An implicit method for
two-electron time-dependent
fairly
large-scale
quantum simulations
Raymond Nepstad
University of Bergen
Time dependent quantum mechanics – analysis and numerics
Oslo 30 April 2010
Motivation
Free-electron laser (Hasylab, DESY)
Mauritsson et al., PRL (2008)
Short pulses
Attosecond electron motion
Ultra strong field
Electron interactions
The crew
Morten Førre
Raymond Nepstad
Sigurd Askeland
Tore Birkeland
Stian Sørngård
Ingjald Pilskog
Outline
Implicit method for two-electron systems
Pyprop and the Helium package
Results
Summary
Simulating two-electron systems
TDSE
Solve the (6D) two-electron time-depedent Schrödinger equation,
i
∂
Ψ(r1 , r2 , t) = HΨ(r1 , r2 , t).
∂t
Semiclassical light-atom interaction Hamiltonian,
H=
2
p2 2
p21 2
1
− + Hf ,1 +
− + Hf ,2 +
.
2
r1
2
r2
|r1 − r2 |
Field term (long-wavelength approximation),
Hfv,i = A(t) · pi
Hfl ,i = E(t) · ri .
TDSE
Solve the (6D) two-electron time-depedent Schrödinger equation,
i
∂
Ψ(r1 , r2 , t) = HΨ(r1 , r2 , t).
∂t
Semiclassical light-atom interaction Hamiltonian,
H=
2
p2 2
p21 2
1
− + Hf ,1 +
− + Hf ,2 +
.
2
r1
2
r2
|r1 − r2 |
Field term (long-wavelength approximation),
Hfv,i = A(t) · pi
Hfl ,i = E(t) · ri .
TDSE
Solve the (6D) two-electron time-depedent Schrödinger equation,
i
∂
Ψ(r1 , r2 , t) = HΨ(r1 , r2 , t).
∂t
Semiclassical light-atom interaction Hamiltonian,
H=
2
p2 2
p21 2
1
− + Hf ,1 +
− + Hf ,2 +
.
2
r1
2
r2
|r1 − r2 |
Field term (long-wavelength approximation),
Hfv,i = A(t) · pi
Hfl ,i = E(t) · ri .
TDSE
Solve the (6D) two-electron time-depedent Schrödinger equation,
i
∂
Ψ(r1 , r2 , t) = HΨ(r1 , r2 , t).
∂t
Semiclassical light-atom interaction Hamiltonian,
H=
2
p2 2
p21 2
1
− + Hf ,1 +
− + Hf ,2 +
.
2
r1
2
r2
|r1 − r2 |
Field term (long-wavelength approximation),
Hfv,i = A(t) · pi
Hfl ,i = E(t) · ri .
Discretization
Continuous PDF → set of ODEs via
ψ(r1 , r2 , t) = ∑ cijk (t)
i,j,k
Bi (r1 ) Bj (r2 ) LM
Yl1 ,l2 (Ω1 , Ω2 ).
r1
r2
Coupled spherical harmonics (k = {L, M, l1 , l2 }),
m
M−m
Yl1LM
(Ω2 ).
l2 = ∑hl1 l2 mM − m|LMiYl1 (Ω1 )Yl2
m
B-splines are non-orthogonal (overlap matrix),
Z ∞
0
dr Bi (r)Bj (r) = Sij
S = Ik ⊗ S 1 ⊗ S 2
Discretization
Continuous PDF → set of ODEs via
ψ(r1 , r2 , t) = ∑ cijk (t)
i,j,k
Bi (r1 ) Bj (r2 ) LM
Yl1 ,l2 (Ω1 , Ω2 ).
r1
r2
Coupled spherical harmonics (k = {L, M, l1 , l2 }),
m
M−m
Yl1LM
(Ω2 ).
l2 = ∑hl1 l2 mM − m|LMiYl1 (Ω1 )Yl2
m
B-splines are non-orthogonal (overlap matrix),
Z ∞
0
dr Bi (r)Bj (r) = Sij
S = Ik ⊗ S 1 ⊗ S 2
Discretization
Continuous PDF → set of ODEs via
ψ(r1 , r2 , t) = ∑ cijk (t)
i,j,k
Bi (r1 ) Bj (r2 ) LM
Yl1 ,l2 (Ω1 , Ω2 ).
