Problems
55
b. The ab initio wave functions of the three states with the largest projection on the
model space are
Ψ 1 = (0.686 − 0.024i)|1, −1+(−0.175 + 0.009i)|1, 0+(−0.685 + 0.046i)|1, 1
Ψ 2 = (−0.664 + 0.197i)|1, −1+(0.036 + 0.123i)|1, 0+(−0.667 + 0.187i)|1, 1
Ψ 3 = (0.110 − 01.08i)|1, −1+(0.957 − 0.147i)|1, 0+(−0.137 − 0.070i)|1, 1
Calculate the overlap matrix of
Ψ i , the projections of Ψ on the model space.
c. The orthonormalized projections are given by
|1, −1| 1, 0| 1, 1
Ψ ′
1
0.695504 −
0.023842i
−0.177003 +
0.008987i
−0.694343 +
0.046723i
Ψ ′
2
−0.672287 +
0.199091i
0.035909 +
0.124483i
−0.675035 +
0.189566i
Ψ ′
3
+0.110493 −
0.109038i
0.964328 −
0.147901i
−0.138092 −
0.070912i
E 1 = 0.00; E 2 = 11.54 cm −1 ; E 3 = 37.55 cm −1 . Construct the effective Hamiltonian and check the consistency of the model Hamiltonian by comparing the
numerical matrix elements of the effective Hamiltonian with the symbolic matrix
elements of the model Hamiltonian.
d. Diagonalize the D-tensor, determine the axial (D) and rhombic (E) anisotropy
parameters from Eq. 2.16 and compare the values with those obtained by extracting D and E from the energies differences (Eq. 2.22).
2.2 Extracting D and E foraCo II complex. The ground state of a slightly distorted
tetrahedral Co II complex has quartet spin multiplicity. The fitting of the magnetic
susceptibility shows that the complex has a rather large magnetic anisotropy, but it
remains unclear whether the complex has an easy plane (D > 0) or an easy axis
(D < 0) of magnetization.
1. Draw a level diagram showing the removal of the degeneracy of the M S sublevels
of the quartet state under the influence of (i) spin-orbit coupling and (ii) spin-orbit
coupling and a small external magnetic field along the z-axis.
2. Can the anisotropy parameters D and E be determined from the energy differences
at zero field? And the sign of D?
3. Construct the matrix representation of ˆ
S · D · ˆ
S in an arbitrary frame in the
{|
3
2 ,
3
2 , |
3
2 ,
1
2 , |
3
2 , −
1
2 , |
3
2 , −
3
2 } basis.
4. Use the following data to construct the numerical effective Hamiltonian and
extract D and E. Decide if this complex has easy-axis or easy-plane magnetism.
55
b. The ab initio wave functions of the three states with the largest projection on the
model space are
Ψ 1 = (0.686 − 0.024i)|1, −1+(−0.175 + 0.009i)|1, 0+(−0.685 + 0.046i)|1, 1
Ψ 2 = (−0.664 + 0.197i)|1, −1+(0.036 + 0.123i)|1, 0+(−0.667 + 0.187i)|1, 1
Ψ 3 = (0.110 − 01.08i)|1, −1+(0.957 − 0.147i)|1, 0+(−0.137 − 0.070i)|1, 1
Calculate the overlap matrix of
Ψ i , the projections of Ψ on the model space.
c. The orthonormalized projections are given by
|1, −1| 1, 0| 1, 1
Ψ ′
1
0.695504 −
0.023842i
−0.177003 +
0.008987i
−0.694343 +
0.046723i
Ψ ′
2
−0.672287 +
0.199091i
0.035909 +
0.124483i
−0.675035 +
0.189566i
Ψ ′
3
+0.110493 −
0.109038i
0.964328 −
0.147901i
−0.138092 −
0.070912i
E 1 = 0.00; E 2 = 11.54 cm −1 ; E 3 = 37.55 cm −1 . Construct the effective Hamiltonian and check the consistency of the model Hamiltonian by comparing the
numerical matrix elements of the effective Hamiltonian with the symbolic matrix
elements of the model Hamiltonian.
d. Diagonalize the D-tensor, determine the axial (D) and rhombic (E) anisotropy
parameters from Eq. 2.16 and compare the values with those obtained by extracting D and E from the energies differences (Eq. 2.22).
2.2 Extracting D and E foraCo II complex. The ground state of a slightly distorted
tetrahedral Co II complex has quartet spin multiplicity. The fitting of the magnetic
susceptibility shows that the complex has a rather large magnetic anisotropy, but it
remains unclear whether the complex has an easy plane (D > 0) or an easy axis
(D < 0) of magnetization.
1. Draw a level diagram showing the removal of the degeneracy of the M S sublevels
of the quartet state under the influence of (i) spin-orbit coupling and (ii) spin-orbit
coupling and a small external magnetic field along the z-axis.
2. Can the anisotropy parameters D and E be determined from the energy differences
at zero field? And the sign of D?
3. Construct the matrix representation of ˆ
S · D · ˆ
S in an arbitrary frame in the
{|
3
2 ,
3
2 , |
3
2 ,
1
2 , |
3
2 , −
1
2 , |
3
2 , −
3
2 } basis.
4. Use the following data to construct the numerical effective Hamiltonian and
extract D and E. Decide if this complex has easy-axis or easy-plane magnetism.