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

9

Approximation methods

In the preceding chapters we solved the time-independent SchroÈdinger equation for a few one-particle and pseudo-one-particle systems: the particle in a box, the harmonic oscillator, the particle with orbital angular momentum, and the hydrogen-like atom. There are other one-particle systems, however, for which the SchroÈdinger equation cannot be solved exactly. Moreover, exact solutions of the SchroÈdinger equation cannot be obtained for any system consisting of two or more particles if there is a potential energy of interaction between the particles. Such systems include all atoms except hydrogen, all molecules, nonideal gases, liquids, and solids. For this reason we need to develop approximation methods to solve the SchroÈdinger equation with suf®cient accuracy to explain and predict the properties of these more complicated systems. Two of these approximation methods are the variation method and perturbation theory. These two methods are developed and illustrated in this chapter.

9.1 Variation method

Variation theorem

The variation method gives an approximation to the ground-state energy E0

(the lowest eigenvalue of the Hamiltonian operator		^
		H) for a system whose
time-independent SchroÈdinger equation is
^	n 0, 1, 2,	. . .	(9:1)
Høn Enøn,

In many applications of quantum mechanics to chemical systems, a knowledge of the ground-state energy is suf®cient. The method is based on the variation theorem: if ö is any normalized, well-behaved function of the same variables as øn and satis®es the same boundary conditions as øn, then the quantity

^
E höjHjöi is always greater than or equal to the ground-state energy E0
^	(9:2)
E höjHjöi > E0

232

9.1 Variation method

233

Except for the restrictions stated above, the function ö, called the trial function, is completely arbitrary. If ö is identical with the ground-state eigenfunction ø0, then of course the quantity E equals E0. If ö is one of the excited-state eigenfunctions, then E is equal to the corresponding excited-state energy and is obviously greater than E0. However, no matter what trial function

öis selected, the quantity E is never less than E0.

To prove the variation theorem, we assume that the eigenfunctions øn form

a complete, orthonormal set and expand the trial function ö in terms of that set

				ö Xn		anøn					(9:3)
where, according to equation (3.28)
				an hønjöi							(9:4)
Since the trial function ö is normalized, we have										n	ak anhøk jøni
höjöi * k		ak øk		n	anøn	+			k
X				X				XX
Xk	Xn								2
			ak anäkn Xn				janj			1

We next substitute equation (9.3) into the integral for E in (9.2) and subtract the ground-state energy E0, giving

E ÿ E0 höjH^ ÿ E0jöi Xk			Xn	ak anhøk jH^ ÿ E0jøni
Xk	Xn	ak an(En ÿ E0)høk jøni Xn			janj2(En ÿ E0) (9:5)

where equation (9.1) has been used. Since En is greater than or equal to E0 and janj2 is always positive or zero, we have E ÿ E0 > 0 and the theorem is proved.

In the event that ö is not normalized, then ö in equation (9.2) is replaced by Aö, where A is the normalization constant, and this equation becomes

		2	^
	E jAj		höjHjöi > E0
	The normalization relation is
	hAöjAöi jAj2höjöi 1
	giving
			^
	E	höjHjöi		> E	0	(9:6)
		höjöi

In practice, the trial function ö is chosen with a number of parameters ë1,

234	Approximation methods

ë2, . . . , which can be varied. The quantity E is then a function of these parameters: E (ë1, ë2, . . .). For each set of parameter values, the corresponding value of E (ë1, ë2, . . .) is always greater than or equal to the true ground-state energy E0. The value of E (ë1, ë2, . . .) closest to E0 is obtained, therefore, by minimizing E with respect to each of these parameters. Selecting a suf®ciently large number of parameters in a well-chosen analytical form for the trial function ö yields an approximation very close to E0.

Ground-state eigenfunction

If the quantity E is identical to the ground-state energy E0, which is usually non-degenerate, then the trial function ö is identical to the ground-state

eigenfunction ø0. This identity follows from equation (9.5), which becomes

X

janj2(En ÿ E0) 0

n(60)

where the term for n 0 vanishes because En ÿ E0 vanishes. This relationship is valid only if each coef®cient an equals zero for n 6 0. From equation (9.3), the normalized trial function ö is then equal to ø0. Should the ground-state energy be degenerate, then the function ö is identical to one of the ground-state eigenfunctions.

When the quantity E is not identical to E0, we assume that the trial function ö which minimizes E is an approximation to the ground-state eigenfunction ø0. However, in general, E is a closer approximation to E0 than ö is to ø0.