r1
r2
Coupled spherical harmonics (k = {L, M, l1 , l2 }),
m
M−m
Yl1LM
(Ω2 ).
l2 = ∑hl1 l2 mM − m|LMiYl1 (Ω1 )Yl2
m
B-splines are non-orthogonal (overlap matrix),
Z ∞
0
dr Bi (r)Bj (r) = Sij
S = Ik ⊗ S 1 ⊗ S 2
B-splines
Piecewise polynomials (order k)
Can handle non-smooth functions
Compact support, sparse
matrices
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Radial box and B-splines
Matrix elements
Single-particle operators
Gauss-Legendre quadrature for radial integrals
Analytical solution of angular integrals
Multipole expansion of r12 ,
lmax
1
≈∑
r1 − r2 l=0
l
l
4π r<
Y ∗ (Ω1 )Yl,m (Ω2 ).
l+1 l,m
m=−l 2l + 1 r>
∑
Matrix-form TDSE,
ıSċ(t) = H(t)c(t)
Matrix elements
Single-particle operators
Gauss-Legendre quadrature for radial integrals
Analytical solution of angular integrals
Multipole expansion of r12 ,
lmax
1
≈∑
r1 − r2 l=0
l
l
4π r<
Y ∗ (Ω1 )Yl,m (Ω2 ).
l+1 l,m
m=−l 2l + 1 r>
∑
Matrix-form TDSE,
ıSċ(t) = H(t)c(t)
Matrix elements
Single-particle operators
Gauss-Legendre quadrature for radial integrals
Analytical solution of angular integrals
Multipole expansion of r12 ,
lmax
1
≈∑
r1 − r2 l=0
l
l
4π r<
Y ∗ (Ω1 )Yl,m (Ω2 ).
l+1 l,m
m=−l 2l + 1 r>
∑
Matrix-form TDSE,
ıSċ(t) = H(t)c(t)
Matrix structure
2D radial B-spline matrix
Matrix structure
Full matrix
Matrix structure
Per 2D radial block:
Ex:
≈ (2NB · k)2
(2 · 280 · 6)2 → 170MB
Total matrix (500 non-zero
angular blocks):
Typically
80GB
Nb k
Memory-intensive approach,
parallelization required
Distributing the wavefunction
Distributing the wavefunction
?
Distributing the wavefunction
Distributing the wavefunction
Distributing the wavefunction
Distributing matrices
P0
P1
P2
=
*
Calculate - receive/send
Distributing matrices
P0
P1
P2
=
*
Calculate - receive/send
Time discretization
Exponential (approximate) propagator,
c(t + ∆t) = exp −ı∆tS−1 H c(t) + O(∆t2 )
Numerical schemes:
1. Approximate exponential in Krylov subspace (Arnoldi/Lanczos)
No go - the equations are too stiff!
2. Padé approximation: Cayley-Hamilton (Crank-Nicolson)
Time discretization
Exponential (approximate) propagator,
c(t + ∆t) = exp −ı∆tS−1 H c(t) + O(∆t2 )
Numerical schemes:
1. Approximate exponential in Krylov subspace (Arnoldi/Lanczos)
No go - the equations are too stiff!
2. Padé approximation: Cayley-Hamilton (Crank-Nicolson)
Time discretization
Exponential (approximate) propagator,
c(t + ∆t) = exp −ı∆tS−1 H c(t) + O(∆t2 )
Numerical schemes:
1. Approximate exponential in Krylov subspace (Arnoldi/Lanczos)
No go - the equations are too stiff!
2. Padé approximation: Cayley-Hamilton (Crank-Nicolson)
Time discretization
Exponential (approximate) propagator,
c(t + ∆t) = exp −ı∆tS−1 H c(t) + O(∆t2 )
Numerical schemes:
1. Approximate exponential in Krylov subspace (Arnoldi/Lanczos)
No go - the equations are too stiff!
2. Padé approximation: Cayley-Hamilton (Crank-Nicolson)
Time discretization
Exponential (approximate) propagator,
c(t + ∆t) = exp −ı∆tS−1 H c(t) + O(∆t2 )
Numerical schemes:
1. Approximate exponential in Krylov subspace (Arnoldi/Lanczos)
No go - the equations are too stiff!
