1. James Brown, Tucker Carrington Jr. Computing vibrational energies with phase-space localized functions and an iterative eigensolver

2. 1D Variational Calculation Expand unknown nuclear wavefunction in a known basis. Wanted wavefunctionUnknown coefficients Basis functions 2

3. Matrix Representation Diagonalize matrix to obtain eigenvalues (energies) and eigenvectors (wavefunctions) 3

4. von Neumann basis functions In 1979 Davis and Heller introduced the idea of using von Neumann (vN) basis function in vibrational energy calculations These vN functions are localized in phase-space. 4

5. What is phase-space? 5

6. Quantum Phase-space 6

7. Using vN basis in calculations Set up a grid (eq. 3x3 grid) x0x0 x2x2 x1x1 p0p0 p2p2 p1p1 This ensures a complete but not overcomplete basis. One basis function per area in phase-space 7

8. Hypothesis Goal: one basis function per wanted eigenfunction Should be possible since both eigenfunctions and phase- space functions are complete in classical region 8

9. Applying the vN basis S is not the identity matrix Generalized eigenproblem 9

10. For more dimensions, it is convenient to represent the wavefunction in a product basis The basis will now have size The matrix elements of More than one dimension 10

11. The Curse of dimensionality Standard Eigenproblem Solved by storing vectors in RAM Generalized Eigenproblem Solved by storing matrices in RAM HV=VE MoleculeVector of CoefficientsHamiltonian Matrix H2H2 80 Bytes50 Kb H2OH2O7.8 KB488 Mb HCHO7.6 MB7.3 Tb CH 4 7.5 GB6.6 Yb C2H4C2H4 7.3 TB… 11 HV=SVE

12. vN Doesn’t work vN doesn’t produce acceptable accuracy with vN functions spaced in boxes of Accuracy increases very slowly with larger vN basis. nExactError (25 vNs)Error (48 vNs) 0-11.50520.0082 6-10.54090.370.33 12-9.63020.300.29 18-8.7552>1 21-7.9219>1 12

13. New formulation Shimshovitz and Tannor introduced using vNs as coefficients to expand N Sinc discrete variable representation (DVR) functions. Expand wavefunction and solve for coefficients New Basis Sinc DVR function nn th DVR point 13

14. Sinc DVR Discrete Variable Representation is unity at and zero at Example: Sinc DVR 14

15. In Matrix form Also a generalized eigenproblem with new basis 15

16. When keeping all basis functions Convergence of 7 th eigenvalue of harmonic oscillator vN is Davis Heller method FGH pvN 16

17. Removing Basis Functions Chopping Matrix acts as a projector of the basis functions we want to keep. 17

18. Doesn’t work Convergence of 24 th eigenvalue of Morse potential Performs worse! 18

19. Next attempt Expand using basis functions. Biorthogonal to Use coefficients Results in eigenproblem 19

20. Why better? Look at the coefficients The coefficients are an approximation to the overlap integral Many coefficients of expansion will be small. Those basis functions can be removed 20

21. Using basis works FGH pvN ST Convergence of 24 th eigenvalue of Morse potential ST 21

22. Convert to a standard eigenproblem is a direct product of matrices is not Invert before applying Brown/Carrington 22 Shimshovitz/Tannor

23. Comparing ST and BC: Memory requirements Hermitian generalized eigenproblem Non-symmetric eigenproblem Shimshovitz/Tannor (ST)Brown/Carrington (BC) MoleculeVector of CoefficientsHamiltonian Matrix H2H2 80 bytes50 Kb H2OH2O7.8 Kb488 Mb HCHO (formaldehyde)7.6 Mb7.3 Tb CH 4 (Methane)7.5 Gb6.6 Yb C 2 H 4 (Ethylene)7.3 Tb… 23 “The curse of dimensionality”

24. Comparing ST and BC: Computational effort Difficult for ST to use iterative methods is direct product but is not Store the full matrix and use operations basis functions per degree of freedom (D) BC can use iterative methods Store vectors and use is the number of matrix-vector products required to converge wanted eigenvalues ST BC 24

25. Comparing ST and BC: Accuracy Error of 22 nd eigenvalue of 1D Morse potential (number of kept functions) 25

26. Performing matrix-vector products Matrix-vector products are performed using the Arnoldi method to calculate eigenvalues This is most easily performed with a sum of products KEO and potential 26

27. Initial basis 27 Before performing a calculation, it is impossible to know what basis functions have large coefficients. Start with a basis that approximates these coefficients. Basis functions with E below some energy

28. Expanding the basis 28 1. Perform a calculation using ARPACK and obtain eigenvectors for wanted eigenvalues 2. Find basis functions with largest values is the eigenvector coefficient for the basis function and the wavefunction. 3. Add new basis functions in area adjacent to largest values. Repeat steps 1-3 until iterations do not change energies to a desired accuracy.

29. Expanding the basis 29

30. Test expanding the basis method on 3D system 30 Bilinearly coupled 3d system nEnergy (Hartree) 12.98295779014893 23.94446877907824 34.90597976800756 44.95910687506793 55.86749075693687 nEnergy (Hartree) 65.92061786399724 76.01121329659847 86.82900174586619 96.88212885292655 106.93525595998693

31. Comparing Energy vs Building Classical energy criteria (8624 basis functions) Iterative basis building (8696 basis functions 31 NError (Hartree) 1627.2 x10 -6 2575.5 x10 -6 3525.4 x10 -6 4563.3 x10 -6 5476.3 x10 -6 6519.2 x10 -6 7428.9 x10 -6 8606.5 x10 -6 9475.9 x10 -6 10502.6 x10 -6 Error (Hartree) 7.4 x10 -6 7.9 x10 -6 8.8 x10 -6 8.9 x10 -6 8.4 x10 -6 9.2 x10 -6 8.8 x10 -6 13.5 x10 -6 9.8 x10 -6 15.5 x10 -6

32. Formaldehyde Calculation 32 Use the potential of Hubert Romanowski, Joel M. Bowman, and Lawrence B. Harding, The Journal of Chemical Physics, 82, 4155-4165 (1985) Optimized basis set for lowest 20 eigenvalues Comparison to calculation with direct product of 10 Gauss- Hermite functions for each degree of freedom. (1,000,000 total functions)

33. Formaldehyde Calculation Results 33 vnvn Gauss-Hermite (cm -1 ) (N=1,000,000) Phase-Space (cm -1 ) (N=814,446) Difference (cm -1 ) 11156.321156.260.06 21248.981249.020.04 31502.821503.070.23 41749.271749.360.09 52307.232307.320.09 102788.692790.361.67 153244.123244.510.39 193667.023667.360.34 Transition frequencies

34. Conclusions Firsts for phase-space functions 1. Formulated as a standard (non-generalized) eigenproblem 2. 4-atom molecule calculated to reasonable accuracy Shows promise in defeating the “curse of dimensionality”. The method is very “hands-off” and could be used for any sum-of-products Hamiltonian with minimal effort A “black box” method 34

35. The Curse of dimensionality: Revisited 35 Direct ProductPhase-Space H2H2 1026 H2OH2O1,000 (10 3 )3,500 (~16 3 ) HCHO1,000,000 (10 6 )800,000 (~10 6 )