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

262	Approximation methods

Calculate the ®rst-order perturbation correction to the ground-state energy level using the particle in a box with V (x) 0 for 0 < x < a, V (x) 1 for x , 0, x . a as the unperturbed system. Then calculate the ®rst-order perturbation correction to the ground-state wave function, terminating the expansion after the term k 5. (See Appendix A for trigonometric identities and integrals.)

9.9Using ®rst-order perturbation theory, determine the ground-state energy of a hydrogen atom in which the nucleus is not regarded as a point charge. Instead, regard the nucleus as a sphere of radius b throughout which the charge e is evenly distributed. The potential of interaction between the nucleus and the

electron is	ÿ2eb		3 ÿ b2 ,		0 < r < b
V(r)
		92		r2
	ÿe	92	,		r . b
	r

The unperturbed system is, of course, the hydrogen atom with a point nucleus. (Inside the nuclear sphere, the exponential

eÿ2r=a0 1 ÿ (2r=a0) (2r2=a20) ÿ

may be approximated by unity because r is very small in that region.)

9.10Using ®rst-order perturbation theory, show that the spin±orbit interaction energy for a hydrogen atom is given by

	1
	21á2jE(0)n j			for j l 21, l 60
		n(l 21)(l 1)
	1
	ÿ21á2jE(0)n j			for j l ÿ 21, l 60
			nl(l 21)		^
	The Hamiltonian operator is given in equation (7.33), where					represents the
					H0

unperturbed system and	^
	Hso is the perturbation. Use equations (6.74) and (6.78)

to evaluate the expectation value of î(r).

10

Molecular structure

A molecule is composed of positively charged nuclei surrounded by electrons. The stability of a molecule is due to a balance among the mutual repulsions of nuclear pairs, attractions of nuclear±electron pairs, and repulsions of electron pairs as modi®ed by the interactions of their spins. Both the nuclei and the electrons are in constant motion relative to the center of mass of the molecule. However, the nuclear masses are much greater than the electronic mass and, as a result, the nuclei move much more slowly than the electrons. Thus, the basic molecular structure is a stable framework of nuclei undergoing rotational and vibrational motions surrounded by a cloud of electrons described by the electronic probability density.

In this chapter we present in detail the separation of the nuclear and electronic motions, the nuclear motion within a molecule, and the coupling between nuclear and electronic motion.

10.1 Nuclear structure and motion

We consider a molecule with Ù nuclei, each with atomic number Zá and mass Má (á 1, 2, . . . , Ù), and N electrons, each of mass me. We denote by Q the set of all nuclear coordinates and by r the set of all electronic coordinates. The positions of the nuclei and electrons are speci®ed relative to an external set of

coordinate axes which are ®xed in space.
The Hamiltonian operator	^	for this system of Ù N		particles may be
	H
written in the form
	^	^	^	(10:1)
	H TQ VQ He

where ^ is the kinetic energy operator for the nuclei

TQ

263

264

Molecular structure

^

2

Ù

=á2

TQ ÿ"

X

(10:2)

á 1 2Má

VQ is the potential energy of interaction between nuclear pairs

Ù

Zá Zâe92

VQ

X

(10:3)

ráâ

á , â

1

^

and He is the electronic Hamiltonian operator

N

e92

^

"2

N

2

Ù N

Záe92

X

=i

XX

X

He ÿ

2me

i 1

ÿ á 1 i 1

rái

i , j 1

rij

(10:4)

The symbols =2á and =2i are, respectively, the laplacian operators for a single nucleus and a single electron. The variable ráâ is the distance between nuclei á and â, rái the distance between nucleus á and electron i, and rij the distance between electrons i and j. The summations are taken over each pair of particles. The quantity e9 is equal to the magnitude of the electronic charge e in CGS units and to e=(4ðå0)1=2 in SI units, where å0 is the permittivity of free space.

