Encyclopedia of chemical physics and physical chemistry(1).pdf

-1-

A1.1 The quantum mechanics of atoms and

molecules

John F Stanton

A1.1.1 INTRODUCTION

At the turn of the 19th century, it was generally believed that the great distance between earth and the stars

would forever limit what could be learned about the universe. Apart from their approximate size and distance

from earth, there seemed to be no hope of determining intensive properties of stars, such as temperature and

composition. While this pessimistic attitude may seem quaint from a modern perspective, it should be

remembered that all knowledge gained in these areas has been obtained by exploiting a scientific technique

that did not exist 200 years ago—spectroscopy.

In 1859, Kirchoff made a breakthrough discovery about the nearest star—our sun. It had been known for some

time that a number of narrow dark lines are found when sunlight is bent through a prism. These absences had

been studied systematically by Fraunhofer, who also noted that dark lines can be found in the spectrum of

other stars; furthermore, many of these absences are found at the same wavelengths as those in the solar

spectrum. By burning substances in the laboratory, Kirchoff was able to show that some of the features are

due to the presence of sodium atoms in the solar atmosphere. For the first time, it had been demonstrated that

an element found on our planet is not unique, but exists elsewhere in the universe. Perhaps most important,

the field of modern spectroscopy was born.

Armed with the empirical knowledge that each element in the periodic table has a characteristic spectrum, and

that heating materials to a sufficiently high temperature disrupts all interatomic interactions, Bunsen and

Kirchoff invented the spectroscope, an instrument that atomizes substances in a flame and then records their

emission spectrum. Using this instrument, the elemental composition of several compounds and minerals were

deduced by measuring the wavelength of radiation that they emit. In addition, this new science led to the

discovery of elements, notably caesium and rubidium.

Despite the enormous benefits of the fledgling field of spectroscopy for chemistry, the underlying physical

processes were completely unknown a century ago. It was believed that the characteristic frequencies of

elements were caused by (nebulously defined) vibrations of the atoms, but even a remotely satisfactory

quantitative theory proved to be elusive. In 1885, the Swiss mathematician Balmer noted that wavelengths in

the visible region of the hydrogen atom emission spectrum could be fitted by the empirical equation

(A1.1.1)

where m = 2 and n is an integer. Subsequent study showed that frequencies in other regions of the hydrogen

spectrum could be fitted to this equation by assigning different integer values to m , albeit with a different

value of the constant b . Ritz noted that a simple modification of Balmer’s formula

(A1.1.2)

-2-

succeeds in fitting all the line spectra corresponding to different values of m with only the single constant R H .

Although this formula provides an important clue regarding the underlying processes involved in

spectroscopy, more than two decades passed before a theory of atomic structure succeeded in deriving this

equation from first principles.

The origins of line spectra as well as other unexplained phenomena such as radioactivity and the intensity

profile in the emission spectrum of hot objects eventually led to a realization that the physics of the day was

incomplete. New ideas were clearly needed before a detailed understanding of the submicroscopic world of

atoms and molecules could be gained. At the turn of the 20th century, Planck succeeded in deriving an

equation that gave a correct description of the radiation emitted by an idealized isolated solid (blackbody

radiation). In the derivation, Planck assumed that the energy of electromagnetic radiation emitted by the

vibrating atoms of the solid cannot have just any energy, but must be an integral multiple of h ν , where ν is the

frequency of the radiation and h is now known as Planck’s constant. The resulting formula matched the

experimental blackbody spectrum perfectly.

Another phenomenon that could not be explained by classical physics involved what is now known as the

photoelectric effect. When light impinges on a metal, ionization leading to ejection of electrons happens only

at wavelengths (λ = c /ν, where c is the speed of light) below a certain threshold. At shorter wavelengths

(higher frequency), the kinetic energy of the photoelectrons depends linearly on the frequency of the applied

radiation field and is independent of its intensity. These findings were inconsistent with conventional

electromagnetic theory. A brilliant analysis of this phenomenon by Einstein convincingly demonstrated that

electromagnetic energy is indeed absorbed in bundles, or quanta (now called photons), each with energy h ν

where h is precisely the same quantity that appears in Planck’s formula for the blackbody emission spectrum.

While the revolutionary ideas of Planck and Einstein forged the beginnings of the quantum theory, the physics

governing the structure and properties of atoms and molecules remained unknown. Independent experiments

by Thomson, Weichert and Kaufmann had established that atoms are not the indivisible entities postulated by

Democritus 2000 years ago and assumed in Dalton’s atomic theory. Rather, it had become clear that all atoms

contain identical negative charges called electrons. At first, this was viewed as a rather esoteric feature of

matter, the electron being an entity that ‘would never be of any use to anyone’. With time, however, the

importance of the electron and its role in the structure of atoms came to be understood. Perhaps the most

significant advance was Rutherford’s interpretation of the scattering of alpha particles from a thin gold foil in

terms of atoms containing a very small, dense, positively charged core surrounded by a cloud of electrons.

