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

534

k. k. liang et al.

Therefore

hZAj qqR rA1 0jZBi

1

2

N

N

X X

1

1

nAþnB

1

1ÞlAþlB

!

1 þ ð 1ÞmAþmB

mAþ1 mB

¼

p

A

B mA

0

þ ð

Þ

þ ð

¼

0 mB

¼

2

2

2

CmAþ1CmB

ðnA þ nBÞ!ðlA þ lBÞ!ðmA þ mBÞ!

RmAþmB mA mBþ1

mA

mB 2nA

nB

lA

lBþmAþmB

nAþnB

!

lAþlB

!

mAþmB

!

1

dt

þ

þ

þ

m

1

2

t2 2

2

aB

mB mB

ð0

aBA mA

þ

1

ðaA þ aB þ t2Þ3=2

aA þ aB þ t2

aA þ aB þ t2

1

ðnAþnBþlAþlBþ2mA mAþmBÞ=2

2

2

2

e aBðaAþt

ÞR

=ðaAþaBþt

Þ

ð137Þ

aA þ aB þ t2

¼

By introducing the new variable x

t=paA þ aB þ t2 as usual, we have

hZAj qqR rA1 0jZBi

1

mA

mB

!

2

N

N

X X

1

1

nAþnB

1

1ÞlAþlB

1 þ ð 1ÞmAþmB

mAþ1 mB

¼ p

A

B

þ ð

Þ

þ ð

¼

0

¼

0

2

2

2

CmAþ1CmB

ðnA þ nBÞ!ðlA þ lBÞ!ðmA þ mBÞ!

RmAþmB mA mBþ1

mA

mB 2nA

þ

nB

þ

lA

þ

lB

þ

mA

þ

mB

nAþnB

!

lAþlB

!

mAþmB !

2

2

2

m

1

m

1

ðnAþnBþlAþlBþ2mAþ2mB mA mBÞ=2

2

aBA mAþ

ð 1Þ

B mB

e aAaBR

=ðaAþaBÞ

aA þ aB

1

ð0 dx x2 1 x2

ðnAþnBþlAþlBþ2mA mAþmBÞ=2 aA þ aBx2 mB mB e aB2 R2x2=ðaAþaBÞ

ð138Þ

The integral over x in Eq. (138) is discussed in the Appendix. From Eq. (A.18), we can find the expression for this integral. Inserting it into Eq. (138), we have

hZAj qqR rA1 0jZBi

1

X X

1

1

nAþnB

1

!

1 þ ð 1ÞmAþmB

2

N

N

mAþ1 mB

1ÞlAþlB

p

B mA

0 mB

0

þ ð 2

Þ

þ ð

2

2

¼

A

¼

¼

the crude born–oppenheimer adiabatic approximation

535

CmAþ1CmB

ðnA þ nBÞ!ðlA þ lBÞ!ðmA þ mBÞ!

!

RmAþmB mA mBþ1

mA

mB 2nA

þ

nB

þ

lA

þ

lB

þ

mA

þ

mB

nAþnB

! lAþlB

! mAþmB

m

1

m

1

ðnAþnBþlAþlBþ2mAþ2mB mA mBÞ=2

2

aBA

mAþ

ð 1Þ

B mB

e aAaBR

=ðaAþaBÞ

aA þ aB

Ix 1;

