It goes far beyond most classic texts in the presentation of the properties of solids and experimentally observed phenomena, along with the basic concepts and theoretical methods used to understand them and the essential features of various experimental techniques. The first volume deals with the atomic and magnetic structure and dynamics of solids, the second with those electronic properties that can be understood in the one-particle approximation, and the third with the effects due to interactions and correlations between electrons.
This volume is devoted to the electronic properties of metals and semiconductors in the independent-electron approximation. After a brief discussion of the free-electron models by Drude and Sommerfeld, the methods for calculating and measuring the band structure of Bloch electrons moving in the periodic potential of the crystal are presented. The dynamics of electrons in applied electric and magnetic fields is treated in the semiclassical approximation.
The effects due to the quantization of the energy levels in strong magnetic field are also discussed. The overview of the transport and optical properties of metals and semiconductors is followed by a phenomenological description of superconductivity. The last chapter deals with the physics of semiconductor devices. This comprehensive treatment provides ample material for upper-level undergraduate and graduate courses.
It will also be a valuable reference for researchers in the field of condensed matter physics. Charge density waves in solids : proceedings of the international conference held in Budapest, Hungary, September , by Gy Hutiray 16 editions published in in English and held by WorldCat member libraries worldwide Neutron and x-ray scattering study on K0.
Sólyom, J. (Jenö) [WorldCat Identities]
In the present volume the electron-electron interaction is treated first in the Hartree-Fock approximation. The density-functional theory is introduced to account for correlation effects. The response to external perturbations is discussed in the framework of linear response theory.
Landau's Fermi-liquid theory is followed by the theory of Luttinger liquids. The calculation of requires special care in the iterative solution of the KS equations at finite. At this initial step of the perturbed run, the variation of the external potential has not yet been screened by the response of the electronic charge density through the Hartree and xc potentials and is thus coincident with the variation of the total potential:.
Thus, from Eq. Therefore, the first diagonalization of the electronic Hamiltonian in the perturbed run must be very precise in order to obtain an accurate noninteracting response matrix. In fact, consistently with the treatment of localized states as isolated atomic orbitals in contact with a bath represented by the rest of the crystal , and with the definition of the Hubbard U as the energy cost associated with the double occupancy of the orbitals of a single atom, it is necessary to isolate the atom with perturbed states and to avoid the interaction with its periodic images. If the separation between perturbed atoms i.
The charge redistribution induced by the perturbation in the external potential, Eq. This is the reason for which U is obtained from inverting the entire response matrices rather than their single diagonal elements. However, this condition would be legitimate to impose only if there was a perfect overlap between the Hilbert spaces spanned by atomic and KS states never exactly the case, in practice. This refinement was found to have beneficial effects on the convergence of the calculation with the size of the chosen supercell.
However, as the total energy is not variational with respect to the magnetization, and the magnetization of a system often reaches its saturation value compatibly with the number of electronic states , a perturbative approach is generally not viable. Also, in these circumstances, n I and m I are not independent variables in fact, only one spin population can be perturbed, the other spin states being fully occupied and typically removed from the Fermi level and only linear combinations of U and J can be obtained from LR, not their separate values.
A possible way around this problem consists in perturbing a ground state whose absolute magnetization has been constrained to be lower than its saturation value so that n I and m I can be varied independently.
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However, calculations of this kind require effective constraints on the atomic magnetization of atoms and turn out to be technically difficult to perform and very delicate to bring to convergence. Similar problems arise when computing U for fully occupied or empty states. In fact, the LR approach discussed above is suitable to calculate the effective electronic coupling of manifolds of states that are either in the vicinity of the Fermi level and thus partially full , or result from the hybridization of atomic orbitals of different atoms as, e.
If the manifold is completely full e. In these cases, the reliability of the obtained U is questionable values of 30 eV or higher are not uncommon. Whether or not a preliminary shift of the manifold closer to the Fermi level could be a solution, depends on the specific material and on the entity of the collateral effects this shift has on its electronic structure and its physical properties.
