A perfectly matched layer approach to a
reactive scattering problem
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
Anna Nissen1 , Hans O. Karlsson2 and Gunilla Kreiss1
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
1
2
Division of Scientific Computing, Department of Information
Technology, Uppsala, Sweden.
Quantum Chemistry, Department of Physical and Analytical
Chemistry, Uppsala, Sweden.
Numerical experiments
Time dependent quantum mechanics
– analysis and numerics, 28-30 April 2010, Oslo
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(1 : 31)
Motivation and background
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
(a) Reaction rates
(b) Femtosecond laser
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(2 : 31)
The nuclear Schrödinger equation
We consider the nuclear time-dependent Schrödinger
equation
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
i~
∂
ψ(X, t) = H(X)ψ(X, t),
∂t
where the influence from the electrons has been modelled
into a potential energy surface V(X) through the
Born-Oppenheimer approximation.
The Hamiltonian is given by
Numerical experiments
H(X) = T + V =
X
i
−
~2 ∇2N,i
2Mi
+ V (X).
X, Mi – Nucleus coordinates andRmass on nucleus i.
2
|ψ|2 – Probability distribution,
Ω |ψ| dΩ = 1.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(3 : 31)
Numerical challenges
I
High-dimensionality (3N-6 DOF for N atoms).
Example: molecule with 5 atoms ⇒ 9 DOF.
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
I
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Highly oscillatory solutions.
Couplings to strong electromagnetic fields.
I
Restrict computational domain to a finite domain
(dissociative problems).
Numerical experiments
Two approaches: Absorbing boundary conditions and
absorbing layers.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(4 : 31)
Numerical methods
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
I
High order finite difference methods in space.
I
Arnoldi algorithm in time, corresponding to the
Lanczos algorithm for general matrices.
I
Currently Matlab implementation ⇒ MPI and
OpenMP parallelized code HAParaNDA [1].
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
[1]. M. Gustafsson and S. Holmgren, Proceedings of ENUMATH 2009,
Uppsala, Sweden.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(5 : 31)
Absorbing layers
Extension of the computational domain.
Introduction
Interior domain [0,x0 ]
Absorbing layer [x0 ,x0 + d]
The nuclear TDSE
Numerical challenges
Numerical methods
0
x0
x0 + d
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
I
Modify the potential operator
V ⇒ V − iW .
Complex absorbing potential (CAP).
I
Modify the kinetic operator T through a coordinate
transformation, x ⇒ F (x).
Exterior complex scaling (ECS), Perfectly matched
layer (PML).
Numerical experiments
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(6 : 31)
Absorbing layers
Introduction
V ⇒ V − iW
I
Complex absorbing potential:
Ref. [2], [3].
I
Complex scaling: x ⇒ F (x) = xe iθ
Ref. [4].
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
[2] D. Neuhauser, M. Baer, J. Chem. Phys. 119:77-89, 2003.
Numerical experiments
[3] U. V. Riss, H-D. Meyer, J. Phys. B 26:4503-4535, 1993.
[4] N. Moiseyev, Physics Reports 302:211-293, 1998.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(7 : 31)
Absorbing layers
I
Exterior complex scaling (ECS)
Ref. [5].
x,
x < x0
F (x) =
,
x0 + e iγ (x − x0 ), x ≥ x0
I
Smooth exterior scaling (SES) / Perfectly matched
layer (PML)
Ref. [6], [7]. x ⇒ F (x) after x = x0 , where dF
dx is
cont. at x0 for a smooth transition.
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
[5] C. W. McCurdy, C. K. Stroud, M. K. Wisinski, Phys. Rev. A
443:5980-5990, 1991.
[6] H. O. Karlsson, J. Chem. Phys. 109:9366-9371, 1998.