2. Padé approximation: Cayley-Hamilton (Crank-Nicolson)
Implicit Cayley-Hamilton
Cayley-Hamilton propagator,
ı∆t
ı∆t
H c(t + ∆t) = S −
H c(t).
S+
2
2
A can be very large, but sparse, direct methods not feasible
However, since A is sparse, an iterative method could work
We use GMRES
Implicit Cayley-Hamilton
Cayley-Hamilton propagator,
ı∆t
ı∆t
H
H c(t)
c(t + ∆t) =
S−
S+
2
2
|
|
{z
} | {z }
{z
}
A
x
=
b
A can be very large, but sparse, direct methods not feasible
However, since A is sparse, an iterative method could work
We use GMRES
Implicit Cayley-Hamilton
Cayley-Hamilton propagator,
ı∆t
ı∆t
H
H c(t)
c(t + ∆t) =
S−
S+
2
2
|
|
{z
} | {z }
{z
}
A
x
=
b
A can be very large, but sparse, direct methods not feasible
However, since A is sparse, an iterative method could work
We use GMRES
Implicit Cayley-Hamilton
Cayley-Hamilton propagator,
ı∆t
ı∆t
H
H c(t)
c(t + ∆t) =
S−
S+
2
2
|
|
{z
} | {z }
{z
}
A
x
=
b
A can be very large, but sparse, direct methods not feasible
However, since A is sparse, an iterative method could work
We use GMRES
Implicit Cayley-Hamilton
Cayley-Hamilton propagator,
ı∆t
ı∆t
H
H c(t)
c(t + ∆t) =
S−
S+
2
2
|
|
{z
} | {z }
{z
}
A
x
=
b
A can be very large, but sparse, direct methods not feasible
However, since A is sparse, an iterative method could work
We use GMRES
GMRES iterative solver
Construct Krylov subspace of A (sparse,
m × m)
Vn ← span{b, Ab, A2 b, . . . , An−1 b} (n m)
Solve (n × (n − 1)) least square problem
min{||AVn y − b||} = min{||Hn y − V∗n+1 d||}
where xn
= Vn y and Vn+1 Hn = AVn .
Problem: converges extremely slowly for typical A
Solution: use preconditioner
GMRES iterative solver
Construct Krylov subspace of A (sparse,
m × m)
Vn ← span{b, Ab, A2 b, . . . , An−1 b} (n m)
Solve (n × (n − 1)) least square problem
min{||AVn y − b||} = min{||Hn y − V∗n+1 d||}
where xn
= Vn y and Vn+1 Hn = AVn .
Problem: converges extremely slowly for typical A
Solution: use preconditioner
GMRES iterative solver
Construct Krylov subspace of A (sparse,
m × m)
Vn ← span{b, Ab, A2 b, . . . , An−1 b} (n m)
Solve (n × (n − 1)) least square problem
min{||AVn y − b||} = min{||Hn y − V∗n+1 d||}
where xn
= Vn y and Vn+1 Hn = AVn .
Problem: converges extremely slowly for typical A
Solution: use preconditioner
GMRES iterative solver
Construct Krylov subspace of A (sparse,
m × m)
Vn ← span{b, Ab, A2 b, . . . , An−1 b} (n m)
Solve (n × (n − 1)) least square problem
min{||AVn y − b||} = min{||Hn y − V∗n+1 d||}
where xn
= Vn y and Vn+1 Hn = AVn .
