142
5 Towards a Quantitative Understanding
5.1.1 Valence Mechanisms
First we focus on the mechanisms that arise from interactions among the configurations in the active space, restricting ourselves to the role played by the two Cu 2+
ions. For this centro-symmetric system, the active orbitals g = (a + b)/
√
2 and
u = (a − b)/
√
2 shown in Fig. 5.1 define four different determinants
CAS ={|gg|, |uu|, |gu|, |ug|}
(5.1)
The diagonalization of the corresponding 4 × 4 matrix produces four eigenstates,
three singlets and one triplet
S g = λ|gg|−µ|uu|
S
′
g = µ|gg|+λ|uu|
T u =
|gu|−|ug|
/
√
2
S u =
|gu|+|ug|
/
√
2( 5 . 2 )
5.1 Demonstrate by substitution that S g is dominated by the neutral determinants when λ ≈ µ. What situation is described for λ ≫ µ?
The energy difference of S g and T u defines J but the analysis is much easier in a
representation with localized orbitals. Therefore, we rewrite the CAS in terms of the
orthogonal localized Cu orbitals a and b, shown in Fig. 5.2. By defining the following
electronic structure parameters
E ref ==ab| ˆ
H |ab=0
K ab ==ab| ˆ
H |ba
t ab ==ab| ˆ
H |aa
U ==aa| ˆ
H |aa−−ab| ˆ
H |ab
(5.3)
Fig. 5.1 Delocalized magnetic orbitals of gerade (left)andungerade (right) symmetry