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The Augmented Plane Wave Method (APW)

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The Augmented Plane Wave Method (APW) New approach to the problem of electron semicore states in LAPW band structure calculations А.V. Nikolaev (Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University and Moscow Institute of Physics and Technology) collaboration with D. Lamoen (EMAT, Department of Physics, UA) and B. Partoens (CMT group, Department of Physics, UA) 30 august 2016 Outline Electric Field Gradient (EFG) 1. Where needed 2. Definitions 3. Relations between the electric field gradient and the expansions of the full potential around a nucleus Ab initio solid state calculation of solids (LAPW method) 1. Augmented plane wave method (APW) 2. Linearized augmented plane wave method (LAPW). 3. Full potential (FLAPW). Treatment of semicore states in the FLAPW method. J. Chem. Phys. v. 145, 014101 (2016) Experimental measurements (TDPAC-spectroscopy): А.V. Tsvyashchenko, L.N. Fomicheva (Vereshchagin Institute for High Pressure Physics, RAS) G.К. Ryasnyj (Skobeltsyn Institute of Nuclear Physics Lomonosov Moscow State University) А.I. Velichkov, А.V. Salamatin, О.I. Kochetov, D.А. Salamatin, (Joint Institute for Nuclear Research, Dubna) M. Budzynski (M.Curie–Sklodowska University, Lublin) Time dependent perturbed angular correlation (TDPAC) method Two -detectors for Nuclear states Propagation vectors two directions of in the - cascade of two -quanta - correlations The emission probability of two -quanta TDPAC method : probe nuclei in crystal Time evolution of the intensity of the -emission in external electric or magnetic field The probe nucleus experiences the external electric field through the electric field gradient (EFG) which splits nuclear states (the quadrupolar hyperfine interaction) The electric field gradient (EFG) at the nucleus The electric field gradient (EFG) is a tensor quantity: 2V V where i,,, j x y z ij, ij 1 3 The nuclear HQV quadrupolar interaction: Q6 i,, j i j ij,1 If = 0 then Vxx = Vyy, and we have the axial symmetry EFG tensor can be reduced to the diagonal form: Vzz,Vxx,Vyy Therefore, EFG is fully defined by the largest component: Vzz and the asymmetry parameter: ()/VVVxx yy zz The quadrupolar frequency in TDPAC eQV/ h is defined as Q zz Electric field gradient (EFG) at a crystal site (nucleus) The tensor of electric field gradient is defined by means of the second derivatives at the nucleus The total electron density is expanded in terms of symmetrized angular functions (harmonics) The symmetrized angular function is a linear combination of spherical harmonics with the same angular index l The full crystal potential (a sum of Coulomb and exchange contribution) can also be expanded in terms of symmetry adapted functions: The electric field gradient (EFG) is related with the full potential coefficient with l = 2 ! Electric field gradient (EFG) at a crystal site The electric field gradient (EFG) can be obtained from the full potential expansion in terms of symmetry adapted functions Notice the following asymptotic behavior of the potential component with l = 2 at the nucleus (at ) Conclusion: To obtain (to compute) the tensor of the electric field gradient at crystal site (nucleus) one has to find first the expansion of the electron density about the nucleus: and to find the expansion of the full potential about the nucleus For that we need ab initio calculations Ab initio Ab initio (or "from first principles") — A calculation is said to be ab initio if it relies on basic and established laws of nature without additional assumptions or special models. The term ab initio was first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene. In solid state physics ab initio implies computational methods based on the density functional theory (DFT). Electronic band structure calculation (general notes) We solve the Schrödinger (Dirac) equation: The periodic potential The translational property of the wave function: Here k is the wave vector, or quasi-momentum The wave function satisfies the Bloch law: The wave function is expanded in a basis set: Varying the coefficients Ci, we obtain the secular equation: Here N is the number of basis functions, Н is the matrix of the Hamiltonian, O is the matrix of the overlap Crystal potential Bound states and full potential of free atoms Core and band states and full potential of atoms in a crystal Partition of the crystal space: 1. Inside spheres around nuclei – a set of atomic spheres (S) 2. Interstitial space between the spheres (I) Construction of basis functions Crystal basis state wave function (satisfies the Bloch theorem): The radial part inside a sphere is atomic-like: where atomic-like solution plane wave The augmented plane wave method (APW) Crystal basis state wave function (satisfies the Bloch theorem): The radial part inside a sphere is atomic-like: The augmented plane wave method was formulated by Slater in 1937. However it has certain shortcomings: 1. The first derivative of the basis function changes discontinously at the sphere surface 2. The basis function depends nonlinearly on