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

242				Approximation methods
applying the hermitian property of						^ (0)	to the ®rst term on the left-hand side,
						H
(1)	(0)	^	(1)	(0)
and writing H kn for høk		jH		jøn	i, we may express equation (9.25) as
		(0)		(0)		(0)	(1)	^ (1)	(9:27)
	(Ek		ÿ En		)høk jøn			i ÿH kn
The orthonormal	eigenfunctions					ø(0)	for	the unperturbed system	are as-
						j

sumed to form a complete set. Thus, the perturbation corrections ø(1)n may be
expanded in terms of the set ø(0)j			X
ø(1)n	X
	anjø(0)j annø(0)j		anjø(0)j
	j		j(6n)
where anj are complex constants given by
	anj hø(0)j jø(1)n	i	(9:28)

If the complete set of eigenfunctions for the unperturbed system includes a continuous range of functions, then the expansion of ø(1)n must include these functions. The inclusion of this continuous range is implied in the summation notation. The total eigenfunction øn for the perturbed system to ®rst order in ë is, then

øn (1 ëann)ø(0)n ë anjø(0)j	(9:29)
j(6n)
X

Since the function ø(0)n is already included in zero order in the expansion of øn, we may, without loss of generality, set ann equal to zero, so that

X

ø(1)n anjø(0)j (9:30) j(6n)

This choice affects the normalization constant of øn, but has no other consequence. Furthermore, equation (9.28) for j n becomes

showing that with a	nn 0,	hø(0)n jø(1)n i 0	(1)	(9:31)
		the ®rst-order correction		is orthogonal to the
		(0)	øn
unperturbed eigenfunction øn .

With the choice ann 0, the total eigenfunction øn to ®rst order is normalized. To show this, we form the scalar product hønjøni using equation (9.29) and retain only zero-order and ®rst-order terms to obtain

X

hønjøni hø(0)n jø(0)n i ë (anjhø(0)n jø(0)j i anjhø(0)j jø(0)n i)

j(6n)

X

1 ë (anj anj)änj 1 j(6n)

where equation (9.26) has been used. Substitution of equation (9.30) into (9.27) gives

ank

(9:32)

9.3 Non-degenerate perturbation theory

243

X

( (0) ÿ (0)) hø(0)jø(0)i ( (0) ÿ (0)) ÿ ^ (1)

Ek En anj k j Ek En ank H kn j(6n)

where again equation (9.26) is utilized. If the eigenvalue E(0)n is non-degen- erate, then E(0)k cannot equal E(0)n for all k and n and we can divide by (E(0)k ÿ E(0)n ) to solve for ank

ÿ ^ (1)

H kn

E(0)k ÿ E(0)n

The situation where E(0) is degenerate requires a more complex treatment,
n
which is presented in Section 9.5. The ®rst-order correction ø(1)						is obtained by
					n
combining equations (9.30) and (9.32)
	^ (1)
(1)	H kn				(0)
øn ÿ					øk	(9:33)
	E(0)	ÿ	E(0)
k(6n)	k			n
X

Second-order corrections

The second-order correction E(2)n to the eigenvalue En is obtained by multi-

plying equation (9.23) by ø(0)n

and integrating over all space

(0)

^ (0)

(0)

(2)

(0)

^ (1)

(1)

(1)

(0)

(1)

i

høn

jH

ÿ En

jøn i høn

jH

jøn i ÿ

En

høn jøn

(0)

^

(2)

(0)

(2)

where the normalization of ø(0)

ÿhøn

jH

jøn i En

has been noted. Application of the hermitian

^ (0)

n

property of

cancels the ®rst term on the left-hand side. The third term on

H

the left-hand side vanishes according to

equation

^ (2)

for

(9.31). Writing H nn

(0)

^

(2)

(0)

høn jH

jøn i and substituting equation (9.33) then give

(2)

^ (2)

(0)

^

(1)

(1)

En

H nn

høn

jH

jøn i

X

ÿ

X

ÿ

(1)

2

^ (1)

^

^ (1)

H^ (2)

H nk

H kn

H^ (2)

jH kn j

(9:34)

nn

ÿ

E(0)

E(0)

nn

ÿ

E(0)

E(0)

k(6n)

k

n

^ (1)

k(6n)

k

n

where we have also noted that

^ (1)

equals

because

^ (1)

is hermitian.

H nk

H kn

H

In

many

applications there

is

no second-order

term in

the

perturbed

Hamiltonian

operator

so that

^

(2)

is zero. In such cases each unperturbed

H nn

eigenvalue

E(0)n

is raised by the terms in the summation corresponding to

eigenvalues E(0) less than E(0)

and lowered by the terms with eigenvalues E(0)

k

n

k

greater than E(0). The eigenvalue E(0)

is perturbed to the greatest extent by the

n

n

terms with eigenvalues E(0) close to E(0). The contribution to the second-order

k

n

correction E(2)

of terms with eigenvalues far removed from E(0) is small. For

n

n

the lowest eigenvalue E(0)0 , all of the terms are negative so that E(2)0 is negative.

244 Approximation methods

We also see that, in these cases, the ®rst-order correction ø(1)n to the eigenfunction determines the second-order correction E(2)n to the eigenvalue.

To obtain the second-order perturbation correction ø(2) to the eigenfunction,

ø(0) 6 n

we multiply equation (9.23) by kn for k n and integrate over all space

(0)	^ (0)	(0)	(2)	i
høk	jH	ÿ En	jøn

(0)	^ (1)	(1)	i ÿ	(1)		(0)		(1)	i
høk	jH	jøn		En høk				jøn
				(0)	^	(2)		(0)	(2)	(0)	(0)	i
		ÿhøk			jH		jøn i En			høk	jøn

As before, we apply the hermitian property of ^ (0), introduce the abbreviation

H

^ (2)

H kn , and use the orthogonality relation (9.26) to obtain

(0)	(0)	(0)	(2)	(0)	^ (1)	(1)	(1)	(0)	(1)	^ (2)	(9:35)
(Ek	ÿ En	)høk	jøn	i høk	jH	jøn	i ÿ En	høk	jøn	i ÿH kn

We next expand the function ø(2)n in terms of the complete set of unperturbed
eigenfunctions ø(0)
j
ø(2)n	bnjø(0)j	(9:36)
	j(6n)
	X

where, without loss of generality, the term j n may be omitted for the same reason that ø(0)n is omitted in equation (9.30). The coef®cients bnj are complex constants given by

bnj hø(0)j jø(2)n i

(9:37)

Substitution of equations (9.24), (9.28), (9.30), and (9.37) into (9.35) gives

(0)

(0)

X

^

(1)

^ (1)

^ (2)

(Ek

ÿ En )bnk

anj H kj

ÿ ank H nn ÿH kn

j(6n)

or

^ (2)

^ (1)

^ (1)

H kn

anj H kj ÿ ank H nn

j(6n)

b

X

(9:38)

ÿ

(E(0)k ÿ E(0)n )

nk

Combining equations (9.32), (9.36), and (9.38), we obtain the ®nal result

X 4

ÿ

(2)

X

ÿ

^ (1) ^ (1)

ÿ

^

(1)

^

(1)

5

H

H

ÿ

(2)

^

kj

jn

(0)

H kn

H kn

H nn

øn

2

ÿ

ÿ

3øk

E(0)

E(0)

j(6n)

(E(0)

E(0))(E(0)

E(0))

(E(0)

E(0))2

k(6n)

k

n

k

n

j

n

k

n

(9:39)

Summary

The non-degenerate eigenvalue En for the perturbed system to second order is obtained by substituting equations (9.24) and (9.34) into (9.20) to give

9.3 Non-degenerate perturbation theory

245

	4	X		ÿ			5
	2H^ (2)nn ÿ			^	(1) 2		3
				H
En E(0)n ëH^ (1)nn ë2			j		kn j			(9:40)
		k(6n)	E(0)		E	(0)
			k			n

The corresponding eigenfunction øn to second order is obtained by combining equations (9.19), (9.33), and (9.39)

X

ÿ

^

(1)

(0)

H

kn

(0)

øn øn

ÿ ë

k(6n)

E(0)

E(0)

øk

k

n

X 4

ÿ

(2)

X

ÿ

^ (1) ^ (1)

ÿ

^

(1)

^

(1)

5

H

H

ÿ

^

kj

jn

(0)

2

H

kn

H

kn

H

nn

ë

2

ÿ

ÿ

3øk

E(0)

E(0)

j(6n)

(E(0)

E(0))(E(0)

E(0))

(E(0)

E(0))2

k(6n)

k

n

k

n

j

n

k

n

(9:41)

While the eigenvalue E(0)n for the unperturbed system must be non-degen- erate for these expansions to be valid, some or all of the other eigenvalues E(0)k for k 6 n may be degenerate. The summations in equations (9.40) and (9.41) are to be taken over all states of the unperturbed system other than the state ø(0)n . If an eigenvalue E(0)i is gi-fold degenerate, then it is included gi times in the summations. If the unperturbed eigenfunctions have a continuous range, then the summations in equations (9.40) and (9.41) must include an integration over those states as well.

Relation to variation method

If we use the wave function ø(0) for the unperturbed ground state as a trial
function ö in the variation			0					^
			method		of Section				equal to
							9.1 and set H
^ (0)	^ (1)	, then we have from equations (9.2), (9.18), and (9.24)
H	ëH
	E	^	(0)	^ (0)	^ (1)	(0)	(0)	(1)
		höjHjöi hø0		jH	ëH	jø0	i E0	ëE0

and E is equal to the ®rst-order energy as determined by perturbation theory. If we instead use a trial function ö which contains some parameters and which equals ø(0)0 for some set of parameter values, then the corresponding energy E from equation (9.2) is at least as good an approximation as E(0)0 ëE(1)0 to the true ground-state energy.

Moreover, if the wave function ø(0)0 ëø(1)0 is used as a trial function ö, then the quantity E from equation (9.2) is equal to the second-order energy determined by perturbation theory. Any trial function ö with parameters which reduces to ø(0)0 ëø(1)0 for some set of parameter values yields an approximate energy E from equation (9.2) which is no less accurate than the second-order perturbation value.

246	Approximation methods

9.4 Perturbed harmonic oscillator

As illustrations of the application of perturbation theory we consider two examples of a perturbed harmonic oscillator. In the ®rst example, we suppose that the potential energy V of the oscillator is

V 12kx2 cx4 12mù2 x2 cx4

where c is a small quantity. The units of V are those of "ù (energy), while the units of x are shown in equation (4.14) to be those of ("=mù)1=2. Accordingly, the units of c are those of m2ù3=" and we may express c as

c ë m2ù3

"

where ë is dimensionless. The potential energy then takes the form

V 21mù2 x2

ë

m2ù3 x4

(9:42)

"

The Hamiltonian operator

^

(0)

for the unperturbed harmonic oscillator is

H

given by equation (4.12) and its eigenvalues E(0)

and eigenfunctions ø(0) are

n

n

^

shown in equations (4.30) and (4.41). The perturbation H9 is

^

^

(1)

ë

m2

ù3 x4

H9 H

"

(9:43)

^

(2)

,

^ (3)

,

. . . in the perturbed Hamiltonian operator do

Higher-order terms H

H

not appear in this example.

To ®nd the perturbation corrections to the eigenvalues and eigenfunctions, we require the matrix elements hn9jx4jni for the unperturbed harmonic oscillator. These matrix elements are given by equations (4.51). The ®rst-order correction E(1)n to the eigenvalue En is evaluated using equations (9.24), (9.43),

and (4.51c)

(1)

^ (1)

ëm2

ù3

4

3

2

1

En

H nn

hnjx

jni

2(n

n

2)ë"ù

(9:44)

"

The second-order correction E(2)

is obtained from equations (9.34), (9.43), and

n

(4.51) as follows

9.4 Perturbed harmonic oscillator

247

E(2)

n

^ (1)

2

^ (1)

2

^ (1)

2

^ (1)

2

ÿ

jH nÿ4,nj

ÿ

jH nÿ2,nj

ÿ

jH n 2,nj

ÿ

jH n 4,nj

E(0)nÿ4 ÿ E(0)n

E(0)nÿ2 ÿ E(0)n

E(0)n 2 ÿ E(0)n

E(0)n 4 ÿ E(0)n

h

4"jù j

i

#

ÿ ë "2ù

"hn

(ÿ 4j"ùj) i

h

(ÿ

2j"ùj)

i

h

2"jù j i

2 m4

6

4 x4 n

2

n

2 x4 n

2

n

2 x4 n

2

n

4 x4

n

2

ÿ

ÿ

ÿ81(34n3 51n2 59n 21)ë2"ù

(9:45)