Example: particle in a box

As a simple application of the variation method to determine the ground-state energy, we consider a particle in a one-dimensional box. The SchroÈdinger equation for this system and its exact solution are presented in Section 2.5. The ground-state eigenfunction is shown in Figure 2.2 and is observed to have no nodes and to vanish at x 0 and x a. As a trial function ö we select

ö x(a ÿ x),			0 < x < a
0,			x , 0, x . a
which has these same properties. Since we have
		a		a5
höjöi …0 x2(a ÿ x)2 dx
				30
the normalized trial function is
p
30
ö	a5•••••=2	x(a ÿ x),	0 < x < a

9.1 Variation method

235

The quantity E is, then

…0

ÿ

2m dx2

E h j

j i a5

ö H^

ö

30

a(ax

x2)

ÿ"2

d2

30"2

a

5"2

…0 (ax ÿ x2) dx

ma5

ma2

The exact ground-state energy

E1

is shown

in

ð2"2=2ma2. Thus, we have

E

10

E1 1:013E1 . E1

ð2

giving a 1.3% error.

(ax ÿ x2) dx

equation (2.39) to be

Example: harmonic oscillator

We next consider an example with a variable parameter. For the harmonic oscillator, discussed in Chapter 4, we select

ö eÿcx2

as the trial function, where c is a parameter to be varied so as to minimize E (c). This function has no nodes and approaches zero in the limits x ! 1.

Since the integral höjöi is

dx

2c

höjöi …ÿ1eÿ2cx

1=2

1

2

ð

where equation (A.5) is used, the normalized trial function is

ö ð

eÿcx

2c

1=4

2

The Hamiltonian operator

^

H for the harmonic oscillator is given in equation

(4.12). The quantity E (c) is then determined as follows

…ÿ1

1=2 "2

…ÿ1

1=2 mù2

2c

2 d2

2

2c

1

2

E (c) ÿ

1 eÿcx

eÿcx

dx

x2eÿ2cx

dx

ð

2m

dx2

ð

2

1=2

…ÿ1

1=2

…ÿ1

ð

m

1

(1 ÿ 2cx2) eÿ2cx

2

dx 2ð

mù2

1

x2eÿ2cx

2

dx

2c

"

c

c

ð

m

"

2c

ÿ c

8c3

#

2ð

2

8c3

1=2

"2 c

ð

1=2

ð

1=2

c

1=2 mù2

1=2

2c

ð

"2 c mù2

2m 8c

236	Approximation methods

where equations (A.5) and (A.7) have been used.

To ®nd the minimum value of E (c), we set the derivative dE =dc equal to zero and obtain

dE		"2			mù2
				ÿ		0
dc			2m		8c2

so that

c m2"ù

We have taken the positive square root because the parameter c must be positive for ö to be well-behaved. The best estimate of the ground-state energy is then

E 2m		2"		8	mù			2
"2		mù		mù2	2"			"ù

which is the exact result.

The reason why we obtain the exact ground-state energy in this simple example is that the trial function ö has the same mathematical form as the exact ground-state eigenfunction, given by equation (4.39). When the parameter c is evaluated to give a minimum value for E , the function ö becomes identical to the exact eigenfunction.

Excited-state energies

The variation theorem may be extended in some cases to estimate the energies of excited states. Under special circumstances it may be possible to select a trial function ö for which the ®rst few coef®cients in the expansion (9.3)

vanish: a0 a1 . . . akÿ1 0, in which case we have

X

ö anøn

n(>k)

and

X

janj2 1

n(>k)

We assume here that the eigenfunctions øn in equation (9.1) are labeled in order of increasing energy, so that

E0 < E1 < E2 <

Following the same procedure used to prove the variation theorem, we obtain

9.2 Linear variation functions		237
E ÿ Ek	nX(
	janj2(En ÿ Ek )
	>k)
from which it follows that
	E > Ek	(9:7)

Thus, the quantity E is an upper bound to the energy Ek corresponding to the state øk . For situations in which ö can be made orthogonal to each exact eigenfunction ø0, ø1, . . . , økÿ1, the coef®cients a0, a1, . . . , akÿ1 vanish according to equation (9.4) and the inequality (9.7) applies.

An example is a one-dimensional system for which the potential energy V(x) is an even function of the position variable x. The eigenfunction ø0 with the lowest eigenvalue E0 has no nodes and therefore must be an even function of x. The eigenfunction ø1 has one node, located at the origin, and therefore must be an odd function of x. If we select for ö any odd function of x, then ö is orthogonal to any even function of x, including ø0, and the coef®cient a0

vanishes. Thus, the integral E höj ^ jöi gives an upper bound to even

H E1 though the ground-state eigenfunction ø0 may not be known.

When the exact eigenfunctions ø0, ø1, . . . , økÿ1 are not known, they may be approximated by trial functions ö0, ö1, . . . , ökÿ1 which successively give upper bounds for E0, E1, . . . , Ekÿ1, respectively. In this case, the function ö1 is constructed to be orthogonal to ö0, ö2 constructed orthogonal to both ö0 and ö1, and so forth. In general, this method is dif®cult to apply and gives increasingly less accurate results with increasing n.

