Relaxation of atomic orbitals in a plane-wave basis

We present a first-principles calculation scheme for crystals that diagonalizes an effective Hamiltonian in a small atomic orbital basis set which is expanded in plane waves, yielding initial trial wave functions. A simple relaxation procedure applied to the trial wave functions expanded in a plane-wave basis can be used to generate a set of correction wave functions. We show that a subsequent diagonalization of the effective Hamiltonian in the subspace of the trial and correction wave functions is sufficient to obtain results quite close from a full converged calculation on the plane-wave basis set used for the projection. The proposed method should be simple to implement in other plane-wave computer programs and leads to substantial gains in computation speed while maintaining reasonable accuracy.

Figure 1

The LDA band structures of silicon calculated with atomic orbitals are compared with a converged plane-wave calculation. In both panels, the bands are depicted using thick (black) lines when obtained in the full plane-wave (PW) procedure, (red) dashed-dot lines for the atomic orbital (AO) procedure, and dashed (blue) lines are used for the atomic orbitals with Jacobian correction (AOJC) procedure. The top panel is obtained using a SZ basis set (four orbitals per atom) and the bottom panel is obtained using the SZP basis set (nine orbitals per atom). The improvement of the AOJC method over the AO method is clear.

Figure 2

The LDA band structures of hexagonal graphite calculated with atomic orbitals are compared with a converged plane-wave calculation. In both panels, the bands are depicted using thick (black) lines when obtained in the full plane-wave (PW) procedure, (red) dashed-dot lines for the atomic orbital (AO) procedure, and dashed (blue) lines are used for the atomic orbitals with Jacobian correction (AOJC) procedure. The top panel is obtained using a SZ basis set (four orbitals per atom) and the bottom panel is obtained using the SZP basis set (nine orbitals per atom). The improvement of the AOJC method over the AO method is clear, in particular for the interlayer conduction bands.

Figure 3

Diamond equation of state calculated with the full plane wave (PW, blue squares), with atomic orbitals (AO, green diamonds), and with the atomic orbitals completed by the Jacobi correction (AOJC, red circles) are compared with the experimental curve and two all-electron calculations.

Figure 4

The time evolution of a coordinate of a Si atom in a microcanonical molecular dynamics simulation is shown for a target temperature of 600 K. That coordinate is plotted against the time step for the full plane-wave calculation (PW, solid black line), the calculation with atomic orbitals (AO, dotted red line), and the calculation with the atomic orbitals completed by the Jacobi correction (AOJC, dashed blue line).

Figure 5

The calculated radial distribution function $g\left(r\right)$ for liquid silicon is shown for three different procedures. The microcanonical simulations were performed at the target temperature 1800 K and a liquid density 2.59 g/cm${}^3$. The radial distribution function as a function of the distance is depicted using thick (black) lines when obtained in the full PW procedure, dashed (blue) lines are used for the AOJC procedure and (red) dashed-dot lines for the AO procedure. In both AO and AOJC procedures a SZP basis set was used.