5.1 Decomposition of the Magnetic Coupling
143
Fig. 5.2 Left and right localized magnetic orbitals
the CAS Hamiltonian matrix can be written as
|ab| ba| aa|bb
ab| 0 K ab t ab t ab
ba| K ab 0 t ab t ab
aa| t ab t ab UK ab
bb| t ab t ab K ab U
(5.4)
Here E ref = h aa + h bb + J ab is the reference energy and has been subtracted from
all diagonal matrix elements. K ab is the direct exchange, U is the on-site repulsion
parameter and t ab is the hopping integral and gives a measure of the probability for
the electron hopping from site a to b and vice-versa.
5.2 Express the energy of the ionic determinants |aa| and |bb| in terms of
the one-electron integrals h and the Coulomb and exchange integrals J and
K . What assumption has been made to reduce the diagonal element of these
determinants to U ?
The diagonalization of the 4 × 4 matrix gives four eigenvectors, equivalent to
those listed above but expressed in local orbitals
S g = λ
|ab|+|ba|
/
√
2 − µ
|aa|+|bb|
/
√
2
S
′
g = µ
|ab|+|ba|
/
√
2 + λ
|aa|+|bb|
/
√
2
T u =
|ab|−|ba|
/
√
2
S u =
|aa|−|bb|
/
√
2
(5.5)
with energy eigenvalues
E(S g ) = K ab +
U −
U 2 + 16t 2
ab
2