its energy Linearization of the augmented plane wave method – the LAPW method Crystal basis state wave function (satisfies the Bloch theorem): The radial part inside the MT sphere: Notice that If one assumes the linear behavior: ul()()() E E u l E Eu l E then This is the procedure of linearization for the APW method ! Linear augmented plane wave method (LAPW) The idea of linearization was put forward by Marcus and Andersen, and also by Koelling and Arbmann in 1975 and 1976. As a result the radial part of basis function has the following form From the continuity of the wave function and its derivative we obtain equations for the coefficients AL and BL: It is also very important that the basis function does not depend on the final energy, that is, there is no non-linearity Linear augmented plane wave method with the general potential or full potential (FLAPW, FP-LAPW) In 1979-1981 on the basis of LAPW a new method was developed which employed the potential of general form (full potential) (FLAPW, FP-LAPW). The expansion of the electron density in the interstitial region The expansion of electron density inside the МТ-spheres Angular symmetrized functions (symmetry adapted surface harmonics) Outline Electric Field Gradient (EFG) 1. Where needed 2. Definitions 3. Relations between the electric field gradient and the expansions of the full potential around a nucleus Ab initio solid state calculation of solids (LAPW method) 1. Augmented plane wave method (APW) 2. Linearized augmented plane wave method (LAPW). 3. Full potential (FLAPW). Treatment of semicore states in the FLAPW method. J. Chem. Phys. v. 145, 014101 (2016) The semicore electron state problem Electron band is described by continuum of electron states (of finite width) Core (atomic) states are usually described as discrete electron states (of zero width) Valence electron band states However, sometimes so called semicore electron states appear Semicore electron band states The semicore electron state problem Electron band is described by continuum of electron states (of finite width) Core (atomic) states are usually described as discrete electron states (of zero width) If we check the orthogonality of valence and core states we get: Therefore, the orthogonality of states implies: The semicore electron state problem The orthogonality of core states implies: The condition is violated for semicore states The problem can be solved if one considers two radial solutions (with the same l) inside the sphere: These radial solutions mergers in a single plane wave in the interstitial region: Solution of the semicore state problem The problem can be solved if one considers two radial solutions (with the same l) inside the sphere: which mergers in a single plane wave in the interstitial region: The condition of continuity of the wave function and its first derivative on the sphere surface: Solution of the semicore state problem The problem can be solved if one considers two radial solutions (with the same l) inside the sphere: The condition of continuity on the sphere surface: Solution with supplementary The general functions at Canonical solution: solution: Solution of the semicore state problem The general solution can be rewritten in the following convenient form: Here the 1st function stands for the standard solution: with two additional (supplementary) functions: Solution of the semicore state problem The general solution: implies the appearance of two supplementary basis functions which satisfies the following boundary conditions: two additional basis functions can be considered as tight binding basis functions: Solution of the semicore state problem two supplemented basis state functions 1st and 2nd radial functions for fcc La: two additional basis functions can be considered as tight binding basis functions: Example: electron band structure of lanthanum The electron band structure of body centered cubic lattice of lanthanum (La) Solution of the semicore state problem, which is used now: FLAPW + LO FLAPW + LO method: LO stands for local orbital. In FLAPW + LO method only the first basis function (LO) is used LAPW + LO method was put forward by D. Singh in PRB 43, 6388 (1991). LAPW+LO method is understood as a procedure giving additional variational freedom through an increase of the number of basisfunctions. In our approach in comparison with LAPW+LO, the number of supplemented basis functions in doubled Example: electron band structure of lanthanum Energy shifts of electron bands of bcc La along high symmetry lines of the Brillouin zone (solid lines stand for FLAPW++, dashed lines for FLAPW+LO). (a) above the Fermi level (EF=0), (b) semicore (5p) band. Example: Cadmium (Cd) The electron band structure of the hexagonal close packed cadmium (Cd) Example: Cadmium (Cd) Energy shifts of 4d-electron bands of hexagonal close packed Cd along high symmetry lines of the Brillouin zone (solid lines stands for FLAPW++, dashed lines for FLAPW+LO, EF=0) Test calculations of fcc-La: FLAPW++ versus FLAPW+LO Total energy of fcc-La (Etot, in eV) for various basis sets relative to the FLAPW++ value for RMTKmax = 10.
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