-sphere ap- of the magnetization density vector. They are computed from proximation by studying the intra-atomic dispersion.29 Eq. 3 through the requirement of vanishing off-diagonal We have carried out an unconstrained structural search by elements as fully optimizing electronic and ionic degrees of freedom. To find the ground state and stable isomers, the ionic positions Im were computed via Monte Carlo sampling applying the tech- r tan 1 r , nique of simulated annealing. After some Metropolis steps, Re r the electronic wave functions are updated with Kohn-Sham iterations. The optimization of the ionic geometry involves a 2 Re minimization of the one-ion energies and is explained in de- r tan 1 r 2 Im r 2 1/2 . 4 tail in Ref. 30. The static Kohn-Sham equations are solved in r r a combined coordinate and momentum space representation These new degrees of freedom complicate the mean-field by using an efficient damped gradient iteration.31 Local op- equations and lead to an exchange-correlation potential V xc erators are applied on coordinate space wave functions while in the form of a complex matrix in spin space the kinetic energy and the action of the spin-orbit operator are computed in momentum space applying fast Fourier 1 1 techniques. The Poisson equation is solved via the FALR V xc xc xc xc xc 2 V V 1 2 V V *d , 5 Fourier analysis under special consideration of long range terms method.31 As it is more convenient for most physical where d is a position-dependent unit vector along the direc- observables, electronic wave functions and densities are tion of the vector Re stored on a three-dimensional coordinate space mesh. We (r ),Im (r ), (r ) (r ) . The presence of the second term in the exchange-correlation perform our calculations in a cubic box with a mesh spacing potential allows a general coupling of the up and down com- of 0.32 a.u. and up to 64 grid points in each direction. We ponents of the spinor wave functions. To interpret the mag- checked that the mesh size was big enough to avoid artifacts netic properties, we compute the vector magnetization den- from the boundaries. A detailed description of our numerics can be found in Ref. 32. sity m (r ) by expressing the spin-density matrix in the form r 0.5 n r 1 m r * . 6 III. RESULTS AND DISCUSSION We associate magnetic moments with individual atoms by Before discussing our results, we mention some of the integrating each component of m (r ) within a sphere centered various tests we performed in order to increase our confi- on the ions, giving us the local magnetic moment vectors dence in the Hamiltonian and its numerical implementation. at . The integration radius is chosen to be one half of the The dimer plays a key role in the description of small chro- smallest interatomic distance in each cluster to avoid overlap mium clusters. It is known that its subtle electronic proper- and the resulting spheres contain about 80­90 % of the mag- ties demand a high accuracy of the Cr-Cr interactions and the netization density. Taking a larger radius for more distant numerical representation.21 By applying the pseudopoten- PRB 60 NONCOLLINEAR MAGNETIC ORDERING IN SMALL . . . 4207 FIG. 1. Geometric and magnetic structures for the energetically lowest noncollinear configurations of CrN , 3 N 6. The local magnetic moments, including their angles with respect to the x and z axes, are indicated by arrows. The interatomic distances are shown in atomic units and the magnitudes of the local magnetic moments at in units B) are given in brackets. tial in the semicore version, our binding energy 1.98 eV figurations of small chromium clusters CrN (3 N 6) are and bonding length (d0 3.25 a.u.) for the antiferromagnetic presented in Fig. 1. All structures except the one of Cr ground state were in good agreement with experimental re- 4 represent ground states. The corresponding total magnetiza- sults (1.56 eV 0.3 eV, dexp 3.19 a.u.) and previous all- tion is shown in Table I. Our geometric and magnetic struc- electron or pseudopotential calculations.21 The correct ener- tures are obtained by performing up to 50 full Monte Carlo getic order of the single-particle levels as a function of the intramolecular distance, the symmetry of the wave functions, and the properties of the ferromagnetic coupling (d TABLE I. Total magnetization per atom Mat in units B) for 5.2 a.u.) could be reproduced as well. The same holds for the ground states of CrN and gain in binding energy Enc in units the bonding length of the CrO molecule that deviated from eV/atom with respect to their collinear counterparts. In case of a collinear ground state, the result for the energetically lowest non- the experimental result by 1.3%. Additionally, we achieved collinear isomer is given in parentheses. The last column shows the degenerate d states up to a level of 1% and the correct ener- corresponding magnetization from the collinear calculation of getic order of 3d and 4s levels in the chromium atom. The Cheng and Wang Ref. 22 . rotationally invariant spin-density theory was checked by let- ting the FE configuration of the dimer relax to the AF ground N state. As in the collinear theory, all spins were initially re- at Type Mat Enc Mat Ref. 22 stricted to point in the z direction. The wave functions and 2 Collinear 0.0 ( ) 0.0 ( ) 0.0 energies of the final result turned out to be identical with the 3 Noncollinear 0.69 0.083 2.0 ground state as computed in a separate collinear approach, 4 Collinear 0.0 1.33 0.0 0.12 0.0 although the quantization axis of both atoms had rotated by 5 Noncollinear 0.53 0.054 0.93 90° during the iteration. This confirms the degeneracy of 6 Noncollinear 0.0 0.022 0.33 the electronic properties with respect to the orientation of 7 Noncollinear 0.13 0.019 0.29 their spin. Furthermore, we have been able to verify the re- 9 Noncollinear 0.09 0.015 0.22 sult of Car and co-workers13 concerning the noncollinear 12 Noncollinear 0.81 0.011 1.67 spin arrangement of Fe3. 