9.2 Linear variation functions

A convenient and widely used form for the trial function ö is the linear variation function

XN

ö ci÷i	(9:8)
i 1

where ÷1, ÷2, . . . , ÷N are an incomplete set of linearly independent functions which have the same variables and which satisfy the same boundary conditions as the exact eigenfunctions øn of equation (9.1). The functions ÷i are selected to be real and are not necessarily orthogonal to one another. Thus, the overlap integral Sij, de®ned as

Sij h÷ij÷ji

(9:9)

is not generally equal to äij. The coef®cients ci are also restricted to real values and are variation parameters to be determined by the minimization of the variation integral E .

238	Approximation methods
If we substitute equation (9.8) into (9.6) and de®ne Hij by
					^			(9:10)
we obtain		Hij h÷ijHj÷ji
	E			XX
				N	N
					cicjHij
				i 1 j 1
				N	N
				XX
				i 1	cicjSij
					j 1
or					XX
XX
N	N				N	N
E			cicjSij				cicjHij	(9:11)
i 1 j 1					i 1 j 1

To ®nd the values of the parameters ci in equation (9.8) which minimize E , we differentiate equation (9.11) with respect to each coef®cient ck (k 1, 2,

. . . , N)				@XX	A			@XX	A
		XX
	@E N N		@	N N			@	N N
	@ck	i 1 j 1 cicjSij E	@ck	0 i 1 j 1 cicjSij	1		@ck	0 i 1 j 1 cicjHij	1

and set (@E =@ck ) 0 for each value of k. The ®rst term on the left-hand side vanishes. The remaining two terms may be combined to give

@	N	N		N		N		@ci		@cj
	0 i 1 j 1		cicj(Hij ÿ E Sij)1	i			1		cj ci		(Hij ÿ E Sij)
@ck					1 j			@ck		@ck
			A
	@XX			XX

XN XN

(äik cj ciäjk )(Hij ÿ E Sij)

i 1 j 1

X	X
N	N
cj(Hkj ÿ E Skj) ci( Hik ÿ E Sik )
j 1	i 1

0

where we have noted that (@ci=@ck ) äik because the coef®cients ci in equation (9.8) are independent of each other. If we replace the dummy index j by i and note that Hik Hki and Sik Ski because the functions ÷i are real, we obtain a set of N linear homogeneous simultaneous equations

XN

ci(Hki ÿ E Ski) 0,

k 1, 2, . . . , N

(9:12)

i 1

9.3 Non-degenerate perturbation theory

239

Equation (9.12) has the form

XN

akixi 0, k 1, 2, . . . , N (9:13)

i 1

for which a trivial solution is xi 0 for all i. A non-trivial solution exists if, and only if, the determinant of the coef®cients aki vanishes

	a11	a12			a1N
aN1		a	N2		aNN			0
	a21	a22			a2N
This determinant or its equivalent			algebraic expansion						is known as the secular

equation. In equation (9.12) the parameters ci correspond to the unknown quantities xi in equation (9.13) and the terms ( Hki ÿ E Ski) correspond to the coef®cients aki. Thus, a non-trivial solution for the N parameters ci exists only if the determinant with elements (Hki ÿ E Ski) vanishes

H11

E S11

H12

E S12

H1N

E S1N

0

(9:14)

H21

ÿ E S21

H22

ÿ E S22

H2N

ÿ E S2N

ÿ

ÿ

ÿ

H N1 E SN1

H N2 E SN2

H NN

E SNN

The

ÿ

ÿ

ÿ

of E . Since

secular equation (9.14) is satis®ed only for certain values

this equation is of degree N in E , there are N real roots

E 0 < E 1 < E 2 < < E Nÿ1

According to the variation theorem, the lowest root E 0 is an upper bound to the ground-state energy E0: E0 < E 0. The other roots may be shown1 to be upper bounds for the excited-state energy levels

E1 < E 1, E2 < E 2, . . . , ENÿ1 < E Nÿ1

9.3 Non-degenerate perturbation theory

Perturbation theory provides a procedure for ®nding approximate solutions to the SchroÈdinger equation for a system which differs only slightly from a system

						^
for which the solutions are known. The Hamiltonian operator H for the system
of interest is given by			X
	2		N	1
^	"				2
H	ÿ 2 i 1 mi =i					V(r1, r2, . . . , rN )

1 J. K. L. MacDonald (1933) Phys. Rev. 43, 830.

240 Approximation methods

where N is the number of particles in the system. We suppose that the SchroÈdinger equation for this Hamiltonian operator

^

Enøn

(9:15)

Høn

cannot be solved exactly by known mathematical techniques.

