Structure of Molecules and the Chemical Bond

Table of contents :
Front Cover
Title Page
PREFACE
CONTENTS
1. THE HYDROGEN ATOM
2. PERIODIC CLASSIFICATION OF THEELEMENTS
3. THE HOMOPOLAR (COVALENT) BOND
4. SATURATION AND DIRECTION OF VALENCY BONDS
5. RESONANCE OF VALENCY STRUCTURES
6. RESONANCE OF COVALENT AND IONIC STATES
7. METHOD OF MOLECULAR ORBITALS
8. SPECTRA OF DIATOMIC MOLECULES
9. VIBRATIONAL FREQUENCIES AND INTER-ATOMIC DISTANCES IN POLYATOMIC MOLECULES
10. DIPOLE MOMENTS
11. BOND ENERGIES
12. INTERMOLECULAR ATTRACTION
13. THE CHEMICAL BOND IN CRYSTALS
14. COMPLEX COMPOUNDS
15. THE STRUCTURE OF THE BORON HYDRIDES
16. THE SOLUTION OF THE THREE ELECTRON PROBLEM USING SLATER'S METHOD
17. THE POLYELECTRON PROBLEM AND RESONANCE ENERGY
18. MATHEMATICAL APPENDIX
AUTHOR INDEX
SUBJECT INDEX
FORMULA INDEX
Back Cvoer

Citation preview

STRUCTURE OF MOLECULES AND THE CHEMICAL BOND

by Y. K SYRKIN and M. E. DYATKINA

T ranslated and revised by M. A. P A R T R ID G E and D. O. JO R D A N

DOVER PUBLICATIONS, INC., NEW YORK

Published in Canada by General Publishing Com­ pany, Ltd., 30 Lesmill Road, Don Mills, Toronto. Ontario. Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London W. C. 2.

This Dover edition, first published in 1964, is an unabridged and unaltered republication of the Eng­ lish translation first published by Butterworths Scien­ tific Publications in 1950. This edition is published by special arrangement with Butterworth & Co. (Publishers) Limited, 88 Kingsway, London W. C. 2, England.

Library of Congress Catalog Card Number: 64-18371

Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York 14, N. Y.

PREFACE D u r in g the past twenty five years our knowledge of the structure and properties of atoms and molecules has been increased considerably by the progress made in both the theoretical and experimental methods of approach to the problems involved. The introduction of wave mechanics has permitted the development of theories of atomic and molecular structure which have placed on a theoretical basis such fundamental concepts as the spatial configuration of atomic orbitals, the nature and directional charac­ teristics of bonds between atoms in molecules and the mechanism of chemical reactions. However, calculations of bond strengths and distances and of activation energies are possible only with the simplest systems and perhaps the greatest contribution of wave mechanics to chemistry up to the present has been the introduction of ideas which, by general application to the experimental data for complex molecules, has increased considerably our understanding of the structure of such molecules and the reactions which they undergo. Therefore it is necessary’ that in any treatise on molecular structure, the theoretical and the experimental aspects of the problem should be considered side by side. This is what the authors have attempted in this book. In the early chapters the wave mechanics of simple atoms and molecules are discussed ; all the mathematical derivations are collected in the last three chapters of the book apart from the Heitler-London treatment of the homopolar bond, which the authors consider should be familiar to all who desire to understand the present position regarding the theory of the chemical bond and which is presented in a form such that the necessary approximations and simplifications may be appreciated readily. The later chapters are confined to a survey of the experimental data relevant to a study of molecular structure and to the theoretical conclusions which may be drawn therefrom.

