Page 273 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

264 Molecular structure
Ù 2
X =
^ 2 á (10:2)
T Q ˆÿ"
2M á
áˆ1
V Q is the potential energy of interaction between nuclear pairs
Ù 2
X Z á Z â e9
V Q ˆ (10:3)
r áâ
á , âˆ1
^
and H e is the electronic Hamiltonian operator
Ù
N
N
N
" 2 X X X Z á e9 2 X e9 2
^ 2
H e ˆÿ = ÿ ‡ (10:4)
i
2m e r ái r ij
iˆ1 áˆ1 iˆ1 i , jˆ1
2
2
The symbols = and = are, respectively, the laplacian operators for a single
á
i
nucleus and a single electron. The variable r áâ is the distance between nuclei á
and â, r ái the distance between nucleus á and electron i, and r ij the distance
between electrons i and j. The summations are taken over each pair of
particles. The quantity e9 is equal to the magnitude of the electronic charge e in
CGS units and to e=(4ðå 0 ) 1=2 in SI units, where å 0 is the permittivity of free
space.
È
The Schrodinger equation for the molecule is
^
HØ(r, Q) ˆ EØ(r, Q) (10:5)
where Ø(r, Q) is an eigenfunction and E the corresponding eigenvalue. The
differential equation (10.5) cannot be solved as it stands because there are too
many variables. However, approximate, but very accurate, solutions may be
found if the equation is simpli®ed by recognizing that the nuclei and the
electrons differ greatly in mass and, as a result, differ greatly in their relative
speeds of motion.

Born±Oppenheimer approximation
È
The simplest approximate method for solving the Schrodinger equation (10.5)
uses the so-called Born±Oppenheimer approximation. This method is a two-
step process. The ®rst step is to recognize that the nuclei are much heavier than
an electron and, consequently, move very slowly in comparison with the
È
electronic motion. Thus, the electronic part of the Schrodinger equation may
be solved under the condition that the nuclei are motionless. The resulting
electronic energy may then be determined for many different ®xed nuclear
È
con®gurations. In the second step, the nuclear part of the Schrodinger equation
is solved by regarding the motion of the nuclei as taking place in the average
potential ®eld created by the fast-moving electrons.
In the ®rst step of the Born±Oppenheimer approximation, the nuclei are all
held at ®xed equilibrium positions. Thus, the coordinates Q do not change with