nA þ nB þ lA þ lB þ 2mA mA þ mB

; mB mB; aA; aB; R

ð139Þ

2

2.Second-Order Derivatives

To calculate the matrix elements of second-order derivatives, we have

2

1

hZAj

q

0jZAi

qR2

rA1

N2

1

ð dt t

zA1

t2

aAþ

2

Þ

2

ð140Þ

¼ pp

ð 1 dt

4

2

xA1yA1zA1 e ð

t

A1

A

2

2n 2l 2m

2

r

Neglecting the subscripts of the coordinates for simplicity, one obtains

2

1

hZAj

q

0jZAi

qR2

rA1

N2

1

1

e ð

aA

þ

2

Þ

2

1

e ð

aA

þ

2

Þ

2

1

þ

e ð

aAþ

2

Þ

2

¼ pp ð 1 dt t

ð 1 dx x

2n

ð 1 dy y

2l

ð 1 dz z

2m

A

4

2

t x

2

t y

2

2

t z

1

t2

1

2n

2aA

t2

x2

1

2l

2aA

t2

y2

1

2m

2aA

t2

z2

ð 1 dt

ð 1 dx x

e ð

þ

Þ

ð 1 dy y

e ð

þ

Þ

ð 1 dz z

e ð

þ

Þ

2

N2

2n ! 2l

! 2m

2 !

1

p

3=2

1

nþlþmþ1

A

ð Þ ð

Þ ð

þ

Þ

dt t4

1ð

t2

3=2

1

nþlþm

þ

t2

2n ! 2l ! 2m

¼ pp "22nþ2lþ2mþ2n!l! m þ

1Þ! ð 1

2aA þ t2

2aA

#

ð 1

þ

þ

ð

Þ ð

Þ ð

Þ

dt

p

22nþ2lþ2mn!l!m!

2 2aA

t2

þ

2aA

t2

þ

ð

ð

Þ

¼ p

N2

ð2nÞ!ð2lÞ!ð2mÞ!

2m

1

1

dt

t4

1

nþlþmþ1

A 22nþ2lþ2mn!l!m! "ð

þ Þ

0 2aA

t2 3=2

2aA

t2

1

dt

t2

1

nþlþm

#

ð0

ð2aA þ t2Þ3=2

2aA þ t2

2

nþlþm 1

ð2nÞ!ð2lÞ!ð2mÞ!

1

dt

t2

1

N2

2m

¼ p

ð0 ð2aA þ t2Þ3=2

2aA þ t2

2aA þ t2

ð

A 22nþ2lþ2mn!l!m! "

1

ð

þ

Þ

þ

þ

nþlþm

2

aA

t2

1

0

dt

2aA

t2

3=2 2aA

t2

2aA

t2

#

ð141Þ

536	k. k. liang et al.

It has to be noted that, when reduced to the pure Gaussian case, that is, when n þ l þ m ¼ 0, m must always be 0, and

q2 1

2

1

2aA

t2

hZAj

0jZAi ¼ pNA

ð

0

dt

ð142Þ

qR2

rA1

ð

2aA

þ

t2

Þ

3=2

2aA

þ

t2

By introducing a new variable

d

t

144

x ¼ p2aA þ t2

ð143Þ

x ¼

2aAdt

ð Þ

ð2aA þ t2Þ3=2

we find that

q2

1

2

1

2

hZAj

0jZAi ¼ pNA

ð0 x

dx

qR2

rA1

¼

pNA2

ð145Þ

3

In the general cases where n þ l þ m > 0, it is

guaranteed

that

n þ l þ m

1 0. Therefore, by using the same new variable, we find

	2			1
hZAj		q				0jZAi
		qR2			rA1
¼ p	N2		ð2nÞ!ð2lÞ!ð2mÞ!
		A ð8aAÞnþlþmn!l!m!

1				1			nþlþm
2m ð0 dx x4 1 x2			nþlþm 1	ð0 dx x2 1 x2
							ð146Þ

where we have made use of the fact that

1 x

2

¼

2aA

ð147Þ

2aA þ t2

By noticing that

1

m

1

ð0 dx x2n 1 x2

m ¼ m

0 Cmmð 1Þmð0 dx x2ðnþmÞ

¼

X

ð 1Þm

m

Cm

148

¼ m

ð

Þ

0

m 2ðn þ mÞ þ 1

¼

X

the crude born–oppenheimer adiabatic approximation

we find that

2

1

hZAj

q

0jZAi

qR2

rA1

X

ð 1Þm

¼ p

N2

ð2nÞ!ð2lÞ!ð2mÞ!

