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

272 Molecular structure

shown in Figure 10.1 is obtained. The value of R for which å0(R) is a minimum represents the equilibrium or most stable nuclear con®guration for the molecule. As the parameter R increases or decreases, the molecular energy å0(R) increases. As R becomes small, the nuclear repulsion term VQ becomes very large and å0(R) rapidly approaches in®nity. As R becomes very large (R ! 1), the molecule dissociates into its two constituent atoms. We assume that equation (10.6) has been solved for the ground-state wave function ø0(r, R) and ground-state energy å0(R) for all values of the parameter R from zero to in®nity.

The potential energy function U0(R) for the ground electronic state is given

			^				2			2
by equations (10.19) and (10.16) with TQ (ÿ"								=2ì)=R as
U0(R)	å0(R) c00(R)		å0(R)	"2		…	ø			(r, R)=2 ø0(r, R) dr
									0
				ÿ 2						R

Within the adiabatic approximation, the term c00(R) evaluates the coupling between the ground-state motion of the electrons and the motion of the nuclei. The magnitude of this term at distances R near the minimum of å0(R) is not negligible3 for the lightweight hydrogen molecule (all isotopes), the hydrogenmolecule ion (all isotopes), and the system He2. However, the general shape of the function U0(R) for these systems does not differ appreciably from the schematic shape of å0(R) shown in Figure 10.1. For heavier nuclei, the term c00(R) is small and may be neglected. For these molecules the Born±

å0(R)

Re

R

Figure 10.1 The internuclear potential energy for the ground state of a diatomic molecule.

3See J. O. Hirschfelder and W. J. Meath (1967) Advances in Chemical Physics, Vol. XII (John Wiley and Sons, New York), p. 23 and references cited therein.

10.2 Nuclear motion in diatomic molecules

273

Oppenheimer and the adiabatic approximations are essentially identical. Since we are interested here in only the ground electronic state, we drop the subscript on U0(R) from this point on for the sake of simplicity.

The functional form of U(R) differs from one diatomic molecule to another. Accordingly, we wish to ®nd a general form which can be used for all molecules. Under the assumption that the internuclear distance R does not ¯uctuate very much from its equilibrium value Re so that U(R) does not deviate greatly from its minimum value, we may expand the potential U(R) in a Taylor's series about the equilibrium distance Re

UU(Re) U (1)(Re)(R ÿ Re) 21! U (2)(Re)(R ÿ Re)2

31! U (3)(Re)(R ÿ Re)3 41! U (4)(Re)(R ÿ Re)4

where

U (l)(Re) dl U(lR) , l 1, 2, . . .

dR R Re

The ®rst derivative U (1)(Re) vanishes because the potential U(R) is a minimum at the distance Re. The second derivative U (2)(Re) is called the force constant for the diatomic molecule (see Section 4.1) and is given the symbol k. We also introduce the relative distance variable q, de®ned as

q R ÿ Re			(10:29)
With these substitutions, the potential takes the form
U(q) U(0) 21kq2 61U (3)(0)q3	1	U (4)(0)q4	(10:30)
	24

Nuclear motion

The nuclear equation (10.21) when applied to the ground electronic state of a diatomic molecule is

^	(10:31)
[TQ U(R)]÷í(R) Eí÷í(R)

where the superscript and one subscript on ÷(1)0í (R) and on E(1)0í are omitted for simplicity. In solving this differential equation, the relative coordinate vector R is best expressed in spherical polar coordinates R, è, j. The coordinate R is the magnitude of the vector R and is the scalar distance between the two nuclei. The angles è and j give the orientation of the internuclear axis relative to the external coordinate axes. The laplacian operator =2R is then given by (A.61) as

274

Molecular structure

"