The SchroÈdinger equation for the molecule is
^	(10:5)
HØ(r, Q) EØ(r, Q)

where Ø(r, Q) is an eigenfunction and E the corresponding eigenvalue. The differential equation (10.5) cannot be solved as it stands because there are too many variables. However, approximate, but very accurate, solutions may be found if the equation is simpli®ed by recognizing that the nuclei and the electrons differ greatly in mass and, as a result, differ greatly in their relative speeds of motion.

Born±Oppenheimer approximation

The simplest approximate method for solving the SchroÈdinger equation (10.5) uses the so-called Born±Oppenheimer approximation. This method is a twostep process. The ®rst step is to recognize that the nuclei are much heavier than an electron and, consequently, move very slowly in comparison with the electronic motion. Thus, the electronic part of the SchroÈdinger equation may be solved under the condition that the nuclei are motionless. The resulting electronic energy may then be determined for many different ®xed nuclear con®gurations. In the second step, the nuclear part of the SchroÈdinger equation is solved by regarding the motion of the nuclei as taking place in the average potential ®eld created by the fast-moving electrons.

In the ®rst step of the Born±Oppenheimer approximation, the nuclei are all held at ®xed equilibrium positions. Thus, the coordinates Q do not change with

10.1 Nuclear structure and motion		265
^	in equation (10.2) vanishes. The
time and the kinetic energy operator TQ
SchroÈdinger equation (10.5) under this condition becomes
^	åk(Q)øk(r, Q)	(10:6)
(He VQ)øk(r, Q)

where the coordinates Q are no longer variables, but rather are constant parameters. For each electronic state k, the electronic energy åk(Q) of the molecule and the eigenfunction øk(r, Q) depend parametrically on the ®xed values of the coordinates Q. The nuclear±nuclear interaction potential VQ is now a constant and its value is included in åk(Q).

We assume in this section and in Section 10.2 that equation (10.6) has been solved and that the eigenfunctions øk(r, Q) and eigenvalues åk(Q) are known for any arbitrary set of values for the parameters Q. Further, we assume that the

eigenfunctions form a complete orthonormal set, so that
…		(r, Q)øë(r, Q) dr		äkë	(10:7)
ø	k

In the second step of the Born±Oppenheimer approximation, the energy åk(Q) is used as a potential energy function to treat the nuclear motion. In this

case, equation (10.5) becomes
^	(10:8)
[TQ åk(Q)]÷kí(Q) Ekí÷kí(Q)

where the nuclear wave function ÷kí(Q) depends on the nuclear coordinates Q and on the electronic state k. Each electronic state k gives rise to a series of nuclear states, indexed by í. Thus, for each electronic state k, the eigenfunc-

tions of [^ å ( )] are ÷ ( ) with eigenvalues . In practice, the nuclear

TQ k Q kí Q Ekí

states are differentiated by several quantum numbers; the index í represents, then, a set of these quantum numbers. In the solution of the differential equation (10.8), the nuclear coordinates Q in åk(Q) are treated as variables. The nuclear energy Ekí, of course, does not depend on any parameters. Most applications of equation (10.8) are to molecules in their electronic ground states (k 0).

In the original mathematical treatment1 of nuclear and electronic motion, M. Born and J. R. Oppenheimer (1927) applied perturbation theory to equation

(10.5) using the kinetic energy operator	^
	TQ for the nuclei as the perturbation.

The proper choice for the expansion parameter is ë (me=M)1=4, where M is the mean nuclear mass

1 XÙ

M Ù á 1 Má

When terms up to ë2 are retained, the exact total energy of the molecule is

1 M. Born and J. R. Oppenheimer (1927) Ann. Physik 84, 457.

ø ( , )^ [÷ ( )ø ( , )] d [å ( ) ÿ ]÷ ( ) 0 k r Q TQ ë Q ë r Q r k Q E k Q

266 Molecular structure

approximated by the energy Ekí of equation (10.8) and the eigenfunction Ø(Q, r) is approximated by the product

Ø(Q, r) ÷kí(Q)øk(Q, r)

(10:9)

Perturbation terms in the Hamiltonian operator up to ë4 still lead to the uncoupling of the nuclear and electronic motions, but change the form of the electronic potential energy function in the equation for the nuclear motion. Rather than present the details of the Born±Oppenheimer perturbation expansion, we follow instead the equivalent, but more elegant procedure2 of M. Born and K. Huang (1954).

