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.

1 INTRODUCTION TO QUANTUM MECIIANICS ー″んノ盟ん c 住〃 0 れ s C ん e 襯なケ BY LINUS PAULING, PH. D. , Sc. D. Professor 可 C ん e 襯 t , C 記ゾ 0 れ 1 れ就 e 0 / Tec ん叩 4 AND . BRIGHT WILSON, 3 , PH. D. ス 880C 観 e ~ 0 / を 880r 可 C ん e 襯け , ″ a の ard U れを ~ e ア 8 リ INTERNATIONAL STUDENT EDITION MCGRAW-HILL BOOK COMPANY, INC. NEW YORK ST. LOUIS SAN FRANCISCO LONDON MEXICO PANAMA SYDNEY TORONTO KöGAKUSHA COMPANY, LTD. TOKYO

428 G 刃Ⅳ刃んユん 7 ' 〃刃 0 必 OF Q UANTUM MECIIANICS [XV-53 whereas from Equation 52 ー 4 we find that M2 OP ・ ん 2 ö2 4 2 0 曽 2 ' so that Equation 52 ー 13 may be written ⅲ the form ん 2 4T ( 52 ー 15 ) ( 52 ー 16 ) The formal similarity of Equations 52 ー 13 and 52 ー 16 with the wave equation p. = ルれ is quite evident. All three equations consist 0f an operator acting upon the wave function equated with wave function multiplied by the quantized value 0f the physical quantity repre- sented by the operator. Furthermore, the operators Hop , 。 , and M2zop will be found to commute with each other; that is, 〃。 p. (M02p x) = M:p.(IIop. x) , etc. , where X is any function Of , and ? 、 . lt is beyond the scope of this b00k t0 discuss this question more thoroughly, but the considerations which we have given above for this special case can be generalized t0 0ther systems and 0ther setS Of coordinates. Whenever the wave equation separated it will be found that the separated parts can be thrown into the form discussed above, involving the operators 0f several physical quantities. These physical quantities will be constants of the motion for the resulting wave functions, and their operators will commute with each other. 53. THE UNCERTAINTY PRINCIPLE The ″ e な e れれ ce れ〃れ el may be stated ⅲ the following way : ん e 0 ん es 可と 0 dy れ 0 襯を c ク 0 れい工 0 れイ 0 可 0 sys 襯 c 佖れ be 佖 cc ? な観 e 襯 ea & リ d the same 襯 e 0 e CO 襯ね ? 、な 名 ero , ・ otherwise these ? れ eas れ ? ℃〃 ~ e れと S C 佖れ be made 0 れリん 0 れ れ c 阨をれ△工△叨ん偽 e 襯 0 れな de 〃 e れ 0 れ the 2 佖ん e 可 the co 襯厄 r. / れの c 厄ら for 佖 ca れ 0 れ c のリ 0 観 e coor 市れ e 1 W. Ⅱ環 s 、お RG , Z. /. P んい . 43 , 172 ( 1927 ).

M / SC 刃んんュル OUS APPLICA TIONS 400 [XIV-49c —WEIGHTS FOR STATES OF INDIVIDUAL OSCILLATORS IN TABLF, 49 ー 2. COUPLED SYSTEM Weight Probability Pn 0 etC. 286 220 165 120 84 56 35 20 10 4 1 0 . 2 .220 .165 . 120 .084 .056 .035 .020 .010 004 1 1 .81 Total . 1 1 stationary states. 1 The result 0f the treatment is the 召 0 〃 2 襯 0 れれ d ななれⅲ its quantum-mechanical form : 〃 0 〃 the 面 c こ / れ c 0 れ s ( の重 ( の 可 0 system co 襯 0 “ d 可 0 e 襯 6 可 3e0 たれ 0 0 s 佖 , ・ are accessible, 腕 e れ the 60 碗襯 4 可市 s しれ可 0 可 the 20 s , S04 佖 , 0 襯 0 れ 4 its states, 〃 se れ d 6 び the 空 0 襯 襯 6 催。 , なれ 6 リ the e 0 0 ( 49 ー 1 ) れ = Ae 流ん ' 。な the e れ e リ可 the 0 佖をれ its 0 04S states 0 れ d the co s ん as such 0 ? e as 襯 0 ん e 0 1 0 編 3 4 -0 ^ 0 8 0 ( 4g2 ) There is considered t0 be one state for every independent wave function ( の . The exponential factor' called the 襯 0 れれ 0 れ e / 7 、 , is the same as ⅲ the classical B01tzmann distribution law, which differs from Equation 49 ー 1 0 Ⅱ IY ⅲ the way the state 0f the system part is described. The constant ん the お 0 〃 2 襯 0 れれ co れ s 厄厩 , with the value 1.3709 X 10 ー 16 e rg deg¯l. The absolute temperature ア occurring ⅲ Equation 49 ー 1 1 That is, among the stationary states for this part Of the system when. isolated from the other patts.

