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

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two-particle problem of Section 6.1, this limit corresponds to the nucleus remaining at a fixed point in space. In most applications, the reduced mass is sufficiently close in value to the electronic mass me that it is customary to replace ì in the expressions for the energy levels and wave functions by me. The parameter aì "2=ìe92 is thereby replaced by a0 "2=mee92. The quantity a0 is, according to the earlier Bohr theory, the radius of the circular orbit of the electron in the ground state of the hydrogen atom (Z 1) with a stationary nucleus. Except in Section 6.5, where this substitution is not appropriate, we replace ì by me and aì by a0 in the remainder of this book. 6.4 Atomic orbitals We have shown that the simultaneous eigenfunctions \u142(r, \u141, j) of the opera- tors H^ , L^2, and L^z have the form \u142nlm(r, \u141, j) jnlmi Rnl(r)Ylm(\u141, j) (6:56) where for convenience we have introduced the Dirac notation. The radial functions Rnl(r) and the spherical harmonics Ylm(\u141, j) are listed in Tables 6.1 and 5.1, respectively. These eigenfunctions depend on the three quantum numbers n, l, and m. The integer n is called the principal or total quantum number and determines the energy of the atom. The azimuthal quantum number l determines the total angular momentum of the electron, while the 6.4 Atomic orbitals 175 magnetic quantum number m determines the z-component of the angular momentum. We have found that the allowed values of n, l, and m are m 0, \ufffd1, \ufffd2, . . . l jmj, jmj 1, jmj 2, . . . n l 1, l 2, l 3, . . . This set of relationships may be inverted to give n 1, 2, 3, . . . l 0, 1, 2, . . . , nÿ 1 m ÿl, ÿl 1, . . . , ÿ1, 0, 1, . . . , l ÿ 1, l These eigenfunctions form an orthonormal set, so that hn9l9m9jnlmi änn9ä ll9ämm9 The energy levels of the hydrogen-like atom depend only on the principal quantum number n and are given by equation (6.48), with aì replaced by a0, as En ÿ Z 2e92 2a0 n2 , n 1, 2, 3, . . . (6:57) To find the degeneracy gn of En, we note that for a specific value of n there are n different values of l. For each value of l, there are (2l 1) different values of m, giving (2l 1) eigenfunctions. Thus, the number of wave functions corre- sponding to n is given by gn Xnÿ1 l0 (2l 1) 2 Xnÿ1 l0 l Xnÿ1 l0 1 The first summation on the right-hand side is the sum of integers from 0 to (nÿ 1) and is equal to n(nÿ 1)=2 (n terms multiplied by the average value of each term). The second summation on the right-hand side has n terms, each equal to unity. Thus, we obtain gn n(nÿ 1) n n2 showing that each energy level is n2-fold degenerate. The ground-state energy level E1 is non-degenerate. The wave functions jnlmi for the hydrogen-like atom are often called atomic orbitals. It is customary to indicate the values 0, 1, 2, 3, 4, 5, 6, 7, . . . of the azimuthal quantum number l by the letters s, p, d, f, g, h, i, k, . . . , respectively. Thus, the ground-state wave function j100i is called the 1s atomic orbital, j200i is called the 2s orbital, j210i, j211i, and |21 ÿ1l are called 2p orbitals, and so forth. The first four letters, standing for sharp, principal, diffuse, and 176 The hydrogen atom fundamental, originate from an outdated description of spectral lines. The letters which follow are in alphabetical order with j omitted. s orbitals The 1s atomic orbital j1si is j1si j100i R10(r)Y00(\u141, j) 1 ð1=2 Z a0 \ufffd \ufffd3=2 eÿZr=a0 (6:58) where R10(r) and Y00(0, j) are obtained from Tables 6.1 and 5.1. Likewise, the orbital j2si is j2si j200i (Z=a0) 3=2 4 2ð p 2ÿ Zr a0 \ufffd \ufffd eÿZr=2a0 (6:59) and so