Electronic structure calculations for magnetic systems Manuel Richter (IFW Dresden) 1. Hohenberg-Kohn-Sham theory & Local Density Approximation 2. Tight-binding approach and chemical binding in a nutshell 3. Exchange, the root of condensed matter magnetism 4. Binding meets exchange: applications – Typeset by Foil T EX – 1
1. Hohenberg-Kohn-Sham theory & Local Density Approximation H. Eschrig, The Fundamentals of Density Functional Theory , Teubner-Texte zur Physik, Vol. 32, Teubner, Stuttgart 1996, ISBN 3-8154-3030-5. M. Richter, Density Functional Theory applied to 4f and 5f Elements and Metallic Compounds , Handbook of Magnetic Materials (Ed. K.H.J. Buschow), Vol. 13, Elsevier, Amsterdam 2001, pp. 87-228, ISBN 0-444-50666-7. odinger Hamiltonian ˆ Our starting point is the non-relativistic Coulomb-Schr¨ H with fixed positions of the nuclei, generating the potential v nuc : N − ∆ i 2 + v nuc ( r i ) + 1 1 ˆ � � = ˆ T + ˆ V + ˆ H = U , 2 | r i − r j | i j � = i ( ˆ H − E ν ) ψ ν ( r 1 , σ 1 ; . . . ; r N , σ N ) = 0 , with many-particle wave function ψ ν for N electrons at coordinates ( r i , σ i ) . There is no spin-dependent interaction. Where do magnetic states come from? – Typeset by Foil T EX – 2
Numerical solutions for ψ ν can be obtained up to N ≈ 10 . For N > 100 , the stationary states ψ ν cannot be resolved in general. Example: an N-particle Ising system with stochastic interaction has 2 N − 1 different levels. Mean level distance: ∆ E ≈ E 0 / 2 N − 1 . For N = 100 and E 0 = 10 eV, resolution of a single level needs 10 6 years. Thus, ψ ν has no meaning for N > 100 . H = � 100 j>i J ij s z i s z j Exception: the ground state, ψ 0 , has a meaning: (i) it serves as reference for quasi -stationary (spin or charge) excitations; (ii) the ground-state energy E 0 [ v nuc [ R s ]] allows to determine stable structures with nuclear positions [ R s ] . – Typeset by Foil T EX – 3
How can E 0 [ v nuc ] be calculated? Hohenberg and Kohn [PR 136 (1964) B864]; Levy [PRA 26 (1982) 1200]; Lieb [Int. J. Quant. Chem. XXIV (1983) 243]: � �� � � ψ � ψ | ˆ ψ n � ψ n | ˆ T + ˆ d 3 r v nuc n + min d 3 r n = N � E 0 [ v nuc ] = min H | ψ � = min U | ψ n � , � n � where ψ n are all wave functions that generate a density n ( r ) . Definition: ψ n � ψ n | ˆ T + ˆ min U | ψ n � =: F [ n ] =: T s [ n ] + E H [ n ] + E xc [ n ] . Here, the exchange-correlation energy E xc [ n ] contains all contributions which are not included in the other two terms, (i) the mean-field Hartree energy: d 3 rd 3 r ′ n ( r ) n ( r ′ ) E H = 1 � � , | r − r ′ | 2 – Typeset by Foil T EX – 4
(ii) the kinetic energy of a model system of non-interacting electrons with density n : N � T s [ n ] = � φ i | − ∆ / 2 | φ i � . i The φ i can be obtained from � − ∆ � 2 + v nuc ( r ) + v H ( r ; [ n ]) + v xc ( r ; [ n ]) φ i = ε i φ i , with n = � N i φ i φ ∗ i and φ i being the N lowest single-particle eigenstates in a corresponding effective potential, � d 3 r ′ n ( r ′ ) / | r − r ′ | + δE xc [ n ] /δn . v eff = v nuc + v H + v xc = v nuc + Kohn and Sham [PR 140 (1965) A1133]. – Typeset by Foil T EX – 5
What does the exchange-correlation potential v xc mean? (i) Exchange “x”: ψ ( r 1 , σ 1 ; . . . ; r 2 = r 1 , σ 2 = σ 1 ; ... ) = 0 for Fermions; (ii) Correlation “c”: ψ ( r 1 , σ 1 ; . . . ; r 2 = r 1 , σ 2 ; ... ) < ψ ( r 1 , σ 1 ; . . . ; r 2 � = r 1 , σ 2 ; ... ) due to Coulomb repulsion. The total energy is lowered due to the xc- hole around each electron. Thus, v xc is always attractive , while v H is repulsive. The xc potential corrects both the mean-field treatment of the Coulomb interaction and the single-particle treatment of the kinetic energy. – Typeset by Foil T EX – 6
Up to this point, the theory is exact. However, v xc is in general not known and has to be approximated. There exists a multitude of different approximations for the xc potential: LDA, GGA, metaGGA, etc. Most of these approximations do not contain free parameters. However, they should not be called “ ab initio ”, since they do not solve the general Coulomb-Schr¨ odinger Hamiltonian (this is attempted by quantum chemical methods). Rather, these methods solve a certain model, approximating ˆ H . For the homogeneous interacting electron gas, v hom xc ( n ) has been obtained numerically by Ceperley and Alder [PRL 45 (1980) 566]. Local Density Approximation (LDA): v xc ( r ; [ n ]) ≈ v hom xc ( n ( r )) . ∝ n 1 / 3 and E hom ∝ n 4 / 3 . This is similar to what is found for the Roughly, v hom xc xc exchange energy in the Hartree-Fock approximation. The mostly used parameterization of LDA has been proposed by Perdew and Wang [PRB 45 (1992) 13244]: v PW92 ( n ) ≈ v hom xc ( n ) . xc – Typeset by Foil T EX – 7
