(Ab initio) path-integral Molecular Dynamics The double slit experiment Sum over paths: Suppose only two paths: interference The double slit experiment ● Introduce of a large number of intermediate gratings, each containing many slits. ● Electrons may pass through any sequence of slits before reaching the detector ● Take the limit in which infinitely many gratings → empty space Do nuclei really behave classically ? Do nuclei really behave classically ? Do nuclei really behave classically ? Do nuclei really behave classically ? Example: proton transfer in malonaldehyde only one quantum proton Tuckerman and Marx, Phys. Rev. Lett. (2001) Example: proton transfer in water + + Classical H5O2 Quantum H5O2 Marx et al. Nature (1997) Proton inWater-Hydroxyl (Ice) Overlayers on Metal Surfaces T = 160 K Li et al., Phys. Rev. Lett. (2010) Proton inWater-Hydroxyl (Ice) Overlayers on Metal Surfaces Li et al., Phys. Rev. Lett. (2010) The density matrix: definition Ensemble of states: Ensemble average: Let©s define: ρ is hermitian: real eigenvalues The density matrix: time evolution and equilibrium At equilibrium: ρ can be expressed as pure function of H and diagonalized simultaneously with H The density matrix: canonical ensemble Canonical ensemble: Canonical density matrix: Path integral formulation One particle one dimension: K and Φ do not commute, thus, Trotter decomposition: Canonical density matrix: Path integral formulation Evaluation of: Canonical density matrix: Path integral formulation Matrix elements of in space coordinates: acts on the eigenstates from the left: Canonical density matrix: Path integral formulation Where it was used: It can be Monte Carlo sampled, but for MD we need momenta! Path integral isomorphism Introducing a chain frequency and an effective potential: Path integral molecular dynamics Fictitious momenta: introducing a set of P Gaussian integrals : convenient parameter Gaussian integrals known: (adjust prefactor ) Does it work? i. change of integration variables: uncouple harmonic term; ii. multiple time scale numerical integration algorithm; inherent time scales of the harmonic and external potential terms treated with appropriate time steps; iii. a highly efficient canonical MD method that rigorously generates a canonical phase space distribution (thermostat). Many body path integral ● Each bead evolves at ● Mapping converges to temperature P´T give the exact quantum ● P determined by how mechanical result as P is ªquantumº the problem is increased Path integrals: expectation values eigenvalue of O acting on a coordinate eigenstate Same formula, but more efficient to evaluate: Path integrals: expectation values Discretized (and finite) evaluation: So that: Estimators for thermodynamic quantities (derivatives of Z): Alternative / complementary approach: Colored thermostats Stochastic velocity rescaling thermostat Langevin dynamics Langevin dynamics Colored noise thermostat Example: liquid water classical Colored thermostat PIMD .