96
3 Two (or More) Magnetic Centers
Hamiltonian, it is common practice to consider the basis of uncoupled determinants
and then transform to the basis of spin-adapted CSFs.
3.12 Perform the matrix multiplication of the anisotropic term in the model
Hamiltonian.
The uncoupled basis is formed by the determinants |αα|, |αβ|, |βα| and |ββ|.
The result of acting with the isotropic part of Hamiltonian on these determinants can
directly be written down with the help of Eqs. 3.79, but the anisotropic part requires a
little more work. Based on the relations given in Eqs. 1.16a and 1.20a, the following
is easily derived for the products of one-electron operators
• ˆ
S 1 A ˆ
S 2 |αα|
A xx ˆ
S x ˆ
S x αα =
1
4
A xx ββ
A xy ˆ
S x ˆ
S y αα =−
1
4i
A xy ββ
A xz ˆ
S x ˆ
S z αα =
1
4
A xz βα
A yx ˆ
S y ˆ
S x αα =−
1
4i
A yx ββ
A yy ˆ
S y ˆ
S y αα =−
1
4
A yy ββ
A yz ˆ
S y ˆ
S z αα =−
1
4i
A yz βα
A zx ˆ
S z ˆ
S x αα =
1
4
A zx αβ
A zy ˆ
S z ˆ
S y αα =−
1
4i
A zy αβ
A zz ˆ
S z ˆ
S z αα =
1
4
A zz αα
(3.91a)
• ˆ
S 1 A ˆ
S 2 |ββ|
A xx ˆ
S x ˆ
S x ββ =
1
4
A xx αα
A xy ˆ
S x ˆ
S y ββ =
1
4i
A xy αα
A xz ˆ
S x ˆ
S z ββ =−
1
4
A xz αβ
A yx ˆ
S y ˆ
S x ββ =
1
4i
A yx αα
A yy ˆ
S y ˆ
S y ββ =−
1
4
A yy αα
A yz ˆ
S y ˆ
S z ββ =−
1
4i
A yz αβ
A zx ˆ
S z ˆ
S x ββ =−
1
4
A zx βα
A zy ˆ
S z ˆ
S y ββ =−
1
4i
A zy βα
A zz ˆ
S z ˆ
S z ββ =
1
4
A zz ββ
(3.91b)
• ˆ
S 1 A ˆ
S 2 |αβ |
A xx ˆ
S x ˆ
S x αβ =
1
4
A xx βα
A xy ˆ
S x ˆ
S y αβ =
1
4i
A xy βα
A xz ˆ
S x ˆ
S z αβ =−
1
4
A xz ββ
A yx ˆ
S y ˆ
S x αβ =−
1
4i
A yx βα
A yy ˆ
S y ˆ
S y αβ =
1
4
A yy βα
A yz ˆ
S y ˆ
S z αβ =
1
4i
A yz ββ
A zx ˆ
S z ˆ
S x αβ =
1
4
A zx αα
A zy ˆ
S z ˆ
S y αβ =
1
4i
A zy αα
A zz ˆ
S z ˆ
S z αβ =−
1
4
A zz αβ
(3.91c)