Electronic Separability Theory: Embedding, thermodynamics and electron density topology

Transcript

1. Lanzhou 2005 Electronic Separability Theory: Embedding, thermodynamics and electron density topology V´ ıctor Lua˜ na Departamento de Qu´ ımica F´ ısica y Anal´ ıtica, Universidad de Oviedo http://web.uniovi.es/qcg/ International Workshop in Computational Materials Science May 23–28, 2005, Lanzhou University, China. c V. Lua˜ na 2005 (1)
2. Lanzhou 2005 Talk Map Conceptual map of the talk R. McWeeny S. Huzinaga Embedding Cluster−in−the−lattice (Evolution of Crystal Field Theory) AIM Quantum Theory of Atoms in Molecules RWF. Bader Crystal properties Topological partitioning of the energy (Equation of state, elastic properties, phase transitions ...) Electronic Separability Theory of TES c V. Lua˜ na 2005 (2)
3. Lanzhou 2005 Theory of Electronic Separability I. The Theory of Electronic Separability We assume that the system can be described as a set of weakly correlated electronic groups, {A, B, C, ...}, each group being represented by an antisymmetric group function, {ΦA(1..NA), ΦB(1..NB), ...}. The electronic wavefunction for the whole system is given by Ψ(1, ...N) = M p ˆ PpΦA(1, ...NA)ΦB(NA+1, ...NA+NB)... (1) where M is a normalization constant, ˆ Pp is a permutation operator that changes the order of electrons, and p runs over all electron permutations that exchange electrons among groups. In other words, p ˆ Pp enforces the antisymmetry of Ψ(1, ...N). Examples of weakly correlated groups: • core versus valence electrons in atoms and molecules, • core vs. extended sea of electrons in alkaline metals, • σ vs π electrons in conjugate polyenes, • each ion can be regarded as a group for very ionic crystals. It must be stressed that weak correlation does no imply a small interaction energy between groups. c V. Lua˜ na 2005 (3)
4. Lanzhou 2005 Theory of Electronic Separability Work with the group functions is much simplified by enforcing strong-orthogonality conditions: Φ∗ R (x1, x2, x3, ...)ΦS(x1, x2 , x3 , ...)dx1 = δRS, (2) xi being the spatial and spin coordinates of electron i. The total energy of the system is then E = R ER + R>S ERS (3) where (a) ER (net energy) contains all energy components internal to the group R, and (b) ERS (interaction energy) contains the Coulomb interactions between the particles in R and those in S. The detailed form of these energies is ER = ΦR| i∈R ( ˆ Ti − α∈R Zαr−1 iα ) + 1 2 i,j∈R r−1 ij |ΦR + 1 2 α,β∈R ZαZβR−1 αβ , (4) ERS = ΦR| − i∈R β∈S Zβr−1 iβ + i∈R [ ˆ V S c (i) + ˆ V S x (i)]|ΦR + ΦS| − j∈S α∈R Zαr−1 jα |ΦS + α∈R β∈S ZαZβR−1 αβ , (5) where i, j are electrons, α, β nuclei, Zα is an atomic number, and rij, riα and Rαβ are distances between particles. c V. Lua˜ na 2005 (4)
5. Lanzhou 2005 Theory of Electronic Separability Let us assume that we are mainly interested in a particular group: the active group A. All energy components in which A appears can be collected to form the effective energy of this group: EA eff = EA + R=A EAR = EA net + EA int . (6) Restricted variational principle: Because EA eff is the only part of the total energy that depends on ΦA, the best group function for the active group can be obtained by minimizing its effective energy while keeping the strong orthogonality with respect to the other, frozen groups. In many instances the frozen groups can be adequately described by single Slater determinants. This is the case for a system of closed-shell frozen groups. The effective energy can then be derived from an effective hamiltonian: EA eff = ΦA| ˆ HA eff |ΦA , (7) ˆ HA eff = i∈A ˆ hA eff + 1 2 i,j∈A r−1 ij + 1 2 α,β∈A ZαZβR−1 αβ + α∈A S=A ZαV S eff (Rα), (8) ˆ hA eff = ˆ T(i) − α∈A Zαr−1 iα + S=A [V S eff (i) + ˆ PS(i)]. (9) Each frozen group contributes an effective potential term, V S eff , plus a projection operator, ˆ PS . c V. Lua˜ na 2005 (5)
