Atoms per Cell

The LMTO method has the computational speed and flexibility needed to perform calculations of electron states in molecules and compounds. Therefore in the present chapter we shall generalise the LMTO formalism purely within the atomic-sphere approximation to include the case of many inequivalent atoms per cell. The LMTO method is based on the variational principle in conjunction with energy-independent muffin-tin orbitals but, in addition to this approach, we have also considered the tail-cancellation principle which led to the KKR-ASA condition (2.8). Since the latter has conceptual advantages, we apply the tail-cancellation principle to the simplest possible case of more than one atom, namely the diatomic molecule. After that, we turn to crystalline solids and generalise or sometimes rederive the important equations of LMTO formalism. Hence, in addition to giving the LMTO equations for many atoms per cell, the present chapter may also serve as a short and compact presentation of the crystal-structure-dependent part of LMTO formalism. The potential-dependent part is treated in Chap.3. In the final sections are listed the modifications needed to calculate ground-state properties for materials with several atoms per cell. 

The present chapter is based on the papers by Andersen and co-workers 

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Given the efficiency of VASP, electronic structure calculations with or without a static optimization of the atomic structure can now be performed on fast workstations for systems with a few hundred inequivalent atoms per cell (including transition-metais and first row elements). Molecular dynamics simulationsextending over several picoseconds are feasible (at tolerable computational effort) for systems with 1000 or more valence electrons. As an example we refer to the recent work on the metal/nonmetal transition in expanded fluid mercury[31]. 

The amount of boron required for BNCT can be estimated using the neutron capture cross sections, which are atomic properties, and thus pertain to the number, and not the mass, of the atoms present. Conservative estimates for successful therapy result in boron concentrations of around 20 ppm in tumor tissue, to at least match the dose liberated by neutron capture reactions in the other elements of biological tissue. This would correspond to around 109 boron-10 atoms per cell, assuming that one cell corresponds to 10-9 g. 

FAU type zeolites exchanged with many different cations (Na, K, Ba, Cu, Ni, Li, Rb, Sr, Cs, etc.) have been extensively studied. The unit cell contents of hydrated FAU type zeolite can be represented as M,j(H20)y [A Sii92 0384] -FAU, where x is the number of A1 atoms per unit cell and M is a monovalent cation (or one-half of a divalent cation, etc.). The number of A1 atoms per cell can vary from 96 to less than 4 (Si/Al ratios of 1 to more than 50). Zeolite X refers to zeolites with between 96 and 77 A1 atoms per cell (Si/Al ratios between 1 and 1.5) and Zeolite Y refers to zeolites with less than 76 A1 atoms per cell (Si/Al ratios higher than 1.5). 

Diffusion coefficients at different Xe loadings were found to be comparable to those found in the work of Pickett et al. (10) and also to NMR measurements (24) (approximately 4 X 10-9 m2/s at 300 K and 4 atoms per cell). Again, the predicted diffusion coefficients decreased with increasing Xe loading. This effect was most noticeable for the simulations at 300 and 400 K. 

Allotrope Stability range, °C Crystal structure Lattice parameters, pm Reference temperature, °C Atoms per cell, Z Density, calculated, g/ cm3 Average M—M distance, pm... 

AIN exists in two types the hexagonal (wurtzite structure) and the cubic (zincblende structure). The former is more stable, and has been investigated in more detail. The wurtzitic AIN has two formula units per unit cell (4 atoms per cell) and 9 optical branches to the phonon dispersion curves [1] ... 

In the crystal structure of erionite, Mg cations can be present up to 0.8 atom per cell. Si-P A1 (-pFe ) should be approximately equal to 36 atoms based upon 72 oxygen atoms in the erionite formula, although the Si/Al ratio alone cannot be used for identification. 

The number of atoms per unit cell in any crystal is partially dependent on its Bravais lattice. For example, the number of atoms per unit cell in a crystal based on a body-centered lattice must be a multiple of 2, since there must be, for any atom in the cell, a corresponding atom of the same kind at a translation of from the first. The number of atoms per cell in a base-centered lattice must also be a multiple of 2, as a result of the base-centering translations. Similarly, the number of atoms per cell in a face-centered lattice must be a multiple of 4. 

The reverse of these propositions is not true. It would be a mistake to assume, for example, that if the number of atoms per cell is a multiple of 4, then the lattice is necessarily face-centered. The unit cell of the intermediate phase AuBe, for example (Fig. 2-20), contains 8 atoms and yet it is based on a simple cubic Bravais lattice. The atoms are located as follows ... 

