Efficient algorithm of the transcorrelated method for periodic systems

Abstract

The transcorrelated (TC) method is one of the promising wave-function-based approaches for the first-principles electronic structure calculations. In this method, the many-body wave function is approximated as the Jastrow-Slater type and one-electron orbitals in the Slater determinant are optimized with a one-body self-consistent-field equation such as that in the Hartree-Fock (HF) method. Although the TC method has yielded good results for both molecules and solids, its computational cost in solid-state calculations, being of order O(N(k)(3)N(b)(3)) with N(k) and N(b) the respective numbers of k-points and bands, has for some years hindered its wide application in condensed matter physics. Although an efficient algorithm was proposed for a Gaussian basis set, that algorithm is not applicable to a plane-wave basis that is suited to and widely used in solid-state calculations. In this paper, we present a new efficient algorithm of the TC method for the plane-wave basis or an arbitrary basis function set expanded in terms of plane waves, with which the computational cost of the TC method scales as O(N(k)(2)N(b) (2)). This is the same as that of the HF method. We applied the TC method with the new algorithm to obtain converged band structure and cell parameters of some semiconductors.

Perspective: Explicitly correlated electronic structure theory for complex systems

“…121 Ochi et al have also shown that integrals for the kernel ∇ 1 u 12 · ∇ 1 u 13 can be efficiently computed using plane wave basis sets. 44 This kernel appears in both TC and explicitly correlated F12 theories. 44,61 In contrast to the direct evaluation of three-electron integrals using atom-centered Gaussian basis sets in GTG theory, plane wave basis sets benefit greatly from the computational efficiency of Fourier transformations and the translational invariance of the employed kernels.…”

“…44 This kernel appears in both TC and explicitly correlated F12 theories. 44,61 In contrast to the direct evaluation of three-electron integrals using atom-centered Gaussian basis sets in GTG theory, plane wave basis sets benefit greatly from the computational efficiency of Fourier transformations and the translational invariance of the employed kernels. Another advantage of using plane wave basis sets is that the Fourier coefficients of the correlation factor in the above expression can be easily modified to further improve the rate of convergence of the wave function expansion.…”

“…This is one reason why there have been relatively few studies on explicitly correlated methods including TC implementations for solids. 42,44,45,120,121,158 Alternatively, it is possible to employ Gaussian basis sets in explicitly correlated theory for solids. 129,159 The main advantage of this approach is that one can exploit the framework developed in F12 theory for many-electron integrals in conjunction with the local correlation techniques described in Sec.…”

“…Many promising approaches to overcome these problems have already been proposed for large or periodic systems, including Monte Carlo techniques, 115,116 local explicitly correlated methods, 136,145,146 and plane wave basis set techniques. 44,121 These methods exhibit a trade-off between computational cost and accuracy that is strongly dependent on the system size, dimensionality, and possibly the band gap. In addition, a significant portion of the total correlation energy comes from the accumulation of long-range dispersion in a solid.…”

“…This has been successfully applied to the calculations of band structures and optical absorption spectra in solids. [44][45][46] Variational quantum Monte Carlo integration using Metropolis sampling 47 is another powerful technique for more complicated correlation factors, as will be discussed in Sec. II.…”