Electronic Structure

CONQUEST can be used to produce a wide variety of information on the electronic structure of different systems, including: density of states (DOS) and atom-projected DOS (or pDOS); band-resolved charge density; band structure; and electronic polarisation. Many of these are produced with the post-processing code using a converged charge density. All of these (at present) require the exact diagonalisation approach to the ground state; linear scaling solutions are not possible.

Converged charge density

In most cases (except polarisation) the data required is produced by a non-self-consistent calculation which reads in a well-converged charge density. The convergence is mainly with respect to Brillouin zone sampling, but also self-consistency (a tight tolerance should be used). The basic procedure is:

1. Perform a well-converged calculation, writing out charge density (ensure that the Brillouin zone is well sampled, the SCF tolerance is tight (minE.SCTolerance) and that the flag IO.DumpChargeDensity T is set)

2. Perform a non-self-consistent calculation for the quantity desired (set minE.SelfConsistent F and General.LoadRho T to read and fix the charge density) using an appropriate Brillouin zone sampling

3. Run the appropriate post-processing to generate the data

However, note that the charge density often converges much faster with respect to Brillouin zone sampling than the detailed electronic structure, so the use of a non-self-consistent calculation is more efficient. Often it is most efficient and accurate to use a very high density k-mesh for the final, non-SCF calculation, but a lower density k-mesh to generate the charge density (which converges faster with respect to Brillouin zone sampling than DOS and other quantities).

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Density of states

The total density of states (DOS) is generated from the file eigenvalues.dat which is written by all diagonalisation calculations. See density of states for details on parameters which can be set.

The atom-projected DOS resolves the total DOS into contributions from individual atoms using the pseudo-atomic orbitals, and can further decompose this into l-resolved or lm-resolved densities of states. It requires the wave-function coefficients, which will be generated by setting IO.write_proj_DOS T; further analysis is performed in post-processing.

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Band structure

The band structure along a series of lines in reciprocal space can be generated. See post-processing for more details.

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Band-resolved densities

A band-resolved density is the quantity \(\mid \psi_n(\mathbf{r}) \mid^2\) for the \(n^{\mathrm{th}}\) Kohn-Sham eigenstate (we plot density because the eigenstates are in general complex). It requires wavefunction coefficients which are generated by setting IO.outputWF T. Full details are found in the band density section of the post-processing part of the manual.

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Electronic Polarisation

The electronic polarisation (the response of a material to an external electric field) can be calculated using the approach of Resta [ES1] by setting the tag General.CalcPol T. The direction in which polarisation is found is set using the tag General.PolDir (choosing 1-3 gives x, y or z, respectively, while choosing 0 gives all three directions, though this is normally not recommended).

The Resta approach is a version of the modern theory of polarisation (MTP) (perhaps better known in the method of King-Smith and Vanderbilt [ES2]) where the polarisation is found as:

\[\begin{split}\mathbf{P} = -\frac{e\mathrm{L}}{\pi V}\mathrm{Im}\mathrm{ln}\mathrm{det}\mathbf{S}\\ \mathrm{S}_{mn} = \langle \psi_{m} \vert \exp{i2\pi \mathbf{r}}/L\vert\psi_{n} \rangle\end{split}\]

where \(\mathrm{L}\) is a simulation cell length along an appropriate direction and \(V\) is the simulation cell volume. This approach is only valid in the large simulation cell limit, with \(\Gamma\) point sampling (e.g. for BaTiO3, a minimum of 3x3x3 formula units is needed, though this is perhaps a little too small).

As with all calculations in the MTP, the only valid physical quantity is a change of polarisation between two configurations. A very common quantity to calculate is the Born effective charge (BEC), which is defined as \(Z^{*}_{k,\alpha\beta} = V\partial P_{\alpha}/\partial u_{k,\beta}\) for species \(k\) and Cartesian directions \(\alpha\) and \(\beta\). It is most easily calculated by finding the change in polarisation as one atom (or one set of atoms in a sublattice) is moved a small amount.

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[ES1]

R. Resta. Macroscopic polarization from electronic wave functions. Int. J. Quantum Chem., 75:599–606, 1999. doi:10.1002/(SICI)1097-461X(1999)75:4/5%3C599::AID-QUA25%3E3.0.CO;2-8.

[ES2]

R. D. King-Smith and D. Vanderbilt. Theory of polarization of crystalline solids. Phys. Rev. B, 47:1651–1654, 1993. doi:10.1103/PhysRevB.47.1651.

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