) V( T ) ( H r r The Finite Element Discrete Variable Method for the Solution of theTime Dependent Schroedinger Equation B. I. Schneider Physics Division National Science Foundation Collaborators Lee Collins and Suxing Hu Theoretical Division Los Alamos National Laboratory High Dimensional Quantum Dynamics: Challenges and Opportunities University of Leiden September 28-October 1, 2005
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B. I. Schneider Physics Division National Science Foundation Collaborators
The Finite Element Discrete Variable Method for the Solution of theTime Dependent Schroedinger Equation. B. I. Schneider Physics Division National Science Foundation Collaborators Lee Collins and Suxing Hu Theoretical Division Los Alamos National Laboratory - PowerPoint PPT Presentation
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)V( T )( H rr
The Finite Element Discrete Variable Method for theSolution of theTime Dependent Schroedinger Equation
B. I. Schneider
Physics Division National Science Foundation
CollaboratorsLee Collins and Suxing Hu
Theoretical DivisionLos Alamos National Laboratory
High Dimensional Quantum Dynamics: Challenges and Opportunities
University of LeidenSeptember 28-October 1, 2005
Basic Equation
t),V( m2
- )t,( Hi i
2i
2
rr
Where PossiblyNon-Local
orNon-Linear
0 )t,( )t,( H )t,(
rrrt
i
Objectives•Flexible Basis (grid) – capability to represent dynamics
on small and large scale
Good scaling properties – O(n)
Time propagation stable and unitary
”Transparent” parallelization
Matrix elements easily computed
Discretizations & Representations Grids are simple -
converge poorly
.quadratureby
done beonly can thesebasis expansionmany For
elements;matrix calculate toneeds One
)( C )(
ji
iii
H
rr
Can we avoid matrix element quadrature, maintain locality and keep global convergence ?
z)y,,x(
z)y,,x(
z)y,,x(
z)y,, x(
8/720 108/720- 1080/720 1960/720-
0 1/12- 16/12 30/12-
0 0 1 2-
* 1
z)y,,x(dx
d
ulasOrder Form-
7
5
3
es; DerivativSecond
)( )( )(
3i
2i
1i
i
2i2
2
i
h
ii r - rrr
Global or Spectral Basis Sets – exponential
convergence but can require complex matrix element evaluation
Properties of Classical Orthogonal Functions
; )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) -(2n order of integrand polynomialany integrateexactly
which found bemay i
w weights,and i
xpoints, quadrature Gauss ofset A
procedure Lanczos theusing computed bemay tscoefficienrecursion The
)x(2n
χ1n
β)x(1n
χ)1n
αx()x(n
χn
β
form; theof iprelationshrecursion term threeasatisfy functions The
mn, (x)
mχ (x)
nχ
b
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mχ
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function. weight positive some .lity w.r.tOrthonorma
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More Properties
p
1i)
i(x
i w)
i(x
qχ (x) (x)
qχ w(x)
b
adx
qc
(x)q
p
1qq
c Ψ(x)
expansion,an Given
Corollary
.quadrature by theexactly
integrated becan which polynomial a is integrand thebecause trueis This
mn,δ
p
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i(x
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i(x
nχ
iw
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pq whereq
allfor squadrature Gausspoint -p
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Boundary Conditions, Singular Potentials and Lobatto Quadrature
Physical conditions require wavefuntion to behave regularlyFunction and/or derivative non-singular at
left and right boundaryBoundary conditions may be imposed using
constrained quadrature rules (Radau/Lobatto) – end points in quadrature rule
ConsequenceAll matrix elements, even for singular
potentials are well definedONE quadrature for all angular momentaNo transformations of Hamiltonian required
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fu
nct
ion
Val
ue
Coordinate
Hat Functions
Often only functions or low order derivatives continuous
Ability to treat complicated geometryMatrix representations are sparse –
discontinuities of derivatives at element boundaries must be carefully handled Matrix elements require quadrature
Discretizations & Representations
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fu
nct
ion
Val
ue
Coordinate
Hat Functions
Finite Element Methods - Basis functions have compact support – they live only in a restricted
region of space
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 Methods• Lie-Trotter-Suzuki
1/3
22222
121
11
21
211
4-4
1p
)tr,( )(p)U(p U))4p-((1 U)(p)U(p U= )+tr,(
Phys.), Math. J. ki,order(Suzufourth To
ncalculatio hesimplify t enormouslycan choice judicious