)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