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

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orbitals are given in Table 6.2. d orbitals The five wave functions for n 3, l 2 are j3d0i j320i 1 81 6ð p Z a0 \ufffd \ufffd7=2 r2eÿ( Zr=3a0)(3 cos2 \u141ÿ 1) (6:63a) j3d\ufffd1i j32\ufffd 1i 1 81 ð p Z a0 \ufffd \ufffd7=2 r2eÿ( Zr=3a0) sin \u141 cos\u141 e\ufffdij (6:63b) j3d\ufffd2i j32\ufffd 2i 1 162 ð p Z a0 \ufffd \ufffd7=2 r2eÿ( Zr=3a0) sin2\u141 e\ufffdi2j (6:63c) The orbital j3d0i is real. Substitution of equation (5.29c) into (6.63a) and a change in notation for the subscript give 6.4 Atomic orbitals 179 j3dz2i 1 81 6ð p Z a0 \ufffd \ufffd7=2 (3z2 ÿ r2)eÿ( Zr=3a0) (6:64a) From the four complex orbitals j3d1i, j3dÿ1i, j3d2i, and j3dÿ2i, we construct four equivalent real orbitals by the relations j3dxzi \ufffd 2ÿ1=2(j3d1i j3dÿ1i) 2 1=2 81ð1=2 Z a0 \ufffd \ufffd7=2 xzeÿ( Zr=3a0) (6:64b) j3d yzi \ufffd ÿi2ÿ1=2(j3d1i ÿ j3dÿ1i) 2 1=2 81ð1=2 Z a0 \ufffd \ufffd7=2 yzeÿ( Zr=3a0) (6:64c) z 1 2 2pz any axis \u2019 z-axis 12 12 any axis \u2019 x-axis any axis \u2019 y-axis 2px 2py x y Figure 6.2 Polar graphs of the hydrogen 2p atomic orbitals. Regions of positive and negative values of the orbitals are indicated by and ÿ signs, respectively. The distance of the curve from the origin is proportional to the square of the angular part of the atomic orbital. 180 The hydrogen atom j3dx2ÿ y2i \ufffd 2ÿ1=2(j3d2i j3dÿ2i) 1 81(2ð)1=2 Z a0 \ufffd \ufffd7=2 (x2 ÿ y2)eÿ( Zr=3a0) (6:64d) j3dxyi \ufffd ÿi2ÿ1=2(j3d2i ÿ j3dÿ2i) 2 1=2 81ð1=2 Z a0 \ufffd \ufffd7=2 xyeÿ( Zr=3a0) (6:64e) In forming j3dx2ÿ y2i and j3dxyi, equations (A.37) and (A.38) were used. Graphs of the square of the angular part of these five real functions are shown in Figure 6.3 and the mathematical expressions are listed in Table 6.2. Radial functions and expectation values The radial functions Rnl(r) for the 1s, 2s, 2p, 3s, 3p, and 3d atomic orbitals are shown in Figure 6.4. For states with l 6 0, the radial functions vanish at the origin. For states with no angular momentum (l 0), however, the radial function Rn0(r) has a non-zero value at the origin. The function Rnl(r) has (nÿ l ÿ 1) nodes between 0 and 1, i.e., the function crosses the r-axis (nÿ l ÿ 1) times, not counting the origin. The probability of finding the electron in the hydrogen-like atom, with the distance r from the nucleus between r and r dr, with angle \u141 between \u141 and \u141 d\u141, and with the angle j between j and j dj is j\u142nlmj2 dô [Rnl(r)]2jYlm(\u141, j)j2 r2 sin\u141 dr d\u141 dj To find the probability Dnl(r) dr that the electron is between r and r dr regardless of the direction, we integrate over the angles \u141 and j to obtain Dnl(r) dr r2[Rnl(r)]2 dr ð 0 2ð 0 jYlm(\u141, j)j2 sin\u141 d\u141 dj r2[Rnl(r)]2 dr (6:65) Since the spherical harmonics are normalized, the value of the double integral is unity. The radial distribution function Dnl(r) is the probability density for the electron being in a spherical shell with inner radius r and outer radius r dr. For the 1s, 2s, and 2p states, these functions are 6.4 Atomic orbitals 181 2 1 1 2 z x 3dxz 2 1 1 2 z y 3dyz 2 1 1 2 y x 3dxy Figure 6.3 Polar graphs of the hydrogen 3d atomic orbitals. Regions of positive and negative values of the orbitals are indicated by and ÿ signs, respectively. The distance of the curve from the origin is proportional to the square of the angular part of the atomic orbital. 