Advanced methods ofmolecular dynamics

1. Monte Carlo methods

2. Free energy calculations

3. Ab initio molecular dynamics

4. Quantum molecular dynamics III

5. Trajectory analysis

Computational costsN degrees of freedom

Classical trajectory (1-dimensional object)

M grid points (or basis functions) for each degree of freedom

MN (exponential) scaling

Quantum wave function: (N-dim.object)t

1N

Compare with...

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

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: (q1,...,qN,t) = ei(t) i i(qi,t)

ihi(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-1i+1...N|V(q1,...,qN)|1...i-1i+1...N>

Classical separable potentialsInstead of:Vi (qi,t) = <1...i-1i+1...N|V(q1,...,qN)|1...i-1i+1...N>

Averaging over auxilliary classical trajectories:Vi

CSP (qi,t) = j V(qj1,..., qj

i-1, qji,qj

i+1,...,qjN) j

Replacing (N-1)dimensional integration by summingOver a set of 100-1000 trajectories - computationally more efficient:

instead of ~10 up to ~1000 atoms

Configurational interactionand 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:varying cj(t) and ji(qi,t)

Semiclassical methods

Expansion of the evolution operator U=e-iH t/ћ with h first “quantum” term (containing the Planck constant)Is proportional to 3V/ x3

Dynamics on a constant, linear, or quadratic potential is “classical”

Most interesting: quadratic potential - harmonic oscillatorSolution - general Gaussian:

(x,t) = exp{(i/ћ)[at(x-xt)2+pt(x-xt)+ct]}

Equations of motion for a Gaussian

dxt/dt=pt/m dpt/dt=-dV(xt)/dxClassical 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 + pt

2/2m - V(xt)“Non-classical” equations for time evolution of the width and phase of the Gaussian

In a quadratic potentialA Gaussian remains a Gaussian

Position, momentum, width, and phaseof the Gaussian changes in time:

B. Thaller, University of Graz

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/mF/q+V/qF/p+O[ћ23V/q33F/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.

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 dtAbsorption spectrum as a Fourier transform of the autocorrelation function.

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Quantum dynamics: SummaryWhere?- 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- spectroskopyHow?-time-dependent vs time-independent solution of the Schrodinger equation -- numerically exact solution for small systems -- approximate methods for larger system