Born±Huang treatment

Under the assumption that the SchroÈdinger equation (10.6) has been solved for the complete set of orthonormal eigenfunctions øk(r, Q), we may expand the

eigenfunction Ø(r, Q) of equation (10.5) in terms of øk(r, Q)

X

Ø(r, Q) ÷ë(Q)øë(r, Q)	(10:10)
ë

where ÷ë(Q) are the expansion coef®cients. Substitution of equation (10.10)

into (10.5) using (10.1) gives

X

(^ ^ ÿ )÷ ( )ø ( , ) 0 (10:11)

TQ VQ He E ë Q ë r Q

ë

where the operators have been placed inside the summation. Since the operator

^ commutes with the function ÷ ( ), we may substitute equation (10.6) into

He ë Q

(10.11) to obtain
X[T^Q åë(Q) ÿ E]÷ë(Q)øë(r, Q) 0	(10:12)
ë

We next multiply equation (10.12) by øk (r, Q) and integrate over the set of

electronic coordinates r, giving

X…

(10:13)

ë

where we have used the orthonormal property (equation (10.7)). The operator

^	acts on both functions in the product ÷ë(Q)øë(r, Q) and involves the
TQ

second derivative with respect to the nuclear coordinates Q. To expand the

expression ^ [÷ ( )ø ( , )], we note that

TQ ë Q ë r Q

2M. Born and K. Huang (1954) Dynamical Theory of Crystal Lattices (Oxford University Press, Oxford), pp. 406±7.

10.1 Nuclear structure and motion

267

=á÷ø ø=á÷ ÷=áø =2á÷ø ø=2á÷ ÷=2áø 2=á÷ : =áø

Therefore, we obtain

Ù

^

X

1

2

2

TQ[÷ë(Q)øë

(r, Q)]

ÿ"

á 1 2Má =á[÷ë(Q)øë(r, Q)]

^

^

øë(r, Q)TQ÷ë(Q) ÷ë(Q)TQøë(r, Q)

Ù

ÿ "2

X

1

=á÷ë(Q) : =áøë(r, Q)

(10:14)

á 1

Má

Substitution of equation (10.14) into (10.13) yields

[T^Q åk(Q) ÿ E]÷k(Q) Xë

(ckë Ë^ kë)÷ë(Q) 0

(10:15)

^

where the coef®cients ckë(Q) and the operators Ëkë are de®ned by

…

^

ckë(Q)

ø

(r, Q)TQøë(r, Q) dr

(10:16)

ÿ

k

á 1 Má …

Ù

^

"

2

X

1

ø

(r, Q)=áøë(r, Q) dr

:

=á

(10:17)

Ëkë

k

and equation (10.7) has been used. Since we have assumed that the electronic eigenfunctions øk(r, Q) are known for all values of the parameters Q, the coef®cients ckë(Q) and the operators Ë^ kë may be determined. The set of coupled equations (10.15) for the functions ÷k(Q) is exact.

The integral I contained in…the operator Ë^ kk is

I øk (r, Q)=áøk(r, Q) dr

For stationary states, the eigenfunctions øk(r, Q) may be chosen to be real

functions, so that this integral can also be written as

…

I 12=á [øk(r, Q)]2 dr

According to equation (10.7), the integral I vanishes and, therefore, we have

Ë^ kk 0.

We now write equation (10.15) as

^	X
	^	(10:18)
[TQ Uk(Q) ÿ E]÷k(Q)	(ckë Ëkë)÷ë(Q) 0

ë(6k)

where Uk(Q) is de®ned by

(1) kí

268	Molecular structure
	Uk(Q) åk(Q) ckk(Q)	(10:19)

The ®rst term on the left-hand side of equation (10.18) has the form of a SchroÈdinger equation for nuclear motion, so that we may identify the expansion coef®cient ÷k(Q) as a nuclear wave function for the electronic state k. The second term couples the in¯uence of all the other electronic states to the nuclear motion for a molecule in the electronic state k.

