Problems
175
sin 2 φ − cos 2 φ = cos(2φ). Hint: Add and subtract
1
2 cos 2 α|ba|+
1
2 sin 2 α|ab|
and split the first two terms of Eq. 5.26 in halfs. Then, order the terms to form the
given trigonometric relations.
3. Calculate Φ BS | ˆ
S 2 |Φ BS using the above derived expression Φ BS = (|ab|+
|ba|)/2 + (|ab|−|ba|) cos(2α)/2 + (|aa|+|bb|) sin α cos α.
5.4 Kinetic exchange by second-order perturbation theory. Make a secondorder estimate of the singlet energy taking into account the interaction between
S =
1
√
2
|ab|+|ba|
and the ionic states I 1 =
1
√
2
|aa|+|bb|
and I 2 =
1
√
2
|aa|−|bb|
. The energy of the triplet, T =
1
√
2
|ab|−|ba|
, equals E ref − K ab
and should be taken as reference.
5.5 Biquadratic exchange versus t 13 /t 24 . Express the perturbative estimate of λ
(Eq. 5.56) in terms of t 13 and t 24 with a common denominator for the two terms.
Determine for which values of t 13 and t 24 the biquadratic exchange vanishes and for
which values it can be expected to be maximal.
5.6 Estimating J r . Accurate calculations on a polynuclear paramagnetic compound
with four S =
1
2 magnetic centers indicate that the only significant interactions
are the following bilinear isotropic interactions: J 12 = J 34 =− 25.1 meV, and
J 23 = J 14 =− 39.5 meV. Nevertheless, the experimental temperature dependence
of the magnetic susceptibility could not be fitted satisfactorily with these values.
Provide a perturbational estimate for the four-center cyclic exchange to improve the
fitting (extra data: U = 5.3eV).
References
1. J. Cabrero, C. de Graaf, E. Bordas, R. Caballol, J.P. Malrieu, Chem. Eur. J. 9, 2307 (2003)
2. C.J. Calzado, J. Cabrero, J.P. Malrieu, R. Caballol, J. Chem. Phys. 116(7), 2728 (2002)
3. C.J. Calzado, J. Cabrero, J.P. Malrieu, R. Caballol, J. Chem. Phys. 116(10), 3985 (2002)
4. C.J. Calzado, C. Angeli, D. Taratiel, R. Caballol, J.P. Malrieu, J. Chem. Phys. 131, 044327
(2009)
5. E. Coulaud, J.P. Malrieu, N. Guihéry, N. Ferré, J. Chem. Theory Comput. 9, 3429 (2013)
6. R. Bastardis, N. Guihéry, C. de Graaf, J. Chem. Phys. 129, 104102 (2008)
175
sin 2 φ − cos 2 φ = cos(2φ). Hint: Add and subtract
1
2 cos 2 α|ba|+
1
2 sin 2 α|ab|
and split the first two terms of Eq. 5.26 in halfs. Then, order the terms to form the
given trigonometric relations.
3. Calculate Φ BS | ˆ
S 2 |Φ BS using the above derived expression Φ BS = (|ab|+
|ba|)/2 + (|ab|−|ba|) cos(2α)/2 + (|aa|+|bb|) sin α cos α.
5.4 Kinetic exchange by second-order perturbation theory. Make a secondorder estimate of the singlet energy taking into account the interaction between
S =
1
√
2
|ab|+|ba|
and the ionic states I 1 =
1
√
2
|aa|+|bb|
and I 2 =
1
√
2
|aa|−|bb|
. The energy of the triplet, T =
1
√
2
|ab|−|ba|
, equals E ref − K ab
and should be taken as reference.
5.5 Biquadratic exchange versus t 13 /t 24 . Express the perturbative estimate of λ
(Eq. 5.56) in terms of t 13 and t 24 with a common denominator for the two terms.
Determine for which values of t 13 and t 24 the biquadratic exchange vanishes and for
which values it can be expected to be maximal.
5.6 Estimating J r . Accurate calculations on a polynuclear paramagnetic compound
with four S =
1
2 magnetic centers indicate that the only significant interactions
are the following bilinear isotropic interactions: J 12 = J 34 =− 25.1 meV, and
J 23 = J 14 =− 39.5 meV. Nevertheless, the experimental temperature dependence
of the magnetic susceptibility could not be fitted satisfactorily with these values.
Provide a perturbational estimate for the four-center cyclic exchange to improve the
fitting (extra data: U = 5.3eV).
References
1. J. Cabrero, C. de Graaf, E. Bordas, R. Caballol, J.P. Malrieu, Chem. Eur. J. 9, 2307 (2003)
2. C.J. Calzado, J. Cabrero, J.P. Malrieu, R. Caballol, J. Chem. Phys. 116(7), 2728 (2002)
3. C.J. Calzado, J. Cabrero, J.P. Malrieu, R. Caballol, J. Chem. Phys. 116(10), 3985 (2002)
4. C.J. Calzado, C. Angeli, D. Taratiel, R. Caballol, J.P. Malrieu, J. Chem. Phys. 131, 044327
(2009)
5. E. Coulaud, J.P. Malrieu, N. Guihéry, N. Ferré, J. Chem. Theory Comput. 9, 3429 (2013)
6. R. Bastardis, N. Guihéry, C. de Graaf, J. Chem. Phys. 129, 104102 (2008)