Effective hamiltonian of crystal field method for periodic systems containing transition metals

Figures & data

Figure 1. Classification of the experimentally observed low-lying transitions in TM oxides.

Figure 2. Schematic representations of the proposed methodology and data exchange between EHCF and GoGreenGo modules.

Figure 3. Periodic EHCF calculations of (a) total (black) and orbital projected density of states for NiO: orange – O 2s, blue – O 2p, green – Ni 4s and red – Ni 4p; (b) resonance term $V_{\mu \mu }^{res}(\epsilon )$ defined in Equation (18(18) $V_{\mu \nu }^{res}\left(\epsilon \right)=\sum _{n,\mathbf{k}}{\beta }_{\mu n\mathbf{k}}{\beta }_{\nu n\mathbf{k}}\delta (\epsilon -{\epsilon }_{n\mathbf{k}}),$(18) ) for both $t_{2g}$ (blue) and $e_g$ (red) d-orbitals.

Table 1. Periodic EHCF parameters of the electronic band structure of l-system: $E_{gap}^{sp}$ is the energy gap between valence and conduction s,p-bands, and $Q(M),Q(O)$ are effective atomic charges on metal and oxygen.

	$E_{gap}^{sp}$, eV		$Q(M)=-Q(O)$
	periodic model	cluster model	periodic model	cluster model
	this work	[25]	this work	[25]
MnO	7.72	9.71	1.21	1.03
FeO	9.30	10.61	1.25	0.96
CoO	11.53	11.25	1.24	0.89
NiO	12.74	11.82	1.23	0.85

Table 2. Periodic EHCF splitting parameter of d-orbitals compared to the experimental data and cluster model calculations [25]; MAE and MPAE for optical spectra lines are calculated based on Tables 3–6.

	$10Dq$, eV
	periodic model	cluster model	experiment	MAE, eV	MPAE, %
	this work	[25]
MnO	1.35	0.84	1.21 [74], 1.25 [75]	0.08	3.67
FeO	1.38	1.04	1.20 [76,77]	0.16	8.17
CoO	1.27	0.90	1.17 [74]	0.15	7.22
NiO	1.13	0.87	1.13 [2]	0.09	3.47

Table 3. Periodic EHCF d-d transitions in MnO compared to the experimental data [74] and cluster model calculations [25].

Term	periodic model	cluster model	experiment
	this work	[25]	[74]
${}^6{A}_{1g}$	0.00	0.00	0.00
${}^4{T}_{1g}$	1.90	2.92	2.03
${}^4{T}_{2g}$	2.47	3.30	2.58
${}^4{A}_{1g}$	2.96	3.45	2.95
${}^4{E}_g$	2.96	3.45	–

Table 4. Periodic EHCF d-d transitions in FeO compared to the experimental data [76,77] and cluster model calculations [25].

Term	periodic model	cluster model	experiment
	this work	[25]	[76,77]
${}^5{T}_{2g}$	0.00	0.00	0.00
${}^5{E}_g$	1.38	1.05	1.20
${}^3{T}_{1g}$	1.42	1.72	–
${}^1{A}_{1g}$	1.40	2.02	–
${}^3{T}_{2g}$	1.92	2.14	–
${}^3{T}_{1g}$	2.58	2.58	2.64
${}^3{T}_{2g}$	2.78	2.79	–
${}^3{E}_g$	2.98	2.94	–
${}^3{T}_{1g}$	3.04	3.00	3.28

Table 5. Periodic EHCF d-d transitions in CoO compared to the experimental data [74] and cluster model calculations [25].

	periodic model	cluster model	experiment
Term	this work	[25]	[74]
${}^4{T}_{1g}$	0.00	0.00	0.00
${}^4{T}_{2g}$	1.13	0.77	0.90−1.03
${}^2{E}_g$	0.84	1.64	1.61
${}^2{T}_{1g}$	1.88	2.32	2.03
${}^2{T}_{2g}$	1.95	2.35	2.05
${}^4{A}_{2g}$	2.38	1.67	2.14
${}^4{T}_{1g}$	2.39	2.45	2.26−2.33
${}^2{T}_{1g}$	2.41	2.91	2.49−2.56
${}^2{A}_{1g}$	2.90	2.93	2.60

Table 6. Periodic EHCF d-d transitions in NiO compared to the experimental data [2], cluster model calculations [25], and recent DMFT calculations of the triplet-triplet transitions [78].

