Multilayer Formulation of the Multi-Configuration Time-Dependent Hartree Theory

1. Multilayer Formulation of the Multi-Configuration Time-Dependent Hartree Theory Haobin Wang Department of Chemistry and Biochemistry New Mexico State University Las Cruces, New Mexico, USA Collaborator: Michael Thoss Support: NSF, NERSC

2. Outline • From convention wave packet propagation to MCTDH: a variational perspective • The multilayer formulation of MCTDH (ML-MCTDH) • Scaling of the ML-MCTDH theory • Generalization to treat identical particles: ML-MCTDH with Second Quantization (ML-MCTDH-SQ)

3. Conventional Wave Packet Propagation • Dirac-Frenkel variational principle • Conventional Full CI Expansion (orthonormal basis) • Equations of Motion • Capability: <10 degrees of freedom (<~n10 configurations)

4. Multi-Configuration Time-Dependent Hartree • Multi-configuration expansion of the wave function • Variations • Both expansion coefficients and configurations are time-dependent Meyer, Manthe, Cederbaum, Chem. Phys. Lett. 165 (1990) 73

5. MCTDH Equations of Motion (Meyer, Manthe, Cederbaum) The “single hole” function • Reduced density matrices and mean-field operators Meyer, Manthe, Cederbaum, Chem. Phys. Lett. 165, 73 (1990) Manthe, Meyer, Cederbaum, J.Chem.Phys. 97, 3199 (1992).

6. Variational Grouping of the Subspaces • Single particle functions (SPFs): full CI expansion within the subspace (and adiabatic basis contraction) • Only a fewSPFs are selected among the full CI subspace, and then build the approximation for the whole space • The “complete active space” strategy: first defines the whole space, then selects a subset • Thus, the philosophy is different!

7. The MCTDH Theory • Capability of the MCTDH theory: ~10×10 = 100 degrees of freedom Worth, Meyer, Cederbaum, J. Chem. Phys. 105, 4412 (1996) Worth, Meyer, Cederbaum, J. Chem. Phys. 109, 3518 (1998) Raab, Worth, Meyer, Cederbaum, J. Chem. Phys. 110, 936 (1999) Mahapatra, Worth, Meyer, Cederbaum. Koppel, J. Phys.Chem. A 105, 5567 (2001) H. Koppel, Doscher, Baldea, Meyer, Szalay, J. Chem. Phys. 117, 2657 (2002) Nest, Meyer, J. Chem. Phys. 117, 10499 (2002) Huarte-Larranaga. Manthe, J. Chem. Phys. 113, 5115 (2000) Huarte-Larranaga, U. Manthe, J. Chem. Phys. 117, 4653 (2002) McCurdy, Isaacs, Meyer, Rescigno, Phys.Rev. A 67, 042708 (2003) Gatti, Meyer, Chem.Phys. 304, 3 (2004) Wu, Werner, Manthe, Science 306, 2227 (2004) Kühn, Chem.Phys.Lett. 402, 48-53 (2005) Markmann, Worth, Mahapatra, Meyer, H. Köppel, Cederbaum. J.Chem.Phys. 123, 204310, (2005) Viel, Eisfeld, Neumann, Domcke, Manthe, J.Chem.Phys.,124, 214306, (2006) Vendrell, Gatti, Meyer, Angewandte Chemie 46, 6918 (2007) ••••••

8. ……. ……. Multilayer Formulation of the MCTDH Theory • Another multi-configuration expansion of the SP functions • More complex way of expressing the wave function Wang, Thoss, J. Chem. Phys. 119 (2003) 1289

9. ML-MCTDH Equations of Motion Wang, Thoss, J. Chem. Phys. 119 (2003) 1289

10. Exploring Dynamical Simplicity Using ML-MCTDH Conventional MCTDH ML-MCTDH • Capability of the two-layer ML-MCTDH: ~10×10×10 = 1000 degrees of freedom • Capability of the three-layer ML-MCTDH: ~10×10×10×10 = 10000 degrees of freedom

11. The Scaling of the ML-MCTDH Theory • f: the number of degrees of freedom • L: the number of layers • N: the number of (contracted) basis functions • n: the number of single-particle functions

12. The Scaling of the ML-MCTDH Theory electronic nuclear coupling • The Spin-Boson Model • Hamiltonian • Bath spectral density

13. Model Scaling of the ML-MCTDH Theory

14. Model Scaling of the ML-MCTDH Theory

15. Model Scaling of the ML-MCTDH Theory

16. Simulating Time Correlation Functions • Examples • Imaginary Time Propagation and Monte Carlo Sampling

17. V Simulating Electric Current M. Galperin, M.A. Ratner, A. Nitzan, J. Phys. Condens. Matter, 19, 103201 (2007)

18. Nuclear DOF's

19. Nuclear DOF's

20. Nuclear DOF's

21. M M- Δq Vibrationally inelastic electron transport • Modeling: • Tight-binding approximation, Wannier states of each lead transformed to Bloch states • additional (or missing) electron in the bridge state results in a change of the potential energy surface • Calculation of the current: Čižek, Thoss, Domcke, Phys. Rev. B 70 (2004) 125406

22. The MCTDHF Approach? Fermi-Dirac Statistics: Anti-symmetric wave function J. Caillat, J. Zanghellini, M. Kitzler, O. Koch, W. Kreuzer, and A. Scrinzi, Phys. Rev. A 71, 012712 (2004) M. Nest, T. Klammroth, and P. Saalfrank, JCP 122, 124102 (2005) T. Kato and H. Kono, CPL 392, 533 (2004)

23. Active Space ……. ……. What Strategy? But how to put identical particles into different groups (and try to distinguish them)?

24. The Concept of Second Quantization Fock Space Each determinant is represented by an occupation-number vector which can be represented by actions of creation operators

25. Empty orbital exposed to creation operator Creation

26. Empty orbital exposed to annihilation operator Annihilation 0

27. Filled orbital exposed to annihilation operator Annihilation

28. Filled orbital exposed to creation operator Creation 0

29. The ML-MCTDH-SQ Theory Fock sub-space within one “single particle” for several states/electrons The multi-configuration combination of the Fock sub-space to formthe whole Fock space The multilayer formulation

30. The ML-MCTDH-SQ Theory Change the identical particle system to “distinguishable particles” Each “particle” defines a Fock subspace with all possible occupations The occupation for each “particle”/subspace is not conserved. However, the total occupation within the whole Fock space is of course conserved. • Second Quantization vs. Slater Determinant: two formal ways of enforcing permutation/exchange symmetry • Slater Determinant: wave function approach, valid for any form of Hamiltonian operators • Second Quantization: operator approach, superior for special form of Hamiltonian The formulation for Bosons is simpler than Fermions

31. Simulating Current Without Nuclear Motion

32. Without Nuclear Motion

33. Absorbing Boundary Condition

34. Effect of Nuclear Motion

35. Summary of the ML-MCTDH Theory • Powerful tool to propagate wave packet in “complex” systems • Can reveal various dynamical information • population dynamics and rate constant • wave packet motions • time-resolved nonlinear spectroscopy • Has been generalized to handle indistinguishable particles • Limitation: can only be implemented for certain class of models • Potentials: two-body, three-body, etc. (but cf. the CDVR) • Product form of the Hamiltonian • Difficulties: • Implementation: somewhat challenging • Long time dynamics: “chaos”