Problem: converges extremely slowly for typical A
Solution: use preconditioner
Preconditioner
Preconditioner M for system of equations,
Ax = b
→
M−1 Ax = M−1 b
Desired properties
B = M−1 A should have clustered eigenvalues (GMRES)
M−1 c should be easier to compute than A−1 b
M somewhere between I and A
Preconditioner
Preconditioner M for system of equations,
Ax = b
→
M−1 Ax = M−1 b
Desired properties
B = M−1 A should have clustered eigenvalues (GMRES)
M−1 c should be easier to compute than A−1 b
M somewhere between I and A
Choosing a preconditioner
Rewrite hamiltonian
H = Hr + Hf (t) + Hmultipole
“Radial” hamiltonian
Hr =
p21 p22 2
2
1
+ − − +
2
2
r1 r2 r>
Use as preconditioner,
operations requires no
communication
M = Hr ,
Choosing a preconditioner
Rewrite hamiltonian
H = Hr + Hf (t) + Hmultipole
“Radial” hamiltonian
Hr =
p21 p22 2
2
1
+ − − +
2
2
r1 r2 r>
Use as preconditioner,
operations requires no
communication
M = Hr ,
Choosing a preconditioner
Rewrite hamiltonian
H = Hr + Hf (t) + Hmultipole
“Radial” hamiltonian
Hr =
p21 p22 2
2
1
+ − − +
2
2
r1 r2 r>
Use as preconditioner,
operations requires no
communication
M = Hr ,
Applying the preconditioner
Solve My = c
Calculate factorization M = LU (only once)
Backsubstitution y = U−1 L−1 c (each time)
Possible methods
Sparse “exact” factorization: superLU
Fill-in problem
Don’t need “exact” solution
Structure-preserving factorization: incomplete LU (ILU)
IFPACK / Trilinos
Applying the preconditioner
Solve My = c
Calculate factorization M = LU (only once)
Backsubstitution y = U−1 L−1 c (each time)
Possible methods
Sparse “exact” factorization: superLU
Fill-in problem
Don’t need “exact” solution
Structure-preserving factorization: incomplete LU (ILU)
IFPACK / Trilinos
Applying the preconditioner
Solve My = c
Calculate factorization M = LU (only once)
Backsubstitution y = U−1 L−1 c (each time)
Possible methods
Sparse “exact” factorization: superLU
Fill-in problem
Don’t need “exact” solution
Structure-preserving factorization: incomplete LU (ILU)
IFPACK / Trilinos
Applying the preconditioner
Solve My = c
Calculate factorization M = LU (only once)
Backsubstitution y = U−1 L−1 c (each time)
Possible methods
Sparse “exact” factorization: superLU
Fill-in problem
Don’t need “exact” solution
Structure-preserving factorization: incomplete LU (ILU)
IFPACK / Trilinos
How well does it work?
Nbsplines = 40 lmax = 3 L = [0, 1, 2]
10-1
No preconditioner
Preconditioner
10-2
10-3
Error estimate
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12 0
5
10
Krylov vectors
15
20
Drawbacks
Max procs given by
(l1 , l2 , L) (50 - 300)
Load balancing not optimal (mem. usage)
Possible improvement: Distribute
one/both radial ranks
Drawbacks
20
(l1 , l2 , L) (50 - 300)
Load balancing not optimal (mem. usage)
15
% of procs
Max procs given by
10
Possible improvement: Distribute
5
one/both radial ranks
0
3500
4000
4500
5000
5500
Memory (MB)
6000
6500
7000
Drawbacks
20
(l1 , l2 , L) (50 - 300)
Load balancing not optimal (mem. usage)
15
% of procs
Max procs given by
10
Possible improvement: Distribute
5
one/both radial ranks
0
3500
4000
4500
5000
5500
Memory (MB)
6000
6500
7000
Distributing two ranks: wavefunction
Distributing two ranks: wavefunction
Distributing two ranks: matrix
Non-radial preconditioner?
Radial preconditioner blocks no longer proc-local
Use only local blocks with new distribution scheme?
ILU(n),
n > 0, some communication, performance penalty
Work in progress
Non-radial preconditioner?
Radial preconditioner blocks no longer proc-local
Use only local blocks with new distribution scheme?
ILU(n),
n > 0, some communication, performance penalty
Work in progress
Non-radial preconditioner?
Radial preconditioner blocks no longer proc-local
Use only local blocks with new distribution scheme?
ILU(n),
n > 0, some communication, performance penalty
Work in progress
Non-radial preconditioner?
Radial preconditioner blocks no longer proc-local
Use only local blocks with new distribution scheme?
ILU(n),
n > 0, some communication, performance penalty
Work in progress
Pyprop
What is Pyprop?
Toolkit for solving the Time Dependent Schrödinger Equation
What is Pyprop?
Goals
Flexibility
Performance
Research tool, not QM@Home
Common tasks automated
Difficult tasks possible
T. Birkeland PyProp - a Python Framework for Propagating the Time
Dependent Schrödinger Equation, Ph.D. thesis (2009)
Free Software (GPL)
http://pyprop.googlecode.com
Development branch on Github (http://github.com/kvantetore/PyProp)
What is Pyprop?
