Editor's ChoiceSecond-order Møller–Plesset perturbation theory for the transcorrelated Hamiltonian applied to solid-state calculations
Graphical abstract
Introduction
Recently, wave function theory, which had been traditionally developed and applied to the first-principles calculations of molecular systems in the field of quantum chemistry, has been actively investigated also for solid-state calculations. Wave function theory has an attractive advantage that its accuracy can be systematically improved, which is difficult for density functional theory [1], [2] because the exact form of the exchange-correlation functional is nontrivial. Despite this great benefit, application of the wave function theory to solid-state calculations has been limited because of its huge computational requirement. The HF method, which is the starting point for most of wave function theories, requires relatively low computational cost and so often has been applied to solid-state calculations [3], [4], [5]. However, the HF method in many cases does not provide enough accuracy for solid-state calculations because it neglects all the electron correlation effects such as the screening effect. The simplest post-HF method to go beyond is the second-order Møller–Plesset (MP2) perturbation theory. MP2 is tractable in terms of computational cost for solid-state calculations and was recently applied to some simple solids [6], [7]. It was shown that MP2 correction successfully improves the HF results in some cases, but also an unfavorable feature was observed: the band structure of narrow-gap systems are not well improved, e.g., the band gap of bulk silicon becomes negative. This seems to be because the unperturbed HF band structure is very bad. In addition, an inaccurate unperturbed state often yields a large correction with the perturbative treatment, which makes it difficult to achieve sufficient convergence with respect to, e.g., the number of conduction bands and k-points, as verified later in this Letter. Therefore, another accurate starting point to describe electron correlation in solids is desired.
The Transcorrelated (TC) method [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] is a promising alternative. In the TC method, the many-body wave function is approximated with a product of a Jastrow factor, , and a Slater determinant, a similar form of which has often been used also in variational Monte Carlo (VMC) calculations [18]. A key idea of the TC method is to perform similarity transformation of many-body Hamiltonian with the Jastrow factor, which changes the Schrödinger equation to an equivalent similarity-transformed eigenvalue equation. This similarity-transformed Hamiltonian is called the TC Hamiltonian. One of the advantageous features of the TC method is that we can optimize one-electron orbitals and obtain the band structure by solving a one-body self-consistent field (SCF) equation constructed with the TC Hamiltonian just like the HF method. Therefore, we can use the TC Hamiltonian and TC orbitals as the starting point of the post-HF theories with partially retrieved correlation effects through the Jastrow factor. Using this strategy, the TC method in combination with the MP2 theory [13], [19] and linearized coupled cluster theory [20] was reported to yield successful accuracy for molecular systems. Also for solid-state calculations, configuration interaction singles was successfully combined with the TC method and verified to provide accurate optical absorption spectra [21]. Moreover, computation time of the TC method is the same order as that for the HF method with respect to the number of k-points, bands, and plane waves to expand one-electron orbitals [22].
In this Letter, we applied the MP2 theory combined with the TC method to solid-state calculations. Because electron correlation effects are quite different between molecules and solids, it is of significant importance to observe how MP2 using the TC method as an unperturbed starting point works to describe the electronic structures in solids. To apply the MP2 theory to the TC method, we adopted bi-orthogonal formulation of the TC method (BiTC method) [19]. This study is the first application not only of the BiTC-MP2 method but also of the BiTC method to solid-state calculations. We verified that the correlation energy retrieved with the Jastrow factor significantly changes the total energies or one-electron energies of the BiTC-MP2 method from the traditional HF-MP2 method. In addition, convergence of the BiTC-MP2 method with respect to the numbers of conduction bands and k-points is preferable to that for the HF-MP2 method because the electron correlation is partially retrieved already at the unperturbed BiTC level. Computational cost of the BiTC-MP2 method is the same order as that of the HF-MP2 method with respect to the numbers of k-points, bands, and plane waves, under some approximations.
Section snippets
(Bi-orthogonal) Transcorrelated method
In the TC and BiTC methods, we rewrite a many-body wave function Ψ0 in the form Ψ0 = FΦ0, where F is the Jastrow factor, a symmetric product of a two-body positive function, with Φ0 being a many-body function formally defined as Φ0 = Ψ0/F. Then the Schrödinger equation,with E being the eigenenergy, is equivalent to the similarity-transformed eigenvalue equation,
For the first-principles Hamiltonian of many-electron systems under the external potential ,
Convergence issues
First, we checked the speed of convergence of our MP2 correlation energy defined in Section 2.2 with respect to the numbers of conduction bands and k-points. Because computation time increases very rapidly as these parameters become larger, how fast convergence will be achieved is a critical issue for practical applications. Figure 1 presents the plot of convergence of the MP2 correlation energy with respect to the number of conduction bands for solid lithium hydride. We used a 3 × 3 ×3 k-point
Conclusion
We applied the BiTC and BiTC-MP2 methods to solid-states calculations, and compared them with other methods such as the HF and HF-MP2 methods. Convergence with respect to the number of k-points and conduction bands for the BiTC-MP2 method is more easily achieved than that for the HF-MP2 method, which means that the BiTC-MP2 method can be applied to solid-state calculations with realistic computational effort. However, the band gaps of several solids calculated by the BiTC-MP2 method are in many
Acknowledgments
This work was supported by a Grant-in-Aid for JSPS Fellows as well as a Grant-in-Aid project from MEXT, Tokodai Institute for Element Strategy, and Computational Materials Science Initiative, Japan. A part of the computation was performed at the supercomputer center of the Institute for Solid State Physics, The University of Tokyo, and K-computer at the RIKEN Advanced Institute for Computational Science (Proposal number hp120086).
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