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Propagating the Time Dependent Schroedinger Equation. B. I. Schneider Division of Advanced Cyberinfrastructure National Science Foundation National Science Foundation September 6, 2013. - PowerPoint PPT Presentation
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)V( T )( H rr
Propagating the Time Dependent Schroedinger Equation
B. I. Schneider Division of Advanced Cyberinfrastructure
National Science Foundation
National Science Foundation September 6, 2013
•Novel light sources: ultrashort, intense pulses Nonlinear (multiphoton) laser-matter interaction
What Motivates Our Interest
Attosecond pulses probe and control electron
dynamicsXUV + IR pump-probe
Free electron lasers (FELs)Extreme intensities
Multiple XUV photons
Basic Equation
t),V( m2
- )t,( Hi i
2i
2
rr
Where PossiblyNon-Local
orNon-Linear
0 )t,( )t,( H )t,(
rrrt
i
Properties of Classical Orthogonal Functions
(x)nχ (x) w )x(n
; )x-(x )x(n )x(n
ssCompletene
ts.coefficien and theof up madematrix al tridiagon the
ingdiagonalizby found bemay weightsand points The
function. weight therespect to with
lessor 1) - (2norder of integrand polynomialany integrateexactly
whichfound bemay i w weights,and ix points, quadrature Gauss ofset A
procedure Lanczos theusing computed bemay tscoefficien recursion The
)x(2nχ1nβ)x(1nχ)1nαx()x(nχnβ
form; theof iprelationsh recursion terma threesatisfy functions The
mn, (x)mχ (x)nχ b
adx w(x) mχnχ
function. weight positive some .lity w.r.tOrthonorma
1/2
n
More Properties
p
1i)i(xi w)i(xqχ (x) (x)qχ w(x)
b
adx qc
(x)q
p
1qqc (x) (x)1/2 w Ψ(x)
expansion, an Given
Corollary
.quadrature by theexactly
integrated be can whichpolynomiala is integrand thebecause trueis This
mn,δ p
1i )i(xm)χi(xnχ iw mn
pq whereqallfor squadrature Gausspoint -p
byexactly performed be can integrallity orthonorma that theNote
iprelationshlity orthonorma discrete A
Matrix Elements
formula. quadraturea like looks this
thatNote, x. esdiagonaliz whichone totionrepresenta
original thefrom ation transform theis where
T )V(x T V
obtained, ismatrix the toionapproximat
excellent an that suggests and sets basis finitefor even
useful quite remains This complete. isset basis theas long as
)(
Then, operator.
position theof tionrepresentamatrix know the weif
evaluated bemay element matrix thisly,Conceptual
V(x)V
potential, theofelement matrix a Consider,
i,qii
iq,qq,
qqqq,
''
''
T
xV
V
Properties of Discrete Variable Representation
ion.approximatexcellent an be toappearsit practice In
true.is that thisASSUMED is its DVR, theIn
mutiply)matrix and expansion seriespower (Think
complete. is quadraturebasis/or theunless
) F(x F
toequal benot will thisgeneral In
uF(x)u F
elementmatrix heConsider t
x uxu
and
(x)(x)uu dx w(x) ; w
)(xu
that,properties thewith
)(x (x) w (x)u
functions, "coordinate" ofset new a Define
ji,iji,
jiji,
ji,iji
ji,ji
b
ai
ji,ji
iqq
p
1qii
Its Actually Trivial
points. quadrature Gauss theare i
x where
ij )j
x-i
x (
) j
x-x (
iW
1 (x)iu
tionrepresenta simple A
i
ji
xx
u2dx
2d )x (iui w ju
2dx
2diu
:scoordinate Cartesian For .itsbut rule, quadrature
theusing evaluated bemay operators derivative theof elementsMatrix
trivial
Multidimensional Problems Tensor Product Basis
Consequences
nk,mj,li,
mj,li,
nk,li,
nk,mj,
nm,l,Vkj,i,
nTk
mTj
lTi nm,l,Hkj,i,
Multidimensional Problems
Two Electron matrix elements also ‘diagonal” using Poisson equation
Finite Element Discrete Variable Representation
11
11 ) )()( (
)(
iin
iini
nww
xfxfxF
• Properties• Space Divided into Elements – Arbitrary
size• “Low-Order” Lobatto DVR used in each
element: first and last DVR point shared by adjoining elements
• Elements joined at boundary – Functions continuous but not derivatives
• Matrix elements requires NO Quadrature – Constructed from renormalized, single element, matrix elements
•
• Sparse Representations – N Scaling
• Close to Spectral Accuracy
Finite Element Discrete Variable Representation • Structure of
Matrix
7776
676665
5655
64
54
464544434241
34333231
24232221
14131211
hh
hhh
hh
h
h
hhhhhh
hhhh
hhhh
hhhh
Time Propagation Method)tr,( )
tH(t)exp(-i = )t+tr,(
Diagonalize Hamiltonian in Krylov basis
• Few recursions needed for short time- Typically 10 to 20 via adaptive time stepping
•Unconditionally stable • Major step - matrix vector multiply, a few scalar products and
diagonalization of tri-diagonal matrix
Putting it together for the He Code
NR1 NR2 Angular
Linear scaling with number of CPUs
Limiting factor: Memory bandwidth
XSEDE Lonestar and VSC Cluster have identical Westmere
processors
Comparison of He Theoretical and Available Experimental Results NSDI -Total X-Sect
Considerable discrepancies!
Rise at sequential threshold
Extensive convergence tests: angular momenta, radial grid, pulse duration (up to 20 fs), time after pulse (propagate electrons to asymptotic region)
error below 1%
Two-Photon Double Ionization in
The spectral Characteristics of the Pulse can be Critical
Can We Do Better ?
How to efficiently approximate the integral is the key issue