218 ″ E SP / ルⅣ / 0 刃ん EC ん 0 Ⅳ LVIII-29b two completely independent sets Of wave functions. TO show that Ⅱ 0 perturbation will cause the system ⅲ a state represented by the symmetric wave function Vs t0 change t0 a state repre- sented by the antisymmetric wave function we need 0 Ⅱ ly show that the integral vanishes ( 〃′ being the perturbation function, involving the spin as well as the positional coordinates 0f the electrons) , inasmuch as it is shown ⅲ Chapter XI that the probability 0f transition from one stationary state t0 another as a result 0f a perturbation is determined by this integral. NOW, if the electrons are identi- cal, the expression 〃早 s is a symmetrical function Of the coordi- nates, whereas is antisymmetric ; hence the integrand will change sign on interchanging the coordinates 0f the two electrons' and since the region Of integration is symmetrical in these coordinates, the contribution Of one element Of configuration space is just balanced bY that 0f the element corresponding t0 the interchange 0f the electrons, and the integral vanishes. 1 The question as t0 which types 0f wave functions actually occur in nature can at present answered only tO expenment. SO far all observations which have been made 0 Ⅱ helium atoms have shown them tO be in antisymmetric states. 2 ア e accordingly make the additional postulate that the 砒 , e / c 0 れ〃 se れれ 0 0 ac s 阨可 0 s e 襯 CO れ 0 れ 00 or 襯 0 electrons 襯れ be CO 襯〃ん 0 れ s 襯襯 e なをれ the coord 田 s 可 the e ん c な 0 れ s , ・観な , 0 れを c ん 0 れれ 0 the coor 市れ可の こ 30 e ん c わ℃れ s 襯 c ん 0 れをな s れ . This is the statement 0f the 0 れ e 工 c ん 0 れれ c をんⅲ wave-mechanical language ・ This is a principle 0f the greatest importance. A universe based 0 Ⅱ some other principle, that represented bY wave functions 0f different symmetry character, would be completely different in nature from our own universe. The properties ⅲ particular 0f substances are determined bY this principle, which, for example, restricts the population 0f the K shell of an atom tO two electrons, and thus makes lithium 1 The same conclusion is reached from the following argument : On inter- changing the subscripts 1 and 2 the entire integral is converted intO itself with the negative SIgn, and hence its value must be zero. 2 The states are identified through the splitting due tO spin-orbit inter- actions neglected in our treatment.