#

=2R

1 @

R2

@

1

1 @

@

1 @2

R2

@R

@R

R2

sin è

@è

sin è

@è

sin2è

@j2

1 @

2

@

1

^2

R2

@R

R

@R

ÿ

"2 R2

L

(10:32)

where

^2

is the square of the orbital angular momentum operator given by

L

equation (5.32). With =2R expressed in spherical polar coordinates, equation

(10.31) becomes

" #

ÿ "2 @ 2 @ 1 ^2 ( ) ÷ ( , è, j) ÷ ( , è, j) 2ìR2 @R R @R 2ìR2 L U R í R Eí í R

(10:33)

The operator in square brackets on the left-hand side of equation (10.33)

^2			^				^2
commutes with the operator L	and with the operator Lz in (5.31c), because L
commutes with itself as well	as with	^	and neither	^2	nor	^	contain the
		Lz		L		Lz

variable R. Consequently, the three operators have simultaneous eigenfunctions. From the argument presented in Section 6.2, the nuclear wave function ÷í(R, è, j) has the form

÷í(R, è, j) F(R)YJm(è, j)

(10:34)

where F(R) is a function of only the internuclear distance R, and YJm(è, j) are the spherical harmonics, which satisfy the eigenvalue equation

^2	2
L	YJm(è, j) J(J 1)" YJm(è, j)
	J 0, 1, 2, . . . ;	m ÿJ, ÿJ 1, . . . , 0, . . . , J ÿ 1, J

It is customary to use the index J for the rotational quantum number. Equation

(10.33) then becomes

dR

ÿ

#

"ÿ 2ìR2 dR

2ìR2

í

"2

d

R2

d

J(J 1)"2

U(R)

E

F(R)

0 (10:35)

where we have divided through by YJm(è, j).

We next replace the independent variable R in equation (10.35) by q as de®ned in equation (10.29). Equation (10.35) has a more useful form if we also

make the substitution S(q) RF(R). Since dq=dR 1, we have

d2 S(q)

F(R)

1 dS(q)

1

d

dF(R)

d

ÿ

S(q),

R2

R

dR

R

dq

R2

dR

dR

dq2

and equation (10.35) becomes

"ÿ

"2 d2

J(J

1)"2

#S(q) 0

q)2

U(q) ÿ Eí

(10:36)

2ì

dq2

2ì(Re

10.2 Nuclear motion in diatomic molecules

275

after multiplication by the variable R.

The potential function U(q) in equation (10.36) may be expanded according to (10.30). The factor (Re q)ÿ2 in the second term on the left-hand side may

also be expanded in terms of the variable q as follows

1

1

1

2q

3q2

1 ÿ

ÿ !

(10:37)

(Re

q)2

2

1

q

2

Re2

Re

Re2

Re

Re

where the expansion (A.3) is used. For small values of the ratio q=Re, equation (10.37) gives the approximation R Re.

If we retain only the ®rst two terms in the expansion (10.30) and let R be

approximated by Re, equation (10.36) becomes
	ÿ"2	d2 S(q)	(1kq2	ÿ	W )S(q)		0	(10:38)
	2ì dq2
			2
where
	W Eí ÿ U(0) ÿ J(J 1)Be							(10:39)
		Be "2=2ìRe2 "2=2I						(10:40)

The quantity I ( ìR2e ) is the moment of inertia for the diatomic molecule with the internuclear distance ®xed at Re and Be is known as the rotational constant (see Section 5.4).

Equation (10.38) is recognized as the SchroÈdinger equation (4.13) for the one-dimensional harmonic oscillator. In order for equation (10.38) to have the same eigenfunctions and eigenvalues as equation (4.13), the function S(q) must have the same asymptotic behavior as ø(x) in (4.13). As the internuclear distance R approaches in®nity, the relative distance variable q also approaches in®nity and the functions F(R) and S(q) RF(R) must approach zero in order for the nuclear wave functions to be well-behaved. As R ! 0, which is equivalent to q ! ÿRe, the potential U(q) becomes in®nitely large, so that F(R) and S(q) rapidly approach zero. Thus, the function S(q) approaches zero as q ! ÿRe and as R ! 1. The harmonic-oscillator eigenfunctions ø(x) decrease rapidly in value as jxj increases from x 0 and approach zero as x ! 1. They have essentially vanished at the value of x corresponding to q ÿRe. Consequently, the functions S(q) in equation (10.38) and ø(x) in (4.13) have the same asymptotic behavior and the eigenfunctions and eigenvalues of (10.38) are those of the harmonic oscillator. The eigenfunctions Sn(q) are the harmonic-oscillator eigenfunctions given by equation (4.41) with x replaced by q and the mass m replaced by the reduced mass ì. The eigenvalues, according to equation (4.30), are

