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the electronic properties of many solids are described reasonably well in the local density approximation ( lda ) of the density functional theory ( dft),@xcite which is essentially a single - particle theory representing a cardinal simplification of the original many - body problem .
however , the lda becomes inadequate in materials exhibiting strong spatial correlations between different electrons in atoms with localized orbitals , temporal correlations ( retardation ) associated with coulomb screening , etc .
the corresponding difficulties are extremely hard to overcome within the dft without sacrificing the first - principles approach .
one of the methods used to correct the deficiencies of lda in the description of excited states in metals and semiconductors is the gw approximation ( gwa).@xcite the gwa usually improves the band gaps in semiconductors and insulators;@xcite in metals it may provide information on the quasiparticle lifetimes and renormalization which is absent in the dft.@xcite however , in all treatments used until recently , the gwa was employed in a non - self - consistent fashion , by using unrenormalized green s functions based on the kohn - sham wave functions obtained in lda .
such an approach is easier to implement , but it is internally inconsistent , does not correspond to any luttinger - ward functional@xcite and hence violates basic conservation laws,@xcite and produces different results depending on the approximation used to solve the kohn - sham equations . on the other hand , studies of the homogeneous electron gas
have shown that gw self - consistency worsens the agreement with experiment at typical metallic densities,@xcite highlighting the limitations of gwa which is strictly correct only in the high - density limit .    recently , two self - consistent realizations of the green s function method were demonstrated .
one of them@xcite was tailored for transition metals and employed the one - site approximation for the self - energy which is based on the assumption that dynamical screening in @xmath2-metals is well described at the intraatomic level .
some spin - selective diagrams beyond the gw set were also included .
the results for fe and ni were quite reasonable , although no improvement was obtained compared to lda .
while the choice of gw diagrams is unjustified for the homogeneous electron gas at typical metallic densities and likewise for semiconductors ( the diagrams that are left out do not contain any small parameter ) , it was noted that the situation may be better in transition metals , because high orbital degeneracy provides an additional enhancement to the diagrams with the largest number of closed electron loops favoring the gw set.@xcite in addition , for transition metals the `` local '' ( one - site ) approximation for the self - energy is quite reasonable due to the short screening length and greatly facilitates self - consistent calculations .
another realization@xcite using gw set with the full @xmath3-dependence of the self - energy was applied to elemental semiconductors .
it was found that self - consistency and accurate treatment of core electrons improve the agreement with experiment for the band gaps in si and ge .
while the adequacy of gwa for the studies of metals and semiconductors has not been firmly established from the point of view of the many - body theory , this method may be regarded as one of the possible steps toward a consistent green s function - based scheme .
therefore , it is important to establish the accuracy of this approximation for different materials , and in particular for transition metals where , as noted above , there are reasons for it to work better than in the homogeneous electron gas . to this end , in this paper we calculate the conduction - band spectra for all elemental @xmath0 and @xmath1 metals using the self - consistent gw approach of ref .   and compare them with the observed ones and with those given by the lda .
the technique used in this paper was introduced in ref .  .
here we describe some points in more detail .
the self - energy @xmath4 is obtained from the luttinger - ward generating functional @xmath5 defined by the set of skeleton graphs .
@xcite the hartree diagram gives the local contribution @xmath6 , and the exchange diagram contributes @xmath7 where @xmath8 is the coulomb potential .
the total contribution of the remaining gw sequence ( the `` correlation term '' ) is @xmath9 here @xmath10 is the effective ( screened coulomb ) interaction defined by the integral dyson equation @xmath11 and the polarization operator @xmath12contour of integration in eqs .
( [ sigmac]),([pi ] ) is directed along the imaginary axis and embraces the cut on the real axis from the fermi energy @xmath13 to the external energy .
the gw approximation may be modified by the inclusion of vertex corrections in the polarization operator ( [ pi ] ) which may improve the results in some cases .
the calculations are drastically simplified by the use of the one - site approximation  @xcite ( osa ) , in which the self - energy is calculated only on one lattice site neglecting all non - diagonal maxrix elements connecting different lattice sites .
thus , in osa the self - energy depends on energy and on the coordinates @xmath14 , @xmath15 belonging to the same unit cell . in order to implement osa
we first need to choose an appropriate on - site basis .
