3.1 Localized Versus Delocalized Description of the Two-Electron/Two-Orbital Problem
63
3.2 (a) Demonstrate that ψ a and ψ b are normalized and orthogonal (b) Consider the example where the delocalized orbitals φ 1 and φ 2 are bonding and
antibonding combinations of atom centered basis functions χ a and χ b :
φ 1 =
χ a + χ b
√
(2(1 + S)
φ 2 =
χ a − χ b
√
(2(1 − S)
with S ==χ a |χ b . Compute the coefficient of the orthogonalization tail of ψ a
on atom B,forS = 0.2 and for S = 0.003.
Using the inverse relations of Eq. 3.10a
φ 1 =
1
√
2
(ψ a + ψ b )φ 2 =
1
√
2
(ψ a − ψ b )
(3.10b)
the exchange integral K 12 can be rewritten as a sum of Coulomb integrals
K 12 =
1
4
(ψ a + ψ b )(ψ a − ψ b )| ˆ
H |(ψ a − ψ b )(ψ a + ψ b )
=
1
4
ψ a ψ a | ˆ
H |ψ a ψ a −2ψ a ψ b | ˆ
H |ψ a ψ b ++ψ b ψ b | ˆ
H |ψ b ψ b
=
1
4
J aa − 2J ab + J bb
(3.11)
Only the term J ab approaches zero for small coupling between A and B. The other
terms are local Coulomb integrals which are both positive and both occur with a
positive coefficient: these two terms do not cancel each other. Note, that since we
have assumed centrosymmetry, J aa = J bb . Figure 3.3 shows the localized orthogonal
orbitals ψ a and ψ b and the product of these as they appear in the Coulomb integrals of
the expression of K 12 . From this pictorial representation it is obvious that J ab —with
ψ a ψ a × ψ b ψ b in the numerator—is small for weak interaction between A and B,
while J aa and J bb do not strongly depend on the distance between A and B.
Hence, there is significant interaction between the configurations ...φ 2
1 and ...φ 2
2 .
Note that the smaller J ab , the larger is K 12 , and therefore, in the case of weak
coupling between A and B we need to use the two-configuration wave function
of Eq. 3.7 instead of simply Φ 11 (0, 0). Clearly, the best two-configuration wave
function is obtained by varying c 1 /c 2 until the energy expectation value is minimal.
Only in case of strong coupling between A and B (remember the case of Li 2 near
equilibrium distance) J ab may become so large that K 12 and therewith c 2 becomes
negligible so that the closed shell determinant Φ 11 (0, 0) is a reasonable ansatz for the
lowest singlet wave function. For intermediate and small couplings the lowest S = 0
and S = 1 states are competing, i.e. close in energy. An approximate yet balanced
treatment is obtained by describing the lowest S = 0 state using the expression of
63
3.2 (a) Demonstrate that ψ a and ψ b are normalized and orthogonal (b) Consider the example where the delocalized orbitals φ 1 and φ 2 are bonding and
antibonding combinations of atom centered basis functions χ a and χ b :
φ 1 =
χ a + χ b
√
(2(1 + S)
φ 2 =
χ a − χ b
√
(2(1 − S)
with S ==χ a |χ b . Compute the coefficient of the orthogonalization tail of ψ a
on atom B,forS = 0.2 and for S = 0.003.
Using the inverse relations of Eq. 3.10a
φ 1 =
1
√
2
(ψ a + ψ b )φ 2 =
1
√
2
(ψ a − ψ b )
(3.10b)
the exchange integral K 12 can be rewritten as a sum of Coulomb integrals
K 12 =
1
4
(ψ a + ψ b )(ψ a − ψ b )| ˆ
H |(ψ a − ψ b )(ψ a + ψ b )
=
1
4
ψ a ψ a | ˆ
H |ψ a ψ a −2ψ a ψ b | ˆ
H |ψ a ψ b ++ψ b ψ b | ˆ
H |ψ b ψ b
=
1
4
J aa − 2J ab + J bb
(3.11)
Only the term J ab approaches zero for small coupling between A and B. The other
terms are local Coulomb integrals which are both positive and both occur with a
positive coefficient: these two terms do not cancel each other. Note, that since we
have assumed centrosymmetry, J aa = J bb . Figure 3.3 shows the localized orthogonal
orbitals ψ a and ψ b and the product of these as they appear in the Coulomb integrals of
the expression of K 12 . From this pictorial representation it is obvious that J ab —with
ψ a ψ a × ψ b ψ b in the numerator—is small for weak interaction between A and B,
while J aa and J bb do not strongly depend on the distance between A and B.
Hence, there is significant interaction between the configurations ...φ 2
1 and ...φ 2
2 .
Note that the smaller J ab , the larger is K 12 , and therefore, in the case of weak
coupling between A and B we need to use the two-configuration wave function
of Eq. 3.7 instead of simply Φ 11 (0, 0). Clearly, the best two-configuration wave
function is obtained by varying c 1 /c 2 until the energy expectation value is minimal.
Only in case of strong coupling between A and B (remember the case of Li 2 near
equilibrium distance) J ab may become so large that K 12 and therewith c 2 becomes
negligible so that the closed shell determinant Φ 11 (0, 0) is a reasonable ansatz for the
lowest singlet wave function. For intermediate and small couplings the lowest S = 0
and S = 1 states are competing, i.e. close in energy. An approximate yet balanced
treatment is obtained by describing the lowest S = 0 state using the expression of