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Table of contents
1 Introduction
2 State of the art
2.1 Formalism of path integrals
2.1.1 Path integrals and the canonical density matrix
2.1.2 Polymer isomorphism and PIMD
2.1.3 Time-independent equilibrium properties in the canonical ensemble from path integral formulation
2.2 Dynamical properties from quantum correlation functions
2.3 Empirical quasi-classical methods: RPMD and CMD
2.3.1 Outline of RPMD
2.3.2 Outline of CMD
2.4 Linearized methods
2.4.1 The Linearized Path Integral (LPI) representation of quantum correlation functions
2.4.2 The FK-LPI method
2.4.3 The Linearized Semiclassical Initial Value Representation (LSC-IVR) method with a Local Gaussian Approximation (LGA)
2.5 Conclusion
3 Wigner densities with the Phase Integration Method
3.1 Wigner densities via the phase integration method (PIM)
3.1.1 The PIM expression for the thermal Wigner density
3.1.2 Edgeworth expansion for the Wigner density
3.2 Structure of the PIM algorithm
3.2.1 Structure of the algorithm
3.2.2 Definition and evaluation of the numerical estimators
3.3 Results
3.3.1 Harmonic oscillator
3.3.2 Morse potential
3.3.3 Proton transfer model
3.4 Conclusions
4 Application of PIM to the Infrared spectroscopy
4.1 Symmetrised correlation function for Infrared spectroscopy
4.2 Kubo correlation functions for Infrared spectroscopy
4.2.1 Operators linear in positions
4.2.2 Operators linear in momentum
4.2.3 Application to the Infrared spectroscopy
4.3 Conclusions
5 Application of PIM to the calculation of rate constants
5.1 General expression for chemical rate constant
5.2 PIM Flux-side correlation function
5.3 Free energy calculations
5.4 Results
5.4.1 Free energy profiles
5.4.2 Rate constants
5.5 Conclusion
6 Conclusion
A Modification of the algorithm for the multidimensional case
B Monte Carlo sampling of the polymer chains
B.1 Staging variables
C Infrared spectroscopy
C.1 Absorption coefficient from the Fermi Golden rule
D Alternative demonstration for the Kubo momentum autocorrelation function
D.1 Notations of Kubo
D.2 Momentum autocorrelation function
D.3 Equipartition of the energy
E Eckart transformation
Bibliography