6. Lanzhou 2005 Theory of Electronic Separability The effective potential due to the frozen group S has nuclear attraction, Coulomb and exchange components: V S eff (i) = − β∈S Zβr−1 iβ + ˆ V S c (i) + ˆ V S x (i). (10) The nuclear attraction and Coulomb parts act on all charged particles (nuclei and electrons) of the active group, but the exchange part acts only on the electrons. If S has a closed-shell structure Coulomb and exchange parts become: ˆ V S c (i) = g∈S 2 ˆ JS g , ˆ V S x (i) = − g∈S ˆ KS g , (11) where g counts occupied orbitals in S. ˆ Jg and ˆ Kg are Coulomb and exchange orbital operators, respectively: ϕA i | ˆ Jg|ϕA j = ϕA i (1)ϕS g (2)|r−1 12 |ϕA j (1)ϕS g (2) , ϕA i | ˆ Kg|ϕA j = ϕA i (1)ϕS g (2)|r−1 12 |ϕS g (1)ϕA j (2) . ˆ PS projects the occupied levels of the S frozen group out from the active group wave function. This operator represents the strong orthogonality to the group S in the effective hamiltonian of the active group. For a closed-shell S group ˆ PS(i) = g∈S |ϕS g (−2 S g ) ϕS g | , (12) where g sums over all occupied orbitals of S, and g is the energy of the ϕg orbital. c V. Lua˜ na 2005 (6)
7. Lanzhou 2005 The ab initio Perturbed Ion method Ia. The ab initio Perturbed Ion method (aiPI) The aiPI method is a very simple application of the TES formalism designed to work on highly ionic materials: • The material is assumed to be formed by ions. • Every ion forms a weakly correlated electronic group. • Large STO (Slater Type functions) bases are employed on each ion, but only the STO’s of a given center are retained to build its group function. • Electrostatic interactions are included through an exact summation of the Madelung series. • A non-local spectral resolution is used for the exchange operator of each frozen group. • The method is further simplified by assuming spherical symmetry en each center group function. This is equivalent to assume that the radial part of each STO depends on the angular quantum number l but not on the azimuthal one m: |alm, S = Ral(rS)Ylm(θS, φS) with rS = r −RS. • Correlation energy can be included on each ion using any of several density functionals. • The aiPI equations, that minimize the effective energy of an ion, are iteratively solved for each different type of ion in the material, until convergence is achieved. • The total energy of a material can be recovered from the additive energies of the different ions. If AaBbCc... is the stoichiometric formula of the material: EA add = EA net + 1 2 EA int =⇒ E = aEA add + bEB add + cEC add + ... (13) c V. Lua˜ na 2005 (7)
8. Lanzhou 2005 Equilibrium properties of MgO aiPI example: Equilibrium properties of MgO Cubic, Fm¯ 3m, a = 4.210 ˚ A Mg 4a 0 0 0 O 4b 1/2 1/2 1/2 PI input: uchf crystal title MgO. Experimental geometry. spg f m -3 m cell 7.9557 7.9557 7.9557 90.0 90.0 90.0 neq 0.0 0.0 0.0 mg.ion mg.int mg.cint mg.lint neq 0.5 0.5 0.5 o2.ion o2.int o2.cint o2.lint endcrystal end The *.ion files contain the basis set, number of electrons, etc. for each different type of ion. The cpu time is ≈44 s/100 geometries on a 1.1 GHz Intel PIV. c V. Lua˜ na 2005 (8)
9. Lanzhou 2005 Equilibrium properties of MgO The aiPI calculation provides E(V ), where E is the internal energy if we neglect the zero-point contribution. The equilibrium properties and 0 K equation of state are immediately obtained: min a E → {ae, Ee}, Elatt = Ee − Evac(Mg+2) − Evac(O−1), (14) A = E + TS ≈ E, p = − ∂A ∂V T , Be = −V ∂p ∂V T ≈ V ∂2E ∂V 2 . (15) −720 −700 −680 −660 −640 −620 −600 −580 −560 3.5 4.0 4.5 5.0 5.5 Elatt (kcal/mol) a (Å) Prop. Unit aiPI Exptal. ae ˚ A 4.212 4.210 Elatt kcal/mol 708 725 Be GPa 156 163 (dB/dp)e — 3.28 4.13 Cp J/mol K 38.16 37.89 α 10−6 K−1 24.8 13.5 Thermal properties (Cp and α = V −1(∂V/∂T)p) have been obtained using a quasi-harmonic Debye model, that uses B(V ) data to get a Debye temperature as a function of volume [Ref. [10]]. c V. Lua˜ na 2005 (9)
10. Lanzhou 2005 B1-B2 transition in alkali halides B1-B2 transition in alkali halides B1 phase 3 2 Cl 4 5 6 7 1 8 60 Fm¯ 3m (a, a, a, 90◦, 90◦, 90◦) A (4a) 0 0 0 X (4b) 1/2 1/2 1/2 Rhombohedral intermediate R¯ 3m (a, a, a, α, α, α) A (1a) 0 0 0 X (1b) 1/2 1/2 1/2 α ∈ [60◦ (B1), 90◦ (B2)] (2D Buerger’s mechanism) B2 phase Cl 3 7 8 4 5 1 2 6 90 Pm¯ 3m (a, a, a, 90◦, 90◦, 90◦) A (1a) 0 0 0 X (1b) 1/2 1/2 1/2 c V. Lua˜ na 2005 (10)
11. Lanzhou 2005 B1-B2 transition in alkali halides Some example results for LiCl: Raw QM results: E(a, α) 5.0 5.5 6.0 6.5 7.0 7.5 8.0 50 60 70 80 90 100 -0.32 -0.30 -0.28 -0.26 -0.24 -0.22 E (hartree) a (bohr) α (deg) E (hartree) The phase transition is determined by the Gibbs en- ergy: G (x; p, T) = E(x) + pV (x) + Avib(x; T), (16) where x collects all geometrical degrees of freedom. A quasiharmonic Debye model can be used to estimate Avib(x; T) or we can ignore the term (static calcula- tion). The B1 and B2 phases will be metastable as far as G (x; p, T) would remain a local minimum for the xB1 or xB2 sites. Of particular concern are the ceff 44 elastic constants, where ceff ij = 1 V ∂G ∂εi∂εj (17) and εi are the Voigt reductions of Lagrange’s defor- mation parameters (ε4 = 2ε23). -100 0 100 200 300 400 500 600 700 0 50 100 150 200 250 300 c44, c44 eff (GPa) P (GPa) B1 B1(eff) B2 B2(eff) c V. Lua˜ na 2005 (11)
12. Lanzhou 2005 B1-B2 transition in alkali halides We can determine a reaction path, (a, α) in Buerger’s mechanism, by minimizing G (x; p, T) for fixed values of α. We can see that the reaction paths have a fixed slope at both the B1 and B2 phases. This can be de- mostrated to be a direct consequence of the symmetry properties of the hessian matrix (H = ∇ ⊗ ∇G ) at both sites. 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 50 60 70 80 90 100 a (bohr) α (grados) 0 GPa 40 GPa 80 GPa 300 GPa c V. Lua˜ na 2005 (12)
13. Lanzhou 2005 B1-B2 transition in alkali halides -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 50 60 70 80 90 100 110 G (hartrees) α (grados) 0.00 GPa 40.00 GPa 80.00 GPa 300.00 GPa Along with the reaction path we also obtain a reaction energy curve, G(α), that determines both the stability of phases and the kinetics of the phase transition. The B1-B2 transition pressure is Ptr ≈ 80 GPa, but there is a significant energy barrier and thus the transformation will occur with hysteresis. c V. Lua˜ na 2005 (13)
