Advanced methods of molecular dynamics

1. Advanced methods of molecular dynamics • Monte Carlo methods • Free energy calculations • Ab initio molecular dynamics • Quantum molecular dynamics III • Trajectory analysis

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

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 Separable approximation: (q1,...,qN,t) = ei(t) i i(qi,t) ihi(qi,t)/t =hi(t)i(qi,t) “separate” Schrödinger equation for each mode hi(t) = Ti + Vi(qi,t) Vi(qi,t) = <1...i-1i+1...N|V(q1,...,qN)|1...i-1i+1...N> Intermode couplings in the self-consistent field approximation - time-dependence OF efective single mode Hamiltonians variationally best one mode approximation

5. Classical separable potentials Instead of: Vi(qi,t) = <1...i-1i+1...N|V(q1,...,qN)|1...i-1i+1...N> Averaging over auxilliary classical trajectories: ViCSP(qi,t) = jV(qj1,..., qji-1, qji,qji+1,...,qjN) j Replacing (N-1)dimensional integrationby 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: (q1,...,qN,t) = j cj(t)i ji(qi,t) Application of time dependent variational principle Configurational interaction: varying only coefs. cj(t) Multiconfigurational methods:varyingcj(t) and ji(qi,t)

7. Semiclassical methods Expansionof the evolution operator U=e-iHt/ћwith h first “quantum” term (containing the Planck constant) Is proportional to 3V/ x3 Dynamicson aconstant, linear, orquadratic potentialis “classical” Most interesting: quadratic potential - harmonic oscillator Solution - general Gaussian: (x,t) = exp{(i/ћ)[at(x-xt)2+pt(x-xt)+ct]}

8. Equations of motion for a Gaussian dxt/dt=pt/m dpt/dt=-dV(xt)/dx Classical Newton equations for time evolution of the mean position of the Gaussian and its mean momentum dat/dt=-2at2/m-d2V(xt)/dx2/2 dct/dt = iћat/m + pt2/2m - V(xt) “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[ћ23V/q33F/p3] Classical equations of motion 1. “Wigner” mapping of the initial wave function onto a distribution of classical initial conditions qi,pi. 2. Propagation of a set of classical rajectories.

11. Wavepacket analysis -“by naked eye” - amplitude and phase. - Calculation of the autocorrelation function: C(t) = <(0)|(t)> Direct connection to spectroscopy: I() ~ 2Re C(t) ei(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