The translation is based on the Russian edition published in 1946 and during the course of the translation it became evident that some revision of the original text was desirable. Such revision has been made only where it is apparent that advances in the subject, made since the publication of the original work, have extended or necessitated a modification of the views put forward by the authors. We feel that it is desirable to state which parts of the original text have been so revised. In Chapter 2, the expanded form of the periodic table has been included along with the Mendeleef form and the section on the synthesis of new elements rewritten. In Chapter 5, the section on cyr/fl-octatetrene has been rewritten. Chapter 7, dealing with molecular orbitals, has been almost entirely rewritten and we wish to acknowledge our indebtedness to the published work of Professor C. A. C oulson which was of great assistance to us. In Chapters 8 and 9, recent data collected by Dr A. G. G aydon *has been incorporated and the discussion extended or rewritten where necessary to conform with the new data. The sections on the hydrogen bond in Chapter 12 have been revised and extended. Chapter 15, on the boron hydrides, has been almost v

THE STRUCTURE OF MOLECULES

entirely rewritten and the structures proposed by Dr K. S. P itzer given in addition to those put forward by the authors. In making these modi­ fications, which we hope will have improved the value of the book, we have always attempted to conform as far as possible with the authors’ general method of treatment, as indicated in the original work. Figure 5 was reproduced from Physical Review by permission of Professor H. E. W hite and Figure 25 was reproduced from Quarterly Revirws by permission of Professor C. A. C oulson and of the Council of the Chemical Society, London. We wish to acknowledge our thanks to Mr G. S tanley S mith for the loan of his copy of the Russian text, to the Commonwealth Fund for a Fellowship to one of us (D. O. J.) during part of the tenure of which the translation was largely carried out and to Mr D. R. R ex w o rth y and the editorial staff of Butterworths Scientific Publications for their most helpful cooperation. November 1949

M onica A. P artridge University of Nottingham

D. O . J ordan Princeton University, N.J. University of Nottingham

vi

CONTENTS PAGE

P refa ce 1

v

T h e H ydrogen A tom

Rutherford-Bohr theory of atomic structure Matter as wave motion The wave equation The Schrodinger wave equation Solution of the wave equation for the hydrogen atom 2

P eriodic C lassification

of t h e

E lements

The problem of atoms containing many electrons Pauli’s exclusion principle and electron orbitals Distribution of electrons in atoms Periodicity of properties 3

S aturation

and

D irection

of

65

68 72 74 76

R e so n a n c e o f V a le n c y S t r u c t u r e s

Benzene Conjugated systems Naphthalene Other aromatic hydrocarbons Free radicals 6

43 45 55 61

V alency B onds

Saturation of valencies.Localized bonds Directional covalency Hybridization of bond orbitals a- and 7r- bonds 6

16 17 19 36

T he H omopolar (C ovalent ) B ond

Development of ideas on the chemical bond The hydrogen molecule ion, H£ The hydrogen molecule, H2 The Pauli principle Calculation of the bond energy and interatomic distance for the hydrogen molecule 4

I 3 4 6 9

81 84 86 91 95

R e so n a n c e o f C o v a l e n t a n d Io n ic S t a t e s

The ionic (heteropolar) bond Bonds of intermediate type Valency structures of the elements Alternating polarity of atoms in valence bond structures vii

100 104 107 126

THE STRUCTURE OF MOLECULES PAGE 7

M ethod

of

M olecular O rbitals

Molecular orbitals of diatomic molecules Distribution of electrons in homonuclear diatomic molecules : Distribution of electrons in heteronuclear diatomic molecules The three electron bond Molecular orbitals of somesimple polyatomic molecules Comparison of the electron pair and molecular orbital treatments 8

V ibrational F requencies P olyatomic M olecules

and

I nteratomic

143 146 149 158 D istances

Raman, Mandelstam and Landsberg effect Characteristic frequencies Interatomic distances Covalent radii 10

163 167 179 189 193 197 204 211 214 219 235

B ond E n e r c ie s

Additivity of bond energies in organic compounds Deviations from the additivity rule Bond energies in inorganic compounds 12

in

D ipo le M oments

Dielectric polarization Refraction of inorganic and organic compounds Dipole moments and the ionic character of bonds Dipole moments of hydrocarbon molecules Dipole moments of aliphatic compounds Dipole moments of aromatic compounds Dipole moments and molecular structure 11

