more appropriate than that of transparent boundary, because electrons can flow in
only one direction.
However, we note an important difference between the two approaches.
Whereas the Green function embedding defines the exact matching conditions at
the boundary of a finite volume A, the scaled (69) acts on a wavefunction defined in
the full space A[B. This makes the size of the simulation box a weakness in
numerical simulations if a wave is capable of reaching the end of the box. The
scaling transformation imposes an asymptotic form which can be efficiently captured by exponential functions e
Àαr
. By employing a finite element approach with
an element at infinity which captures the exponential tail, it was numerically shown
by Scrinzi [38] that exterior complex scaling indeed provides perfectly absorbing
conditions for numerical precision.
Restricting (69) to A otherwise implies a truncation which irrevocably breaks its
perfect properties. In this case the scaling transformation reduces to an absorbing
boundary which can be regarded as a simple CAP and, as such, presents reflections
[74, 91]. We should mention that the use of (69) restricted to A in combination with
a smooth exterior complex scaling in the literature has been going under the
misleading name of reflection-free CAP, in spite of presenting a certain degree of
reflection [39, 91–93].
In the context of TDDFT, exterior complex scaling has been applied purely as an
absorbing boundary [53, 94].
6 Electron Photoemission
We focus here on the approaches that can be employed in the description of multielectron ionization initiated by external electromagnetic fields within TDDFT. As
in previous sections, we are interested only in electronic processes, neglecting any
ionic motion, and we restrict ourselves to the class of methods that requires
knowledge of the wavefunctions only on a bounded region of space A much as in
Fig. 8.
We are interested in the family of problems characterized by time-dependent
electronic Hamiltonians with the structure
^
H t
ð Þ ¼
1
2
Ài∇À
A t
ð Þ
c
! 2
þ v ext þ v ee ;
ð71Þ
where ν ee is the electron–electron Coulomb interaction, ν ext is the external potential
which generally consists of a static potential produced by the nuclei, A t
ð Þ is the
vector potential of the external field, and c is the speed of light. In writing (71) we
implied the choice of the velocity gauge to describe the action of the field. The
associated electric field can easily be obtained as a time derivative:
ℰ t
ð Þ ¼ À ∂ t A t
ð Þ. Typically, one would want to perform a simulation by choosing
254
A.H. Larsen et al.
only one direction.
However, we note an important difference between the two approaches.
Whereas the Green function embedding defines the exact matching conditions at
the boundary of a finite volume A, the scaled (69) acts on a wavefunction defined in
the full space A[B. This makes the size of the simulation box a weakness in
numerical simulations if a wave is capable of reaching the end of the box. The
scaling transformation imposes an asymptotic form which can be efficiently captured by exponential functions e
Àαr
. By employing a finite element approach with
an element at infinity which captures the exponential tail, it was numerically shown
by Scrinzi [38] that exterior complex scaling indeed provides perfectly absorbing
conditions for numerical precision.
Restricting (69) to A otherwise implies a truncation which irrevocably breaks its
perfect properties. In this case the scaling transformation reduces to an absorbing
boundary which can be regarded as a simple CAP and, as such, presents reflections
[74, 91]. We should mention that the use of (69) restricted to A in combination with
a smooth exterior complex scaling in the literature has been going under the
misleading name of reflection-free CAP, in spite of presenting a certain degree of
reflection [39, 91–93].
In the context of TDDFT, exterior complex scaling has been applied purely as an
absorbing boundary [53, 94].
6 Electron Photoemission
We focus here on the approaches that can be employed in the description of multielectron ionization initiated by external electromagnetic fields within TDDFT. As
in previous sections, we are interested only in electronic processes, neglecting any
ionic motion, and we restrict ourselves to the class of methods that requires
knowledge of the wavefunctions only on a bounded region of space A much as in
Fig. 8.
We are interested in the family of problems characterized by time-dependent
electronic Hamiltonians with the structure
^
H t
ð Þ ¼
1
2
Ài∇À
A t
ð Þ
c
! 2
þ v ext þ v ee ;
ð71Þ
where ν ee is the electron–electron Coulomb interaction, ν ext is the external potential
which generally consists of a static potential produced by the nuclei, A t
ð Þ is the
vector potential of the external field, and c is the speed of light. In writing (71) we
implied the choice of the velocity gauge to describe the action of the field. The
associated electric field can easily be obtained as a time derivative:
ℰ t
ð Þ ¼ À ∂ t A t
ð Þ. Typically, one would want to perform a simulation by choosing
254
A.H. Larsen et al.