Filter-Diagonalization

1. Matrix Diagonalization &

Quantum Dynamics: circumventing

2. Signal Processing!

3. Examples:Experimental signals

Semiclassical:[Trajectory-dependent cellularization (traj.-dep. Filinov)]

QMC (DMC)

E t

( )IKC t

Groups interested in extracting eigenstates (or Density-Matrices) using “filters”Mandelshtam, Shaka, Chen (Irvine)Taylor (USC)Baer (Jerusalem) (Density-Matrices)Rabani (Tel-Aviv) (Density-Matrices)Wyatt (Houston)Head-Gordon (Berkeley) (Density-Matrices)Moiseyev (Haifa)Guo (New-Mexico)Meyer, Cederbaum, Beck (Heidelberg)Ruchman&Gershgoren, Labview implementation

for condensed phases signals (Jerusalem)

D.N., Mike Wall, Johnny Pang, Sybil Anderson, Jaejin Ka

Emily Carter, Antonio, de Silva, E. Fattal, Peter Felker, Julie Feigon, Wousik Kim(UCLA)

Existing Approaches for eigenstates:

Non-sepearable H:

Lowest state: ITERATE.

General: LANCOSZ

H Tridiagonal

Eigenvalues simple for REAL H’s

Converges fastest near gaps.

Too democratic.

Filter-Diagonalization:

Extends FFT

Bridges FFT and other approaches,

Trick:Connect Q.M. Signal Processing

H ( ) exp( )n nn

C t a i t

Signal processing can be recast (mapped) as aQM problem.

To see connection: start from QM.

H given, need , ( )n n

Simplest approach:

0 0( ) exp( )C t iHt

FFT C(t).

Expensive! Need long time to resolve closely-spaced eigenvalues

E

Usually: for resolution, width: 1/T

121120

Filter-Diagonalization:

2 2

0

( ) exp( ) ( )T

E iE t t dt

1) Filter the same w.p. at 2 (or more) energies

Resulting in energy-localized states,

even if T is short!

1 1

0

( ) exp( ) ( )T

E iE t t dt

EE

Filter-Diagonalization:Short time (wide width) and…

E1

e121e120

EE

E2

e121e120

E1

…and use the filtered vectors as an energy selected basis!

Practically:• Orthogonalize

• Diagonalize small matrix.

1( )E 2( )E

1 1 1 2

2 1 2 2

( ) | ( ) ( ) | ( )

( ) | ( ) ( ) | ( )

E H E E H E

E H E E H E

h

Filter – short time throws contribution of most eigenstates.

Diagonalization: separates contribution of closely-spaced eigenvalues.

Method: as is useful for extracting eigenstatesFrom a short time filter;

Or in general diagonalizing matrices in selectedenergy ranges

(Especially if multiple initial vectors are used).

Combined Approach: First:

0( ) exp( )t iHt

0

( ; ) ( ; ) exp( )T

j jx x t i t dtE E

k

jE

Then: Orthogonalize the ( ; )jx E

Finally: diagonalize the small matrix:

( ; ) ( ; )jl j lh x E H x E

Time-dependent propagation. First: general methods:

Spectral Propagation:

1 2

exp( ) ( ) ( )

2

( )

n nn

n n n

n n o

iHt a t T H

H

T H

Split-Operator:

exp ( ) exp exp exp2 2A B A B A

t ti H H t iH iH t iH

Pre-conditioning+Filter-Diagonalization:

(Wyatt; Carrington)

Pre-conditioning:

H=H0+V

00

1

( )jjE H

Basis-set localized around Ej !

Diagonalize H in basis

k

jE

DFT: Divide and ConquerRenormalization Group—Baer and Head-Gordon.

( ) ( )x x H x

( ) ( () )smooth HH D H

D: concentrated around , sojust few e.functions are enough.

Surprising feature of Filter-Diagonalization:

can be recast as a:

Signal processing application!

And now to : Signal Processing:

From C(t) t=0,dt,2dt,3dt,…,T

Get C(t) all t

OR:( ) exp( )n n

n

C t a i t

Signal Processing: Not trivial.

1) “Classical” “MUSIC”, Linear-Prediction, Maximum-Entropy: work usually increases for long signals

2) FFT:

• Handles easily long signals.

But:• Handles only a single signal at a time • Long propagation time

( ) ( ) exp( )t

C C t i t

1995: Wall and Neuhauser.Do not orthogonalize.Solve instead Generalized-eigenvalue problem

hB SB

' ( ) ( ')

exp( )ex( ') p( ' ') '

EES E E

iEt iE tC t tdtt d

( ) ( ')t t

hB SB

' ( ) ( ')

exp( )ex( ') p( ' ') '

EES E E

iEt iE tC t tdtt d

• Single C(t) needed for all energy-ranges!