It is useful, at this point, to study the analytic expression of the Hubbard U , obtained, as detailed in Appendix A, from the linear response of atomic occupations to a perturbation in the potential acting on localized orbitals that is a generalization of the one given in Eq. The matrix , that is obtained from the inversion of this matrix and its noninteracting analog , as indicated in Eq. This expression might surprise for the lack of screening. A similar result was obtained in Ref. A 24 for the expression of U in terms of with explicit sums over state and site indexes.
It is instructive, at this point, to compare the effective U obtained from the LR method outlined above, Eq. The difference between the two results is in the way the screening is performed. If all the electronic states were treated explicitly, a bare i. This case has been discussed in Appendix A for LR, and would correspond to putting i. As described earlier, within cRPA the kernel of the effective interaction is computed as , through the screening operated by all the electronic degrees of freedom not treated explicitly in the model Hamiltonian e.
An analogous approach in LR would require writing as the product of two contributions, from localized d and delocalized s states or, equivalently, to write as the sum of d and s terms,. In LR, an effective screening of the electronic interaction is operated by the matrix multiplications in Eq. These summations lead to a significant contraction of the computed interactions whose value decreases from 15—30 eV, typical of the unscreened quantity, to the 2—6 eV range of the effective one.
It also introduces the possibility to compute the values of these interactions in consistency with the choice of the localized basis set, the crystal structure, the magnetic phase, the crystallographic position of atoms, and so forth. The procedure usually reaches convergence in few cycles less than five in most cases.
Very often this choice is dictated by the specific implementation of DFT being used e. As, in practice, the first three conditions and often the forth too are never verified, some care must be used in the selection of the localized orbitals. In fact, when basis sets are finite, switching from one to another only generates an equivalent description if the two span the same Hilbert space. The choice of atomic orbitals e. This approximation remains indeed well justified even for systems where electronic localization occur on more general orbitals centered, e.
In these cases, however, the correction loses its atomic character and the effective interactions should be recomputed accordingly. These orbitals are represented by wave functions centered on the single atoms and decaying with the distance from its nucleus:. In this expression, is the position of the I th atom in each unit cell of the crystal or in the molecule , designates the unit cell. The atomic orbital occupations, Eq. However, the periodicity of the crystal allows us to drop this index from the expression of the occupation matrices and to use the definition in Eq.
Consequently, the Hubbard energy per unit cell does not depend on and can be computed from a single unit cell. The expression in Eq. Therefore, the calculation of atomic occupations in Eq.
This observation will be important when computing the derivatives of the occupation matrices to obtain, for example, forces and stresses see Energy Derivatives section. A problem that arises with atomic wave functions is the finite overlap between orbitals belonging to neighbor atoms. This fact compromises the summation rules of atomic occupations in Eq. Within this scheme an orthonormal basis set can be obtained as 34 where is the overlap matrix between orbitals of the original basis set the low case indexes i and j are comprehensive of site and state labels.
The mixing of orbitals from different sites through the overlap matrix in Eq. It is important to stress that the use of an orthogonalized basis set makes the calculation of energy derivatives significantly more challenging. In fact, the overlap matrix in Eq. As a consequence, when derivatives of the energy are needed, a nonorthogonal basis set is generally used. It is also useful to keep in mind that the effective interaction parameter to be used in the Hubbard functional is sensitive to the specific localized basis set used to define the atomic occupations and the difference between orthogonalized and nonorthogonalized wave functions is sufficient to cause an appreciable variation in its value.
Therefore, the Hubbard U should be recomputed consistently, for example, using the LR technique discussed in Computing the Hubbard U section, with the same basis set used in the construction of the functional. Another possible way to eliminate or significantly alleviate the orthogonalization problem consists in truncating the atomic wave function at the core radius of the pseudopotential of the atoms they belong to.
