3.1 Localized Versus Delocalized Description of the Two-Electron/Two-Orbital Problem
65
which in terms of φ 1 and φ 2 lead to Ψ 12 (1, 0) of Eq. 3.2c. The other two S = 1
components are
Ψ
cov (1, 1) =|ψ a ψ b |
(3.13b)
and
Ψ
cov (1, −1) =|ψ a ψ b |
(3.13c)
which written in terms of φ 1 and φ 2 yield the familiar Slater determinants |φ 1 φ 2 |
(Eq. 3.2a) and |φ 1 φ 2 | (Eq. 3.2b).
3.3 Demonstrate the equivalence of the wave functions of Eqs. 3.12a and
3.12b, and those of Eqs. 3.2 and 3.13.
This simple Valence Bond ansatz with a common set of localized orthonormal
orbitals for both states leads to a separation of the energy expectation values for
singlet and triplet that reads
E
cov
S − E
cov
T = 2ψ a ψ b |
1
r 12
|ψ b ψ a =2K ab
(3.14)
which is positive because the exchange integral
K ab ==ψ a (1)ψ b (1)|
1
r 12
|ψ a (2)ψ b (2) 0
3.4 Calculate the energy expectation values of the wave functions given in
Eqs. 3.12a and 3.13a to demonstrate that E cov
S − E cov
T = 2K ab .
2K ab is traditionally called the direct exchange. It favours high spin states. In this
two electron case, treated with only covalent VB determinants, it favours S = 1 over
S = 0. It is interesting to note that this positive energy difference between singlet
and triplet spin states can be seen as a manifestation of Hund’s rule for two electrons
in two orbitals, be it in this case not for two degenerate orbitals on one site, but for
two degenerate orbitals at two separate sites. In the single site case, i.e. for atoms,
this Hund rule is almost always correct, however, in the two site case the sign of
E S − E T is often wrong. The simple covalent VB model using localized orthogonal
orbitals is simply too crude.
The singlet VB wave function can be improved variationally by mixing in an ionic
term
Ψ
ion (0, 0) =
1
√
2
|ψ a ψ a |+|ψ b ψ b |
(3.15)
65
which in terms of φ 1 and φ 2 lead to Ψ 12 (1, 0) of Eq. 3.2c. The other two S = 1
components are
Ψ
cov (1, 1) =|ψ a ψ b |
(3.13b)
and
Ψ
cov (1, −1) =|ψ a ψ b |
(3.13c)
which written in terms of φ 1 and φ 2 yield the familiar Slater determinants |φ 1 φ 2 |
(Eq. 3.2a) and |φ 1 φ 2 | (Eq. 3.2b).
3.3 Demonstrate the equivalence of the wave functions of Eqs. 3.12a and
3.12b, and those of Eqs. 3.2 and 3.13.
This simple Valence Bond ansatz with a common set of localized orthonormal
orbitals for both states leads to a separation of the energy expectation values for
singlet and triplet that reads
E
cov
S − E
cov
T = 2ψ a ψ b |
1
r 12
|ψ b ψ a =2K ab
(3.14)
which is positive because the exchange integral
K ab ==ψ a (1)ψ b (1)|
1
r 12
|ψ a (2)ψ b (2) 0
3.4 Calculate the energy expectation values of the wave functions given in
Eqs. 3.12a and 3.13a to demonstrate that E cov
S − E cov
T = 2K ab .
2K ab is traditionally called the direct exchange. It favours high spin states. In this
two electron case, treated with only covalent VB determinants, it favours S = 1 over
S = 0. It is interesting to note that this positive energy difference between singlet
and triplet spin states can be seen as a manifestation of Hund’s rule for two electrons
in two orbitals, be it in this case not for two degenerate orbitals on one site, but for
two degenerate orbitals at two separate sites. In the single site case, i.e. for atoms,
this Hund rule is almost always correct, however, in the two site case the sign of
E S − E T is often wrong. The simple covalent VB model using localized orthogonal
orbitals is simply too crude.
The singlet VB wave function can be improved variationally by mixing in an ionic
term
Ψ
ion (0, 0) =
1
√
2
|ψ a ψ a |+|ψ b ψ b |
(3.15)