• No Hamiltonian necessary!!!!!

Route “eventual”:

H

Eigenvalues from C(t)

H not needed (need not exist)

Route “completed”:Eigenvalues from C(t)

Sig. Proc. Algorithm (automated):Choose frequency rangeChoose # of vectors (2-10)Calculate h,S from C(t)Diagonalize to get poles.

Cheap! (Single FFT)

Extends FFT to a matrix method(FFT: L=1!)

Applicable to MATRIX signals Cik(t)

Developments: Mandelshtam;Taylor;GuoShaka)

• Discrete nature of signals.

• Multiple time-scales.

• Avoiding Diagonalization.

(long time spectrum directly from short-t.)

Applications:

NMR -- Multiple time-dimensions

t

Semiclassical correlation functions (He-aromatic clusters;He2-aromatics next.)

Excited states in DMC

Extracting frequencies from short-time segments – Mass spectra

Classical frequencies from < v(t)v(0)>

1’st Example: Use with an Experimental Signal (absorption in I3

-). (Gershgoren and Ruchman, Jerusalem, 2000.)

0 1000 2000 3000 4000 5000-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

C(t) vs. anexp(-int)A

bsor

ptio

n [a

.u]

Time [fsec]

50 100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0

Line resolution in I3

-

absorption

| |

[cm-1

]

Matrix-correlation functions: help disentangle eigenvalues

( ) exp( )IJ I JC t iHt

hB SBStill apply:

But now:

, ' exp( )exp( ' ') '( ')IE JE IJS iC Et iE t dtdtt t

Semi-classical signal with Filter-Diagonalization

(Anderson, Ka, Felker, Neuhauser, 1999-2001)

• Semiclassical – excellent at short times.• Cross-correlation: helps!.

• Example: He+Naphthalene (3D system),

[Developed: Trajectory-dependent Filinov]

2nd example:

He+Naphthalene

(Earlier simulations:He+Benzene)

Comparison between single correlation function and

5x5 cross correlation function (Benzene)

-39

-38

-37

-36

-35

1 2 3 4

Single Correlation Function

5x5

t (psec)

E(c

m-1)

Benzene: Converged results from a 5x5 cross-correlation analysis vs. exact results for different symmetries.

Symmetry Semiclassical Exact

A1 -56.5 -56.57

E1 -46.0 -45.46

A1 -38.8 -38.77

E2 -37.7 -36.96

E1 -32.6 -32.45

B2 -31.4 -30.92

B1 -30.2 -29.39

A1 -28.0 -27.82

E2 -27.2 -26.99All energies are in wavenumbers

Insert: Trajectory-Dependent Cellularization.

Herman-Kluck

21( ) ,2

N

N

C t d A t

y (p,q)

y exp(B(y,t)) y

Problems: (related) – Weights increasing;Trajectory chaotic –

cellularization (Filinov) problematic

det

12

NN d e

y y η y yηy

Filinov-Transform (Filinov, Freeman, Doll, Coalson; Manolopolous).

Problem – B may be steep in certain directions

Solution: make ηtime-dependent and trajectory-dep. matrix.

22

22

( (

( )

,N N y y

C t

d d A t

2B 1

B(y,t)+ y)+ y)y

B

y2y y y exp

We find the 2’nd derivative matrix,

2

2T

AλA

B

yset

Tη AφAAnd REQUIRE

And condition so that the overall integrand is well-behaved and not large.

21( ) ,2

N

N

C t d A t

y exp(iB(y,t)) y

Trajectory-dependent cellularization: “details”

Trajectory-Dependent Cellularization.

Work-in-progress: Naphthalene, Effect of Trajectory-Dependent

Cellularization (single C(t), few trajectories)

Other-improvements (in progress):

Backward-forward propagation(Makri):

0 0

0 0 0 0 0

exp( )

exp( ) exp( )exp( )

iHt

iH t iH t iHt

known semiclassical

3-rd example: Eigenvalues in DMC(with Chen and Mandelshtam)

0 0( ) exp( )IK I KC t A Ht A

(see Whaley too).Suitable for Filter-Diagonalization.

FDG for QMC vs. exact results –2D He-Be. work in progress;

Will implement: better initial guesses,smaller dt, more trajectories

0 0( ) exp( )IK I KC t A iHt A

Potential for automatically many degrees of freedom (even if ground-state unknown):

0 0

( ; )

exp( ) exp( ) exp( )

IK

I K

C t T

HT A Ht A HT

Ground-stateIf T is large.

Conclusions:

Filter-Diagonalization:

* Handles large signals* Applicable when long-times expensive/difficult* Is general extension of Fourier-Transforms

Trajectory-dependent cellularization effective.