In this way, the integration of on site occupations is restricted within the regions around the atomic cores and the Hubbard potential amounts to a renormalization of the coefficients of the nonlocal pseudopotential: Although Eq. A similar method was also used to construct an atomic SIC and to effectively embed it in the pseudopotential. A possible solution to this problem consists in generalizing the expression of the Hubbard corrective functional to include interaction terms e.
An alternative approach to this problem consists in adopting a basis set of orbitals particularly suitable to capture the localization of electrons in the considered system. In the case of elemental band semiconductor, e. Wannier functions have indeed become a quite popular choice in recent years to define corrective functionals and computational schemes to improve the description of electronic localization in strongly correlated systems. In Refs. Although MLWF are a popular choice for the definition of many of these functionals based on Wannier functions, other schemes have also been used in literature.
The Wannier function WF basis obtained in this way thus results optimally adapted to capture electronic localization and to produce a density matrix as close as possible to be idempotent. This redistribution of charge is indeed in better agreement with experiments and chemical intuition than a metalic state with excess electrons spread among the f orbitals of all the Ce atoms surrounding the vacancy, as predicted by non corrected DFT functionals.
With all the occupations equal to either 0 or 1, the total energy of the system does not depend on the value of the Hubbard U , as it can be easily understood from Eq. It is important to notice that the effective interaction U still controls the position of the Hubbard bands the Ce f states in this case with respect to the conduction and valence manifolds. In fact, the Hubbard potential [Eq. Therefore, even in cases where the energy does not depend on U , its calculation is still important to accurately describe the electronic properties of the system and its chemical reactivity.
The lack of an explicit expression of the xc energy makes it difficult to model how electronic correlation is accounted for in approximate DFT energy functionals. As a result, simple dc functionals, like the ones in Eqs. The idea that inspired this formulation is quite different from the one behind the FLL atomic limit.
This latter approach corresponds to considering the approximate DFT total energy as containing a mean field approximation of the electron—electron interaction. This is easily seen from the identity 36 :. In both flavors, H Hub contains electron—electron interactions as modeled in the Hubbard Hamiltonian: 38 rotational invariance is neglected here where the asterisk has the same meaning as in Eq. In fact, one can recover the two corrective functionals using, in Eq. An exhaustive discussion about the main ideas at the basis of both the FLL and the AMF formulations, their theoretical framework within DFT, and their specific characteristics, has been also presented in Ref.
A comparison between these two flavors was also offered in Ref. An attempt to obtain a general correction that bridges the AMF and the FLL formulations and is able to treat a broad range of systems with intermediate degrees of electronic localization has been made in Ref. This approach has been used to study intermetalic and selected rare earth compounds 58 showing promising results and a significant improvement with respect to either functional.
In spite of the desirable capability to improve the prediction of properties related to electronic localization such as, e. In fact, this scheme offers a more rigorous treatment of dynamical effects particularly important for metalic system and is able to capture, within the same theoretical framework, the physics of systems and phases characterized by widely different levels of electronic localization. In view of its ability to increase the separation in energy between full and empty states, the FLL corrective functional could still be useful to selectively correct the energetics of localized states i.
In bulk transition metals generally cubic and often magnetic , for example, it was recognized that the e g 2 2 Figure 6 was kindly contributed by Dr. Yang, one of the developers of the AFLOW framework [23,,], that was employed in the structural optimization of the material. Similar results have also been obtained for a variety of metalic systems , with an analogous correction on localized d states.
The transitions this material undergoes between AFM, ferromagnetic FM , and canted orders have not yet been completely clarified see Refs. This is in contrast with the electrical resistance experiments on the material, suggesting at least at temperatures above K a semiconducting behavior. As a side effect, the Fermi level is pushed downward, toward and partly within the occupied p bands.
A third case will be illustrated in the next section, focusing on the intermetalic Heusler alloy Ni 2 MnGa. This example will provide a more precise physical interpretation for the shift in the single particle energies promoted by the Hubbard correction and will illustrate its consequences on the strength of magnetic interactions and on the relative stability of different structural phases.
