Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

544	k. k. liang et al.
and	jw1s;Ai ¼ sp e zjr RAj
			ð175Þ
		z3

Here, we shall replace w1s;A with a single Gaussian wave function jgAðiÞi as defined earlier. That is, we have used the approximation

jsðiÞi ¼	1	ðjgAðiÞi þ jgBðiÞiÞ	ð176Þ
	p
	2ð1 þ SABÞ

For H2, let us write down the zeroth-order electronic Hamiltonian (in atomic unit):

H^ ð0Þ

r12

1

1

r22

1

1

1

1

177

rA1 rB1

þ

2 rA2

rB2 þ r12 þ R

ð

Þ

0

¼ 2

Let

^

1

^

1

h0

R

;

h2

r12

h^

h^

1

h^

2

r12

1

1

r22

1

1

1

2 rA1 rB1

þ

rA2

rB2

1ð Þ þ

1

ð Þ ¼

2

we have

^ ð0Þ

j gi

h gjH0

1

þ

þ

^ ð0Þ þ

þ

¼

2

hsð1Þ sð2Þ sð1Þ sð2ÞjH0

jsð1Þ sð2Þ sð1Þ sð2Þi

1

þ

^ ð0Þ þ

þ

^ ð0Þ

þ

¼

hhsð1Þ sð2ÞjH0

j sð1Þ sð2Þi þ hsð1Þ sð2ÞjH0

jsð1Þ sð2Þi

2

þ

^ ð0Þ þ

þ

^ ð0Þ þ

ð178Þ

hsð1Þ sð2ÞjH0

jsð1Þ sð2Þi hsð1Þ sð2ÞjH0

j sð1Þ sð2Þii

The last two terms in (178) with negative signs vanish after integrating out the spin part, and, consequently,

^ ð0Þ

^

^

^

^

jsð1Þsð2Þi

h gjH0

j gi ¼ hsð1Þsð2Þj h0

þ h1

ð1Þ þ h1

ð2Þ þ h2

¼

^

0

þ hsð Þj ð Þjsð Þi þ hsð Þsð

Þj

^

2jsð

1

Þsð

2

Þi

ð

179

Þ

h

h

2 1 h1 1

1

1

the crude born–oppenheimer adiabatic approximation

Next, we expand them into the atomic orbitals

	^		ð1Þjsð1Þi
hsð1Þjh1
			1	^
¼		2ð1 þ SABÞ		hgAð1Þ þ gBð1Þjh1	ð1ÞjgAð1Þ þ gBð1Þi
¼			1	hgAjh^1jgAi þ hgBjh^1jgBi þ 2hgAjh^1jgBi
	2ð1 þ SABÞ

where

^	1
hgAjh1jgAi ¼ EgðHÞ hgAj	rB1	jgAi
^	1
hgBjh1jgBi ¼ EgðHÞ hgBj	rA1	jgBi
^	1

hgAjh1jgBi ¼ EgðHÞSAB hgAjrA1jgBi

Therefore,

545

ð180Þ

ð181Þ

ð182Þ

ð183Þ

	^	ð1Þjsð1Þi
hsð1Þjh1
			1	1		1		1
¼	EgðHÞ			hgAj		jgAi þ hgBj		jgBi þ 2hgAj		jgBi
			2ð1 þ SABÞ		rB1		rA1		rA1

The quantity hsð1Þsð2Þj1=r12jsð1Þsð2Þi can also be expanded but we do not show the final result here. In Figure 4, we show the calculated result of

^ ð0Þ

the quantity h gjH0 j gi 2EgðHÞ. We have simply taken z ¼ 1. The corres-

˚

ponding equilibrium position is marked in the figures, which is 0:9112 A. This value is not realistic, however, the magnitude and the feature of the PES are reasonable. We believe that the more realistic value can be obtained by using larger basis set.

To obtain the force constant for constructing the equation of motion of the nuclear motion in the second–order perturbation, we need to know about the excited states, too. With the minimal basis set, the only excited-state spatial orbital for one electron is

js i ¼	1	jw1s; Ai jw1s; Bi	ð184Þ
	p
	2ð1 SABÞ

546	k. k. liang et al.

Figure 4. The ground-state potential energy surface with z ¼ 1.

