32 M. M. Woolfson and G. J. Pert, An Introduction to Computer Simu-

lation (1999) is not on models but on methods, from solving partial

di¬erential equations to particle simulation, with accessible mathe-

matics.

33 A. R. Leach, Molecular Modelling, Principles and Applications (1996)

aims at the simulation of molecular systems leading up to drug dis-

covery. Starting with quantum chemistry, the book decribes energy

minimization, molecular dynamics and Monte Carlo methods in de-

tail.

34 C. J. Cramer, Essentials of Computational Chemistry (2004) is the

second edition of a detailed textbook of modern computational chem-

istry including quantum methods, simulation, optimization and reac-

tion dynamics.

2

Quantum mechanics: principles and relativistic

e¬ects

Readers who are not sensitive to the beauty of science can skip this en-

tire chapter, as nothing is said that will help substantially to facilitate the

solution of practical problems!

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

2.1 The wave character of particles

Textbooks on quantum mechanics abound, but this is not one of them.

Therefore, an introduction to quantum mechanics is only given here as a

guideline to the approximations that follow. Our intention is neither to be

complete nor to be rigorous. Our aim is to show the beauty and simplicity

of the basic quantum theory; relativistic quantum theory comprises such

19

20 Quantum mechanics: principles and relativistic e¬ects

subtleties as electron spin, spin-orbit and magnetic interactions in a natural

way. For practical reasons we must make approximations, but by descending

down the hierarchy of theoretical models, we unfortunately lose the beauty

of the higher-order theories. Already acquired gems, such as electron spin,

must be re-introduced at the lower level in an ad hoc fashion, thus muting

their brilliance.

Without going into the historical development of quantum mechanics, let

us put two classes of observations at the heart of quantum theory:

• Particles (such as electrons in beams) show di¬raction behavior as if they

are waves. The wavelength » appears to be related to the momentum

p = mv of the particle by » = h/p, where h is Planck™s constant. If we

de¬ne k as the wave vector in the direction of the velocity of the particle

and with absolute value k = 2π/», then

p= k (2.1)

(with = h/2π) is a fundamental relation between the momentum of a

particle and its wave vector.

• Electromagnetic waves (such as monochromatic light) appear to consist

of packages of energy of magnitude hν, where ν is the frequency of the

(monochromatic) wave, or ω, where ω = 2πν is the angular frequency

of the wave. Assuming that particles have a wave character, we may

generalize this to identify the frequency of the wave with the energy of

the particle:

E = ω. (2.2)

Let us further de¬ne a wave function Ψ(r, t) that describes the wave. A

homogeneous plane wave, propagating in the direction of k with a phase

velocity ω/k is described by

Ψ(r, t) = c exp[i(k · r ’ ωt)]

where c is a complex constant, the absolute value of which is the amplitude

of the wave, while its argument de¬nes the phase of the wave. The use of

complex numbers is a matter of convenience (restriction to real numbers

would require two amplitudes, one for the sine and one for the cosine con-

stituents of the wave; restriction to the absolute value would not enable us

to describe interference phenomena). In general, a particle may be described

by a superposition of many (a continuum of) waves of di¬erent wave vector

and frequency:

dωG(k, ω) exp[i(k · r ’ ωt)],

Ψ(r, t) = dk (2.3)

2.1 The wave character of particles 21

where G is a distribution function of the wave amplitude in k, ω space. Here

we recognize that Ψ(r, t) and G(k, ω) are each other™s Fourier transform,

although the sign conventions for the spatial and temporal transforms dif-

fer. (See Chapter 12 for details on Fourier transforms.) Of course, the

transform can also be limited to the spatial variable only, yielding a time-

dependent distribution in k-space (note that in this case we introduce a

factor of (2π)’3/2 for symmetry reasons):

Ψ(r, t) = (2π)’3/2 dk g(k, t) exp[i(k · r)]. (2.4)

The inverse transform is

g(k, t) = (2π)’3/2 dr Ψ(r, t) exp(’ik · r). (2.5)

The next crucial step is one of interpretation: we interpret Ψ— Ψ(r, t) as

the probability density that the particle is at r at time t. Therefore we

require for a particle with continuous existence the probability density to be

normalized at all times:

dr Ψ— Ψ(r, t) = 1, (2.6)

where the integration is over all space. Likewise g — g is the probability den-

sity in k-space; the normalization of g — g is automatically satis¬ed (see Chap-

ter 12):

dk g — g(k, t) = 1 (2.7)

The expectation value, indicated by triangular brackets, of an observable

f (r), which is a function of space only, then is

dr Ψ— Ψ(r, t)f (r)

f (r) (t) = (2.8)

and likewise the expectation value of a function of k only is given by

dk g — g(k, t)f (k).

f (k) (t) = (2.9)

If we apply these equations to de¬ne the expectation values of the variances

of one coordinate x and its conjugate k = kx :

σx = (x ’ x )2 ,

2

(2.10)