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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.

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References

  1. 1.

    Adler, S.L.: Quantum theory of the dielectric constant in real solids. Phys. Rev. 126, 413–420 (1962)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Anderson, D.G.: Iterative procedures for nonlinear integral equations. J. Assoc. Comput. Mach. 12, 547–560 (1965)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Baroni, S., Giannozzi, P., Testa, A.: Green’s-function approach to linear response in solids. Phys. Rev. Lett. 58, 1861–1864 (1987)

  4. 4.

    Baroni, S., de Gironcoli, S., Dal Corso, A., Giannozzi, P.: Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys. 73, 515–562 (2001)

    Google Scholar 

  5. 5.

    Becke, A.D.: Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 38, 3098–3100 (1988)

    Google Scholar 

  6. 6.

    Bowler, D.R., Miyazaki, T.: O(N) methods in electronic structure calculations. Rep. Prog. Phys. 75, 036503 (2012)

    Google Scholar 

  7. 7.

    Cancès, E., Deleurence, A., Lewin, M.: A new approach to the modeling of local defects in crystals: the reduced Hartree-Fock case. Commun. Math. Phys. 281, 129–177 (2008)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Cancès, E., Deleurence, A., Lewin, M.: Non-perturbative embedding of local defects in crystalline materials. J. Phys.: Condens. Matter. 20, 294213–294218 (2008)

    Google Scholar 

  9. 9.

    Cances, E., Mourad, N.: A mathematical perspective on density functional perturbation theory. Nonlinearity 27, 1999 (2014)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Ceperley, D.M., Alder, B.J.: Ground state of the electron gas by a stochastic method. Phys. Rev. Lett. 45, 566–569 (1980)

    Google Scholar 

  11. 11.

    Chan, T.F., Hansen, P.C.: Computing truncated singular value decomposition least squares solutions by rank revealing QR-factorizations. SIAM J. Sci. Stat. Comput. 11, 519–530 (1990)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Cheng, H., Gimbutas, Z., Martinsson, P.G., Rokhlin, V.: On the compression of low rank matrices. SIAM J. Sci. Comput. 26, 1389–1404 (2005)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Foerster, D.: Elimination, in electronic structure calculations, of redundant orbital products. J. Chem. Phys. 128, 034108 (2008)

    Google Scholar 

  14. 14.

    Frenkel, D., Smit, B.: Understanding Molecular Simulation: From Algorithms to Applications. Academic Press, New York (2002)

    MATH  Google Scholar 

  15. 15.

    Giustino, F., Cohen, M.L., Louie, S.G.: GW method with the self-consistent Sternheimer equation. Phys. Rev. B 81, 115105 (2010)

    Google Scholar 

  16. 16.

    Goedecker, S.: Linear scaling electronic structure methods. Rev. Mod. Phys. 71, 1085–1123 (1999)

    Google Scholar 

  17. 17.

    Gonze, X., Lee, C.: Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory. Phys. Rev. B 55, 10355 (1997)

    Google Scholar 

  18. 18.

    Gu, M., Eisenstat, S.: Efficient algorithms for computing a strong rank-revealing qr factorization. SIAM J. Sci. Comput. 17(4), 848–869 (1996)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Hohenberg, P., Kohn, W.: Inhomogeneous electron gas. Phys. Rev. 136, B864–B871 (1964)

    MathSciNet  Google Scholar 

  20. 20.

    Kleinman, L., Bylander, D.M.: Efficacious form for model pseudopotentials. Phys. Rev. Lett. 48, 1425–1428 (1982)

    Google Scholar 

  21. 21.

    Knyazev, A.V.: Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comp. 23, 517–541 (2001)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Kohn, W.: Density functional and density matrix method scaling linearly with the number of atoms. Phys. Rev. Lett. 76, 3168–3171 (1996)

    Google Scholar 

  23. 23.

    Kohn, W., Sham, L.: Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965)

    MathSciNet  Google Scholar 

  24. 24.

    Lee, C., Yang, W., Parr, R.G.: Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 37, 785–789 (1988)

    Article  Google Scholar 

  25. 25.

    Lin, L., Lu, J., Ying, L.E.W.: Pole-based approximation of the Fermi-Dirac function. Chin. Ann. Math. 30B, 729 (2009)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Lin, L., Saad, Y., Yang, C.: Approximating spectral densities of large matrices. SIAM Rev. 58, 34 (2016)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Lin, L., Xu, Z., Ying, L.: Adaptively compressed polarizability operator for accelerating large scale ab initio phonon calculations. Multiscale Model. Simul. 15, 29–55 (2017)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Lu, J., Sogge, C.D., Steinerberger, S.: Approximating pointwise products of Laplacian eigenfunctions (2018). Preprint arXiv:1811.10447

  29. 29.