14. Lanzhou 2005 B1-B2 transition in alkali halides 0 500 1000 1500 2000 2500 0 50 100 150 200 T(K); ∆G‡ (K) P (GPa) B1 → B2 B2 → B1 Hysteresis can be qualitatively studied with the help of a very simple kinetics model: dnB1 dP = ωB1e−∆G‡ B1→B2 /kBT nB1 − ωB2e−∆G‡ B2→B1 /kBT nB2, (18) where nB1 and nB2 are the concentrations of both phases, ∆G‡ are the energy barriers (different for the B1 → B2 and B2 → B1 ways except on equilibrium), and the ωi are parameters of this simple model. Equilibrium transition pressures at 300 K (GPa, experimental values in parenthesis) for the B1-B2 transition in alkali halides. Ptr F Cl Br I Li 252 79 94 113 Na 12 (23–24) 21 (26–30) 16 16 K 5.6 (1.73) 2.0 (1.9–2.1) 1.64 (1.77) 2.7 (1.8–1.9) Rb −0.2 (0.9–3.3) 0.3 (0.5) 0.1 (0.45) 0.8 (0.3–0.4) Cs −2.6 (2) −1.3 (B2) −1.5 (B2) −0.7 (B2) c V. Lua˜ na 2005 (14)
15. Lanzhou 2005 crystal adapted pseudopotentials (caPS) Crystal adapted pseudopotentials (caPS) The aiPI description can be used to produce a pseudopotential that represents frozen ions in any embedding problem. The form of the aiPI operators due to S is ˆ V S aiPI = ˆ V S eff + ˆ PS = − qS riS + V S nc (riS) − ˆ V S x + ˆ PS, (19) − ˆ V S x + ˆ PS = l l m=−l a,b |alm, S [−A(lab, S) + P(lab, S)] blm, S| , (20) where |alm, S is a STO function. The ˆ V S aiPI operator must be converted into a form able to be used in other electronic structure codes. The ECP form (effective core potential) is accepted in most quantum chemical codes: ˆ U = − qS riS + UL(r) + L−1 λ=0 λ µ=−λ |Yλµ Uλ−L(r) Yλµ| (21) where Ul(r) is a linear combination of GTO’s (Gaussian Type functions). The conversion from ˆ V S aiPI to ˆ U involves a delicate fit to pass from a combination of many STO’s to a combination of a few GTO’s. The details can be seen in Ref. [16]. c V. Lua˜ na 2005 (15)
16. Lanzhou 2005 crystal adapted pseudopotentials (caPS) The caPS Ul(r) functions for Mg+2 and O−2 in MgO are shown in the figures below. −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 arctan[r Uλ−L (r)] r (bohr) Mg+2:MgO Numerical Us−d Up−d Ud −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 arctan[r Uλ−L (r)] r (bohr) O−2:MgO Numerical Us−d Up−d Ud • The Ul(r) functions have very large positive and negative values. • Notice the use of the arctan(rUl) transformation to map the (−∞, +∞) range into [−π/2, +π/2]. • The ripples in the mid-distance range tell the atomic shell structure. • caPS can be used as an outer embedding component of a cluster-in-the-lattice model to prevent the fugue of electrons from the inner cluster to the outer ionic lattice. c V. Lua˜ na 2005 (16)
17. Lanzhou 2005 Cluster-in-the-lattice calculations Ib. Cluster-in-the-lattice calculations This is a technique for obtaining the electronic structure of defects on ionic materials. The system is divided into: • A cluster, C, of atoms surrounding the defect site. The quantum mechanical calculation corresponds to this region, that contains explicit electrons and basis sets. The size of the cluster dominates the computational effort. C can be further divided into – C1, an inner cluster whose geometry will be relaxed to accomodate the defect. – C2, a electronic buffer formed by atoms that keep their geometry fixed as in the host material. • The lattice, L, is formed by a collection of TES frozen groups. The lattice part close to the cluster can be represented with caPS. The Madelung potential created by the lattice ions is of very large range, but quantum chemical codes do not include the special techniques needed to converge the Madelung series. This can be replaced by a carefully designed set of point charges. Defect Geom. response Electronic buffer caPS Point charges Cluster C1 C2 Lattice M O O O O Mg Mg Mg Mg Mg Mg Mg Mg O O O O O O O O Mg O Mg O Mg Mg O Mg O Mg O Mg O O Cluster + caPS = Quantum region c V. Lua˜ na 2005 (17)
