TOMBO Ver.2 Manual

1.2 Mixed basis formulation 4
Many-body perturbation theory (such as the GW calculations) or spectral method (such
as the expansion of the wave packet in terms of the eigenstates of the Hamiltonian in the
TDDFT dynamics simulations) requires summing over large number of empty states. The
PW basis set can most accurately describe the empty states. In contrast, to describe the
electrons in the core region accurately, the AO basis set works better then the PW basis
set. The all-electron mixed basis approach, using both PWs and AOs as a basis set in a
combined way, is able to meet the requirements to describe both spatially extended and
localized states. In our code, AOs are numerically described inside the non-overlapping
atomic spheres and the radial part is treated using the logarithmic mesh.
The all-electron mixed basis approach has the following advantages:
1. The number of basis functions can be significantly reduced.
2. In Hamiltonian matrix elements, it is not necessary to store PW-PW part because it is
given simply by the Fourier components V (G − G ′ ).
3. It is possible to accurately treat core states because we determine AOs by using
Herman–Skillman code with logarithmic radial mesh.
4. There is no complexity to generate and treat pseudoptentials. There is also no problem
of transferability.
5. The overlap between AOs and PWs is calculated accurately by first performing angular
integral analytically and then performing radial integral of spherical Bessel functions
numerically in logarithmic radial mesh.
6. Because AOs are confined inside non-overlapping atomic spheres, there is no BSSE
problem, and it is not necessary to calculate overlap integrals between AOs centered
at different atoms, which might produce unnecessary computational errors. Simultaneously,
this reduces the overcompleteness problem.
Since the basis functions (PWs and AOs), f ξ (r), are not orthogonal each other, the
eigenvalue ε ν is obtained by solving the following generalized eigenvalue problem:
∑H ξ ξ ′c ν,ξ ′ = ε ν ∑S ξ ξ ′c ν,ξ ′, (1.4)
ξ ′ ξ ′
where H ξ ξ ′ = ⟨ f ξ |H| f ξ ′⟩ and S ξ ξ ′ = ⟨ f ξ | f ξ ′⟩ are, respectively, the Hamiltonian and overlap
matrix elements between the ξ ’th and ξ ′ ’th basis functions. They looks like
H =
(
⟨PW|H|PW⟩
⟨AO|H|PW⟩
⟨PW|H|AO⟩
⟨AO|H|AO⟩
)
, S =
(
⟨PW|PW⟩
⟨AO|PW⟩
⟨PW|AO⟩
⟨AO|AO⟩
)
. (1.5)