In perturbation theory we assume that

^

H may be expanded in terms of a

small parameter ë

^

^ (0)

^ (1)

2

^ (2)

^

(0)

^

(9:16)

H

H

ëH

ë

H

H

H9

where

^

^ (1)

2

^ (2)

(9:17)

^ (0)

H9 ëH

ë

H

is the unperturbed Hamiltonian operator whose orthonormal

The quantity H

eigenfunctions ø(0) and eigenvalues E(0) are known exactly, so that

n

n

^

(0)

(0)

(0)

(0)

(9:18)

H

øn

En

øn

^

^

The operator H9 is called the perturbation and is small. Thus, the operator H

differs only slightly from

^

(0)

and the eigenfunctions and eigenvalues of

^

H

H do

not differ greatly from those of the unperturbed Hamiltonian operator

^ (0)

H .

The parameter ë is introduced to facilitate the comparison of the orders of magnitude of various terms. In the limit ë ! 0, the perturbed system reduces

to the unperturbed system. For	many	systems		there are no terms in the
perturbed Hamiltonian operator	higher	than	^	(1)	and for convenience the
			H

parameter ë in equations (9.16) and (9.17) may then be set equal to unity.

The mathematical procedure that we present here for solving equation (9.15) is known as Rayleigh±SchroÈdinger perturbation theory. There are other procedures, but they are seldom used. In the Rayleigh±SchroÈdinger method, the eigenfunctions øn and the eigenvalues En are expanded as power series in ë

øn ø(0)n ëø(1)n ë2ø(2)n

(9:19)

En E(0)n ëE(1)n ë2 E(2)n

(9:20)

The quantities ø(1)n and E(1)n

are the ®rst-order corrections to øn

and En, the

quantities ø(2)n and E(2)n

are the second-order corrections, and so forth. If the

^

perturbation H9 is small, then equations (9.19) and (9.20) converge rapidly for

all values of ë where 0 < ë < 1.

We next substitute the expansions (9.16), (9.19), and (9.20) into equation

(9.15) and collect coef®cients of like powers of ë to obtain

^

(0) (0)

^ (1)

(0)

^ (0)

(1)

2

^ (2)

(0)

^ (1)

(1)

^ (0)

(2)

H

øn ë( H

øn H

øn )

ë

(H

øn H

øn

H

øn

)

E(0)n ø(0)n

ë(E(1)n ø(0)n

E(0)n ø(1)n

) ë2(E(2)n ø(0)n

E(1)n ø(1)n

E(0)n ø(2)n

)

(9:21)

9.3 Non-degenerate perturbation theory						241
This equation has the form
f (å) Xk				bk åk 0
where
1			@k f
bk					å 0
	k!		@åk

Since f (å) is identically zero, the coef®cients bk all vanish. Thus, the coef®- cients of ëk on the left-hand side of equation (9.21) are equal to the coef®cients of ëk on the right-hand side. The coef®cients of ë0 give equation (9.18) for the unperturbed system. The coef®cients of ë yield

		^ (0)	(0)	(1)		^ (1)	(1)	(0)		(9:22)
		(H	ÿ En	)øn	ÿ(H		ÿ En	)øn
while the coef®cients of ë2 give
^ (0)	(0)	(2)	^ (1)		(1)	(1)	^ (2)	(2)	(0)	(9:23)
( H	ÿ En	)øn (H		ÿ En		)øn ÿ(H		ÿ En	)øn

and so forth.

First-order corrections

To ®nd the ®rst-order correction E(1)n to the eigenvalue En, we multiply equation (9.22) by the complex conjugate of ø(0)n and integrate over all space to obtain

(0)	^ (0)	(1)	(0)		(0)	(1)		(0)	^		(1)		(0)	(1)
høn	jH	jøn	i ÿ En		høn jøn i ÿhøn				jH			jøn		i En
			(0)	is normalized. Since					^		(0)	is hermitian, the ®rst
where we have noted that øn									H
integral on the left-hand side takes the form
	(0)	^ (0)	(1)		^ (0)	(0)	(1)		(0)			(0)		(1)
høn jH			jøn i hH			øn	jøn	i En		høn			jøn i

and therefore cancels the second integral on the left-hand side. The ®rst-order

(1)

is, then, the expectation value of the perturbation

^

(1)

in the

correction En

H

unperturbed state

(1)

(0)

^ (1)

(0)

^ (1)

(9:24)

En

høn

jH

jøn i

H nn

The ®rst-order correction ø(1)

to the eigenfunction is obtained by multi-

n

k 6 n and

plying equation (9.22) by the complex conjugate of ø(0)k

for

integrating over all space to give

(0)

^ (0)

(1)

(0)

(0)

(1)

(0)

^

(1)

(0)

(1)

(0)

(0)

høk

jH

jøn

i ÿ En høk jøn i ÿhøk

jH

jøn i En høk jøn

i

(9:25)

Noting that the unperturbed eigenfunctions are orthogonal

hø(0)k jø(0)n i äkn

(9:26)