1. Advanced methods of molecular dynamics 1.Monte Carlo methods 2.Free energy calculations 3.Ab initio molecular dynamics 4.Quantum molecular dynamics III 5.Trajectory analysis

2. Computational costs N degrees of freedom Classical trajectory (1-dimensional object) M grid points (or basis functions) for each degree of freedom M N (exponential) scaling Quantum wave function: (N-dim.object) t 1N1N Compare with...

3. Time-dependent Schrodinger equation: Exact vs approximate solution Numerically exactly for <4 atoms (up to 6 degrees of freedom) Larger systems: APROXIMATIONS - self-consistent field methods or - semiclassical and quasiclassical methods

4. Self-consistent field method Intermode couplings in the self-consistent field approximation - time-dependence OF efective single mode Hamiltonians variationally best one mode approximation Separable approximation:  (q 1,...,q N,t) = e i  (t)  i  i (q i,t) ih  i (q i,t)/  t =h i (t)  i (q i,t) “separate” Schrödinger equation for each mode h i (t) = T i + V i (q i,t) V i (q i,t) =

5. Classical separable potentials Instead of: V i (q i,t) = Averaging over auxilliary classical trajectories: V i CSP (q i,t) =  j V(q j 1,..., q j i-1, q j i,q j i+1,...,q j N )  j Replacing (N-1)dimensional integration by summing Over a set of 100-1000 trajectories - computationally more efficient: instead of ~10 up to ~1000 atoms

6. Configurational interaction and multiconfigurational methods Wave function in the form of a sum of products:  (q 1,...,q N,t) =  j c j (t)  i  ji (q i,t) Application of time dependent variational principle Configurational interaction: varying only coefs. c j (t) Multiconfigurational methods:varying c j (t) and  ji (q i,t)

7. Semiclassical methods Expansion of the evolution operator U=e -iH  t/ћ with h first “quantum” term (containing the Planck constant) Is proportional to  3 V/  x 3 Dynamics on a constant, linear, or quadratic potential is “classical” Most interesting: quadratic potential - harmonic oscillator Solution - general Gaussian:  (x,t) = exp{(i/ћ)[a t (x-x t ) 2 +p t (x-x t )+c t ]}

8. Equations of motion for a Gaussian dx t /dt=p t /m dp t /dt=-dV(x t )/dx Classical Newton equations for time evolution of the mean position of the Gaussian and its mean momentum da t /dt=-2a t 2 /m-d 2 V(x t )/dx 2 /2 dc t /dt = iћa t /m + p t 2 /2m - V(x t ) “Non-classical” equations for time evolution of the width and phase of the Gaussian

9. In a quadratic potential A Gaussian remains a Gaussian Position, momentum, width, and phase of the Gaussian changes in time: B. Thaller, University of Graz

10. Quasiclassical methods Wigner transform: F(q,p,t) = (1/  ћ)  dx e -2ipx/ћ  * (q-x,t)  (q+x,t) Classical phase variables q, p Equation of motion:  F/  t=-p/m   F/  q+  V/  q   F/  p+O[ћ 2  3 V/  q 3   3 F/  p 3 ] Classical equations of motion 1. “Wigner” mapping of the initial wave function onto a distribution of classical initial conditions q i,p i. 2. Propagation of a set of classical rajectories.

11. Wavepacket analysis -“by naked eye” - amplitude and phase. - Calculation of the autocorrelation function: C(t) = Direct connection to spectroscopy: I(  ) ~   2Re  C(t) e i(E + ћ  ) t dt Absorption spectrum as a Fourier transform of the autocorrelation function.  0

12. Quantum dynamics: Summary Where? - Quantum effects not only for electrons but also for the nuclei - Low temperatures, light atoms (H, He,...) What? -Zero point vibtaions, tunneling, resonance energy transfer - non-adiabatic interactions with electrons - spectroskopy How? -time-dependent vs time-independent solution of the Schrodinger equation -- numerically exact solution for small systems -- approximate methods for larger system