[7] J. P. Berenger J. Comput. Phys. 114:185-200, 1994.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(8 : 31)
Perfectly matched layer
2D Schrödinger equation with a potential
∂ψ
∂2ψ ∂2ψ
=− 2 −
+ V (y ) in Ω = [0, ∞) × [−L, L],
∂t
∂x
∂y 2
∂2
2
j = 1, ..., ∞.
where − ∂y
2 + V (y ) ψj = κj ψj
Assume compactly supported initial conditions ψ0 (x),
homogeneous boundary conditions at y = ±L, a Dirichlet
condition at x = 0
i
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
ψ(0, y , t) = 0,
Numerical experiments
and a decay condition
lim ψ(x, y , t) = 0.
x→∞
Want to reduce the computational domain to
Ω = [0, x0 +
d] × [−L, L].
, Time dependent
quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(9 : 31)
Find bounded solutions
Laplace-transform outside compact support of initial data
is ψ̂j = −
Introduction
The nuclear TDSE
Numerical challenges
∂ 2 ψ̂j
+ κ2j ψ̂j
∂x 2
in
Ω = [x0 , ∞) × [−L, L]
yields ODE in x with modal solutions
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
ψ̂j (x) = Aj e λ+ (x−x0 ) + Bj e λ− (x−x0 ) ,
q
λ± = ± −is + κ2j .
(1)
Consider Re(s) ≥ 0 and choose
q
−is + κ2j ≥ 0 for Re(s) ≥ 0,
Re
ψ̂j (x) = Bj e
−
q
−is+κ2j (x−x0 )
.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(10 : 31)
Modify solutions
Substitute
q
−is + κ2j (x − x0 ) against
Z x
q
iγ
2
−is + κj ((x − x0 ) + e
σ(ω)dω)
Introduction
The nuclear TDSE
Numerical challenges
x0
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
in the PML.
Absorption function σ(ω) – smooth, non-negative
function in ω.
σ(x0 ) = 0 for perfect matching.
The modified solutions look like
ψ̂j
PML
(x) = Bj e
−
q
R
−is+κ2j ((x−x0 )+e iγ xx σ(ω)dω)
0
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
.
(11 : 31)
PML equation
Introduction
The nuclear TDSE
Numerical challenges
Transforming modal PML solutions back into physical
space yields PML equation
2
1
∂
1
∂ψ
∂ ψ
∂ψ
=−
− 2 +V (y )ψ,
i
iγ
iγ
∂t
1 + e σ(x) ∂x 1 + e σ(x) ∂x
∂y
Numerical methods
Absorbing layers
in
Ω = [0, x0 + d] × [−L, L].
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
Can be viewed as a coordinate change from real
coordinate x in the interior domain to complex coordinate
F (x) in the layer, where
Z x
F (x) = x + e iγ
σ(ω)dω.
x0
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(12 : 31)
Well-posedness
Introducing ϕ =
i
Introduction
f ψ yields
∂ϕ
1 ∂2 1
=−
ϕ + Vk (x)ϕ + V (y )ϕ,
∂t
f (x) ∂x 2 f (x)
The nuclear TDSE
Numerical challenges
Numerical methods
√
where Vk (x) =
3f 2 −2f 00 f
4f 4
,
f (x) =
dF (x)
dx
(2)
= 1 + e iγ σ(x).
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
Well-posedness for (2) for
0≤γ≤
π
,
2
follows from the bound
||ϕ(·, t)||2H p ≤ Kp (T )||ϕ(·, 0)||2H p ,
where Kp depends on derivatives of f up to order p for
p ≥ 2, and up to order 2 for p = 0.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(13 : 31)
Continuity requirements
Introduction
The nuclear TDSE
Numerical challenges
Finite difference discretization in space of order m yields
1D error equation
(
1 ∂2 1
iet = − f (x)
e + Vk e + ∆x m Te ,
∂x 2 f (x)
e(x, 0) = 0,
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
where ej (t) = ϕ(xj , t) − uj (t).
Here m is assumed to be even and
cm ∂ m+2 1
ϕ .
