Pressure-induced magnetic transitions with change of the orbital configuration in dimerised systems

We suggest a possible scenario for magnetic transition under pressure in dimerised systems where electrons are localised on molecular orbitals. The mechanism of transition is not related with competition between kinetic energy and on-site Coulomb repulsion as in Mott-Hubbard systems, or between crystal-field splitting and intra-atomic exchange as in classical atomic spin-state transitions. Instead, it is driven by the change of bonding-antibonding splitting on part of the molecular orbitals. In the magnetic systems with few half-filled molecular orbitals external pressure may result in increase of the bonding-antibonding splitting and localise all electrons on low-lying molecular orbitals suppressing net magnetic moment of the system. We give examples of the systems, where this or inverse transition may occur and by means of ab initio band structure calculations predict that it can be observed in α−MoCl4 at pressure P ~ 11 GPa.


General Treatment
We start with an isolated dimer. Transition metals are in the ligand octahedra and these octahedra share their edges or faces (metal-metal distances in the "common corner" geometry are usually too large for the formation of molecular orbitals). There are two types of orbitals in such geometries, which we will denote as c and d. The c orbitals have a direct overlap, characterized by hopping parameter t c , with neighbouring transition metal (the a 1g orbitals in the "common face" 14,15 and the xy orbitals in the "common edge" geometry 16 ), which results in a large bonding-antibonding splitting, 2t c . If there are more than one electron per site and t c is large enough (with respect to Hund's rule coupling J H ), the bonding molecular orbital is fully occupied and corresponding electrons do not contribute to the total magnetic moment of the dimer 17 . So that the magnetization is defined by other, d, electrons, localised on the π e g (for face-sharing) or xz/yz (for edge-sharing) orbitals, as shown in Fig. 1(a). These two d orbitals ( π e g for face-sharing and xz/yz for edge-sharing geometry) are not directed to each other, so that corresponding hoppings (t d1 , t d2 ) ≪ t c . It is important for us here that t d1 and t d2 can also be very different, since typically the TMO 6 octahedra are strongly distorted in the dimerised systems. Applying external pressure one increases all the bonding-antibonding splittings, given by hopping parameters t d1 , t d2 , and t c , and may suppress net magnetization, even if there was nonzero magnetic moment at ambient conditions. In a some sense this effect reminds classical atomic spin-state transition 1 , but instead of the crystal-field splitting here bonding-antibonding splitting between molecular orbitals competes with the Hund's rule coupling.
It is easier to illustrate this general picture on a particular example. Let us consider dimerised system with two electrons per site, t d1 ≠ t d2 , and (t d1 , t d2 ) ≪ t c . Then two out of four electrons will occupy c bonding orbitals, while the rest two electrons provide magnetic moment, see Fig. 1(a). If these d1 and d2 orbitals are molecular orbitals then there is a gain in intra-atomic exchange energy for spin triplet (ferromagnetic) state with respect to spin singlet (antiferromagnetic) one. In the ionic limit taking into account intra-atomic Hund's rule as and neglecting on-site Coulomb repulsion U (and hence modification of the ground state wave function from molecular orbital-like to Heitler-London) one may find that the energy of this state will be Applying external pressure we increase all the hopping parameters t c , t d1 , and t d2 , so that finally one may end up with the situation, when not only c, but also one of the d molecular orbitals is completely filled, as shown in Fig. 1(b). The energy of this nonmagnetic state will be Comparing last two equations one finds that the transition to the nonmagnetic state is expected, when In real materials the situation, however, can be much more complicated. Mentioned above effect of the Hubbard U does not simply renormalize J H , but changes energetics of the bonding orbitals, which is defined solely by corresponding hopping parameter t in the absence of U and by t 2 /U in the large U limit. In addition the It is supposed that there are two different sets of orbitals in the system: c orbitals (green) with larger bonding-antibonding splitting, given by hopping parameter t c , and d orbitals (blue). If hopping for one of the d orbitals is much larger than for another t d2 ≫ t d1 , when nonmagnetic state is realized in the system. on-site energies of the d orbitals can be very different due to strong distortions of the TMO 6 octahedra. However, qualitative picture is rather general: having magnetic dimerised system with few degenerate or nearly degenerate half-filled d molecular orbitals one may expect to have a transition to nonmagnetic state under external or due to internal (chemical) pressure. In order to check this effect we performed ab initio band structure calculations for α− MoCl 4 , which fulfills aforementioned conditions.

Pressure-induced Magnetic Transition in α− MoCl 4
The α− MoCl 4 crystalizes in the NbCl 4 structure consisting of Mo-Mo dimers 18 , see Fig. 2. Mo 4+ has 4d 2 electronic configuration and at ambient conditions this material is paramagnetic with positive Curie-Weiss temperature ~220 K 17 , which presumes net ferromagnetic exchange coupling. The effective magnetic moment is ~0.85 − 0.93μ B 18,19 , much smaller than μ eff = 2.82 μ B expected for isolated Mo 4+ ion having S = 1. Suppression of the magnetic moment is related to orbital-selective effect in dimerised systems 17,20 . Each Mo is in the Cl 6 octahedron and two neighbouring octahedra share their edges forming a dimer. As a result there has to be a strong bonding-antibonding splitting for the xy orbitals, which play a role of the c orbitals (here and below all notations are with respect to the local coordinate system, where axis are directed to Cl, and x and y are in the plane of common edge and short Mo-Mo bond). Bonding xy orbitals are fully occupied and this explains experimentally observed partial suppression of the magnetic moment in this system at ambient conditions. This strong splitting ~3.2 eV is clearly seen from the nonmagnetic band structure (Fig. 3(a)), obtained in the generalized gradient approximation (GGA). In Fig. 4(a) we plotted the charge density corresponding to these bonding xy orbitals.
The xz and yz orbitals also form molecular orbitals. These are d1 and d2 orbitals in the notations of the previous section. This is clear that effective d − d hopping via Cl p z orbital is the same for xz and yz orbitals centered on different sites, but one may also maximize direct d − d hopping constructing the xz + yz orbital, see Fig. 4(b), so that systems gains maximum kinetic energy localising electrons on these xz + yz and xz − yz orbitals. Very similar situation is observed in Li 2 RuO 3 21 . The bonding-antibonding splitting for the xz + yz molecular orbitals is ~1.6 eV, while for xz − yz it is much smaller, ~0.2 eV.
In order to take into account strong Coulomb correlations we performed the GGA + U calculations. Constrained RPA (cRPA) calculations for metallic Mo give U − J H ~ 3 eV 22 . One may think that metallic Mo is very different from MoCl 4 and Hubbard U in a chloride can be much larger resulting to Mott-Hubbard physics. However, an estimation of U using constrained GGA method within the Wannier function formalism 23 for α − MoCl 4 gives U~2.9 eV, so that one may use the cRPA result for the GGA + U calculation. The local magnetic moment in the GGA + U was found to be m tot = 0.85 μ B /Mo. Analysis of the occupation matrix shows that this moment is mainly due to the xz + yz and xz − yz orbitals: m xz+yz = 0.29μ B and m xz−yz = 0.36μ B . Because of a large spatial extension of the Mo 4d orbitals substantial portion of the spin density is on the ligands, m Cl = 0.07μ B /Cl.
Increasing pressure we induce magnetic transition, as it was described above in details. We studied this transition by the total energy (E) GGA + U calculations for ferromagnetic and nonmagnetic configurations for several volumes (V). Corresponding E(V) dependencies are shown in Fig. 5. The first order transition with collapse of the volume was found at critical pressure P c = 11.2 GPa, which was estimated by fitting E(V) with the fifth order polynomial and finding its derivative 10,24,25 . Analysis of the occupation matrix shows that orbitals configuration indeed changes in the nonmagnetic phase, where four electrons occupy xy and xz + yz bonding orbitals, as it is shown in Fig. 1(b).
Thus, we see that the magnetic transition proposed in the previous section basing on quite general arguments does occurs in the GGA + U calculation for the real material, α− MoCl 4 . One may also argue that such transition can be realized in many other different systems, e.g. in WCl 4 26 or Nb 2 O 2 F 3 13 . Moreover, an inverse transition from nonmagnetic to ferromagnetic state under tensile stress is also possible. It would be interesting to study whether such transition can be observed, e.g. in MoO 2 or WO 2 films grown on the substrates with larger inter-atomic distances.
It is also exciting that very similar transition seems to occur in famous half-metallic CrO 2 . At ambient conditions this compound is ferromagnetic and has the rutile crystal structure, where neighbouring CrO 6 octahedra share their edges 27 . Main mechanism of the ferromagnetism is double exchange, when itinerant xy electrons make localised xz/yz electrons to have the same spin projection 28 . On a language of an isolated dimer this would correspond to the situation when xy is c and xz/yz are d orbitals and having two electrons per Cr site we fully occupy xy (c) orbital and leave xz/yz (d) orbitals half-filled to fulfill Hund's rule. The LDA + U calculations show that this is exactly what is going on in CrO 2 28 . However, detailed band structure calculations shows that CrO 2 undergoes structural phase transition to dimerised phase at P~70 GPa and turns out to be nonmagnetic 29 . This strongly reminds pressure-induced transition in α− MoCl 4 discussed in the present paper.

Conclusions
To sum up in the present paper we considered the dimerised transition metal compounds with degenerate (or nearly degenerate) half-filled magnetic molecular orbitals and showed that the pressure-induced magnetic transition is possible in this case. This transition to nonmagnetic state is related to the change of the orbital configuration and results in a strong suppression of the magnetic moment of the system. Using band structure calculations we checked that this transition does occurs in one of such systems: α− MoCl 4 and argue that it can be related to stabilization of nonmagnetic state in CrO 2 under high pressure.

Methods
All calculations in this work were performed with Quantum-ESPRESSO package 30 that implements the ultrasoft pseudopotential formalism in plane-waves basis. The exchange-correlation potential was taken in the form proposed in ref. 31. A kinetic energy cutoff for the plane-wave expansion of the electronic states was set to 45 Ry. Reciprocal space integration were done on a regular 8 × 8 × 8 k-points grid in the irreducible part of the Brillouin zone. In order to check reliability of pseudopotential method we calculated δE = E FM − E NM at V = 0.8 V exp using  our ultrasoft pseudopotentials and the projector augmented-wave (PAW) method 32 . The difference in δE in these two calculations was found to be less than 1%. The crystal structure was taken from ref. 18.