# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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1 12 (4ÿ r)reÿr=2 S41 3 40 r r d dr ÿ r 2 4 \ufffd \ufffd 1 12 (4ÿ r)reÿr=2 1 8 30 p (20ÿ 10r r2)reÿr=2 .. . The functions S31, S41, . . . are automatically normalized as specified by equation (6.25). The normalized eigenfunctions Snl(r) for l 2, 3, 4, . . . with n > (l 1) are obtained by the same procedure. A general formula for Snl involves the repeated application of B^k for k l 2, l 3, . . . , nÿ 1, n to Sl1, l in equation (6.49). The raising operator must be applied (nÿ l ÿ 1) times. The result is Snl (bnl)ÿ1(bnÿ1, l)ÿ1 . . . (bl2, l)ÿ1 B^n B^nÿ1 . . . B^l2Sl1, l (l 1)(2l 1)! n(n l)!(nÿ l ÿ 1)!(2l 2)! \ufffd \ufffd1=2 r d dr ÿ r 2 n \ufffd \ufffd 3 r d dr ÿ r 2 nÿ 1 \ufffd \ufffd \ufffd \ufffd \ufffd r d dr ÿ r 2 l 2 \ufffd \ufffd r leÿr=2 (6:50) Just as equation (6.46) can be used to go \u2018up the ladder\u2019 to obtain Sn, l from 170 The hydrogen atom Snÿ1, l, equation (6.44) allows one to go \u2018down the ladder\u2019 and obtain Snÿ1, l from Snl. Taking the positive square root in going from equation (6.43) to (6.44) is consistent with taking the positive square root in going from equation (6.45) to (6.46); the signs of the functions Snl are maintained in the raising and lowering operations. In all cases the ladder operators yield normalized eigen- functions if the starting eigenfunction is normalized. The radial factors of the hydrogen-like atom total wave functions \u142(r, \u141, j) are related to the functions Snl(r) by equation (6.23). Thus, we have R10 2 Z aì \ufffd \ufffd3=2 eÿr=2 R20 1 2 2 p Z aì \ufffd \ufffd3=2 (2ÿ r)eÿr=2 R30 1 9 3 p Z aì \ufffd \ufffd3=2 (6ÿ 6r r2)eÿr=2 .. . R21 1 2 6 p Z aì \ufffd \ufffd3=2 reÿr=2 R31 1 9 6 p Z aì \ufffd \ufffd3=2 (4ÿ r)reÿr=2 R41 1 32 15 p Z aì \ufffd \ufffd3=2 (20ÿ 10r r2)reÿr=2 .. . and so forth. A more extensive listing appears in Table 6.1. Radial functions in terms of associated Laguerre polynomials The radial functions Snl(r) and Rnl(r) may be expressed in terms of the associated Laguerre polynomials L j k(r), whose definition and mathematical properties are discussed in Appendix F. One method for establishing the relationship between Snl(r) and L j k(r) is to relate Snl(r) in equation (6.50) to the polynomial L j k(r) in equation (F.15). That process, however, is long and tedious. Instead, we show that both quantities are solutions of the same differential equation. 6.3 The radial equation 171 Table 6.1. Radial functions Rnl for the hydrogen-like atom for n 1 to 6. The variable r is given by r 2Zr=naì R10 2(Z=aì)3=2eÿr=2 R20 (Z=aì) 3=2 2 2 p (2ÿ r)eÿr=2 R21 (Z=aì) 3=2 2 6 p reÿr=2 R30 (Z=aì) 3=2 9 3 p (6ÿ 6r r2)eÿr=2 R31 (Z=aì) 3=2 9 6 p (4ÿ r)r eÿr=2 R32 (Z=aì) 3=2 9 30 p r2 eÿr=2 R40 (Z=aì) 3=2 96 (24ÿ 36r 12r2 ÿ r3)eÿr=2 R41 (Z=aì) 3=2 32 15 p (20ÿ 10r r2)r eÿr=2 R42 (Z=aì) 3=2 96 5 p (6ÿ r)r2 eÿr=2 R43 (Z=aì) 3=2 96 35 p r3 eÿr=2 R50 (Z=aì) 3=2 300 5 p (120ÿ 240r 120r2 ÿ 20r3 r4)eÿr=2 R51 (Z=aì) 3=2 150 30 p (120ÿ 90r 18r2 ÿ r3)r eÿr=2 R52 (Z=aì) 3=2 150 70 p (42ÿ 14r r2)r2 eÿr=2 R53 (Z=aì) 3=2 300 70 p (8ÿ r)r3 eÿr=2 R54 (Z=aì) 3=2 900 70 p r4 eÿr=2 R60 (Z=aì) 3=2 2160 6 p (720ÿ 1800r 1200r2 ÿ 300r3 30r4 ÿ r5)eÿr=2 172 The hydrogen atom We observe that the solutions Snl(r) of the differential equation (6.24) contain the factor r leÿr=2. Therefore, we define the function Fnl(r) by Snl(r) Fnl(r)r leÿr=2 and substitute this expression into equation (6.24) with º n to obtain r d2 Fnl dr2 (2l 2ÿ r) dFnl dr (nÿ l ÿ 1)Fnl 0 (6:51) where we have