boundary conditions, fixing the value of a function on a surface, into Neumann
boundary conditions, fixing the normal derivative over a surface, and vice versa.
The resulting time-dependent equations for ψ A (r,t) can be written in the same way
as (59) with an embedding operator defined as [67, 70]
^
H ℰ ψ A
½ t
ð Þ ¼ δ r À r S
ð
Þ
1
2
∂ψ A r S ; t
ð
Þ
∂n S
þ
ð
S
ð t
0
G
À1
0 r S , r
0
S , t À t
0
ð
Þ
∂ψ A r
0
S ; t
0
ð
Þ
∂t 0
dt
0 dr
0
S
!
;
ð61Þ
where ∂/∂n s denotes the directional derivative out of A and perpendicular to S, and
G
À1
0 r S ; r
0
S ; t
ð
Þ¼
1
2π
ð 1
À1
e
Àiεt G
À1
0 r S ; r
0
S ; ε
ð
Þ dε:
ð62Þ
Here G
À1
0 r S ; r
0
S ; ε
ð
Þis the inverse of the Green function defined by (57) evaluated on
the boundary surface S with r S , r
0
S 2 S. Because G 0 (r,r
0 ,t À t
0 ) depends only on time
differences it is conveniently expressed in the energy domain ε with a Fourier
transform over the time domain. Because of the presence of the δ(r À r S ), the
embedding operator (61) is non-zero only on the boundary surface and involves
normal and time derivatives of ψ A (r,t) over that surface.
Because of the equivalence of H ˆ ℰ and H ˆ Σ defined in (60) and (61), we refer in the
following to an embedding operator with the symbol ^
ℰ ψ A
½ t
ð Þ for simplicity. We are
now in the position to comment on the most characteristic features of ^
ℰ ψ A
½ t
ð Þ. In
general, it involves complex quantities which make it an explicitly non-Hermitian
operator. This fact implies that the total number of electrons is no longer conserved
during the propagation. Furthermore, it contains a memory term in the form of a
time integral. In Frensley [73] it was postulated that transparent boundary conditions should break time reversal symmetry. The presence of a memory term in (59)
turns the time propagation into a non-Markovian process and precisely breaks this
symmetry.
The extension to the many-electron case is straightforward using the same 2 Â 2
block structure of (56) with the difference that the entries must be interpreted as
operators acting on the N-body Hilbert space. The previous steps of the derivation
hold in a completely equivalent way up to (59) and (60) provided the interacting
many-body Green function G is used in place of G 0 .
Formulating this in the language of TDDFT, the OQS-TDDFT theory establishes
a one-to-one connection between potential and density for non-unitary dynamics
[5–7]. The evolution from an initial state is uniquely defined if we find a way to
write the coupling with the environment as a functional ν
B [n] of the total density n.
Once again the equations retain the block structure of (56) with entries interpreted
as multi-index tensors, each index being associated with a Kohn–Sham orbital. The
result is a set of equations equivalent to (59) for each orbital, where the exact
embedding operator ^
ℰ n
½ depends on the total density of the system (i.e., in A[B)
248
A.H. Larsen et al.
boundary conditions, fixing the normal derivative over a surface, and vice versa.
The resulting time-dependent equations for ψ A (r,t) can be written in the same way
as (59) with an embedding operator defined as [67, 70]
^
H ℰ ψ A
½ t
ð Þ ¼ δ r À r S
ð
Þ
1
2
∂ψ A r S ; t
ð
Þ
∂n S
þ
ð
S
ð t
0
G
À1
0 r S , r
0
S , t À t
0
ð
Þ
∂ψ A r
0
S ; t
0
ð
Þ
∂t 0
dt
0 dr
0
S
!
;
ð61Þ
where ∂/∂n s denotes the directional derivative out of A and perpendicular to S, and
G
À1
0 r S ; r
0
S ; t
ð
Þ¼
1
2π
ð 1
À1
e
Àiεt G
À1
0 r S ; r
0
S ; ε
ð
Þ dε:
ð62Þ
Here G
À1
0 r S ; r
0
S ; ε
ð
Þis the inverse of the Green function defined by (57) evaluated on
the boundary surface S with r S , r
0
S 2 S. Because G 0 (r,r
0 ,t À t
0 ) depends only on time
differences it is conveniently expressed in the energy domain ε with a Fourier
transform over the time domain. Because of the presence of the δ(r À r S ), the
embedding operator (61) is non-zero only on the boundary surface and involves
normal and time derivatives of ψ A (r,t) over that surface.
Because of the equivalence of H ˆ ℰ and H ˆ Σ defined in (60) and (61), we refer in the
following to an embedding operator with the symbol ^
ℰ ψ A
½ t
ð Þ for simplicity. We are
now in the position to comment on the most characteristic features of ^
ℰ ψ A
½ t
ð Þ. In
general, it involves complex quantities which make it an explicitly non-Hermitian
operator. This fact implies that the total number of electrons is no longer conserved
during the propagation. Furthermore, it contains a memory term in the form of a
time integral. In Frensley [73] it was postulated that transparent boundary conditions should break time reversal symmetry. The presence of a memory term in (59)
turns the time propagation into a non-Markovian process and precisely breaks this
symmetry.
The extension to the many-electron case is straightforward using the same 2 Â 2
block structure of (56) with the difference that the entries must be interpreted as
operators acting on the N-body Hilbert space. The previous steps of the derivation
hold in a completely equivalent way up to (59) and (60) provided the interacting
many-body Green function G is used in place of G 0 .
Formulating this in the language of TDDFT, the OQS-TDDFT theory establishes
a one-to-one connection between potential and density for non-unitary dynamics
[5–7]. The evolution from an initial state is uniquely defined if we find a way to
write the coupling with the environment as a functional ν
B [n] of the total density n.
Once again the equations retain the block structure of (56) with entries interpreted
as multi-index tensors, each index being associated with a Kohn–Sham orbital. The
result is a set of equations equivalent to (59) for each orbital, where the exact
embedding operator ^
ℰ n
½ depends on the total density of the system (i.e., in A[B)
248
A.H. Larsen et al.