Basis Set and Pseudopotentials

Basis Set

ABACUS supports both PW and LCAO basis set, controlled by keyword basis_type in INPUT file.

The default value of basis_type is pw. The size of pw basis set is controlled by imposing an upper bound for the kinetic energy cutoff of the plane wave.

When choosing lcao basis set, users need to prepare a set of atomic orbitals. Such files may be downloaded from the official website. For more information, also check the NUMERICAL_ORBITAL section in the specification of the STRU file.

The angular part of orbitals are real spherical harmonics defined (in terms of conventional spherical harmonics in quantum mechanical literature) as

\[\begin{split}Y_{lm} = \left\{\begin{matrix}\sqrt{2}~\text{Im} Y_l^{|m|} & m \lt 0 \\[6pt] Y_l^0 & m = 0 \\[6pt] \sqrt{2}~\text{Re}Y_{l}^{|m|} & m \gt 0\end{matrix}\right. \end{split}\]

Note that real spherical harmonics adopted by ABACUS differ from some other definition, e.g. Table of spherical harmonics - Wikipedia, by a factor of \((-1)^m\).

Inside ABACUS, orbitals in LCAO basis are arranged lexicographically by species-position-l-zeta-m except for the intra-m ordering. Specifically, orbitals are first ordered by their atomic species in accordance with the ATOMIC_SPECIES section of the STRU file. For orbitals of the same species, orbitals belonging to each atom are put together, with their overall order following the ATOMIC_POSITIONS section of the STRU file. Orbitals on each atom are further ascendingly ordered by their angular momentum (s,p,d,f,…), followed by an order based on their their zeta number. Finally, m is ordered as 0, 1, -1, 2, 2, \(\ldots\), l, -l, which is the only exception of the lexicographic order.

Generating atomic orbital bases

Users may also choose to generate their own atomic obitals. In ABACUS, the atomic orbital bases are generated using a scheme developed in the paper. A detailed description of the procedure for generating orbitals will be provided later.

BSSE Correction

For treating BSSE(Basis Set Superposition Error), we allow for the inclusion of “empty” or “ghost” atoms in the calculation. Namely, when expanding the Hamiltonian, basis sets on the atoms are used, while the ionic potentials on those atoms are not included when constructing the Hamiltonian.

An empty atom is defined in the STRU file when an element name contains the “empty” suffix, such as “H_empty”, “O_empty” and so on. Here we provide an example of calculating the molecular formation energy of \(H_2O\) with BSSE correction.

In the example, we provide four STRU files:

STRU_0 : used along with ntype = 2;normal calculation of water molecule (\(E(\text{H}_2\text{O})\))

obtained total energy of -466.4838149140513 eV
STRU_1 : used along with ntype = 2;calculation of single O atom (\(E_O\))

obtained total energy of -427.9084406198214 eV
STRU_2 : used along with ntype = 3;calculation of 1st H atom (\(E_{H1}\))

obtained total energy of -12.59853381731160 eV
STRU_3 : used along with ntype = 3;calculation of 2nd H atom (\(E_{H2}\))

obtained total energy of -12.59853378720844 eV

Note : Remember to adjust the parameter ntype in INPUT file

Thus, the formation energy is given by:

\[ \Delta E(\text{H}_2\text{O}) = E(\text{H}_2\text{O}) - E(\text{O}) - E(\text{H}^1) - E(\text{H}^2) \approx -13.38 eV \]

Pseudopotentials

In ABACUS, we support norm-conserving and ultrasoft pseudopotentials. For norm-conserving pseudopotentials, we support four different formats of the pseudopotential files: UPF, UPF2, VWR, and BLPS. For ultrasoft pseudopotentials, currently we support only one format of the pseudopotential files: UPF2.

For more information, check the ATOMIC_SPECIES section in the specification of the STRU file.

Here we list some common sources of the pseudopotential files: