IV PREFACE learn something Of these subjects as he studies quantum mechan- ics. ln order that he may d0 and that he may follow the discussions given without danger Of being deflected from the course Of the argument bY inability tO carry through some minor step, we have avoided the temptation t0 condense the various discussions int0 shorter and perhaps more elegant forms. After introductory chapters on classical mechanics and the 01d quantum theory, we have introduced the Schrödinger wave equation and its physical interpretation on a postulatory basiS' and have then gwen in great detail the solution 0f the wave equation for important systems (harmonic oscillator' hydrogen atom) and the discussion 0f the wave functions and their proper- ties, omitting none 0f the mathematical steps except the most elementary. A similarly detailed treatment has been given in the discussion 0f perturbation theory, the variation method' the structure of simple molecules, and, in general' in every important section 0f the b00k. ln order to limit the size 0f the book, we have omitted from discussion St1Ch advanced tOPiCS as transformation theory general quantum mechanics (aside from brief mention in the last chapter) , the Dirac theory 0f the quantization of the electromagnetic field, etc. We have also omitted several subjects which are ordinarily considered as part 0f elementary quantum mechanics, but which are 0f minor importance t0 the chemist, such as the Zeeman effect and magnetic interactions general, the dispersion 0f light and allied phenomena, and rnost 0f the theory 0f aperiodic processes. The authors are severally indebted t0 Professor A. Sommerfeld and Professors . U. Condon and Ⅱ . P. Robertson for their own introduction tO quantum mechanics. The constant adVice of Professor R. C. ToIman is gratefully acknowledged, as well as the aid of Professor P. M. Morse, Dr. L. E. Suttcn, Dr. G. W. Wheland, Dr. L. O. Brockway, Dr. 工 Sherman, Dr. S. Weinbaum, Mrs. Emily Buckingham Wilson, and Mrs. Ava HeIen PauIing. LINUS PAULING. E. BRIGHT WILSON, R. PASADENA, CALIF. , CAMBRIDGE, MASS. , J リ , 1935.

CHAPTER XI PERTURBATION THEORY INVOLVING THE TIME, THE EMISSION AND ABSORPTION OF RADIATION, AND THE RESONANCE PHENOMENON 39. THE TREATMENT OF A TIME-DEPENDENT PERTURBATION BY THE METHOD 0 VARIATION OF CONSTANTS There have been developed two essentially different wave- mechanical perturbation theories. The first Of these' due tO Schrödinger, provides an approximate method Of calculating energy values and wave functions for the stationary states 0f a system under the influence 0f a constant (time-independent) perturbation. We have discussed this theory ⅲ Chapter VI. The second perturbation theory, which we shall ・ treat ⅲ the following paragraphs, deals with the time behavior 0f a system under the influence Of a perturbation ; it permits us t0 discuss such questions as the probability 0f transition 0f the system from one unperturbed stationary state t0 another as the result 0f the perturbation. ()n Section 40 we shall apply the theory t0 the problem 0f the emission and absorption 0f ・ radiation. ) The theory was developed by Dirac. 1 lt is often called the ! ぎ 可 e 。 - ? 血 0 れ一可 - ℃ 0 れ s 厄 the reason for this name will be evident from the following discussion. Let us consider an unperturbed system with wave equation including the time んö 0 2 れ = 0 the normalized general solution 0f which is ( 39 ー 1 ) ( 39-2) 2 P. A. M. DIRAC, proc. 0 Soc. A 112 , 661 ( 1926 ) ; A 114 , 243 ( 1927 ). Less general discussions were also given bY Schrödinger in his fourth 1926 paper and by J. C. Slater' proc. ル観 . Acad. S . 13 , 7 ( 1927 ). 294

T 刃 S UCTURE OF COMPLEX MOLECULES [XlII-46e 374 Approximate discussions Of the interaction 0f a hydrogen atom and hydrogen molecule have been given by Eyring and PolanY1' and a more accurate treatment for some configurations has been carried out by C001idge and James. 2 46e. GeneraIization of the Meth0d of Va1ence-bond Wave Functions. —The procedure which we have described above for discussing the interaction 0f three hydrogen atoms can be generalized t0 provide an analogous treatment 0f a system con- sisting Of many atoms. Many investigators have contributed t0 the attack 0 Ⅱ the problem 0f the electronic structure 0f complex molecules, and several methods 0f approximate treatment have been devised. ln this section we shall outline a method 0f treat- ment (due ⅲ large part t0 Slater) which may be called the 襯ん od 可 0 れ c ← 6 ひ〃 d 0 0 れ s , without giving proofs Of the pertinent theorems. The method is essentially the same that used above for the three-hydrogen-atom problem. Let us now restrict our discussion t0 the singlet states 0f molecules with spin degeneracy only. For a system involving 2 れ electrons and 2 れ stable orbitals (such as the ls orbitals ⅲ 2 れ hydrogen atoms) , there are ( 2 れ ) ! / 2 " 卍 different ways ⅲ which valence bonds can be drawn between the orbitals in pairs. Thus for the case of four orbitals 0 , 6 , c, and d the bonds can be drawn in three ways, namely, 0 d C 0 d ( 2 の ! 6 C 佖 d 6 C independent singlet wave There are, however, only が ( 0 十 1 ) ! functions which can be constructed from the 2 れ orbitals with one electron assigned tO each Örbital (that with neglect Of lt was shown by Slater that wave functions ionic structures). can be set up representing structures 4 , and and that only 2 A. S. COOLIDGE and Ⅱ . M. JAMES, 工 C ん e 襯 . P ん . , 2 , 811 ( 1934 ). Z. /. ん . C ん e 襯 . B12 , 279 ( 1931 ). 1 Ⅱ . EYRING and M. POLANYI, Ⅳ観瀝なル sc ん可れ , analogous t0 that described ⅲ Section 46 し . two 0f them are independent. The situation is very closely 18 , 914 ( 1930

VII-27a] OT 刃ん pp OX 工 4 」 . ア刃 METIIODS 27. OTHER APPROXIMATE METHODS 191 There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers- BriIIouin method, the method of numerical integration, the method Of difference equations, and an approximate second-order perturbation treatment, are discussed ⅲ the following sections. Another method which has been of some importance is based on the polynomial method used ⅲ Section 110 to solve the harmonic oscillator equation. ()nly under special circumstances does the substitution Of a series for lead to a two-term recursion formula for the coeffcients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly ⅲ Section 42C. 27a. A Genera1ized Perturbation Theory. —A method of approximate (and in some cases exact) solution Of the wave equation which has been found useful in many problems was developed by Epstein1 ⅲ 1926 , immediately after the publication of Schrödinger's first papers, and applied by him ⅲ the complete treatment 0f the first-order and second-order Stark effects of the hydrogen atom. The principal feature of the method is the expansion 0f the wave function in terms of a complete set of orthogonal functions which are not necessarily solutions of the wave equation for any unperturbed system related to the system under treatment, nor even necessarily orthogonal functions in the same configuration space. CloseIy related discussions of perturbation problems have since been given by a number of authors, including SIater and Kirkwood2 and Lennard-Jones. 3 ln the following paragraphs we shall first discuss the method ⅲ general, then its application t0 perturbation problems and its relation to ordinary perturbation theory (Chap. (I) , and finally an illustration its application t0 the second-order Stark effect for the normal hydrogen atom. ln applying this method ⅲ the discussion of the wave equation ( 27 ー 1 ) 1 P. S. EPSTEIN, んい . e 既 28 , 695 ( 1926 ). 3 工 . LENNARD-JONES, Proc. 0 . Soc. A 129 , 598 ( 1930 ). 2 工 C. SLATER and 工 G. KIRKWOOD, んい . . 37 , 682 ( 1931 ).

