Concise Physical Chemistry

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

274 THE VARIATIONAL METHOD: ATOMS

17.4 HELIUM

The helium atom is similar to the hydrogen atom, with the critical difference that there

are two electrons moving in the potential field of the nucleus. The nuclear charge is

+2. The Hamiltonian for the helium atom is

Hˆ =−^1

2 ∇

2

1 −

1

2 ∇

2

2 −

2

r 1

−

2

r 2

+

1

r 12

Regrouping, we obtain

Hˆ =

(

−^12 ∇ 12 −

2

r 1

)

+

(

−^12 ∇ 22 −

2

r 2

)

+

1

r 12

The first two terms on the right replicate the hydrogen case, except for a different

nuclear charge. The third term on the right, 1/r 12 , is due to electrostatic repulsion

of the two electrons acting over the interelectronic distancer 12. This term does

not exist in the hydrogen Hamiltonian. The sum of two nuclear and one repulsion

Hamiltonians is

HˆHe=Hˆ 1 +Hˆ 2 +^1

r 12

If this Hamiltonian were to operate on an exact, normalized wave function for helium,

the exact energy of the system would be obtained:

EHe=

∫∞

0

(r 1 ,r 2 )HˆHe(r 1 ,r 2 )dτ

The helium atom, however, is a three-particle system for which we cannot obtain an

exact solution. The orbital and the total energy must, of necessity, be approximate.

As a naive orzero-orderapproximation, we can simply ignore the “r 12 term” and

allow the simplified Hamiltonian to operate on the 1sorbital of the H atom. The

result is

EHe=−

22

2

−

22

2

=− 4. 00 Eh

which is 8 times the exact energy of the hydrogen atom (−^12 Eh). The 2 in the

numerators are the nuclear chargeZ=2. In general, the energy of any hydrogen-like

atom or ion is−Z^2 /2 hartrees per electron, provided that we ignore interelectronic

electrostatic repulsion.

We can compare this result with the first and secondionization potentials(IP) for

helium, which are energies that must go into the system to bring about ionization:

He→He++e−→He^2 ++e−