18. Lanzhou 2005 Cluster-in-the-lattice calculations Cluster models for MMg:MgO defect centers. The models for XO:MgO centers are equivalent through an exchange of the Mg and O ions. Ions position M-4.1.1 M-6.2.1 0 1M (0, 0, 0) C1 C1 1 6O (1/2, 0, 0) C1 C1 2 12Mg (1/2, 1/2, 0) caPS caPS 3 8O (1/2, 1/2, 1/2) caPS caPS 4 6Mg (1, 0, 0) caPS C2 5 24O (1, 1/2, 0) L L 6 24Mg (1, 1/2, 1/2) L L 7 12Mg (1, 1, 0) L L 8 6O (3/2, 0, 0) L caPS ... .... ... ... ... 14 6Mg (2, 0, 0) L caPS We will examine two different models. The M-6.2.1 model (image on the right) includes a C2 electronic buffer that is absent in the M-4.1.1 model. c V. Lua˜ na 2005 (18)
19. Lanzhou 2005 Cluster-in-the-lattice calculations The performance of a model can be checked against the self-embedding consistency test. In other words, a cluster-in-the-lattice model of the pure host lattice should reproduce the geometry of the crystal as faithfully as possible. -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 0.35 0.40 0.45 0.50 0.55 0.60 0.65 E-E(x1 =0.5) (hartree) x1 OO -4.1.1 model STO-3G 6-31G 6-311++G 6-311++G(d,p) The lack of a C2 electronic buffer produces a collapse of the Mg cluster ions towards the lat- tice region in this OO self-defect. 0.0 0.5 1.0 1.5 2.0 0.35 0.40 0.45 0.50 0.55 0.60 0.65 E-E(x1 =0.5) (hartree) x1 6.2.1 cluster model MgMg center OO center The self-embedding is attained on both, MgMg and OO, defects when the M-6.2.1 model is used. Furthermore, the small lattice relaxation observed in the self-embedding test can be used to correct the relaxation of true defect calculations. c V. Lua˜ na 2005 (19)
20. Lanzhou 2005 Cluster-in-the-lattice calculations Center ∆Rrelax (˚ A) Erelax (eV) ωa1g (cm−1) Mg× Mg +0.030 −0.060 607 Al• Mg −0.147 −0.901 638 Be× Mg −0.041 −0.006 551 Ca× Mg +0.079 −1.085 700 Sr× Mg +0.133 −2.826 758 Ba× Mg +0.193 −6.734 855 Na Mg +0.145 −2.191 656 Cation site defect: • Charge rule: positive charged defects tend to relax inwards and negative ones outwards. • Impurity size rule: as the impurity cation radius increases an outwards push is added. Anion site defect: • Neutral F centers suffer a negligible relax- ation. A strong outwards shift occurs when electrons are removed from the F center. • Charge rule: positive defects tend to relax outwards and negative ones inwards. • Impurity size rule: as the impurity cation radius increases an outwards push is added. Center ∆Rrelax (˚ A) Erelax (eV) ωa1g (cm−1) O× O −0.006 −0.005 678 F+2 +0.208 −5.667 763 F+ +0.094 −1.196 768 F• O +0.100 −1.112 686 F +0.000 −0.003 597 S× O +0.093 −1.193 756 Se× O +0.122 −2.162 761 N O −0.072 −0.730 688 c V. Lua˜ na 2005 (20)
21. Lanzhou 2005 The Quantum Theory of Atoms in Molecules II. The Quantum Theory of Atoms in Molecules (AIM) • The topology of the electron density, ρ(r), provides a unique partition of real space into atomic basins. • Basins are separated by zero flux surfaces: ∇ρ(rs) · n(rs) = 0 . • Each basin contains a single maximum of ρ(r) that coincides with the nucleus of an atom and it is the source of all ∇ρ(r) gradient lines. • The only gradient line crossing a zero flux surface (Inter Atomic Surface or IAS) represents the bond path between the two nuclei that the IAS separates. ∇ρ(r) vector field for the C2H4 molecule. Field lines are in red, bond paths in blue, and IAS lines are the thick green lines. c V. Lua˜ na 2005 (21)
