243 IX-30d] S. カ T 刃 ' T 記刃 -4 一刃Ⅳ 0 COMPLEX TOMS ( 21 十 1 ) ( I ーの ! ( 21 ′十 1 ) ( ′ー lmfl)! 2(1 十の ! 2 ( ′十 lmfl)! 「 {PPi ・ l(cos の } ( 。去 i 。み記 and ( 30 ー 23 ) ( れ I ; が′ ) = ( 4 2e2 2 た十 1 ・ 2 0 The a's are obtained from the angular parts Of the wave functions, which are the same as for the hydrogen atom (Tables 21 ー 1 and 21 ー 2 , Chap. V). Some of these are given ⅲ TabIe 30 ー 3 , taken TABLE 3g3. —VALUES 0 を ( 襯′襯 : ) ()n cases with two 士 signs, the two can be combined ⅲ any 0f the four possible ways ) ( 3g22 ) 0 4 士 士 士士 士士土士 Electrons 士士 0 0 0 1 0 1 ・ 1 SS 1 1 ー 5 1 1 1 ・ 1 -1 -1 0 0 編 0 0 0 編 0 臨ワ 1 0 臨 01 り 1 0 み 0 0 01 り 0 0 0 1 1 1 ー 5 ー % 5 ー % 5 ー % 9 ー % 9 % 9 1 -1 1 1 -1 1 % 41 ー % 41 % 41 12 イ 41 ー 2 少 41 39441 1 一 1 一 1 1 -1 1
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.
XIV-49 引 87 Ⅵア / S ア / C ん QU Ⅳア U 」一 MECIIANICS 403 pletely antisymmetric ones, or those with the various inter- mediate symmetry characters. lt is only the two extreme types which have been observed ⅲ nature. There are 30 completely symmetric wave functions for れ = 10 ; they are formed from the successive sets ⅲ Table 49 ー 1 by addition, the first being { ( 10.0.0.0.0 ) 十 ( 0.10.0.0.0 ) 十 ( 0.0.10.0.0 ) 十 ( 0.0.0.10.0 ) 十 ( 0.0.0.0.10 ) } and the last being ( 2.2.2.2.2 ). From these we can obtain weights for the successive values, similar t0 those given in TabIe 49 ー 2 ; these weights will not be identical with those of the table, however, and so will correspond to a new statistics. This is very clearly seen for the case that only the completely anti- symmetric functions are accessible. The only wave function with = 10 which is completely antisymmetric is that formed by suitable linear combination of the 120 product func- tions ( 4.3.2.1.0 ) , etc. , marked ⅲ Table 49 ー 1 (the other functions violate PauIi's principle, the quantum numbers not being all different). Hence even at the lowest temperatures only one of the five oscillators could occupy the lowest vibrational state, whereas the BoItzmann distribution law would ⅲ the limit → 0 place all five in this state. / / 0 the co 襯〃んんのな襯襯 c 社怩 / れれ c 0 れ s are accessü)le ん 0 s い襯 composed 可 0 印 e ? 襯しげ可んん rac れ 0 佖 r な , the s いん襯 ar な c イ br 襯 the 襯 D ac な cs ; 1 ヴ 0 れ the CO 襯 ete 襯襯 e な社 , 0 / c 0 れ s 0 acc い s 仂ん , e び co れ / br 襯 the お e - 刃をれ靃れ観な cs. 2 ん e 催襯乞 - の ac 市 s なれⅲ the forms analogous to 1 5 Equations 49 ー 1 , 49 ー 3 , and 49 ー 4 is 1 e 紅十Ⅳ ( 49 ー 6 ) 1 E. FERMI, み /. P ん . 36 , 902 ( 1926 ) ; p. A. M. DIRAC, proc. 記 0 び . soc. A112 , 661 ( 1926 ). This statistics was first developed by Fermi, 0 Ⅱ the basis of the Pauli exclusion prmciple, and was discovered independently by Dirac, usmg antisyrnmetric wave functions. 2 S. N. BOSE, み /. P んい . 26 , 178 ( 1924 ) ; A. EINSTEIN, 催 . p s. ん ad. ル s. p. 261 , 1924 ; p. 3 , 1925. Bose developed this statistics to obtain a formal treatment of a photon gas, and Einstein extended it the case of materi al gases.
