Fitts D.D. - Principles of Quantum Mechanics[c] As Applied to Chemistry and Chemical Physics (1999)(en)

172 The hydrogen atom

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

( Z=aì)3=2

(2 ÿ r)eÿr=2

R20

2p2

R

( Z=aì•••)3=2

r

eÿr=2

21

p6

2

•••

( Z=aì)3=2

(6 ÿ 6r r2)eÿr=2

R30

9p3

R

( Z=aì•••)3=2

(4

)

eÿr=2

( Z=aì•••)3=2

R

2 eÿr=2

31

9p6

ÿ r r

32

p30

r

9

•••••

R40

( Z=aì)3=2

(24 ÿ 36r 12r2 ÿ r3)eÿr=2

96

( Z=aì)3=2

(20 ÿ 10r r2)r eÿr=2

R41

32p15

R

( Z=aì•••••)3=2

(6

)

2 eÿr=2

( Z=aì)•••3=2

R

3 eÿr=2

42

96p5

ÿ r r

43

p35

r

96

•••••

( Z=aì)3=2

(120 ÿ 240r 120r2 ÿ 20r3 r4)eÿr=2

R50

300p5

R

( Z=aì)•••3=2

(120

90

18

2

3)

eÿr=2

( Z=aì)•••••3=2

R

(42

14

r

2)

2 eÿr=2

r

51

150p30

ÿ

r

ÿ r

52

150p70

ÿ

r r r

R

( Z=aì)•••••3=2

(8

)

3 eÿr=2

( Z=aì)•••••3=2

R

4 eÿr=2

53

300p70

ÿ r r

54

p70

r

900

•••••

( Z=aì)3=2

(720 ÿ 1800r 1200r2 ÿ 300r3 30r4 ÿ r5)eÿr=2

R60

2160

p6

•••

6.3 The radial equation

173

Table 6.1. (cont.)

( Z=aì)3=2

(840 ÿ 840r 252r2 ÿ 28r3 r4)r eÿr=2

R61

432p210

R

( Z=aì••••••••)3=2

(336

168

24

2

3) 2 eÿr=2

( Z=aì••••••••)3=2

R

(72 18

2)

3 eÿr=2

r

62

864p105

ÿ

r

r

ÿ r

R

( Z=aì)3•••••=2

(10

)

4 eÿr=2

63

2592p35

ÿ

r r

r

R

(Z=aì)3•••=2 5 eÿr=2

64

12 960p7

ÿ r r

65

p77 r

12 960

•••••

We observe that the solutions Snl(r) of the differential equation (6.24) contain the factor rleÿr=2. Therefore, we de®ne the function Fnl(r) by

Snl(r) Fnl(r)rleÿr=2

and substitute this expression into equation (6.24) with ë n to obtain

r	d2 Fnl	(2l 2 ÿ r)	dFnl	(n ÿ l ÿ 1)Fnl 0	(6:51)
	dr2		dr

where we have also divided the equation by the common factor r.

The differential equation satis®ed by the associated Laguerre polynomials is given by equation (F.16) as

r d2 Lkj (j 1 ÿ r) dr2

dLkj (k ÿ j)Lj 0 dr k

If we let k n

d2 L2l 1 r n l

dr2

l and j 2l 1, then this equation takes the form

dL2l 1

(2l

2

)

n l

(n

ÿ

l

ÿ

1)L2l 1

0

(6:52)

ÿ r

dr

n l

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 L2nl 11. Thus, the function Snl(r) is related to the polynomial L2nl l1(r) by

S	(	r	)		c	nlr	leÿr=2 L2l 1	(	r	)	(6:53)
	nl						n l

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

174

The hydrogen atom

1

c2nl…0

r2 l 1eÿr[L2nl l1(r)]2 dr 1

á n l and

The value of the

integral

is given

by equation (F.25) with

j 2l 1, so that

2n[(n l)!]3

c2

1

nl (n

ÿ

l

ÿ

1)!

and Snl(r) in equation (6.53) becomes

S

(

)

(n ÿ l ÿ 1)!

1=2

leÿr=2 L2l 1

(

)

(6:54)

r

nl r

ÿ 2n[(n l)!]3

n l

r

Taking the negative square root maintains the sign of Snl(r).

Equations (6.39) and (F.22), with Snl(r) and Lkj (r) related by (6.54), are

identical. From equations (F.23) and (F.24), we ®nd
	1Snl(r)Sn			1,l(r)r2 dr	1	(n ÿ l)(n l 1)
…	0				ÿ s••••••••••••••••••••••••••••••••••••
	1
	…0	Snl(r)Sn9,l(r)r2 dr 0,				n9 6n, n 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)
R	(r)	ÿ	4(n ÿ l ÿ 1)!Z3		2Zr		leÿ Zr=na0 L2l 1	(2Zr=na		)	(6:55)
nl			n4[(n l)!]3 a3ì		naì		n l		ì
			s••••••••••••••••••••••••••••••••

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)			2ìE		R(r) 0
	dr2			"2
for which the solution is

R(r) ce i(2ìE)1=2 r="

where c is a constant of integration. This solution has oscillatory behavior at in®nity 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

6.4 Atomic orbitals

175

unbound or scattering states and are useful in the study of electron±ion 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.