Te =
f ∂x m+2 f
Numerical experiments
Thus, the bound
||ϕ(·, t)||2H m+2 ≤ Km+2 (T )||ϕ(·, 0)||2H m+2
is needed for optimal convergence, implying that f should
have m + 2 bounded derivatives.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(14 : 31)
Illustration of PML errors
Introduction
Two main types of errors from the PML:
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
I
Modeling error – due to solving equation on
truncated domain [0,x0 + d].
I
Numerical reflections – discretization destroys
perfect matching at interface x0 .
I
MOVIE
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(15 : 31)
Modeling error
Introduction
I
Error due to truncation of the computational
domain.
I
Can be estimated in terms of the modal PML
solutions in Laplace-space.
I
Modeling error ∝ e
I
k can be estimated as the dominating wavenumber
of a wave packet.
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
−k
Rx
0 +d
x0
σ(ω)dω
.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(16 : 31)
Numerical reflections
Introduction
I
Depends on the discretization and of the absorption
function σ(x).
I
High order central finite difference methods, order m
(m = 8 typically).
I
Truncation error Te ∝
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
cm ∂ (m+2)
f ∂x (m+2)
1
fϕ
.
Numerical experiments
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(17 : 31)
Numerical reflections
Introduction
I
Largest effect from the vicinity of the interface
⇒ Te ∝ f α f (m+2) ϕ.
I
For polynomial profile of order p, p = m + 2,
σmax
p
σ(x) = σdmax
p (x − x0 ) , Te ∝ d p .
I
Smoothing effect since equation is parabolic in the
PML (gain at least one derivative).
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(18 : 31)
Error matching
Explicit error formulas for polynomial absorption profile
p
σ(x) = σdmax
p (x − x0 ) .
Goal: For error level ε
Introduction
The nuclear TDSE
Numerical challenges
modeling error ε1 ≈ e
−2ksin(γ)σmax d
p+1
≤ ε,
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
m ≤ ε.
numerical reflections ε2 ≈ C σdmax
p ∆x
Procedure:
I
Given ∆x and error level ε.
I
Determine C from numerical reflections numerically,
|ε2 | = ε gives optimal M = σmax /d p .
I
|ε1 | = ε gives optimal width d.
I
Optimal σmax determined from σmax = Md p .
Numerical experiments
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(19 : 31)
H2 molecule on a solid surface
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
Dissociative adsorbtion (H2 molecule dissociates along
the surface) and associative desorption (H2 molecule
leaves surface).
r - Internuclear distance in H2 molecule, z - Distance
between molecule and surface.
Potential energy surface in [8].
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
10
8
r
PML errors
Modeling error
Numerical reflections
Error matching
6
Numerical experiments
4
2
2
4
6
z
8
10
[8] R. C. Mowrey, . Chem. Phys. 94:7098-7105, 1991.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(20 : 31)
Comparison of absorbing layers
Numerical reflections for different absorbing layers
(dissociative adsorbtion).
0
Introduction
10
PML
CAP
TCAP
ECS
8th order
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
PML errors
Modeling error
Numerical reflections
Error matching
−5
10
l2 error
Perfectly matched layer
Derive PML equation
Well-posedness
−10
10
Numerical experiments
−15
10
2
10
Number of points along r−axis
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(21 : 31)
Comparison of absorbing layers
Simulations with PML parameters optimized for certain
tolerance TOL (dissociative adsorbtion).
−1
Introduction
10
The nuclear TDSE
Numerical challenges
−2
Numerical methods
10
Absorbing layers
PML errors
Modeling error
Numerical reflections
Error matching
−3
l2 error
Perfectly matched layer
Derive PML equation
Well-posedness
10
−4
10
PML
CAP
TCAP
TOL
Numerical experiments
−5
10
−6
10 −5
10
−4
10
−3
Tolerance
10
−2
10
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(22 : 31)
Outgoing flux
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
The conservation law for probability in quantum
mechanics is given by
Z
Z
∂
2
|Ψ| dΩ + j · dS = 0,
∂t Ω
S
(2)
where S is the boundary of Ω, and the definition of the
flux in direction x, jx , is
~
∗ ∂Ψ
= Ψ
.
