Quantum Mechanical Insights into Biological Processes at the Electronic Level - Computational Modeling of Biological Systems - page 121

coordinates of two electrons and represents electron-electron interactions. These
terms make TISE inseparable. Hence, approximations are invoked when solving
the TISE.
The first key approximation that is made is the BOA. The BOA is based on the
fact that electrons are much lighter than nuclei, and so the motion of electrons and
nuclei happens on very different time scales. Because of this, electrons are thought
to adjust instantaneously to the positions of the nuclei in a molecule. The often
used analogy here is the motion of flies on top of a garbage truck: the truck moves
so slowly that the flies do not even notice its movement and adjust their positions
instantaneously to the position of the truck. The BOA is a good approximation in
most cases of chemical relevance. BOA implies that the total wave function of a
molecule can be written as a product of the nuclear and electronic parts, and the
total Hamiltonian as a sum of the nuclear and electronic Hamiltonians:

total D nuclear electronic ; (4)

H total H nuclear C H electronic :

D (5)

Then, the TISE is separable within the BOA, and in particular the electronic TISE
can be solved with the nuclear coordinates included as parameters:

H el el D E el el ;

(6)

where

X X X X X

1
2

Z a 1

H el D

i

r C : (7)

j R a r i j j r i r j j

i a i i

j ¤ i

The first term in the electronic Hamiltonian represents the sum of kinetic energies
of electrons, the second term is the potential energy term for the attraction of
electrons to the now stationary nuclei, and the third term is the interelectronic
repulsion. The nuclear kinetic energy is zero, and the internuclear repulsion is a
constant, within the BOA. The solution of the TISE is the total energy and the wave
function itself. For the exact wave function, the energy can be found as

D el ˇ ˇ ˇ H el ˇ ˇ ˇ el E
el ˇ ˇ ˇ ˇ

el

E el D : (8)

As was mentioned already, solving the TISE exactly is impossible for any system
with more than one electron. However, there is a helpful variational principle that
tells us that any approximate wave function would return an eigenenergy that is an
upper-bound of the true energy:

D el ˇ ˇ ˇ H el ˇ ˇ ˇ el E
el ˇ ˇ ˇ ˇ

el

E approximate E el D (9)