If the coef®cients ckk(Q) and ckë(Q) and the operators Ë^ kë are suf®ciently small, the summation on the left-hand side of equation (10.18) and ckk(Q) in (10.19) may be neglected, giving a zeroth-order equation for the nuclear motion

^	(0)	(0)	(Q) 0	(10:20)
[TQ åk(Q) ÿ Ekí		]÷kí

where ÷(0)kí (Q) and E(0)kí are the zeroth-order approximations to the nuclear wave functions and energy levels. The index í represents a set of quantum numbers which determine the nuclear state. The neglect of these coef®cients and operators is the Born±Oppenheimer approximation and equation (10.20) is identical to (10.8). Furthermore, the molecular wave function Ø(r, Q) in equation (10.10) reduces to the product of a nuclear and an electronic wave function as shown in equation (10.9).

When the coupling coef®cients ckë for k 6ë and the coupling operators Ë^ kë are neglected, but the coef®cient ckk(Q) is retained, equation (10.18) becomes

^	(1)	(1)	(Q) 0	(10:21)
[TQ Uk(Q) ÿ Ekí		]÷kí

where ÷(1)kí (Q) and E are the ®rst-order approximations to the nuclear wave functions and energy levels. Since the term ckk(Q) is added to åk(Q) in this

approximation,		equation (10.21) is different from (10.20) and, therefore,
÷(1)	(Q) and E(1)	differ from ÷(0)	(Q) and E(0)	. In this ®rst-order approximation,
kí	kí	kí	kí

the molecular wave function Ø(r, Q) in equation (10.10) also takes the form of (10.9). The factor ÷(1)kí (Q) describes the nuclear motion, which takes place in a potential ®eld Uk(Q) determined by the electrons moving as though the nuclei were ®xed in their instantaneous positions. Thus, the electronic state of the molecule changes in a continuous manner as the nuclei move slowly in comparison with the electronic motion. In this situation, the electrons are said to follow the nuclei adiabatically and this ®rst-order approximation is known as the adiabatic approximation. This adiabatic feature does not occur in higher-order approximations, in which coupling terms appear.

An analysis using perturbation theory shows that the in¯uence of the coupling terms with ckë(Q) and Ë^ kë is small when the electronic energy levels åk(Q) and åë(Q) are not closely spaced. Since the ground-state electronic energy of a molecule is usually widely separated from the ®rst-excited

10.2 Nuclear motion in diatomic molecules

269

electronic energy level, the Born±Oppenheimer approximation and especially the adiabatic approximation are quite accurate for the electronic ground state. The in¯uence of the coupling terms for the ®rst few excited electronic energy levels may then be calculated using perturbation theory.

10.2 Nuclear motion in diatomic molecules

The application of the Born±Oppenheimer and the adiabatic approximations to separate nuclear and electronic motions is best illustrated by treating the simplest example, a diatomic molecule in its electronic ground state. The diatomic molecule is suf®ciently simple that we can also introduce center-of- mass coordinates and show explicitly how the translational motion of the molecule as a whole is separated from the internal motion of the nuclei and electrons.

Center-of-mass coordinates

The total number of spatial coordinates for a molecule with Ù nuclei and N electrons is 3(Ù N), because each particle requires three cartesian coordinates to specify its location. However, if the motion of each particle is referred to the center of mass of the molecule rather than to the external spaced-®xed coordinate axes, then the three translational coordinates that specify the location of the center of mass relative to the external axes may be separated out and eliminated from consideration. For a diatomic molecule (Ù 2) we are left with only three relative nuclear coordinates and with 3N relative electronic coordinates. For mathematical convenience, we select the center of mass of the nuclei as the reference point rather than the center of mass of the nuclei and electrons together. The difference is negligibly small. We designate the two nuclei as A and B, and introduce a new set of nuclear coordinates de®ned by

X	MA	QA	MB	QB	(10:22a)
	M		M
R QB ÿ QA					(10:22b)

where X locates the center of mass of the nuclei in the external coordinate system, R is the vector distance between the two nuclei, and M is the sum of the nuclear masses

