Methods: Ab Initio Downfolding and Model-Calculation Techniques

Abstract

We describe computational methods which are used in the present calculations. Our scheme relies on a combination of ab initio and model-calculation techniques. Considering that a low-energy physics is governed by the subspace near the Fermi level, we construct a realistic electron-phonon-coupled low-energy Hamiltonian from first principles (ab initio downfolding) and solve it accurately by model-calculation methods. In this chapter, we first describe a general framework of the present scheme. Then, we move onto the details of the ab initio downfolding. To derive phonon-related terms in the low-energy Hamiltonian, we develop a novel ab initio downfolding scheme, which we call constrained density-functional perturbation theory (cDFPT). The analysis of the low-energy Hamiltonian is done by the extended dynamical mean-field theory, for which we also give a detailed description. Finally, we discuss interfaces between the two steps (ab initio downfolding and analysis of the model).

Appendix: Supplemental Information for DFPT

2.1.1 A. Expression for Interatomic Force Constants with Non-local Pseudopotential

In Sect. 2.2.4, the expression of the interatomic force constants is based on the locality of the ionic potential. However, in the actual DFPT calculations with the plane-wave basis, the true ionic potential, which is local, is replaced by the pseudopotential. When the pseudopotential has non-local components, i.e., depends on two coordinates $\mathbf{r}$ and $\mathbf{r'}$,¹⁶ we need some modifications in the equations. The generalized expression for the second derivative of the energy with respect to parameters $\{ \lambda _i \}$ reads

$$\begin{aligned} \frac{\partial ^{2} E_{\lambda }}{\partial \lambda _{i} \partial \lambda _{j}} = \sum _{\mathbf{k}} \sum _{n}^{\mathrm{occ.}} \Biggl [ \Bigl \langle \frac{\psi _{n \mathbf{k}}}{\partial \lambda _{i} }\Bigl |\frac{\partial V_{\lambda }}{\partial \lambda _{j}} \Bigr | \psi _{n \mathbf{k}} \Bigr \rangle +\Bigl \langle \psi _{n \mathbf{k}}\Bigl |\frac{\partial V_{\lambda }}{\partial \lambda _{j}} \Bigr | \frac{\psi _{n \mathbf{k}}}{\partial \lambda _{i}} \Bigr \rangle + \Bigl \langle \psi _{n \mathbf{k}}\Bigl |\frac{\partial ^{2} V_{\lambda }}{\partial \lambda _{i}\partial \lambda _{j}} \Bigr | \psi _{n \mathbf{k}} \Bigr \rangle \Biggr ] \end{aligned}$$

(2.277)

with $\psi _{n \mathbf{k}}$ being the one-particle wave function with the band n and the wave vector $\mathbf{k}$. Then, the electronic contribution to the interatomic force constant $^{el}C_{\kappa \kappa ^{\prime }}^{\alpha \alpha ^{\prime }}(\mathbf{q})$, i.e., the contribution other than the ionic contribution $\partial E_\mathrm{N} ( \{ \mathbf R \}) / \partial u_{\kappa }^{*\alpha } (\mathbf{q}) \partial u_{\kappa ^{\prime }}^{\alpha ^{\prime }}(\mathbf{q})$, is given by

$$\begin{aligned} ^{el}C_{\kappa \kappa ^{\prime }}^{\alpha \alpha ^{\prime }}(\mathbf{q})= & {} \Biggl [ \sum _{\mathbf{k}} \! \sum _{n}^\mathrm{occ.} \ \Biggl ( \frac{4}{N} \Bigl \langle \frac{\psi _{n \mathbf{k}}}{\partial u_{\kappa }^{\alpha } (\mathbf{q})} \Bigl |\frac{\partial V_\mathrm{ion}}{\partial u_{\kappa ^{\prime }}^{\alpha ^{\prime }}(\mathbf q)} \Bigr | \psi _{n \mathbf{k}} \Bigr \rangle + \! \frac{2}{N} \Bigl \langle \psi _{n \mathbf{k}}\Bigl |\frac{\partial ^{2} V_\mathrm{ion}}{\partial u_{\kappa }^{*\alpha } (\mathbf{q}) \partial u_{\kappa ^{\prime }}^{\alpha ^{\prime }}(\mathbf{q})} \Bigr | \psi _{n \mathbf{k}} \Bigr \rangle \Biggr ) \ \Biggr ]_{u=0} \nonumber \\= & {} \Biggl [ \sum _{\mathbf{k}} \sum _{n}^\mathrm{occ.} \ \Biggl ( \frac{4}{N} \Bigl \langle \frac{\psi _{n \mathbf{k}}}{\partial u_{\kappa }^{\alpha } (\mathbf{q})} \Bigl |\frac{\partial V_\mathrm{ion}}{\partial u_{\kappa ^{\prime }}^{\alpha ^{\prime }}(\mathbf q)} \Bigr | \psi _{n \mathbf{k}} \Bigr \rangle \nonumber \\&+ \, \delta _{\kappa \kappa ^{\prime }}\frac{2}{N} \Bigl \langle \psi _{n \mathbf{k}}\Bigl |\frac{\partial ^{2} V_\mathrm{ion}}{\partial u_{\kappa }^{\alpha } (\mathbf{q}\!=\!0) \partial u_{\kappa }^{\alpha ^{\prime }}(\mathbf{q}\!=\!0)} \Bigr | \psi _{n \mathbf{k}} \Bigr \rangle \Biggr ) \ \Biggr ]_{u=0} \end{aligned}$$

