BONDING AND ELECTRONIC STRUCTURE OF MINERALS

Transcription

1 BONDING AND ELECTRONIC STRUCTURE OF MINERALS RONALD E. COHEN Carnegie Institution of Washington Geophysical Laboratory and Center for High Pressure Research 5251 Broad Branch Rd., N.W. Washington, D.C USA Abstract. Minerals are crystalline solids, and their properties are governed by quantum mechanics. Density functional theory in the local density approximation or the generalized gradient approximation gives accurate predictions for energetic properties of closed shell systems, as well as ionic/covalent crystals, and open-shelled transition metals and transition metals oxides. The electronic structure and phase transitions in transition metal oxides are forefront problems in solids state physics and high pressure physics, and much progress is being made. 1. Introduction Minerals are Crystalline Solids The first thing that one must realize in order to understand minerals, is that minerals are solids, and the same physical laws that govern mineral behavior are those for all other solids. It is quantum mechanics that governs the behavior of electrons and nuclei in solids, and it is quantum mechanics that rules mineral behavior. The atomic positions in minerals have spacegroup symmetry, and the behavior of the electrons and nuclei, and thus the properties of minerals depend on symmetry. Thus one must study solid state physics and crystallography to understand mineral behavior. No attempt will be made here to duplicate the manifold excellent texts on solid state physics and crystallography. The student is pointed to texts such as Ashcroft and Mermin [1], Kittel [2], and Harrison [3] for the general background needed to understand minerals. It used to be that important minerals were generally more complex than the important materials of solid state physics. Mineralogists were interested in feldspars, and physicists were

2 2 interested in NaCl or Cu. This is not generally true anymore, with the great interest in cuprate superconductors, complex ferroelectrics and multicomponent and complicated alloys in physics, and the greater interest in simple materials such as MgO in mineralogy. Many of the problems are also the same, predicting and understanding phase transitions, equations of state, electrical and chemical transport, etc. The goals remain somewhat different however. The goal of mineralogy is to understand natural materials in order to interpret better the way the world works, without regard to utility or applications. In this brief review the underlying physics of minerals will be touched on, and bonding in minerals will be discussed. No attempt is made here to thoroughly review different theoretical methods, to study any systems in detail, or to provide a practical tutorial on computation. Also, this is not meant to be a balanced review of all methods and work done on theory of materials; rather it is a biased picture and represents just one point of view. 2. Minerals are made of atoms, or electrons and nuclei No attempt will be made here to review solid state theory in any significant way. Instead the flavor of our current understanding of the fundamental physics of minerals will be presented, and some key points and milestones in our understanding of the theory of minerals will be discussed. First one must understand that minerals are made out of atoms, and that all mineral properties are governed by the atoms that make up a mineral, and the interactions between the atoms. As a graduate student I remember telling my advisor, J.B. Thompson, Jr. that I didn t want to have to deal with electrons, but he said that you may have to worry about the electrons! And indeed, if you are interested in the fundamentals of mineral behavior, you will have to worry about the electrons, because it is the interactions between electrons in atoms in a material that are responsible for everything about the mineral. Quantum theory tells us that the time-independent ground state of a system is given by a complex antisymmetric many body wave function, whose square gives the probability of finding a particle in each point in space ρ(r)= dr 2 dr 3 dr 4...Ψ (r, r 2, r 3, r 4, r 5...)Ψ(r, r 2, r 3, r 4, r 5...) (1) In principle, the wave function can be represented as a sum of determinants of orthogonal single particle states, φ i, of the system. In practice this approach, known as Configuration Interaction (CI) is intractable for solids, and even for molecules and many-electron atoms converges very slowly. In an atom or molecule, the eigenvalues are the well known energy levels of

3 the electronic system. In a crystal, these states are modified by the other atoms in the crystal. The core states, or deep levels for each atom, remain sharp delta function-like states, which may be raised or lowered in energy relative to their positions in isolated atoms. These core-level shifts are due largely to the screened Coulomb potential from the rest of the atoms in the crystal. The occupied valence and empty conduction states no longer look like atomic states, but are broadened into energy bands (fig. 1). There are often intermediate states between the core levels and the valence states called semi-core states which are slightly broadened at low pressures, but which become broader and more different from atomic states with increasing pressure. Atoms in crystals can also donate or accept electrons from 3 3p 3s 2p 2s 2p 2s 1s Si molecule O 1s 3p 3s 2p 2s 2p 2s 1s Si crystal O 1s Figure 1. Schematics of electronic energy states in (a) an Si-O molecule and (b) a SiO 2 crystal. The atom level hybridize, and are broadened into energy bands rather than discrete levels in the crystal. The three hatched bands in the crystal indicate bonding, non-bonding, and conduction (anti-bonding) bands from bottom to top. other atoms, or in other words the states can be filled differently than in atoms, leading to the formation of ions. This ionicity is very important in the bonding of minerals, and is driven by the increased stability of an ion when it has a filled shell of electrons. For example, oxygen has a nuclear charge Z=8, which means that it would have 2 1s, 22s, and42p electrons, but a filled p shell has 6 electrons. Thus the O atom is highly reactive, and

4 4 in the gas combines to form O 2 or other molecules with other atoms. In an oxide or silicate crystal, the O grabs two more electrons to form an O 2 anion from another atom, which becomes a positively charged cation. This leads to a strong attractive electrostatic, or Madelung, interaction between the ions which greatly enhances the crystal stability. The band states in a crystal can be characterized by a continuous quantum number k, so that the eigenvalues are ɛ(k), and the eigenfunctions can represented as Bloch states φ i (k, r)=u(r)exp ik r (2) where u(r) is a periodic function of position r. The eigenvalues as a function of k are known as the band structure. The density of states gives the density of eigenvalues around each energy E. Each eigenfunction also has a certain amount of weight around each atom, and a partial density of weights can be found that indicates the amount each atom contributes to each energy range. By studying the band structure and densities of states (figs. 2,3,4), and the charge densities, one can understand the nature of bonding and how it changes with chemistry, distortions, and pressure. (a) (b) Figure 2. Computed band structures for MgO. (a) MgO B1 V =18.1Å 3, GPa. (b) MgO B1 V =10Å 3, 290 GPa. States from the bottom are Mg 2p, O2s, O2p, Mg3s, O 3s at zero pressure. The band gap is above the O 2p states. Note that the O 2s band has significant width even at zero pressure, and a large width at high pressures, indicating it is not really an atomic-like state. Such a state is called a semi-core state. Note that the gap widens and changes from direct at Γ to indirect Γ to X with increasing pressure. From Ref. [4].

5 5 Figure 3. Band structures for CaO. (a) CaO B1 V = 26Å 3, 1 GPa. (b) CaO B1 V =22Å 3, 33 GPa. In CaO the bands are for Ca 3p, O2s, O2p, Ca3d. Note that the gap is already indirect in CaO. The gap narrows with pressure in CaO. From Ref. [4]. 10 E (ev) 0 Figure 4. Band structures for hcp iron at V =23.7Å 3 /primitive cell, -10 GPa. The Fermi level is shown with a horizontal dashed line. Only the 3d and 4s bands are shown.

6 6 Figure 5 shows the density of states for MgO compared with XPS spectroscopy (injecting x-ray photons and measuring the energies of electrons that are ejected) [5]. The close agreement between LDA band energies and experimental quasiparticle energies is evident. Matrix elements are not included in the density of states calculation, and the experiment is surface sensitive and instrumentally broadened, contributing to the differences in intensities shown. Figure 5. MgO density of states computed with the LDA compared with XPS measurements. From Ref. [5]. 3. Bonding in crystals Crystals must have a lower energy, or be bound, relative to separated atoms or molecules, or the atoms would just fly off into space. The binding forces that hold the atoms together are generally electrostatic in origin. Secondly, there must be some force that keeps the atoms from collapsing into each other. This force is largely due to the increased kinetic energy as atoms get closer, the Pauli exclusion principle that keeps electrons apart, the electrostatic repulsion between electrons, and ultimately as the atoms are brought closer and closer together, the electrostatic repulsion of the nuclei. Crystals can be characterized by the primary source of their binding energy.

7 In ionic crystals, the primary source of binding is from the electrostatic attraction among ions. Examples are NaCl (Na + Cl ) and MgO (Mg 2+ O 2 ). Covalent materials primarily bind through hybridization (sometimes called sharing ) of valence electrons, which is a way to lower the energy of the electrons, good examples being diamond and the semiconductor Si. The potential energy is lowered since the electrons can see the attractive positive potential of two nuclei or atom cores rather than just one. The electron distribution is modified, and charge builds up between the atoms. The lowering of the potential energy more than overcomes the increase in kinetic energy due to the increased electron density in the region between the atoms. In metals, the primary binding comes from embedding atom cores in a sea of itinerant electrons, for example in Na metal. Most materials have combinations of these interactions. Thus silicates are about half ionic and half covalent. Transition metals such as Fe are both metallic and covalent. Another source of bonding is from dispersion, or van der Waals forces, fluctuating dipoles on separated atoms or molecules. Although these forces are important at long range, where atoms do not overlap, these forces are quenched as the atoms or molecules overlap. So called van der Waals solids are probably not held together by dispersion forces, but rather by local many-body exchange and correlation interactions among electrons. Usually we are not asking whether a crystal will be bound at all, but rather questions like what crystal structure will be stable over another, and what are the physical properties, such as elasticity and equation of state, of a given mineral. Many of these properties, including all thermodynamic properties (at ordinary temperatures, i.e. most all geophysically relevant conditions) can be found by computing ground state properties alone, for example the energy as a function of moving an atom or straining the lattice. Excited state properties are more difficult to obtain accurately, and involve the energetics of exciting electrons out of their ground state configurations. Examples of such properties include optical spectra Band theory The eigenstates of a crystal can be characterized by a quantum number k (eq. 2). The energy bands are dispersive as functions of k, and the energy versus k is known as the band structure. Much can be learned from studying band structures and how they change as the crystal structure or chemistry changes. Much of solid state electronics is based on relative simple ideas based on band structures, occupancy of different bands with doping, and how bands line up across interfaces. Bands can also be studied experimentally using photoemission, for example the study of the relative energies of emitted electrons as functions of input photon energy and wavevector,

