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In view of the growing interest in molecular orbitals (MOs) encountered in certain complex oxides, we review some of their properties from the band theory perspective and provide detailed examples based on real materials. Our discussion includes some technical aspects of identifying MOs in electronic structure calculations and considers cases when MOs can be both orthogonal and non-orthogonal. We also describe orthonormalization of MOs, a procedure converting them into Wannier functions, and discuss the problem of Wannier functions possibly being rather spatially extended and how using MO, rather than atomic orbital, based effective Hamiltonians might be a better choice in describing certain strongly correlated systems as well as systems with strong electron-phonon coupling. Furthermore, we address the problem of strongly correlated MOs and how it can be treated in band theory calculations.

Ligand molecular orbitals (MOs) have been recognized as important players in the physics of transition-metal compounds since the introduction of the Zhang-Rice singlet back in 1988 [

Let us consider the prototypical negative charge-transfer insulator NaNiO_{2} as our first example. Its monoclinic C2/m crystal structure consists of triangular layers of edge-sharing cooperatively elongated NiO_{6} octahedra intercalated with Na^{+} ions, with one formula unit per primitive unit cell [^{2+}L, where L is an oxygen hole [^{3+} ( e g 3 ) with spin 1/2, which is quite contrary to what one would expect from Hund’s rules if it really were Ni^{3+}. Due to the strong hybridization between the Ni—e_{g} orbitals and the oxygen MOs of respective symmetry, the oxygen holes select to occupy the O—(x^{2}-y^{2}) MOs, one hole per oxygen octahedron, and form a Zhang-Rice-like spin singlet state with the hole in the Ni—(x^{2}-y^{2}) orbital. This selective occupation of the (x^{2}-y^{2}) MOs by the holes, which results in the mentioned cooperative elongation of the NiO_{6} octahedra, constitutes a molecular orbital (or band) Jahn-Teller effect.

A way of verifying and visualizing this picture using band theory is to perform a projection of the NaNiO_{2} electronic structure on a MO-based basis set. We note, however, that band theory is unable to fully capture the multi-determinant character of the spin singlet state in question and will underestimate the energy

of the singlet as 1 4 J p d rather than 3 4 J p d , where J_{pd} is the spin exchange coupling

between interacting atomic d and molecular p states. Still, it provides a fair description of on-site symmetries and spin densities, worth exploring with the projection technique.

_{2} density of states (DOS) projected onto the Ni—3d orbitals (left) and the oxygen molecular orbitals (right) for the majority and minority spin channels. This is a spin-polarized local density

approximation + U (LDA + U) [_{H} = 1 eV for Ni—3d electrons [^{2}-y^{2}) atomic orbital and the oxygen—(x^{2}-y^{2}) molecular orbital, which is a result of strong hybridization between these orbitals. This hole state is the anti-bonding combination of the two orbitals pushed up in energy above the Fermi level, while the bonding combination lands at about −5 eV below the Fermi level. An isolated oxygen—(x^{2}-y^{2}) molecular orbital ψ R M ( r ) centered at a Ni site R is shown in

ψ k M ( r ) = ∑ R ψ R M ( r ) e − i k R , (1)

where M indicates an MO character of the wave function and k is the crystal wave vector. To give a visual example, we show the oxygen—(x^{2}-y^{2}) MO Bloch function at the Γ point [k = (0, 0, 0)] in

Generally speaking, MO Bloch functions ψ k M i ( r ) can be obtained from the atomic orbital Bloch functions ψ k a j ( r ) via a unitary transformation U(k):

ψ k M i ( r ) = ∑ j U i j ( k ) ψ k a j ( r ) . (2)

Allowing for the k-dependence of the transformation matrix elements U_{ij}(k) reflects the fact that an MO can be built from atomic orbitals nominally belonging to different unit cells. This is, for instance, the case for the oxygen—(x^{2}-y^{2}) MO shown in _{σ} orbitals to the left of the Ni atom and those to its right belong to different unit cells. _{σ} orbitals on a given NiO_{6} octahedron are in fact the | p x 〉 , | p y 〉 , and | p z 〉 orbitals of the two inequivalent oxygens O1 and O2 in a unit cell, but each coming from an oxygen belonging to a different unit cell. In a way, using a k-dependent unitary transformation matrix U(k) broadens the technical definition of a unit cell as used in band-structure calculations for the purpose of setting up a Bloch basis set, where now atomic orbitals from the same atom can belong to different unit cells as in the case of NaNiO_{2}.