M MA MB

The kinetic energy operator ^ for the two nuclei, as given by equation

TQ

(10.2), is

270	Molecular structure
	"2			=2A		=2B
	T^Q	ÿ					!	(10:23)
		2		MA		MB

The laplacian operators in equation (10.23) refer to the spaced-®xed coordinates Qá with components Qxá, Qyá, Qzá, so that

=á2		@2		@2		@2	,	á A, B
		@Q2		@Q2		@Q2
		xá		yá		zá

However, these operators change their form when the reference coordinate system is transformed from space ®xed to center of mass.

To transform these laplacian operators to the coordinates X and R, with components Xx, Xy, Xz and Rx, Ry, Rz, respectively, we note that

@

@Xx @

@Rx @

MA @

ÿ

@

@QxA

@QxA

@Xx

@QxA @Rx

M

@Xx

@Rx

@

@Xx @

@Rx @

MB @

@

@QxB

@QxB

@Xx

@QxB @Rx

M

@Xx

@Rx

from which it follows that

@2

MA

2

@2

@2

ÿ

2MA

@2

@Q2xA

M

@X 2x

@R2x

M @Xx@Rx

@2

MB

2

@2

@2

2MB

@2

@Q2xB

M

@X 2x

@R2x

M @Xx@Rx

Analogous expressions apply for QyA, QyB, QzA, and QzB. Therefore, in terms of the coordinates X and R, the operators =2A and =2B are

		MA	2		2MA
	=2A		=2X	=2R ÿ		=X : =R	(10:24a)
		M			M
		MB	2		2MB
	=2B		=2X	=2R		=X : =R	(10:24b)
		M			M

where =2X =X : =X and =2R =R : =R are the laplacian operators for the vectors X and R and where =X and =R are the gradient operators. When the

^

transformations (10.24) are substituted into (10.23), the operator TQ becomes

"2

1

1

T^Q

ÿ

=2X

=2R

(10:25)

2

M

ì

where ì is the reduced mass of the two nuclei

1

1

1

ì

MA

MB

or

10.2 Nuclear motion in diatomic molecules			271
ì	MAMB		(10:26)
	MA MB

The cross terms in =X : =R cancel each other.

For the diatomic molecule, equations (10.1), (10.3), (10.5), and (10.25)

combine to give
"ÿ	"2	=2X ÿ	"2	=2R	ZAZBe92	H^ e ÿ Etot#Øtot 0	(10:27)
	2M		2ì		R

where R rAB is the magnitude of the vector R and where now the laplacian

operator =2 in ^ of equation (10.4) refers to the position of electron relative i He i

to the center of mass. The interparticle distances rAB R, rAi, rBi, and rij are independent of the choice of reference coordinate system and do not change as a result of the transformation from external to internal coordinates. If we write Øtot as the product

Øtot Ö(X)Ø(R, r)

and Etot as the sum

Etot Ecm E

then the differential equation (10.27) separates into two independent differential equations

				ÿ	"2	=2X Ö(X) EcmÖ(X)		(10:28a)
					2M
and	"ÿ						H^ e ÿ E#Ø(R, r) 0
		"2	=2R	ZAZBe92				(10:28b)
		2ì			R

Equation (10.28b) describes the internal motions of the two nuclei and the electrons relative to the center of mass. Our next goal is to solve this equation using the method described in Section 10.1. Equation (10.28a), on the other hand, describes the translational motion of the center of mass of the molecule and is not considered any further here.

Electronic motion and the nuclear potential function

The ®rst step in the solution of equation (10.28b) is to hold the two nuclei ®xed in space, so that the operator =2R drops out. Equation (10.28b) then takes the form of (10.6). Since the diatomic molecule has axial symmetry, the eigenfunc-

tions and eigenvalues of ^ in equation (10.6) depend only on the ®xed value

He

R of the internuclear distance, so that we may write them as øk(r, R) and åk(R). If equation (10.6) is solved repeatedly to obtain the ground-state energy å0(R) for many values of the parameter R, then a curve of the general form