Goals
Flexibility
Performance
Research tool, not QM@Home
Common tasks automated
Difficult tasks possible
T. Birkeland PyProp - a Python Framework for Propagating the Time
Dependent Schrödinger Equation, Ph.D. thesis (2009)
Free Software (GPL)
http://pyprop.googlecode.com
Development branch on Github (http://github.com/kvantetore/PyProp)
What is Pyprop?
Goals
Flexibility
Performance
Research tool, not QM@Home
Common tasks automated
Difficult tasks possible
T. Birkeland PyProp - a Python Framework for Propagating the Time
Dependent Schrödinger Equation, Ph.D. thesis (2009)
Free Software (GPL)
http://pyprop.googlecode.com
Development branch on Github (http://github.com/kvantetore/PyProp)
What is Pyprop?
Goals
Flexibility
Performance
Research tool, not QM@Home
Common tasks automated
Difficult tasks possible
T. Birkeland PyProp - a Python Framework for Propagating the Time
Dependent Schrödinger Equation, Ph.D. thesis (2009)
Free Software (GPL)
http://pyprop.googlecode.com
Development branch on Github (http://github.com/kvantetore/PyProp)
Flexibility
Choose dimensionality and discretization
Several discretization schemes built in
Calculate inner products, load/save wavefunctions
Supply potentials (hamiltonian)
PyProp takes care of a lot of repetitive code
Operator-wavefunction multiplications
Choose propagator
Several propagators built in
Perform analysis and data exploration
High level code is written in Python
All the propagation tools can be used interactively
Flexibility
Choose dimensionality and discretization
Several discretization schemes built in
Calculate inner products, load/save wavefunctions
Supply potentials (hamiltonian)
PyProp takes care of a lot of repetitive code
Operator-wavefunction multiplications
Choose propagator
Several propagators built in
Perform analysis and data exploration
High level code is written in Python
All the propagation tools can be used interactively
Flexibility
Choose dimensionality and discretization
Several discretization schemes built in
Calculate inner products, load/save wavefunctions
Supply potentials (hamiltonian)
PyProp takes care of a lot of repetitive code
Operator-wavefunction multiplications
Choose propagator
Several propagators built in
Perform analysis and data exploration
High level code is written in Python
All the propagation tools can be used interactively
Flexibility
Choose dimensionality and discretization
Several discretization schemes built in
Calculate inner products, load/save wavefunctions
Supply potentials (hamiltonian)
PyProp takes care of a lot of repetitive code
Operator-wavefunction multiplications
Choose propagator
Several propagators built in
Perform analysis and data exploration
High level code is written in Python
All the propagation tools can be used interactively
Pyprop Framework Design
Core Routines
Independent Modules
User Code
Pyprop Framework Design
Core Routines
Independent Modules
User Code
Pyprop Framework Design
Core Routines
Independent Modules
User Code
Pyprop Framework Design
Core Routines
Independent Modules
User Code
What can Pyprop do
Two-electron quantum dot molecules
Ion-molecule collisions (Kr34+ + H2 )
Parallel redistribution of multidimensional data
Dynamics in two-electron atoms
Life on Proc(0,0)
x
k
k
k
b
j
z
j
S={1,2}
S={0,1}
y
i
S={0,2}
i
i
Helium package layout
CayleyPropagator
GMRESSolver
Pyprop
Bsplines
CoupledSphericalHarmonics
Pyprop.Helium
Core
Eigenstates
Potentials
Preconditioner
Project-specific wrappers
Find eigenstates
Basic eigenstate operations
Propagation
Propagation flow
Propagation tasks
Postprocessing
Analysis
Projectors
Observables
Differential prob.
Namegenerator
Results
Workhorse: Cray XT4 @Bergen
Cray XT4
The XT4 at Parallab, Bergen
Two-photon double ionization of Helium
Energy (a.u.)
DI
SI
He++
0
−0.5
He+ (2p)
−2.0
He+ (1s)
−2.9
He
Two-photon double ionization of Helium
10
Pyprop (4 fs)
Feist et al. (4 fs)
Foumouo et al. (NC)
Foumouo et al. (WC)
Nikolopoulos et al.
DI
SI
He++
0
−0.5
He+ (2p)
−2.0
He+ (1s)
−2.9
He
TPDI cross section (10−52 cm4 s)
Energy (a.u.)