XIV-47c] 第Ⅳの刃んル A 石 S 0 C 刃 S 387 47b. Van der Waa1s Forces fo て He1ium. —ln treating the dipole-dipole interaction of two helium atoms, the expression for 〃 ' consists of four terms like that of Equation 47-4 , correspond- ing tO taking the electrons ⅲ pairs (each pair consisting of an electron on one atom and one 0 Ⅱ the other atom). The variation function has the form Hassé1 has considered five variation functions of this form, shown with their results ⅲ Table 47 ー 2. The success of his similar treatment of the polarizability of helium (function 6 of TabIe 2g3 ) makes it probable that the value ー 1.413e2 / 6 for ル″ is not ⅲ error by more than a few per cent. Slater and Kirk- W00d1 obtained values 1.13 , 1.78 , and 1.59 for the coeffcient of ー e2 / 6 by the use of variation functions based on their helium atom functions mentioned ⅲ section 29C. An approxl- mate discussion 0f dipole-quadrupole and quadrupole-quadrupole interactions has been given by Margenau. TABLE 47 ー 2. ーー、 5 、沢 IATION TREATMENT OF VAN D ・ AALS INTERACTION OF 1 TWO HELIUM ATOMS / 0 マ 2 ) A 十 B 2 は十 Br1r2 十 C ドド 十 Br1r2 ー 2 % ( 1 十の e →′ 3 ( 1 十の 刃ール 0 ー 1.079e20g / 6 ー 1 .225 ー 1 .226 ー 1 .280 ー 1 .413 47C. The Estimation of van der Waals Forces from M01ec 猷 P01arizab ilitie s. ーー Lo Ⅱ do Ⅱ 2 h as suggested a rough method of estimating the van der Waals forces between two atoms or mole- cules, based 0 Ⅱ the approximate second-order perturbation treatment Of Section 27e. We obtain by this treatment (see Secs. 27e and 29 の the expression 2 れ e2 名 2 1 oc. れ 2 F. LONDON, 2. /. P ん . 63 , 245 ( 1930 ). ( 47 ー 8 )
228 71 〃刃 SPINNING ELEC 7 0 [VIII-29e field (assumed to lie along the 名 axis) is ( 21 十名 2 ) , 21 and 22 being the 名 coordinates of the two electrons relative to the nucleus. The argument of Section 27C suggests that the variation function be of the form = 0 { 1 十 ( 21 十名 2 ゾ@1 , 41 , 21 , 昀 , 22 ) } , ( 29 ー 11 ) in which 0 is an approximate wave function for zero field. Variation functions of this form ()r approximating (t) have been discussed by Hassé, Atanasoff, and SIater and Kirkwood, 1 whose results are given ⅲ TabIe 29 ー 3. TABLE 29ー3. ー-ー・ VARIATION FUNCTIONS OR THE CALCULATION OF THE POLARIZABILITY 0 を THE 、・ ORMAL HELIUM 、 TOM Experimental value : 夜 = 0.205 ・ 10 ー 24 cm3 1 ヨー r2 00 Variation function e-Z'• { 1 十ス ( 21 十 2 ) } . ー 2 ' を { 1 十は ( 21e ー 2 " 十名 2e ー 2 " つ } 3. @1r2 ) 0 ・ 255e ー幻 8 { 1 十 ( 名 le ー 2 " ア 1 十 22e ー 2 " つ } 4. e ー 2 ' 8 { 1 十 ( 21 十司十 B ( 2 産 1 十名 2 ) } . ー 2 % { 1 十 ( 21 十名 2 ) 十 terms to quartic) } ー 2 ′ 8 ( 1 十の { 1 十 ( 名 1 十 32 ) 十 B ( 名産 1 十 2r2 ) } 7. e ー 2 1 十十ド十 ( 箚十名 2 ) } . 8. e ー 2 ' ・ { 1 十社十ド十い十 Bs)(Z1 十 22 ) 十夜 ( 21 ー 22 ) } 9. e ー 2 ' 8 { 1 十 CI 十 C2 卩十 C3S 十 C4S2 十 C5 2 十い十 Bs)(21 十 22 ) 十夜 ( 名 1 名 2 ) 十 1 十 22 ) } 10. e ー 2 % ( 1 十 cru 十ド ) { 1 十ス ( 21 十名 2 ) 十 B ( 1 十名 2 2 ) } 11. A non-algebraic function. 1 Ⅱ = Ⅱ s き , A = Atanasoff, SK = Slater and Kirkwood. Of these functions, 1 , 2 , 4 , and 5 are based 0 Ⅱ the simple screening-constant function 2 of Table 29 ー 1 ; these give 10W values Of 伐 , the experimental value (from indices of refraction extrapolated tO large wave length Of light and from dielectric 1 Ⅱ . R. HASSE, Proc. Ca 襯い記 ge Phil. Soc. 26 , 2 ( 1930 ) , 27 , 66 ( 1931 ) ; 工 V. ATANASOFF, P んい . 側 . 36 , 1232 ( 1930 ) ; 工 C. SLATER and 工 G. KIRK- WOOD, P んい . e 37 , 682 ( 1931 ). R. eferences 1 . 0.1 測 ・ 10 ー 24cm3 .164 SK .222 SK . 182 . 183 .201 . 127 . 182 . 194 .231 . 210