The perturbed energy En to second order is, then

En E(0)n

E(1)n E(2)n

(n 12)"ù 32(n2 n 12)ë"ù ÿ 18(34n3 51n2 59n 21)ë2"ù (9:46)

In the expression (9.45) for the second-order correction E(2)n , the summation on the right-hand side includes all states k other than the state n, but only for

the states (n ÿ 4), (n ÿ 2), (n 2), and (n 4) are the contributions to the summation non-vanishing. For the two lowest values of n, giving E(2)0 and E(2)1 ,

only the two terms k (n 2) and k (n 4) should be included in the summation. However, the terms for the meaningless values k (n ÿ 2) and

k (n ÿ 4) vanish identically, so that their inclusion in equation (9.45) is valid. A similar argument applies to E(2)2 and E(2)3 , wherein the term for the

meaningless value k (n ÿ 4) is identically zero. Thus, equation (9.46) applies to all values of n and the perturbed ground-state energy E0, for example, is

E0 (12 34ë ÿ 218 ë2)"ù

The evaluation of the ®rstand second-order corrections to the eigenfunctions is straightforward, but tedious. Consequently, we evaluate here only the

®rst-order correction ø(1) for the ground state. According to equations (9.33),

0

(9.43), and (4.51), this correction term is given by

^ (1)

^ (1)

(1)

H

20

(0)

H40

(0)

ø0

ÿ

ø2

ÿ

ø4

E2(0) ÿ E0(0)

E4(0) ÿ E0(0)

ÿ

"

h

2j"ùj

i ø2(0) h

4j"ùj i ø4(0)

ëm2ù3

2 x4

0

4 x4

0

ÿ

4 2

(0)

•••

(0)

ë

(0)

p

(0)

p

[6ø2

3ø4

]

(9:47)

eigenfunctions

ø2

and

ø4

as given by equation (4.41) are

If the unperturbed

•••

248 Approximation methods

explicitly introduced, then the perturbed ground-state eigenfunction ø0 to ®rst order is

		mù	1=4		ë			2
ø0 ø0(0) ø0(1)				1 ÿ		(4î4	12î2	ÿ 9) eÿî	=2 (9:48)
		ð"			16

As a second example, we suppose that the potential energy V for the perturbed harmonic oscillator is

V 21kx2 cx3 21mù2 x2 ë	m3ù5		1=2
			x3	(9:49)
	"

where c ë(m3ù5=")1=2 is again a small quantity and ë is dimensionless. The

^
perturbation H9 for this example is
H^ 9 H^ (1) ë						m3ù5	1=2 x3			(9:50)
						"
9	3	jni for the unperturbed						harmonic oscillator are
The matrix elements hn jx									(1)
given by equations (4.50). The ®rst-order correction term									En	is obtained by
substituting equations (9.50) and (4.50e) into (9.24), giving the result
E(1)n ë			m3ù5		1=2
						hnjx3jni 0				(9:51)
			"

Thus, the ®rst-order perturbation to the eigenvalue is zero. The second-order

term E(2) is evaluated using equations (9.34), (9.50), and (4.50), giving the

n

result

En E(2)n

^ (1)

2

^ (1)

2

^ (1)

2

^ (1)

2

ÿ

jH nÿ3,nj

ÿ

jH nÿ1,nj

ÿ

jH n 1,nj

ÿ

jH n 3,nj

E(0)nÿ3 ÿ E(0)n

E(0)nÿ1 ÿ E(0)n

E(0)n 1 ÿ E(0)n

E(0)n 3 ÿ E(0)n

h

3"jù j i

#

ÿ ë

" ù

"hn

(ÿ 3j"ùj) i

h

ÿ( "jù)j i

h

"ùj

j

i

2 m3

5

3 x3 n

2

n

1 x3 n

2

n

1 x3

n

2

n

3 x3 n

2

ÿ

ÿ

ÿ81(30n2 30n 11)ë2"ù

(9:52)

9.5 Degenerate perturbation theory

The perturbation method presented in Section 9.3 applies only to non-degen- erate eigenvalues E(0)n of the unperturbed system. When E(0)n is degenerate, the denominators vanish for those terms in equations (9.40) and (9.41) in which E(0)k is equal to E(0)n , making the perturbations to En and øn indeterminate. In

9.5 Degenerate perturbation theory

249

this section we modify the perturbation method to allow for degenerate eigenvalues. In view of the complexity of this new procedure, we consider only the ®rst-order perturbation corrections to the eigenvalues and eigenfunctions.

