any discontinuities of wavefunctions, their derivatives, or the Jacobian, and standard methods such as finite-difference stencils can be applied straightforwardly as
long as F is adequately differentiable.
How do the different types of complex scaling discussed above compare computationally? The basic equations of exterior and smooth exterior complex scaling
are clearly more complicated than those for uniform scaling. However, as mentioned, the purpose of exterior complex scaling is that it admits potentials that are
not analytic within the interior region. This includes any strictly localized function
such as most atomic pseudopotentials – a major advantage for advanced selfconsistent field methods such as DFT. One other advantage of exterior scaling is
that. within the interior region, quantities such as the density retain their true
physical values rather than a difficult-to-interpret complex continuation which is
also numerically difficult to rotate back to real space.
For real-space methods, an advantage of smooth exterior complex scaling is that
one can transparently use finite-difference stencils as per (23). Standard finitedifference stencils, representing, for instance, the kinetic operator, do not work on
a non-differentiable contour although one can derive special stencils for this case
[40]. Basis sets should also make sure to take the discontinuity into account. Finiteelement representations involving some kind of basis are commonly used; see, for
example, Rescigno et al. [41] and Scrinzi and Elander [42]. Rescigno and
co-workers have reported that finite-element calculations with a basis set which
properly takes the discontinuity of “sharp” exterior scaling into account require less
functions than a purely analytic basis set using smooth scaling [41]. A more detailed
discussion of the numerical representations and basis sets can be found in work by
McCurdy et al. [24], who also argue that grid-based methods enjoy a similar
advantage with sharp exterior complex scaling, provided the scaling onset is exactly
on a grid point.
3
3.6 Example: Resonance in One Dimension
Let us perform an analytic calculation of a resonance using complex scaling to see
how exactly the resonance emerges. We consider a barrier formed by the piece-wise
constant potential
3 This would be less of an advantage in Cartesian 3D calculations where a smooth scaling could be
applied spherically, whereas the sharp scaling would need a cube to align its boundary with
the grid.
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
233
long as F is adequately differentiable.
How do the different types of complex scaling discussed above compare computationally? The basic equations of exterior and smooth exterior complex scaling
are clearly more complicated than those for uniform scaling. However, as mentioned, the purpose of exterior complex scaling is that it admits potentials that are
not analytic within the interior region. This includes any strictly localized function
such as most atomic pseudopotentials – a major advantage for advanced selfconsistent field methods such as DFT. One other advantage of exterior scaling is
that. within the interior region, quantities such as the density retain their true
physical values rather than a difficult-to-interpret complex continuation which is
also numerically difficult to rotate back to real space.
For real-space methods, an advantage of smooth exterior complex scaling is that
one can transparently use finite-difference stencils as per (23). Standard finitedifference stencils, representing, for instance, the kinetic operator, do not work on
a non-differentiable contour although one can derive special stencils for this case
[40]. Basis sets should also make sure to take the discontinuity into account. Finiteelement representations involving some kind of basis are commonly used; see, for
example, Rescigno et al. [41] and Scrinzi and Elander [42]. Rescigno and
co-workers have reported that finite-element calculations with a basis set which
properly takes the discontinuity of “sharp” exterior scaling into account require less
functions than a purely analytic basis set using smooth scaling [41]. A more detailed
discussion of the numerical representations and basis sets can be found in work by
McCurdy et al. [24], who also argue that grid-based methods enjoy a similar
advantage with sharp exterior complex scaling, provided the scaling onset is exactly
on a grid point.
3
3.6 Example: Resonance in One Dimension
Let us perform an analytic calculation of a resonance using complex scaling to see
how exactly the resonance emerges. We consider a barrier formed by the piece-wise
constant potential
3 This would be less of an advantage in Cartesian 3D calculations where a smooth scaling could be
applied spherically, whereas the sharp scaling would need a cube to align its boundary with
the grid.
Dynamical Processes in Open Quantum Systems from a TDDFT Perspective:. . .
233