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Self-consistent Green function approach for calculations of electronic structure in transition metals

We present an approach for self-consistent calculations of the many-body Green function in transition metals. The distinguishing feature of our approach is the use of the one-site approximation and the self-consistent quasiparticle wave function basis set, obtained from the solution of the Schrodinger equation with a nonlocal potential. We analyze several sets of skeleton diagrams as generating functionals for the Green function self-energy, including GW and fluctuating exchange sets. Their relative contribution to the electronic structure in 3d-metals was identified. Calculations for Fe and Ni revealed stronger energy dependence of the effective interaction and self-energy of the d-electrons near the Fermi level compared to s and p electron states. Reasonable agreement with experimental results is obtained.

Density-functional theory (DFT), in particular the local-density approximation (LDA), has proven to be a rather successful ab-initio approach to describe physical properties of many materials. Nevertheless numerous applications of this method have revealed a number of shortcomings related to the inadequate treatment of both excited (energy gap) and strongly correlated states. For a long time the many-body Green function (GF) approach was considered as a possible ab-initio alternative to DFT, but its current applicability is usually restricted to the homogeneous electron gas (HEG) model [1,2,3] or semiconductors in the GW approximation. However, realization of the full self-consistent GW scheme is complicated due to enormous computational difficulties. As a result, for the transition metals (TM) the GW approach was applied only to Ni [4,5] and these calculations used the wave functions obtained within the LDA as the basis set, i.e. they were not self-consistent. Thus, it is quite unlikely that a universal GF method can be formulated and tailoring the approximations to specific properties of a given class of materials is desirable. The localized character of the d-wave function in TM often implies the well-known one-site approximation [6] which in real systems acquires certain material specific features.
Already the calculations of Refs. [4,5] revealed a strong energy-dependence of the effective interaction in TM, even in the vicinity of the Fermi level. For example in Ni the effective Coulomb interaction W is nearly zero at 6 eV and increases to an unscreened value of 20-25 eV at 20 eV. Another important observation is related to a proper intra-atomic scale treatment. Elementary analysis shows that the d-function in bcc Fe is significantly altered at a distance as small as 0.8 a.u., while s and p wave functions are altered at distances of about the Wigner-Seitz radius (∼ 2.6 a.u.). The "local" screening length, λ = κ −1 (κ 2 (r) = 4πe 2 Π 0 (ρ(r)), with Π 0 (ρ(r)) = (3/πρ) 1/3 /π), which varies from 0.5 a.u. in the region of the maximum of the d−wave function to a value of 1.0 a.u. at the atomic sphere (AS) boundary, provides an initial justification for the one-site approximation and suggests that the correct material-specific implementation must include a spatially resolved representation of the polarization operator (PO) and other two-channel operators, at least inside the AS.
We believe that the energy dependence and intra-atomic resolution are both crucial for the quantitative description of TM and must be included explicitly in any reliable technique.
On the other hand, we will show below that this approximation can describe the essential physics of TM, greatly reducing the computational efforts.
In the present paper we incorporate the effects described above into the self-consistent GF technique using the quasiparticle wave function basis set and the Luttinger-Ward functional [7] (LWF) approach for the self-energy calculations. We tested several sets of diagrams, mainly from the fluctuating-exchange subset (FLEX) [8]. All diagrams beyond these sets were taken in the local approximation. The use of the LWF guarantees the equivalence of one-particle properties calculated with the GF and with the corresponding total energy variation [9]. The LWF formalism naturally leads to the integration over the imaginary axis, greatly improving the numerical accuracy of the integration. On the imaginary axis both the GF and PO quickly reach their quasiclassical limits, are sufficiently smooth, and are determined by the local potential only [10]. The rotation of the integration contour leads to a very convenient separation of the structural and local density dependent degrees of freedom for both self-energy and total energy [11,12]. Also, in contrast with the numerical method [3] we analytically select the contributions from both the GF cut on the real axis and the integration along the imaginary axis. An energy linearization of the quasiparticle wave function similar to the one in linear density functional methods was used.