This picture of atoms is fundamentally correct, and is now learned each year by millions of elementary school

students.

Like the photoelectric effect, the atomic model developed by Rutherford in 1911 is not consistent with the

classical theory of electromagnetism. In the hydrogen atom, the force due to Coulomb attraction between the

nucleus and the electron results in acceleration of the electron (Newton’s first law). Classical electromagnetic

theory mandates that all accelerated bodies bearing charge must emit radiation. Since emission of radiation

necessarily results in a loss of energy, the electron should eventually be captured by the nucleus. But this

catastrophe does not occur. Two years after Rutherford’s gold-foil experiment, the first quantitatively

successful theory of an atom was developed by Bohr. This model was based on a combination of purely

classical ideas, use of Planck’s constant h and the bold assumption that radiative loss of energy does not occur

provided the electron adheres to certain special orbits, or ‘stationary states’. Specifically, electrons that move

in a circular path about the nucleus with a classical angular momentum mvr equal to an integral multiple of

Planck’s constant divided by 2π (a quantity of sufficient general use that it is designated by the simple symbol

) are immune from energy loss in the Bohr model. By simply writing the classical energy of the orbiting

electron in terms of its mass m , velocity v , distance r from the nucleus and charge e ,

-3-

(A1.1.3)

invoking the (again classical) virial theorem that relates the average kinetic (〈 T 〉) and potential (〈 V 〉) energy of

a system governed by a potential that depends on pairwise interactions of the form r k via

(A1.1.4)

and using Bohr’s criterion for stable orbits

(A1.1.5)

it is relatively easy to demonstrate that energies associated with orbits having angular momentum in the

hydrogen atom are given by

(A1.1.6)

with corresponding radii

(A1.1.7)

Bohr further postulated that quantum jumps between the different allowed energy levels are always

accompanied by absorption or emission of a photon, as required by energy conservation, viz.

(A1.1.8)

or perhaps more illustratively

(A1.1.9)

precisely the form of the equation deduced by Ritz. The constant term of equation (A1.1.2) calculated from

Bohr’s equation did not exactly reproduce the experimental value at first. However, this situation was quickly

remedied when it was realized that a proper treatment of the two-particle problem involved use of the reduced

mass of the system µ ≡ mm proton /( m + m proton ), a minor modification that gives striking agreement with

experiment.

-4-

Despite its success in reproducing the hydrogen atom spectrum, the Bohr model of the atom rapidly

encountered difficulties. Advances in the resolution obtained in spectroscopic experiments had shown that the

spectral features of the hydrogen atom are actually composed of several closely spaced lines; these are not

accounted for by quantum jumps between Bohr’s allowed orbits. However, by modifying the Bohr model to

allow for elliptical orbits and to include the special theory of relativity, Sommerfeld was able to account for

some of the fine structure of spectral lines. More serious problems arose when the planetary model was

applied to systems that contained more than one electron. Efforts to calculate the spectrum of helium were

completely unsuccessful, as was a calculation of the spectrum of the hydrogen molecule ion ( ) that used a

generalization of the Bohr model to treat a problem involving two nuclei. This latter work formed the basis of

the PhD thesis of Pauli, who was to become one of the principal players in the development of a more mature

and comprehensive theory of atoms and molecules.

In retrospect, the Bohr model of the hydrogen atom contains several flaws. Perhaps most prominent among

these is that the angular momentum of the hydrogen ground state ( n = 1) given by the model is ; it is now

known that the correct value is zero. Efforts to remedy the Bohr model for its insufficiencies, pursued

doggedly by Sommerfeld and others, were ultimately unsuccessful. This ‘old’ quantum theory was replaced in

the 1920s by a considerably more abstract framework that forms the basis for our current understanding of the

detailed physics governing chemical processes. The modern quantum theory, unlike Bohr’s, does not involve

classical ideas coupled with an ad hoc incorporation of Planck’s quantum hypothesis. It is instead founded

upon a limited number of fundamental principles that cannot be proven, but must be regarded as laws of

nature. While the modern theory of quantum mechanics is exceedingly complex and fraught with certain

philosophical paradoxes (which will not be discussed), it has withstood the test of time; no contradiction

between predictions of the theory and actual atomic or molecular phenomena has ever been observed.