8 8 or from x-ray spectroscopy, or combinations of x-ray and electron spectroscopy. Crystals with band gaps between occupied and unoccupied states should be insulators, and those with partially filled bands should be metals. (If the gap is small or if the material is useful in electronic applications by chemical doping to inject electrons in the conduction bands, or holes in the valence bands, a crystal is called a semiconductor.) In a non-magnetic system, each band holds two electrons, and thus a crystal with an odd number of electrons in the unit cell should be a metal since it will have at least one partially filled band. This is not always true, as will be discussed below in the section on Mott insulators. A most important concept to understand band theory of crystals is the idea of a quasiparticle. The real energy states in a crystal are not single particle eigenstates at each value of k; rather there is an energy spectrum which has more or less strong peaks at the quasiparticle energies. If there are strong peaks in the spectrum, the energies coincide with objects that behave like independent particles that are dressed by all of the interactions with other real particles in the system. These objects are known as quasiparticles, and these are what are generally observed in an experiment. There is often some confusion by what one means by a band structure, especially when comparing experimental and theoretical results. Usually one calls the eigenvalue spectrum of whatever theory one is using the band structure. However, in Density Functional Theory (DFT, see below) the eigenvalue spectrum has no fundamental relationship to what one would observe in an experiment, or to true quasiparticle energies. Nevertheless, DFT band structures are often in good agreement with experiment (fig. 5) and are widely used to give insight into bonding in crystals. The experimental picture is also not straightforward, and band structures obtained using different methods may have different meanings. In angle resolved photoemission, for example, the observed band structure is the energy spectrum for removing electrons from the surface of the crystal, which is sometimes a complex phenomenon, and may be broadened or shifted from the intrinsic energy levels in the interior of the crystal. 5. Density Functional Theory By far the most computations for solids have been based on the density function theory of Hohenberg and Kohn [6] and Kohn and Sham [7]. The Hohenberg-Kohn theorem states that all ground state properties of a system can be obtained from the ground state charge density, and Kohn and Sham showed a practical way to do this, as described below. An alternative approach is based on Hartree-Fock theory [8]. Traditionally quantum chemistry has been based on Hartree-Fock as a first approximation, followed

9 by further approximations to include correlation effects. Hartree-Fock includes exchange exactly for a single Slater determinantal wave function, but neglects a large part of the energy which is called correlation from electron-electron interactions, culminating eventually in full configuration interaction (CI) in which the wave function is a sum over all possible determinants formed from the basis. The correlated methods are extremely computationally intensive, and cannot be applied completely to crystals, in the sense that one can never exhaust all of the possible correlated states since there are an infinite number of many-body states one can form even from a limited basis by coupling different k-vectors. Instead of relying on the traditional Hartree-Fock to CI path, many quantum chemists are also moving towards using density functional theory (DFT), which has been so successful for crystals, for molecules. 6. Total energy Following the Kohn-Sham (KS) procedure, the total energy is given by E = T 0 (ρ)+e n n + E e n + E h + E xc, (3) where T 0 is the kinetic energy of a non-interaction gas of Fermions of the same density ρ as the real system, E n n is the nuclear-nuclear repulsion, E e n electrostatic electron-nuclear interaction, E h is the Hartree energy, that is the electrostatic energy for the electron charge density, and E xc is the exchange-correlation energy. E xc is defined such that equation 3 is exact; thus it contains the correction from the non-interacting kinetic energy as well as the exchange and correlation interactions between the electrons. One solves the single-particle Schrödinger-like equation: ( h2 2m 2 + V KS )Ψ i = ε i Ψ i (4) where the Kohn-Sham potential V KS is the functional derivative of the total energy with respect to the density, V KS = δe(ρ) δρ, (5) and V xc = δe xc(ρ). (6) δρ Thechargedensityρ is given by: ρ = Ψ i Ψ i f(ɛ i E F ). (7) 9

10 10 where f is 1 for ɛ i E F < 0 and zero otherwise (or the Fermi function at finite T). This set of equations is solved self-consistently (i.e. iteratively). One starts with an initial guess for ρ, computes V KS, solves equation 4 for Ψ i, computes a new ρ from equation 7, and iterates until the input and output densities ρ agree. Equation 4 is solved by using a basis φ for Ψ i = j c ij φ ij, and the secular equation H ij Ψ j (k) = ε i (k) O ij Ψ j, (8) j j must be diagonalized, where the Hamiltonian matrix H is given by the overlap matrix O is given by and the Hamiltonian operator is H ij (k) = Ψ i (k) H Ψ j (k), (9) O ij (k) = Ψ i (k) Ψ j (k), (10) H (r) = 2 + V n + V h (n (r)) + V xc (n (r)), (11) where the first term is the kinetic energy operator, the second is the nuclear potential, the third is the classical electrostatic Hartree potential, and the last term is the quantum mechanical exchange correlation potential which accounts for the many-body electron-electron interactions. The total energy is given by E tot (n) =E bs E h + E xc + E Ewald, (12) where the band structure energy is given by E bs = i ɛ i f(ɛ i E F ). (13) The term E h corrects for double counting of the Hartree energy in E bs, E xc = dr [ε xc V xc ] ρ (r) is the difference in exchange-correlation energy functional and the potential, and E Ewald is the nuclear-nuclear (or core-core) electrostatic energy. If the functional E xc (ρ) were known, the Kohn-Sham procedure would be exact, and the computed energy and density would be exact. As discussed below, even in the exact theory, the KS eigenvalues of equations 4 or 8 are not the actual quasiparticle energies, and the band structure constructed from these eigenvalues is not the band structure that would be measured experimentally. Empirically, it is often found that that the KS band structure is a good approximation to the real band structure.

11 The exact E xc (ρ) (orv xc (ρ)) is not known, although excellent approximations have been developed that work well for most materials. In the Local Density Approximation (LDA), V xc (ρ) is the exchange-correlation potential for the homogeneous electron gas of density ρ(r), for each point r). This assumption is similar to the idea of local equilibrium in metamorphic petrology. It works surprisingly well, and has been the backbone of computation solid state theory for the past thirty years. Several different parameterizations of V xc (ρ) for the homogeneous electron gas exist, with most being very similar, and based on quantum Monte Carlo simulations for the electron gas. The LDA systematically produces volumes that are too small compared with experiment, and thus at zero pressure it tends to give elastic constants such as the bulk modulus accordingly too high. The Wigner approximation to the LDA exchange-correlation potential gives volumes that are slightly too high, and is significantly different from other LDA s. It was derived in a different way from other LDA s, as an interpolation formula between known high density and low density behavior. The Generalized Gradient Approximation (GGA) includes gradients of the charge density from the linear response of the electron gas [9, 10, 11]. Although the exact density functional is unknown, many sum rules and constraints on the exact functional are known. The GGA was developed such that sum rules that are obeyed in the LDA continue to be obeyed (thus the term generalized ); a simple addition of gradient terms in a Taylor expansion to V xc actually results in a worse approximation than the LDA, due to the loss of these sum rules. The GGA generally results in better agreement with experiment than the LDA, though in many cases it overcorrects, and in some cases the volume is larger than experiment by the same amount that LDA underestimates the volume. There are a number of cases, such as the stability of quartz versus stishovite, or magnetic bcc iron versus non-magnetic hcp, where the GGA gives accurate results and LDA fails. One way of understanding the exchange-correlation potential is through the exchange-correlation hole. The exchange correlation hole is the region carved out around an electron due to the Pauli principle and electrostatics, and integrates to one-electron. Most work assumes that the exchangecorrelation hole is local (i.e. short-ranged), but it can be demonstrated to have long-range behavior that is not properly included in LDA or GGA. One general failure of the LDA and GGA is their behavior when an electron is removed from an atom or a crystal surface. The exchange-correlation hole should stay behind where the rest of the electrons are, but in the LDA and GGA the xc-hole follows the electron that is being removed. This is also related to self-interaction errors. An electron cannot interact with itself, yet in self-consistent field methods, a Hartree interaction is computed for 11

12 12 the change density, so that a spurious interaction is included between an electron and itself. If the exchange were computed exactly, the spurious self-interaction would cancel out. This is very clear in a one-electron system. There should be no Hartree-potential in a one-electron system, or it should be completed canceled in the exchange potential. This cancellation is not complete in the LDA and GGA. There should also be no correlation energy in a one-electron system, but the LDA and GGA include a false self-correlation. Self-interaction corrections (SIC) attempt to correct for this, and though they work very well in atoms and show promise for crystals [12, 13, 14], they are not generally used due to some arbitrariness in how they are applied. There are some other improved methods which show great promise. The Weighted Density Approximation (WDA) evaluates the exchange-correlation energy for a local system with density of that in the local exchange-correlation hole[15, 16]. The WDA has shown great promise with very accurate predictions of the volume and other cohesive properties. Unfortunately, a spin-polarized WDA has not yet been developed so magnetic materials cannot be treated. Other improved methods involve using exact (i.e. Hartree-Fock) exchange with density functionals for the correlation energy [11]. There are not many applications of WDA or exact exchange plus correlation corrections methods to minerals so they will not be considered further here, but the reader should expect to see such work in the future. A hybrid method that is discussed below is the LDA+U method, where LDA (or GGA) is supplemented with a local orbital dependent potential to correct problems in transition metal materials. Different types of bonding in crystals are now discussed: van der Waals, ionic, covalent, covalent/ionic, and metallic. Finally Mott insulators will be discussed, which are metallic according to band theory, but are actually ionic insulators. 7. Rare gas and van der Waals solids Crystals made of rare gas atoms and/or weakly bound molecules are often called van der Waals solids, and are considered as if the atoms or molecules were bound by dispersion forces that arise from fluctuations in the charge densities (often called fluctuating dipoles) of the atoms or molecules. Separated, non-overlapping charge densities do indeed have such an attractive force between them, and it varies as C 6 /r 6 at large distances. The constant C 6 is known as a van der Waals coefficient, and it can be obtained from studying the scattering of atoms or molecules in the gas phase. The LDA or GGA do not give 1/r 6 behavior at long distances, but rather an exponential decay. Exact DFT, however, should give proper van der Waals behavior since the energy is a ground state property, and recently Kohn et

13 al. [17] presented a formalism, and obtained very accurate results for simple pairs of atoms. Some potential models include 1/r 6 terms, assuming that van der Waals continues to be important in crystals. Such forces have also been called on to explain the binding of neutral planar units in some minerals, such as graphite. However, the importance of van der Waals forces in solids is unclear, especially at elevated pressures, and in general these forces are probably quenched. Unlike the gas phase, there is appreciable overlap of charge densities in a crystal, especially at high pressures, but this is also true at low pressure. This is reflected in the band structures, which do show some dispersion even at low pressures (fig. 6a). Thus the effects attributed to van der Waals, such as the binding energy of rare gas crystals, is more properly ascribed to correlation forces, such as those present in LDA or GGA. In fact, such computations do quite well for the crystalline properties of rare gas solids [18] and molecules. For the Ne 2 dimer, for example, the PBE GGA [11] gives a bond length of 5.83 bohr and anharmonic frequency of 11 cm 1 compared with 5.84 and 14 from experiment [19]. If one were to add unquenched, long-range 1/r 6 terms in the interactions in crystals, agreement could only degrade. On the other hand, it is probably true that LDA and GGA leave out some of the long-range correlation, but generally speaking these effects must be small or not very dependent on atomic configurations, because all insulating materials could have such interactions in principle. It must be that the charge density fluctuations are strongly screened, even in insulators, and thus die off rapidly than 1/r 6 in crystals, leaving the more local correlation effects dominant [20]. One approach has been to take accurate pair potentials from atomic scattering experiments, and apply these potentials to the crystalline phase [21]. The resulting crystalline properties and structures are not accurate unless one includes many-body forces [22]. At high pressures, the attractive terms in molecular solids become less and less important, and one could rather think of the atoms as soft spheres. Much of the chemistry and phase diagrams of such materials can be understood as the packing of spheres of various sizes. However, as pressure is increased, the electronic structure also changes. This happens first in the crystals of heavier atoms, such as Xe. Figure 6 shows the computed band structure of Xe at three pressures, where it is an atomic-like van der Waals solid, a semiconductor and a metal. The metallization of Xe has been studied experimentally [23, 24] and using various electronic structure methods, including the GW approach which is an approximation to compute quasiparticle energies or excitation spectra, and thus gives a more accurate gap than the KS eigenvalues [25]. A complex stacking phase transition appears in Xe in the range of the metallic transition [26]. It is typical to find changes in the crystal structure when the bonding type changes, such 13

14 14 (a) (b) (c) Figure 6. Band structures for Xe over five-fold compression. (a) 0 GPa, V=500 bohr 3 /atom (b) 20 GPa, V=200 bohr 3 /atom (c) 250 GPa, V=100 bohr 3 /atom. At low pressures, Xe has almost atomic-like valence states, with very narrow band widths. Note that the conduction bands have significant width even at low pressures. At intermediate compression, Xe is a semiconductor, and at high pressures a metal. The ordinate is energy with each tick being 1 ev. From Ref. [18].

15 as at a metal-insulator transition, because it is the nature of the bonding that gives rise to the stability of one crystal structure over another. 15 NeAr Kr Xe hcp GK GK GK LDA GK LDA GGA fcc Figure 7. Equations of state of rare gas solids. Gordon-Kim (GK) results use quasiharmonic lattice dynamics at 300 K. The LAPW LDA and GGA-PBE results are for static lattice. Points are from experiment. Both fcc and hcp are shown for Xe, and the other results are for fcc. Agreement is quite good with the self-consistent results. Note that there is a small but significant error in the Gordon-Kim results, due to the pair approximation, the Thomas-Fermi overlap kinetic energy, and, with increasing pressure, covalency and metallization. Van der Waals interactions are not included in any of the computations, and are apparently small effects for the crystal. However, Ne in GGA-PBE appears to be slightly unbound without van der Waals. Figure 7 shows the equations of state of Ne, Ar, Kr and Xe computed using the LAPW method and the LDA (for Xenon) and PBE-GGA (for Neon), compared with experiments. Agreement is good, and there is no evidence that some large piece of the interactions is missing. On the other hand, it is difficult to reach definitive conclusions around zero pressure, since the crystals are so soft, zero point and thermal vibrational corrections are very important, and the binding energies are very small. As mentioned above, indications for molecules are that the GGA is quite accurate [19]. It remains to be seen if this is true for crystals. Preliminary computations using GGA for Ne show anomalous behavior at zero pressure, with a net repulsive potential between the atoms at large volume leading to expansion of the crystal without bound, but it remains to be seen if this is a numerical problem due to the tiny energy differences in the low pressure regime, or a real failure of the PBE-GGA. LDA does not show this, but rather shows a shallow minimum in energy.

16 16 One theme in research on rare gas solids has been an interest in the importance of many-body forces between three or more atoms [29]. These forces arise in three main ways. Firstly, when a crystal is compressed, regions of density where three atoms overlap significantly introduce threebody terms, since the exchange-correlation functional is not linear in density. These contributions appear to be small, on the order of a few percent or less of the energy [30], at least at pressures that are not extreme. A second contribution comes from multiple atom dispersion forces, the lowest order being the attractive Axelrod-Teller forces which are from fluctuating dipoles on three atom neighbors [31]. The expansion in terms of higher order multipoles and dispersion interactions is only slowly convergent. It was thought for some time that many-body forces would stabilize the fcc phase over the hcp structure. Pair potentials that accurately describe molecular beam scattering fail to give the fcc phase as the ground state structure, but Ne, Ar, and Kr are fcc at zero pressure, and transform to hcp with increasing pressure. However, many-body interactions do not fix this apparent problem. Rather it is zero point and thermal energy that stabilizes the fcc phase. Quantum anharmonicity has been another area of much interest in rare gas solids, and the most extreme cases are 3 He and 4 He, the former a Fermion system, and the latter a Bose system. Quantum Monte Carlo simulations have been used to study the solid and liquid phase diagram, and the properties of the solid and the liquid [32]. The liquid state exhibits superfluidity, which is a quantum state in which the viscosity vanishes. 4 He is also unusual in that it remains liquid down to 0 K. Only pressure stabilizes the crystalline phase, zero point motions being so large. The bcc structure of the low pressure crystalline phase in 4 He is also stabilized by zero point energy. This is not a review of methods, but if it were, quantum Monte Carlo would have to be described in some detail, since in the future such computations are expected to be much more common. Gordon and Kim [27] developed a model for rare-gas solids, that turned out to be more generally useful and has lead to significant advances in our understanding of thermal properties of solids. They developed a model that is really the prototypical example of LDA, and shows the power of the LDA even when not used self-consistently. In Gordon-Kim models, the total charge density is modeled by overlapping atomic or ionic charge densities, and then the total energy is computed for that charge density using the LDA. The method is less accurate than the self-consistent KS method, even if the model density is good, because the local density form for the kinetic energy, T 0 = ρ 5/3 is not accurate enough in many cases (for example it does not give the proper shell structure for atoms). The KS approach does not make this approximation for the kinetic energy even in the LDA and the ki-

17 netic energy derives from the occupied orbitals. The electrostatic energies are computed exactly, and the LDA is used for the exchange-correlation energy. Gordon and Kim also modified the kinetic and correlation interactions (leading to the term modified electron gas, or MEG) to give better results for atoms. Gordon-Kim models will be discussed more generally in the next section on ionic models, but the prototypical application was rare gas solids. The MEG model generally gave equations of state that were too stiff for the rare gas solids, but was quite successful considering the simplicity of the model. The main problem with the simple Gordon-Kim model is resolved by modifying the rare gas density in response to the embedding crystal potential, so that the rare gas atom is compressed with increasing pressure [28]. These studies have also shown that the dispersion forces are damped or screened as the charge densities overlap [20]. The Gordon-Kim model is the prototypical ab initio model. An ab initio model is a way to compute ground state properties without performing full self-consistent computations, but without fitting experimental data. Ab initio models are generally based on the DFT. The total energy is a functional of the charge density, so given the crystal charge density the total energy is in principle constrained, and the total energy is variational with respect to the changes in the density (i.e. the total energy is a minimum with respect to variations in the density). In Gordon-Kim models, the charge density is approximated as overlapping ions or atoms. The overlapping ion model appears to be an excellent model for the charge density compared with self-consistent computations for system of closed shell ions Ionic solids Attractive forces in ionic solids are dominated by the electrostatic, or Madelung forces between charged ions. The most easily understood ionic crystals are those made of closed-shell ions, such as Mg 2+,O 2,Na +,Cl etc. Thus the alkali halides and alkaline earth oxides are prototypical ionic solids. One can study these systems using the full machinery of DFT, and results have been obtained for equations of state [33, 34, 35, 36], elasticity [37], and a range of defect [38, 39, 40], surface [41, 42], electronic [43, 44], and optical [45] properties of these materials using self-consistent methods. Since the physics of closed-shell systems is straightforward, more efficient models have been developed, based on first-principles, which allow the computation of time-dependent properties such as transport properties and the performance of long molecular dynamics simulations not possible using self-consistent methods. Most of these methods are based on the Gordon- Kim model discussed above. In ionic crystals, one must add the Madelung energy to the overlap energies. A further important complication arises in

18 18 oxides, since the O 2 ion is unstable in the free state. Rather it is stabilized by the crystal field in an oxide. An O 2 charge density can be obtained by including a stabilizing potential in the atomic computation, and most studies have used a Watson sphere [46], which is a charged sphere, usually of opposite charge of the ion. Thus for O 2 a sphere of charge 2 is included as well as the nuclear charge Z = 8 [47]. Then when an electron moves far from the atom it sees an object of positive charge +1 behind it, the electron is bound and the configuration remains stable. The remaining question is how to choose the radius of the sphere. Muhlhausen and Gordon [47] stabilized the ion with a sphere whose radius was chosen so that the electrostatic potential in the sphere equals the Madelung potential at the site at a given volume, a potential appropriate to that charge density was found, and then rigid-ion calculations were performed with that potential. Figure 8. Difference in charge density of MgO generated with overlapping ions using the PIB model and computed self-consistently using the LAPW method. The contour interval is e /bohr 3.FromRef.[4]. In the Potential Induced Breathing (PIB) model, the Watson-sphere radius is given by the Madelung potential, V mad, as atoms are displaced or the lattice strained, R wat = Z wat /V mad, giving a non-rigid ion, many-body, potential, where Z wat is the charge on the sphere (2+ for O 2 ). Figure 8

19 shows the difference in charge density of MgO computed with overlapping PIB ions and computed self-consistently using the LAPW method. The agreement is excellent. Furthermore, the bands computed from the crystal potential generated from the PIB charge density is in excellent agreement with the self-consistent band structure (figs. 9, 10, and 11). Thus the PIB charge density is a good approximation to the self-consistent charge density for ionic materials such as MgO. Presumably similar agreement would be found with Gordon-Kim rare gas solids, though such a comparison has not been made. In the PIB model, the total energy consists of the Madelung energy, the overlap energy (which contains the Thomas-Fermi kinetic energy approximation, the LDA exchange and correlation energies, and the local electrostatic energies among electrons and between electrons and nuclei), and the self-energy. The self-energy is the energy of each ion density as generated with the Watson sphere, with the Watson-sphere interaction energy removed. The self-energy for each ion thus depends on the Watson sphere radius. The dependence of the energy on the Watson sphere radii gives rise to effective many-body non-pair-wise interactions between the atoms that give improved elastic properties [48]. In the rocksalt structure (or any structure with inversion symmetry at each site), if all atoms interact through central pairwise forces, the elastic constant C 44 = C 12 at zero pressure (the Cauchy relation) [49], and yet the experimental elastic constants deviate from this relation in the alkaline earth oxides (the Cauchy violation). Mehl et al. [48] found reasonable values of the Cauchy violation for the alkaline earth oxides, and the correct trend from MgO to BaO. This promising result lead to the development of the lattice dynamics of the PIB model [50] and reasonable dispersion curves were obtained for the alkaline earth oxides using the PIB model. All was not well, though. In the PIB model, the Watson sphere radii are given by the Madelung potential, but the Madelung potential is not well behaved in the long-wave limit. A longitudinal wave exp[ q r] gives rise to a potential wave, and as q 0 the Madelung potentials on O sites approach a linear slope from + to across the infinite crystal. Since the self-energy depends on the value of the Madelung potential, it diverges as q 0, and the longitudinal optic (LO) mode diverges [51]. It is possible to remove this divergent part and recover reasonable results, but the resulting model is still not completely satisfactory since it predicts spherical breathing of atoms in a linear field. Nevertheless, many useful results were obtained with the PIB model. It is fast, and often nearly as accurate as self-consistent calculations for ionic materials. The key effect of PIB is that the anion changes size with crystal environment, shrinking with increasing pressure or local compression. A better procedure, though several times slower, is to optimize the total 19

20 20 Figure 9. MgO band structure at V=18.1 Å(zero pressure) computed using the LAPW method (lines) compared with that from the potential generated by the overlapping ion PIB charge density. From Ref. [4]. energy with respect to Watson sphere radii rather than to chose the radius using the Madelung potential [52]. This gives a Watson sphere radius close to that of PIB at zero pressure. However, it changes more rapidly with compression than PIB due to the compression of the atom by short-range forces, in addition to the electrostatic crystal field (fig. 11). This model is known as the VIB, or variationally induced breathing, model. The anomalous behavior shown by the PIB model is absent in the VIB model. In the VIB model, the LO-TO splitting is the same as given by a rigid ion model, since all atomic deformations are spherical. There is no dipolar charge relaxation. In spite of the absence of atomic polarizability, the VIB model is very accurate and gives results that compare quite well with self-consistent results and experiment for non-polar materials and for properties for which dielectric behavior is not crucial. It is clear that the spherical breathing included in PIB and VIB is more important than dipolar polarizability, and one should not neglect spherical breathing in models for oxides and minerals containing other unstable ions such as S 2. Even in alkali halides, the spherical breathing effect is important [53, 54]. The advantage of fast methods such as PIB or VIB is that one can

21 21 Figure 10. Differences in eigenvalues from the LAPW band structure and those derived from PIB as functions of the Watson sphere potential (V wat =Z wat/r wat) (a) at zero pressure (18.1 Å 3 ) and at (b) 290 GPa (10 Å 3 ). The Madelung potential is shown by an arrow. At low pressures PIB gives an excellent approximation for the band structure (almost as good as self-consistent). At high pressures, the best fit Watson sphere potential would be larger (i.e. a smaller Watson sphere radius, and a smaller O 2 ion) due to short-range forces that compress the oxygen ion. do lattice dynamics and long molecular dynamics simulations on reasonable sized systems, and study thermodynamic properties, phase transitions, transport properties, etc. For example, Isaak et al. performed lattice dynamics on MgO as a function of lattice strain, going beyond the normal quasiharmonic approximation, and studied the high effects of temperature and pressure on elasticity and the equation of state [55]. In that study, the dynamical matrix was found throughout the Brillouin zone in order to obtain the free energy, and this was repeated for different lattice strains. In spite of increased computational power in the last eight years, no such study has yet been done self-consistently. The Isaak et al. study gave what are still one of the few estimates of cross derivatives of pressure and temperature on elasticity that are available. The results were then used to help understand the increase in seismic parameter d ln V s /d lnv p with depth in the Earth [56], a quantity that has been difficult to constrain experimentally. Going beyond lattice dynamics, Inbar and Cohen [57] determined the thermal equation of state of MgO using molecular dynamics and the PIB model. Such studies are just becoming possible using self-consistent methods, and still have not been performed. Using molecular dynamics and the

22 22 Figure 11. Behavior of best fit, VIB, and PIB Watson sphere potentials as functions of compression and composition in alkaline earth oxides. (a) The Watson sphere potentials V wat that give band structures that best fit self-consistent LAPW band structures versus Madelung potential for different alkaline earth oxides versus pressure. Pressure increases to the right for each curve. PIB (represented by the 1:1 dashed line) works well at low pressures, but less well with increasing pressure. (b) Watson sphere potential obtained using VIB (minimizing the total energy with respect to the Watson sphere radii) versus Madelung potential. VIB gives a better approximation to the slope seen in (a). (c) VIB potential versus best-fit potential. PIB is represented again as the 1:1 heavy dashed line. The VIB results parallel the best fit results, giving the correct compressive behavior. There is a small offset to lower potentials for VIB, which may compensate for the kinetic energy approximation in VIB. VIB model, it is also possible to study complex phenomena, such as thermal conductivity [58] and diffusion [59, 60]. The diffusivity of O in MgO, for example, is obtained in agreement with measurements within experimental error. Ab initio models can also be used to study the stability of a variety of structures to search for possible phase transitions. This was done successfully for Al 2 O 3, where the PIB model showed a high pressure elastic instability [61], detailed PIB computations for different structures showed a phase transition at high pressures to the Rh 2 O 3 II structure [62], and the transition was confirmed and pressure computed accurately using the

23 LAPW method with the PIB structural parameters [63]. These computations predicted a phase transition at 90 GPa, in excellent agreement with later experiments [64]. Ab initio models have undergone further development by including the crystal potential in the atomic calculation, and performing a self-consistent cycle between the atomic densities and the crystal potential [28, 65]. In the Self-Consistent Charge Deformation model (SCAD) [66, 67, 68] atomic densities are computed in the crystal potential, and states are occupied in order of energy, allowing charge flow between the atoms. The inclusion of non-spherical charge deformations has increased the accuracy of the models, but at the cost of much increased complexity. Perhaps fast implementations of these new self-consistent model methods will be developed that will allow molecular dynamics and other types of simulations for crystals where non-spherical charge distortions are important. Whether covalency can be properly modeled by these methods remains to be seen. Ab initio models also give insights into bonding and electronic structure that are not obvious from self-consistent computations alone. One example is the relationship among and meaning of ionicity, covalency, and band width. One might think for example that a purely ionic model would have atomic-like energy levels, and that band width arises from hybridization or covalency. However, band structures computed from the potential generated from overlapping ionic charge densities (fig. 9) not only have width, but are in excellent agreement with self-consistent computations for ionic crystals such as MgO. On the other hand, if one were to ask the origin of the band width in a tight-binding representation, one would find that the O 2p band width in MgO, for example, comes primarily from O-O ppσ interactions [69]. Thus we see that even a purely ionic charge density, generated by overlapping spherical ions, has a charge density that generates a potential, that when used in the KS equations implies a band width consistent with hybrid electronic states. Thus there is a sort of duality in the description of ionic materials, and they can be described from a charge density or tightbinding (or LCAO) perspective. In either case, there must be long-range Madelung terms in the total energy, that gives rise to LO-TO splitting in the lattice dynamics. Therefore, it would not be correct to say that MgO could be treated as an ionic or as a covalent crystal. It is an ideal ionic crystal (as indicated by the LO-TO splitting with effective charges near the nominal values [68, 70]), but this ionic charge density implies a band width in the KS eigenvalues from its potential. Note that the SCAD model does indeed have atomic like eigenvalues in its spectrum, but presumably if one generated the band structure from the SCAD charge density one would find reasonable agreement with the self-consistent band structure. This would be a useful test for SCAD and other improved ab initio models. 23

24 24 Not all ionic solids are formed from closed shell ions. For example, FeO (wüstite) and solid solutions between MgO and FeO (magnesiowüstite) behave like ionic solids, yet Fe 2+ is a d 6 ion and is not closed shell. FeO is also a Mott insulator, and is discussed below. Such materials are very difficult to treat, and in spite of the importance of Fe in minerals, there is not yet a good method for obtaining first-principles results that are completely correct. Ab initio models such as PIB fail to give accurate predictions for the equation of state and other properties for these materials as well, even if one sphericalizes the Fe, and treats it as ionic. Nevertheless, much can be learned about these materials from self-consistent computations as described below. Perhaps simple and accurate ab initio models can be developed for these materials, but this has not yet been done. The hydrogen bond is really another type of ionic bond, but usually classified separately. A proton has of course a 1+ charge, and in an oxide or silicate, hydrogen almost always occurs as the (OH) 1 ion or bound as H 2 O. In either case there is a residual positive charge around the proton, even though it is enveloped by the polarized oxygen charge density, and this residual charge can bind with an excess negative charge on another atom or molecular unit. This binding strongly influences the H vibrational frequencies and may also influence the crystalline configuration if there are floppy structural units attached to either the positive or negative side of the bond. Whether hydrogen bonds are favorable at a site or not may also strongly influence the position of H in the crystal. 9. Covalent solids In covalent solids the dominant bonding interaction is caused by hybridization among the states on different atoms. Such interactions can be very strong, and are responsible for the strong bonds between C s in diamond and less so in Si. In a covalent bond, charge concentrates in the bonding region, increasing the potential and kinetic energy of interaction between electrons, but reducing the energy through the electron-nuclear interactions (since each electron is now on average close to both nuclei or atomic cores, which have a net positive charge to the electrons). Atoms that form strong covalent bonds have decreased repulsion relative to atoms that do not. The Pauli exclusion principle says that two electrons cannot be in the same state, so as electron density increases, electrons must be in higher energy states. In strong covalent solids, orbitals are not fully occupied on the constituent atoms. Hybrid states can form without occupying higher energy levels, and thus the total energy decreases due to the favorable electron-core interactions. For example, in H 2, each H atom separately has one 1s electron, and when brought close, there is no problem forming one molecular bonding

25 orbital out of the two 1s atomic orbitals, which is then doubly occupied. In contrast, in He, the 1s states are filled, and as two He atoms approach, electrons must be boosted into higher energy states. Thus H forms strong covalent bonds, and He does not. Covalent bonds tend to be very directional since they are formed from linear combinations of directional orbitals on the two atoms. Self-consistent computations can of course be done for these materials, and many such studies of all kinds of properties of electronic and optical semiconductors have been performed. It is also straightforward to develop tight-binding models for such covalent materials, and such models have ranged from models with empirical parameters to those with parameters obtained from first-principles. The advantage of tight-binding models is that they are very fast compared with self-consistent methods, and one can perform molecular dynamics and other simulations on large systems for relatively long times. Tight-binding models also give insight into the bonding interactions, especially in covalent solids. Tight-binding models give a complementary picture to ab initio models, the former being based on matrix elements and band structures, and the latter focused entirely on the charge density. Since the formalism is instructive, and may give some further understanding of bonding and energetics of crystals it will be described in detail. In tight-binding total energy models (TBTE) the total energy E as a function of the atomic or nuclear positions r is represented as a sum of a band structure term, E bs, 25 E bs = i ε i f (ε i (k) E F,T), (14) where ε i are the band eigenvalues and f is the Fermi function, with E F chosen to give the correct number of electrons, and additional structure dependent term F, which is often represented as a pair potential, so that: E = E bs + F. (15) In tight-binding methods [71], the radial parts of the integrals (eqs. 9-10) are parameterized, and there is no self-consistency loop. In the twocenter approximation, only integrals between pairs of atoms are included; integrals that involve a potential on one atom and orbitals on two others are neglected. Thus the Hamiltonian and overlap matrices can be written as: H ijαβ = exp [ik (R jl R il )] h ijαβ dωφ iα φ jb (16) l O ijαβ = l exp [ik (R jl R il )] o ijαβ dωφ iα φ jb (17)

26 26 where the integrals are the angular integration for orbital α on atom i with orbital β on atom j, and the sums are over lattice vectors l. Angular momentum is quantized along z, the vector from atom jl to atom i, sothat the indices αβ are contracted to a total of ten bonding states up to d-states, ssσ, spσ, ppσ, ppπ, sdσ, pdσ, pdπ, ddσ, ddπ,andddδ. The parameters h and o are fit to results of first-principles band structures or are fit empirically for individual interactions in a given crystal structure. The on-site parameters h iiαα are fit or set to atomic energies, and a set of parameters h ssσ,etc.are obtained, either by coordination shell (i.e. first-neighbor, second-neighbor, etc.) or as functions of distance. If O is assumed to be the identity matrix the method is orthogonal tight-binding, otherwise it is non-orthogonal. A non-orthogonal fit gives a more accurate band structure at the expense of large correlations among parameters, but generally a non-orthogonal model is required to fit a large range of structures and/or volumes. Non-orthogonal models sometimes are problematic when used outside the range of parametrization, since the overlap matrix may become non-positive definite, which makes solution of the secular equations impossible (or unphysical). In standard tight-binding models there is a large source of ambiguity that arises from separating and fitting separately E bs from the function F (eq. 15). The problem is that for a periodic crystal the zero of energy, V 0, for a band structure is arbitrary, so that the band structure energy (eq. 13) contains an arbitrary and unknown structure dependent term. This term arises from the G=0 term in Fourier expansion of the Coulomb potential, and is canceled by an identical contribution in F in self-consistent calculations. This cancellation is not guaranteed in tight-binding methods where the band structures and F are fit separately. Figure 12 illustrates the severity of the problem. This problem is solved by shifting the eigenvalues so that the band structure energy is the total energy, and F = 0 [18, 72]. An alternative approach is to fit eigenvalue differences, rather than absolute eigenvalue energies, let the zero of energy float, and then simultaneously fit total energy differences [73]. The problem is then reduced to finding a parametrization that accurately reproduces not only the band structure, represented by the shifted eigenvalues, but also the total energies. Such a parametrization was found and tested for several transition and noble metals [72]. Cohen et al. [18, 72] parameterized the Hamiltonian h and overlap o parameters as a functions of distance: P i =(a i + b i r)exp[ c 2 i r]f(r), (18) where f(r) =(exp[(r r 0 )/l]+1) 1 (19)

27 27 V = V 0 mt E total V 0 = E F Figure 12. Band-structure energy obtained for hcp Fe using LAPW for three different choices of V 0,theFermilevel,E F, the LAPW muffin-tin zero (such that the average potential in the interstitial is chosen as the energy zero), or the value such that E bs = E total+c, where C is a constant chosen such that E(P = 0) = 0. From Ref. [18]. is a universal cutoff function to guarantee that interactions vanish smoothly with distance. Cohen et al. took r 0 =16.5bohr and l =0.5bohr. The treatment of the on-site terms is crucial to obtaining good fits to the shifted eigenvalues and total energies. Two different methods were used in [18] and [72]. In the first parameterization, the on-site terms vary as a function of a local atom density around each atom, given by ρ k = j exp[ d r]f(r), (20) 2 j k where d j k depends on atom types k and j. ( j symbolizes the type of atom j). The on-site terms D l k for l = s, p, andd for each atom k are then fit to a finite strain polynomial D lk = e l k + g l k ρ 2/3 k + h l k ρ 4/3 k + i l kρ 2 k. (21) This parameterization worked very well for the transition and noble metals, Xe and Fe, but it did not work well for Si. A formulation similar to that of [74] and [75] works well for Si, as well as for other materials. In this parameterization, the on-site terms in the Hamiltonian are given by

28 28 D lml m k = q ll k + j ) (s lm k + t lm kr jk exp [u 2lm kr ] jk Y lm ( r jk ) Y l m ( r jk) (22) This form arises from the onsite integrals Ψ ilm V j Ψ il m for interactions of an orbital on atom i with the same or another orbital on i with a spherical potential on atom j. This can be reduced to similar angular representations as the hopping and overlap terms to ssσ, spσ, ppσ, ppπ, sdσ, pdσ, pdπ, ddσ, ddπ, andddδ if off-diagonal m m are excluded. For pp onsite terms, equation 22 reduces to [74, 75] D pzk = q pp + j l 2 jk ( r jk ) I ppσ (r jk )+(1 l 2 jk ( r jk ))I ppπ (r jk ) (23) where ljk 2 ( r jk) =z 2 /r 2, for example, and I ppσ (r jk ) are the distance dependent terms, represented in our model by ) (s σ k + t σ kr jk exp [ u 2σ kr ] jk. Excellent fits were found for Si using this formalism, for which we included s, p,and d interactions using only the diagonal terms corresponding to ssσ, ppσ, andddσ, giving twelve onsite parameters: D s = q s + I ssσ (24) D px = q p + x2 r 2 I ppσ D py = q p + y2 r 2 I ppσ D pz = q p + z2 r 2 I ppσ D dxz = q d + x2 z 2 r 4 D dyz = q d + y2 z 2 r 4 D dxy = q d + x2 y 2 D d(x 2 y 2 ) = q d + I ddσ I ddσ r 4 I ddσ ( x 2 y 2) r 4 I ddσ D d(3z 2 r 2 ) = q d + 3z2 r 2 r 4 I ddσ Figure 13 shows the fitted parameters versus distance. This gives a graphic depiction of how the bonding interactions vary in Si with distance.

29 sss sps pps ppp 0.2 sss sps pps ppp E (Ryd) 0 0 E (Ryd) r (bohr) r (bohr) Figure 13. Non-orthogonal tight-binding parameters obtained for Si versus distance. (a) Hamiltonian (b) overlap. Only the s and p parameters are shown. Si is a complex covalent material, with a complex phase diagram. The tight-binding model does a good job predicting the energetics of both the close-packed and open structures (fig. 14). At zero pressure Si forms in the open diamond structure, which is stabilized entirely because of strong directional sp hybrid bonds. However, when Si melts, it forms a dense closepacked metallic melt with an average of 12 neighbors. The strong covalent network cannot be maintained in the liquid state. At high pressures, crystalline silicon also becomes unstable with respect to close-packed metallic structures, with several intermediate stages in-between. It is interesting to compare the behavior of Si with that of SiO 2. Bonds in SiO 2 and other silicates are not as covalent as Si (see next section), and a close-packed metallic state is not stable, even at extreme pressures. Thus silica and silicates form polymeric melts at low pressures, and with increasing pressure structures become more ionic rather than metallic. This difference in behavior occurs because Si is open-shelled in Si, and in silica and silicates, Si and O are closer to the closed-shell ionic configurations Si 4+ and O 2. The energy it would take to transfer the electrons from O to Si is too large to occur under geophysical conditions.

30 Energy (Ry) bcc fcc hcp sc sh β-sn Diamond Volume (Bohr 3 ) Figure 14. Energy versus volume for Si. The lines were from the tight-binding model, and the points are from LAPW computations. The model does a good job with both the open diamond structure, the close-packed phases, and the intermediate distorted β-tin phase. From Ref. [18]. Covalency is very important in many classes of minerals, including sulfides, phosphates, carbonates, etc. In many cases, it is a molecular group that contains strong covalent bonds, but which is ionically bound to other ions in the crystal. An example is CaCO 3, where the C-O bonds are strong and directional, forming a planar triangle in the CO 2 3 carbonate group, which is bound ionically to Ca 2+ ions. 10. Ionic/covalent solids-silicates Most of the common rock-forming minerals are silicates, and silicates seem to fall about half way between the ionic and covalent endmembers, according to Pauling [76]. This is no problem for self-consistent methods they can treat materials of any bonding type. It is, however, difficult to design fast models that are accurate. Both approaches will be discussed SILICA The chemically simplest silicate is silica, not surprisingly, and much work has been done on quartz and stishovite and some other SiO 2 phases [77]. Figure 15 shows the computed valence charge density for the upper valence band ( O 2p bands) for stishovite. The charge around the oxygen is not

31 spherical, but rather is polarized towards the Si atoms. Since there are three planar Si s around each oxygen, the O s have a triangular-prismatic shape. They are polarized strongly towards the oxygens. Most of the charge is around the oxygen. All of it would be around the oxygen if it were O 2 and the Si were Si 4+. There is no unique way to extract a static ionicity from the charge density (though the dynamic effective are well defined, as described below for CaSiO 3.) One way to extract an effective ionicity is to compare the selfconsistent charge density with that of a model system, such as overlapping ions. Model charge densities made of ions with different valence can be compared with the self-consistent density (for example by comparing the amount of charge in a sphere around each atom with that of the model overlapping ion system). Alternatively, one can compute the band structure from the potential generated by model charge densities with different ionicities, and compare this with the self-consistent band structure. Both procedures were performed for stishovite and gave similar results, with a best fit of between O 1.4 and Si 2.8+ and O 1.2 and Si 2.4+ [78], which supports the idea that the Si-O bond is a mix of ionic and covalent bonding. There was no evidence for changes in these values with pressure. The same exercise has not been performed for quartz but it would be interesting to see if there is any conspicuous difference in the ionicity of tetrahedral and octahedrally coordinated Si using this measure. The LDA did a very good job predicting the charge density of stishovite (rutile-structured SiO 2 ) compared with experiment [79](fig. 16), and the equation of state, structure parameters, and Raman frequencies computed were also in excellent agreement with experiment [78]. Further computations [80] showed an elastic instability in c 11 -c 12 and a phase transition at 45GPatotheCaCl 2 structure, and suggested that the transition could be detected by following the B 1g Raman mode, which would decrease in frequency with pressure until the phase transition, and then increase. Such experiments were done, and were in excellent agreement with the theoretical predictions [81]. This illustrates the great power of accurate computations within the LDA for making reliable predictions. LDA computations for quartz also showed good agreement with experiment [83, 84, 85, 86, 87], suggesting that LDA should be predictive for a variety of SiO 2 phases. Pseudopotential computations showed energies of stishovite lower than quartz, making stishovite the ground state, but this was attributed to inaccuracies in the pseudopotentials [85]. It was quite surprising, therefore, when more accurate computations showed that indeed LDA predicts stishovite to be the ground state structure for SiO 2, contrary to experiment [88]. The energy error is not huge (0.6 ev), but clearly methods with smaller errors are needed to predict phase diagrams. Fortunately, 31

32 32 Figure 15. Isosurfaces of the valence density of stishovite. The ionic and covalent character are evident. The isosurface densities are at 0.06 and 0.08 electrons/bohr 3. the GGA is a great improvement, and gives quartz as the correct ground state phase [88]. Interestingly, the Wigner LDA gives the correct ordering of quartz and stishovite, but the energy difference is much too small (0.01 ev). Analysis of the charge density differences between GGA and LDA for quartz and stishovite show that the main differences are in quartz, and are greater in the atomic rather than bonding regions (fig. 18). In other words, the sizes of the atoms in GGA are slightly different than in LDA. The bonding doesn t change at all between GGA and LDA. This is evident by comparing the densities of states computed using the LDA and GGA, which are identical (fig. 19). Unfortunately these rather subtle changes lead to significant energy differences when comparing different phases. Note that within a given phase, these errors can cancel out, for example, in computing an equation of state. In fact the LDA equation of state for stishovite is somewhat better than the GGA. Another way of looking at the stability problem is to realize that the accuracy of predicting phase transitions between considerably differently structured phases may exceed 10 GPa in LDA (and perhaps GGA), yet the quartz-stishovite transition is at 9 GPa, and the LDA errors are such that the transition is predicted at about 0

33 33 Figure 16. Deformation density (total density minus spherical atoms) of stishovite from (a) theory (Ref. [78]) and (b) experiment (Ref. [79]) Si is in the center. The experiment refined only the non-spherical deformations, not the ionicity. The non-spherical density is in very good agreement with the experimental model. Note particularly the non-bonding charge density, perpendicular to the Si-O bond, and the bonding charge. The contour interval is 0.05 e /Å 3. From Ref. [78]. GPa instead. If one had a 10 GPa error in a 100 GPa phase transition it would be considered a success, but a 10 GPa error for a 9 GPa transition seems unacceptable. This is why so much theoretical work has centered on ultrahigh pressure phase diagrams. Fortunately, the development of more accurate exchange-correlation

34 34 Figure 17. Raman frequencies for stishovite predicted using LAPW (solid) (Ref. [80]) compared with experiment (Ref. [81,82]). Agreement is excellent. This is one of the few phase transitions predicted theoretically before being observed. An ionic model (PIB++, dashed) fit to the LAPW results is also shown; the difference in transition pressure of PIB++ compared to LAPW, to which it was fit, indicates the sensitivity of the transition to details of the interactions. functionals is an active field, and we should have methods that are more accurate for everything in the near future. See for example the promising results of the weighted density approximation (WDA) for a wide range of materials [15, 16] MgSiO 3 PEROVSKITE Probably the most abundant mineral in the Earth is silicate perovskite, with composition close to MgSiO 3, which is believed to form the bulk of

35 35 Figure 18. Isosurface of the difference in charge density of quartz computed using GGA and LDA. Isosurface level is electrons/bohr 3. The lumpy objects are oxygen ions, and the spheres are silicons. the Earth s lower mantle. Extensive computations have been performed for MgSiO 3, although it is difficult to study because there are 20 atoms in the primitive unit cell. MgSiO 3 appears to be more ionic than stishovite [89, 90], but the fully charged ionic model is not as successful as it is for simple minerals such as MgO. Nevertheless, the predictions of single crystal elastic constants for MgSiO 3 [91] were in reasonable agreement with later experiments [92, 93], and high pressure elastic constants are still not available experimentally. More recently, large scale computations of elastic constants have been performed for MgSiO 3 using a plane wave basis with pseudopotentials [94]. Agreement with available experiments is excellent, and these computations give elasticity data for perovskite for pressures throughout the mantle. These computations will be benchmarks for models that allow computation of thermoelasticity, to obtain properties at mantle temperatures.

36 36 (a) (b) (c) Figure 19. Density of state of quartz in the (a) Hedin-Lunqvist LDA, (b) GGA, and (c) Wigner LDA. The valence band densities of states are essentially identical. An accurate and fast model for MgSiO 3 would still be very useful. Recent results show that if one reduces the amount of charge transfer, using fixed ion charges Mg 2+,Si 3.4+,andO 1.8, consistent with the LAPW charge density, excellent agreement with LAPW computations can be obtained (Ita

37 and Cohen, unpublished) (fig. 20). Preliminary results also show excellent agreement with the experimental thermal equation of state. These results suggest that the primary effect of covalency in silicate perovskite may be the reduction of the effective ionic charges. This implies that ionic charges are different for oxygen, say, in perovskite and magnesiowüstite. If this is true, then energetics and electrical properties of interfaces between silicate perovskite may be more interesting than expected. 37 Figure 20. Energy versus M-point and R-point rotations in perovskite. The points are from LAPW computations, and the lines are from VIB with reduced ionic charges as described in the text LINEAR RESPONSE COMPUTATIONS Computation of phonon frequencies allows investigation of crystal stability, and derivation of the thermal free energy in the quasiharmonic approximation. Computations are quite fast for ab initio models, once the expressions for the dynamical matrix are derived, and one can compute the phonon frequencies for any wavevector q throughout the Brillouin zone. In self-consistent methods the computational burdens are much higher. The straightforward procedure is to do frozen phonon calculations, in which the energy is computed as a function of displacement of the atoms along symmetry or normal mode directions. This method becomes less tractable for wavevectors other than Γ (q = 0) since supercells must be used, and the computational burden goes as N 3 using conventional diagonalization tech-

38 38 niques, where N is the number of atoms in the supercell. Linear response theory [95] avoids this by expanding the total energy expression and Kohn- Sham equations for second-order changes in external potential, V (q), which allows direct determination of phonon frequencies, the dielectric constant, and transverse effective charges with a computational burden that is constant for arbitrary wavevector. The dynamical matrix elements are derived from the perturbing potential being displacements of atoms (i.e. nuclei or atom cores). For the high frequency (electronic) dielectric constant the perturbing potential is a potential (electric field) wave. For the transverse effective charges the perturbing potential is an atomic displacement and an electric field wave. With linear response, the computational complexity involves only N atoms, where N is the number of atoms in the primitive cell, regardless of the wavevector q. It is still a large job to derive the phonon dispersion curves with self-consistent linear response theory, since in general there are 3N phonon modes at each k-point, so one must perform 3N separate linear response computations at a general symmetry k-point for an atom at the general position (i.e. with no symmetry). Each computation still scales at N 3 using conventional diagonalization techniques, so this procedure rapidly becomes intractable for large low symmetry structures. However, for high symmetry phases and high symmetry k-points only a few computations are necessary to find the dynamical matrix. Typically, one does a small number of such computations, and interpolates the frequencies by finding short-range force constants from the dynamical matrix at a few high symmetry points in conjunction with the effective charges and dielectric constant, also obtained from linear response computations. Thus the entire set of phonon dispersion curves throughout the Brillouin zone can be obtained. The transverse, or Born, effective charges are important for understanding the degree of ionicity and type of bonding in crystals. The Born effective charges Q iαβ = 2 F/ u iα E β = P β / u iα where F is the free energy, E is the electric field and u iα is the displacement of atom i in the α direction, and P is the polarization (dipole moment per volume). As the notation indicates, the effective charge is a tensor. It indicates how much charge moves as an atom is displaced. In an ideal rigid ion crystal, the effective charges would be independent of direction (i.e. scalars), and would equal the nominal ionic charges. Deviations from this behavior indicate covalency, and give information on the directionality and nature of the bonding CaSiO 3 PEROVSKITE Stixrude et al. [96] performed linearresponse computationswiththe LAPW method on CaSiO 3 perovskite in the cubic perovskite structure. Unlike

39 MgSiO 3, which is orthorhombic, experimental studies have found CaSiO 3 to be cubic. Unexpectedly, however, the linear response computations showed cubic CaSiO 3 to be unstable at the M- and R-points. Frozen phonon calculations confirmed this result. The results suggest strongly that CaSiO 3 is not cubic but tetragonal or lower symmetry. The small strain predicted is consistent with the x-ray studies that were interpreted as showing a cubic structure for CaSiO 3. Perhaps more important, from the point of view of this survey of bonding in minerals, are the effective charges (Table 1). The effective charges in CaSiO 3 are close to the nominal ionic charges, and are only slightly affected by pressure, indicating that the degree of covalency does not vary over the pressure range of the Earth s mantle. This makes clear that the much reduced numbers that come from Mulliken populations, for example, are not indicative of the amount of charge that moves when a nucleus is displaced, nor are they good measurements of the amount of static charge on each atom since in reality the charge densities strongly overlap, so that atomic basis functions represent states that extend from one atom, for example, O 2 onto another. 39 TABLE 1. Born effective charges for CaSiO 3 (at a volume of 310 bohr 3, -8 GPa) (Ref. [97]) and PbTiO 3 (Ref. [98,99]) at approximately zero pressure. Note that the effective charges can differ greatly from their nominal values, reflecting covalency, and that they can be directional, and strongly dependent on structural distortions. For tetragonal PbTiO 3,O 1 is the O is closest to Ti, along [001], and O 2 is the O on the xz face, farther from the Ti. CaSiO 3 Ca Si O (x)/-3.06(z) PbTiO 3 Pb Ti O 1 O 2 (cubic) (x,y)/-5.51(z) (tetragonal) (x)/5.18(z) -2.15(x)/-4.38(z) -2.61(x)/-5.18(y)/-2.16(z) Effective charges have only been computed for a few materials, and very few minerals. Although PbTiO 3 is not a mineral, the effective charges computed for it are instructive as compared with CaSiO 3.PbTiO 3 is much more covalent than CaSiO 3, and this is evident in the effective charges. Though some values are close to the nominal values, such as for O motions transverse to the Ti-O bond, the effective charges parallel to the Ti-O bond are greatly enhanced, as are the Ti effective charges. This is a general feature of effective charges in more covalent, but still ionic materials. When a nucleus or atom core is displaced, much more charge redistributes than the nominal value due to the bonding states formed from hybridization between the atoms. The sensitivity of the effective charges to structural

40 40 distortions in PbTiO 3 suggests a large amount of covalency, consistent with other diagnostics such as the partial density of states. The large effective charges in PbTiO 3 contribute to its ferroelectric properties and piezoelectric response. Detailed study of how effect charges change with distortion in minerals is liable to shed more quantitative details on bonding and how it changes with structural charges, but little such work has been done. 11. Metals Metals are distinguished by the fact that they easily conduct electricity, which is due to the fact that they have partially occupied states at the Fermi level, the highest occupied energy level in a crystal. One can understand the insulating behavior of materials with fully filled bands by considering what happens when one applies an electric field. An electric field raises the potential at one part of the sample relative to another, and one would think that electrons would then flow down that potential gradient. In fact both the electrons move and the nuclei are displaced in response to the applied field; the material becomes polarized. The electrons do not continue to flow however. The material becomes polarized by mixing in part of the excited state, or conduction band, eigenvectors into the valence band. Only so much can mix in, however, due to the finite gap, and since the bands are filled in an insulator, there can be no continual flow under a small field since electrons would have to be excited above the gap, which takes a finite energy. Thus the materials has a finite susceptibility and the material is insulating. In a metal, the partially filled states at the Fermi level means that current will flow for any applied field, since there is no energy gap between occupied and unoccupied states, and the susceptibility is infinite. Metals are not often considered as in mainstream of mineralogy or mineral physics, though there are natural metals, and much of the mass of the Earth, that is the Earth s core, is metallic iron. Recent progress in understanding metals such as iron at high pressures is thus reviewed briefly here. In self-consistent computations, metals are treated in the same way as insulators or semiconductors. The same methods work regardless of the different type of bonding. In metals, however results can be more sensitive to convergence parameters such as the type of basis and basis set size, and the number of k-points used to integrate over the Brillouin zone. Many k- points are needed since different electronic states are occupied at different k-points, due to the fact that bands cross the Fermi level. Also, the bonding in metals is not localized; rather one can think of atoms or ions embedded in an inhomogeneous sea of electrons. Since there is significant charge density everywhere, localized basis sets such as Gaussians are not optimal. This becomes even more important when considering metallic surfaces, where

41 the charge density fades off gradually into the vacuum. The LAPW basis is ideal for metals, as for other materials, in that it uses a plane basis in the region between atoms, which can represent any density. Plane wave bases with pseudopotentials are also ideal, as long as the pseudopotential is accurate, which is sometimes difficult for 3d metals. The LDA works quite well for metals in general, but the GGA is a marked improvement for magnetic 3d metals such as Fe. Iron is magnetic and forms in the bcc structure at low pressures. The LDA however, incorrectly, gives a non-magnetic hcp structure as the ground state. It was thought that the GGA overstabilized magnetic phases, but LAPW computations showed that GGA gives not only the proper magnetic bcc ground state, but also an accurate transition pressure to non-magnetic hcp at 11 GPa, compared with an experimental bracket of GPa, indicating that the magnetic stabilization is accurately predicted in GGA[99]. Theory shows that it is magnetism that stabilizes the bcc structure, and that otherwise bcc iron would not form. There had been much discussion of bcc as the possible structure for iron in the Earth s inner core, but calculations show that bcc iron is mechanically unstable at high pressures due to the loss of magnetism with pressure [100]. Models have also been developed for metals that work quite well. Embedded atoms models are similar to Gordon-Kim models for closed shell systems, except that the main term is the embedding energy for putting an atom into an electron gas [101, 102]. Embedded atom models have not yet been much applied to high pressure or mineralogical problems. Tightbinding models work well for metals, and both non-empirical and empirical models have been developed. Very accurate models have been developed for iron, for example, that are fit to self-consistent calculations, and that work well over a factor of two of compression [18]. Hybrid models have been developed that appear to be quite accurate for transition metal systems [103, 104, 105, 106]. High temperature properties are difficult to obtain self-consistently, so Wasserman et al. applied the particle-in-a-cell model to compute thermodynamic properties of iron at high pressures and temperatures in conjunction with a new fast and accurate tight-binding model which allowed computations for large unit cells [107]. Excellent agreement with shock compression data was obtained, up to ultrahigh pressures and temperatures. Metals have contributions to thermodynamic properties from thermal electronic excitations in addition to phonons at finite temperatures. Figure 21 shows the computed thermal pressure for iron, and shows the phonon and electronic contributions. Metallization of iron bearing oxides, if it occurs in the mantle, will significantly change the density as well as transport properties due to the electrons at the Fermi surface. 41

42 42 Figure 21. Contribution to thermal pressure of hcp Fe at high pressures and temperatures. The phonon contribution and total thermal pressure are shown for Fe at 2000 K. The metallic electronic contribution is clearly significant. From Ref. [107]. Molecular dynamics simulations have also been performed for iron liquid at outer core conditions to study it s properties using the tight-binding model [108] and self-consistently [109]. Estimates of the viscosity of liquid iron, a quantity which has been uncertain to 13 orders of magnitude were obtained. Experiments under such conditions are extremely difficult, so theory will remain an important source of information about the probable behavior of iron in the Earth s core. Elasticity is a crucial property for mantle minerals, since most of what we know about the deep Earth comes from seismology, which measures acoustic velocities versus depth. Seismological observations had shown that the Earth s inner core is highly acoustically anisotropic, with sound waves traveling much faster in polar directions than in equatorial directions. However, there were no experimental elastic constants for iron available under core conditions. Stixrude et al. computed the elastic constants of fcc and hcp iron as functions of pressure (at zero temperature, static conditions) using the tight-binding model and found excellent agreement with the ob-

43 43 TABLE 2. Elastic constants for non-magnetic hcp Fe at V=60 bohr 3. From Ref. [110]. C 11 C 33 C 12 C 13 C 44 C 66 (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) Iron GGA Iron LDA Iron Tight-Binding[18] Iron FP-LMTO GGA[111] Diamond Anvil Cell Strain Exp.[112] 640(55) 650(85) 300(55) 255(40) 420(25) 170(55) served anisotropy if the inner core is a large hcp single crystal, or oriented smaller crystals [113]. Self-consistent computations using the full-potential LMTO method gave similar elastic constants [111]. The situation has become more complicated, in that recent diamond anvil experiments find very different anisotropy from what has been predicted [112], and seismologists find much more complicated patterns of anisotropy in the inner core which are difficult to reconcile with experimental or theoretical elastic constants. It may take some time until this is sorted out, because experiments are extremely difficult, and the interpretation of the seismic data is also fraught with problems. In order to better understand the accuracy of predictions of high pressure elasticity in hcp metals, we have performed detailed self-consistent LAPW computations for hcp Co and Re in addition to Fe, and find very good agreement with experimental data for Co and Re (Table 3) [110]. 12. Magnetism Perhaps this is a good place to mention some general ideas about magnetism. Magnetism is due to the presence of unpaired electrons. Electrons have magnetic moments, but if they are paired up in states as up and down pairs, there will be no net magnetism in the absence of a magnetic field. However, in open shelled atoms there may be a net moment. For example, ferrous iron, Fe 2+, has 6 d-electrons, and two endmember situations can be considered. In low-spin ferrous iron, the 6 d-electrons are paired up to fill 3 t 2g states, and there is no net-magnetic moment. In highspin ferrous iron, five up d-states are split in energy from five down

44 44 TABLE 3. Elastic constants for hcp Co and Re. After Ref. [111]. Volume C 11 C 33 C 12 C 13 C 44 C 66 (Bohr 3 ) (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) Cobalt Experiment[114] Cobalt GGA Cobalt LDA Rhenium Experiment[115] Rhenium GGA Rhenium LDA d-states by the exchange energy, and 5 d-electrons go into the 3 t 2g and 2 e g lower-energy, majority spin states, and one electron into a high-energy t 2g state, giving a net magnetic moment component of 4 µ B (Bohr magnetons) (fig. 22). According to Hund s rules, atoms (or isolated ions) will maximize their net magnetic moment, which lowers their total energy due to a decrease in electrostatic energy. In a crystal, interactions with other atoms, and formation of energy bands (hybrid crystalline electronic states) may lead to intermediate or low spin magnetic structures. In the Stoner model, the effect of magnetism on the total energy is given by: E = M 2 I 2 + M 2 2N(0), (25) where M is the magnetic moment or magnetization, the Stoner Integral I is an atomic property, and N(0) is the density of states at the Fermi level (or top of the valence band). The first-term is the magnetic energy and the second is the change in the band energy with magnetic moment. Minimization of E gives the Stoner criterion: IN(0) > 1 (26) for a magnetic state (M >0) to be stable. In the more sophisticated extended Stoner model, the average density of states is introduced, Ñ(m) =M/ ɛ, (27)

45 45 Figure 22. iron. Schematic of occupancies of states in (a) high-spin and (b) low-spin ferrous where ɛ is the spin (exchange) splitting, Ñ(0) = N(0) = M/ ɛ, (28) and the instability criterion is IÑ(M) > 1. The Stoner model works quite well in predicting magnetic states. It will be discussed further below. As pressure is increased, band widths increase due to increasing hybridization, so that in the absence of new bands crossing the Fermi level or changes in band topology at the Fermi level or top of the valence band, the effective density of states will decrease, and magnetism will decrease and disappear with increasing pressure. 13. Transition metal ions and Mott insulators One might not infer that there was anything special about transition metal ions such as ferrous iron (Fe 2+ ) and ferric iron (Fe 3+ ) from examining many physical properties of minerals. There is complete solid solution between Mg and Fe endmembers for most minerals, and in general thermodynamic mixing properties and thermoelastic properties are smooth and even close to ideal. This evident simplicity hides a problem that is at the frontier of solid state physics. Whereas Mg 2+ is a simple closed-shell ion, and it forms simple ionic bonds in oxides and silicates, Fe 2+ is an opened-shell, magnetic, and very non-spherical ion that forms complex ionic/covalent bonds. Although the transition metal oxides such as FeO, MnO, NiO, and CoO form simple rocksalt (or distorted rocksalt) structures, and seem like ordinary ionic insulators in some ways, they are actually very complex and poorly understood

46 46 materials, experimentally as well as theoretically. Wüstite, Fe 1 x O, has additional complications since it is non-stoichiometric, and contains some ferric iron even in equilibrium with iron metal. The vacancies in wüstite form complex defect clusters[116]. With increasing pressure, more stoichiometric wüstite can be stabilized. Theoretical work on wüstite has mostly concentrated on the difficult enough problem of pure FeO, and that is the focus here ELECTRONIC STRUCTURE The band structures computed for these materials using conventional band theory (LDA or standard GGA methods) give a metallic ground state at zero pressure for FeO and CoO, and a small band gap for MnO and antiferromagnetic NiO, whereas these materials are believed to be wide gap insulators. Magnetic crystals that are insulators by virtue of local magnetic moments are known as Mott insulators [117]. The failure of conventional band theory to predict proper band gaps is not a reason by itself to believe that band theory is completely wrong, since there are other normal materials for which LDA has problems. For example, Ge is found to be metallic in LDA, rather than a semiconductor [118], and LDA always gives band gaps that are too small by a large factor compared with experiments. However, this failure to predict correct gaps in ordinary materials is well understood, and is really not a failure of the theory, but a failure of a fundamentally unjustified interpretation of the Kohn-Sham eigenvalues as quasiparticle energies, which they are not. We are used to the Kohn-Sham eigenvalues being good approximations for the band structure, but there is no fundamental justification for this correspondence, except that the Kohn-Sham equations look similar to the quasiparticle equations. Actually, the root cause of the failure of band theory for the transition metal oxides may be similar to the failure to predict accurate band gaps in other materials, but it is also believed that there is a more specific failure of LDA-like theories for the transition metal oxides, which is that LDA does consider energetic differences for a electron at a given point in space depending on what orbital it is in. The Hohenberg-Kohn theorem says that such orbital dependent potentials should not be necessary to find the energy and ground state charge density of a system, but the exact functional that would give this behavior is unknown, and would likely be extremely complex in order to give the proper charge density of transition metal oxide compounds, especially those that involve orbital ordering [119, 120]. The main problem with LDA-like theories is believed to be the meanfield treatment of the local Coulomb repulsion U, whichisameasureofthe increase in energy when an electron is added to an atom. For example, Fe 2+

47 has 6 d-electrons per atom, and a conducting electron for d-states at the Fermi level, as predicted by LDA, must hop from Fe to Fe. However, there is actually a large increase in energy when an electron hops from one Fe to another. The simplest model that contains local interactions of this sort is called the Hubbard model. Even this simple model, though, has not been solved, even in two-dimensions (it has been solved in infinite dimensions, in the dynamical mean field model [121, 122], but no one yet knows if this model is accurate for smaller numbers of dimensions). Even numerical simulations of the Hubbard model have proven to be very difficult. Recent Monte Carlo results for the Hubbard model [123] are shown in fig. 23 and will be discussed further below. The critical parameter that indicates the importance of U is U/W, wherew is the effective single-particle band width (i.e. not including magnetism or hybridization with O 2p states.) 47 Figure 23. Monte Carlo results for the Hubbard model. Pressure increases to the left on the absicca. The transition at the right (at low pressures) is T N,theNéel transition, (antiferromagnetic ordering). The high pressure transition is the high-spin low-spin transition. Between is a metal insulator transition which is discontinuous at low temperatures, and is continuous out at high temperatures, above T N. From Ref. [123]. Hartree-Fock calculations include an unscreened orbital dependent U,

48 48 and thus give large band gaps for these materials. they also give reasonable energetics [124, 125]. However, they give band gaps that are way too large, and values of U that are much too high. Hartree-Fock seems to be a good way to study crystals at low pressures in the high U limit, and it gives proper orbital ordering in cases where LDA fails completely. On the other hand, Hartree-Fock cannot be used for metals, where it gives pathological results, and the huge overestimation of band gaps makes it unsuitable to study metal-insulator transitions, or materials at high pressure which may be properly described as metallic by LDA. A promising method for realistic computations for Mott insulators, but which needs further investigation is the LDA+U model [126, 127, 128]. In this method an orbital dependent potential is added to simulate U. U can be computed by using constrained LDA computations, where one estimates U by computing the change in energy with orbital occupancy. LDA+U appears to be an excellent approximation at zero pressure in the high U/W limit, but it is unclear whether it will give reasonable results at high pressures, where U/W is much smaller. The parameter U/W decreases with pressure since U is relatively insensitive to pressure, but the band width W increases rapidly with pressure due to increased hybridization ENERGETICS In spite of the problems with the electronic structure of transition metal oxides, energetic properties can be predicted with accuracy similar to simple materials using the LDA and GGA, with the exception of distortions in orbitally ordered structures. The energy versus magnetic moment must also be reasonable, because magnetic moments are well predicted. The GGA equation of state is superior to the LDA, in that the LDA predicts significantly too small volumes. Nevertheless, even the LDA gives many properties correctly, such as the qualitative change in rhombohedral angle with pressure in FeO [129]. This result indicates the power of theory, because we were able to study why the rhombohedral angle changes with pressure, which would be very difficult to understand from experiments alone. Visualization of the charge density as a function of rhombohedral angle [129, 130] indicated that Fe-Fe bonding causes the rhombohedral strain, and the increase in Fe-Fe bonding with pressure is responsible for the increased strain with pressure. The rhombohedral strain brings Fe ions closer together, and thus is favored with increasing Fe-Fe interactions. It is dangerous, however, to extrapolate such conclusions to other materials. NiO has the opposite sense rhombohedral strain, and Isaak et al. suggested, without doing computations for NiO, that the sign of that strain would change with pressure

49 due to increased Ni-Ni bonding with pressure, and the strain would change in a similar way to FeO. Actually, the strain in NiO does not change sign with pressure, and increases in the sense that Ni ions move apart due to the strain [131]. Evidently the Ni-Ni interactions are antibonding in nature, compared with the interactions in FeO MAGNETISM AND MAGNETIC COLLAPSE Isaak et al. found that LDA predicted that the magnetic moment in FeO collapsed at pressures above 100 GPa [129]. They suggested that the magnetic collapse may lead to a, possibly simultaneous, metal-insulator transition. The predicted magnetic collapse is a type of high-spin low-spin transition, which has been observed in other materials such as NiI 2 [132, 133] and MnS [134, 135]. High-spin low-spin transitions can be either continuous higherorder phase transitions, or first-order phase transitions. A more comprehensive study of the transition metals oxides MnO, FeO, CoO, and NiO was then performed using the GGA and the LMTO-ASA method [136]. The GGA LMTO equation of state results for FeO were better than LDA at low pressures, and the magnetic moments versus volume were quite similar to the earlier LAPW results, except that it appeared that LMTO could access metastable high-spin states in the low-spin stability field, and low-spin states in the high spin stability field. The LMTO computations gave a 200 GPa high-spin low-spin transition for FeO, and at the time we believed that difference from the 100 GPa transition predicted earlier was the difference between GGA and LDA, and that the LMTO computations gave good agreement with the earlier LAPW LDA computations of Isaak et al. Publication of the new results was followed by experiments that showed a 100 GPa transition in FeO [137], which lead us to question whether the agreement with the earlier LDA calculations was fortuitous, whether LDA could be more accurate than GGA, or whether the observed transition was different from that predicted. Detailed LAPW GGA calculations [108] have now shown that the LMTO results were in error, and they give a transition pressure of 100 GPa in agreement with the Pasternak et al. experiments. Figure 24 shows the predicted magnetic moments versus volume for FeO, and figure 25 shows the equation of state. A discontinuous transition is predicted for AFM FeO, and a continuous transition for ferromagnetic FeO. There is a large V for the high-spin low-spin transition. Published diffractiondata for FeO of Yagi et al. showed no discontinuity in the equation of state [138], so either the sample examined by Yagi et al. did not transform for some reason, or the transition was spread out over a large pressure range so that no discontinuity was seen, perhaps due to

50 50 non-hydrostatic stresses, non-stoichiometry, or finite temperatures. One might question whether band theory would be expected to give valid predictions for high-spin low-spin transitions in Mott insulators, when it does not give the proper ground state electronic structure. However, earlier work suggests that it should. The Mott insulator NiI 2 has a high-spin low-spin transition that has been observed by Mössbauer at 19 GPa [133]. LDA computations give 20 GPa, and GGA gives 25 GPa [132]. Thus it appears that theory can give good results for such transitions. One explanation for the success of band theory for these high pressure transitions is that the key parameter is U/W, as discussed above, and since U/W decreases with pressure due to the increasing band width with pressure, and theory becomes more applicable and predictive with increasing pressure. The story continues to evolve, however, and very recent results of Mao et al. (pers. comm.) show no high-spin low-spin transition in FeO up to 130 GPa using x-ray spectroscopy. New LAPW computations show the high-spin low-spin transition in FeO broadens into a continuous change in moment with pressure with inclusion of the rhombohedral strain. It will require both further theory and experiments together to unravel the complexities of the transition metal oxides. Figure 24. Magnetic moment of AFM and FM FeO versus volume in FeO computed using different methods. From Ref. [143].

51 The study of Mott insulators such as FeO is fascinating since no one knows what to expect;even the generic behavior of these systems is unknown. Even the simplest pertinent model, the Hubbard model, has never been solved. Quantum Monte Carlo simulations for the Hubbard model [123] have shown behavior similar to what has been observed in FeO and other systems studied so far. Figure 23 shows a proposed phase diagram from Hubbard model simulations designed to study f-electron systems, such as Ce, which also show strong volume collapse transitions. There has been some controversy as to the nature of these transitions, but one view is that these transitions are essentially high-spin low-spin transitions. Figure 23 shows ordered and disordered high-spin magnetic phases, a non-magnetic low-spin phase, and a metal insulator transition. The metal-insulator transition is particularly interesting, in that it occurs not at the high-spin low-spin transition boundary, but before this transition. Furthermore the metal-insulator transition is sharp at low temperatures, but becomes continuous at high temperatures, with a funny triple-peaked density of states. Conductivity may still onset discontinuously at a Mott-Anderson transition when there are sufficient densities of states at the Fermi-level to have bulk conduction, the so-called mobility edge [117]. Secondly, note that the magnetic collapse boundary at high pressures may be the same as the Neèl transition boundary at low pressures, and one may be able to go continuously from a phase with disordered local moments, to a phase with no local moments, since these phases have the same symmetry. Thus the Neèl temperature initially increases, and then decreases with increasing pressure due to the decreasing local moments. Now it is possible to better understand the results of Pasternak et al. [137] The Mössbauer experiments showed a transition with increasing pressure to a low spin phase, and also a transition to a non-magnetic phase with increasing temperature. This would be difficult to understand if the moments were ordered in all cases, that is if the temperature was well below T N as in their proposed phase diagram, because the high-spin phase should be the high entropy phase; thus increasing temperature should promote the high-spin magnetic phase, rather than the low-spin non-magnetic phase. However, if one understands that the transition with increasing temperature is a disordering of the local moments, rather than a loss of local moments, all is well, since temperature should promote disordering. Mössbauer does not distinguish between loss of local moments, and disordering of moments. So it appears that the experiments on FeO are consistent with what is known about the behavior of the Hubbard model except that the location of the metal insulator transitions in FeO and other transition metal oxides are unknown. At about 100 GPa the B1 phase appears to be insulating, not metallic, which suggests that the insulator metal transition should be 51

52 52 Figure 25. Equation of state of high-spin and low-spin FeO computed using LAPW and the GGA. From Ref. [108]. between 100 and 120 GPa if the single-band Hubbard model results are any indication of the behavior to expect in FeO (fig. 23). A further complication is that angle resolved photoemission data [139] for NiO do not agree well with the Hubbard picture, but are closer to the LDA band model, except around the Fermi level (fig. 26). In contrast, LDA+U, which opens a band gap, gives a very different band structure than LDA [128]. Structural phase transitions further complicate the transition metal oxides. At room temperature and pressure, FeO is in the cubic rocksalt (B1) structure. As temperature is lowered, it passes T N at 200 K and becomes magnetically ordered, and simultaneously assumes a distorted rhombohedral structure. As pressure is increased, T N increases [140], so that pressure promotes the rhombohedral phase. It was thought that the rhombohedral distortion was due entirely to magnetostriction, but Isaak et al. showed pressure promotes the rhombohedral distortion even in the absence of magnetism. [129] At high temperatures, however, another phase is found, which is conducting, [141] and which is hexagonal and was originally identified

53 53 Figure 26. Angle resolved photoemission data for NiO compared with the LDA band structure. (a) Photoemission data. The marked peaks are interpreted as quasiparticle states. The breadth of peaks is due to instrumental resolution, surface effects, multiple bands, and lifetime broadening. (b) Comparison with the antiferromagnetic band structure. Agreement is quite good except for the band crossing the Fermi level. From Ref. [139]. from in situ diffraction studies to be the NiAs (B8) structure [142]. This phase has not yet been seen at room temperatures, although the phase diagram seems to suggest that it should occur at low temperatures as well; perhaps the transition is kinetically inhibited. Although the positions of the diffraction peaks agreed with the B8 structure, the intensities did not. By reanalyzing the diffraction data, Mazin et al. [143] concluded that the most likely interpretation of the data was that of a polytype or superlattice between B8 and anti-b8, with Fe in the As-site and O in the Ni-site. The B8 and anti-b8 structures can be joined together smoothly, and the boundary between them is the rhombohedrally-distorted B1 structure. Thus one can join together all three phases into a single coherent structure (fig. 27). This phase could form either due to lack of equilibrium, or could even form a unique continuous structure transition between the different phases. The inverse-b8 structure is predicted to be an insulator (the only known LDA

54 54 Figure 27. Normal and inverse NiAs superlattice proposed for FeO. From Ref. [108]. band insulator in FeO) whereas normal B8 is predicted to be a metal. This is due to the fact that the Fe-Fe distances are very short in normal B8, and much farther apart in inverse-b8. GGA band theory predicts similar energies for normal and inverse B8 in the non-magnetic or ferromagnetic cases, but when allowed to be anti-ferromagnetic, inverse B8 is stabilized so much that it is actually predicted to be the ground state phase, i.e. lower in energy than B1. This is very unlikely to be correct, although it isn t entirely impossible. B1 could be found due to a combination of non-stoichiometry, which would stabilize B1, and kinetic constraints, since inverse-b8 is only more stable in the low-temperature antiferromagnetically ordered phase, which would likely not form spontaneously at low temperatures from B1. The fact that the high pressure and temperature phase has not been successfully quenched to room conditions argues against its stability relative to B1, though. Finally, we consider the origin of high-spin low-spin magnetic transitions, and the likely magnetic structure of transition metal ions in solid solution in other phases. The conventional, local atomic, picture of highspin low-spin transitions is that the crystal field splitting between e g and t 2g states gets larger than the exchange splitting between spin-up and spindown states, thus leading to a transition to a low spin electronic structure