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Register a new userVariational projector augmented-wave method: theoretical analysis and preliminary numerical results
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In Kohn-Sham electronic structure computations, wave functions have singularities at nuclear positions. Because of these singularities, plane-wave expansions give a poor approximation of the eigenfunctions. In conjunction with the use of pseudo-potentials, the PAW (projector augmented-wave) method circumvents this issue by replacing the original eigenvalue problem by a new one with the same eigenvalues but smoother eigenvectors. Here a slightly different method, called VPAW (variational PAW), is proposed and analyzed. This new method allows for a better convergence with respect to the number of plane-waves. Some numerical tests on an idealized case corroborate this efficiency.
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In this article, a numerical analysis of the projector augmented-wave (PAW) method is presented, restricted to the case of dimension one with Dirac potentials modeling the nuclei in a periodic setting. The PAW method is widely used in electronic ab initio calculations, in conjunction with pseudopotentials. It consists in replacing the original electronic Hamiltonian $H$ by a pseudo-Hamiltonian $H^{PAW}$ via the PAW transformation acting in balls around each nuclei. Formally, the new eigenvalue problem has the same eigenvalues as $H$ and smoother eigenfunctions. In practice, the pseudo-Hamiltonian $H^{PAW}$ has to be truncated, introducing an error that is rarely analyzed. In this paper, error estimates on the lowest PAW eigenvalue are proved for the one-dimensional periodic Schrodinger operator with double Dirac potentials.
In solid-state physics, energies of crystals are usually computed with a plane-wave discretization of Kohn-Sham equations. However the presence of Coulomb singularities requires the use of large plane-wave cut-offs to produce accurate numerical results. In this paper, an analysis of the plane-wave convergence of the eigenvalues of periodic linear Hamiltonians with Coulomb potentials using the variational projector-augmented wave (VPAW) method is presented. In the VPAW method, an invertible transformation is applied to the original eigenvalue problem, acting locally in balls centered at the singularities. In this setting, a generalized eigenvalue problem needs to be solved using plane-waves. We show that cusps of the eigenfunctions of the VPAW eigenvalue problem at the positions of the nuclei are significantly reduced. These eigenfunctions have however a higher-order derivative discontinuity at the spheres centered at the nuclei. By balancing both sources of error, we show that the VPAW method can drastically improve the plane-wave convergence of the eigenvalues with a minor additional computational cost. Numerical tests are provided confirming the efficiency of the method to treat Coulomb singularities.
The so-called local density approximation plus the multi-orbital mean-field Hubbard model (LDA+U) has been implemented within the all-electron projector augmented-wave method (PAW), and then used to compute the insulating antiferromagnetic ground state of NiO and its optical properties. The electronic and optical properties have been investigated as a function of the Coulomb repulsion parameter U. We find that the value obtained from constrained LDA (U=8 eV) is not the best possible choice, whereas an intermediate value (U=5 eV) reproduces the experimental magnetic moment and optical properties satisfactorily. At intermediate U, the nature of the band gap is a mixture of charge transfer and Mott-Hubbard type, and becomes almost purely of the charge-transfer type at higher values of U. This is due to the enhancement of the oxygen 2p states near the top of the valence states with increasing U value.
We present a computational scheme for orbital-free density functional theory (OFDFT) that simultaneously provides access to all-electron values and preserves the OFDFT linear scaling as a function of the system size. Using the projector augmented-wave method (PAW) in combination with real-space methods we overcome some obstacles faced by other available implementation schemes. Specifically, the advantages of using the PAW method are two fold. First, PAW reproduces all-electron values offering freedom in adjusting the convergence parameters and the atomic setups allow tuning the numerical accuracy per element. Second, PAW can provide a solution to some of the convergence problems exhibited in other OFDFT implementations based on Kohn-Sham codes. Using PAW and real-space methods, our orbital-free results agree with the reference all-electron values with a mean absolute error of 10~meV and the number of iterations required by the self-consistent cycle is comparable to the KS method. The comparison of all-electron and pseudopotential bulk modulus and lattice constant reveal an enormous difference, demonstrating that in order to assess the performance of OFDFT functionals it is necessary to use implementations that obtain all-electron values. The proposed combination of methods is the most promising route currently available. We finally show that a parametrized kinetic energy functional can give lattice constants and bulk moduli comparable in accuracy to those obtained by the KS PBE method, exemplified with the case of diamond.
We present valence electron Compton profiles calculated within the density-functional theory using the all-electron full-potential projector augmented-wave method (PAW). Our results for covalent (Si), metallic (Li, Al) and hydrogen-bonded ((H_2O)_2) systems agree well with experiments and computational results obtained with other band-structure and basis set schemes. The PAW basis set describes the high-momentum Fourier components of the valence wave functions accurately when compared with other basis set schemes and previous all-electron calculations.
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