2m nþlþm 1 Cnþlþm 1

1

A 22nþ2lþ2mn!l!m! "

m

0

m

2 m

þ

2

Þ þ

¼

ð

X

1

m

#

nþlþm

0

Cmnþlþm

2

ð

Þ

1

m

ð

m

1

¼

þ

Þ þ

In the second case,

2

1

hZAj

q

0jZAi

qR2

rB1

N2

1

t

zB1

t2

e

aA

2

2

2

¼ pp ð 1 dt ð dtxA1yA1zA1

2

A1

t r

B1

A

2n 2l 2m

4

2

2

r

537

ð149Þ

ð150Þ

By using

the relations: xB1 ¼ xA1,

yB1 ¼ yA1,

and

zB1 ¼ zA1 R,

we shall

simplify Eq. (150) by letting x ¼ xA1, y ¼ yA1, and z ¼ zA1. Thus

2

1

hZAj

q

0jZAi

qR2

rB1

N2

1

1

e ð aAþ

2

Þ

2

1

e ð

aAþ

2

Þ

2

¼ pp

ð 1 dt

ð 1 dx x

2n

x

ð 1 dy y

2l

y

A

2

t

2

t

1

2m

4

2

t2

2aA

t2

z2

2t2Rz

t2R2

ð 1 dz z

t

ðz

RÞ

e ð

þ

Þ

þ

2

ð

Þ

¼ pp

22nþ2ln!l!

ð 1

2aA

þ t2 2aA þ t2

nþle 2aAt2R2

NA2

ð2nÞ!ð2lÞ!

1

dt

p

1

=

2aAþt2

1

2m

4

2

t2

2aA

t2

z

t2R= 2aA

t2

2

ð 1 dz z

t

ðz

RÞ

e ð

þ

Þ½

ð

þ

Þ&

ð151Þ

2

By making a change of the variable z0 ¼ z t2R=ð2aA þ t2Þ and by ignoring the prime sign in the dummy index results in

2

1

hZAj

q

0jZAi

qR2

rB1

2aA þ t2 2aA þ t2

ð

Þ

¼ pp

22nþ2ln!l!

ð 1

nþle 2aAt2R2

NA2

ð2nÞ!ð2lÞ!

1

dt

p

1

=

2aAþt2

1

t2R

2m

4

2aAR

2

t2

2aA

t2

z2

ð 1

"t

#e ð

þ Þ

þ

2aA

þ

2

ð152Þ

dz

z þ 2aA

t2

z

t2

538

k. k. liang et al.

Note that

þ

þ

ð 1

2m

2

1

t2R

2

R

2

2

2

Iz

ð 1

dz z þ

2aA

þ

t2

"t4 z

2aA

t2

2

#e ð2aAþt

Þz

2m

þ

þ

1

t2R

4

R

2

R

2

2

2

2

¼

dz z þ

2aA

t2

"t4z2 t4

2aA

t2

z þ t4

2aA

t2

2

#e ð2aAþt

Þz

m

t2R

2m 2m 1

2 2

X

¼ t4 m¼0 C22mm

2aA þ t2

ð 1 dz z2ðmþ1Þe ð2aAþt Þz

4 R

m 1

t2R

2m 2m 1 1

2 2

t

4

2aA

þ t2

X

2aA þ t2

ð 1 dz z

ðmþ Þe ð aA

þ

Þ

z

m 0 C2mþ1

aA

2m

2

1

2

t

¼

2 t2

t2R

2m 2m 1

4

2aAR

m

2m

2

t2 z2

X

þ "t

2aA þ t2

2

# m 0 C2m

2aA þ t2

ð 1 dz z2me ð aAþ Þ

r

¼

2m 2m ð2m þ

mþ1

m

C2m

t2R

2Þ!