also divided the equation by the common factor r. The differential equation satisfied by the associated Laguerre polynomials is given by equation (F.16) as r d2 L j k dr2 ( j 1ÿ r) dL j k dr (k ÿ j)L jk 0 If we let k n l and j 2l 1, then this equation takes the form r d2 L2 l1n l dr2 (2l 2ÿ r) dL 2 l1 n l dr (nÿ l ÿ 1)L2 l1n l 0 (6:52) We have already found that the set of functions Snl(r) contains all the solutions to (6.24). Therefore, a comparison of equations (6.51) and (6.52) shows that Fnl is proportional to L 2 l1 n1 . Thus, the function Snl(r) is related to the polynomial L2 l1n l (r) by Snl(r) cnlr leÿr=2 L2 l1n l (r) (6:53) The proportionality constants cnl in equation (6.53) are determined by the normalization condition (6.25). When equation (6.53) is substituted into (6.25), we have Table 6.1. (cont.) R61 (Z=aì) 3=2 432 210 p (840ÿ 840r 252r2 ÿ 28r3 r4)r eÿr=2 R62 (Z=aì) 3=2 864 105 p (336ÿ 168r 24r2 ÿ r3)r2 eÿr=2 R63 (Z=aì) 3=2 2592 35 p (72ÿ 18r r2)r3 eÿr=2 R64 (Z=aì) 3=2 12 960 7 p (10ÿ r)r4 eÿr=2 R65 (Z=aì) 3=2 12 960 77 p r5 eÿr=2 6.3 The radial equation 173 c2nl 1 0 r2 l1eÿr[L2 l1n l (r)] 2 dr 1 The value of the integral is given by equation (F.25) with Æ n l and j 2l 1, so that c2nl 2n[(n l)!]3 (nÿ l ÿ 1)! 1 and Snl(r) in equation (6.53) becomes Snl(r) ÿ (nÿ l ÿ 1)! 2n[(n l)!]3 \ufffd \ufffd1=2 r leÿr=2 L2 l1n l (r) (6:54) Taking the negative square root maintains the sign of Snl(r). Equations (6.39) and (F.22), with Snl(r) and L j k(r) related by (6.54), are identical. From equations (F.23) and (F.24), we find 1 0 Snl(r)Sn\ufffd1, l(r)r2 dr ÿ12 (nÿ l)(n l 1) n(n 1) s 1 0 Snl(r)Sn9, l(r)r2 dr 0, n9 6 n, n\ufffd 1 The normalized radial functions Rnl(r) may be expressed in terms of the associated Laguerre polynomials by combining equations (6.22), (6.23), and (6.54) Rnl(r) ÿ 4(nÿ l ÿ 1)!Z3 n4[(n l)!]3a3ì s 2Zr naì \ufffd \ufffd l eÿZr=na0 L2 l1n l (2Zr=naì) (6:55) Solution for positive energies There are also solutions to the radial differential equation (6.17) for positive values of the energy E, which correspond to the ionization of the hydrogen-like atom. In the limit r!1, equations (6.17) and (6.18) for positive E become d2 R(r) dr2 2ìE "2 R(r) 0 for which the solution is R(r) ce\ufffdi(2ìE)1=2 r=" where c is a constant of integration. This solution has oscillatory behavior at infinity and leads to an acceptable, well-behaved eigenfunction of equation (6.17) for all positive eigenvalues E. Thus, the radial equation (6.17) has a continuous range of positive eigenvalues as well as the discrete set (equation (6.48)) of negative eigenvalues. The corresponding eigenfunctions represent 174 The hydrogen atom unbound or scattering states and are useful in the study of electron\u2013ion collisions and scattering phenomena. In view of the complexity of the analysis for obtaining the eigenfunctions and eigenvalues of equation (6.17) for positive E and the unimportance of these quantities in most problems of chemical interest, we do not consider this case any further. Infinite nuclear mass The energy levels En and the radial functions Rnl(r) depend on the reduced mass ì of the two-particle system ì mN me mN me me 1 me mN where mN is the nuclear mass and me is the electronic mass. The value of me is 9:109 39 3 10ÿ31 kg. For hydrogen, the nuclear mass is the protonic mass, 1:672 62 3 10ÿ27 kg, so that ì is 9:1044 3 10ÿ31 kg. For heavier hydrogen-like atoms, the nuclear mass is, of course, greater than the protonic mass. In the limit mN !1, the reduced mass and the electronic mass are the same. In the classical