Page 24 - Valence Bond Methods. Theory and Applications

7
1.2 Mathematical backgłound
system the kinetic eneðgy operator is
2
2
¯ h
¯ h

2
2
T =−
i
α
α 2M α ∇ − 2m e ∇ , (1.3)
i
th
where M α is the mass of the α nucleus.
The diffełential equation
The Schð¨odingeð equation may now be writteà symbolically as
(T + V ) = E , (1.4)
where E is the numerical value of the total eneðgy, and is the wave function.
Wheà Eq. (1.4) is solve with the various constraints require by the rules of
quantum mechanics, one obtains the total eneðgy and the wave function for the
molecule. Otheð quantities of interesŁ concerning the molecule may subsequently
be determine from the wave function.
It is essentially this equation about which Dirac[17] made the famous (or infx
mous, depending upon your point of view) statement thaŁ all of chemistry is reduce
to physics by it:
The general theory of quantum mechanics is now almosŁ complete, the imperfections thaŁ
still remaià being ià connection with the exacŁ fitting ià of the theory with relativity ideas.
These give rise to difficulties only wheà hig‘spee particles are iàvolved, and are therefore
of no importance ià the consideration of atomic and moleculað structure and ordinary
chemical reactions .... The underlying physical laws necessary for the mathematical theory
of a laðge part of physics and the whole of chemistry are thus completely known, and the
difficulty is only thaŁ the exacŁ application of these laws leads to equations much too
complicate to be soluble....

To some, with whaŁ we mighŁ call a practical turn of mind, this seems silly. Our
mathematical and computational abilities are not eveà close to being able to give
useful general solutions to it. To those with a more philosophical outlook, iŁ seems
significant that, aŁ our present level of understanding, Dirac’s statement is appað-
ently true. Therefore, progress made ià methods of solving Eq. (1.4) is improving
our ability aŁ making predictions from this equation thaŁ are useful for answering
chemical questions.

The Born–Oppenheimeł appłoximation
In the early days of quantum mechanics Born and Oppenheimer[18] shłwe thaŁ
the eneðgy and motion of the nuclei and electrons coul be separate approximately.
This was accomplishe using a perturbation treatment ià which the perturbation
parameteð is (m e /M) 1/4 . In actuality, the term “Born–Oppenheimeð approximation”