1

p

(

t4

m

¼

X

2m

0

2aA

þ

t2

2aA

þ

t2

22mþ2

ð

m

þ

1 ! 2aA

þ

t2

¼

Þ

mþ1

t4 4aAR

m 1

C2m

t2R

2m 2m 1 ð2m þ 2Þ!

1

X

2mþ1

2aA þ t2

m

0

2aA þ t2

22mþ2ðm þ 1Þ!

2aA þ t2

t4

¼

2 t2

C2m

t2R

2m 2mð2mÞ!

1

m

2aAR

m

X

þ"

2aA þ t2

2

# m 0

2m

2aA þ t2

22mm!

2aA þ t2

)

r

¼

2m 2mð2mÞ!

m

m

C2m

t2R

1

2m þ

1

t2

p

(

t2

m

¼

X

2m

2aA

þ

t2

¼

0

2aA

þ

t2

22mm!

2aA

þ

t2

2

2aA

þ

t2

4

m 1

C2m

t2R

2m 2mð2mÞ!

1

m

m

t2

X

2m

ð mÞ

aA m 0

2aA þ t2

22mm!

2aA þ t2

2aA þ t2

t4

¼

2 t2

C2m

t2R

2m 2mð2mÞ!

1

m

2aAR

m

X

þ"

2aA

þ

t2

2

# m 0

2m

2aA

þ

t2

22mm!

2aA

þ

t2

)

¼

ð153Þ

In the second summation, we find that, since m m ¼ 0 when m ¼ m, we can safely extend the upper limit to m. Thus,

I

rþ

m

m

C2m

þ

2m 2m

þ

m

þ

p

t2

t2R

ð2mÞ!

1

2m þ 1

t2

z ¼

(

X

2m

2aA

t2

¼

0

2aA

t2

22mm!

2aA

t2

2

2aA

t2

4

m 1

C2m

t2R

2m 2mð2mÞ!

1

m

m

t2

X

2m

ð mÞ

aA m 0

2aA þ t2

22mm!

2aA þ t2

2aA þ t2

¼

the crude born–oppenheimer adiabatic approximation

t4

2aAR

2 t2

m

C2m

t2R

2m 2mð2mÞ!

1

m

X

)

þ"

2aA

þ

t2

2

# m 0

2m

2aA

þ

t2

22mm!

2aA

þ

t2

¼

2m 2m

m

þ

m

rþ m

0

þ

p

m

C2m

t2R

ð2mÞ!

1

¼

X

2m

2aA

t2

2aA

t2

22m !

2aA

t2

¼

þ

þ

þ

2m þ

1

t2

t2

4

m

t2

t4

2aAR

2

t2

"

2

2aA

t2

aA

ð mÞ

2aA

t2 þ

2aA

t2

2

r m

0

t2R

2m 2m

m

1

m

p

m

C2m

ð2mÞ!

¼

X

2m

2aA

þ

t2

2aA

þ

t2

22m !

2aA

þ

t2

¼

2

t2

t2

(4aA2 R2

þ mt2 4aAðm mÞ aA

)

2aA þ t2

2aA þ t2

539

#

ð154Þ

By inserting Eq. (154) into Eq. (152), we found that

2

1

hZAj

q

0jZAi

qR2

rB1

2n ! 2l !

m

2

! 1

dt

1

nþlþm

t2R

2m 2m

2 N2

ð

Þ ð Þ

C2m

ð

mÞ

¼

m¼0

22mm! ð0

2aA

t2

3=2 2aA þ t2

2aA þ t2

p

A 22nþ2ln!l!

2m

þ

Þ

X

2

ð

t2

t2

2

2

2

(4aA2 R2

þ mt2 4aAðm mÞ aA

)e 2aAt

R

=ð2aAþt

Þ

2aA þ t2

2aA þ t2

By introducing the new variable as in Eq. (143), one obtains

ð155Þ

2

1

hZAj

q

0jZAi

qR2

rB1

ð

Þ

m

ð2mÞ!

