B. I. Schneider Physics Division National Science Foundation Collaborators
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Transcript of B. I. Schneider Physics Division National Science Foundation Collaborators
)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
adx w(x)
mχ
nχ
function. weight positive some .lity w.r.tOrthonorma
n
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
1i )
i(x
m)χ
i(x
nχ
iw
mn
pq whereq
allfor squadrature Gausspoint -p
byexactly performed becan integrallity orthonorma that theNote
iprelationshlity orthonorma discreteA
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 practiceIn
true.is that thisassumed is its DVR, In the
mutiply)matrix andexpansion seriespower (Think
complete. is quadraturebasis/or theunless
)F(x F
toequal benot will thisgeneralIn
uF(x)u F
elementmatrix heConsider t
x uxu
and
w )(xu
hat,property t with the
)(x (x) w (x)u
functions, "coordinate" ofset new a Define
ji,iji,
jiji,
ji,iji
i
ji,ji
iqq
p
1qii
More Properties
points. quadrature Gauss theare i
xwhere
ij )j
x-i
x(
) j
x- x(
iW
1 (x)iu
tionrepresenta simpleA
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
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
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hhh
hh
h
h
hhhhhh
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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
abut arbitrary isn Hamiltonia theof breakup theNote
)tr,( )2
Ht )exp(-i
Ht exp(-i )
2
Ht exp(-i
)tr,( )2
()U2
( U= )+tr,(
accuracy,order second then to,H H H
)H exp(-i )H exp(-i )(U
Let
7776
676665
5655
64
54
464544434241
34333231
24232221
14131211
hh
hhh
hh
h
h
hhhhhh
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hhhh
Time Propagation Methods
FEDVR Propagation
)2
Hexp(-i )
2
Hexp(-i )
Hexp(-i )
2
Hexp(-i )
2
Hexp(-i
)2
Hexp(-i )
)H (Hexp(-i )
2
Hexp(-i )(U
matrices. goverlappin diagonal,
are H and H and dependence time
theof all contains and diagonal theis H where
H H H H
into,matrix n Hamiltonia theDecompose
dabad
dbad2
ba
d
bad
block
t)ψ(r,2
t)ψ(r,t
i 22
m
Imaginary Time PropagationProblem Order
tMatrix Size Eigenvalue
Well 2 .005 20 5.2696
Well 4 .005 20 4.9377
Well 2 .001 20 4.9356
Well 4 .001 20 4.9348
Fourier 2 .003 80 49.9718
Fourier 4 .003 80 49.9687
Coulomb 2 .005 120 -.499998
Coulomb 4 .005 120 -.499999
Coulomb 4 .01 120 -.499997
Propagation of BEC on a 3D Lattice
Almost linear speeding-up up to n=128 CPUs. It breaks down from n=128 to n=256 CPUs for this data set.
Ground State Energy of BEC in 3D Trap
Method No. Regions(X Basis)
Points Energy
RSP-FEDVR 20( x 3) (41)3 19.85562355
RSP-FEDVR 20 ( x 4) (61)3 19.84855573
RSP-FEDVR 20 ( x 6) (101)3 19.84925147
RSP-FEDVR 20 ( x 8) (141)3 19.84925687
3D Diagonalization 19.847
Ground State Energy of 3D Harmonic Oscillator
Method No. Regions(X Basis) Points Energy
RSP-FD 1 (104)3 1.496524844
RSP-FD 1 (144)3 1.498189365
RSP-FD 1 (200)3 1.499061950
RSP-FD 1 (300)3 1.499632781
RSP-FD 1 (500)3 1.499907103
RSP-FEDVR 20( x 3) (41)3 1.497422285
RSP-FEDVR 20 ( x 4) (61)3 1.499996307
RSP-FEDVR 20 ( x 6) (101)3 1.500000028
RSP-FEDVR 20 ( x 8) (141)3 1.500000001
Exact 1.500000000
Solution of TD Close Coupling Equations for He Ground State E=-2.903114138 (-2.903724377) 160 elements x 4 Basis