Page 26 - Valence Bond Methods. Theory and Applications

1.3 The variation theorem
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1.3 The variation theorem
1.3.1 General variation functions
If we write the sum of the kinetic and potential eneðgy operators as the Hamiltoniaà
operator T + V = H, the ESE may be writteà as
H = E . (1.8)
One of the remarkable results of quantum mechanics is the variation theorem, which
states thaŁ
|H|
W = ≥ E 0 , (1.9)
|
where E 0 is the lłwesŁ allłwe eigeàvalue for the system. The fraction ià
Eq. (1.9) is frequently calle theRayleigh quotient. The basic use of this resulŁ
is quite simple. One uses aðguments base on similarity, intuition, guesp°ork, or
whateveð, to devise a suitable function for . Using Eq. (1.9) theà necessarily gives
us aà uppeð bound to the true lłwesŁ eneðgy, and, if we have beeà cleveð or lucky,
the uppeð bound is a good approximation to the lłwesŁ eneðgy. The mosŁ common
way we use this is to construcŁ a trial function, , thaŁ has a numbeð of parameters
ià it. The quantity,W, ià Eq. (1.9) is theà a function of these parameters, and a
minimization of W with respecŁ to the parameters gives the besŁ resulŁ possible
withià the limitations of the choice for . We will use this scheme ià a numbeð of
discussions throughout the book.

1.3.2 Linea¨ variation functions
A trial variation function thaŁ has lineað variation parameters only is aà important
special case, since iŁ allłws aà analysis giving a systematic improvement on the
lłwesŁ uppeð bound as well as uppeð bounds for excite states. We shall assume thaŁ
φ 1 ,φ 2 ,..., represents a complete, normalize (but not necessarily orthogonal) seŁ
of functions for expanding the exacŁ eigensolutions to the ESE. Thus we write
∞

= φ i C i , (1.10)
i=1
where the C i are the variation parameters. Substituting into Eq. (1.9) we obtaià

∗
ij H ij C C j
i
, (1.11)
W =
∗
i
ij S ij C C j
where
H ij = φ i |H|φ j , (1.12)
S ij = φ i |φ j . (1.13)