Transcript es2002 1442

Filter-Diagonalization
1. Matrix Diagonalization &
Quantum Dynamics: circumventing
Et
2. Signal Processing!
3. Examples:
Experimental signals
CIK (t )
Semiclassical:
[Trajectory-dependent cellularization (traj.-dep. Filinov)]
QMC (DMC)
Groups interested in extracting eigenstates (or DensityMatrices) 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.
H  
Signal Processing
C (t )   an exp(i nt )
n
Signal processing can be recast (mapped) as a
QM problem.
To see connection: start from QM.
H given, need
Simplest approach:
 n , (n )
C(t )   0 exp(iHt )  0
FFT C(t).
Expensive! Need long time to resolve closely-spaced
eigenvalues
120 121
E
Usually: for resolution,
width: 1/T
Filter-Diagonalization:
T
 ( E1 )   exp(iE1t ) (t )dt
0
T
 ( E2 )   exp(iE2t ) (t )dt
0
1) Filter the same w.p. at 2 (or more) energies
Resulting in energy-localized states,
even if T is short!
E1
e120
e121
E
Filter-Diagonalization:
Short time
(wide width) and…
E1 E2
e120
e121
E
…and use the filtered vectors as an
energy selected basis!
Practically:
• Orthogonalize
 ( E1 )  ( E2 )
• Diagonalize small matrix.
  ( E1 ) H |  ( E1 )
h
  ( E2 ) H |  ( E1 )
 ( E1 ) H |  ( E2 ) 

 ( E2 ) H |  ( E2 ) 
Filter – short time throws contribution of most
eigenstates.
Diagonalization: separates contribution of
closely-spaced eigenvalues.
Method: as is useful for extracting eigenstates
From a short time filter;
Or in general diagonalizing matrices in selected
energy ranges
(Especially if multiple initial vectors are used).
Combined Approach:
First:
k
 (t )  exp(iHt ) 0
T
 ( x; E j )    ( x; t ) exp(iE j t )dt
0
Then:
Orthogonalize the
( x; E j )
Finally: diagonalize the small matrix:
h jl  ( x; E j ) H ( x; El )
Ej
Time-dependent propagation.
First: general methods:
Spectral Propagation:
exp( iHt )   an (t )Tn ( H )
n
 n  2 H n1   n 2
 n  Tn ( H ) o
Split-Operator:
exp  i ( H A  H B )t 
t
t


exp  iH A  exp  iH B t  exp  iH A 
2
2


Pre-conditioning+Filter-Diagonalization:
k
(Wyatt;
Carrington)
Pre-conditioning:
H=H0+V
1
j 
0
( E j  H0 )
Basis-set localized around Ej !
Diagonalize H in basis
Ej
DFT: Divide and Conquer
Renormalization Group—Baer and Head-Gordon.
 ( x)  x  (  H ) x
 (   H )   smooth (   H )  D(   H )
D: concentrated around , so
just 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:
C (t )   an exp(i nt )
n
Signal Processing: Not trivial.
1) “Classical” “MUSIC”, Linear-Prediction, Maximum-Entropy:
work usually increases for long signals
C ( )   C (t ) exp(it )
2) FFT:
•
t
Handles easily
long signals.
But:
• Handles only a single signal at a time
•
Long propagation time
1995: Wall and Neuhauser.
Do not orthogonalize.
Solve instead Generalized-eigenvalue problem
hB  SB
S EE '   ( E )  ( E ') 
 C (t  t ') exp(iEt ) exp(iE ' t ')dtdt '
 (t )  (t ')
hB  SB
S EE '   ( E )  ( E ') 
 C (t  t ') exp(iEt ) exp(iE ' t ')dtdt '
• 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 range
Choose # 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;
Guo
Shaka)
•
Discrete nature of signals.
•
Multiple time-scales.
•
Avoiding Diagonalization.
(long time spectrum directly from short-t.)
Applications:
NMR -- Multiple time-dimensions
t
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.)
C(t) vs. anexp(-int)
0.6
Absorption [a.u]
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
0
1000
2000
3000
Time [fsec]
4000
5000
3
Line resolution in I absorption
1.0
0.8
||
0.6
0.4
0.2
0.0
50
100
150
200
250
-1
[cm ]
300
350
Matrix-correlation functions:
help disentangle eigenvalues
CIJ (t )   I exp(iHt )  J
Still apply:
hB  SB
But now:
SIE , JE '   CIJ (t  t ') exp(iEt )exp(iE ' t ')dtdt '
2nd example:
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]
He+Naphthalene
(Earlier simulations:He+Benzene)
Comparison between single correlation
function and
5x5 cross correlation function (Benzene)
-35
Single Correlation Function
5x5
E(cm-1)
-36
-37
-38
-39
1
2
3
t (psec)
4
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
-26.99
All energies are in wavenumbers
Insert: Trajectory-Dependent Cellularization.
Herman-Kluck
 1
C (t )  
2


y  (p,q)




N
 d y exp(B(y,t)) A y, t 
2N
Problems: (related) –
Weights increasing;
Trajectory chaotic –
cellularization (Filinov) problematic
Filinov-Transform
(Filinov, Freeman, Doll, Coalson;
Manolopolous).
1
det η
 2 
N

 y y  η y y 
N
d ye
C (t ) 
 d
2N
y d y A  y, t 
2N



 B 
1  2 B 
2
exp  B(y, t) +   ( y  y) +  2     ( y  y) 
2  y 
 y 



Problem – B may be steep in certain directions
Solution: make
η
time-dependent and trajectory-dep. matrix.
Trajectory-dependent cellularization: “details”
 1
C (t )  
2






N
 d y exp(iB(y,t)) A y, t 
2N
We find the 2’nd derivative matrix,
set
And REQUIRE
 2 B 
T

AλA
 y 2 


η  AφAT
And condition  so that the overall integrand is
well-behaved and not large.
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 exp( iHt ) 0 
exp( iH 0 t0 ) 0 exp(iH 0 t ) exp( iHt ) 0
known
semiclassical
3-rd example: Eigenvalues in DMC
(with Chen and Mandelshtam)
CIK (t )   0 AI exp( Ht ) AK  0
(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
CIK (t )   0 AI exp(iHt ) AK  0
Potential for automatically many degrees of
freedom (even if ground-state unknown):
CIK (t; T ) 
 0 exp( HT ) AI exp( Ht ) AK exp( HT ) 0
Ground-state
If 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.