Split representation of adaptively compressed polarizability operator

Abstract

The polarizability operator plays a central role in density functional perturbation theory and other perturbative treatment of first principle electronic structure theories. The cost of computing the polarizability operator generally scales as ${\mathcal {O}}(N_{e}^4)$ where $N_e$ is the number of electrons in the system. The recently developed adaptively compressed polarizability operator (ACP) formulation [L. Lin, Z. Xu and L. Ying, Multiscale Model. Simul. 2017] reduces such complexity to ${\mathcal {O}}(N_{e}^3)$ in the context of phonon calculations with a large basis set for the first time, and demonstrates its effectiveness for model problems. In this paper, we improve the performance of the ACP formulation by splitting the polarizability into a near singular component that is statically compressed, and a smooth component that is adaptively compressed. The new split representation maintains the ${\mathcal {O}}(N_e^3)$ complexity, and accelerates nearly all components of the ACP formulation, including Chebyshev interpolation of energy levels, iterative solution of Sternheimer equations, and convergence of the Dyson equations. For simulation of real materials, we discuss how to incorporate nonlocal pseudopotentials and finite temperature effects. We demonstrate the effectiveness of our method using one-dimensional model problem in insulating and metallic regimes, as well as its accuracy for real molecules and solids.

Appendix A

Using the Cauchy contour integral formulation, the density matrix at finite temperature can be represented as

$$\begin{aligned} P_0 = \frac{1}{ 2\pi i } \oint _{\mathcal {C}} f(z) (z - H)^{-1} \,\mathrm {d}z. \end{aligned}$$

(69)

When the Hamiltonian is perturbed to $H_{\varepsilon } = H_{0} + \varepsilon \mathfrak {g}$, and when $\varepsilon$ is small enough, the perturbed density matrix $P_\varepsilon$ can still be computed as

$$\begin{aligned} P_\varepsilon = \frac{1}{ 2\pi i } \oint _{\mathcal {C}} f(z) (z - H_{\varepsilon })^{-1} \,\mathrm {d}z. \end{aligned}$$

(70)

Then we have

$$\begin{aligned} \begin{aligned} P_{\varepsilon } - P_0&= \frac{1}{2\pi i} \oint _{\mathcal {C}} f(z) \left[ (z - H_{\varepsilon })^{-1} - (z - H)^{-1} \right] \,\mathrm {d}z \\&= \frac{1}{2\pi i} \oint _{\mathcal {C}} f(z) \left[ (z - H_{\varepsilon })^{-1} \varepsilon \mathfrak {g} (z - H)^{-1} \right] \,\mathrm {d}z \\&= \frac{1}{2\pi i} \oint _{\mathcal {C}} f(z) \left[ (z - H)^{-1} \varepsilon \mathfrak {g} (z - H)^{-1} \right] \,\mathrm {d}z + {\mathcal {O}}(\varepsilon ^2). \end{aligned} \end{aligned}$$

(71)

Hence by the definition of $\mathfrak {X}_{0}$, we have

$$\begin{aligned} \mathfrak {X}_{0} \mathfrak {g} = \frac{1}{2\pi i} \oint _{\mathcal {C}} f(z) \left[ (z - H)^{-1} \mathfrak {g} (z - H)^{-1} \right] \,\mathrm {d}z. \end{aligned}$$

(72)