With symmetry considerations, we can write down all of the possible spin orbitals:

		þ þ			ji			ð	185	Þ
j	1i ¼ k s s
				ji				ð	186	Þ
j	2i ¼ ks s
	1		k sþ s ji þ ksþ sji
j 3i ¼		p2						ð187Þ
		1			þ	þ		ð	188	Þ
j	4i ¼ p2 k s s					ji ks	sji
j 5i ¼ ksþ s ji								ð189Þ

Nonvanishing matrix element only exists between j 3i and the ground state for

^	^
any one-electron operator O1	that does not involve spin. Obviously, h 1jO1j gi

^
and h 2jO1j gi are zero after integrating over the spin part. It is then clear that all
	^
of the triplet states will result in vanishing matrix elements. Therefore h 4jO1j gi
is also zero. For the doubly excited singlet state, however,	we have
^	and j 5i.
h 5jO1j gi ¼ 0 because both spatial orbitals are different in j gi

The only excited state that we need to calculate, therefore, is the singly excited singlet state j ei ¼ j 3i. Although these can be easily demonstrated, we shall neglect the algebra here.

the crude born–oppenheimer adiabatic approximation									547
Consequently, the term Unð2Þ is reduced to
					^ ð1Þ		2
Uð2Þ	H^ ð2Þ	j		gi þ	jh ejH0	j gij		ð	190	Þ
g	¼ h gj 0				Ugð0Þ Ueð0Þ

where we have changed the subscripts to g and e for convenience. We have to

e

0

h

gj

^ ð2Þ

j

gi

h

ej

^ ð1Þ

j

gi

2

Þ,

0

, and

H

0

g

calculate Uð

H

to obtain Uð Þ. The zeroth-order

excited state energy Ueð0Þ is

0

^ ð0Þ

j ei

Ueð Þ ¼ h ejH0

1

þ

þ

^ ð0Þ

þ

þ

¼

hðj s s

j þ js s jÞjH0

jðj s s j þ js sjÞi

2

1

þ

^ ð0Þ

þ

þ

^ ð0Þ

þ

¼

ðhjs s jjH0

jjs s ji þ hj s s jjH0

jj s s ji

2

þ

^ ð0Þ

þ

þ

^ ð0Þ

þ

ð191Þ

þ hj s

s jjH0

jj s s ji þ hj s s jjH0

jj s s jiÞ

Expanding the Slater determinants and integrating out the spin part and collecting terms that are the same under exchange of electron indices, we have

0			^ ð0Þ	^ ð0Þ	js si
Ueð	Þ ¼ hss jH0			jss i þ hss jH0
	¼	1	þ hsjh^1jsi þ hs jh^1js i þ hss jh^2jss i þ hss jh^2js si ð192Þ
		R

By expanding the spatial orbitals into atomic orbitals and manipulating them properly, we have

	1		1			1
	hwAj		jwAi þ hwBj		jwBi þ 2hwAj		jwBi
^		rB1		rA1		rA1
hsjh1jsi ¼ EgðHÞ			2ð1 þ SABÞ
	1		1			1
	hwAj		jwAi þ hwBj		jwBi 2hwAj		jwBi
^		rB1		rA1		rA1
hs jh1js i ¼ EgðHÞ			2ð1 SABÞ

hss j^ jss i þ hss j^ js si h2 h2

ð193Þ

ð194Þ

1

1

1

1

¼

hwAwAjr12jwAwAi hwAwAjr12jwAwBi hwAwBjr12jwAwAi

2 1 SAB2

þ hwAwBj

1

1

1

jwAwBi þ hwBwAj

jwAwAi hwBwAj

jwAwBi

r12

r12

r12

hwBwBj

1

1

jwAwBi!

jwAwAi þ hwBwBj

ð195Þ

r12

r12

548	k. k. liang et al.

Figure 5. The vertical axis is the potential energy minus 2Eg (H) in hartree, and the x axis is R in a0. Both Ugð0Þ and Ueð0 were plotted for comparison.

Putting them all together, and letting z ¼ 1, we can obtain the excited-state potential energy surface (see Fig. 5). We wish to emphasize again that we calculate the energy for a range of the nuclear coordinate because we need to find the minima. In crude BOA, further properties of the molecules, such as vibrational mode frequency (determined by the force constant), anharmonicity, and so on, can be calculated in higher order perturbation, instead of being extracted from the PES curve.