Remarks: LDA, GGA and similar approxiations are rather successful in the calculation of structural properties, elasticity, phonons. The interpretation of Kohn-Sham single-particle energies in terms of charge excitations is justified only in so-called “weakly correlated” materials. This notation is only loosely related with the types of correlations we discussed. What is meant is that charge excitations in such materials are well screened by valence electrons. v A number of codes is available which solve the non-linear integro-differential equations with different Kohn−Sham methods. They contain typically 10 5 lines of source self−consistent n code and need 10-20 person years development. iterations Total energy deviations between good codes: ≈ 1 meV/atom. φ, ε – Typeset by Foil T EX – 8
2. Tight-binding approach and chemical binding in a nutshell J. Singleton, Band Theory and Electronic Properties of Solids , Oxford Master Series in Condensed Matter Physics, Oxford University Press, Oxford 2006, ISBN 0-19-850644-9. C. Cohen-Tannoudji, B. Diu, and F. Lalo¨ e, Quantum Mechanics , Vol. II, Hermann, Paris 1977, ISBN 0-471-16435-6. Up to now, nothing has been told about the character of the Kohn-Sham single-particle orbitals φ i . We therefore proceed from the most simple case of atomic orbitals via molecular orbitals to spatially extended Bloch states. – Typeset by Foil T EX – 9
The H 2 molecule, considered in LDA and adiabatic approximation: v H 2 nuc = − 1 / | r − R 1 | − 1 / | r − R 2 | ; N = 2 . Minimalistic Ansatz for φ : φ = c 1 ϕ H 1 s ( | r − R 1 | ) + c 2 ϕ H 1 s ( | r − R 2 | ) =: c 1 ϕ 1 + c 2 ϕ 2 ( − ∆ / 2 + v H 2 eff − ε )( c 1 ϕ 1 + c 2 ϕ 2 ) = 0 This yields the 2 × 2 matrix equation � c j ( − ∆ / 2 + v H 2 eff − ε ) ij = 0 j with ( ϕ i | ˆ A | ϕ j ) =: A ij . – Typeset by Foil T EX – 10
Separate the atomic state energy, ε H 1 s , from the “crystal field”, ∆ ε : ( − ∆ / 2 + v H 2 eff ) ii + ( v H 2 eff ) ii = ( − ∆ / 2 + v H eff − v H eff ) ii =: ε H 1 s + ∆ ε ; ( i � = j ) . Tight-binding approximation: ( − ∆ / 2 − ε ) ij ≈ 0 ; ( i � = j ) . Hopping integral: t =: ( v H 2 eff ) ij ; ( i � = j ) . ε This yields the solutions: ε σ * ε H 2 = ε H 1 s + ∆ ε − t , c 1 = c 2 ; bonding ε H 2 antibonding = ε H 1 s + ∆ ε + t , c 1 = − c 2 . ∆ε t ε 1s ε 1s Remarks: ε σ v eff needs self-consistent calculations. As a result, the binding energy (6.6 eV per H H 2 H H 2 ) is smaller than 2( t − ∆ ε ) = 7 . 5 eV. – Typeset by Foil T EX – 11
Toward extended systems: Bloch’s theorem. X-ray diffraction shows that the charge distribution in single crystals is periodic. This means, that also v LDA is periodic: v LDA ( r + R ) = v LDA ( r ) , if R is a eff eff eff lattice vector. Then, the Kohn-Sham states can most conviniently be chosen as Bloch states in this periodic potential: φ k ( r + R ) = e i kR φ k ( r ) . The charge density evaluated from this set of states has the same symmetry as the initial potential. Thus, the initial symmetry is always kept in the Kohn-Sham self-consistent cycle. (This does not always hold, if spin-orbit coupling is considered.) – Typeset by Foil T EX – 12
Infinite H-chain: v nuc ( r ) = � integer ( − 1 / | r − R m | ) ; R m = m (0 , 0 , a ) . m One-band ansatz: integer integer 1 1 � � e − i kR m ϕ 1 s ( | r − R m | ) =: e − ikma ϕ m . √ √ φ k ( r ) = N N m m Nearest-neighbor tight-binding: − ( v eff ) m,m +1 =: t ; ( v eff ) m,n ≈ 0 ( | m − n | > 1) ; ( − ∆ / 2 − ε k ) m,n ≈ 0 ( m � = n ) provides ∆ ε = ( v chain − v H ε k = ε 1 s + ∆ ε − 2 t cos( ka ) ; eff ) 00 . eff This dispersion has a period of 2 π/a and a band width of 2 t . – Typeset by Foil T EX – 13
3. Exchange, the root of condensed matter magnetism S. Blundell, Magnetism in Condensed Matter , Oxford Master Series in Condensed Matter Physics, Oxford University Press, Oxford 2006, ISBN 0-19-850591-4. J. K¨ ubler, Theory of Itinerant Electron Magnetism , International Series of Monographs on Physics, Vol. 106, Oxford Science Publications, Clarendon Press, Oxford 2000, ISBN 0-19-850028-9. We step back to the case of a free atom and consider an incompletely filled atomic shell with l � = 0 . One electron: trivial case with s = 1 / 2 . Two electrons: Is the ground state a singlet S = 0 ( ↑↓ ) or a triplet S = 1 ( ↑↑ ) ? – Typeset by Foil T EX – 14
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