V-21aJ T ル 4 VE U Ⅳ C ア 70 Ⅳ 0 T ″刃 } ' D OG Ⅳ 4 TO 137 The wave functions O ( の given ⅲ TabIe 21 ー 2 are the asso- ciated Legendre functions 君に (cos ) normalized to unity. The functions 戸 (cos ) as usually written and as defined by Equations 19 ー 1 and 19 ー 7 consist of the term sinlm!t} and the ー 1 polynomial ⅲ cos multiplied by the factor (l 十 ) ! or (l 十ー十 1 ) ! 卩十回卩一 2. 1 + 回 + 1 , (1 一回 2 2 2 2 as 襯十 is even or 0dd. Expressions for additional associated Legendre functions are given in many bOOkS, as, for example, by Byerly. 1 NumericaI tables for the Legendre polynomials are given by Byerly and by Jahnke and Emde. 2 Following Mulliken, we shall occasionally refer to one-electron orbital wave functions such as the hydrogenlike wave functions 0f this chapter as 碗厄な . In accordance With spectroscopic practice, we shall also use the symbols s, 〃 , d, / , ・ to refer t0 states characterized by the values 0 , 1 , 2 , 3 , 4 , respectively, 0f the azimuthal quantum number l, speaking, for example, 0f an s orbital tO mean an orbital with I ln the table of hydrogenlike radial wave functions the poly- Ⅱ 01 れ ial contained in parentheses represents for each function the associated Laguerre polynomial も物 1 ( p ) , as defined by Equations 20 ー 1 and 20 ー 5 , except for the factor which has been combined with the normalizing factor and reduced tO the simplest form. lt is t0 be borne in mind that the variable p is related tO ⅲ different ways for different values Of . The complete wave functions 協。 ( ららの for the first three shells are given ⅲ Table 217. Here for convenience the variable p = 2Z ツれ 00 has been replaced by the new variable び , such that 00 tafeln, ” B. G. Teubner, Leipzig, 1933. 2 W. . BYERLY, 記 . , pp. 278 ー 281 ; JAHNKE and EMDE, "Funktionen- 159 , 198 , Ginn and Company, Boston, 1893. W. . BYERLY, "Fourier's Series and Spherical Harmonics,' pp. 151
238 」一 A Ⅳ Y - ん EC ア 0 Ⅳ TOMS [IX-30c tOtal resultant orbital angular momentum Of the atom, and the letter S for the total spin angular momentum ; see section 296. ) ln the approximationl we are using, the states of an atom may be labeled by giving the configuration and the quantum numbers ん , S, 」一石 = ) こ , and 」一 & = ) 襯 . , the last two having Ⅱ 0 effect 0 Ⅱ the energy. Just as for one electron, the allowed values of 石 are , 秀一 1 , restricted to S, S ー 1 , 一十 1 , ーん ; 」ム is similarly —S 十 1 , ー all of these values 0f 」一 & and 」一 belonging tO the same degenerate energy level and corresponding tO different orientations in space 0f the vectors L and S. We shall now apply these ideas to the solution of the secular equation, taking the configuration 〃 2 as an example. From TabIe 30 ー 2 we see that ″ 11 ール is a linear factor of the equation, since 1 alone has ) ? = 2 and ) 襯 8 = 0. A state with 」一石 = 2 must from the above considerations have 石 > 2. Since 2 is the highest value 0f ユ I 石ⅲ the table, it must correspond t0 石 = 2. Furthermore the state must have S = 0 , because otherwise there would appear entries in the table with ユー石 = 2 and Ms / 0. This same root must appear five times ⅲ the secular equation, corresponding to the degenerate states 秀 = 2 , S = 0 , 」ム = 0 , 」一 = 2 , 1 , 0 , ー 2. From this it is seen that this root (which can be obtained from the linear factors) must occur in two 0f the linear factors い一石 = 2 , ー 2 ; 〕 s = 0 ) , ⅲ two 0f the quadratic factors い一石 = 1 , ー 1 ; ユム = 0 ) , and ⅲ the cubic factor ()L = 0 , ユム = 0 ). The linear factor 〃 22 ール with ML = 1 , Ms = 1 must belong to the level = 1 , S = 1 , because Ⅱ 0 terms with higher values 0f 」石 and Ms appear ⅲ the table except those already accounted for. This level will correspond t0 the nine states with 」一ん = 1 , 0 , ー 1 , and Ms = 1 , ー 1. Six of these are roots of linear factors ()L = 士 1 , 0 , ム = 士 1 ; ん = 0 ; 」ム = 士 1 ) , two of them are roots of the quadratic factors ()L = 士 1 , 〕ム = 0 ) , and one is a root 0f the cubic factor ()L = 0 , Ms = 0 ). Without actually solving the quadratiO equations or evaluating the integrals involved in them, we have determined their roots, since all the roots 0f the quadratics occur also ⅲ linear factors. 1 This approximation, called (LS) or Russell-Saunders coupling, 1S valid for light atoms. Other approximations must be made for heavy atoms ⅲ which the magnetic effects are more important.
XIII-46] SLA 7 ' 刃 ' S TREA TMENT OF CO P ん刃 X MO ん刃 CU. ん刃 S 367 functions are exactly analogous t0 the functionsl used ⅲ Section 300 ⅲ the treatment Of the electronic structure Of atoms. lt may be possible t0 construct for a complex molecule many such functions with nearly the same energy, all Of which would have t0 be considered ⅲ any satisfactory approximate treatment. Thus if we consider one atom to have the configuration 1S22S22 〃 , we must consider the determinantal functions involving all three 2 functions for that atom. A system 0f th.s type, ⅲ which there are a large number 0f available orbitals, is said t0 involve 碗阨 I 虎 e れ e c . ven ⅲ the absence Of orbital degeneracy, the number 0f determinantal functions t0 be considered may be large because 0f the variety 0f ways ⅲ which the spin functions 伐 and 0 can be associated with the orbital functions. This s がれ市れ e c び has been encountered ⅲ the last chapter ; ⅲ the simple treatment 0f the hydrogen molecule we considered the two functions corresponding t0 associating positive spin with the orbital ? and negative spin with uB, and then negative spin with ? and positive spin with ぉ (Sec. 45 ). The four wave functions described ⅲ Section 44 佖 for the helium molecule-ion might be represented by the scheme 0f Table 46 ー 1. The plus TABLE 46 ー 1. —WAVE FUNCTIONS FOR THE HELIUM MOLECULE-ION, HEÉ Function Ⅱ III IV uB 十 十 十 十 % 十 % and minus signs ShOW WhiCh spin function 伐 or 0 iS tO be a,SS(F ciated with the orbital functions ? and uB ()n this case ls func-• tions 0 Ⅱ the atoms and お , respectively) ⅲ building up the determinantal wave functions. Thus row 1 of TabIe 4 1 corresponds t0 the function given ⅲ Equation 44 ー 1. The column labeled ) 襯 . has the same meaning as ⅲ the atomic problem ; namely, it is the sum 0f the z-components 0f the spin angular momentum 0f the electrons (with the factor ん / 2 . Just ⅲ the atomic case, the wave functions which have different 1 ln Section 300 the convention was adopted that the symbol 社。@) should include the spin function a(i) or (@). ln this section we shall not use the convenüon, instead writing the spin function 夜 or 0 explicitly each time.