forth for higher values of the quantum number n. The expressions for jnsi for n 1, 2, and 3 are listed in Table 6.2. All the s orbitals have the spherical harmonic Y00(\u141, j) as a factor. This spherical harmonic is independent of the angles \u141 and j, having a value (2 ð p )ÿ1. Thus, the s orbitals depend only on the radial variable r and are spherically symmetric about the origin. Likewise, the electronic probability density j\u142j2 is spherically symmetric for s orbitals. p orbitals The wave functions for n 2, l 1 obtained from equation (6.56) are as follows: j2p0i j210i (Z=a0) 5=2 4 2ð p reÿZr=2a0 cos \u141 (6:60a) j2p1i j211i 1 8ð1=2 Z a0 \ufffd \ufffd5=2 reÿZr=2a0 sin\u141 eij (6:60b) j2pÿ1i j21ÿ1i 1 8ð1=2 Z a0 \ufffd \ufffd5=2 reÿZr=2a0 sin \u141 eÿij (6:60c) The 2s and 2p0 orbitals are real, but the 2p1 and 2pÿ1 orbitals are complex. Since the four orbitals have the same eigenvalue E2, any linear combination of them also satisfies the Schro¨dinger equation (6.12) with eigenvalue E2. Thus, we may replace the two complex orbitals by the following linear combinations to obtain two new real orbitals 6.4 Atomic orbitals 177 Table 6.2. Real wave functions for the hydrogen-like atom. The parameter aì has been replaced by a0 State Wave function Spherical coordinates Cartesian coordinates 1s (Z=a0) 3=2 ð p eÿZr=a0 2s (Z=a0) 3=2 4 2ð p 2ÿ Zr a0 \ufffd \ufffd eÿZr=2a0 2pz (Z=a0) 5=2 4 2ð p reÿZr=2a0 cos \u141 (Z=a0) 5=2 4 2ð p zeÿZr=2a0 2px (Z=a0) 5=2 4 2ð p reÿZr=2a0 sin \u141 cosj (Z=a0) 5=2 4 2ð p xeÿZr=2a0 2p y (Z=a0) 5=2 4 2ð p reÿZr=2a0 sin \u141 sinj (Z=a0) 5=2 4 2ð p yeÿZr=2a0 3s (Z=a0) 3=2 81 3ð p 27ÿ 18 Zr a0 2 Z 2 r2 a20 ! eÿZr=3a0 3pz 2(Z=a0) 5=2 81 2ð p 6ÿ Zr a0 \ufffd \ufffd reÿZr=3a0 cos \u141 2(Z=a0) 5=2 81 2ð p 6ÿ Zr a0 \ufffd \ufffd zeÿZr=3a0 3px 2(Z=a0) 5=2 81 2ð p 6ÿ Zr a0 \ufffd \ufffd reÿZr=3a0 sin \u141 cosj 2(Z=a0) 5=2 81 2ð p 6ÿ Zr a0 \ufffd \ufffd xeÿZr=3a0 3p y 2(Z=a0) 5=2 81 2ð p 6ÿ Zr a0 \ufffd \ufffd reÿZr=3a0 sin \u141 sinj 2(Z=a0) 5=2 81 2ð p 6ÿ Zr a0 \ufffd \ufffd yeÿZr=3a0 3dz2 (Z=a0) 7=2 81 6ð p r2eÿZr=3a0 (3 cos2\u141ÿ 1) (Z=a0) 7=2 81 6ð p (3z2 ÿ r2)eÿZr=3a0 3dxz 2(Z=a0) 7=2 81 2ð p r2eÿZr=3a0 sin \u141 cos \u141 cosj 2(Z=a0) 7=2 81 2ð p xzeÿZr=3a0 3d yz 2(Z=a0) 7=2 81 2ð p r2eÿZr=3a0 sin \u141 cos \u141 sinj 2(Z=a0) 7=2 81 2ð p yzeÿZr=3a0 3dx2ÿ y2 (Z=a0) 7=2 81 2ð p r2eÿZr=3a0 sin2\u141 cos 2j (Z=a0) 7=2 81 2ð p (x2 ÿ y2)eÿZr=3a0 3dxy (Z=a0) 7=2 81 2ð p r2eÿZr=3a0 sin2\u141 sin 2j 2(Z=a0) 7=2 81 2ð p xyeÿZr=3a0 178 The hydrogen atom j2pxi \ufffd 2ÿ1=2(j2p1i j2pÿ1i) 1 4(2ð)1=2 Z a0 \ufffd \ufffd5=2 reÿZr=2a0 sin\u141 cosj (6:61a) j2p yi \ufffd ÿi2ÿ1=2(j2 p1i ÿ j2pÿ1i) 1 4(2ð)1=2 Z a0 \ufffd \ufffd5=2 reÿZr=2a0 sin \u141 sinj (6:61b) where equations (A.32) and (A.33) have been used. These new orbitals j2pxi and j2p yi are orthogonal to each other and to all the other eigenfunctions jnlmi. The factor 2ÿ1=2 ensures that they are normalized as well. Although these new orbitals are simultaneous eigenfunctions of the Hamiltonian operator H^ and of the operator L^2, they are not eigenfunctions of the operator L^z. If we now substitute equations (5.29a), (5.29b), and (5.29c) into (6.61a), (6.61b), and (6.60a), respectively, we obtain for the set of three real 2p orbitals j2pxi 1 4(2ð)1=2 Z a0 \ufffd \ufffd5=2 xeÿZr=2a0 (6:62a) j2p yi 1 4(2ð)1=2 Z a0 \ufffd \ufffd5=2 yeÿZr=2a0 (6:62b) j2pzi 1 ð1=2 Z 2a0 \ufffd \ufffd5=2 zeÿZr=2a0 (6:62c) The subscript x, y, or z on a 2p orbital indicates that the angular part of the orbital has its maximum value along that axis. Graphs of the square of the angular part of these three functions are presented in Figure 6.2. The mathema- tical expressions for the real 2p and 3p atomic