141

S p e c t r a o f D ia to m ic M o le c u le s

General characteristics of molecular spectra Potential energy curves Results of spectrographic measurements Ortho and para hydrogen 9

129 133 136 138 139

237 243 255

I nterm olecular A ttraction

Van der WaaPs forces Molecular compounds involving van der WaaPs forces Molecular compounds involving chemical bonds The hydrogen bond Intramolecular hydrogen bonds Internal rotation 13 T he C hemical Bond in C rystals Crystal structure of the elements Molecular crystals Coordination lattices Ionic crystals Interionic distances in crystals viii

262 269 269 273 282 286 294 3°® 3 12 315 319

CONTENTS PAGE

Halogen compounds Hydrides, oxides, hydroxides andthe anions of the oxyacids Silicates Sulphides and some othercompounds 14

C omplex C ompounds

Valency states of atoms involving d electrons Magnetic criterion of bond type Metallic carbonyls Nitrosyl and nitro compounds Cyanides Halogen compounds Ammino compounds and salt hydrates Inner coordination compounds Conclusions 15

324 329 334 339

S tructu re

of t h e

343 349

358 366 369 379 383 389

393

B oron H ydrides

Diborane B2H6 Higher boron hydrides

393 399

16 S o lu tio n o f t h e T h r e e E l e c t r o n P ro b le m u sln o S l a t e r ’s M ethod

Reaction of atoms with molecules Slater’s method Energy of activation 17

T h e P o ly electron P roblem

and

R esonance E neroy

The four electron problem Evaluation of integrals of secular equation Calculation of energies for molecules of butadiene, benzene and fulvene Calculation of resonance energy by the molecular orbital method Comparison of calculated and experimental values of resonance energy Colour of chemical compounds IS

408 409 429 434 436 439 447 449 450

M athem atical A ppen d ix

Solution of the Schrodinger equation for the hydrogen atom Directional covalency Hybrid electron functions Vibrational and rotational states The Clausius-Mosotti equation The Langevin-Debve equation Dipole moments of bonds Deformation, orientation and dispersion effects Energy states of an electron gas A u th or

452 458 461 466 469 472 474 478 482 487

index

S ubject index F ormula index

491

500 IV

1

THE HYDROGEN ATOM RUTHERFORD-BOHR

THEORY

OF

ATOMIC

STRUCTURE

progress in atomic physics that occurred at the end of the nineteenth and at the beginning of the twentieth centuries led to the conclusion that the atom is not an indivisible unit, but consists of a positively charged nucleus, having dimensions of the order of io~12 to io-13 cm, in which is concentrated the greater portion of the mass of the atom, and electrons which are particles possessing unit negative charge e, equal to 4779 X io -10 e.s.u and a mass of 9-1 X io"28 gm. The atoms of the various elements differ from one another in mass, in the magnitude of the nuclear charge and, since the atom is electrically neutral, in the number of electrons. The suggestion that the positive charge on the nucleus, expressed as an integral multiple of the unit charge e, represented the atomic number was first made by V an den B r o ek . The importance of this suggestion was quickly realized and the first experimental evidence in its favour was given by M osely ’s investigation of the x-ray spectra of the elements ; it was directly tested by G eiger and M arsden and later and more accurately, by C had w ick , using R u th erfo r d ’s law for the scattering of a-particles by thin metal foils. The nuclear theoiy of atomic structure, put forward by Rutherford, regarded the electrons as moving in orbits round the nucleus. The dy­ namical theory' of this system was developed by B o h r , who found it necessary to supplement classical mechanics by the quantum mechanics of P lanck . According to classical theory, a system consisting of an electron moving in a circular orbit round a nucleus, to which it is attracted according to Coulomb’s law, would lose energy, with the result that the electron would approach and finally collide with the nucleus. Thus on the basis of classical theory, the Rutherford atom would only be stable for about io-11 seconds, after which time the electron would have fallen into the nucleus. The experimental foundation of the quantum theory of atomic structure as put forward by Bohr, lies in the stability of the atom and in the existence of discrete energy levels and the ability of the atom to absorb and emit energy only in quanta, as demonstrated by the discontinuous nature of atomic spectra and by the critical potential measurements of F ranck and H e r t z . Bohr postulated that the atom could only exist in a limited number of orbits or stationary states, which were defined by the quantum condition that the angular momentum mvr can assume only certain limited values which are given by the expression mvr = nhl2n , . . . (1.1) where m is the mass and v the angular velocity of the electron, r is the radius of the orbit, h is Planck’s constant equal to 6-6 X io-27 erg sec and n is a positive integer. The rotation of the electron in the closed orbits does not involve the loss of energy and for these selected orbits, to account for the stability of the atom, the inward coulombic attractive force is equated to the outward centrifugal force e2/r2 — mv2/r (1.2) From equations 1.1 and 1.2 we have the following expression for the radius of the permitted orbits r = n2h2j47T2e2m 1