Term	periodic model	cluster model	DFT + DMFT	experiment
	this work	[25]	[78]	[2]
${}^3{A}_{2g}$	0.00	0.00	0.00	0.00
${}^3{T}_{2g}$	1.13	0.87	0.93	1.13
${}^3{T}_{1g}$	1.86	1.48	1.55	1.75
${}^1{E}_g$	1.94	2.12	–	1.95
${}^1{T}_{2g}$	3.02	2.92	–	2.75
${}^1{A}_{1g}$	3.11	3.04	–	2.95
${}^3{T}_{1g}$	3.25	3.28	2.91	3.25
${}^1{T}_{1g}$	3.60	–	–	3.52

Figure 4. Periodic EHCF calculations of (a) total (black) and orbital projected density of states for MgO: orange – O 2s, blue – O 2p, green – Mg 3s and red – Mg 3p; (b) perturbed (red) and unperturbed (blue) density of states on the impurity site in Ni:MgO; (c) same as (b) but for the oxygen sites surrounding the TM point defect.

Table 7. Periodic EHCF atomic charges induced by the point defect in Co:MgO and Ni:MgO compared to the corresponding values in the ideal MgO crystal.

	impurity site	1${}^{\mathrm{st}}$ neighbours	2${}^{\mathrm{nd}}$ neighbours
MgO	0.958	$-0.958$	0.958
Co:MgO	1.083	$-0.983$	0.954
Ni:MgO	1.043	$-0.978$	0.953

Table 8. Splitting parameter of d-orbitals for Co:MgO and Ni:MgO compared to the experimental data.

		$10Dq$ (eV)
	periodic EHCF	corrected value	experiment
Co:MgO	1.67	1.49	1.20 [80]
Ni:MgO	1.21	1.07	1.07 [81]; 1.00 [82]

Note: Corrected values correspond to the shifted 2p O band as discussed in the text.

Table 9. Periodic EHCF d-d transitions in Co:MgO compared to the experimental data [80].

Term	periodic EHCF	corrected value	experiment [80]
${}^4{T}_{1g}$	0.00	0.00	0.00
${}^4{T}_{2g}$	1.50	1.34	1.05
${}^2{E}_g$	0.46	0.63	1.13
${}^2{T}_{1g}$	1.86	1.87	2.13
${}^2{T}_{2g}$	1.95	1.95	2.13
${}^4{A}_{2g}$	2.43	2.42	2.32
${}^4{T}_{1g}$	2.75	2.58	2.43
${}^2{T}_{1g}$	3.18	2.83	2.54
${}^2{A}_{1g}$	3.28	3.11	3.05

Table 10. Periodic EHCF d-d transitions in Ni:MgO compared to experimental data [81].

Term	periodic EHCF	corrected value	experiment [81]
${}^3{A}_{2g}$	0.00	0.00	0.00
${}^3{T}_{2g}$	1.20	1.07	1.07
${}^1{E}_g$	1.95	1.78	1.68
${}^3{T}_{1g}$	1.97	1.94	1.83
${}^1{T}_{2g}$	3.10	2.96	2.69
${}^1{A}_{1g}$	3.13	3.08	3.04
${}^3{T}_{1g}$	3.36	3.16	3.21
${}^1{T}_{1g}$	3.67	3.54	3.50

Figure A1. Total and orbital projected densities of states for MnO (a), FeO (b), CoO (c) and NiO (d). Low lying wide 2s bands located in the interval from are not shown for clarity. Color code is the same as in Figure 3(a).

M. Razumov and A.L. Tchougréeff, Russ. J. Phys. Chem. 74, 87–93 (2000).

 Google Scholar

G.W. Pratt and R. Coelho, Phys. Rev. 116 (2), 281–286 (1959). doi:10.1103/PhysRev.116.281

 Google Scholar

D.R. Huffman, R.L. Wild and M. Shinmei, J. Chem. Phys. 50 (9), 4092–4094 (1969). doi:10.1063/1.1671670

 Web of Science ®Google Scholar

K.W. Blazey, Jo. Phys. Chem. Solids 38 (6), 671–675 (1977). doi:10.1016/0022-3697(77)90236-0

 Web of Science ®Google Scholar

P. Cox, Transition Metal Oxides (Clarendon Press, Oxford, 1992).

 Google Scholar

R. Newman and R.M. Chrenko, Phys. Rev. 114 (6), 1507–1513 (1959). doi:10.1103/PhysRev.114.1507

 Google Scholar

S. Acharya, C. Weber, D. Pashov, M. van Schilfgaarde, A.I. Lichtenstein and M.I. Katsnelson, A Theory for Colors of Strongly Correlated Electronic Systems, arXiv 2022. doi:10.48550/arXiv.2204.11081

 Google Scholar

W. Low, Phys. Rev. 109 (2), 256–265 (1958). doi:10.1103/PhysRev.109.256

 Google Scholar

W. Low, Phys. Rev. 109 (2), 247–255 (1958). doi:10.1103/PhysRev.109.247

 Google Scholar

J.E. Ralph and M.G. Townsend, J. Phys. C: Solid State Phys. 3 (1), 8–18 (1970). doi:10.1088/0022-3719/3/1/002

 Google Scholar