    Lu, J., Ying, L.: Compression of the electron repulsion integral tensor in tensor hypercontraction format with cubic scaling cost. J. Comput. Phys. 302, 329 (2015)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Martin, R.: Electronic Structure: Basic Theory and Practical Methods. Cambridge Univ. Pr, West Nyack, NY (2004)

    MATH  Google Scholar 

  31. 31.

    Mermin, N.: Thermal properties of the inhomogeneous electron gas. Phys. Rev. 137, A1441 (1965)

    MathSciNet  Google Scholar 

  32. 32.

    Moussa, J.E.: Minimax rational approximation of the fermi-dirac distribution. J. Chem. Phys. 145(16), 164108 (2016)

    Google Scholar 

  33. 33.

    Nakatsukasa, Y., Sète, O., Trefethen, L.N.: The aaa algorithm for rational approximation. SIAM J. Sci. Comput. 40(3), A1494–A1522 (2018)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Nguyen, H.V., Pham, T.A., Rocca, D., Galli, G.: Improving accuracy and efficiency of calculations of photoemission spectra within the many-body perturbation theory. Phys. Rev. B 85, 081101 (2012)

    Google Scholar 

  35. 35.

    Niklasson, A.M.N., Challacombe, M.: Density matrix perturbation theory. Phys. Rev. Lett. 92, 193001 (2004)

    Google Scholar 

  36. 36.

    Onida, G., Reining, L., Rubio, A.: Electronic excitations: density-functional versus many-body Greens-function approaches. Rev. Mod. Phys. 74, 601 (2002)

    Google Scholar 

  37. 37.

    Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Perdew, J.P., Burke, K., Ernzerhof, M.: Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996)

    Google Scholar 

  39. 39.

    Perdew, J.P., Zunger, A.: Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23, 5048–5079 (1981)

    Google Scholar 

  40. 40.

    Ren, X., Rinke, P., Blum, V., Wieferink, J., Tkatchenko, A., Sanfilippo, A., Reuter, K., Scheffler, M.: Resolution-of-identity approach to Hartree-Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions. New J. Phys. 14, 053020 (2012)

  41. 41.

    Sodt, A., Subotnik, J.E., Head-Gordon, M.: Linear scaling density fitting. J. Chem. Phys. 125, 194109 (2006)

    Google Scholar 

  42. 42.

    Solovej, J.P.: Proof of the ionization conjecture in a reduced Hartree-Fock model. Invent. Math. 104, 291–311 (1991)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Umari, P., Stenuit, G., Baroni, S.: Optimal representation of the polarization propagator for large-scale GW calculations. Phys. Rev. B 79(20), 201104 (2009)

    Google Scholar 

  44. 44.

    Umari, P., Stenuit, G., Baroni, S.: GW quasiparticle spectra from occupied states only. Phys. Rev. B 81, 115104 (2010)

    Google Scholar 

  45. 45.

    Weigend, F.: A fully direct RI-HF algorithm: implementation, optimised auxiliary basis sets, demonstration of accuracy and efficiency. Phys. Chem. Chem. Phys. 4, 4285–4291 (2002)

    Google Scholar 

  46. 46.

    Wiser, N.: Dielectric constant with local field effects included. Phys. Rev. 129, 62–69 (1963)

    MATH  Google Scholar 

  47. 47.

    Woolfe, F., Liberty, E., Rokhlin, V., Tygert, M.: A fast randomized algorithm for the approximation of matrices. Appl. Comput. Harmon. Anal. 25(3), 335–366 (2008)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Yang, C., Meza, J.C., Lee, B., Wang, L.W.: KSSOLV-a MATLAB toolbox for solving the Kohn-Sham equations. ACM Trans. Math. Software 36, 10 (2009)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the National Science Foundation under Grant No. DMS-1652330 (D. A. and L. L.), the U.S. Department of Energy under Contract No. DE-SC0017867 (L. L. and Z. X.), and the U.S. Department of Energy under the Center for Applied Mathematics for Energy Research Applications (CAMERA) program (L. L.). We thank Berkeley Research Computing for the computational resources.

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Appendix A

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.

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An, D., Lin, L. & Xu, Z. Split representation of adaptively compressed polarizability operator. Res Math Sci 8, 51 (2021). https://doi.org/10.1007/s40687-021-00285-0

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Keywords

  • Density functional perturbation theory
  • Phonon calculations
  • Vibration properties
  • Adaptive compression
  • Split representation
  • Polarizability operator
  • Sternheimer equation
  • Dyson equation

Mathematics Subject Classification

  • 65F10
  • 65F30
  • 65Z05