XIV-49] STA TISTICAL Q U Ⅳ U 外耄刃 c Ⅳ / 395 have nOt treated the spin moment vector of the electrons, which combines with the angular momentum vectors and K in various ways t0 form resultants ; the details of this can be found ⅲ the treatises 0 Ⅱ molecular spectroscopy listed at the end of Chapter X. Let us now for simplicity consider transitions among 1 ) states, assuming that the nuclei have Ⅱ 0 spins, and that the existent complete wave functions are symmetric in the nuclei ()s for helium). The allowed rotational states are then those with K even for 1 ) + and 1 ) 「 , and those with K odd for 1 ) 一 and 1 ) + and the transitions allowed by 1 and 3 are the following : 1 ) + K = 0 4 1 ) + K = 1 ) + K = 0 1 ) ー K = 0 1 ) ー K = 1 ) + K = 1 ) ー K = 1 ) 一 K = 0 49. STATISTICAL QUANTUM MECHANICS. SYSTEMS IN THERMODYNAMIC EQUILIBRIÜM The subject of statistical mechanics is a branch of mechanics which has been found very useful ⅲ the discussion of the proper- ties of complicated systems, such as a gas. ln the following sections we shall give a brief discussion of the fundamental theorem Of statistical quantum mechanics (Sec. 49 の , its applica- tion t0 a simple system (Sec. 49 の , the BoItzmann distribution law (Sec. 49C ) , Fermi-Dirac and Bose-Einstein statistics (See. 49 の , the rotational and vibrational energy of molecules (Sec. 49 の , and the dielectric constant of a diatomic dipole gas (Sec. 49 / ). The discussion in these sections is mainly descriptive and elementary ; we have made Ⅱ 0 effort to carry through the diffcult derivations or to enter int0 the refined arguments needed ⅲ 2 etc. 1 3 0 木ーー↓ 2 4 木ーー↓ 4 etc. ワひ个ーー↓ 00 00 etc. etc. 2 4

XII 42 al 〃霽〃 D OGE ノ 0 ん刃 CU ん刃 -70 Ⅳ 42. THE HYDROGEN MOLECULE-ION 327 The simplest of all molecules is the hydrogen molecule-ion, Ⅱオ , composed 0f two hydrogen nuclei and one electron. This mole- cule was one of the stumbling blocks for the 01d quantum theory, for, like the helium atom, it permitted the treatment to be carried 。 hrough ()y Pauli1 and Niessen2) to give results ⅲ disagreement with experiment. lt was accordingly very satisfying that within a year after the development 0f wave mechanics a discussion Of the normal state of the hydrogen molecule-ion ⅲ complete agreement with expenment was carried out by Burrau by numerical integration of the wave equation. This treatment, together with somewhat more refined treatments due to Hylleraas 0 rAB alone ⅲ the field of two stationary nuclei. Using the symbols equation is the solution of the wave equation for the electron Section 34 , the first step ⅲ the treatment of the complete wave 42a. A Very Simple Discussion. 3—Following the discussion of for the sake of the ease with which they can be interpreted. less accurate methods are described ⅲ sections 420 and 426 , and Jaffé, is described ⅲ section 42C. Somewhat simpler and FIG. 42ー1. —Coordinates used for the hydrogen molecule-ion. 0f Figure 42 ー 1 , the electronic wave equation is 8 27 〃 0 2 ル十生十一 ▽早十 ん 2 ( 42 ー 1 ) ⅲ which ▽ 2 refers to the three coordinates of the electron and 襯 0 is the mass of the electron. 4 1 W. PAULI, スル d. p ん . 68 , 177 ( 1922 ). 2 K. F. NIESSEN, Dissertation, Utrecht, 1922. 3 L. PAULING, C 襯 . 究側 . 5 , 173 ( 1928 ). 4 We have included the mutual energy of the two nuclei e2 / ″スぉⅲ this equation. This is not necessary, masmu ch as the term appears unchanged ⅲ the final expression for Ⅳ , and the same result would be obtained by omitting it in this equation and adding it later.

