precision with which it estimates the ionization potential, i.e., which energy it
assigns to the highest occupied state.
LDA is well known to overestimate this energy, and therefore calculates too high
ionization rates for low fields. This problem is attributed to the wrong asymptotic
decay of the LDA potential [54]. Meanwhile, Hartree–Fock is known to produce
accurate orbital energies, and the decay of the exact exchange potential has the
correct asymptotic form. EXX also yields results that are close to the reference by
Scrinzi. This all suggests that a good XC functional for DFRT resonance lifetime
calculations is one retaining the correct asymptotic form of the potential, such as the
previously mentioned LB94 functional.
4.4 Time-Dependence in Complex Scaling
In this section we consider the extension of complex scaling to time-dependent
simulations. Most obviously, one could simulate the dynamics of a system whose
initial state is derived from a resonance. However, the method has been found
useful for another practical reason, namely that complex scaling can be used to
avoid the effects of waves reflecting from the boundaries. An early approach by
Parker and McCurdy [56] showed that a complex basis set, with properties closely
related to the complex scaling method, reduced the amount of basis functions
necessary to represent properly a Gaussian wave packet under time evolution.
The authors found that the representation avoided reflection effects produced by
incompleteness of the basis sets as the wave packet moved away from the central
region.
Exterior complex scaling is now widely used as a practical absorber to prevent
reflections of waves because of the finite size of the simulation box. Details of its
use in this context are given in Sect. 5.6.
Let us go back to the basic question of how to time evolve complex-scaled states.
Bengtsson and co-workers [57, 58] have considered this problem in detail. The time
evolution of a state vector and its corresponding functional (or bra) are determined
by
i
∂ψ rt
ð Þ
∂t
¼ ^
H ψ rt
ð Þ:
ð53Þ
We apply the complex rotation operator and get
i
∂ψ θ rt
ð Þ
∂t
¼ i ^
R θ
∂ψ rt
ð Þ
∂t
¼ ^
R θ ^
H ^
R
À1
θ
^
R θ ψ rt
ð Þ ¼ ^
H θ ψ θ rt
ð Þ:
ð54Þ
A general state ψ θ (rt) can be time-evolved according to its expansion in
eigenstates. If ϕ
0
θ (r) is an eigenstate with energy ε θ , then
242
A.H. Larsen et al.
assigns to the highest occupied state.
LDA is well known to overestimate this energy, and therefore calculates too high
ionization rates for low fields. This problem is attributed to the wrong asymptotic
decay of the LDA potential [54]. Meanwhile, Hartree–Fock is known to produce
accurate orbital energies, and the decay of the exact exchange potential has the
correct asymptotic form. EXX also yields results that are close to the reference by
Scrinzi. This all suggests that a good XC functional for DFRT resonance lifetime
calculations is one retaining the correct asymptotic form of the potential, such as the
previously mentioned LB94 functional.
4.4 Time-Dependence in Complex Scaling
In this section we consider the extension of complex scaling to time-dependent
simulations. Most obviously, one could simulate the dynamics of a system whose
initial state is derived from a resonance. However, the method has been found
useful for another practical reason, namely that complex scaling can be used to
avoid the effects of waves reflecting from the boundaries. An early approach by
Parker and McCurdy [56] showed that a complex basis set, with properties closely
related to the complex scaling method, reduced the amount of basis functions
necessary to represent properly a Gaussian wave packet under time evolution.
The authors found that the representation avoided reflection effects produced by
incompleteness of the basis sets as the wave packet moved away from the central
region.
Exterior complex scaling is now widely used as a practical absorber to prevent
reflections of waves because of the finite size of the simulation box. Details of its
use in this context are given in Sect. 5.6.
Let us go back to the basic question of how to time evolve complex-scaled states.
Bengtsson and co-workers [57, 58] have considered this problem in detail. The time
evolution of a state vector and its corresponding functional (or bra) are determined
by
i
∂ψ rt
ð Þ
∂t
¼ ^
H ψ rt
ð Þ:
ð53Þ
We apply the complex rotation operator and get
i
∂ψ θ rt
ð Þ
∂t
¼ i ^
R θ
∂ψ rt
ð Þ
∂t
¼ ^
R θ ^
H ^
R
À1
θ
^
R θ ψ rt
ð Þ ¼ ^
H θ ψ θ rt
ð Þ:
ð54Þ
A general state ψ θ (rt) can be time-evolved according to its expansion in
eigenstates. If ϕ
0
θ (r) is an eigenstate with energy ε θ , then
242
A.H. Larsen et al.