THE 〃 YD 究 OG 刃Ⅳ」イ 0 ん刃 CU ん刃 XII-43c] 349 orbital treatment is retained also for molecules containing atoms Of larger atomic number remains an open question. SO far we have not considered polarization of one atom by the other ⅲ setting up the variation function. An interesting attempt t0 d0 this was made by Rosen,1 by replacing 1. ( 1 ) in the HeitIer-London-Wang function by 術 . ( 1 ) 十び ( 1 ) (with a similar change ⅲ the other functions), as ⅲ Dickinson's treatment of the hydrogen molecule-ion (which was suggested by Rosen's work). The effective nuclear charges in 術 . and Z"e ⅲ切 were assumed to be related, with z ″ = 2Z ′ . lt was found that this leads to an improvement of 0.26 v. e. ⅲ the value of の。 over Wang's treatment, the minimum in the = 0.77 Å , の。 = 4.02 v. e. , energy corresponding to the values Z ′ = 1.19 , andc = 0.10. A still more general function, obtained by adding ionic terms ()s ⅱ 1 43 ー 13 ) to the Rosen function, was discussed by Weinbaum, who obtainedDe = 4.10 v. e. , 絜 = 1.190 , び = 0.07 , and c = 0.176. The results of the various calculations described ⅲ this section are collected ⅲ TabIe 43 ー 1 , together with the final results of James and Coolidge (see following section). TABLE 43ー1. —R. ESULTS OF APPROXIMATE TREATMENTS OF T 日、・ ORMA ム HYDROGEN MOLECULE De 80 Å 48 cm—l Heitler-London-Sugiura. 3 .14 v. e. 0 . MolecuIar-orbitaI treatment. 3 . 47 0 . 73 Wang. 3 . 76 0 .76 4900 Weinbaum (ionic) . 4 .00 0 .77 47 測 Rosen (polarization) . 4 02 0 . 77 4260 Weinbaum (ionic-polarization) . 4 .10 James-Coolidge. 4.722 0 . 74 Expenment. . 4 .72 0.7395 4317.9 43C. The Treatment of James and Coolidge. —ln none of the variation functions discussed in the preceding section does the interelectronic interaction find suitable expression. A major advance ⅲ the treatment Of the hydrogen molecule was made by James and C001idge2 by the explicit introduction of th€ 1 N. ROSEN, P ん . 側 . 38 , 2099 ( 1931 ). 2 Ⅱ . M. JAMES and A. S ・ C00 DG 工 C . 戸んい . 1 , 825 ( 1933 ). っ 6 ワむ 1 -1 -1 1 1 一 1 1 1 人 1 ・ 1
368 T ″刃 STRUCTURE OF' COMPLEX MO ん刃ん刃 s [X111-46a values Of ) 襯 . d0 not combine with One another, so that we were justified ⅲ Section 44 佖ⅲ considermg only and VII. Problem 46 ー 1. Set up tables similar to Table 46 ー 1 for the hydrogen molecule using the following choices orbital func tions : ( の 1 s orbitals on the two atoms, allowing only one electron ⅲ each. ( の The same orbitals but allowing two electrons to occur ⅲ a single orbital also ; i. e. , allowing ionic functions. (c) The same as ( の with the addition of func- tions 2 ル 0 Ⅱ each atom. ( の The molecular orbital (call it の obtained by the accurate treatment of the normal state of the hydrogen molecule-ion. 46a. Approximate Wave Functions for th e System of Thre e Hydrogen Atoms. —ln the case of three hydrogen atoms we can set up a similar table, restricting ourselves to the three ls func- tions 。 , 協 , and 0 Ⅱ three atoms 0 , わ , and c, respectively, and neglecting ionic structures (Table 4 2 ). TABLE 46 ー 2.- ー - ー、 VAV 刃 FUNCTIONS FOR THE SYSTEM OF THREE HYDROGEN ATOMS 十 十 % 十 III 十 % IV 十 十 VI 十 VII VIII The wave function corresponding to row Ⅱ of Table 46 ー 2 is, for illustration, ( 1 ) 伐 ( 1 ) ( 1 ) 伐 ( 1 ) れ。 ( 1 ) 0 ( 1 ) ②。②②伐②。 ( 2 ) ② . ( 4 い 1 ) ( 3 ) 住 ( 3 ) ( 3 ) 伐 ( 3 ) ( 3 ) 0 ( 3 ) Each of the functions described in Table 46 ー 2 is an approxl- mate solution 0f the wave equation for three hydrogen atoms ; it is therefore reasonable to consider the sum of them with undetermined as a linear variation function. The determination of the coeffcients and the energy values then requires the solution of a secular equation (Sec. 26 の of eight rows and columns, a typical element of which is 1 ー△Ⅱ } 仏 ( 4 2 ) Function 十一十十一一十一 十十十一十一一 1