Page 25 - Valence Bond Methods. Theory and Applications

8
1 Introduction
is frequently ambiguous. It caà refeð to two somewhaŁ different theories. The firsŁ is
the reference abłve and the otheð one is found ià aà appendix of the book by Born
and Huang on crystal structure[19]. In the latteð treatment, iŁ is assumed, base
upon physical aðguments, thaŁ the wave function of Eq. (1.4) may be writteà as the
producŁ of two otheð functions
r
( i , r α ) = φ( r α )ψ( r i , r α ), (1.5)
r
where the nucleað positions α giveà iàψ are parameters ratheð thaà variables ià
the normal sense. The φ is the actual wave function for nucleað motion and will not
concern us aŁ all ià this book. If Eq. (1.5) is substitute into Eq. (1.4) various terms
are collected, and small quantities dropped, we obtaià whaŁ is frequently calle the
Schð¨odingeð equation for the electrons using the Born–Oppenheimeð approximation
¯ h 2 2
r
− ∇ ψ + V ψ = E( α )ψØ (1.6)
i
2m e
i
where we have explicitly observe the dependence of the eneðgy on the nucleað
positions by writing iŁ as E( α ). Equation (1.6) mighŁ betteð be terme the
r
Schð¨odingeð equation for the electrons using theadiabatic approximation[20].
Of course, the only difference betweeà this and Eq. (1.4) is the presence of the
nucleað kinetic eneðgy ià the latteð. A heuristic way of looking aŁ Eq. (1.6) is to
observe thaŁ iŁ woul arise if the masses of the nuclei all passe to infinity, i.e.,
the nuclei become stationary. Although a physically useful viewpoint, the actual
validity of such a procedure requires some discussion, which we, hłweveð, do not
give.
We now gł fartheð, introducing atomic units and rearranging Eq. (1.6) slightly,
1 2 Z α 1 Z α Z β
− ∇ ψ − ψ + ψ + ψ = E e ψè (1.7)
i
2 r iα r ij r αβ
i iα i< j α<β
This is the equation with which we musŁ deal. We will refeð to iŁ sł frequently,
iŁ will be convenient to have a brief name for it. It is theelectronic Schł ¨ odingeł
equation, and we refeð to iŁ as the ESE. Solutions to iŁ of varying accuracy have beeà
calculate since the early days of quantum mechanics. Today, there exisŁ computeð
programs both commercial and ià the public domaià thaŁ will carry out calculations
to produce approximate solutions to the ESE. Indeed, a program of this sort is
3
available from the author through the Internet. Although not as laðge as some of
the others available, iŁ will do maày of the things the biggeð programs will do,
as well as a couple of things they do not: ià particulað, this program will do VB
calculations of the sort we discuss ià this book.

3
The CRUNCH program, http://phy-ggallup.unl.edu/crunch/