2 2 1 1 y x 3dx22y2 z 1 1 3dz2 any axis \u2019 z-axis 2 2 182 The hydrogen atom Figure 6.4 The radial functions Rnl(r) for the hydrogen-like atom. 0 1 2 3 4 5 r/aì0 0.5 1 1.5 2aì 3/2R10 0 2 4 6 8 10 20.2 0 0.2 0.4 0.6 0.8 l 5 0 l 5 1 r/aì aì 3/2R2l 0 2 4 6 8 10 12 14 16 18 20 r/aì aì 3/2R3l 20.1 0.0 0.1 0.2 0.3 0.4 l 5 0 l 5 1 l 5 2 6.4 Atomic orbitals 183 D10(r) 4 Z a0 \ufffd \ufffd3 r2eÿ2 Zr=a0 D20(r) 1 8 Z a0 \ufffd \ufffd3 r2 2ÿ Zr a0 \ufffd \ufffd2 eÿZr=a0 (6:66) D21(r) 1 24 Z a0 \ufffd \ufffd5 r4eÿZr=a0 Higher-order functions are readily determined from Table 6.1. The radial distribution functions for the 1s, 2s, 2p, 3s, 3p, and 3d states are shown in Figure 6.5. The most probable value rmp of r for the 1s state is found by setting the derivative of D10(r) equal to zero dD10(r) dr 8 Z a0 \ufffd \ufffd3 r 1ÿ Zr a0 \ufffd \ufffd eÿ2 Zr=a0 0 which gives rmp a0=Z (6:67) Thus, for the hydrogen atom (Z 1) the most probable distance of the electron from the nucleus is equal to the radius of the first Bohr orbit. The radial distribution functions may be used to calculate expectation values of functions of the radial variable r. For example, the average distance of the electron from the nucleus for the 1s state is given by hri1s 1 0 rD10(r) dr 4 Z a0 \ufffd \ufffd3 1 0 r3eÿ2 Zr=a0 dr 3a0 2Z (6:68) where equations (A.26) and (A.28) were used to evaluate the integral. By the same method, we find hri2s 6a0 Z , hri2p 5a0 Z The expectation values of powers and inverse powers of r for any arbitrary state of the hydrogen-like atom are defined by hrkinl 1 0 rkDnl(r) dr 1 0 rk[Rnl(r)] 2 r2 dr (6:69) In Appendix H we show that these expectation values obey the recurrence relation k 1 n2 hrkinl ÿ (2k 1) a0 Z hr kÿ1inl k l(l 1) 1ÿ k 2 4 \ufffd \ufffd a20 Z2 hr kÿ2inl 0 (6:70) 184 The hydrogen atom Figure 6.5 The radial distribution functions Dnl(r) for the hydrogen-like atom. 0 2 4 6 8 10 12 14 16 18 20 0.00 0.10 0.20 0.30 0.40 0.50 0.60a0D10 r/a0 0 2 4 6 8 10 12 14 16 18 20 0.00 0.10 0.05 0.10 0.15 0.20a0D2l r/a0 D21 D20 0 2 4 6 8 10 12 14 16 18 20 0.00 0.02 0.04 0.06 0.08 0.10 0.12a0D3l r/a0 D30 D31 D32 6.4 Atomic orbitals 185 For k 0, equation (6.70) gives hrÿ1inl Z n2a0 (6:71) For k 1, equation (6.70) gives 2 n2 hrinl ÿ 3a0 Z l(l 1) a 2 0 Z2 hrÿ1inl 0 or hrinl a0 2Z [3n2 ÿ l(l 1)] (6:72) For k 2, equation (6.70) gives 3 n2 hr2inl ÿ 5a0 Z hrinl 2[l(l 1)ÿ 34] a20 Z2 0 or hr2inl n 2a20 2Z2 [5n2 ÿ 3l(l 1) 1] (6:73) For higher values of k, equation (6.70) leads to hr3inl, hr4inl, . . . For k ÿ1, equation (6.70) relates hrÿ3inl to hrÿ2inl hrÿ3inl Z l(l 1)a0 hr ÿ2inl (6:74) For k ÿ2, ÿ3, . . . , equation (6.70) gives successively hrÿ4inl, hrÿ5inl, . . . expressed in terms of hrÿ2inl. Although the expectation value hrÿ2inl cannot be obtained from equation (6.70), it can be evaluated by regarding the azimuthal quantum number l as the parameter in the Hellmann\u2013Feynman theorem (equation (3.71)). Thus, we have @En @ l @ H^ l @ l \ufffd \ufffd (6:75) where the Hamiltonian operator H^ l is given by equation (6.18) and the energy levels En by equation (6.57). The derivative @ H^ l=@ l is just @ H^ l @ l " 2 2ìr2 (2l 1) (6:76) In the derivation of (6.57), the quantum number n is shown to be the value of l plus a positive integer. Accordingly, we have @n=@ l 1 and @En @ l ÿ Z 2e92 2a0 @ @ l nÿ2 ÿ Z 2e92 2a0 @n @ l @ @n nÿ2 Z 2"2 ìa20 nÿ3 (6:77) where aì "2=ìe92 has been replaced by a0 "2=mee92. Substitution of equations (6.76) and (6.77) into (6.75) gives the desired result 186 The hydrogen atom hrÿ2inl Z 2 n3(l 1 2 )a20 (6:78) Expression (6.71) for the expectation value of rÿ1 may be used to calculate the average potential energy of the electron in the state jnlmi. The potential energy V (r) is given by equation (6.13). Its expectation value is hV inl ÿZe92hrÿ1inl ÿ Z 2e92 a0 n2 (6:79) The result depends only on the principal quantum