# Recent Progress in First-Principles Methods for Computing the Electronic Structure of Correlated Materials

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background and Historical Development

#### 2.1. LDA+U

#### 2.2. The $GW$ Method

#### 2.3. Dynamical Mean-Field Theory

#### 2.4. LDA+DMFT

## 3. Recent Theoretical Progress

#### 3.1. $GW$+EDMFT

#### 3.1.1. Functional Approach

#### 3.2. Multitier Self-Consistent $GW$+EDMFT

- Subspace 1: The correlated subspace in which self-consistent GW + EDMFT is applied.
- Subspace 2: The intermediate subspace in which self-consistent GW is active.
- Subspace 3: The remaining subspace in which a one-shot GW is active.

#### 3.3. Illustrative Example: SrVO${}_{3}$

#### 3.3.1. One-Shot GW+DMFT

#### 3.4. Effect of Self-Consistency: Multitier $GW$+EDMFT

- Are the local vertex corrections from DMFT sufficient to counteract the detrimental effects of self-consistent $GW$?
- What is the role of the long-range screening?
- What is the nature of the satellites in SrVO${}_{3}$? Are they plasmons as indicated by the $GW$ calculations and the cumulant expansion or are they Hubbard bands as DMFT-based calculations suggest?

#### 3.5. Different Levels of Self-Consistency

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Construction of a Low-Energy Model

#### Appendix A.1. Constrained RPA

#### Appendix A.2. Downfolding the Self-Energy

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**Figure 1.**LDA+U band structure, density of states (DOS) and partial-f DOS for gadolinium. The calculations were done using the parameters U = 12.4 eV and J = 1.0 eV. The displayed directions are 1/2(1,1,1) → $\mathsf{\Gamma}$ → (1,0,0). For comparison, we also show partial-f DOS from a spin-polarized LDA-calculation. The LDA+U figures are taken from Ref. [45] and the LDA-calculation was done using the full potential linearized augmented planewave (FLAPW) code FLEUR [46]. The experimental exchange splitting is approximately 12–13 eV [7,44].

**Figure 3.**Spectral function of Ce in the $\alpha $-phase (

**left**) and $\gamma $-phase (

**right**) compared to experiment. The dashed lines show the total spectral function and the solid lines the projected 4f spectral function (for details, see Ref. [61]). The $GW$-approximation fails to capture the reduction of the quasiparticle weight in going from the $\alpha $- to the $\gamma $-phase. The figure is taken from Ref. [61].

**Figure 4.**Schematic figure showing the dynamical mean-field theory (DMFT) (black), extended-DMFT (EDMFT) (black+blue) and $GW$+EDMFT (black+blue+red) self-consistency cycles.

**Figure 5.**Crystal structure (

**left**) and LDA band structure (

**right**). The red bands show the ${t}_{2g}$ conduction bands.

**Figure 6.**(

**top**): Quasiparticle band structure of SrVO${}_{3}$ within LDA, $GW$, LDA+DMFT and (one-shot) $GW$+DMFT. The LDA+DMFT results were computed with a frequency-dependent U calculated with the cRPA. (

**bottom**): $GW$+DMFT bands compared to scaled $GW$-bands showing the kink structure in $GW$+DMFT, taken from Ref. [86].

**Figure 7.**Partial ${t}_{2g}$ spectral function of SrVO${}_{3}$ in (one-shot) $GW$+DMFT, compared to LDA, $GW$ and LDA+DMFT. The LDA+DMFT results were computed with a frequency-dependent U calculated with the cRPA, taken from Ref. [86].

**Figure 8.**Partial ${t}_{2g}$ spectral function of SrVO${}_{3}$ computed with the multitier $GW$+EDMFT method compared with one-shot $GW$ (${G}^{0}{W}^{0}$) and self-consistent $GW$ ($GW$), taken from Ref. [38].

**Figure 9.**Onsite projection of the fully screened interaction W for the ${t}_{2g}$-states for SrVO${}_{3}$ computed with the multitier $GW$+EDMFT method compared with one-shot $GW$ (${G}^{0}{W}^{0}$) and self-consistent $GW$ ($GW$), taken from Ref. [38].

**Figure 10.**$\mathbf{k}$-resolved spectral function of SrVO${}_{3}$ computed with the multitier $GW$+EDMFT method compared with one-shot $GW$ (${G}^{0}{W}^{0}$) and LDA+DMFT with cRPA U. Each figure has the quasiparticle peaks of the other methods superimposed, taken from Ref. [38].

**Figure 11.**LDA band structure (purple) and Wannier interpolated band structure for sodium with the experimental lattice constant (${a}_{0}$) as well as increased lattice constants $1.4{a}_{0}$ and $1.6{a}_{0}$. The color coding shows the s-character of the bands, as defined by the s-like Wannier function. Taken from Ref. [39].

**Figure 12.**Bare interaction (U) computed with the constrained random-phase approximation (cRPA) compared with the $GW$+EDMFT effective impurity interaction $\mathcal{U}$ for sodium with the different lattice constants, taken from Ref. [39].

**Figure 13.**Spectral function of sodium with the different lattice constants. Comparison between multitier $GW$+EDMFT and one-shot $GW$ (${G}^{0}{W}^{0}$), taken from Ref. [39].

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Nilsson, F.; Aryasetiawan, F.
Recent Progress in First-Principles Methods for Computing the Electronic Structure of Correlated Materials. *Computation* **2018**, *6*, 26.
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**AMA Style**

Nilsson F, Aryasetiawan F.
Recent Progress in First-Principles Methods for Computing the Electronic Structure of Correlated Materials. *Computation*. 2018; 6(1):26.
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**Chicago/Turabian Style**

Nilsson, Fredrik, and Ferdi Aryasetiawan.
2018. "Recent Progress in First-Principles Methods for Computing the Electronic Structure of Correlated Materials" *Computation* 6, no. 1: 26.
https://doi.org/10.3390/computation6010026