T

he

THE STRUCTURE OF MOLECULES

Thus r is proportional to n2 and the radius a of the first orbit, n = i, is given by a = h2l±TT2e2m = 0*528 X io”8 cm . . . . (1-4) The radius of the second orbit, n = 2 will be 4a, the radius of the third orbit, n = 3 will be 9a etc. The potential energy V of the system is the work done in bringing the electron from infinity to a distance r from the nucleus, thus

The kinetic energy T of the atom will be T = mv2j2 (1.6) which in combination with equation 1.2 gives T = e il2r (1.7) Thus the total energy of the atom is given by E = V + T = — e2/2r = — 27t2exmjn2h2 . . .. ( i .8) On passing from an outer orbit with energy E2 to an inner orbit with energy Ex an electron will emit a quantum of radiation with energy E2 — Ex = hv = 27T2e*m (1/nf - 1/nl)/h2 . . . . (1.9) where v is the frequency of the emitted radiation. We may therefore write 1/A = v = /?H (1 jn\ - l/fi|) . . . .(1.10) where /?H = 2TT2eAmjh2c and is known as the Rydberg constant, c is the velocity and A the wavelength of light, and v is termed the wave number. The success of the Bohr theory lay in the fact that formulae 1.9 and 1.10 were in excellent agreement with experimental data. Thus, for example, the theoretical value of the Rydberg constant is. 109677 *76 cm*1, compared with the experimental value of 109678*18 cm-1. The application of the quantum theory to the hydrogen atom leads to the introduction of the quantum number n, on the value of which the energy of the atom depends. The treatment of Bohr was developed by S ommerfeld to account for the fine structure of the lines in the spectrum of hydrogen and other elements. As a result of this treatment, the electrons were considered to be rotating in elliptical as well as circular orbits, which could be oriented with respect to each other in a magnetic field. In place of the single quantum number used in the Bohr treatment, the atom was defined in terms of three quantum numbers, n, I and m, where n is now termed the principal quantum number, I the azimuthal quantum number which determines the value of the angular momentum, and m defines the orientation of the orbit relative to the direction of a magnetic field. The theory of Bohr and Sommerfeld has been applied extensively to atomic spectra with considerable success, but it has proved of little use when applied to molecules and the problem of the chemical bond ; attempts by P a u l i 1 and N iessen 2 to apply the theory to the simplest molecular system i.e. the hydrogen molecule ion were unsuccessful. This stable molecule, which has a bond energy of 61 keal/gm mol, was shown on the basis of the Bohr-Sommerfcld theory to be unstable. More recently, defects have become apparent in the application of this theory to atomic spectra. 2