8
6
4
2
038
40
42
44 46 48 50
Photon energy (eV)
52
54
56
Ionization Probability
Ultra-strong laser fields
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12 -2
10
Total
Double
Single
1-photon
2-photon
>2-photon
1.0
0.5
0.0
10-1
0
5
10 15 20 25 30
101
100
Field Strength (a.u.)
Ultra-strong laser fields
1.0
Ionization Probability
0.8
0.6
0.4
0.2
0.00.0
0.8
E0 /ω2 (a.u.)
1.6
0.0
0.8
E0 /ω2 (a.u.)
1.6
Summary
Summary
1. Two-electron method
2 x Bspline + coupled spherical harmonics
Implicit propagator (Cayley-Hamilton)
GMRES
Precondition with ILU
Implemented as a Python package for Pyprop (C++ used where needed)
Improvements possible (in progress)
2. Pyprop
Flexible toolkit for TDSE-problems
3. Future: Circular polarization, H2 , . . .
Summary
1. Two-electron method
2 x Bspline + coupled spherical harmonics
Implicit propagator (Cayley-Hamilton)
GMRES
Precondition with ILU
Implemented as a Python package for Pyprop (C++ used where needed)
Improvements possible (in progress)
2. Pyprop
Flexible toolkit for TDSE-problems
3. Future: Circular polarization, H2 , . . .
Summary
1. Two-electron method
2 x Bspline + coupled spherical harmonics
Implicit propagator (Cayley-Hamilton)
GMRES
Precondition with ILU
Implemented as a Python package for Pyprop (C++ used where needed)
Improvements possible (in progress)
2. Pyprop
Flexible toolkit for TDSE-problems
3. Future: Circular polarization, H2 , . . .
Summary
1. Two-electron method
2 x Bspline + coupled spherical harmonics
Implicit propagator (Cayley-Hamilton)
GMRES
Precondition with ILU
Implemented as a Python package for Pyprop (C++ used where needed)
Improvements possible (in progress)
2. Pyprop
Flexible toolkit for TDSE-problems
3. Future: Circular polarization, H2 , . . .
Summary
1. Two-electron method
2 x Bspline + coupled spherical harmonics
Implicit propagator (Cayley-Hamilton)
GMRES
Precondition with ILU
Implemented as a Python package for Pyprop (C++ used where needed)
Improvements possible (in progress)
2. Pyprop
Flexible toolkit for TDSE-problems
3. Future: Circular polarization, H2 , . . .
Summary
1. Two-electron method
2 x Bspline + coupled spherical harmonics
Implicit propagator (Cayley-Hamilton)
GMRES
Precondition with ILU
Implemented as a Python package for Pyprop (C++ used where needed)
Improvements possible (in progress)
2. Pyprop
Flexible toolkit for TDSE-problems
3. Future: Circular polarization, H2 , . . .
Summary
1. Two-electron method
2 x Bspline + coupled spherical harmonics
Implicit propagator (Cayley-Hamilton)
GMRES
Precondition with ILU
Implemented as a Python package for Pyprop (C++ used where needed)
Improvements possible (in progress)
2. Pyprop
Flexible toolkit for TDSE-problems
3. Future: Circular polarization, H2 , . . .
Summary
1. Two-electron method
2 x Bspline + coupled spherical harmonics
Implicit propagator (Cayley-Hamilton)
GMRES
Precondition with ILU
Implemented as a Python package for Pyprop (C++ used where needed)
Improvements possible (in progress)
2. Pyprop
Flexible toolkit for TDSE-problems
3. Future: Circular polarization, H2 , . . .
Summary
1. Two-electron method
2 x Bspline + coupled spherical harmonics
Implicit propagator (Cayley-Hamilton)
GMRES
Precondition with ILU
Implemented as a Python package for Pyprop (C++ used where needed)
Improvements possible (in progress)
2. Pyprop
Flexible toolkit for TDSE-problems
3. Future: Circular polarization, H2 , . . .
Summary
1. Two-electron method
2 x Bspline + coupled spherical harmonics
Implicit propagator (Cayley-Hamilton)
GMRES
Precondition with ILU
Implemented as a Python package for Pyprop (C++ used where needed)
Improvements possible (in progress)
2. Pyprop
Flexible toolkit for TDSE-problems
3. Future: Circular polarization, H2 , . . .
Thank you
for your attention