Page 233 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

224 Systems of identical particles
á(1)á(2)
â(1)â(2)
2 ÿ1=2 [á(1)â(2) ‡ á(2)â(1)]

and one antisymmetric spin wave function
2 ÿ1=2 [á(1)â(2) ÿ á(2)â(1)]
where the factors 2 ÿ1=2 are normalization constants. When the spatial and spin
wave functions are combined, there are four antisymmetric combinations: a
singlet state (S ˆ 0)
1 [ö a (1)ö b (2) ‡ ö a (2)ö b (1)][á(1)â(2) ÿ á(2)â(1)]
2
and three triplet states (S ˆ 1)
8
á(1)á(2)
<
2 ÿ1=2 [ö a (1)ö b (2) ÿ ö a (2)ö b (1)] â(1)â(2)
: ÿ1=2
2 [á(1)â(2) ‡ á(2)â(1)]
These four antisymmetric wave functions are normalized if the single-particle
spatial wave functions ö n (r i ) are normalized. If the two fermions are in the
same state ö a (r i ), then only the singlet state occurs
2 ÿ1=2 ö a (1)ö a (2)[á(1)â(2) ÿ á(2)â(1)]
The helium atom serves as a simple example for the application of this
construction. If the nucleus (for which Z ˆ 2) is considered to be ®xed in
^
space, the Hamiltonian operator H for the two electrons is
" 2 Ze9 2 Ze9 2 e9 2
^ 2 2
H ˆÿ (= ‡ = ) ÿ ÿ ‡ (8:54)
2
1
2m e r 1 r 2 r 12
where r 1 and r 2 are the distances of electrons 1 and 2 from the nucleus, r 12 is
the distance between the two electrons, and e9 ˆ e for CGS units or
e9 ˆ e=(4ðå 0 ) 1=2 for SI units. Spin±orbit and spin±spin interactions of the
electrons are small and have been neglected. The electron±electron interaction
is relatively small in comparison with the interaction between an electron and a
nucleus, so that as a crude ®rst-order approximation the last term on the right-
^
hand side of equation (8.54) may be neglected. The operator H then becomes
the sum of two hydrogen-atom Hamiltonian operators with Z ˆ 2. The
corresponding single-particle states are the hydrogen-like atomic orbitals ø nlm
discussed in Section 6.4. The energy of the helium atom depends on the
principal quantum numbers n 1 and n 2 of the two electrons and is the sum of
two hydrogen-like atomic energies with Z ˆ 2
2
m e Z e9 4 1 1 1 1
ˆÿ ‡ ˆÿ54:4eV ‡
E n 1 ,n 2 2 2 2 2
2" 2 n n n n
1 2 1 2