THE HYDROGEN ATOM

The weakness of the Bohr treatment lies in the attempt to combine quantum and classical theories and in particular in the attempt to apply the laws obeyed by macroparticles to the very small particles that occur in atoms. Instead the quantum condition of discrete energy levels should be derived directly from a general theory ; such a theory has been produced on the basis of wave and quantum mechanics through the work of de B r o g lie , S c h ro d in g er , H eisenberg and D ir a c . MATTER AS WAVE MOTION

The development of wave mechanics has been made possible through the introduction by de Broglie of a new principle dealing with the wave character of matter. The basis of this principle is the recognition that different interpretations are appropriate to different kinds of measurements ; thus atoms and electrons which have hitherto been regarded as discrete particles are considered to possess a dual character, in the sense that they may possess both corpuscular and wave properties. A duality of a similar kind had been revealed earlier in studies on the propagation of light. The theory of the propagation of light has passed through several stages during its development. Newton first conceived light as consisting of a collection of corpuscles obeying the laws of mechanics, but studies of such phenomena as reflection, dispersion, refraction, interference and diffraction led to rejection of this theory in favour of the electromagnetic wave theory of light. This concept assumes that radiation consists of a wave motion with electric and magnetic vectors at right angles to each other and to the direction of the wave, and that matter contains particles which are electrical in character and in which electrical dipoles are pro­ duced by the passage of the wave. Other phenomena, however, which were first investigated at the beginning of the twentieth century and which were associated with the absorption and emission of light by matter, were found to be incompatible with a wave theory. Examples of such phenomena are the photoelectric and Compton effects which can be explained only by assuming that radiation consists of a stream of particles or light quanta, photons, the energy of each quantum being h v where h is Planck’s constant and v is the frequency of the radiation. A survey of all the available experimental data thus shows that some phenomena, such as diffraction and interference, can only be explained on the basis of the electromagnetic wave theory', whereas other phenomena, the most important being the photoelectric effect, require a corpuscular theory of light. Although this duality was recognized for phenomena associated with the propagation and absorption of light, it was not until de Broglie put forward his views, that electrons and other atomic particles, the corpuscular nature of which was supported by a considerable amount of physical and chemical experimental evidence, were considered also to possess wave properties. De Broglie suggested that with every moving particle there is associated a wave motion of wavelength A, which is related to the mass m and velocity v of the particle by the relation A = hjmv .. . . ( i . 11) Experimental evidence was obtained by D avisson and G ermer in 1927 and later by T homson which supported the de Broglie relationship. These authors showed that crystals diffracted a beam of electrons in exactly the 3

THE STRUCTURE OF MOLECULES Table I.

Comparison of Values for same way as a beam of x-rays is diffracted Interatomic Distances and, furthermore, that a comparison oflattice

distances, calculated from electron diffrac­ tion data assuming equation i.i i with values Substance obtained by x-ray diffraction, showed excel­ X-ray Electron diffraction diffraction lent agreement ( Table /). These data clearly indicate the wave character of electrons as Aluminium 4063 4*035 well as confirm the correctness of the de Broglie 4 06 Gold 3 -9 9 - 4*20 relationship. More recent experiments by Platinum 3^9 3 9 1 E s te rm a n and S t e r n have shown that this Lead 4*92 499 2 85 Iron 2O7 relationship also applies to the diffraction of Silver 4-08 411 a-particles (helium ions, He++) the mass Copper 3 -6 6 3 *6 ° of which is 7,360 times that of the electron. 286 286 Tin Thus it would appear that the de Broglie relationship is applicable to all particles, but it is seen in Table II that the wavelengths for particles with large masses are extremely small and for a particle of mass 1 gm the de Broglie wavelength is 6-6 X io~ 27 cm which would require for its ex­ Table II. Wavelengths Associated with Various Particles perimental determina­ Calculatedfrom the de Broglie Equation tion a diffraction grating with rulings io ~ 27 cm deavH toghe V lojsceity ParticIf I Mass cem c w ecle gm apart, whereas in crys­ mngth tals the distance between 1 9 x icr*" Slowelectron 7-27 the atoms is of the order Slowelectron 9 X IO-i® loo O0727 ctro n a c c e le r a te d b y p o te n ­ of 10'8 cm. It is thus Eletia lof t volt 9 X IO-2' 5 94 * »o* I 22 IO*’ evident that the wave Electronacceleratedbypoten­ 9 X I0"!® 5 04 IO® 1 22 IO~® tia l of to o v o lt properties of large par­ leloajclo ecolervaofeltd by 6 6 x 10-14 6 94 0 143 x IO’1* ticles are never observed. a-P paortic tentia 6 6 > to"14 I*51 10s 656 IO"*J It is therefore neces­ a-Particlefromrodiuni lecityofl icm gfsmecmovingat sary to formulate a Partic 6 6 x io-n 1 1 v e lo 2,500 46 571 IO'12 Tennisball way as to in such conform with the w-ave properties of atomic particles and at the same time to be applicable to large particles, being converted into classical mechanics in the process. Interatomic distance A