276	Molecular structure
where	Wn (n 21)"ù,			n 0, 1, 2, . . .
				p•••••••••
In this approximation, the nuclear		ù
				k=ì
		energy levels are
	EnJ U(0) (n 21)"ù J(J 1)Be				(10:41)
and the nuclear wave functions are
		1
	÷nJm(R, è, j)			Sn(R ÿ Re)YJm(è, j)	(10:42)
			R

Higher-order approximation for nuclear motion

The next higher-order approximation to the energy levels of the diatomic molecule is obtained by retaining in equation (10.36) terms up to q4 in the expansion (10.30) of U(q) and terms up to q2 in the expansion (10.37) of (Re q)ÿ2. Equation (10.36) then becomes

ÿ"2

d2 S(q)

[1kq2

B J(J

1)

V9]S(q)

[E

ÿ

U(0)]S(q) (10:43)

2ì dq2

í

2

e

where

V 9

ÿ

2Be J(J 1)

q

3Be J(J 1)

q2

1

U (3)

(0)q3

1

U (4)(0)q4

Re

Re2

6

24

b1 q b2 q2 b3 q3 b4 q4

(10:44)

For simplicity in subsequent evaluations, we have introduced in equation (10.44) the following de®nitions

b	1		ÿ			2Be J(J 1)			(10:45a)
							Re
b	2			3Be J(J 1)					(10:45b)
							Re2
b3			1		U		(3)(0)		(10:45c)
				6
b4			1			U (4)(0)			(10:45d)
				24

Since equation (10.43) with V 9 0 is already solved, we may treat V 9 as a perturbation and solve equation (10.43) using perturbation theory. The unperturbed eigenfunctions S(0)n (q) are the eigenkets jni for the harmonic oscillator. The ®rst-order perturbation correction E(1)nJ to the energy EnJ as given by equation (9.24) is

10.2 Nuclear motion in diatomic molecules

277

E(1)nJ hnjV 9jni b1hnjqjni b2hnjq2jni b3hnjq3jni b4hnjq4jni

(10:46)

The matrix elements hnjqk jni are evaluated in Section 4.4. According to equations (4.45c) and (4.50e), the ®rst and third terms on the right-hand side of (10.46) vanish. The matrix elements in the second and fourth terms are given by equations (4.48b) and (4.51c), respectively. Thus, the ®rst-order correction in equation (10.46) is

	"						3		"				2
E(1)nJ b2					ÿn	21	b4								n2 n 21
		ìù						2		ìù
	6B2					1			1						"		2	(4)		1	2		1
		e
			ÿn			2 J(J 1)											U		(0)hÿn	2			4i	(10:47)
	"ù										16			ìù

where equations (10.40), (10.45b), and (10.45d) have been substituted.

Since the perturbation corrections due to b1 q and b3 q3 vanish in ®rst order, we must evaluate the second-order corrections E(2)nJ in order to ®nd the in¯uence of these perturbation terms on the nuclear energy levels. According

to equation (9.34), this second-order correction is

E(2)

ÿ

hkjb1 q b3 q3jni2

nJ

k(6n)

E(0)

ÿ

E(0)

k

n

X

b2

hkjqjni2

2b

b

hkjqjnihkjq3jni

ÿ

b2

hkjq3jni2

ÿ

(k ÿ n)"ù ÿ

(k ÿ n)"ù

k( n) (k ÿ n)"ù

1 k( n)

