Filter-Diagonalization

1. Filter-Diagonalization • 1. Matrix Diagonalization & • Quantum Dynamics: circumventing • 2. Signal Processing! • 3. Examples: • Experimental signals • Semiclassical: • [Trajectory-dependent cellularization (traj.-dep. Filinov)] • QMC (DMC)

2. 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)

3. 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.

4. Filter-Diagonalization: Extends FFT Bridges FFT and other approaches, Trick: Connect Q.M. Signal Processing Signal processing can be recast (mapped) as a QM problem.

5. To see connection: start from QM. H given, need Simplest approach: FFT C(t). Expensive! Need long time to resolve closely-spaced eigenvalues

6. e120 e121 E Usually: for resolution, width: 1/T

7. Filter-Diagonalization: • Filter the same w.p. at 2 (or more) energies • Resulting in energy-localized states, • even if T is short!

8. E1 E E e120 e121 Filter-Diagonalization: Short time (wide width) and…

9. E2 E1 E E e120 e121 …and use the filtered vectors as an energy selected basis!

10. Practically: • Orthogonalize • Diagonalize small matrix.

11. Filter – short time throws contribution of most eigenstates. Diagonalization: separates contribution of closely-spaced eigenvalues.

12. 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).

13. Combined Approach: First: Then: Orthogonalize the Finally: diagonalize the small matrix:

14. Time-dependent propagation. First: general methods: Spectral Propagation: Split-Operator:

15. Pre-conditioning+Filter-Diagonalization: (Wyatt; Carrington) Pre-conditioning: H=H0+V Basis-set localized around Ej ! Diagonalize H in basis

16. DFT: Divide and ConquerRenormalization Group—Baer and Head-Gordon. D: concentrated around m, so just few e.functions are enough.

17. Surprising feature of Filter-Diagonalization: can be recast as a: Signal processing application!

18. And now to : Signal Processing: From C(t) t=0,dt,2dt,3dt,…,T Get C(t) all t OR:

19. Signal Processing: Not trivial. • “Classical” “MUSIC”, Linear-Prediction, Maximum-Entropy: • work usuallyincreases for long signals • 2) FFT: • Handles easily long signals. • But: • Handles only a single signal at a time • Long propagation time

20. 1995: Wall and Neuhauser. Do not orthogonalize. Solve instead Generalized-eigenvalue problem

21. Single C(t) needed for all energy-ranges! • No Hamiltonian necessary!!!!!

22. Route “eventual”: H Eigenvalues from C(t)  H not needed (need not exist) Route “completed”: Eigenvalues from C(t)

23. 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)

24. Developments: Mandelshtam; • Taylor; • Guo • Shaka) • Discrete nature of signals. • Multiple time-scales. • Avoiding Diagonalization. • (long time spectrum directly from short-t.)

25. 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)>

26. 1’st Example: Use with an Experimental Signal (absorption in I3-). (Gershgoren and Ruchman, Jerusalem, 2000.)

28. Matrix-correlation functions: help disentangle eigenvalues Still apply: But now:

29. 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]

30. He+Naphthalene (Earlier simulations:He+Benzene)

31. Comparison between single correlation function and 5x5 cross correlation function (Benzene)

32. Benzene: Converged results from a 5x5 cross-correlation analysis vs. exact results for different symmetries. All energies are in wavenumbers

33. Insert: Trajectory-Dependent Cellularization. Herman-Kluck Problems: (related) – Weights increasing; Trajectory chaotic – cellularization (Filinov) problematic

34. Filinov-Transform (Filinov, Freeman, Doll, Coalson; Manolopolous). Problem – B may be steep in certain directions Solution: make time-dependent and trajectory-dep. matrix.

35. Trajectory-dependent cellularization: “details” We find the 2’nd derivative matrix, set And REQUIRE And condition f so that the overall integrand is well-behaved and not large.

36. Trajectory-Dependent Cellularization.

37. Work-in-progress: Naphthalene, Effect of Trajectory-Dependent Cellularization(single C(t), few trajectories)

38. Other-improvements (in progress): Backward-forward propagation(Makri): known semiclassical

39. 3-rd example: Eigenvalues in DMC(with Chen and Mandelshtam) (see Whaley too). Suitable for Filter-Diagonalization.

40. FDG for QMC vs. exact results –2D He-Be. work in progress;Will implement: better initial guesses,smaller dt, more trajectories

42. Potential for automatically many degrees of freedom (even if ground-state unknown): Ground-state If T is large.

43. Conclusions: Filter-Diagonalization: * Handles large signals * Applicable when long-times expensive/difficult * Is general extension of Fourier-Transforms Trajectory-dependent cellularization effective.