XV-531 T 〃 U C 刃ん / Ⅳ T 一Ⅳ CIP. カ刃 429 ク 0 〃 d 襯 0 襯 e れ襯 the れれ ce れ△ 9 △〃な可 the d 催可ル卿れ乞ル de 可厄れ c ん ' s co れ s 厄ん , as な△ ' △ t 和 7 , the e れ e の襯 e. TO prove the first part of this principle, we investigate the conditions under which two dynamical quantities / and 0 can be simultaneously represented by diagonal matrices. Let these matrices be f' and g ′ , xn' being the corresponding representation. The product f ′ g ′ of these two diagonal matrices is found on evaluation t0 be itself a diagonal matrix, its n'th element being the product 0f the n'th diagonal elements ム , and 0 が of the diagonal matrices f' and g ′ . SimiIarIy g'f' is a diagonal matrix, its diagonal elements being identical with those of f'g'. Hence the co 襯襯 0 0f f' and g' vanishes : f g = 0. The value Of the right side of this equation remains zero for any transformation 0f the set of wave functions, and consequently the commutator of f and g vanishes for any set of wave functions ; it is invariant t0 all linear orthogonal transformations. We accordingly state that,in ord / のワ 30 d リれ 0 襯佖 s / 0 d 0 可 0 s ク s 襯 be acc ん襯 60 06 ん観 the same 襯 % 〃肥 co 襯 - 襯阨 r 襯れ 0 れなん , ・観な , the e ? れ観んれ fg ー = 0 り砌ん 0 . ( 53 ー 1 ) A proof 0f the second part of the uncertainty principle åiffcult ; indeed, the statement itself is vague (the exact meaning Df △ / , etc. , not being given). We shall content ourselves with the discussion of a simple case which lends itself to exact treat- ment, namely, the translational motion in one dimension Of a free p article. The wave functions for a free particle with coordinate の are 2 冠、 / 2m ル ( ェーエ 0 ) 2 ル ~ ん (Sec. 13 ) , the positive sign ⅲ the first Ne e exponential corresponding t0 motion ⅲ the direction and the negative sign ⅲ the ー direction. 0 Ⅱ replacing ア by 2 襯 2 をェ ( 工ーエ 0 ) 2T を〃 thiS expression becomes Ne ん 2 襯ん ⅲ which positive and negative values Of the momentum ェ refer t0 motion ⅲ the direction and the ーエ direction, respectively. A single wave function of this type corresponds t0 the physical condition ⅲ which the momentum and the energy are exactly known, that is, t0 a stationary state 0f the system. We have then no knowl- edge Of the position Of the particle, the uncertainty △工ⅲ the

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

IX-30cJ S. T 刃ん ' 7 ' E 7 IE Ⅳ T 0 CO 」イ p. 刃 X TO s 235 number 0f wave functions which must be combined, it is only necessary t0 consider electrons outside of completed shells, because there can be 0 y one set of functions “ ・ for the completed shells. TabIe 30 ー 2 gives the allowed sets of quantum numbers for two equivalent electrons, i. e. , two electrons with the same value of れ and with I ProbIem 3g1. Construct tables similar to Table 30 ー 2 for the configura- tions ? 3 and れ d2. 30C. Factorization and S01ution of the secular Equation. —we have now determined the unperturbed wave functions which must be combined ⅲ order to get the correct zeroth-order wave functions for the atom. The next step is to set up the secular equation for these functions as required by perturbation theory, the form given at the end of section 24 being the most con- 〃 11 ール〃 12 venient. This equation has the form 丑 22 ール ⅲ which ( 30 ー 8 ) ( 3g9 ) れ is an antisymmetric normalized wave function of the form of Equation 30 ー 6 or 30 ー 7 , the functions composing it correspond- ing t0 the nth row of a table such as Table 3g1 or 30 ー 2. 〃 is the true Hamiltonian for the atom, including the interactions 0f the electrons. This equation is of the kth degree, ん being the number of allowed sets of functions 伐・ ・盻 . Thus for the configura- tion 1S22 ん is equal to 6 , as is seen from Table 3g1. However, there is a theorem which greatly simplifies the solution of this equation : the 0 川〃。。な名 0 e 協。 0 d 協。ん砌 e the same 20 ん e 可 ) 襯 . 0 れ d the 覊襯 e 20 ん e 可 ) 襯与 these 40 s 6 0 the sums 可リ 0 れ襯れ必 s 襯 . の記襯 ~ of the / れ c 0 れ s 襯 0 ん 4 。 0 協。 . We shall prove this theorem ⅲ Section 30a ⅲ connection with the evaluation of the integrals 〃。。 , and ⅲ the meantime we shall employ the result to factor the secular equation.