22. Lanzhou 2005 The Quantum Theory of Atoms in Molecules • All the sources and drains of the gradient lines are the critical points ∇ρ(rc) = 0. Those points can be classified according to the eigenvalues of the curvature matrix: H (Hξζ = ∂2ρ/∂ξ∂ζ; ξ, ζ = {x, y, z}). In other words, critical points can be classified according to the dimensions of its attraction and repulsion basins: Type Where attrac. repul. Maximum n: nucleus 3D 0D Saddle-1 b: bond cp. 2D 1D Saddle-2 r: ring cp. 1D 2D Minimum c: cage cp. 0D 3D • Atomic basins in condensed matter are isomorphic to convex polyhedra: each face is an IAS created by a single bond cp, edges are the attraction basins of ring cp’s, and vertices correspond to minima. • Topological restrictions: n−b+r −c = 0, n, c ≥ 1, b, r ≥ 3 per unit cell (Morse relationships); b + c = r + 2 for each atomic basin (Euler relationship). LiI basins KCaF3 basins c V. Lua˜ na 2005 (22)
23. Lanzhou 2005 The Quantum Theory of Atoms in Molecules The zero flux surface condition defining the atomic basins guarantees that all quantum mechanical operators are locally well defined, are hermitian, and can be integrated on each basin to produce the atomic contribution to the properties of the whole system. For instance: • electronic population and fluctuation: N Ω = Ω ρ(r) dr, σ2 N (Ω) = N2 Ω − N 2 Ω ; • multipolar moments: Nlm Ω = Ω rlSlm(θ, φ)ρ(r) dr; • electron kinetic energy: − 2 4me Ω dr [∇2+∇ 2] ρ1(r ; r) = K(Ω) = 2me Ω dr [∇ ·∇] ρ1(r ; r) = G(Ω) − 2 4me S(Ω) dS ∇ρ(r)· n(r). = 0 This equation is a consequence of ∇2(Φ∗Φ) = (∇ 2+∇2+2∇ ·∇)(Φ∗Φ). Notice that K(Ω) = G(Ω) unless Ω is delimited by zero flux surfaces. Basin contributions are strictly additive, O = Ω O Ω , and they can be transferred from one compound to another as far as the shape of the basin do not change much. c V. Lua˜ na 2005 (23)
24. Lanzhou 2005 Some crystal prototypes Some crystal prototypes Diamond: shared-shell interactions. ρb = 0.2659 e/bohr3 ∇2ρb = −0.9044 e/bohr5 qC = 0 Li2O: closed-shell bonding. RLi = 1.402 bohr ρb = 0.0246 e/bohr3 ∇2ρb = +0.2076 e/bohr5 qLi = 0.90 qO = −1.80 Li: metal with NNM. 2 Li , 12 NNM in the unit cell Volume: 24% Li, 76% NNM q: +0.825 Li, −0.137 NNM Radio: 2.023 Li, 1.293 NNM Li-NNM nnm-NNM ρb 0.0072 0.0075 ∇2ρb 0.00526 −0.00024 c V. Lua˜ na 2005 (24)
25. Lanzhou 2005 Classification of covalent, ionic and metallic solids Classification of covalent, ionic and metallic solids • van Arkel-Ketelaar diagram is based on electronegativity: |χA−χB| vs. χi . • Appropriate ρ indices? – charge transfer: QΩ/OSΩ , – flatness: ρmin/ρmax b , – molecularity: µ = (ρmax b − ρmin b )/ρmax b (if ∇2ρmax b × ∇2ρmin b < 0 and µ = 0 otherwise). • Figure based on HF-LCAO crystal calcula- tions. f c 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 C I M Al Ar,C BAs BP Be CS2 Cl2 ,N2 ,Ne Cu,α−Fe γ−Fe GaAs GaP GaN Ge K Li Mg MgCu2 MgO MgS N2 O4 Na ZnS 0.00 0.05 0.10 0.15 0.60 0.70 0.80 0.90 1.00 AlAs AlN AlP CaF2 CaO Li2 O Mg2 Si NaCl NaF SO2 Si Si3 N4 TiO2 ZnO ZrO2 c V. Lua˜ na 2005 (25)