Using the spectral decomposition of H, and use the contour integral formulation

$$\begin{aligned} \begin{aligned} \mathfrak {X}_{0} \mathfrak {g}&= \frac{1}{2\pi i} \oint _{\mathcal {C}} \sum _{j,k = 1}^{\infty } f(z) \left[ \frac{\psi _j \psi _j^* \mathfrak {g} \psi _k \psi _k^*}{(z-\varepsilon _j)(z-\varepsilon _k)} \right] \,\mathrm {d}z \\&= \frac{1}{2\pi i} \sum _{j,k = 1}^{\infty } \oint _{\mathcal {C}} \,\mathrm {d}z\frac{f(z) }{(z-\varepsilon _j)(z-\varepsilon _k)} \left[ \psi _j \psi _j^* \mathfrak {g} \psi _k \psi _k^*\right] \\&= \sum _{j\ne k}^{\infty } \frac{f_j - f_k }{\varepsilon _j-\varepsilon _k} \left[ \psi _j \psi _j^* \mathfrak {g} \psi _k \psi _k^*\right] + \sum _{j}^{\infty } f'_j \left[ \psi _j \psi _j^* \mathfrak {g} \psi _j \psi _j^*\right] \\&= \sum _{j, k}^{\infty } \frac{f_j - f_k }{\varepsilon _j-\varepsilon _k} \left[ \psi _j \psi _j^* \mathfrak {g} \psi _k \psi _k^*\right] , \end{aligned} \end{aligned}$$

(73)

where the $\frac{f_j - f_k}{\varepsilon _j - \varepsilon _k}$ is interpreted as the derivative when $j=k$.

For the purpose of computing singular part with contour representation, we have

$$\begin{aligned} \begin{aligned} \mathfrak {X}_{0}^{(s)} \mathfrak {g}&=\sum _{i=1}^{N_{\mathrm {cut}}} \sum _{a=N_{\mathrm {cut}} +1}^{\widetilde{N}_{\mathrm {cut}}} \frac{f_{a}-f_{i}}{\varepsilon _{a}-\varepsilon _{i}} \psi _{a}(\psi _{a}^{*} \mathfrak {g} \psi _{i})\psi _{i}^{*} + \mathrm {h.c.} \\&\quad + \sum _{i=1}^{N_{\mathrm {cut}}} \sum _{a=1}^{N_{\mathrm {cut}}} \frac{f_{a}-f_{i}}{\varepsilon _{a}-\varepsilon _{i}} \psi _{a}(\psi _{a}^{*} \mathfrak {g} \psi _{i})\psi _{i}^{*} \\&= \frac{1}{2\pi \imath } \oint _{\mathcal {C}} \,\mathrm {d}z \sum _{i=1}^{N_{\mathrm {cut}}} \sum _{a=N_{\mathrm {cut}}+1}^{\widetilde{N}_{\mathrm {cut}}} \frac{f(z)}{(z-\varepsilon _a)(z-\varepsilon _i)} \left[ \psi _a \psi _a^* \mathfrak {g} \psi _i \psi _i^*\right] + \mathrm {h.c.}\\&\quad + \frac{1}{2\pi \imath } \oint _{\mathcal {C}} \,\mathrm {d}z \sum _{i=1}^{N_{\mathrm {cut}}} \sum _{a=1}^{N_{\mathrm {cut}}} \frac{f(z)}{(z-\varepsilon _a)(z-\varepsilon _i)} \left[ \psi _a \psi _a^* \mathfrak {g} \psi _i \psi _i^*\right] \\&= \frac{1}{2\pi \imath } \oint _{\mathcal {C}} f(z) (z-H_{c,2})^{-1}\mathfrak {g}(z-H_{c,1})^{-1} \,\mathrm {d}z + \mathrm {h.c.} \\&\quad + \frac{1}{2\pi \imath } \oint _{\mathcal {C}} f(z) (z-H_{c,1})^{-1}\mathfrak {g}(z-H_{c,1})^{-1} \,\mathrm {d}z, \end{aligned} \end{aligned}$$

(74)

where $H_{c,1}=\sum _{i=1}^{N_{\mathrm {cut}}} \psi _{i} \varepsilon _{i}\psi _{i}^{*}, H_{c,2}=\sum _{i=N_{\mathrm {cut}}+1}^{\widetilde{N}_{\mathrm {cut}}} \psi _{i} \varepsilon _{i}\psi _{i}^{*}$ are the Hamiltonian operators projected to the subspace spanned by the first $N_{\mathrm {cut}}$ states, and to the subspace spanned by the following $(\widetilde{N}_{\mathrm {cut}} - N_{\mathrm {cut}})$ states, respectively.