Concise Physical Chemistry

c17 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

276 THE VARIATIONAL METHOD: ATOMS

The same treatment produces a similar operator for electron 2:

Hˆ 2 =−^1

2 r 22

d

dr 2

r 22

d

dr 2

−

2

r 2

+

∫∞

0

φ 1

1

r 12

φ 1 dτ

We do not know theorbitalsof the electrons either. We can reasonably assume

that the ground state orbitals of electrons 1 and 2 are similar but not identical to the

1 sorbital of hydrogen:

φ 1 =

√

a^3

π

e−αr^1

and

φ 2 =

√

b^3

π

e−αr^2

The integral inHˆ 1 , representing the Coulombic interaction between electron 1 atr 1

and electron 2 somewhere in orbitalφ 2 , has been evaluated for Slater-type orbitals

(Rioux, 1987; McQuarrie, 1983) and is

V 1 =

∫∞

0

φ 2

1

r 12

φ 2 dτ=

1

r 1

[

1 −(1+br 1 )e−^2 br^1

]

Now the approximate Hamiltonian for electron 1 is

hˆ 1 =−^1

2 r 12

d

dr 1

r^21

d

dr 1

−

2

r 1

+

1

r 1

(

1 −( 1 +br 1 )e−^2 br^1

)

with a similar expression forhˆ 2 involvingar 2 in place ofbr 1 in the Slater orbital.

The orbital is normalized, so the energy of electron 1 is

E 1 =

∫∞

0

φ 1 hˆ 1 φ 1 dτ

with a similar expression forE 2.

CalculatingE 1 requires solution of three integrals:

E 1 =

∫∞

0

φ 1

(

−^12 ∇ 12

)

φ 1 dτ−

∫∞

0

φ 1

(

Z

r 1

)

φ 1 dτ+

∫∞

0

φ 1 (V 1 )φ 1 dτ

They yield (Rioux, 1987) three terms for the energy of the electron in orbitalφ 1 :

E 1 =

a^2

2

−Za+

ab

(

a^2 + 3 ab+b^2

)

(a+b)^3