inspired by the good description of the band structure of densely packed solids achieved in the atomic sphere approximation within the linear muffin - tin orbital method ( lmto - asa ) , we used a very similar , minimalistic approach . for the basis set
we also use just one function per each angular momentum @xmath16 and its projection @xmath17 ( below we denote @xmath18 ) , along with its energy derivative . the radial basis functions @xmath19 satisfy the `` schrdinger - type '' equation @xmath20 \phi_l(r)=0   \label{schroed}\ ] ] where @xmath21 is the radial part of the laplasian , and @xmath22 is an integral radial operator whose kernel is obtained from @xmath23 by projecting onto the @xmath16 subspace : @xmath24 where we denoted @xmath25 , @xmath26 are the spherical harmonics , and integration is over the directions of @xmath14,@xmath27 . the operator @xmath22 may be represented as the sum of a local part @xmath28 similar to an external potential and a nonlocal part @xmath29 which ( after acting on @xmath30 ) gives a linear combination of @xmath31 with @xmath32 .    similarly to the lmto - asa method , the solutions of eq .
( [ schroed ] ) are only found at one energy @xmath33 for each @xmath16 which is fixed at the center of gravity of the corresponding band .
we may safely discard the imaginary part of the self - energy operator because it is small in the vicinity of @xmath13 where @xmath33 is usually chosen .
thus , our radial basis functions are real .
the non - local equation ( [ schroed ] ) is solved by iterations .
if @xmath34 is the solution after @xmath35 iterations , then the solution on iteration @xmath36 is obtained using auxiliary functions @xmath37 and @xmath38 defined as @xmath39 f_l(r)&= & \widehat\sigma_l^{(n)}(\epsilon_l)\phi_l^n , \\
\left[\epsilon_l+\frac{\delta_l}{2}-v_{\mathrm{h}}\right ] g_l(r)&=&0\end{aligned}\ ] ] according to @xmath40 , where the coefficient @xmath41 is found by normalizing @xmath42 .
just as in the lmto method , in addition to @xmath30 we compute its energy derivative @xmath43 : @xmath20 \dot\phi_l=-\phi_l+\frac{\partial\widehat\sigma_l(% \epsilon_l)}{\partial\epsilon_l}\phi_l   \label{schroed1}\ ] ]    to stabilize the solution of the schrdinger equation ( [ schroed ] ) , we added the exchange with the nearest - neighbor cells to @xmath44 .
the corresponding matrix element was subtracted from green s function ( gf ) below .    with the on - site self - energy @xmath45 defined for @xmath46 within the single unit cell
we can calculate its matrix elements between the muffin - tin `` eigenfunctions '' @xmath47 , which in turn are the linear combinations of @xmath30 and @xmath48 . due to the @xmath3-dependence of @xmath47 the self - energy also acquires @xmath3-dependence , just as the local potential in lda .
finally , on - site green s function is found by integration using the tetrahedron method : @xmath49 where @xmath50 and @xmath51 are right and left eigenvectors , and @xmath52 the eigenvalues of the non - hermitian operator @xmath53 where @xmath54 is the hartree hamiltonian .
( [ gf ] ) imposes the locality condition and is equivalent to the self - consistency condition of the dynamical mean - field theory.@xcite
using the technique described above ( below referred to as `` gw '' ) we calculated the spectral densities @xmath55 for elemental @xmath0 and @xmath1 metals , where lda has long been the only appropriate approximation .
we used the fcc structure for hcp elements ( sc , ti , y , zr , tc , ru , co ) and the bcc structure for mn .
all atomic volumes were taken from experiment .
cr was treated within the single bcc cell , and hence was non - magnetic .
the iterations were started from the lda potential , and at each iteration the non - local self - energy was mixed to this potential with a gradually increasing weight , so that in the end only @xmath44 was left . in the final state the magnitude of @xmath44 differs by about 40% from its initial value based on the lda wave functions .
the iterational procedure is rather stable in all metals except ni where the magnetic moment is very sensitive to the details of the calculation .
the results are shown in figs .