In®nite 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			me
	mN 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 two-particle problem of Section 6.1, this limit corresponds to the nucleus remaining at a ®xed point in space.

In most applications, the reduced mass is suf®ciently 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 ø(r, è, j) of the opera-

^ ^2	^
tors H, L	, and Lz have the form
	ønlm(r, è, j) jnlmi Rnl(r)Ylm(è, j)	(6:56)

where for convenience we have introduced the Dirac notation. The radial functions Rnl(r) and the spherical harmonics Ylm(è, 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

176 The hydrogen atom

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, 1, 2, . . .

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, . . . , n ÿ 1

m ÿl, ÿl 1, . . . , ÿ1, 0, 1, . . . , l ÿ 1, l

These eigenfunctions form an orthonormal set, so that hn9l9m9jnlmi änn9äll9ämm9

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	Z2 e92	,	n 1, 2, 3, . . .	(6:57)
	2a0 n2

To ®nd the degeneracy gn of En, we note that for a speci®c 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	X	X
X
nÿ1	nÿ1	nÿ1
gn (2l 1) 2	l 0	l 1
l 0		l 0

The ®rst summation on the right-hand side is the sum of integers from 0 to (n ÿ 1) and is equal to n(n ÿ 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(n ÿ 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 ®rst four letters, standing for sharp, principal, diffuse, and

6.4 Atomic orbitals

177

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

	1		Z		3=2
j1si j100i R10(r)Y00(è, j)					eÿ Zr=a0	(6:58)
	ð1=2		a0

where R10(r) and Y00(0, j) are obtained from Tables 6.1 and 5.1. Likewise, the orbital j2si is

	(Z=a0)3=2	2 ÿ	Zr	eÿ Zr=2a0
j2si j200i	4p2ð					(6:59)
			a0
and so forth for higher values of the ••••••					n. The expressions for
	quantum number

jnsi for n 1, 2, and 3 are listed in Table 6.2.

All the s orbitals have the spherical harmonic Y00(è, j) as a factor. This

spherical harmonic is independent of the angles è and j, having a value p•••

(2 ð)ÿ1. Thus, the s orbitals depend only on the radial variable r and are spherically symmetric about the origin. Likewise, the electronic probability density jøj2 is spherically symmetric for s orbitals.

p orbitals

The wave functions for n 2, l 1 obtained from equation (6.56) are as follows:

(Z=a0)5=2

reÿ Zr=2a0 cos è

j2p0i j210i

p

••••••

Z

(6:60a)

1

5=2

4

2ð

j2p1i j211i

reÿ Zr=2a0 sin è eij

(6:60b)

8ð1=2

a0

Z

5=2

j2pÿ1i j21 ÿ1i

1

reÿ Zr=2a0 sin è eÿij

(6:60c)

8ð1=2

a0

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 satis®es the SchroÈdinger 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

178	The hydrogen atom

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	eÿ Zr=a0
	pð
	•••

2s

2pz

2px

2py

3s

3pz

3px

3py

3dz2

3dxz

3dyz

3dx2ÿ y2

3dxy

( Z=a0)3=2			Zr
4p2ð	2 ÿ				eÿ Zr=2a0
			a0
( Z=a••••••0)5=2	reÿ Zr=2a0 cos è
4p2ð
( Z=a••••••0)5=2	reÿ Zr=2a0 sin è cos j
4p2ð
( Z=a••••••0)5=2	reÿ Zr=2a0 sin è sin j
4p2ð
( Z=a••••••0)3=2							Zr		Z2 r2
81p3ð		27 ÿ		18				2			!eÿ Zr=3a0
							a0		a02
2( Z=a••••••0)5=2		6 ÿ		Zr		reÿ Zr=3a0				cos è
81p2ð
				a0
2( Z=a••••••0)5=2		6 ÿ		Zr		reÿ Zr=3a0				sin è cos j
81p2ð
				a0
2( Z=a••••••0)5=2		6 ÿ		Zr		reÿ Zr=3a0				sin è sin j
81p2ð
				a0
( Z=a0)••••••7=2	r2eÿ Zr=3a0 (3 cos2è ÿ 1)
81p6ð
2( Z=a••••••0)7=2		r2eÿ Zr=3a0 sin è cos è cos j
81p2ð
2( Z=a••••••0)7=2		r2eÿ Zr=3a0 sin è cos è sin j
81p2ð
( Z=a0)••••••7=2	r2eÿ Zr=3a0 sin2è cos 2j
81p2ð
( Z=a0••••••)7=2	r2eÿ Zr=3a0 sin2è sin 2j
81p2ð
••••••