Let us also note that if a MO character of an electronic state shows very little variation with k-vector this should be regarded as a sign of the robustness of the molecular nature of this state in real space. As one can see in _{2}, where indeed the majority spin hole state has a strong oxygen—(x^{2}-y^{2}) character at every k-vector.

Although in the charge- and negative charge-transfer systems it is very typical for the ligand holes to occupy MOs of the same symmetry as that of the cation atomic orbitals that they hybridize with [like the oxygen—(x^{2}-y^{2}) MO and the Ni—(x^{2}-y^{2}) atomic orbital in NaNiO_{2}], there are a number of notable exceptions. For example, in the iron disulfide FeS_{2} (pyrite structure) and the recently discussed iron dioxide FeO_{2} [_{σ} orbitals. The same is the case in the superoxides like KO_{2}, where oxygen dimers have a net spin of 1/2 and which order magnetically below the ordering temperature.

As a final technical remark for this section, our MO projections are performed using atomic-like functions | u l α , σ ( E 1 l ) Y m l 〉 , which are the solutions of the Schrödinger equation within the muffin-tin sphere of atom α at the linearization energy E 1 l and where | Y m l 〉 are spherical harmonics, to construct molecular orbitals. This is in some contrast with the approach adopted in Wien2k, where atomic orbital projections are done onto the spherical harmonics | Y m l 〉 inside muffin-tin spheres. In practice, the two approaches give very similar results for the projected DOS, but having also the radial part u l α , σ ( E 1 l , r ) means that projections are being done on a true atomic-like orbital (or molecular orbital combination), which, in particular, enables their visualization in real space.

For certain lattice structures—including those of the cuprates, with their characteristic two-dimensional square CuO_{2} planes, as well as of the bismuth and nickel perovskites—construction of MOs may also require their proper orthonormalization performed on top of the unitary basis transformation discussed above. In this regard, let us consider the example of the barium bismuth perovskite BaBiO_{3}. In a perovskite lattice structure of the ABO_{3} type [_{6} octahedra share their corners such that the O-B-O angle is equal or close to 180˚. As a result, the oxygen—p_{σ} orbitals, typically involved in the formation of MOs, have to be shared between neighboring B cations. In the case of BaBiO_{3}, the MO of most importance is oxygen—a_{1g}, shown in _{σ} orbital, neighboring oxygen—a_{1g} MOs are non-orthogonal to each other. If no orthonormalization is done to such MOs, the Bloch functions constructed out of them will have a k-vector dependent overlap integral, 〈 ψ k M | ψ k M 〉 , and may even completely vanish at certain k-points. As demonstrated pictorially in _{1g} MO in BaBiO_{3} at the Γ point.

In order to be suitable for use in effective models, MOs of this kind need to be orthonormalized first. As was originally shown by Zhang and Rice [

| β k | 2 〈 ψ k M | ψ k M 〉 = 1 , (3)

may be ill-defined at certain k-points. For the oxygen—a_{1g} MO in BaBiO_{3}, for example, it diverges at the Γ point:

β k = [ 1 − 1 3 ( cos k x + cos k y + cos k z ) ] − 1 / 2 , (4)

as obtained using the oxygen—a_{1g} MO given by

| ψ k a 1 g 〉 = 1 6 [ | p k x 〉 ( 1 − e − i a k x ) + | p k y 〉 ( 1 − e − i a k y ) + | p k z 〉 ( 1 − e − i a k z ) ] , (5)

where | p k y 〉 , i = x, y, z, are the Bloch functions of orthonormal oxygen—p_{σ} orbitals and a is the cubic lattice constant. Let us note that an MO with contributions from both the oxygen—a_{1g} MO and the Bi—6s atomic orbital | s k 〉 ,

| ψ k s − a 1 g 〉 = 1 α 2 + β 2 ( α | s k 〉 ± β | ψ k a 1 g 〉 ) , (6)

which was used by us and our co-authors in [_{3} effective hopping parameters, does not have this divergence problem.