When determined in this way, the number of atoms per cell is always an integer, within experimental error, except for a very few substances which have defect structures. In these substances, atoms are simply missing from a certain fraction of those lattice sites which they would be expected to occupy, and the result is a nonintegral number of atoms per cell. FeO and the P phase in the Ni-Al system are examples. 

Our initial assumption that p, q, and r are integers apparently excludes all crystals except those having only one atom per cell, located at the cell corners. For if the unit cell contains more than one atom, then the vector OA from the origin to any atom in the crystal may have nonintegral coordinates. However, the... 

In modern powder diffraction the measurement delivers a raw-file of some thousand step-scan data of counted X-ray photons per step. This raw file contains all the needed information to carry out a crystallographic analysis, but in a way that requires follow up. More informative is a list of distinguishable reflections that includes the position (mostly in the form of f-values) and intensity of each reflection. This dif-file (d-values and intensities) contains some tens to hundreds of reflections. The number of reflections depends on the complexity of the structure and the crystal symmetry the more atoms per cell and the lower the symmetry the more reflections can be identified. But the number of detectible reflections also depends on the resolving power of the equipment, best documented by the half-width of the reflections (more accurately half-width at half-maximum, FWHM). Reflections nearer together than this half-width (or even two half-widths) cannot be resolved. In a second step, very often the Miller indices of the originating lattice planes are added to the dif-file. For this the knowledge of the unit cell is necessary (though not of the crystal structure itself). The powder diffraction file PDF of the International Centre for Diffraction Data (ICDD) contains over 100000 such dif-files for the identification and discrimination of solid state samples. 

The structure of our result is identical to that obtained in eqn (4.64) in the absence of any consideration of crystal symmetry. On the other hand, it should be remarked that the dimensionality of the matrix in this case is equal to the number of orbitals per atom, n, rather than the product nN (our statements are based on the assumption that there is only one atom per cell). As a result of the derivation given above, in order to compute the eigenvalues for a periodic solid, we must first construct the Hamiltonian matrix H(k) and then repeatedly find its eigenvalues for each -point. Thus, rather than carrying out one matrix diagonalization on a matrix of dimension nN x nN, in principle, we carry out N diagonalizations of an n x n matrix. 

In eqn (4.68), we developed the tight-binding formalism for periodic systems in which there was only one atom per unit cell, with each such atom donating n basis functions to the total electronic wave function. Generalize the discussion presented there to allow for the presence of more than one atom per cell. In concrete terms, this means that the basis functions acquire a new index, i, J, a), where the index i specifies the unit cell, / is a label for the atoms within a unit cell and a remains the label of the various orbitals on each site. 

Whereas fee contains 4 and bcc 2 atoms per unit cell, y-brass is rather complex with 52 atoms per cell. Z became close to 1.8 e/a, and the Jones zone is... 

Because of the charge neutrality requirements for extended systems discussed above, literal transcription of variational Coulomb fitting for finite systems [by replacing p(r) everywhere by e(r)] clearly will not work. The solution is to do the charge fitting with neutralized fimctions. Here it is useful to allow explicitly for multiple atoms per cell. 

A comparison of a simple crystal with one atom per cell, 25 primitive ba-... 

Subject to kinetic limitations, equilibrium can always be established between the oxygen of the bulk (usually O ) and of the gas-phase (O2) and this equilibrium will fix Ho for any conditions of temperature and pressure. Using this po, Eq. (2) (section 3) allows us to compare the stability of oxide surfaces of various stoichiometries, albeit with neglect of slab volume and entropy changes [76]. For the example of a rutile surface with a deficiency of m O atoms per cell (i.e. a Tin02 -m surface) ... 

In Chaps.7 and 8 it is shown how the LMTO method and the physically simple concepts contained in linear theory may be used in self-consistent calculations to estimate ground-state properties of metals and compounds. Here we treat the local-density approximation to the functional formalism of Hohenberg3 Kohn, and Sham, and the force relation derived by Andersen together with an accurate and a first-order pressure relation. In addition, the LMTO-ASA and KKR-ASA methods are generalised to the case of many atoms per cell. 

To introduce the subject of many atoms per cell, we apply the tail-cancellation theorem, Sect.2.1, to a collection of atoms. In the derivation it is convenient to consider the simplest case, i.e. a diatomic molecule, but the results will be valid for any molecule or cluster. Our starting point is the energy-independent muffin-tin orbitals (2.1) in the atomic-sphere approximation, i.e. 

The basic input to STR is the translational vectors spanning the unit cell of the crystal, and the basis vectors giving the positions of the individual atoms in the cell. With this information STR may in principle be used to calculate canonical structure constants of any crystal structure, the only limitation being that central processor time grows rapidly as the number of atoms per cell is increased. 

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