# Self-consistency

### Purpose

Self-consistency is a concept closely linked with mean-field theories. In mean-field treatment of the Schrodinger equation, interacting particles give off a field, and interact with the (mean) field other particles generate. The field a particle senses controls its motion, which in turn affects the field it generates. Mean-field theories link the field a particle generates to the field it senses in a self-consistent manner. For the field to be time-independent and the effective field must be some average field. Thus in the mean-field approximation a particle moves in an average effective external field *V*^{eff}(**r**) which determines the particles’ motion, which in turn determines *V*^{eff}(**r**). Each feeds the other; the condition for which both are satisfied is the self-consistency condition.

### Table of Contents

- Self-Consistency in Density-functional Theory
- Self-Consistency in noncollinear magnetism
- Self-Consistency in the Coherent Potential Approximation
- Self-Consistency in nonequilibrium transport
- Self-Consistency in the Quasiparticle Self-Consistent GW Approximation
- Self-Consistency in tight-binding models
- References

### Self-Consistency in Density-functional Theory

Practical implementations of DFT, such as the LDA are instances of a mean-field theory. Formally the density *n* is determined by minimization of a total energy functional *E*[*n*]. In principle it could be done through iterating trial densities *n*_{in} until the point of minimum energy is found. In practice this is not possible because realistic schemes must calculate the kinetic energy contribution to *E*[*n*] via an effective non-interacting Schrodinger equation. An effective one-particle potential *V*^{eff}(**r**) is generated from the functional derivative *V*^{eff}(**r**)=δ*E*[*n*]/δ*n*(**r**). *V*^{eff}(**r**) defines a one-particle Hamiltonian which generates a new density *n*_{out}, which differs from *n*_{in}. Self-consistency is reached when an *n*_{out}=*n*_{in}. This density is also the one for which *E*[*n*] is minimum, or at least stationary, for certain formulations. The codes here implement the standard Hohenberg Kohn (HK) functional, for which the energy is a minimum, and also the Harris-Foulkes or HF functional [1], for which the energy is stationary at self-consistency, though typically it is a maximum point.

Approaching self-consistency can be a tricky business, especially in magnetic systems (or whenever there is a degree of freedom with a very low energy scale associated with it). This page describes how the Questaal suite manages iterations to self-consistency.

LDA+*U* theory, an extension to LDA, is also a mean-field theory. At sites where *U* is added there is a self-consistency condition on the site **density matrix**, which must be incorporated together with the standard charge self-consistency.

### Self-Consistency in noncollinear magnetism

In noncollinear extensions to standard DFT, the density turns into a 2×2 matrix, for which at any point **r** the density may be decomposed into linear combinations of Pauli matrices. The decomposition defines the x,y,z components of the spin density. Using the connection between Pauli matrices and Cartesian coordinates, the spin density represented as spin amplitude and direction. This is how it is represented externally. Internally it is represented at times either as (amplitude, direction), or (when constructing the hamiltonian or charge density) as a 2×2 matrix. Noncollinear magnetism is implemented at present in the ASA only, and the spin orientation a site is assumed to be constant for the whole site. Self-consistency in this context means that the spin at each site is rotated so that the off-diagonal part of the 2×2 spin density matrix vanish.

### Self-Consistency in the Coherent Potential Approximation

Alloys consist of different kinds of atoms sitting at a given lattice site, or in the magnetic version, an atom of a particular type (e.g. Fe) with different spin configurations. The CPA replaces a statistical ensemble of atoms with which is taken to be an average, or composite of the true atoms. CPA theory is a way to construct the “average” properties of a composite atom, in an optimal manner, by constructing an the average scattering properties of a site. The theory is naturally formulated in the language of Green’s functions. The CPA has been implemented in the ASA for both magnetic and chemical disorder, in program **lmgf**. Thus **lmgf** requires two kinds of self-consistency: an outer loop for charge and an inner loop for the CPA condition. See here for a description of Questaal’s implementation of CPA.

### Self-Consistency in nonequilibrium transport

**lmpg** is a Green’s function code, similar to **lmgf** but designed for layered systems (periodic in two dimensions but not in the third). One of the best uses of this method is its ability to calculate transport in the Landauer-Buttiker framework. Outside the small bias regime, the potential must be determined self-consistently under nonequilibrium conditions. Questaal’s implementation is described in Ref [2].

### Self-Consistency in the Quasiparticle Self-Consistent GW Approximation

Self-consistency has a different purpose in the quasiparticle self-consistent GW approximation (QS*GW*) [3]. QS*GW* is based on a kind of self-consistent perturbation theory, where the self-consistency is used to construct an optimal non-interacting hamiltonian by minimizing the difference between it and the many-body hamiltonian, within the *GW* approximation. Self-consistency enables QS*GW* to describe optical properties in a wide range of materials rather well, including cases where the local-density and LDA-based *GW* approximations fail qualitatively. Self-consistency dramatically improves agreement with experiment, and is sometimes essential. QS*GW* avoids some formal and practical problems encountered in conventional self-consistent *GW*, which is more akin to the mean-field self-consistency described earlier. QS*GW* handles both itinerant and correlated electrons on an equal footing, without any ambiguity about how a localized state is defined, or how double-counting terms should be subtracted. Weakly correlated materials such as Na and *sp* semiconductors are described with uniformly high accuracy. Discrepancies with experiment are small and systematic, and can be explained in terms of the approximations made.

### Self-Consistency in tight-binding models

Typically empirical tight-binding models, for which the form of matrix elements is postulated, ignore self-consistency. However, the effects of charge transfer can be important, and the quality tight-binding models can be vastly improved with minimal complexity added to the hamiltonian [4].

This package’s implementation of empirical tight-binding hamiltonians, **tbe**, has a facility for including electrostatic interactions in a self-consistent manner. Site charges are accumulated from eigenvectors, which is used to make a Madelung potential

Typically tight-binding models ignore self-consistency; however, **tbe** has a capability to include potential changes from charge transfer in a self-consistent manner. The site charges generate an electrostatic potential through a Madelung matrix, and must be determined self-consistently.

### References

[1] M. Foulkes and R. Haydock, *Tight-binding models and density-functional theory*, *Phys. Rev.* B39, 12520 (1989). See also comment 3 on this page.

[2] S. V. Faleev, F. Leonard, D. A. Stewart, and M. van Schilfgaarde, *Ab initio tight-binding LMTO method for nonequilibrium electron transport in nanosystems*, *Phys. Rev.* B71, 195422 (2005).

[3] Takao Kotani, M. van Schilfgaarde, S. V. Faleev, *Quasiparticle self-consistent* GW *method: a basis for the independent-particle approximation*, *Phys. Rev.* B 76, 165106 (2007).

[4] M. W. Finnis, A. T. Paxton, M. Methfessel and M. van Schilfgaarde, *Crystal Structures of Zirconia from First Principles and Self-Consistent Tight Binding*, *Phys. Rev. Lett.* **81**, 5149 (1998)

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