1 3 k( n)

3

6

6

6

X

X

X

(10:48)

where the unperturbed energy levels are given by equation (4.30). The matrix elements in equation (10.48) are given by (4.45) and (4.50), so that E(2)nJ becomes

E(2)

b12

(n 1)"

n"

nJ

ÿ

ÿ

"ù

2ìù

2ìù

ÿ

2b1 b3

"

(n 1)" 1=23

(n 1)"

3=2

ÿ

n" 1=2

3

n"

3=2

#

"ù

2ìù

2ìù

2ìù

2ìù

ÿ

b32

"

3

(n

1)(n 2)(n 3)

9(n

1)3

"ù

2ìù

3

1)(n

2)

ÿ9n3 ÿ

n(n ÿ

3

ÿ

This equation simpli®es to

278 Molecular structure

(2)

b2

3b b "

1

b2

"2

2

1

1

3

3

EnJ

ÿ

ÿ

ÿn

2 ÿ

(30n

30n

11)

2ìù2

ì2ù3

8ì3ù4

Substitution of equations (10.40), (10.45a), and (10.45c) leads to

EnJ ÿ "2

ù2 J (J 1)

ì"ù3

ÿn

2 J(J 1)

(2)

4B2

2

2B2 R U (3)

(0)

1

e 2

e

e

ÿ

"2[U (3)(0)]2

1

2

7

ÿ

h30

n

2

2i

(10:49)

288ì3ù4

The nuclear energy levels in this higher-order approximation are given to second order in the perturbation by combining equations (10.41), (10.47), and (10.49) to give

EnJ E(0)nJ E(1)nJ E(2)nJ

(0)

"ù n

1

"ùxe

n

1

2

Be J(J

1)

2 2

2

U

2

ÿ

ÿ

1

ÿ

(10:50)

ÿ DJ

(J 1)

ÿ áe

n 2

J(J 1)

where we have de®ned

ÿ

"

5[

U (3)(0)]2

xe

ÿ U (4)(0)!

(10:51a)

16ì2ù3

3ìù2

4B2

D

e

(10:51b)

"2ù2

B2

R U (3)(0)

áe

ÿ6 e

1

e

!

(10:51c)

"ù

3ìù2

1

"

2

7[U (3)(0)]2

U(0) U(0)

U (4)

(0) ÿ

!

(10:51d)

64

ìù

9ìù2

The approximate expression (10.50) for the nuclear energy levels EnJ is

	terms of a power series expansion in (n				1) and
observed to contain the initial		1 2	2		2
J(J 1 1). Only terms up to (n		2)	and [J(J 1)]	and the cross term in
(n 2)J(J 1) are included. Higher-order terms in				the expansion	may be

found from higher-order perturbation corrections.

The second term on the right-hand side of equation (10.50) is the energy of a harmonic oscillator. Since the factor xe in equation (10.51a) depends on the third and fourth derivatives of the internuclear potential at Re, the third term in equation (10.50) gives the change in energy due to the anharmonicity of that potential. The fourth term is the energy of a rigid rotor with moment of inertia I. The ®fth term is the correction to the energy due to centrifugal distortion in

Problems

279

this non-rigid rotor. As the rotational energy increases, the internuclear distance increases, resulting in an increased moment of inertia and consequently a lower energy. Thus, this term is negative and increases as J increases. The magnitude of the centrifugal distortion is in¯uenced by the value of the force constant k as re¯ected by the factor ùÿ2 in D. The last term contains both quantum numbers n and J and represents a direct coupling between the vibrational and rotational motions. This term contains two contributions: a change in vibrational energy due to the centrifugal stretching of the molecule and a change in rotational energy due to changes in the internuclear distance from anharmonic vibrations. The constant term U(0) merely shifts the zeropoint energy of the nuclear energy levels and is usually omitted completely.