XI-40a] EMISSION ル D ユ BSO ん 2T70 Ⅳ 0 ス D ー TION 299 instead, we assign tO the system a new wave function compatible with our new knowledge Of the result of the expenment. A more de tailed discussion of these points will be given ⅲ Chapter XV. Equation 3g10 shows that ⅲ case だ is small the probability of finding the system ⅲ the stationary state 襯 as a result Of transi- tion from the original state is am 佖 4 ド ん 2 こー ( 39 ー 12 ) being thus proportional t0 the square of the time だ rather than to the first power as might have been expected. ln most cases the nature Of the system is such that experiments can be designed tO measure not the probability 0f transition tO a single state but rather the integrated probability of transition to a group of adj acent states ; it is found 0 Ⅱ carrymg out the solution Of the fundamental equations 39 ー 6 and subsequent integration that for small values Of だ the integrated probability Of transition is pro- portional tO the first power 0f the time だ . An example Of a calculation of a related type will be given ⅲ Section 40 し . 40. T EMISSION AND ABSORPTION OF RADIATION lnasmuch as a thoroughly satisfactory quantum-mechanical theory Of systems containing radiation as well as matter has not yet been developed, we must base our discussion Of the emission and absorption Of radiation by atoms and molecules 0 Ⅱ an approximate method Of treatment, drawing upon classical electro- magnetic theory for aid. The most satisfactory treatment of this type is that of Dirac, 1 which leads directly to the formulas for spontaneous emission as well as absorption and induced emission Of radiation. Because 0f the complexity 0f this theory, however, we shall give a simpler one, ⅲ which only absorption and induced emission are treated, prefacing this by a general discussion Of the Einstein coeffcients Of emission and absorption 0f radiation ⅲ order tO show the relation that spontaneous emission bears tO the 0ther two phenomena. 40a. The Einstein Transition Probabi1ities. —According to classical electromagnetic theory, a system of accelerated electri- cally charged particles emits radiant energy. ln a bath of 6. A. M. DIRAC, proc. 0 . Soc. A112 , 661 ( 1926 ) ; A114 , 243 ( 1927 ) ; J. C. SLATER, Proc. Ⅳ観 . cad. Sci. 13 , 7 ( 1927 ).

192 T ″刃 VARIA 770 Ⅳ METHOD [VII-27a in which 工 is used t0 represent all Of the independent variables for the system, we express (@) ⅲ terms of certain functions 。 ( の , writing ( 27 ー 2 ) we assume that they satisfy the normalization and orthogonality figuration space as that for the system under discussion. lnstead' necessary, however, that they be orthogonal ⅲ the same con- complete set 0f orthogonal functions Of the variables 写 it is not The functions Fn@) are conveniently taken as the members 0f a conditions with 1 for 襯 = れ , 0 for 襯 / ( 27 ー 3 ) ⅲ which p ( の may be different from the volume element corresponding t0 the wave equation 27 ー 1. (@) is called the we / åct 催 1 for the functions 。 ( ェ ). On substituting the expression 27 ー 2 ⅲ Equation 27 ー 1 , we 0btain ( 277 ) which on multiplication by Fr*n(c)p@)dr and integration becomes ル 6 。 n) = 0 , 襯 = 1 , 2 , ⅲ which ( 27 ー 5 ) ( 27 ー 6 ) 1 ln case that the fun ctions (@) satisfy the differential equation ー〆ェ十 xp@)F = 0 , BerIin, 1931. D. Hilbert, " Methoden der mathematischen Physik"' Julius Springer' equations Of the Sturm-Liouville type see' for example' R. Courant and factor (@). For a discussion 0f this point and other properties Of differential a complete set 0f functions which are 0 れ hogonal with respect tO the weight in which 入 is the characteristic-value parameter' they are known tO form