In Figure 5, it can be seen that there is a minimum in the excited state as well. In more realistic calculations, such minimum was not observed. Note that the crude BOA is based on expanding the total wave function in terms of the basis functions obtained at the equilibrium position. In the expression, it seems that we have to find the equilibrium position for each electronic level. This is not practical because if we choose different center of expansion for different electronic levels, we will have to calculate a lot of matrix elements with different centers. Further, this might be impossible because there will be excited states that do not have an equilibrium geometry. Therefore, we chose to expand everything in terms of the basis function at the equilibrium position of the ground electronic state. The zeroth-order electronic energy Ueð0Þ is also calculated under the equilibrium geometry of the ground state. In other words, the vertical energy difference is used in Eq. (190).

the crude born–oppenheimer adiabatic approximation

549

It can be shown that

2 1 SAB

0jwBi

h ej qR

0j gi ¼

1

2

"hwAj qqR rB1 0jwAi hwBj qqR rA1

qV

1

1

q

ð196Þ

þ hwAj qqR rA1 0jwAi hwBj

qqR rB1 0jwBi#

1

1

1

q

1

but both hwAj

q

0jwAi and hwBj

0jwBi are zero, and

qR

rA1

qR

rB1

þ

"hwAj

q2

h gj

q2V

2

1

1

0jwAi

qR2

0j gi ¼

R3

1 SAB

qR2

rA1

2

1

2

1

þ 2hwAj

q

0jwBi þ hwBj

q

0jwBi

qR2

rA1

qR2

rA1

2

1

2

1

þ hwAj

q

0jwAi þ 2hwAj

q

0jwBi

qR2

rB1

qR2

rB1

2

1

0jwBi#

þ hwBj

q

ð197Þ

qR2

rB1

By replacing all of the atomic orbitals with Gaussian functions, we calculate the matrix elements

q

1

0

1

1

p

R

p2aA

2

gA

gA

erf

2aA

R

p

e 2aAR

198

R r

R2

2

h j

q

B1

j i ¼

a

2aBR

ð

Þ

q

p

p

p

0

1

1

1

2

B

R

gB

R r

gB

R2

2

erf

2aB

R

p

e

199

h j

q2

1

j i ¼

2aA

3=2

ð

Þ

q

A1

p

hgAj

0jgAi ¼

ð

3pÞp

ð200Þ

qR2

rA1

q2

1

2aB

3=2

2

1

1

e

aAaBR2= aAþaBÞ

p

2

hgBj qR2 rB1

0jgBi ¼

ð

3pÞp

"

ð201Þ

0

p

B

R2

s

hgAj

q

jgBi ¼ p

ð4aAaBÞ3=4

ð

p

erf

0

aB

R1

qR2

rA1

a2

3=2

2

aA aB

@

þ

A

aAþaB

a

a

s

a2

þ

3=2

#

þ

2

2

aB

Re

B

R2

aB

R2

e aB2 R2=aAþaB

aA

aB

A

B

ð202Þ

550

k. k. liang et al.

2

1

p

p

1

hgAj

q

0jgAi ¼ ppR3

"

p

erf

2aAR

qR2

rB1

2

#

ð203Þ

2aARe 2aAR2

ð2aAR2Þ3=2e 2aAR2

p

We found that

Ugð2Þ ¼ 0:179874

ð204Þ

in the atomic unit. This is the force constant of the oscillator. To demonstrate that this number agrees reasonably with that extracted from the potential energy curve obtained in other quantum chemical calculation, we wish to show that the parabolic curve defined by

U ¼ Ugð2ÞðR R0Þ2þU0

ð205Þ

matches the shape of the potential energy curves obtained in other calculations near the bottom of the potential. For this purpose, we chose to compare our result to that from the simple MO calculation done by Slater [14]. This comparison is shown in Figure 6. The equilibrium position is shifted to coincide with that calculated by Slater. Our force constant value appears to be reasonable.

Figure 6. Matching the calculated harmonic potential to the potential curve obtained by Slater with simple MO theory.

the crude born–oppenheimer adiabatic approximation

551

VI. CONCLUSIONS

In the crude BO approximation, the problem of PES crossing can be avoided. However, the price is to pay. First, there will be degeneracy instead of crossing that we would encounter; second, all of the molecular properties have to be obtained by carrying out the perturbation calculation, which involves the computation of a huge number of matrix elements in realistic cases; third, since the expansion is around one nuclear configuration, the speed of convergence of the perturbation series might be a problem when the nuclear motion is significant.