Ni 2 MnGa is one of the prototype representative of magnetic Heusler alloys. The particular appeal of systems in this family exhibiting a FM ground state consists in the possibility to couple structural martensitic transitions with magnetic ones e. The precise knowledge of the electronic mechanisms controlling both types of transitions could thus greatly facilitate the search for a material with optimal coupling between these transitions and a high degree of reversibility.
Although some controversy still exists in literature, DFT calculations, performed with GGA exchange correlation functionals, generally predict the minimum of the energy in correspondance of a nonmodulated tetragonal structure that has only been observed for nonstoichiometric alloys. As evident from the figure, the GGA functional achieves the minimum of the energy for an elongated tetragonal cell while experiments - report a martensitic phase consisting of a modulated tetragonal structure with.
In fact, the magnetic moments of Mn, Ni and Ga atoms are, respectively, and. The discussion at the end of the previous section should clarify the reason why in Ref. This strategy results in a more accurate description of the material, with larger magnetic moments on Mn atoms see Table I and the austenite phase more stable than the nonmodulated martensite one. Figure 7 compares the energy vs. In particular, computing from constrained DFT calculations the interatomic magnetic couplings J in dependence of the tetragonal elongation and subtracting the corresponding Heisenberg term from the total energy, the minimum corresponding to the nonmodulated martensite phase is eliminated.
In fact, the Anderson model results in RKKY effective interactions between Mn magnetic moments that have a clear dependence on the Hubbard U the parameter controlling the energy splitting between full and empty localized Hubbard bands. The Hubbard U as imposed by the Hubbard correction suppresses magnetic interactions and reduces the energetic advantage associated with the deformation of the cubic austenite phase into a non modulated tetragonal martensite. As shown in Ref.
In these cases a scheme to automatically select localized states or able to treat localized and delocalized orbitals with equal accuracy would be highly desirable. The extended formulation of the Hubbard Hamiltonian has been considered since the early days of this model 33 , 34 and can be expressed as follows: 40 where V is the effective interaction between electrons on neighbor atomic sites.
The interest on the extended Hubbard model has been revamped in the last decades by the discovery of high T c superconductors and the intense research activity focusing around them. Whether the intersite coupling V has a determinant role in inducing superconductivity is, however, still matter of debate. More recently, the extended Hubbard model has been used to study the conduction and the structural properties of polymers and carbon nanostructures and it was shown that the interplay between U and V controls the dimerization of graphene nanoribbons.
Our motivation to include intersite interactions in the formulation of the corrective Hubbard Hamiltonian was the attempt to define a more flexible and general computational scheme able to precisely account for rather than just suppress the possible hybridization of atomic states on different atoms. It is important to notice that the occupation matrix defined in Eq. Within this hypothesis, and assuming that the intersite exchange couplings K IJ can be neglected, it is easy to derive the following expression : 43 where the star in the second summation operator reminds that for each atom I , index J covers all its neighbors up to a given shell.
Subtracting Eq. To better understand the role of the intersite part of the energy functional, it is convenient to derive the correction it contributes to the generalized KS potential: From Eq.
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Thus, the two interactions give rise to competing tendencies, and the character of the resulting ground state depends on the balance between them. Fortunately, the LR calculation of the effective interactions, discussed in Computing the Hubbard U section and in Ref. In the most general case, the differentiation between the interaction parameters would require full orbital dependence for the corrective functional to be invariant.
In the implementation of Eq. The motivation for this extension consists in the fact that different manifolds of atomic states may require to be treated on the same theoretical ground in cases where hybridization is relevant as, e. The choice of these systems was made to test the ability of the new functional to bridge the description of the two kinds of insulators. In fact, as discussed in previous sections [see Eq. Because of the balance between crystal field and exchange splittings of the d states of Ni, nominally occupied by eight electrons, the material has a finite KS gap with oxygen p states occupying the top of the valence band.