The eigenvalues and eigenfunctions for the unperturbed system are given by equation (9.18), but now the eigenvalue E(0)n is gn-fold degenerate. Accordingly, there are gn eigenfunctions with the same eigenvalue E(0)n . For greater clarity, we change the notation here and denote the eigenfunctions corresponding to E(0)n as ø(0)ná, á 1, 2, . . . , gn. Equation (9.18) is then replaced by the equivalent expression

H(0)ø(0)ná E(0)n ø(0)ná, á 1, 2, . . . , gn (9:53)

Each of the eigenfunctions ø(0)ná is orthogonal to all the other unperturbed
(0)	for	k n, but is not necessarily orthogonal to the other
eigenfunctions øká	(0)		6
eigenfunctions for En . Any linear combination öná of the members of the set
ø(0)
ná			X
öná			gn	á 1, 2, . . . ,
			cáâø(0)nâ,		gn	(9:54)
			â 1
is also a solution	of	equation (9.53)		with the same	eigenvalue	E(0). As
						n

discussed in Section 3.3, the members of the set ø(0)ná may be constructed to be orthonormal and we assume that this construction has been carried out, so that

hø(0)nâjø(0)nái äáâ, á, â 1, 2, . . . , gn (9:55)

By suitable choices for the coef®cients cáâ in equation (9.54), the functions öná may also be constructed as an orthonormal set

hönâjönái äáâ,				á, â 1, 2, . . . , gn			(9:56)
Substitution of equation (9.54) into (9.56) and application of (9.55) give
X
gn
ã 1	câãcáã		äáâ,	á, â		1, 2, . . . , gn	(9:57)
The SchroÈdinger equation for the perturbed system is
^	øná Enáøná,			á 1, 2, . . . , gn			(9:58)
H
			^	is given by equation (9.16),			Ená are the
where the Hamiltonian operator H

eigenvalues for the perturbed system, and øná are the corresponding eigen-

functions. While the unperturbed eigenvalue E(0)n is gn-fold degenerate, the
perturbation	^	(0)
	H9 in the Hamiltonian operator often splits the eigenvalue	En

into gn different values. For this reason, the perturbed eigenvalues Ená require the additional index á. The perturbation expansions of Ená and øná in powers of ë are

(0) ná

250	Approximation methods
Ená E(0)n	ëE(1)ná ë2 E(2)ná ,	á 1, 2, . . . , gn	(9:59)
øná ø(0)ná ëø(1)ná ë2ø(2)ná ,		á 1, 2, . . . , gn	(9:60)

Note that in equation (9.59) the zero-order term is the same for all values of á.

	limit	ë !	0, the Hamiltonian operator	^
In the				H approaches the unperturbed
	^ (0)
operator	H	and the perturbed eigenvalue Ená approaches the degenerate

unperturbed eigenvalue E(0)n . The perturbed eigenfunction øná approaches a function which satis®es equation (9.53), but this limiting eigenfunction may not be any one of the initial functions ø(0)ná. In general, this limiting function is some linear combination of the initial unperturbed eigenfunctions ø(0)ná, as expressed in equation (9.54). Thus, along with the determination of the ®rstorder correction terms E(1)ná and ø(1)ná, we must ®nd the set of unperturbed eigenfunctions ö(0)ná to which the perturbed eigenfunctions reduce in the limit ë ! 0. In other words, we need to evaluate the coef®cients cáâ in the linear combinations (9.54) which transform the initial set of unperturbed eigenfunctions ø(0)ná into the `correct' set ö . Equation (9.60) is then replaced by