The molecular constants ù, Be, xe, D, and áe for any diatomic molecule may be determined with great accuracy from an analysis of the molecule's vibrational and rotational spectra.4 Thus, it is not necessary in practice to solve the electronic SchroÈdinger equation (10.28b) to obtain the ground-state energy å0(R).

Problems

10.1Derive equation (10.47) as outlined in the text.

10.2Derive equation (10.49) as outlined in the text.

10.3Derive equation (10.50) as outlined in the text.

10.4An approximation to the potential U(R) for a diatomic molecule is the Morse potential

U(R) ÿDe(2eÿa(RÿRe) ÿ eÿ2a(RÿRe)) ÿDe(2eÿaq ÿ eÿ2aq)

where a is a parameter characteristic of the molecule. The Morse potential has the general form of Figure 10.2.

(a)Show that U(Re) ÿDe, that U(1) 0, and that U(0) is very large.

(b)If the Morse potential is expanded according to equation (10.30), relate the parameter a to ì, ù, and De

(c)Relate the quantities xe, áe, and U (0) in equation (10.50) to ì, ù, and De for the Morse potential.

10.5Another approximate potential U(R) for a diatomic molecule is the Rydberg potential

U(R) ÿDe[1 b(R ÿ Re)]eÿb(RÿRe) ÿDe(1 bq)eÿbq where b is a parameter characteristic of the molecule.

(a) Show that U(Re) ÿDe, that U(1) 0, and that U(0) is very large.

4Comprehensive tables of molecular constants for diatomic molecules may be found in K. P. Huber and G. Herzberg (1979) Molecular Spectra and Molecular Structure: IV. Constants of Diatomic Molecules (Van Nostrand Reinhold, New York).

280	Molecular structure

U (R)

0 Re

R

2De

Figure 10.2 The Morse potential for the ground state of a diatomic molecule.

(b)If the Rydberg potential is expanded according to equation (10.30), relate the parameter b to ì, ù, and De.

(c)Relate the quantities xe, áe, and U (0) in equation (10.50) to ì, ù, and De for the Rydberg potential.

10.6Consider a diatomic molecule in its ground electronic and rotational states. Its

energy levels are given by equation (10.50) with J 0. The value of U(R) at

R Re is ÿDe.

(a)If the anharmonic factor xe is positive, show that the spacing of the energy levels decreases as the vibrational quantum number n increases.

(b)When the vibrational quantum number n becomes suf®ciently large that the difference in energies between adjacent levels becomes zero, the molecule dissociates into its constituent atoms. By setting equal to zero the derivative of En0 with respect to n, ®nd the value of n in terms of xe at which dissociation takes place.

(c)Relate the well depth De to the anharmonic factor xe and compare with the corresponding expressions in problems 10.4 and 10.5.

Appendix A

Mathematical formulas

Useful power series expansions

eaz X1 (az)n

n 0 n!

Binomial expansions

X

n

n!

u nÿává

(u v)n

á

0

á!(n ÿ á)!

Xn

(u v)ÿn

1

(ÿ1)á(n á ÿ 1)!

uÿ(n á)vá

á 0

á!(n

ÿ

1)!

dn uv

X

n!

dáu dnÿáv

á

dzn

0 á!(n ÿ á)! dzá dz nÿá

Useful integrals

1

2

1

2

…ÿ1eÿz

dz

2…0

eÿz

dz ð1=2

1

2

…0

zeÿz

dz 21

1

2

1

2

…ÿ1z2eÿz

dz 2…0

z2 eÿz

dz 21ð1=2

1

2

2

ð1=2

2

2

…ÿ1eÿa

z

bz dz

eb

=4a

a

…ÿ1

r•••••••••••

ÿ

1

eÿaz2 ec(bÿz)2 dz

ð

eacb2=(aÿc)

a

c

1 sin2 z

…ÿ1

dz ð

z2

(A:1)

(A:2)

(A:3)

(A:4)

(A:5)

(A:6)

(A:7)

(A:8)

(A:9)

(A:10)

281