This gap, however, severely underestimates the one obtained from photoemission experiments of about 4. The corrective functional included interactions between the d states of Ni, between Ni d and O p states, and between d and s states of the Ni atoms. Other interactions were found to have a negligible effect on the results and were neglected.
The numerical values of the interaction parameters, all determined through the LR approach discussed above, can be found in Ref. On the contrary, the GGA band gap is too small compared with experiments and also has Ni d states at the top of the valence band. As expected, the intersite interactions between Ni and O electrons also results in a more pronounced overlap in energy between d and p states.
In Table 2 a comparison is made between experiments and calculations on the equilibrium lattice parameter, bulk modulus, and energy gap. Although this is an expected result, corrective methods able to enforce the discontinuity to the xc potential and to improve the size of the fundamental gap, are also beneficial for predicting other properties, and the same can be expected from using Hubbard corrections. The LR approach to calculate the effective Hubbard U , described in Computing the Hubbard U section, allowed to reliably compute all these interaction parameters and to capture their dependence on the volume of the crystal.
Due to the suppression of the interatomic hybridization, in both cases, the energy band gap is lowered compared to GGA, further worsening the agreement with experiments. The use of the intersite correction results in a systematic improvement for the evaluation of all these quantities. In fact, encouraging the occupations of hybrid states, the intersite interactions not only enlarge the splittings between full and empty levels which increases the size of the band gap , but also make bonds shorter so that hybridization is enhanced and stronger, thus tuning both the equilibrium lattice parameter and the bulk modulus of these materials to values closer to the experimental data.
Calculations on GaAs explicitly included Ga 3 d states in the valence manifold as suggested by Ref. These results confirm that the extended Hubbard correction is able to significantly improve the description of band insulators and semiconductors with respect to GGA, providing a more accurate estimate of structural and electronic properties. The inclusion of the intersite interaction was found to be crucial to predict the electronic configuration, the equilibrium structure and its deformations in agreement with experiments.
Approximate DFT functionals yield a poor description of the electronic properties of these systems due to the localization of electrons on Ir d orbitals. Consequently, an accurate calculation of excited state energies requires that the functional used is able to capture the localization of electrons on the d orbitals of Ir as well as their possible hybridization with the organic ligands.
These observations may provide valuable informations to tune the performance of these molecules through the screening of substitutional impurities in their ligand complexes. Although invariance is unanimously recognized as a necessary feature of the corrective functional, whether to use the full rotational invariant correction, Eqs. In fact, the two corrective schemes give very similar results for a large number of systems in which electronic localization is not critically dependent on Hund's rule magnetism.
However, as mentioned in A simpler formulation section, in some materials that have recently attracted considerable interest, this equivalence does not hold and the explicit inclusion of the exchange interaction J in the corrective functional appears to be necessary. The precise account of exchange interactions between localized d electrons beyond the simple approach of Eq. Comparing Eqs. It is important to notice that this term is genuinely beyond HF. In fact, a single Slater determinant containing the four states would produce no interaction term like the one above.
Thus, the expression of the J term given in Eq. However, the J term in Eq. Therefore, its formulation and use in corrective functionals are legitimate. Similar terms in the corrective functional have already been proposed in literature, 10 , 56 , 57 , , although within slightly different functionals. Unlike other transition metal monoxides all rhombohedral , CuO has a monoclinic unit cell.
Although it is not the equilibrium structure of this system, the perfectly cubic crystal has been considered as a limiting case of the tetragonal phase grown on selected substrates or as a proxy system to study the electronic properties of cuprate superconductors. As explained in Theoretical background and practical remarks section, these degeneracies need to be lifted if a gap is to appear in the KS spectrum of the material. It is important to notice that the two degeneracies are mutually reinforcing: if spin states have the same energy the material is not magnetic and the symmetry of the crystal is perfectly cubic with an exact degeneracy between e g states.