Nevertheless, the examination of the applicability of the crude BO approximation can start now because we have worked out basic methods to compute the matrix elements. With the advances in the capacity of computers, the test of these methods can be done in lower and lower cost. In this work, we have obtained the formulas and shown their applications for the simple cases, but workers interested in using these matrix elements in their work would find that it is not difficult to extend our results to higher order derivatives of Coulomb interaction, or the cases of more-than-two-atom molecules.

APPENDIX A: USEFUL INTEGRALS

First, it will be very useful to remember that

1

2

I0 ð1 e ax dx

1

2p

2

1

2

1

2

I02 ¼ ðr¼0 ðy¼0 e ar

r dr dy ¼ 2p ð0

e ar

r dr ¼ p ð0

e ar

dr2

ðA:1Þ

p

0

2

p

¼

ð1

de ar

¼

a

a

¼ r

I0

p

a

With a positive integer n, we find

1

2

I2n 1 ð1 dx x2n 1e ax

¼ 0

ðA:2Þ

1

2

1

2

1

2

I2n ð1 dx x2ne ax

¼ 2 ð0

dx x2ne ax

¼

ð0

x2n 1e ax

dx2

1

1

1

1

1

2n 1

ax2

2n 1

ax2

2 n 1

ax2

¼

a

ðx¼0 x

de

¼

a

x

e

x¼0 ð2n 1Þ ð0

dx x

ð Þe

2n

1

1

2

2n

1

¼

ð0

dx x2ðn 1Þe ax

¼

I

ð

A:3

Þ

a

2a

2ðn 1Þ

	552		k. k. liang et al.
	By doing this iteratively, we can show that
	I	2n ¼	ð2n 1Þ!!	I	ð2n 1Þ!!
			2nan		2nan
				0 ¼

Note that

r

p

ðA:4Þ

a

ð

2n

1

Þ

!!

¼

ð2nÞ!

¼

ð2nÞ!

ð

A:5

Þ

ð2nÞ!!

2nn!

we have

r

¼

2

n!a

ð

Þ

I2n

ð2nÞ!

p

A:6

n

2n

a

The following integral is often encountered

I2nþ1

ð 1 dt 1 þ t2 ðnþ

Þ

ðA:7Þ

1

3=2

Letting

2n

¼

tan y, and

noticing that

dt

¼

dy=cos 2y, and

ð

1

þ

t2

Þ

ðnþ3=2Þ

¼

t

ðcos yÞ

þ

3

, we obtain

I2nþ1 ¼ ðp=2

ðcos yÞ2nþ1dy

y¼ p=2

ðp=2

¼ðcos yÞ2nd siny

y¼ p=2

		p=2		p=2
				ðy¼ p=2		ðcos yÞ2n 1sin2y dy
¼ sinyðcos yÞ2n y¼ p=2			þ2n
p=2					p=2
¼ 2n ðy¼ p=2	ðcos yÞ2nþ1dy þ 2n				ðy¼ p=2		ðcos yÞ2n 1dy ðA:8Þ
¼ 2nI2nþ1 þ 2nI2n 1							ðA:9Þ

Therefore,

I2nþ1 ¼	2n	ðA:10Þ
	2n þ 1 I2n 1

the crude born–oppenheimer adiabatic approximation

553

Consequently,

2n

I2nþ1 ¼ 2n þ 1 I2n 1

2n 2

I2n 1 ¼ 2n 1 I2n 3

.

.

.

2

I3 ¼ 3 I1

and

I1 ¼	ðp=2
	cos y dy

y¼ p=2

ðp=2

¼dsiny

y¼ p=2

	¼ 2								ðA:11Þ
Thus
I				ð2nÞ!!			I
	2nþ1	¼ ð2n þ 1Þ!!						1
		¼	2	ð2nÞ!!ð2nÞ!!
				ð	n þ	1	Þ	!
				2
		¼	2	4nðn!Þ2				ð		A:12	Þ
				ð2n þ 1Þ!

Next, we shall consider the integral basically corresponding to the Rys’ polynomial problem [15,16]. Letting

1

we find that

JnðaÞ ¼ ð0 dx x2ne ax2

ðA:13Þ

1

1

2

1

1

2

Jn ¼

ð0 x2n 1e ax dx2

¼

ðx¼0 x2n 1de ax

2

2a

x2n 1e ax2

1

1

x2ðn 1Þe ax2 dx

1Þ ð0

1

a

¼ 21a

x¼0 ð2n

e

2n 1

x2ðn 1Þe ax2 dx

¼ 2a

þ

2a ð0

¼

e a

þ

2n 1

J

n 1

ð

A:14

Þ

2a

2a