# 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:

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)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 samplingRun 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:

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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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.

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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