Electron gases

An electron gas contains interacting electrons in some geometry with a constant compensating positive charge.

Uniform electron gas

ueg {
    -- options,
}

Returns:

a system object.

ueg creates a system object for the (conventional) electron gas:

\[H = -\frac{1}{2} \sum_i \nabla_i^2 + \sum_{i<j} \frac{1}{r_{ij}}\]

(including an appropriate uniform background potential to counteract the charge), using a single-particle basis of plane waves, \(\psi_{\bf k} = e^{i {\bf k}.{\bf r}}\).

Options

sys

type: system object produced by a previous call.

Optional.

If provided, a previously created system object is updated with the new settings supplied, otherwise a new system object is created.

electrons

type: integer.

Required.

Number of electrons in the unit cell.

ms

type: integer.

Required.

Set the spin polarisation of the system in units of electron spin (i.e. a single electron can take values 1 or -1).

sym

type: integer or string.

Optional. Default: aufbau.

Set the symmetry (i.e. crystal momentum) of the system if a reference determinant is not provided. This can be set to:

• An integer specifying the index of a specific wavevector; see the output produced by creating a system for possible values and their corresponding wavevectors.
• aufbau. Uses the symmetry of a determinant selected using the Aufbau principle.
• tot_sym. Uses the totally symmetric representation, whatever its index may be.

rs

type: float.

Optional. Default: 1.

Set the density, \(r_s\), of the UEG.

cutoff

type: float.

Optional. Default: 3.

Set the maximum kinetic energy of the orbitals included in the basis set.

Note that this is in scaled units of \((2\pi/L)^2\), where \(L\) is the dimension of simulation cell defined by electrons and rs and is compared to the kinetic energy of each plane-wave without the twist angle included. In this way the cutoff can be kept constant whilst the twist is varied and the basis set used will remain consistent.

dim

type: integer.

Optional. Default: 3.

Set the dimension of the electron gas. 2- and 3-dimensional gases are implemented.

twist

type: \(N\)-dimensional vector

Optional. Default: 0 in each dimension.

Apply a twist to the wavevector grid. The twist is an ndim-dimensional vector in terms of the reciprocal lattice vectors. The twist angle is chosen to be within the primitive reciprocal cell, and hence the components should be between -0.5 and +0.5.

verbose

type: boolean.

Optional. Default: true.

Print out the single-particle basis set.

Ringium

ringium {
    -- options,
}

Returns:

a system object.

Ringium [Loos13], is a 1D system of electrons confined to a ring of radius \(R\):

\[H = -\frac{1}{2R^2} \sum_i \frac{\partial^2}{\partial\theta_i^2} + \sum_{i<j} \frac{1}{r_{ij}}\]

where \(r_{ij} = R\sqrt{2-2\cos(\theta_i-\theta_j)}\), using a single-particle basis of functions \(\psi_n = e^{i n \theta}\). As it is 1D, the different spin polarisations are degenerate, so without loss of generality all electrons are forced to be spin up.

Options

sys

type: system object produced by a previous call.

Optional.

If provided, a previously created system object is updated with the new settings supplied, otherwise a new system object is created.

electrons

type: integer

Required.

Number of electrons in the system.

radius

type: float

Required.

The radius of the ring.

maxlz

type: integer

Required.

The maximum angular momentum of the orbitals used in the basis set.

Note that this is in units of \(\frac{\hbar}{2}\) and must have opposite parity to the number of electrons.

verbose

type: boolean.

Optional. Default: true.

Print out the single-particle basis set.