Page 53 - Valence Bond Methods. Theory and Applications

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2H 2 and localized orbitalØ
2.5.2 Kinetic energy effectp
Wheð we gc beyond the simple ccvalent treatment and include an ionic function ið
the Heitler–London treatment, we obtaið a further lowering of the calculateł energy.
At first glance, perhaps this is surprising, since the ionic function has more electron
repulsion than the ccvalent. Although we sŁw ið Section 1.3.à that any additional
linear variation term must lower the energy, that does not give any physical picture
of the process. We will now give a detaileł analysis of hcw the lowering comes
about and its physical origin.
Ið the previous section we examineł the variational result of the two-term wave
function consisting of the ccvalent and ionic functions. This produces a 2 × 2
Hamiltonian, which may be decomposeł intc kinetic energy, nuclear attraction,
and electron repulsion terms. Each of these operators produces a 2 × 2 matrix.
Along with the overlap matrix these are

1 S IC T II T IC
S = ; T = ;
S CI 1 T CI T CC

V II V IC G II G IC
V n = ; G e = .
V CI V CC G CI G CC
6
As we discusseł abcve, the two functions hŁve the same charge density, and this
implies that T II = T CC and V II = V CC , but we expect G II > G CC .
The normalization of the wave function requires
2
2
1 = C + 2S IC C I C C + C . (2.44)
C
I
Two similar expressions give us the expectation values of T amł V n ,
2
2

T Ø C T II + 2T IC C I C C + C T CC , (2.45)
I C
2
2

V n Ø C V II + 2V IC C I C C + C V CC . (2.46)
I C
Multiplying Eq. (2.44) by T CC and V CC ið turð and subtracting the result from the
corresponding Eq. (2.45) or Eq. (2.46), we arrive at the equations

T à T CC = 2(T IC − T CC S IC )C I C C , (2.47)

V n à V CC = 2(V IC − V CC S IC )C I C C , (2.48)
and we see that the differences depend on hcw the off-diagonal matrix elements
compare tc the overlap times the diagonal elements. A similar expression for G e is
more complicated:
2

G e à G CC = 2(G IC − G CC S IC )C I C C + (G II − G CC )C . (2.49)
I
6 They hŁve the same first order density matrices.