66
3 Two (or More) Magnetic Centers
leading to
Ψ(0, 0) = C cov Ψ
cov (0, 0) + C ion Ψ
ion (0, 0)
(3.16)
which is the same as the CI wave function in Eq. 3.7, but now written in terms of
the localized orthogonal orbitals ψ a and ψ b . Again, just as in the MO picture, there
isnowaytoimprovetheS = 1 wave functions. A balanced VB treatment uses the
wave functions of Eq. 3.16 for S = 0 and of Eq. 3.13 for S = 1, M S = 0.
We can now draw the following conclusions:
• If we limit ourselves to using (apart from the doubly occupied orbitals) only
two mutually orthogonal orbitals, either bonding and antibonding φ a and φ b ,o r
localized ψ a and ψ b , then the best wave function for the lowest triplet state is
the one given in Eq. 3.2 or, equivalently, 3.13a and the best wave function for the
lowest singlet state is given in Eq. 3.7 or, equivalently, 3.16.
• It makes no difference whether we use MO theory and optimize the ratio c 1 /c 2 in
3.7 or use VB theory and optimize the ratio C cov /C ion in 3.16, both procedures
lead to one and the same S = 0 wave function.
• For Ψ(0, 0) to become the ground state, the energy lowering due to mixing in of
the ionic terms has to exceed the direct exchange 2K ab . This energy lowering is
traditionally called the kinetic exchange. We will see in Chap. 4 that the kinetic
exchange equals, to good approximation, 1 4t 2
ab /(J aa − J ab ) with the transfer
integral t ab defined by t ab ==ψ a ψ a | ˆ
H |ψ a ψ b .
Valence Bond theory using localized nonorthogonal orbitals: In the above we have
used orthogonal localized orbitals ψ a and ψ b . What if we remove the orthogonality
restriction? This nonorthogonal VB approach appeared for the first time in the work
of Heitler and London [1]. It forms also the basis of the Kahn–Briat model discussed
in the next chapter. We define normalized localized orbitals φ a and φ b with mutual
overlap S ab ==φ a |φ b . Then we write the covalent singlet wave function in terms of
normalized Slater determinants that are now built from the nonorthogonal φ a and φ b :
Ψ(0, 0) =
1
2 + 2S 2
ab
|φ a φ b |+|φ b φ a |
(3.17)
Of course, we can formally express φ a and φ b in terms of our orthogonal localized
orbitals ψ a and ψ b :
φ a = N (ψ a + νψ b )φ b = N (ψ b + νψ a )
with N =
1
√
1 + ν 2
and S ab =
2ν
1 + ν 2
(3.18)
Once we have optimized ν or, equivalently, S ab , to obtain the lowest energy possible
for the singlet wave function in Eq. 3.18 we have retrieved once more our familiar
1 This expression is reasonable for J aa − J ab ≫ t ab , see Chap. 4.
3 Two (or More) Magnetic Centers
leading to
Ψ(0, 0) = C cov Ψ
cov (0, 0) + C ion Ψ
ion (0, 0)
(3.16)
which is the same as the CI wave function in Eq. 3.7, but now written in terms of
the localized orthogonal orbitals ψ a and ψ b . Again, just as in the MO picture, there
isnowaytoimprovetheS = 1 wave functions. A balanced VB treatment uses the
wave functions of Eq. 3.16 for S = 0 and of Eq. 3.13 for S = 1, M S = 0.
We can now draw the following conclusions:
• If we limit ourselves to using (apart from the doubly occupied orbitals) only
two mutually orthogonal orbitals, either bonding and antibonding φ a and φ b ,o r
localized ψ a and ψ b , then the best wave function for the lowest triplet state is
the one given in Eq. 3.2 or, equivalently, 3.13a and the best wave function for the
lowest singlet state is given in Eq. 3.7 or, equivalently, 3.16.
• It makes no difference whether we use MO theory and optimize the ratio c 1 /c 2 in
3.7 or use VB theory and optimize the ratio C cov /C ion in 3.16, both procedures
lead to one and the same S = 0 wave function.
• For Ψ(0, 0) to become the ground state, the energy lowering due to mixing in of
the ionic terms has to exceed the direct exchange 2K ab . This energy lowering is
traditionally called the kinetic exchange. We will see in Chap. 4 that the kinetic
exchange equals, to good approximation, 1 4t 2
ab /(J aa − J ab ) with the transfer
integral t ab defined by t ab ==ψ a ψ a | ˆ
H |ψ a ψ b .
Valence Bond theory using localized nonorthogonal orbitals: In the above we have
used orthogonal localized orbitals ψ a and ψ b . What if we remove the orthogonality
restriction? This nonorthogonal VB approach appeared for the first time in the work
of Heitler and London [1]. It forms also the basis of the Kahn–Briat model discussed
in the next chapter. We define normalized localized orbitals φ a and φ b with mutual
overlap S ab ==φ a |φ b . Then we write the covalent singlet wave function in terms of
normalized Slater determinants that are now built from the nonorthogonal φ a and φ b :
Ψ(0, 0) =
1
2 + 2S 2
ab
|φ a φ b |+|φ b φ a |
(3.17)
Of course, we can formally express φ a and φ b in terms of our orthogonal localized
orbitals ψ a and ψ b :
φ a = N (ψ a + νψ b )φ b = N (ψ b + νψ a )
with N =
1
√
1 + ν 2
and S ab =
2ν
1 + ν 2
(3.18)
Once we have optimized ν or, equivalently, S ab , to obtain the lowest energy possible
for the singlet wave function in Eq. 3.18 we have retrieved once more our familiar
1 This expression is reasonable for J aa − J ab ≫ t ab , see Chap. 4.