Propagating the Time Dependent
Schroedinger Equation
B. I. Schneider
Division of Advanced Cyberinfrastructure
National Science Foundation
H (r )  T  V( r )
National Science Foundation
September 6, 2013
What Motivates Our Interest
•Novel light sources: ultrashort, intense pulses
 Nonlinear (multiphoton) laser-matter interaction
Attosecond pulses
Free electron lasers (FELs)
 probe and control electron
 Extreme intensities
dynamics
 XUV + IR pump-probe
 Multiple XUV photons
Basic Equation

i
 ( r , t )  H ( r , t )  (r , t )  0
t
Where
2

H (r , t )  2
2
i
m
i
i
Possibly
Non-Local
or
Non-Linear
 V( r, t)
Properties of Classical
Orthogonal Functions
 Orthonorma lity w.r.t . some positive weight function.
b
χ χ
  dx w(x) χ (x) χ (x)  
n m
n
m
n, m
a
 The functions satisfy a three term recursion relationsh ip of the form;
β χ (x)  (x  α
)χ
(x)  β
χ
(x)
n n
n 1 n 1
n 1 n  2
 The recursion coefficien ts may be computed using the Lanczos procedure
 A set of Gauss quadrature points, x and weights, w may be found which
i
i
exactly integrate any polynomial integrand of order (2n - 1) or less
with respect to the weight function.
 The points and weights may be found by diagonaliz ing
the tridiagon al matrix made up of the  and  coefficien ts.
 Completene ss
 n (x ) n (x)   (x - x) ;
n
 n ( x )  w1/2 (x) χ n (x)
More Properties
 A discrete orthonorma lity relationsh ip
Note that the orthonorma lity integral can be performed exactly by
p - point Gauss quadrature s for all  where q  p
q
p
 n  m   wi χ n (x i )χ m (x i )  δn, m
i 1
This is true because the integrand is a polynomial which can be integrated
exactly by the quadrature .
 Corollary
Given an expansion,
p
1/2
Ψ(x)  w (x)  (x)   c  (x)
q q
q 1
b
p
c   dx w(x) χ (x)  (x)   χ (x ) w  (x )
q
q
q i i
i
i

1
a
Matrix Elements
Consider, a matrix element of the potential,
Vq,q'   q V(x)  q'
Conceptual ly, this matrix element may be evaluated
if we know the matrix representa tion of the position
operator. Then,
V  V (x)
as long as the basis set is complete. This remains quite useful
even for finite basis sets and suggests that an excellent
approximat ion to the matrix is obtained,
Vq,q'   Tq,i V(x i ) Tq' ,i
i
where T is the transform ation from the original
representa tion to one which diagonaliz es x. Note, that
this looks like a quadrature formula.
Properties of Discrete Variable
Representation
 Define a new set of " coordinate " functions,
p
u i (x)  w i   q (x)  q (x i )
q 1
with the properties that,
u i (x j ) 
 i, j
wi
b
;  dx w(x) u i (x)u j (x)   i, j
a
and
u i x u j  x i i, j
 Consider t he matrix element
Fi, j  u i F(x) u j
In general this will not be equal to
Fi, j  F(x i )  i, j
unless the basis/or quadrature is complete.
(Think power series expansion and matrix mutiply)
In the DVR, its is ASSUMED that this is true.
In practice it appears to be an excellent approximat ion.
Its Actually Trivial
 A simple representa tion
(x-x )
1
j
u i (x) 

