Physical Chemistry , 1st ed.

where the other variables have been defined in the previous chapter. The 2 in
the numerator of each of the first two terms is due to the 2charge on the
helium nucleus. The first two terms are negative, indicating an attraction, and
the final term is positive, indicating a repulsion. The complete Hamiltonian
operator for the helium atom is

Hˆ

2





2

^21 

2





2

^22 

4

2



e

0

2

r 1



4

2



e

0

2

r 2



4

e



2

0 r 12

 (12.4)

This means that for the helium atom, the Schrödinger equation to be solved is

 2





2

^21 

2





2

^22 

4

2



e

0

2

r 1



4

2



e

0

2

r 2



4

e



2

0 r 12

Etot(12.5)

where Etotrepresents the total electronic energy of a helium atom.
The Hamiltonian (and thus the Schrödinger equation) can be rearranged by
grouping together the two terms (one kinetic, one potential) that deal with
electron 1 only and also grouping together the two terms that deal with elec-
tron 2 only:

Hˆ

2





2

^21 

4

2



e

0

2

r 1



2





2

^22 

4

2



e

0

2

r 2



4

e



2

0 r 12

(12.6)

This way, the Hamiltonian resembles two separate one-electron Hamiltonians
added together. This suggests that perhaps the helium atom wavefunction is
simply a combination of two hydrogen-like wavefunctions. Perhaps a sort of
“separation of electrons” approach will allow us to solve the Schrödinger equa-
tion for helium.
The problem is with the last term:e^2 /4
 0 r 12. It contains a term,r 12 , that
depends on the positions ofbothof the electrons. It does not belong only with
the terms for just electron 1, nor does it belong only with the terms for just
electron 2. Because this last term cannot be separated into parts involving only
one electron at a time, the complete Hamiltonian operator is not separableand
it cannot be solved by separation into smaller, one-electron pieces. In order for
the Schrödinger equation for the helium atom to be solved analytically, it
either must be solved completely or not at all.
To date, there is no known analytic solution to the second-order differen-
tial Schrödinger equation for the helium atom. This does not mean that there
is no solution, or that wavefunctions do not exist. It simply means that we
know of no mathematicalfunction that satisfies the differential equation. In
fact, for atoms and molecules that have more than one electron, the lack of
separability leads directly to the fact that there are no known analytical solutions
to any atom larger than hydrogen.Again, this does not mean that the wave-
functions do not exist. It simply means that we must use other methods to un-
derstand the behavior of the electrons in such systems. (It has been proven
mathematically that there is no analytic solution to the so-called three-body
problem, as the He atom can be described. Therefore, we must approach multi-
electron systems differently.)
Nor should this lack be taken as a failure of quantum mechanics. In this text,
we can only scratch the surface of the tools that quantum mechanics provides.
Quantum mechanics does provide tools to understand such systems. Atoms
and molecules having more than one electron can be studied and understood
by applying such tools to more and more exacting detail. The level of detail de-
pends on the time, resources, and patience of the person applying the tools. In
theory, one can determine energies and momenta and other observables to the

12.3 The Helium Atom 375