The use of a triclinic super cell of the cubic structure, depicted in Figure 12 , and an AFII magnetic order are sufficient to lift both these symmetries. However, no insulating state is obtained if J is set to 0. In fact, the presence of the p states of oxygen at the top of the valence band together with Cu d states makes the partial occupation of both manifolds more energetically favorable than the localization of holes on the d states of Cu atoms as necessary to obtain a finite magnetization.
In particular, a magnetic ground state with an inbalance of population between Cu d states of opposite spin and a approximately complete O p manifold has a slightly higher energy than a non magnetic ground state with a larger number of electrons on Cu d states thus closer to complete its d shell equally distributed between the two spin, and a hole spread between d and p levels that results in a metallic ground state. A finite Hund's coupling J favors the magnetic ground state, also resulting in the complete localization of the hole on the Cu d states and in the formation of a band gap.
The total energy of the cubic phase insulating ground state as a function of the tetragonal distortion, shown in Figure 13 , presents a monotonic profile although the material shows a different orbital ordering with a hole occupying the z 2 state of Cu for and the for. The J , instead, was computed [through a generalization of the LR technique based on the perturbative potential of Eq. Furthermore, the smooth variation of U sc about 1 eV from to , shown in Figure 8 of Ref.
These are crucial quantities to identify and characterize the equilibrium structure of materials under different physical conditions, to study the vibrational properties, to perform molecular dynamics calculations, and to account for finite ionic temperature effects typically dominant in insulators. In the derivation of the Hubbard contribution to total energy derivatives, we will also assume a corrective functional based on atomic orbitals as localized basis sets. This derivation can be easily generalized to other basis sets, provided that the derivative of their localized orbitals can be computed analytically.
The Hubbard forces are defined as the negative derivatives of the Hubbard energy with respect to the atomic displacements. Taking the derivative of E U in Eq. Based on the definition in Eq. The problem is then reduced to calculate the quantities As the KS state is a Bloch function, the only nonzero Fourier components are at , where is a vector of the Brillouin zone BZ and is a reciprocal lattice vector. Therefore, the reciprocal space representation of the product between atomic orbitals and KS wave functions reads: 52 where and are, respectively, the atomic orbital and Fourier components.
The explicit dependence on the atomic positions is simply obtained via a change of variable in the integral that defines the Fourier transform, and it can be easily demonstrated that 53 where is the Fourier component of the atomic wave function of the atom I when centered at the origin of the Cartesian system of reference. The structure factor is what determines the dependence of the product in Eq. Due to the Kronecker , this derivative is non zero only if the involved atomic function is centered on the atom which is displaced. This special case is not explicitly treated in this review.
From the expression of the Hubbard energy functional in Eq. This quantity can be defined from the deformation of the crystal as follows: 56 where is the space coordinate internal to the unit cell. The calculation of the stress proceeds along the same steps as for the Hubbard forces [Eqs. The problem is thus reduced to evaluating the derivative This calculation will follow the procedure presented in Ref. The functional dependence of atomic and KS wave functions on the strain can be determined by deforming the lattice according to Eq.
The mathematical details of this calculation can be found in Ref. The final expression of the derivative in Eq. The derivative of the Fourier components of the atomic wavefunctions depends on the particular definition of the atomic orbitals. As its expression can vary according to different implementations, it will not be detailed here. It is, therefore, important to have the capability to compute second and higher order energy derivatives from first principles. This task has represented a considerable challenge when studying correlated systems, for which corrective schemes beyond standard DFT approximations have to be usually used.
In most of these schemes, due to the complexity of the corrective functional and the consequent difficulty in computing derivatives analytically, frozen phonon techniques are normally used. For these types of calculations affordable LR approaches are highly needed. We refer to Ref.
In the following, we specifically treat total energy second derivatives with respect to atomic positions for the calculation of phonons, but the results can be extended to derivatives with respect to any couple of parameters the Hamiltonian depends on. The Hubbard potential , also responds to the atomic displacements and its variation, to be added to , reads: 59 where Once is obtained, the dynamical matrix of the system can be computed to calculate the phonon spectrum and the vibrational modes of the crystal.