13 Noncollinear 0.60 0.008 1.06 Our results for the energetically lowest noncollinear con- 4208 C. KOHL AND G. F. BERTSCH PRB 60 FIG. 2. Three-dimensional plot of the polar rotation angle in degrees from Eq. 4 within the plane defined by the ionic coordinates of Cr3 see Fig. 1 . We also show the corresponding contour lines for with a step size of 10°. The ionic positions are at the center of the dashed squares. runs per cluster starting from arbitrary ionic coordinates. A AF coupling and frustration. This can be seen easily with the few thousand Kohn-Sham iterations are usually necessary to very simple Hamiltonian H 3 completely relax the electronic degrees of freedom. This is i j i* j for three spins on an equilateral triangle. Here is negative for AF coupling. because the numerical convergence with respect to the direc- In a collinear restriction, the lowest-energy state formed by a tion of the local moments governed by competing inter- atomic exchange interactions is much slower than with re- product wave function is ( ) with an energy expectation spect to their magnitude, which is determined by stronger H . Taking instead the state with 120° angles between intra-atomic interactions. the spin directions gives a lower energy of H 3/2 . For The principle effect that leads to noncollinear arrange- higher atomic spins as they occur in our numerical compu- ments in chromium clusters can best be demonstrated in Cr tations the preference of the noncollinear configuration 3. Our calculation restricted to collinear spins gives a trimer would, of course, become more pronounced due to the larger that basically consists of a dimer plus a loosely attached third number of exchange interactions. atom, very similar to the result of Cheng and Wang.22 It is In Fig. 2 we show the rotation angle (r ) in the x-z plane obviously impossible for the atoms to couple antiferromag- of the trimer, including its contour lines in steps of 10°. The netically with all their neighbors at a time so that a FE inter- azimuthal angle (r ) is zero at any grid point due to the action between two atoms becomes inevitable-a situation magnetic anisotropy energy although this quantity is rather that we refer to in the following as frustration. The repulsive small. But it seems that even a difference of a few meV per effect of the FE coupling at smaller distances pushes one atom is enough to keep the magnetization inside the plane of atom away and results in a large interatomic distance of 4.97 the trimer, an observation that agrees with the findings of a.u. If the noncollinear channel is accessible, however, the Ref. 11. The orientation of the magnetization density vector trimer is free to achieve the closest to AF-like coupling it can is remarkably uniform in the regions surrounding the ions: by rotating two local magnetic moments in the x-z plane see 74° for the ions at x 1.645, z 1.64, and 0° Fig. 1 . As the frustration is now reduced, the third atom for the third ion at x 0,z 1.64. Besides a small oscillation comes much closer (d 3.67 a.u.) and enables a buildup of of 10° in these regions, varies significantly only molecular orbits that enhances the total binding inside the right between them where the charge density is essentially trimer significantly. The associated reduction of zero. This is why a smaller integration radius only influences at to 2.29 B for the formerly isolated atom results in a decrease of the magnitude of at and not its orientation. The rapid Mat from 2 B to 0.69 B . The energy difference with re- change from 90° to 90° indicated by a very high spect to the collinear ground state Enc is 0.083 eV per atom density of the contour lines is related to a spin flip in these see Table I , which amounts to 7.8% of the binding energy interatomic regions. We find an intra-atomic spin dispersion of Cr3. A look at the interatomic distances and the tilted local of around 6° that partly comes from spin-orbit coupling. But moments suggests that the dimer lost its dominant role. the dispersion is also induced by the trend to an AF coupling However, the fact that the ground state is not an equilateral with the neighbors. The same especially holds for the varia- triangle with angles of 120° between the local magnetic mo- tion of the spin direction close to the domains of other atoms. ments this would equal the best possible AF-like coupling The change of at the ionic positions dashed squares is an indicates that some trace of the strong dimer binding from artifact of the pseudopotential approximation. A detailed the collinear calculation still persists. Unlike the situation in analysis further shows that these features are common to all Fe3, the linear isomer of Cr3 did not favor noncollinear spins. investigated chromium clusters. The noncollinear spin structure of Cr3 can be understood The collinear ground state of Cr4 was found to have a as a compromise between the energetically very favorable rectangular structure not shown with bonding lengths of PRB 60 NONCOLLINEAR MAGNETIC ORDERING IN SMALL . . . 