Supplementary materials.
Details of theoretical calculations
It is known that for tetrahedron ligands the energy of a particular multielectron d 5 term may be
obtained from the term in octahedral position by the crystal field parameter 10Dq sign reversal, and
reduction in magnitude because of weaker d-orbital–ligand interactions 32.
In the Kanamory model, which takes into account only the intraatomic Coulomb repulsion of two
electrons at the same orbital (Hubbard U), the energies of different spin states can be easily written
following Ref. 33:
For HS : S = 5/2
EHS = Ec(d5) – 10J,
(S1)
For IS:
EIS = Ec(d5) – 6J – 10Dq ,
(S2)
S = 3/2
For LS: S = 1/2
ELS = Ec(d5) – 4J – 20Dq ,
(S3)
Here HS is the high-spin state of Fe3+ ion (S=5/2), IS is the intermediate spin state (S=3/2), and LS is
the low-spin state (S=1/2), J is the intraatomic Hund exchange coupling, and 10Dq is the crystal
splitting. The value Ec(d5) is the spin-independent part of the Coulomb energy.
Under high pressure the intraatomic parameters are assumed to be pressure independent while the
crystal field energy linearly increases with pressure, 10Dq(P) = 10Dq(0) +P.
The analysis of the energy levels evolution under pressure reveals that at small pressure the HS state is
the lowest in energy (ground state). The HS and LS terms crossover takes place at critical pressure P =
Pc ,
Pc = (3J – 10Dq(0)) /  .
(S4)
The IS term energy is above the ground state energy for all values of parameters in non-distorted
tetrahedral environment.
If we take the parameters of Fe3+ found in tetrahedral sites of iron garnet
15
Dq(0) = 620 cm-1,
10Dq(0) = 0.77eV,  = 1.6 meV/GPa, and J = 0.7eV, we calculate the critical pressure to be Pc = 830
GPa.
For octahedral sites the critical pressure is given by the same formula (Eq. S4), but with different
parameters. For Fe+3 Dq = 1220 cm-1, 10Dq(0) = 1.5 eV, and  = 8.5meV/GPa 15, which gives Pc = 70
GPa.
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To calculate the effective Hubbard parameter Ueff at different pressures, we take UeffHS = EHS(d4) +
EHS(d6) - 2EHS(d5) at P < Pc, and UeffLS = ELS(d4) + ELS(d6) - 2ELS(d5) at P > Pc . For tetrahedral sites,
we calculate the HS and LS terms of the d4 and d6 configurations in the same way. Finally:
UeffHS = U + 4J - 10Dq(P) at P < Pc,
UeffLS = U
at P > Pc
(S5)
.
Thus, UeffHS linearly decreases down to (U+J) at P = Pc, then it drops down to UeffLS = U at P > Pc.
These calculations show that for the tetrahedral sites one has the effect of decreasing correlations
under pressure similar to that in the octahedral sites.
2