26. Lanzhou 2005 Electron density topology in alkaline metals Electron density topology in alkaline metals • Electron density is very flat: f = 96% (Li), 95% (Na, K), 91% (Rb), and 88% (Cs). • Common tendency towards topological change. • The topology is labile, at difference from ionic and covalent crystals. • The many topologies can be classified: B2 : first and second metal neighbors bonded. B1 : only first metal neighbors bonded. ML : non-nuclear maxima (NNM) at the M-M midpoint. Mg : twin NNM on the M-M line. Mi : NNM on interstitial positions. • Topologies follow a common sequence on compression: B2 → B1 (→ M) → B2 · · · . • Li shows NNM on a wide range of geometries. • Upon small compression Na and K show NNM for a small range of geometries. • Rb and Cs lack NNM. • The promolecular model or HF calculations on M2 and M4 clusters explain the results. c V. Lua˜ na 2005 (26)
27. Lanzhou 2005 Electron density topology in alkaline metals The topology of Li is quite labile c V. Lua˜ na 2005 (27)
28. Lanzhou 2005 Electron density topology in alkaline metals −7.55 −7.54 −7.53 −7.52 −7.51 −7.50 −7.49 −7.48 −7.47 −7.46 4 5 6 7 8 9 84 86 88 90 92 94 96 98 100 Total Energy (hartree) flatness, f (%) a (bohr) topologies Mi ML Mg B1 B2 aeq Li c V. Lua˜ na 2005 (28)
29. Lanzhou 2005 Electron density topology in alkaline metals 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 0 20 40 60 80 100 Q (|e|) V / V0 (%) a / aeq QLi QNa QK VLi VNa VK c V. Lua˜ na 2005 (29)
30. Lanzhou 2005 Universal sequence of bonding regimes Universal sequence of bonding regimes • Every couple of atoms follows a universal sequence of bonding regimes which is entirely controlled by the interatomic distance. R ↓: closed valence shell → shared val. shell → (NNM) → closed inner shell . . . • This sequence is caused by the intrinsic electronic shell structure of atoms, which is largely conserved in general compounds. • Diatomic molecules serve well to predict the behavior of a bonding pair of atoms in a larger molecular or cristalline environment. • The promolecular model and, to a lesser extent, its exponential tails simplification provide specific predictions for the bond CP electron density and Laplacian, that agree qualitatively and explain the trends actually observed on state-of-the-art quantum mechanical calculations. (Ex.) At the A-A midpoint: ρ = 2ρA (rA ), ρ⊥ = 2ρA (rA )/rA , ∇2ρ = 2ρA (rA ) + 4ρA (rA )/rA c V. Lua˜ na 2005 (30)
31. Lanzhou 2005 Universal sequence of bonding regimes 0.001 0.01 0.1 1 10 100 1000 1 2 3 4 5 |∇2ρb | (e/bohr5) d (bohr) N2 ∇2ρb >0 ∇2ρb <0 ∇2ρb >0 Promolecular HF CISD DFT-BPW91 0.01 0.1 1 10 1 2 3 4 5 ρb (e/bohr3) c V. Lua˜ na 2005 (31)
32. Lanzhou 2005 Universal sequence of bonding regimes fpLAPW/GGA (wien): diamond, graphite, CaC2. B3LYP/6-311G(3df,p) (gaussian): C2, ethane, ethylene, acetylene, benzene, anthracene, alene. 0.01 0.1 1 2 3 4 5 6 7 bond density, ρb (e/bohr3) interatomic distance, d (bohr) C−C bond NNM ∇2 ρb < 0 ∇2 ρb > 0 C2 (3Πu ) crystals molecules −4 −3 −2 −1 0 1 2 3 4 5 6 7 ∇2 ρb (e/bohr5) d (bohr) c V. Lua˜ na 2005 (32)
33. Lanzhou 2005 Partition of thermodynamical properties Partition of thermodynamical properties • The partition of atomic compressibility is simple: κ = 1 B = − 1 Ω ∂Ω ∂p , Ω = i Ωi, fi = Ωi/Ω, κi = 1 Bi = − 1 Ωi ∂Ωi ∂p =⇒ κ = i fiκi. • This provides a very practical analytical tool. A well known rule of thumb says that oxide spinels have a bulk modulus of ≈ 200 GPa. Why? AB2O4 B (GPa) BA BB BO fO MgAl2O4 215.2 282.1 331.9 201.6 0.8127 MgGa2O4 211.2 261.2 283.9 196.1 0.7486 ZnAl2O4 214.8 246.0 335.2 203.3 0.7690 ZnGa2O4 213.3 241.2 308.6 195.7 0.7070 Most of the crystal volume is occupied by the oxide ion. Therefore κ ≈ κO. • This has geophysical consequences, as most crust and mantle minerals are oxide-rich. Is the average compressibility of Earth mantle dominated by κO(p, T)? c V. Lua˜ na 2005 (33)