4209 3.35 a.u. and 4.62 a.u. The double dimer structure of this geometry appears to be resistant to noncollinear effects as the distance of two atoms with parallel moments is very large 5.7 a.u. . But in the case of the lowest isomer, a rhom- bus, frustration becomes important again resulting in noncol- linear spins see Fig. 1 . The argumentation follows the case of Cr3, the triangles in Cr4 are very similar and have only slightly larger bonding lengths. A higher total magnetization of Mat 1.33 B leads to an energy gain of 0.12 eV per atom with respect to the corresponding collinear state of the rhom- bus and halves the energy difference to the rectangular ground state. The isomer of Cr4 is unique in that the ener- getically favored noncollinear configuration has a bigger to- tal magnetization than its collinear counterpart. The magni- tudes of at , however, were similar in both cases. It is also noteworthy that the rhombus structure provides angles be- FIG. 3. Geometric and magnetic structure of the noncollinear tween the local moments of almost exactly 120°, although ground state of Cr12 . The notation follows the one from Fig. 1. The the bondings are not equivalent. This seems to be related magnetic moments of the lighter-colored atoms show a dispersion with the larger moments of the furthermost atoms ( in the positive z direction, whereas the darker atoms almost exactly at 0.5 point in the negative z direction. We also display the absolute val- B). The properties of the noncollinear ground state of Cr ues of the azimuthal and polar angles as well as the magnitudes of 5 can be understood if one considers the pentamer as consisting of at for all four groups of atoms. three triangles. Although the bonding lengths are bigger, each of these triangles exhibits the same basic features as Cr gradually smaller with rising cluster size. Further calcula- 3 and the local moments are again arranged in such a way that tions for Cr7 and Cr9 confirm this trend and predict that the the best possible AF-like coupling is achieved. We observe noncollinear geometries essentially equal the collinear ones that the magnitude of for Nat 6. On the other hand, our results so far show that at decreases with increasing coordi- nation number, as can be seen from the center atom with noncollinear spin configurations considerably reduce the to- tal magnetization of the ground states although it might not at 1.91 B . Note that the reflection symmetry of the ge- be reflected too much in the binding energy. This can be ometry is the same as the symmetry of the magnetization. understood in terms of the competing interatomic exchange The gain in binding energy with respect to the best collinear interactions that involve only small energy changes during state is 0.054 eV per atom ( 3.7% of the binding energy , the rotation of local moments see above . which is less than for the trimer. But the total magnetization We shall finally discuss how noncollinear effects might is again clearly reduced from 1.03 to 0.53 B per atom. The provide a way to ameliorate the disagreement between the lowest isomer is a bipyramid with a noncollinear spin struc- measured total magnetic moments of Cr ture as well and M N (N 9) by Bloom- at 0.79 B . Its energy difference of 0.6 field and co-workers23 and previous theoretical results.22,24 eV to the ground state is 0.25 eV higher than in the collinear The Stern-Gerlach experiment extracted an upper bound for case. Cr5 is the only cluster we found where the noncollinear M and collinear geometries differ by more than just a variation at , M at 0.77 B , assuming a superparamagnetic behavior of the chromium clusters. Earlier calculations, however, re- of the bonding lengths. Our lowest collinear state looks simi- port values for the magnetization of some clusters that are lar to the geometry of Cheng and Wang and has a 2v sym- much higher.24 The values from Cheng and Wang for Cr metry. However, its total magnetization of 1.05 12 B is some- (M what higher than their result of 0.93 at 1.67 B) and Cr13 ( M at 1.06 B) also exceed the ex- B . perimental limit. The trend from smaller clusters gives one The shape of Cr6, on the other hand, resembles the col- the hope that noncollinear effects might reduce the differ- linear one very closely. Cr6 is the smallest cluster that dis- ence. However, an unconstrained simultaneous optimization plays a fully three-dimensional geometric and magnetic of electronic and ionic degrees of freedom surpasses our structure in the ground state Fig. 1 . It consists of three computational resources for clusters as large as Cr dimers distributed over two triangles in which frustration 12 . In- stead, we start the optimization procedure from the geom- sets in. Each of the at is exactly antiparallel to the moment etries of Cheng and Wang. This is a reasonable approach in of its partner atom in the dimer. The azimuthal angles are the light of the very good agreement with our collinear struc- 19° for the atoms in the foreground and 0° for the tures and the small changes in geometry that are induced by ones in the background that are slightly closer. The bonding noncollinear spins. Our final magnetic and geometric con- lengths in the triangles are about 6% shorter than in the col- figuration of Cr12 is shown in Fig. 3. The free relaxation linear case and the dimer distances are somewhat bigger. It is leads to a shortening of the bonding lengths between the important to note that in spite of a