It is important to recognize the fact that orthonormalization of a molecular orbital ψ R M ( r ) of any kind via multiplication of its corresponding Bloch function ψ k M ( r ) by β k converts this molecular orbital into a Wannier function [

| w k 〉 = ∑ n < ν < m 〈 Ψ k , ν | ψ k 〉 | Ψ k , ν 〉 . (7)

Here, | ψ k 〉 can be either an atomic or a molecular orbital Bloch function and ν is a band index running from band n to band m. On top of this construction, one may also apply the maximal localization [^{n}^{-1} state is inside the O—2p band while the unoccupied d^{n}^{+1} state is very well above. In this situation, there would be no easy way to define properly the Hubbard interaction parameter U.

As an example of how much spatial extension a nominally maximally localized atomic Wannier function | w k 〉 can have and how strongly it can be affected by covalence effects, let us consider Ni—3d orbitals in the charge-transfer gap insulator NiO. _{H} = 1 eV and U = 2 eV [panel (a)], U = 12 eV [panel (b)], and U = 6 eV [panels (c)-(e)] applied to Ni—3d electrons. Fat bands indicate the strength of the Ni—3d character, while the dashed lines are the eigenvalues of an effective Hamiltonian in the basis of maximally localized Ni—t_{2g} Wannier functions obtained using the Wannier90 package [_{2g} bands lie either above or below the oxygen—2p bands, respectively, which reduces their hybridization and makes it possible to obtain a reasonable effective Hamiltonian with eigenvalues closely matching the LDA + U bands. However, the resulting Wannier functions in real space (shown below the corresponding band-structure plots in _{2g} and oxygen—2p bands are

energetically very close and so get strongly hybridized. As a result, wannierization may give even more ambiguous results in this case. Depending on whether certain constraints are applied [

The cuprates are another good example where we can define the upper Hubbard band very well because it is well separated from the oxygen—2p band, while the lower Hubbard band, involving also all kinds of multiplets spread out over about 8 eV, is for a large part inside the oxygen—2p band. Also, the lowest energy d^{8} state without the hybridization switched on is a triplet F state and not a singlet state, as one assumes in a Hubbard model. In fact, the singlet A_{1g}d^{8} state hybridizes so strongly that it pushes out the Zhang-Rice singlet state from the top of the O band and this state has more oxygen—2p than Cu—3d character in it. This is very similar to the BaBiO_{3} “bound two-hole state” pushed up above the Fermi level that we discussed in Section 3. If the Zhang-Rice state is well enough pushed out of the top of the O band, then we could actually define a single site Wannier function for this but it would have more density on O than on Cu.

In this case of correlated oxides where transition metal—d orbitals are strongly localized yet also subject to hybridization with oxygen—p orbitals, a way to improve on the localization of Wannier functions would be to also include oxygen—2p orbitals into the Wannier basis. This, however, may significantly increase the size of the Hilbert space required in model calculations. On the other hand, the increase can be much less dramatic if only the most important molecular combinations of oxygen—p orbitals are considered, such as the oxygen—(x^{2}-y^{2}) or oxygen—a_{1g} in our earlier examples of NaNiO_{2} and BaBiO_{3}. Recently, this promising MO based approach has been successfully applied to calculate resonant X-ray spectral responses in rare-earth nickelates [_{3} where the A_{1g}—symmetric (so-called “breathing”) oxygen phonon mode is coupled to the hybridization strength between the Bi-6s atomic orbital and the oxygen—a_{1g} MO. The role of this coupling in the bismuthates’ superconductivity has recently been a subject of intense theoretical research, both at the band theory [

Since molecular orbitals are localized objects, electrons or holes occupying them may be subject to strong on-site Coulomb repulsion, where a site can now be a molecular orbital one and technically comprise more than one atomic site. From the band theory point of view, it would be very desirable to have a mean-field method, perhaps similar to the conventional LDA + U, in order to describe local correlations in systems with dominant MO character of the valence bands. Interestingly, application of the conventional LDA + U method to such systems may have an unphysically detrimental effect on their MO nature.

As an illustration, let us consider sodium iridium oxide Na_{2}IrO_{3}, a famous candidate Kitaev system [_{6} octahedra. _{2}IrO_{3} DOS projected onto Ir—5d MOs, as obtained from a generalized gradient approximation [_{2}IrO_{3} electronic structure and therefore neglect spin-orbit coupling effects, which one would otherwise need to also take into account. In this calculation, we assume a zig-zag order of Ir magnetic moments, shown in _{2}IrO_{3} [_{1g} + E_{2u} MO and slightly filled A_{1g} − E_{2u} MO in one spin channel and an opposite situation in the other spin channel, with A_{1g} ± E_{2u} standing for linear combinations of MOs with the A_{1g} and E_{2g} symmetries. The connection between the zig-zag magnetic order and such peculiar MO occupations becomes

clear if one inspects the shapes of these two MOs in real space. Indeed, as shown in _{2u}, which is slightly empty in both spin channels. If a method like GGA + U but designed to act in the basis of MOs were applied to Na_{2}IrO_{3}, it would push up in energy and completely empty the A_{1g} − E_{2u} and A_{1g} + E_{2u} MO states and would push down and complete fill the E_{2g} MO states. This is not what happens when the conventional GGA + U method (U = 2.7 eV and J_{H} = 0.7 eV on Ir-5d orbitals) is applied [_{1g} − E_{2u}, A_{1g} + E_{2u}, and E_{2u} MO states, thus destroying the MO nature of the Na_{2}IrO_{3} valence bands. Similar damage is also done to the lower-lying B_{1u} and E_{1g} MOs.

For the local MO correlations to be treated properly (at least, in a mean-field sense), two approaches seem to be in place. In the first approach, one can first construct MOs via the unitary basis transformation of Equation (2) and then use the conventional LDA + U expressions for total energy and potential but written in the basis of MOs. We see three problems associated with this approach. First, it is not always obvious in which way MOs should be constructed. Second, this approach is intrinsically rotationally non-invariant, i.e., would produce different results depending on the choice of the basis. Third, calculation of the Coulomb and exchange interaction matrix elements U M i M j and J M i M j , with M_{i} and M_{j} denoting MOs, may pose a serious challenge.

The second and, as we believe, more promising approach consists in extending the atomic orbital LDA + U method by additionally taking into account Coulomb interactions between different atomic sites. Also referred to as extended DFT + U + V, this approach was first discussed by Campo and Cococcioni in [_{2}IrO_{3}. Indeed, although GGA suffers from the self-interaction problem, it at least treats both on-site and inter-site interactions on the same footing. This allows GGA to capture qualitatively all MO energy splittings in Na_{2}IrO_{3} [

In summary, we presented our band theory perspective on some important aspects of molecular orbitals in complex oxides. In particular, we first discussed technical issues related with the implementation of MO-based projection of electronic states, using NaNiO_{2} as an example. Then, this discussion was extended by considering non-orthogonal MOs, as observed in BaBiO_{3}, and the way of performing their orthonormalization. This led us to the concept of atomic and molecular orbital Wannier functions, where we discussed the problem of the Wannier functions being typically rather extended in real space and how this property may restrict their usage as basis functions in model Hamiltonians, using NiO as an example. A suggestion was then made that MO based model Hamiltonians might be a good approach to describing correlated transition metal systems with strong cation-anion orbital hybridization and also systems with strong electron-phonon coupling. Our final point concerned the problem of a proper band theory treatment of strong correlations between electrons or holes occupying molecular rather than atomic orbitals. In this context, we discussed the failure of the conventional atomic LDA + U method to capture the MO nature of the electronic states in Na_{2}IrO_{3} and outlined possible pathways towards a more adequate band theory based description. As the interest in better understanding of the role of molecular orbitals in complex oxides is growing, we expect that our current review will provide a useful reference for future studies.

This work was supported by NSERC, CIfAR, and the Max Planck—UBC Stewart Blusson Quantum Matter Institute.

The authors declare no conflicts of interest regarding the publication of this paper.

Foyevtsova, K. and Sawatzky, G.A. (2019) A Band Theory Perspective on Molecular Orbitals in Complex Oxides. Journal of Modern Physics, 10, 953-965. https://doi.org/10.4236/jmp.2019.108062