In terms of an exact GF the thermodynamic potential is written as [7] where Σ is the self-energy, G and G 0 are exact and 'bare' GF and Φ is the Luttinger generating functional, which is represented by the set of skeleton graphs. Minimizing Ω over The expressions for Φ for the most important two-particle (hole) channels are summarized in Ref. 8. The set from Fig.1b with 'bare' loops corresponds to the GW-approximation. After variation of Φ over the GF, which we consider as a variational variable, one can obtain the usual GW expression for the self-energy: where V c is the Coulomb interaction and W is the effective It is convenient to rotate the contour of integration in Eq.(3) in the complex plane, using the fact that Π(z) has no singularities in the first and third quadrants [13,14], whereas G has a singularity (cut) in the third quadrant for ε < E F and in the first one for ε > E F . In the case of the quasiparticle the singularities of the GF are simple poles. After rotation the only additional contribution comes from the cut of the GF (Fig 1e), and for ε < E F this contribution is where g(ω) = − Im G(ω)sgnω/π. For the term corresponding to the integration along the imaginary axis we obtain At E F , ε = 0 and the pole term disappears. The term Σ i ca is usually negative in the HEG model. On the real axis Π(ω) has an imaginary part, which is important for accurate calculations of W (ω) in TM. The summation over k (one-site approximation) transforms all the above formulas into matrix equations on the local wave function basis, greatly reducing the was obtained self-consistently from Eq.(3). Due to the large value of ∂Σ/∂ε in TM, the final was also obtained self-consistently with the full G.
Let us discuss the choice of skeleton graphs for the LWF. According to perturbation theory with Coulomb parameter α = e 2 m/p f the exchange diagram (Fig 1a) gives the largest contribution. The set of empty bubble diagrams (Fig.1b) gives the next term α 2 ln 1/α. The summation of these terms is necessary to take into account the long-distance character of the Coulomb interaction. In addition, the proper treatment of the "dressed" GF in the bubble approximation must include vertex corrections [15] (Fig.1b). The next term (Fig.1c) is just the exchange diagram in the second-order approximation, which has an order of α 2 .
It can be important in magnets, as the corresponding self-energy depends only on the GF with the same spin, while in bubble diagrams both spins are averaged. In principle, the are considered. But it seems desirable to include also the ladder sequence of Fig.1d, as it is also highly spin dependent. This set does not contain the first (α 2 ) term [8]. Such a sum is reduced by the effective screening of the Coulomb potential, which in turn strongly depends on the energy. The static value of the effective interaction is rather small, but it quickly increases as a function of energy (Fig.2b). We evaluated this sum of lattice-type T-matrix diagrams (Fig.1d)  For d electrons this is a small parameter and the main terms with 1/N are again FLEX diagrams [16]. In summary, we believe, that the set of diagrams in Fig.1 is the minimal set which must be included in the calculations. On the other hand, our calculations revealed that it can be sufficient for description of magnetic properties and electronic structure of 3d metals.
The rotation of the integration contour allows us to deal only with the states lying in the vicinity of E F [12], as the PO is smooth and decreases when Im ε is increased, contrary to its behavior on the real axis. This is shown in Fig.2a,b for ferromagnetic Fe. For the states near E F we can use the GF constructed with low lying LMTO states Ψ ν k (r) = L a να k φ α l (r)Y L ( n), where φ 0 l (r) and φ 1 l (r) are the corresponding solutions of equations with V (r, r ′ ) = Σ(ε l , r, r ′ ). ε l is the center of gravity of the band with orbital moment l and a ν0 k are the eigenvectors of the generalized eigenvalue problem with the Hamiltonian which is a matrix in the space of LMTO states φ + hφ, where h LL ′ are the usual LMTO coefficients. This procedure takes into account the energy-independent exchange potential Σ x = 1/2πi V c ImGdε exactly.
The important problem in the nonlocal calculations is the inclusion of the valence-core interaction. In our method we included the core-valence exchange term exactly, whereas the core-valence correlation term was added using the approximation developed in Ref. [17].
The latter term in TM such as Fe, Ni or Cu is small. The set of Eqs.(4),(5) was solved iteratively [12]. The integral equation for the screened interaction D(ω, r, r ′ ) was solved using the product basis introduced in Ref. [18].
The procedure above was applied to the 3d transition metals Fe and Ni. The main contribution to the self-energy comes from the exchange diagram (Fig 1a), The small value of Z d obtained above leads to a narrowing of the quasiparticle band. The rest of the one-electron states take part in the formation of the satellite structure, which usually appears when the hole-hole interaction is taken into account [20]. Consequently our DOS (Fig.4)  with l = 2 on the real axis (a). The same for the polarization operator and effective interaction V ef f = l D l Z l /Z on the imaginary axis (b).