øná ö(0)ná ëø(1)ná ë2ø(2)ná , á 1, 2, . . . , gn (9:61) The ®rst-order equations (9.22) and (9.24) apply here provided the additional

index á and the `correct' unperturbed eigenfunctions are used

^ (0)	(0)	(1)	^	(1)	(1) (0)	(9:62)
(H	ÿ En	)øná ÿ( H			ÿ Ená)öná
	(1)	(0)	^ (1)		(0)	(9:63)
	Ená hönájH			jönái

However, equation (9.63) for the ®rst-order corrections to the eigenvalues

cannot be used directly at this point because the functions ö(0)ná are not known.
To ®nd E(1)	we multiply equation (9.62) by ø(0) , the complex conjugate of
ná	nâ

one of the initial unperturbed eigenfunctions belonging to the degenerate

eigenvalue E(0)	, and integrate over all space to obtain
n
	(0) ^ (0)	(0)	(1)		(0) ^	(1)		(1)	(0)
hønâjH		ÿ En	jønái ÿhønâjH				ÿ Enájönái
Applying the	hermitian		^	(0)	, we	see		that	the left-hand side
		property of H

vanishes. Substitution of the expansion (9.54) for ö(0)ná using ã as the dummy expansion index gives

Xgn

(0) ^ (1)	(1) (0)
cáãhønâjH	ÿ Enájønãi 0,

ã 1

If we introduce the abbreviation

^ (1)	(0) ^ (1)	(0)
H nâ,nã	hønâjH	jønãi,

á, â 1, 2, . . . , gn

â, ã 1, 2, . . . , gn

and apply the orthonormality condition (9.55), this equation takes the form

9.5		Degenerate perturbation theory			251
gn
^ (1)		(1)	á, â 1, 2,	. . . , gn	(9:64)
cáã(H nâ,nã	ÿ Enáäâã) 0,
ã 1
X		^ (1)
Note that the integrals				known initial	set of
		H ná,nã are evaluated with the

unperturbed eigenfunctions, in contrast to the integrals in equation (9.63), which require the unknown functions ö(0)ná. For a given eigenvalue E(1)ná, the expression (9.64) is a set of gn linear homogeneous simultaneous equations, one for each value of â (â 1, 2, . . . , gn)

â 1:	^ (1)	(1)	^ (1)	cá3	^ (1)	^ (1)		0
	cá1( H n1,n1 ÿ Ená) cá2 H n1,n2				H n1,n3	cá g n H n1,ng n
â 2:	^ (1)	^ (1)	(1)		^ (1)	^ (1)		0
	cá1 H n2,n1	cá2(H n2,n2	ÿ Ená) cá3		H n2,n3	cá g n H n2,ng n
.
.
.
	^ (1)	^ (1)		^ (1)		^ (1)		(1)
â gn: cá1 H ng n,n1 cá2 H ng n,n2 cá3 H ng n,n3 cá g n (H ng n,ng n							ÿ Ená)
								0

Equation (9.64) has the form of (9.13) with the coef®cients cáã correspond-

^ (1)

(1)

ing to the unknown quantities xi and the terms ( H nâ,nã ÿ Enáäâã) correspond-

ing to the coef®cients aki. Thus, a non-trivial solution for the

gn

coef®cients

cáã

(ã

^ (1)

1, 2, . . . , gn) exists only if the determinant with elements (H

nâ,nã ÿ

(1)

Enáäâã) vanishes

H(1)n1,n1 ÿ E(1)ná

H(1)n1,n2

H(1)n1,ng n

H(1)n2,n1

H(1)n2,n2

ÿ

E(1)ná

H(1)n2,ng

n

0

(9:65)

(1)

(1)

(1)

(1)

H ng

n

,n1

H ng

n

,n2

H ng

n

,ng

n ÿ

En

á

(1)

is the secular

Only

for some values of the ®rst-order correction term Ená

(1)

equation (9.65) satis®ed. This secular equation is of degree gn in

Ená, giving

gn roots

E(1), E(1)

, . . . , E(1)

n1

n2

ng n

^ (1)

is hermitian. The perturbed eigenvalues to

all of which are real because H

®rst order are, then

En1 E(0)n ëE(1)n1

.

.

.

Eng n

E(0)n ëE(1)ng n

If the gn roots are all different, then in ®rst order the gn-fold degenerate unperturbed eigenvalue E(0)n is split into gn different perturbed eigenvalues. In this case, the degeneracy is removed in ®rst order by the perturbation. We