06 # returns [r,v,F,V] after step

07 v=v+0.5*delt*invmass*F

08 r=r+v*delt

09 FV=force(r)

10 v=v+0.5*delt*invmass*FV[0]

11 return [r,v,FV[0],FV[1]]

Comments

As mentioned in the Preface (page xiii), it is assumed that scipy has been imported. The initial

values of r, v, F, V are valid at the time before the step, and normally available from the output of

the previous step. To start the run, the routine force(r) must have been called once to initiate

F . The returned values are valid at the end of the step. The arguments are not modi¬ed in place.

1.2 A modeling hierarchy

The behavior of a system of particles is in principle described by the rules of

relativistic quantum mechanics. This is “ within the limitation of our system

choices “ the highest level of description. We shall call this level 1. All other

levels of description, such as considering atoms and molecules instead of

nuclei and electrons, classical dynamics instead of quantum dynamics, or

continuous media instead of systems of particles, represent approximations

to level 1. These approximations can be ordered in a hierarchical sense from

¬ne atomic detail to coarse macroscopic behavior. Every lower level loses

detail and loses applicability or accuracy for a certain class of systems and

questions, but gains applicability or e¬ciency for another class of systems

and questions. The following scheme lists several levels in this hierarchy.

LEVEL 1 relativistic quantum dynamics

System Rules

Atomic nuclei (mass, charge, spin), Relativistic time-dependent quan-

electrons (mass, charge, spin), pho- tum mechanics; Dirac™s equation;

tons (frequency) (quantum) electrodynamics

e

No Go

Approximation e

e

Electrons close to heavy nuc-

Particle velocities small comp-

¡

lei; hot plasmas

ared to velocity of light

¡

e ¡

e¡ ¡

e¡

LEVEL 2 quantum dynamics

System Rules

Atomic nuclei, electrons, photons Non-relativistic time-dependent

Schr¨dinger

o equation; time-

independent Schr¨dinger equation;

o

Maxwell equations

10 Introduction

No Go

e

Approximation Electron dynamics (e.g., in

e

semiconductors); fast elec-

Born“Oppenheimer approx.: e

tron transfer processes; dy-

electrons move much faster ¡

¡

e ¡ namic behavior of excited

than nuclei

e¡ ¡

states

e¡

LEVEL 3 atomic quantum dynamics

System Rules

Atoms, ions, molecules, (photons) Atoms move in e¬ective potential

due to electrons; atoms may be-

have according to time-dependent

Schr¨dinger equation

o

No Go e

Proton transfer; hydrogen e

Approximation e

and helium at low tempera-

Atomic motion is classical ¡

tures; fast reactions and high-

¡

e ¡

e¡ frequency motions

¡

e¡

LEVEL 4 molecular dynamics

System Rules

Condensed matter: (macro)molec- Classical mechanics (Newton™s equa-

ules, ¬‚uids, solutions, liquid crystals, tions); statistical mechanics; molec-

fast reactions ular dynamics

e

No Go e

Approximation

e

Details of fast dynamics,

Reduce number of degrees of

¡

transport properties

freedom

¡

e ¡

e¡ ¡

e¡

LEVEL 5 generalized langevin dynamics on reduced system

System Rules

Condensed matter: large molecu- Superatoms, reaction coordinates;

lar aggregates, polymers, defects in averaging over local equilibrium,

solids, slow reactions constraint dynamics, free energies

and potentials of mean force.

e

Approximation No Go e

Neglect time correlation e

Correlations in motion, short-

and/or spatial correlation in ¡

time accuracy

¡

e ¡

¬‚uctuations

e¡ ¡

e¡

1.2 A modeling hierarchy 11

LEVEL 6 simple langevin dynamics

System Rules

“Slow” dynamic (non-equilibrium) Accelerations given by systematic

processes and reactions force, friction, and noise; Fokker“

Planck equations

e

Approximation e

No Go e

Neglect inertial terms: coarse