X

X

X

X

X

!

6

X

X

the

w ave

e q u a t io n

The'idea of a wave may be illustrated by considering a stretched string or the surface of a liquid : if any part of the string is displaced, then a w ave is propagated along the string and the string vibrates ; similarly if a stone is dropped into w'ater, water waves will travel outwards radially from the centre of the disturbance. All wave motions have tw'o important properties in common : the disturbance travels through the medium without giving to the medium as a w'hole any permanent displacement, and energy is transferred from one point to another. Thus if at a given point there is a fluctuating particle, then this fluctuation is passed on from one particle to another, so producing the wave. In order to study the laws of wfave propagation it is necessary to con­ sider not only the disturbing force, but also the restoring force which tends to return the oscillator to its original position. The simplest kind of wave is that inw'hich the restoring force is directly proportional to the displacement, F = — k*p (1.12) 4

THE HYDROGEN ATOM

and is termed the harmonic wave. In equation 1.12, ip is the displacement and k the proportionality constant which is numerically equal to — F for unit displacement. The negative sign indicates that the restoring force opposes the displacing force. According to Newton’s laws of motion tncPipfdt2 = F ....(1 .1 3 ) where m is the mass of the particle being displaced. It follows from equations 1.12 and 1.13 that m dm dt2 = -k (i-i4) In a harmonic wave the wave profile is a sine or cosine curve, thus we may write ip = a sin tot . . . . (1.15) where to is the angular velocity, t is the time and a is a constant. From equation 1.15 we obtain dip/dt = ato cos tot ....(1 .1 6 ) and d2ip/dt2 = — ato2 sin tot . . . . (1.17) From relationships 1.14, 1.15 and 1.17 it follows that md2ipldt2 — — mato2 sin cot — — kip = — ka sin tot . .. . (1.18) and hence w = (k/my (1.19) When equation 1.19 is substituted in formula 1.15 the expression obtained ip = a sin (k/m) * / . . . . (1.20) must satisfy the condition for harmonic motion. In a similar manner we have the solution for the cosine curve that ip = b cos tot . . . . (1.21) The linear combination of the solutions 1.15 and 1.21 given above will also be a solution of equation 1.14. This is an example of the principle of superposition, which states that when all the relevant equations are linear we may superpose any number of individual solutions to form new functions which are themselves also solutions. The use of this principle is fundamental in the wave mechanical treatment of molecules and it will be discussed later. The linear combination here is ip = a sin wt + b cos tot . . . . (1.22) which, on differentiation with respect to t, gives d2ipjdt2 — — to2 [a sin tot + b cos tot) Substitution for to from equation 1.19 and combining with 1.22 gives md2ipldt2 = — kip . . .. (1.14) This solution may be written in the alternative form ip = A exp (itot) or ip = B exp (— itot) or in the general form ip = A exp (itot) + B exp ( — itot) . . . . (1.23) That this is so may be shown by following the same steps as employed in the solution of equation 1.15. The introduction of this form of solution, in­ volving the unit of imaginaries i = \ / — 1, is in certain circumstances more convenient.