1,2 along with the lda densities of states ( dos ) and the experimental xps and bis spectra taken from refs .  .
strictly speaking , comparison with experiment requires the calculation of the corresponding matrix elements , but we believe that in the present context some qualitative conclusions can be drawn based on the spectral densities .    the main differences between the lda dos and gw spectral density may be summarized as follows .
the conduction band widens as all dos features are pushed away from @xmath13 ; this outward shift is roughly proportional to the distance from @xmath13 .
moreover , all dos features are increasingly smeared out due to the decreasing quasiparticle lifetime as the distance from @xmath13 increases .
substantial spectral weight is transferred from the quasiparticle states to the incoherent `` tail '' extending far below @xmath56 .
the lda dos for @xmath2-metals is typically too small to account for experimentally measured electronic specific heat @xmath57 . as seen from figs .  1,2
, the gw spectral density at @xmath13 is generally smaller compared to lda .
although to obtain @xmath58 we have to remove the renormalization factor @xmath59 from @xmath60 , the gw method does not improve the overall agreement with @xmath58 measurements .
the differences between the gw and lda spectra are most notable for early transition metals ( treated here in the fcc structure ) where the strongest unoccupied peaks given by lda are almost completely smeared out in gw .
notably , the worst agreement with experiment for the lda spectra is observed for the same elements .
in particular , the position of the fermi level obtained in lda for hcp sc and y appears to be off by nearly 1  ev.xps    the agreement between lda and gw improves as we progress to later transition metals and as the fcc structure is replaced by the bcc one .
already in v and nb the gw and lda curves are quite similar except for the shift of the unoccupied @xmath2-peak by 1.52  ev to higher energies .    as in lda , the @xmath0 metals from mn to ni were magnetic in our calculation . for simplicity
, we used the bcc structure for mn , and also the fcc structure for co ( this structure is often realized in thin films ) . as it is seen from fig .  1 ,
the general features of the gw spectral density described above are observed in these metals as well . in general , the gw description also gives a reasonable exchange splitting and magnetic moment .
we obtained the moments of 0.9  @xmath61 for mn compared to 1.03  @xmath61 in lda , 2.3  @xmath61 for fe compared to 2.25  @xmath61 in lda , and 1.85  @xmath61 for fcc co compared to 1.62  @xmath61 in lda . from these results
it is clear that there is no systematic trend for gw to give larger or smaller magnetic moments compared to lda .
the magnetic moment in ni is rather sensitive to various details of the calculation , and proper convergence turned out to be problematic .
we believe that the approach based on matsubara green s functions is necessary to stabilize this problem . apart from the exchange
splitting , the shape of the spectrum is quite stable .
we also note that the magnetic moment is expected to be sensitive to the choice of the skeleton graph set .    as an example of the general trend of band dilatation off the fermi level ,
the distance from @xmath13 to the upper edge of the fully occupied @xmath2-band in cu and ag is notably larger in gw compared to lda , which results in a better agreement with experimental xps spectra . on the other hand
, we also observe a rather strong upward shift of the unoccupied peak in v , nb , fe and mo in obvious disagreement with the bis spectra .
this shift is more pronounced compared to the downward shift of the occupied states at a similar distance from @xmath13 .
the results presented above demonstrate that self - consistent gf approach with one - site approximation can be used for ground state studies . for @xmath0 and @xmath1 systems we have shown that the gw spectral density is generally similar to the lda density of states , while the gf approach includes typical fermi - liquid effects such as finite quasiparticle lifetime and self - consistent renormalization .
the preliminary comparison with experiment is rather satisfactory and clearly indicates the problems that need improvement beyond our simple gw - osa technique are similar to those in lda  the exchange splitting in ferromagnets , value of @xmath62 , and the unoccupied peak position which is too high for some metals .
in general , the presented technique based on the luttinger - ward functional is a practical alternative to other methods based on the density functional theory and can serve as a reliable starting point for more sophisticated methods ..
this work was carried out at the ames laboratory , which is operated for the u.s .
department of energy by iowa state university under contract no .
w-7405 - 82 .
this work was supported by the director for energy research , office of basic energy sciences of the u.s .
department of energy .
nez acknowledges support from the council for the support of leading scientific schools of russia under grant ns-1572.2003.2 .
using the recently developed version of the gw method employing the one - site approximation and self - consistent quasiparticle basis set we calculated the electronic structure of @xmath0 and @xmath1 transition metals at experimental atomic volumes . the results are compared with traditional local density approach and with experimental xps and bis spectra . 
 our results indicate that this technique can be used as a practical starting point for more sophisticated many - body studies of realistic electronic structures .