CHAPTER XV GENERAL THEORY 0 QUANTUM MECHANICS The branch of quantum mechanics to which we have devoted our attention ⅲ the preceding chapters, based 0 Ⅱ the Schrödinger wave equation, can be applied ⅲ the discussion 0f most questions WhiCh arise in physics chemistry. It is sometimes conven— ient, however, to use somewhat different mathematical methods ・ and, moreover, it has been found that a thoroughly satisfactory general theory 0f quantum mechanics and its physical inter- pretation require that a considerable extension 0f the simple theory be made. ln the following sections we shall give a brief discussion 0f matrix mechanics (Sec. 51 ) , the properties of angular momentum (Sec. 52 ) , the uncertainty principle (Sec. 53 ) , and transformation theory (Sec. 54 ). 51. MATRIX MECHANICS ln the first paper written 0 Ⅱ the quantum mechanicsl Heisen- berg formulated and successfully attacked the problem of calcu- lating values 0f the frequencies and intensities of the spectral lines which a system could emit or absorb ; that is, of the energy levels and the electric-moment integrals which we have been discussing. He did use wave functions wave equations, however, but instead developed a formal mathematical method for calculating values of these quantities. The mathematical method is one with which most chemists and physicists are not familiar ()r were not, ten years ago), some of the operations involved being surprisingly different from those of ordinary algebra. Heisenberg invented the new type of algebra as he needed it ; it was immediately pointed out by Born and Jordan,2 however, that in his new quantum mechanics Heisenberg was 1 W ・ . HEISENBERG, 2. /. p んい . 33 , 879 ( 1925 ). M. BORN and P. JORDAN, . 34 , 858 ( 1925 ). 6

CO Ⅳ 7 Ⅳ TS 18C. The Solution of the Equation. 18d. The Solution of the 『 Equation. 1 &. The Energy Levels 19. Legendre Functions and Surface Harmonics. 190. The Legendre Functions or Legendre Polynomials. 196. The Associated Legendre Functions. 20. The Laguerre PoIynomials and Associated Laguerre Functions 200. The Laguerre PoIynomials. 206. The Associated Laguerre Polynomials and Functions 21. The Wave Functions for the Hydrogen Atom. 210. Hydrogen-like Wave Functions. 216. The Normal State of the Hydrogen Atom 21C. Discussion of the Hydrogen-Iike RadiaI Wave Functions. SECTION VII PAGE . 118 . 121 . 124 . 125 . 126 . 127 . 129 . 129 . 131 . 132 . 132 . 139 . 142 21a. Discussion of the Dependence of the Wave Functions 0 Ⅱ the AngIes and CHAPTER VI PERTURBATION THEORY 22. Expansions in Series of Orthogonal Functions. 23. First-order Perturbation Theory for a Non-degenerate Level 230. A SimpIe Example : The Perturbed Harmonic ()scillator. 236. An ExampIe : The Normal Helium Atom. 24. First-order Perturbation Theory for a Degenerate Level 146 . 151 . 156 . 1 . 162 . 165 240. An ExampIe : Application of a Perturbation to a Hydrogen Atom . 25. Second-order Perturbation Theory. 250. An Example : The Stark Effect of the PIane Rotator CHAPTER VII . 172 176 177 THE VARIATION METH()D AND OTHER APPROXIMATE METHODS 26. The Variation Method. 26 佖 . The VariationaI lntegral and its Properties. 266. An ExampIe : The Normal State of the Helium Atom 26C. Application of the Variation Method to ()ther states 26a. Linear Variation Functions. 26e. A More GeneraI Variation Method 27. Other Approximate Methods 27 広 A GeneraIized Perturbation Theory. 276. The Wentzel-Kramers-Brillouin Method. 27C. Numerical lntegration. 27d. Approximation by the Use of Difference Equations 27e. An Approximate Second-order Perturbation Treatment . 1 . 1 . 184 . 186 . 186 . 189 . 191 . 191 . 198 201 . 202 4