34. Lanzhou 2005 Acknowledgements Acknowledgements This work has been done with support from the Spanish Ministerio de Educaci´ on y Ciencia under project BQU2003-06553. Oviedo’s team: Lorenzo Pueyo Casaus, Margarita Bermejo Villanueva, V´ ıctor Lua˜ na Cabal, Evelio Francisco Migu´ elez, Manuel Fl´ orez Alonso, Jos´ e Manuel Recio Mu˜ niz, ´ Angel Mart´ ın Pend´ as, Miguel ´ Alvarez Blanco, Aurora Costales Castro, Paula Mori-S´ anchez. Coworkers: Ravi Pandey (Michigan Tech. Univ., Houghton, MI, USA), Lev N. Kantorovich (King’s College, London, UK), W. H. Adams (Rutgers Univ., NJ, USA), R. F. W. Bader (McMaster Univ., Ontario, Canada). c V. Lua˜ na 2005 (34)
35. Lanzhou 2005 References References [1] R. McWeeny, Proc. R. Soc. London Ser. A 253, 242 (1959). [2] S. Huzinaga and A. A. Cantu, J. Chem. Phys. 55, 5543 (1971). [3] S. Huzinaga, D. McWilliams, and A. A. Cantu, Adv. Quantum Chem. 7, 187 (1973). [4] V. Lua˜ na, M. Bermejo, M. Fl´ orez, J. M. Recio, and L. Pueyo, J. Chem. Phys. 90, 6409 (1989). [5] V. Lua˜ na and L. Pueyo, Phys. Rev. B 39, 11093 (1989). [6] V. Lua˜ na and L. Pueyo, Phys. Rev. B 41, 3800 (1990). [7] E. Francisco, A. Mart´ ın Pend´ as, and W. H. Adams, J. Chem. Phys. 97, 6504 (1992). [8] A. M. Pend´ as, E. Francisco, and J. M. Recio, J. Chem. Phys. 97, 452 (1992). [9] V. Lua˜ na, J. M. Recio, and L. Pueyo, Phys. Rev. B 42, 1791 (1990). [10] M. A. Blanco, E. Francisco, and V. Lua˜ na, Comput. Phys. Commun. 158, 57 (2004), source code distributed by the CPC program library: http://cpc.cs.qub.ac.uk/summaries/ADSY. [11] J. M. Recio, A. M. Pend´ as, E. Francisco, M. Fl´ orez, and V. Lua˜ na, Phys. Rev. B 48, 5891 (1993). [12] A. M. Pend´ as, V. Lua˜ na, J. M. Recio, M. Fl´ orez, E. Francisco, M. A. Blanco, and L. N. Kantorovich, Phys. Rev. B 49, 3066 (1994). [13] M. ´ Alvarez Blanco, Tesis doctoral, Universidad de Oviedo, 1997, director: V´ ıctor Lua˜ na Cabal. [14] M. A. Blanco, A. Costales, A. M. Pend´ as, and V. Lua˜ na, Phys. Rev. B. 62, 12028 (2000). [15] M. Fl´ orez, J. M. Recio, E. Francisco, M. A. Blanco, and A. M. Pend´ as, Phys. Rev. B 66, 144112/1 (2002). [16] V. Lua˜ na, J. M. Recio, A. M. Pend´ as, M. A. Blanco, L. Pueyo, and R. Pandey, Phys. Rev. B 64, 104102 (2001). [17] M. A. Blanco, A. Costales, V. Lua˜ na, and R. Pandey, Appl. Phys. Lett. 85, 4376 (2004). [18] R. F. W. Bader, Atoms in Molecules. A Quantum Theory (Oxford University Press, Oxford, 1990). [19] P. Mori-S´ anchez, Tesis doctoral, Universidad de Oviedo, 2002, directores: V´ ıctor Lua˜ na Cabal y Angel Mart´ ın Pend´ as. [20] V. Lua˜ na, A. Costales, P. Mori-S´ anchez, M. A. Blanco, and A. Mart´ ın Pend´ as, Acta Cryst. A. 60, 434 (2004). [21] P. Mori-S´ anchez, A. Mart´ ın Pend´ as, and V. Lua˜ na, J. Am. Chem. Soc. 124, 14721 (2002). [22] V. Lua˜ na, P. Mori-S´ anchez, A. Costales, M. A. Blanco, and A. Mart´ ın Pend´ as, J. Chem. Phys. 119, 6341 (2003). [23] A. Costales, M. A. Blanco, P. Mori-S´ anchez, V. Lua˜ na, and A. Mart´ ın Pend´ as, J. Phys. Chem. A 108, 2794 (2004). [24] A. M. Pend´ as, A. Costales, M. A. Blanco, J. M. Recio, and V. Lua˜ na, Phys. Rev. B. 62, 13970 (2000). c V. Lua˜ na 2005 (35)