1

X

2

2

¼

2 N

2

ð2nÞ!ð2lÞ!

C2m

R2m 2m

ð0

d

1

x

nþlþm

2m 2m

p

A 8aA

nþln!l! m¼0

2m ð8aAÞmm!

x

x

4aA2 R2

x2

2þ 2maA

x2

4aAðm mÞ aA x2 e 2aAR2x2

1 x2

ð

m

Þ

X

ð2mÞ!

¼

2

N

2

ð2nÞ!ð2lÞ!

C2m

R2m 2m

p

A 8aA

nþln!l! m¼0

2m ð8aAÞmm!

1

1

4aA2 R2 ð0 dx 1 x2 nþlþm x2 2m 2mþ2e 2aAR2x2

1

þ 2maA ð0 dx 1 x2

nþlþm 1

x2

2m 2mþ2e 2aAR2x2

aA½4ðm mÞ þ 1& ð

nþlþm

ð156Þ

dx 1 x2

x2

2m 2mþ1e 2aAR2x2

0

540	k. k. liang et al.

The second integral in Eq. (155) seemed to be singular when n þ l þ m ¼ 0. However, in this case, m must be zero, and consequently this term will never contribute to the final result for being suppressed by the prefactor. With the definition in Eq. (132), we can write

2

1

hZAj

q

0jZAi

qR2

rB1

X

ð

Þ

ð2nÞ!ð2lÞ!

m

ð2mÞ!

¼

2

p

N2

C2m

R2m 2m

A 8aA

nþln!l! m

¼

0

2m ð8aAÞmm!

2

2

2m 2mþ2

n4aAR

Xnþlþm;2m 2mþ2;0;2aA ðRÞ=ð2aAÞ2m 2m

þ

2

þ 2maAXnþlþm 1;2m 2mþ2;0;2aA ðRÞ=ð2aAÞ

aA½4ðm mÞ þ 1&Xnþlþm;2m 2mþ1;0;2aA ðRÞ=ð2aAÞ2m 2mþ1o

ð157Þ

The last kind of second-order derivative considered is of the following form:

2

1

hZAj

q

0jZBi

qR2

rA1

¼

NANB

1

ð dt xAnA1xBnB1yAlA1yBlB1zAm1A zBm1B

t4zA21

t2

2

2

2

ð158Þ

pp

ð 1 dt

2

e ðaAþt

ÞrA1 e aBrB1

With the specific geometry, we can let xA1 ¼ xB1 ¼ x, yA1 ¼ yB1 ¼ y, zA1 ¼ z, and zB1 ¼ z R. After changing these variables, we obtain

2

1

hZAj

q

0jZBi

qR2

rA1

NANB

1

1

2

2

1

2

2

¼

pp

ð 1 dt

ð 1 dx xnAþnB e ðaAþaBþt

Þx

ð 1 dy ylAþlB e ðaAþaBþt

Þy

mA

mB 4 2

aA

aB

t

z

2aBRz aBR

1

t2

2

2

2

ð 1 dz z ðz RÞ

t zA1

e ð

þ þ

Þ

þ

2

ð 1

2

2

NANB

1

dt

1

1 nAþnB

1 þ ð 1ÞlAþlB

nA þ nBÞðlA þ lB

Þ

p

þ ð 2 Þ

1

ðnAþnB

þlAþlB

Þ=2

2

2

2

¼ pp

2

!