The purpose of this chapter is to provide an introduction to the basic framework of quantum mechanics, with

an emphasis on aspects that are most relevant for the study of atoms and molecules. After summarizing the

basic principles of the subject that represent required knowledge for all students of physical chemistry, the

independent-particle approximation so important in molecular quantum mechanics is introduced. A significant

effort is made to describe this approach in detail and to communicate how it is used as a foundation for

qualitative understanding and as a basis for more accurate treatments. Following this, the basic techniques

used in accurate calculations that go beyond the independent-particle picture (variational method and

perturbation theory) are described, with some attention given to how they are actually used in practical

calculations.

It is clearly impossible to present a comprehensive discussion of quantum mechanics in a chapter of this

length. Instead, one is forced to present cursory overviews of many topics or to limit the scope and provide a

more rigorous treatment of a select group of subjects. The latter alternative has been followed here.

Consequently, many areas of quantum mechanics are largely ignored. For the most part, however, the areas

lightly touched upon or completely absent from this chapter are specifically dealt with elsewhere in the

encyclopedia. Notable among these are the interaction between matter and radiation, spin and magnetism,

techniques of quantum chemistry including the Born–Oppenheimer approximation, the Hartree–Fock method

and electron correlation, scattering theory and the treatment of internal nuclear motion (rotation and vibration)

in molecules.

-5-

A1.1.2 CONCEPTS OF QUANTUM MECHANICS

A1.1.2.1 BEGINNINGS AND FUNDAMENTAL POSTULATES

The modern quantum theory derives from work done independently by Heisenberg and Schrödinger in the

mid-1920s. Superficially, the mathematical formalisms developed by these individuals appear very different;

the quantum mechanics of Heisenberg is based on the properties of matrices, while that of Schrödinger is

founded upon a differential equation that bears similarities to those used in the classical theory of waves.

Schrödinger’s formulation was strongly influenced by the work of de Broglie, who made the revolutionary

hypothesis that entities previously thought to be strictly particle-like (electrons) can exhibit wavelike

behaviour (such as diffraction) with particle ‘wavelength’ and momentum ( p ) related by the equation λ = h / p .

This truly startling premise was subsequently verified independently by Davisson and Germer as well as by

Thomson, who showed that electrons exhibit diffraction patterns when passed through crystals and very small

circular apertures, respectively. Both the treatment of Heisenberg, which did not make use of wave theory

concepts, and that of Schrödinger were successfully applied to the calculation of the hydrogen atom spectrum.

It was ultimately proven by both Pauli and Schrödinger that the ‘matrix mechanics’ of Heisenberg and the

‘wave mechanics’ of Schrödinger are mathematically equivalent. Connections between the two methods were

further clarified by the transformation theory of Dirac and Jordan. The importance of this new quantum theory

was recognized immediately and Heisenberg, Schrödinger and Dirac shared the 1932 Nobel Prize in physics

for their work.

While not unique, the Schrödinger picture of quantum mechanics is the most familiar to chemists principally

because it has proven to be the simplest to use in practical calculations. Hence, the remainder of this section

will focus on the Schrödinger formulation and its associated wavefunctions, operators and eigenvalues.

Moreover, effects associated with the special theory of relativity (which include spin) will be ignored in this

subsection. Treatments of alternative formulations of quantum mechanics and discussions of relativistic

effects can be found in the reading list that accompanies this chapter.

Like the geometry of Euclid and the mechanics of Newton, quantum mechanics is an axiomatic subject. By

making several assertions, or postulates, about the mathematical properties of and physical interpretation

associated with solutions to the Schrödinger equation, the subject of quantum mechanics can be applied to

understand behaviour in atomic and molecular systems. The first of these postulates is:

1. Corresponding to any collection of n particles, there exists a time-dependent function Ψ( q 1 ,

q 2 , . . ., q n ; t ) that comprises all information that can be known about the system. This function

must be continuous and single valued, and have continuous first derivatives at all points where

the classical force has a finite magnitude.

In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle

coordinates r i and velocities d r i /d t at any given time. These evolve according to Newton’s equations of

motion. In principle, one can write down equations involving the state variables and forces acting on the

particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t ′,

provided one knows the precise state of the classical system at time t . In quantum mechanics, the state of the

system at time t is instead described by a well behaved mathematical function of the particle coordinates q i

rather than a simple list of positions and velocities.

-6-

The relationship between this wavefunction (sometimes called state function ) and the location of particles in

the system forms the basis for a second postulate:

2. The product of Ψ ( q 1 , q 2 , . . ., q n ; t ) and its complex conjugate has the following physical

interpretation. The probability of finding the n particles of the system in the regions bounded by

the coordinates

at time t is proportional to the integral

(A1.1.10)

and

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: