c17 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come
270 THE VARIATIONAL METHOD: ATOMSThe differencesH 11 −ES 11 , and so on, are scalar energies, so we are left with nothing
more than a simple simultaneous equation pair like the ones we solved in high
school:ax+by=p
cx+dy=qThe only difference is that the case ofp=q=0 is considered an unusual case
in elementary presentations, whereas it is the case of interest here. We wish to
solve the simultaneous equations in theHandSintegrals for the solution setc 1
andc 2.
Alas, we cannot find unique values forc 1 andc 2 because equations withp=q= 0
areinhomogeneous. Without an extra piece of information, we can only get the ratio
ofc 1 toc 2. This is not as much of a limitation as it may seem because for larger sets
ofNequations, we can get the ratios ofN−1 coefficients to each other, whereN
may be very large. In other words, we can get almost all of the information we want.
We can obtain the ratios of nonzero solutions we seek if the second equation is a
linear combinationof the first. For a linearly dependent equation pair, it must be
possible to multiply one equation by a constantkand obtain the other. In this case,
the equations are linearly related ifka=candkb=d. We can assure ourselves that
this is true and that the ratio ofc 1 toc 2 exists if we stipulate that the determinant of
the coefficients be zero∣
∣
∣
∣
∣ab
cd∣
∣
∣
∣
∣
=
∣
∣
∣
∣
∣
ka kb
cd∣
∣
∣
∣
∣
=kad−ckb=cd−cd= 0In the case of the simultaneous equations we have(H 11 −ES 11 )c 1 +(H 12 −ES 12 )c 2 = 0
(H 12 −ES 12 )c 1 +(H 22 −ES 22 )c 2 = 0The stipulation for a linear pair is∣
∣
∣
∣
∣H 11 −ES 11 H 12 −ES 12
H 12 −ES 12 H 22 −ES 22
∣
∣
∣
∣
∣
= 0
This is called thesecular determinant. When expanded, it leads to a quadratic equation
with two roots,E 1 andE 2 , a pair of energy estimates. Upon imposing the normal-
ization condition, which is the extra piece of information necessary to complete the
solution, we take the lower of the two roots as the ground state energy for the system.