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

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) Outline

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

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

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

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

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, (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, (2003) Gatti, Meyer, Chem.Phys. 304, 3 (2004) Wu, Werner, Manthe, Science 306, 2227 (2004) Kühn, Chem.Phys.Lett. 402, (2005) Markmann, Worth, Mahapatra, Meyer, H. Köppel, Cederbaum. J.Chem.Phys. 123, , (2005) Viel, Eisfeld, Neumann, Domcke, Manthe, J.Chem.Phys.,124, , (2006) Vendrell, Gatti, Meyer, Angewandte Chemie 46, 6918 (2007)

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 …….

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

Exploring Dynamical Simplicity Using 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 = degrees of freedom Conventional MCTDH ML-MCTDH

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

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

Model Scaling of the ML-MCTDH Theory

Simulating Time Correlation Functions Examples Imaginary Time Propagation and Monte Carlo Sampling

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

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 : ΔqΔq M-M- M Čižek, Thoss, Domcke, Phys. Rev. B 70 (2004)

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, (2004) M. Nest, T. Klammroth, and P. Saalfrank, JCP 122, (2005) T. Kato and H. Kono, CPL 392, 533 (2004)

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

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

Creation Empty orbital exposed to creation operator

Annihilation 0 Empty orbital exposed to annihilation operator

Annihilation Filled orbital exposed to annihilation operator

Creation 0 Filled orbital exposed to creation operator

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 form the whole Fock space The multilayer formulation

The ML-MCTDH-SQ Theory Change the identical particle system to “distinguishable particles” Each “particle” defines a Fock subspace with all possible occupations 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 occupation for each “particle”/subspace is not conserved. However, the total occupation within the whole Fock space is of course conserved. The formulation for Bosons is simpler than Fermions

Simulating Current Without Nuclear Motion

Without Nuclear Motion

Absorbing Boundary Condition

Effect of Nuclear Motion

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”