5

THE STRUCTURE OF MOLECULES

In a general harmonic wave the displacement if* is not necessarily zero when t = o, as indicated by equations 1.15 and 1.21. Thus the general solution will be tjt = a cos (wt — ) . . . .(1.24) where is a constant. When t — /a>, cos (a>t — ) will be unity and t/t will have a maximum value equal to a. This maximum value of the dis­ placement is termed the amplitude. Since w is the angular velocity, the frequency v will be given by v = co/27r. The reciprocal of the frequency, r = 1fv is termed the period of the wave. If the velocity of propagation of the wave is c, then a point situated a distance A = c T from the initial displacement will vibrate in the same phase. The distance A is termed the wavelength. M< In order to obtain the equation of wave motion let us consider a lateral vibration with v amplitude a which is transmitted from A in the direction B (Figure /). A point M, situated a distance x from A, will after a period of time A_________ _________ * M ty be at Mj. The distance MMX= ift. If the Figure /. Propagation of a time taken for the wave to travel from A to M is t , then the time of oscillation of the point M is t — t and the equation for the wave becomes i/f = a sin 27t v (t — r) . . . . (1.25) but r = x/Cy hence $ = a sin 2n ( vt — vx/c) = a sin 27t (vt — x/X) . . . . (1.26) which on differentiation gives d2^r/dx2 = — 47t2^/A2 , d20/d/2 = — 47T2 v2ip . . . . (1.27) and *V/*2 = (1/c2) V//2 . . . . (1.28) It is possible to generalize equations 1.27 and 1.28 to deal with plane waves in three dimensions. We then obtain t^20 . .. .( 1. 29) '\>x2 'by1 bz2. and b^ip . .. .( 1. 30) This is the equation of wave motion. Laplacian operator and represents

V 2 (read as ‘ del squared ’) is the

to2 + Sr2 + 1 * 2 THE SCHRODINGER WAVE EQUATION

The wave mechanical treatment of atoms may be carried out in various ways, differing in mathematical complexity, but leading to the same general conclusions. We shall employ the methods of Schrodinger, not because they are superior to the treatments of Heisenberg and Dirac, but because of their relative mathematical simplicity. The basis of wave mechanics is the wave equation of Schrodinger which can be obtained by the combination of the de Broglie relationship A = hjmv . . . . (i.i i) 6

THE HYDROGEN ATOM

with the equation of wave motion 1.30. The velocity v in equation 1.11 is first expressed in terms of the total energy E and the potential energy V Thus the kinetic energy is given by 7"= £ - F = / r o 2/2 ••••(>•30 whence it follows that S' 1

(M| 5

II

---- (1.3a)

which on substitution in relationship 1.11 gives

h V2m (E - V) ••••(>•33) Substitution of this expression for A in the equation of wave motion 1.30 then gives the Schrodinger wave equation

Sx* +

7>2tp

)2\js

S n 2m

c

+

t/\

~W (E - V)'t‘ -

/

0

a

/v

• • • • (>-34)

The steps leading to equation 1.34 must not be regarded as a rigorous derivation of the Schrodinger equation, since we have employed an equation of classical mechanics to obtain a wave mechanical expression from which the quantum condition of discrete energy levels may be devised. In the equation of wave motion 1.30 the quantity \ft referred to the amplitude of a wave ; no such simple idea can be attached to \jt in the Schrodinger wave equation 1.34, however, and before proceeding further we must consider the physical significance of this quantity. The wave equation may be applied to the hydrogen atom by substituting the approxi­ mate value of the potential energy V in equation 1.34 ; the hydrogen atom consists of one electron with a charge —