( Z=a0)5=2
4p2ð	zeÿ Zr=2a0
( Z=a••••••0)5=2
4p2ð	xeÿ Zr=2a0
( Z=a••••••0)5=2
4p2ð yeÿ Zr=2a0
••••••
2( Z=a0)5=2		6 ÿ	Zr	zeÿ Zr=3a0
81p2ð
			a0
2( Z=a••••••0)5=2		6 ÿ	Zr	xeÿ Zr=3a0
81p2ð
			a0
2( Z=a••••••0)5=2		6 ÿ	Zr	yeÿ Zr=3a0
81p2ð
			a0
( Z=a0)••••••7=2

p•••••• (3z2 ÿ r2)eÿ Zr=3a0

81 6ð

2( Z=a••••••0)7=2 xzeÿ Zr=3a0 p

81 2ð

2( Z=a••••••0)7=2 yzeÿ Zr=3a0 p

81 2ð

( Z=a0)7=2 2 2 ÿ Zr=3a

p•••••• (x ÿ y )e 0

81 2ð

2( Z=a••••••0)7=2 xyeÿ Zr=3a0 p

81 2ð

6.4	Atomic orbitals									179
	1					Z	5=2
j2pxi 2ÿ1=2(j2p1i j2pÿ1i)										reÿ Zr=2a0 sin è cos j (6:61a)
		4(2ð)1=2			a0
	1							Z		5=2
j2pyi ÿi2ÿ1=2(j2 p1i ÿ j2pÿ1i)										reÿ Zr=2a0 sin è sin j
			4(2ð)1=2					a0
										(6:61b)

where equations (A.32) and (A.33) have been used. These new orbitals j2pxi and j2pyi 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

^	^2	^
H and of the operator L		, they are not eigenfunctions of the operator Lz.

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

	1					Z			5=2
j2pxi									xeÿ Zr=2a0	(6:62a)
	4(2ð)1=2					a0
	1					Z			5=2
j2p yi									yeÿ Zr=2a0	(6:62b)
	4(2ð)1=2					a0
	1		Z				5=2
j2pzi								zeÿ Zr=2a0		(6:62c)
	ð1=2		2a0

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 mathematical expressions for the real 2p and 3p atomic orbitals are given in Table 6.2.

d orbitals

The ®ve wave functions for n 3, l 2 are

j3d0i j320i

1

Z

7=2

(6:63a)

81p6ð

a0

r2eÿ( Zr=3a0)(3 cos2 è ÿ 1)

••••••

7=2

1

Z

j3d 1i j32 1i

81pð

r2eÿ( Zr=3a0) sin è cos è e ij

(6:63b)

a0

1

•••

Z

7=2

j3d 2i j32 2i

162pð

r2eÿ( Zr=3a0) sin2è e i2j

(6:63c)

a0

The orbital j3d0i is real.

Substitution of equation (5.29c) into (6.63a) and a

•••

change in notation for the subscript give

180		The hydrogen atom
		z
		1
			any axis '
			z-axis
		2
		2pz
	any axis '		any axis '
	x-axis		y-axis
2	1	2	1
		x	y
	2px		2py

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.

j3dz2 i	1				Z		7=2	(6:64a)
	81p6ð			a0			(3z2 ÿ r2)eÿ( Zr=3a0)
		••••••

From the four complex orbitals j3d1i, j3dÿ1i, j3d2i, and j3dÿ2i, we construct four equivalent real orbitals by the relations

j3dxzi 2ÿ1=2(j3d1i j3dÿ1i)	21=2				Z			7=2		xzeÿ( Zr=3a0)	(6:64b)
	81ð1=2			a0
	21=2						Z			7=2
j3d yzi ÿi2ÿ1=2(j3d1i ÿ j3dÿ1i)										yzeÿ( Zr=3a0)	(6:64c)
		81ð1=2					a0

6.4 Atomic orbitals

181

1

Z

7=2

j3dx2ÿ y2 i 2ÿ1=2(j3d2i j3dÿ2i)

(x2 ÿ y2)eÿ( Zr=3a0)

81(2ð)1=2

a0

(6:64d)

21=2

Z

7=2

j3dxyi ÿi2ÿ1=2

(j3d2i ÿ j3dÿ2i)

xyeÿ( Zr=3a0)

(6:64e)

81ð1=2

a0

In forming j3dx2ÿ y2 i and j3dxyi, equations (A.37) and (A.38) were used. Graphs of the square of the angular part of these ®ve 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
(n ÿ l ÿ 1) nodes between 0 and 1, i.e., the function crosses	the r-axis

(n ÿ l ÿ 1) times, not counting the origin.

The probability of ®nding the electron in the hydrogen-like atom, with the distance r from the nucleus between r and r dr, with angle è between è and è dè, and with the angle j between j and j dj is

jønlmj2 dô [Rnl(r)]2jYlm(è, j)j2 r2 sin è dr dè dj

To ®nd the probability Dnl(r) dr that the electron is between r and r dr regardless of the direction, we integrate over the angles è and j to obtain

ð	2ð
Dnl(r) dr r2[Rnl(r)]2 dr…0	…0	jYlm(è, j)j2 sin è dè 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