2nAþðnBþlA

þlB

nAþnB !

lAþlB !

e aBðaþt Þ=ðaAþaBþt ÞR Iz

aA þ aB þ t2

aA þ aB þ t2

ð159Þ

the crude born–oppenheimer adiabatic approximation

541

where

Iz ¼

¼

¼

1 dz z

aB

R

mA

z

aA þ t2

R

mB

þ aA

þ aB þ t2

aA þ aB þ t2

ð 1

2

aB

t2

2

2

"t4 z þ

#e ðaAþaBþt

Þz

aA þ aB þ t2

2

mB mB

mA

mB

CmA CmB

aBR

mA mA

aA þ t2

R

mA

¼ 0 mB

¼ 0

mA

mB

aA þ aB þ t2

aA þ aB þ t2

X X

1

2

2

2aBRt4

1

2

2

t4 ð 1 dz zmA þmB þ2e ðaA þaB þt

Þz

þ

ð 1 dz zmAþmBþ1e ðaAþaBþt

Þz

aA þ aB þ t2

"

t2aBR

2

t2

#

1

2

2

)

þ

dz zmA þmB e ðaAþaBþt

Þz

þ

þ

ð 1

mA mA

mB mB

mA

mB

CmA CmB

aBR

aA þ t2

R

p

mA

0 mB

0

mA

mB

A

aB

þ

t2

aA

þ

aB

þ

t2

aA

aB

t2

¼

¼

X X

8

1

1

mA þmB

mA

mB

2 !

1

ðmA þmB þ2Þ=2

þ ð

Þ

t4

ð

þ þ

Þ

2

2 mA mB

2

t2

<

2mA þmB þ

þ2

þ

!

aA

þ aB þ

:1

mA þmB

t2aBR

2

mA

mB !

ðmA þmB Þ=2

1

t2

1

þ

þ ð

Þ

"

#

ð þ Þ

2

aA aB

t2

2

2mAþmB

mA þmB

!

aA aB

t2

þ 4

þ

2

þ þmA

mA

mB

þ

mB

1 =2

9

1

1

þ

2aBRt

m

mB

1 !

1

ð

þ Þ

ð

Þ

ð

A

þ

þ

Þ

þ

2

aA

t2

1

mA

mB

1

t2

þ aB þ

2mAþmB

þ

þ2

þ

!

aA þ aB þ

=

160;

ð

Þ

Note that the first two integrals in Eq. (160) have nonvanishing prefactors when mA þ mB is even, while the last integral has nonvanishing prefactor when mA þ mB is odd. They do not contribute in the final form at the same time. By inserting Iz into Eq. (159), the familiar forms of integral over t as in the calculation

of other matrix elements appear. Again, by changing into the variable p

x ¼ t= aA þ aB þ t2 and integrating, it can be shown that

2

1

hZAj

q

0jZBi

qR2

rA1

!

2

2

¼

2 N

N

1 þ ð 1ÞnAþnB

1 þ ð 1ÞlAþlB

ðnA þ nBÞ!ðlA þ lBÞ!

!

p

A B

2

2

2nAþnBþlAþlB

nAþnB !

lAþlB

mA

mA

aAaB 2

X X

CmA CmA e

R mA mA

1 mB mB RmAþmB mA mB

mA mA

aB

ð Þ

mA¼0 mB¼0

2mAþmB

542	aA þ aB		k. k. liang et al.
			ðnAþnBþlAþlBþ2mAþ2mB mA mBÞ=2		þ ð 2	Þ
	1			1	1		mAþmB

mA þ mB þ 1 ð1 dx x2 2 1 x2 ðnAþnBþlAþlB þ2mA mA þmBÞ=2 1

2 0

aA þ aBx2 mB mB e a2BR2x2=ðaAþaBÞ

ðmA þ mBÞ!

mA þmB !

2

þ

1

1

a2 R2

2

nA

nB

lA

lB

2mA

mA

mB

=2

þ

B

ð0 dx x2

1 x2 ð

þ

þ

þ

þ

þ

Þ

aA

aB

B

þaBÞ ð0 dx x2

mA

aA þ aBx2

mB

mB e aB2 R2x2=ðaA

1 x2

m

ðnAþnBþlAþlBþ2mA mAþmBÞ=2 1

aA aBx2 mB mB e aB2 R2x2=ðaA

þaBÞ

1 þ ð 1Þ

þ

ðmA þ mB þ 1Þ!