T ″ S UCTURE 0 S ーん E 0 ん刃 CU. ん刃 S [XII-43b 346 was considered by Weinbaum, 1 using an effective nuclear charge Z'e ⅲ all the ls hydrogenlike functions. On varying the param- eters, he found the minimum 0f the energy curve (shown ⅲ 0.77 Å , and to correspond to the Figure 43 ー 4 ) t0 lie at ぉ value 4.00 v. e. for the dissociation energy De 0f the molecule. This is an appreciable improvement' 0f 0.24 v. e. , over the value given by Wang's function. The parameter values minimizing the energy2 were found t0 be -216 絜 = 1.193 and c = 0.256. lt may be 0f interest t0 consider the hydrogen-mole- cule problem from another point 0f view. SO far we have attempted t0 build a wave func- tion for the molecule from atomic orbital functions, a pro- cedure which is justified as a L25 first approximation When 『ス B is large. This procedure, as FIG. 43ー4. —Energy curves for the generalized t0 complex mole- hydrogen molecule ()n units e2 / 2 佖 0 ) : for an extreme molecular—orbital cules, is called the ん 0d 可 funct,ion,• . B, for 0 んれ ce 0 れ d 00 / れ c 0 れ s , the bond wave function; and C, for a valence-bond function with partial ionic name sometimes being used character ( Ⅳれし 0 m ). the restricted sense 0f implying neglect 0f the ionic terms. Another way 0f considering the structure 0f complex molecules, called the 忸 et ん 0d 可襯 0 ん c 碗な , 3 can be applied to the hydrogen molecule ⅲ the following way. Let us consider that for small values 0f 、以 B the interaction of the two electrons with each Other is small compared with their interaction with the two nuclei. Neglecting the term e2 12 in the potential energy, the wave equation separates int0 equa- tions for each electron ⅲ the field 0f the two nuclei' as ⅲ the hydrogen-molecule-ion problem, and the unperturbed wave function for the normal state 0f the molecule is seen tO be the 1 S. WEINBAUM, 工 C ん e 襯 . P んい . 1 , 593 ( 1933 ). 2 Weinbaum also considered a mo re general function wi th different effective nuclear charges for the atomic and the ion ic terms and found that this reduced to 4 13 on variation ・ 3 F. HUND, Z. /. P ん . 51 , 759 ( 1928 ) ; 73 , 1 ( 1931 ) ; etc. ; R. S. MULI. IKEN' P んい . . 32 , 186 , 761 ( 1928 ) ; 41 , 49 ( 1932 ) ; etc. ; M. DUNKEL' み /. ん・ C 襯 . B7, 81 ; 10 , 434 ( 1930 ) ; . HÜCKEL, Z. /. 2 んい . 60 , 423 ( 1930 ) ; etc. ZZ ロ 220 W ー 2.24 -228 -232 ロ 5 L50 1.00

l-ld] Ⅳ刃ルア 0 Ⅳ ' S 刃 QU アー 0 Ⅳ S 0 盟 0 0 Ⅳ 9 ()t is important t0 note that must be expressed as a function 0f the coordinates and their first time-derivatives. ) Since the above derivation could be carried out for any value 0f の there are 3 れ such equations, one for each coordinate の . They are called the 40 0 れ s 可襯 0 0 れ the 秀 00m れ 0 れ / 襯 and are 0f great importance. The metho d by whi ch they were derived shows that they are independent of the coordinate system. We have so far rather lirnited the types of systems considered, but Lagrange's equations are much more general than we have indicated and 6 0 r 夘 choice 可腕 e 女れ c 石れ e I び 0 〃面 0 襯 cal ble 襯 s ca れ treated ん e use. These equations are therefore frequently chosen as the fundamental postulates of classical mechanics instead 0f Newton's laws. ld. An Examp1e : The lsotropic Harmonic OsciIIator 血 P01 Coordinates. —The example which we have treated ⅲ Section 1 佖 can equally well be solved by the use of polar coordinates ら ら and (Fig. 1 ー 1 ). The equations of transformation correspond- ing t0 Equation 1 ー 18 are = rsin+cosp, = r sin Sin , COS . 名 ( 1 ー 3 の With the use of these we find for the kinetic and potential energles Of the isotropic harmomc oscillator the following expressions : T = ー襯@2 十の十を 2 ) = (fr2 十ド 2 十ド s ⅲ 2 2 ) , 2 = 2 籠 2 襯 2 , and 秀 = T ー = ー ( テ 2 十ド十ド sin2 2 ) The equations of motion are 2 襯 2. d öL öL d öL öL öp öL öL ー ( 襯ド sin2 の 0 , ー ( 襯ド ) ー襯ド sin cos 2 = 0 , ( 1 ー 34 ) ( 1 ー 31 ) ( 1 ー 32 ) ( 1 ー 33 ) ー ( 襯一川れ一襯 8 ⅲ 2 ら 2 十 4g2 = 0. ( 1 ー 35 )