vanishing total magneti- corner atoms of about 5% and slightly bigger distances of the zation and although the dimer seems to recover a certain capping atoms but the bulklike bcc structure of the collinear influence, tilted spins are still energetically favored. How- geometry clearly persists. The z components of the local ever, the gain of Enc 0.022 eV per atom only accounts for magnetic moments vary on alternating x-y planes, but only 1.4% of the binding energy of Cr6. This indicates that the the moments of the corner atoms have significant x and y impact of noncollinear effects on the energetics becomes components. All the spins of the lightest-colored corner at- 4210 C. KOHL AND G. F. BERTSCH PRB 60 oms point inside the cube towards the central atom, whereas iterations that update the wave functions simultaneously. the spins of the slightly darker corner atoms point outside Our collinear configurations, which we use in order to and away from the next atom. This dispersion is related to extract the effect of noncollinear spins, agree very well with some frustration of the corner atoms in connection with the the results of Cheng and Wang.22 We find that all investi- preferred AF coupling to the moments of their nearest neigh- gated chromium clusters show a pronounced trend to noncol- bors darker atoms . A magnetic arrangement like that can be linear spin configurations. This is caused by a subtle inter- seen as a precursor to the bulk behavior in form of a spin- play between the preferred magnetic order and frustration, a density wave that is achieved by an almost antiparallel order situation that can in principle occur in all clusters of ele- between neighboring layers. It should be noted that more ments that favor antiferromagnetic spins. Therefore our con- compact and thus more frustrated structures different from siderations appear to be of a more general nature although the bcc-like geometry might result in a very different mag- the special properties of chromium indicate that noncollinear netic behavior. This question is presently being investigated. effects could be less dramatic in other transition metals. The The tilted spins of the corner atoms ( 43°, results for Nat 13 show that the influence of noncollinearity 34° and 45°, 30°) have pairwise opposite x on various observables becomes gradually smaller with ris- and y components so that the sum of their net moments in ing cluster size. Induced changes of the cluster geometry are positive z direction is reduced. The moments of the darker generally restricted to an alteration of the bonding lengths, atoms, however, show almost no dispersion in negative z with the exception of Cr5. However, we observe a universal direction. All this results in a much smaller total magnetiza- reduction of the total magnetization that is significant even in tion of Mat 0.81 B , which is now very close to the experi- those clusters for which a variation of the magnetic distribu- mental limit although the geometry is almost identical to the tion is not clearly reflected in the binding energy any more collinear case . The remarkable reduction of Mat (Cr12 and Cr13). This effect is related to the small energetic 0.86 B is associated with Enc 0.011 eV per atom, changes that occur during the rotation of local moments, an which equals only 0.5% of the binding energy of Cr12 . A aspect that makes a proper convergence of the Kohn-Sham very similar situation leads to Mat 0.60 B in the case of iteration very time consuming. Furthermore, our findings Cr13 , a value that is even below the experimental limit. The show that tilted spins due to frustration can even be favored noncollinear gain amounts to Enc 0.008 eV per atom un- when the total magnetization vanishes (Cr6) or the corre- derlining the trend to a gradually decreasing influence of sponding ground state is collinear like in Cr4. The free varia- noncollinear effects on the binding energy. We can conclude tion of the spin quantization axis finally leads to a better from our analysis that the total magnetization of Cr12 and agreement with the experiment concerning the total magne- Cr13 as obtained with the general LSDA represents a consid- tization of Cr12 and Cr13 . It can be concluded that noncol- erable improvement with respect to the experiment. linear effects appear to be an important ingredient for a deeper understanding of the subtle magnetic properties in IV. CONCLUSIONS transition-metal clusters. We present a study of noncollinear effects in antiferro- magnetically coupled clusters by applying the general, rota- ACKNOWLEDGMENTS tionally invariant LSDA for the electronic degrees of free- dom. Their interaction with the ions is described in terms of One of the authors C.K. has been supported by the a relativistic, nonlocal pseudopotential that has been thor- DAAD German Academic Exchange Service , Grant No. oughly tested. The magnetic and geometric structures are D/98/14581. We also thank Ana Proykova, Sanjay Reddy, obtained by employing a simulated annealing technique for Paul-Gerhard Reinhard, Louis Bloomfield, and Lai-Sheng the ionic optimization together with interlaced Kohn-Sham Wang for many useful and encouraging discussions. 1 D. C. Douglass, A. J. Cox, J. P. Bucher, and L. A. Bloomfield, 7 J. Guevera, F. Parisi, A. M. Llois, and M. Weissmann, Phys. Rev. Phys. Rev. B 47, 12 874 1993 ; S. E. Apsel, J. W. Emmert, and B 55, 13 283 1997 ; Q. Sun, G. Wang, J. Z. Yu, Z. Q. 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