þ

þ

2

2mAþmB þ1

mAþ2Bþ

1

!

m

þ

1

2

R

2

n n l

l

2m

1

2

paAaB

aB

ð0 dx x2

1

x2 ð A þ Bþ

Aþ

Bþ

A mAþmB

Þ=

2

mB

mB

e

aB2 R2x2= aA

þ

aB

Þ

ð161Þ

aA þ aBx

ð

These integrals over x are discussed in the appendix. Inserting the formulas for these integrals obtained from Eq. (A.18) into Eq. (161), we will obtain an expression for computation.

V.HYDROGEN MOLECULE: MINIMUM BASIS

SET CALCULATION

To calculate the matrix elements for H2 in the minimal basis set, we approximate the Slater 1s orbital with a Gaussian function. That is, we replace the 1s radial wave function

fSðrÞ ¼ zp3 1=2e zr

with a Gaussian function jgi defined earlier. It can be shown that

þ

2p

AaB

3=2

2

SAB ¼ hgAð1ÞjgBð1Þi ¼

a

e aAaBR

=ðaAþaBÞ

aA

aB

hgAð

1

ÞgAð Þjr12jgAð

1

ÞgAð

2

Þi ¼ pp

2

1

2

paA

ð162Þ

ð163Þ

ð164Þ

the crude born–oppenheimer adiabatic approximation

543

hgBð

ÞgBð

Þjr12jgBð

ÞgBð

Þi ¼ pp

ð

Þ

1

2

1

1

2

2

paB

165

1

1

hgAð1ÞgAð2Þj

jgBð1ÞgBð2Þi ¼ hgBð1ÞgBð2Þj

jgAð1ÞgAð2Þi

r12

r12

r

3=2

2

aA þ aB

4aAaB

e ðaA2 þaB2 ÞR2=ðaAþaBÞR2

166

1

2

ð

aA

þ

aB

Þ

2#

1

ð Þ

¼ pp

"

hgAð1ÞgBð2Þj

jgAð1ÞgBð2Þi ¼ hgBð1ÞgAð2Þj

jgBð1ÞgAð2Þi

r12

r12

r

þ

¼

1

erf

2aAaB

R

ð167Þ

R

aA

aB

1

1

hgAð1ÞgBð2Þj

jgBð1ÞgAð2Þi ¼ hgB

ð1ÞgAð2Þj

jgAð1ÞgBð2Þi

r12

r12

r

3=2

2

aA þ aB

4aAaB

e aAaBR2=ðaAþaBÞ

168

2

ð

aA

þ

aB

Þ

2#

ð Þ

¼ pp

"

and

p

1

1

1

1

erf

R

hgAð1Þj

rB1

jgAð1Þi ¼

R

2aA

ð169Þ

erf

p3=4

R

hgBð1Þj

rA1

jgBð1Þi ¼

R

2aB

ð170Þ

1

4aAaB

Þ

aB e aAaBR

2

=ðaAþaBÞ

1

aB

aB

R

ð171Þ

hgAð1ÞjrA1jgBð1Þi ¼ aðBpaA

R erf

paA

a

R

þ

þ

1

4

AaB

Þ

3=4

2

=ðaAþaBÞ

1

aA

aB

R

ð172Þ

hgAð1ÞjrB1jgBð1Þi ¼ aðApaA

aB e aAaB

R erf

paA

þ

þ

Now we can calculate the ground-state energy of H2. Here, we only use one basis function, the 1s atomic orbital of hydrogen. By symmetry consideration, we know that the wave function of the H2 ground state is

		þ	ð173Þ
	j gi ¼ k s s ji
where
jsi ¼	1	jw1s;Ai þ jw1s;Bi	ð174Þ
	p
	2ð1 þ SABÞ