Again, in Eq. In ionic insulators and semiconductors a nonanalytical term must be added to the dynamical matrix to account for the coupling of longitudinal vibrations with the macroscopic electric field generated by the ion displacement. These quantities can be computed from the transition amplitude , where c and v indicate conduction and valence bands, respectively. Due to its nonlocality, the Hubbard potential contributes to this quantity with the following term: 62 where are Bloch sums of atomic wave functions.
As evident from the figure, the Hubbard correction produces an overall increase in the phonon frequencies of both materials, significantly improving the agreement with available experimental results.
These results demonstrate that electronic correlations have significant effects on the structural and vibrational properties and that the calculation of quantities involving phonons and their interaction with other excitations e. In all the previous sections, while discussing the contribution of the Hubbard functional to energy derivatives, the Hubbard U was kept fixed.
This is, of course, an approximation, whose validity should be tested carefully, case by case. In fact, some recent works have shown that accounting for the variation of U with the ionic positions and with lattice parameters can be important to obtain quantitatively predictive results. The complexity of the analytic expression of the Hubbard U makes it difficult to account for its variation with the atomic position and lattice parameters.
However, a recent article has introduced a method to efficiently compute the derivative that allows to capture at least at first order the variation of U with the ionic position. Starting from Eq. The derivative of the response functions can be also evaluated starting from their definition: This is the quantity used in the calculation of the Hubbard forces and expressed in Eq. The promising results obtained in Ref. The analysis was based on the review of the theoretical foundation of this method, a description of its most common formulations and implementations, and a discussion of its framing in the context of DFT, highlighting the typical problems it aims to address, and the quality of the correction it provides.
We also pointed out many difficulties this method encounters in describing the properties of metals or, in general, systems with more delocalized electrons. Particular emphasis was put on the necessity to have a method to compute the effective electronic interactions from first principles. A LR approach to this problem was described in detail and compared with other methods present in literature.
At the same time, the extension of the Hubbard corrective Hamiltonian to explicitly include magnetic Hund's rule couplings was discussed highlighting the very significant improvements it brings to the treatment of materials where the onset of a magnetic ground state favors or competes with electronic localization or where spin and charge degrees of freedom are intimately coupled in determining the physical behavior of electrons on strongly localized orbitals.
Notwithstanding the inherent limits of this approach such as, e. The automatic calculation of the effective electronic interactions [e. A more flexile expression of the corrective functional including, e. Authors thank Dr. Fabris and Dr. Yang for providing Figures 2 and 6 , respectively. In this appendix, LR theory will be used to derive the explicit expression of the Hubbard U , as computed in Eq.
To this purpose, it is useful to start from a generalization of the perturbing potential in Eq. To proceed, we will assume that the atomic basis set is orthogonal and complete. Therefore, the identity can be resolved as: A5. Also, a generalized occupation matrix, analogous to the one defined in Eq. For the sake of simplicity, the spin index is dropped for KS states or can be considered included in their comprehensive index n.
Using Eq. A 2 the response of the occupation matrix to the external perturbation can be expressed as follows: A7. If the system were non interacting. In this case, the interacting and noninteracting response matrices coincide:. In fact, in these circumstances, the electronic system responds as an independent electron gas of the same density as the real one.
Substituting with in Eq. A 7 , is obtained. Based on the discussion above, the last equality can be used to evaluate at the first iteration of the perturbed run. From Eqs. A 1 , A 5 , A 7 , and A 8 the following expression can be easily derived: A9. In order to obtain the explicit expression of the Hubbard U it is convenient to rewrite Eq. A8 : A Using the completeness of the localized orbital basis set [Eq. A 5 ], it is convenient to rewrite Eq.
A 7 as follows: A Using Eqs. At this point, we need the explicit expression of. Using again the completeness condition, Eq. A 2 can be rewritten as: A A 12 we easily obtain: A Inserting this expression in Eq. A 13 we obtain: A