W j  i ( xi - x j )
i
where x are the Gauss quadrature points.
i
Matrix elements of the derivative operators may be evaluated using the
quadrature rule, but its trivial . For Cartesian coordinates :
 d2 
d2
ui
u j  w iui ( x i ) 
u j
2
2
dx
 dx 
x  xi
Multidimensional Problems
 Tensor Product Basis
 Consequences
i, j, k H l, m, n  i T l
 j,m k,n 
j T m  i,l k,n 
k T n  i,l j,m 
i, j, k V l, m, n  i,l j,m k,n
Multidimensional Problems
out
Vi,j,k
=
å
i,j,k H l,m,n
in
Vl,m,n
=
l,m,n
å
iTl
in
Vl,j,k
l
+å j T m
in
Vi,m,k
m
+å k T n
in
Vi,j,n
n
in
+ i,j,k V i,j,k Vi,j,k
Nested sums.
Two Electron matrix
elements also
‘diagonal” using
Poisson equation
Finite Element Discrete Variable
Representation
•
Properties
•Space Divided into Elements – Arbitrary
size
•“Low-Order” Lobatto DVR used in each
element: first and last DVR point shared
by adjoining elements
F ( x) 
i
n
i 1
1
( f ( x)  f
i
n
i 1
1
w  w
i
n
( x) ) •
•
Sparse Representations
– N Scaling
Close to Spectral
Accuracy
•Elements joined at boundary – Functions
continuous but not derivatives
•Matrix elements requires NO Quadrature
– Constructed from renormalized, single
element, matrix elements
Finite Element Discrete Variable
Representation
•
Structure of
Matrix
h11
h21
h12
h22
h13
h23
h14
h24
h31
h41
h32
h42
h33
h43
h34
h44
h45
h46
h54
h64
h55
h65
h56
h66
h67
h76
h77
Time Propagation Method
H(t)t
 (r, t + t ) = exp(-i
)  (r, t )

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%
R (a.u.)
R (a.u.)
FIG. 4. Left : Ionizat ion probabilit ies (left scale) for H +2 in laser pulses at phot on energies of 1.50 and 1.18 a.u. for t ime
durat ions of 10 and 5 cycles, respect ively . T he molecular axis is chosen along t he linear laser polarizat ion vect or. T he peak
int ensity is 1014 W / cm2 . T he elect ronic energy (dot -dot -dashed line) versus t he int ernuclear dist ance R is shown on t he right
scale. Right : Transit ion moment s (left scale) in t he parallel geomet ry from t he channels ` = 1, 3, and 5 at t he phot on energy of
1.50 a.u. T he t ot al squared cont ribut ion from all t hree channels is shown on t he right scale (solid curve). T he vert ical dashed
line indicat es t he R value where M 1, 0 vanishes.
E 2 (eV)
20
10
15
10
5
5
0
0
0
5
10
E 1 (eV)
15
2
✓1 =
30◦
TDCS (10− 55cm4 s/ sr2 eV)
25
15
TDCS (10− 55cm4 s/ sr2 eV)
Two-Photon Double Ionization in
FE-DVR
ECS (/ 2)
TDCC (⇥2)
1
0
0
60
120
180 240
✓2 (deg)
300
360
40
✓1 = 30◦
30
20
10
0
0
60
120
180 240
✓2 (deg)
300
360
FIG. 5. Left : Energy probability dist ribut ion of two eject ed elect rons when t he laser polarizat ion axis is perpendicular t o t he
molecular axis for a sine-squared laser pulse of 10 opt ical cycles, a cent ral energy of 30 eV , and a peak int ensity of 1014 W / cm2 .
T he color bars correspond t o mult iples of 10− 7 eV − 2 . A lso shown are t he coplanar T DCS for two-phot on double ionizat ion of
H 2 at equal-energy sharing (E 1 = E 2 = 4.3 eV ) of t he two eject ed elect rons in t he parallel (cent er) and again t he perpendicular
geomet ry (right ) for a det ect ion of one elect ron being 30◦ relat ive t o t he laser polarizat ion. T he T DCS result s of Colgan et
al. [69] and M orales et al. [70] for t he parallel geomet ry (cent ral panel) were mult iplied by t he scaling fact ors indicat ed in t he
legend.
our resultspectral
s and t hose from ot her
recent calculat ions by Colgan et al.of
[69] the
and Morales
et al. [70]. can
Our analysis [8]
The
Characteristics
Pulse
suggest s t hat t he reason may not be t he numerics (albeit t he st rong emission of t he second elect ron seen in [69]
in t he vicinity of t he eject ion angle of t he
first Critical
one seems quest ionable), but inst ead a consequence of t he t imebe
dependent t reat ment wit h a relat ively broad spect rum of phot on energies vs. t he e↵ect ively t ime-independent
Can We Do Better ?
How to efficiently approximate the integral is the key issue