154
5 Towards a Quantitative Understanding
Fig. 5.9 Ladder-like
structure of SrCu 2 O 3 .The
open circles represent Cu 2+
ions, the small grey circles
are oxygens, only the
oxygens belonging to the
Cu 2 O 7 cluster are shown.
The Sr ions are below and
above this plane
the sixteen lowest roots of a DDCI calculation on an embedded Cu 2 O 7 cluster (see
Sect. 6.3.1), the roots 1, 2, 7 and 8
Ψ 1 =−0.9224
|ab|+|ba|
− 0.1223
|aa|+|bb|
+···
Ψ 2 =−0.6626
|ab|−|ba|
+···
(5.19)
Ψ 7 = 0.4159
|aa|−|bb|
+···
Ψ 8 = 0.1704
|ab|+|ba|
− 0.5324
|aa|+|bb|
have the largest norm after projection on the model space. The normalized projections, denoted
Ψ ′
1 ...
Ψ ′
4 , will be used for the construction of the effective Hamiltonian. The relative energies are 0.000, 0.158, 6.489 and 6.547 eV, respectively. The
Bloch effective Hamiltonian uses biorthogonal vectors, which are obtained from
Eq. 5.19 by
Ψ
′†
1 =
1
√
1 − s 2
Ψ
′
1 − s
Ψ
′
4
=−0.9524
|ab|+|ba|
− 0.3048
|aa|+|bb|
Ψ
′†
2 =
Ψ
′
2 =
|ab|−|ba|
/
√
2
(5.20)
Ψ
′†
3 =
Ψ
′
3 =
|aa|−|bb|
/
√
2
Ψ
′†
4 =
1
√
1 − s 2
− s
Ψ
′
1 +
Ψ
′
4
= 0.1315
|ab|+|ba|
− 0.9913
|aa|+|bb|
where s ==
Ψ ′
1 |
Ψ ′
4 .