(2.278)

2.1.2 B. Confirmation of the Equality $\varSigma = \varSigma _t + \varSigma _r$ in Sect. 2.2.5.2

Here, we show that the equality $\varSigma = \varSigma _t + \varSigma _r$ in Sect. 2.2.5.2 indeed holds. In principle, the self-energy $\varSigma$, the electron-phonon coupling g, the polarization function $\chi ^0$, and so on, are expressed as matrices. In this section, for the sake of simplicity, we treat them as if they were scalar quantities. One can easily extend the proof to the case where they are matrices. $\varSigma _t = | g^{(p)} | ^2 \chi ^{t}_\mathrm{{LDA}}$ is rewritten as

$$\begin{aligned} \varSigma _t= & {} | g^{(p)}|^2 \frac{\chi ^0_t}{ 1 - \tilde{W}^{(p)} \chi ^0_t} \nonumber \\= & {} | g^{(b)}|^2 \frac{1}{1 - \tilde{v} \chi ^0_r } \frac{\chi ^0_t}{ 1 - \tilde{W}^{(p)} \chi ^0_t} \frac{1}{1 - \tilde{v} \chi ^0_r } \nonumber \\= & {} | g^{(b)}|^2 \left( 1 + \frac{\tilde{v} \chi ^0_r}{1 - \tilde{v} \chi ^0_r } \right) \frac{\chi ^0_t}{ 1 - \tilde{W}^{(p)} \chi ^0_t} \left( 1 + \frac{\tilde{v} \chi ^0_r}{1 - \tilde{v} \chi ^0_r } \right) \nonumber \\= & {} | g^{(b)}|^2 \left( 1 + \chi ^0_r\tilde{W}^{(p)} \right) \frac{\chi ^0_t}{ 1 - \tilde{W}^{(p)} \chi ^0_t} \left( 1 + \tilde{W}^{(p)} \chi ^0_r \right) \nonumber \\= & {} | g^{(b)}|^2 \biggl [ \ \frac{\chi ^0_t}{ 1 - \tilde{W}^{(p)} \chi ^0_t} + \chi ^0_r \frac{\tilde{W}^{(p)} }{ 1 - \tilde{W}^{(p)} \chi ^0_t} \chi ^0_t \nonumber \\&+\, \chi ^0_t \frac{\tilde{W}^{(p)} }{ 1 - \tilde{W}^{(p)} \chi ^0_t}\chi ^0_r + \chi ^0_r \tilde{W}^{(p)} \frac{\chi ^0_t}{ 1 - \tilde{W}^{(p)} \chi ^0_t} \tilde{W}^{(p)} \chi ^0_r \ \biggr ] \nonumber \\= & {} | g^{(b)}|^2 \biggl [ \ \chi ^0_t + \chi ^0_t \tilde{W}^{(f)} \chi ^0_t + \chi ^0_r \tilde{W}^{(f)} \chi ^0_t + \chi ^0_t \tilde{W}^{(f)} \chi ^0_r + \chi ^0_r \tilde{W}^{(p)} \frac{\chi ^0_t}{ 1 - \tilde{W}^{(p)} \chi ^0_t} \tilde{W}^{(p)} \chi ^0_r \ \biggr ]. \nonumber \\ \end{aligned}$$

(2.279)

Similarly, $\varSigma _r = | g^{(b)} | ^2 \chi ^{r}_\mathrm{{LDA}}$ is rewritten as

$$\begin{aligned} \varSigma _r = | g^{(b)}|^2 \frac{\chi ^0_r}{ 1 - \tilde{v} \chi ^0_r} = | g^{(b)}|^2\biggl [ \ \chi ^0_r + \chi ^0_r \tilde{W}^{(p)} \chi ^0_r \ \biggr ]. \end{aligned}$$

(2.280)

Using the equality

$$\begin{aligned} \tilde{W}^{(p)} + \tilde{W}^{(p)} \frac{\chi ^0_t}{ 1 - \tilde{W}^{(p)} \chi ^0_t} \tilde{W}^{(p)} = \frac{\tilde{W}^{(p)} }{ 1 - \tilde{W}^{(p)} \chi ^0_t} = \tilde{W}^{(f)}, \end{aligned}$$

(2.281)

one can show that $\varSigma _t + \varSigma _r$ is expressed as

$$\begin{aligned} \varSigma _t +\varSigma _r= & {} | g^{(b)}|^2\biggl [ \ \chi ^0_t + \chi ^0_r + \bigl ( \chi ^0_t + \chi ^0_r \bigr ) \tilde{W}^{(f)} \bigl ( \chi ^0_t + \chi ^0_r \bigr ) \ \biggr ] \nonumber \\= & {} | g^{(b)}|^2\biggl [ \ \chi ^0 + \chi ^0 \tilde{W}^{(f)} \chi ^0 \ \biggr ] \nonumber \\= & {} | g^{(b)}|^2 \chi _\mathrm{{LDA}}, \end{aligned}$$

(2.282)

which agrees with the expression for $\varSigma$ in Eq. (2.122).