(3)
jx =
mx
∂x
Integrating equation (2) with respect to time yields
Z
|Ψ|2 dΩ +
Ω
since
R
Ω |Ψ0 |
2 dΩ
Z
T
Z
j · dS = 1,
0
(4)
S
= 1.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(23 : 31)
Outgoing flux
Introduction
The nuclear TDSE
Numerical challenges
After a sufficiently long time T ,
Z
T
Z
j · dS = 1,
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
0
(5)
S
when all probability density has vanished from the
computational domain.
Equation (5) should be monotonously increasing, and this
can be used as a measure of the accuracy of the
boundary conditions.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(24 : 31)
Discrete flux
An analogue expression to the flux (3) which is exact in
space for the discrete approximation can be derived from
the semi-discrete Schrödinger equation in 1D
Introduction
The nuclear TDSE
Numerical challenges
i~
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
∂uj (t)
~2
=−
Dm,j uj (t).
∂t
2M
Dm,j – mth order central finite difference operator
corresponding to a second derivative.
uj (t) – semi-discrete solution in grid point xj .
Numerical experiments

Dm,j uj =
1 
α0 uj +
∆x 2
m/2
X

αk (uj+k + uj−k ) ,
k=1
where the α’s are the weights of the finite difference
stencil.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(25 : 31)
Discrete flux
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
A general expression for the discrete flux using a central
finite difference discretization of order m, jDm (t), is
derived, so that
Z TZ
Z T
j · dS ≈
jDm (t)dt.
0
S
0
This expression is exact in space for the finite difference
approximation and only the temporal integral needs to be
approximated.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(26 : 31)
Integrated flux
1
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
Integrated flux
Introduction
0.995
0.99
PML
CAP
TCAP
0.985
0.98
700
800
900
1000
Time
1100
1200
Figure: Integrated outgoing flux in the z-direction. The flux for
the PML and the TCAP converge to one, while numerical
reflections causes an unphysical decrease for the CAP flux.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(27 : 31)
Integrated flux
1
Introduction
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
1
Integrated flux
The nuclear TDSE
Numerical challenges
1
1
PML
CAP
TCAP
1
850
900
950
1000 1050
Time
1100
1150
1200
Figure: Integrated outgoing flux in the z-direction, closer view.
Here it is visible that the flux decreases also for the TCAP.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(28 : 31)
Integrated flux
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
Boundary condition
reference PML
optimized PML
non-optimized PML
CAP
TCAP
fluxz
0.011364
0.011389
0.011392
0.084520
0.018261
fluxr
0.98863
0.98857
0.98860
0.91469
0.98134
Table: Integrated flux in r - and z-direction, corresponding to
fractions of transmitted and reflected parts of the wave packet,
respectively. PML, CAP and TCAP boundary conditions with
the same number of points are used.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(29 : 31)
Integrated flux
1
Introduction
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
0.95
Integrated flux
The nuclear TDSE
Numerical challenges
0.9
CAP
PML
reference
TCAP
0.85
0.8
0
0.5
1
1.5
Time
2
2.5
3
4
x 10
Figure: Integrated outgoing flux in the r -direction. The
accuracy of the boundary conditions affect the results to a
large extent.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(30 : 31)
Concluding remarks
I
Accuracy of the absorbing boundary conditions may
affect the measured quantity considerably.
I
Both continuous and discrete reflection properties
need to be taken into account in the boundary
treatment.
I
Systematic procedure of choosing parameters saves
time and leads to efficient absorbing boundary
treatment. [9]
Introduction
The nuclear TDSE
Numerical challenges
Numerical methods
Absorbing layers
Perfectly matched layer
Derive PML equation
Well-posedness
PML errors
Modeling error
Numerical reflections
Error matching
Numerical experiments
[9]. A. Nissen, G. Kreiss, An optimized perfectly matched layer for the
Schrödinger equation, To appear in Communications in Computational
Physics.
, Time dependent quantum mechanics – analysis and numerics, 28-30 April 2010, Oslo
(31 : 31)