Matrix primitive

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerfiil algoritlun for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

Next, we shall consider four kinds of integrals. The first is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at that nucleus. The second is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at a different point (usually another nucleus). Then, we will consider the matrix element of a Coulomb term between two primitive basis functions at different centers. The third case is when one basis function is centered at the nucleus considered. The fourth case is when both basis functions are not centered at that nucleus. By that we mean, for two Gaussian basis functions defined in Eqs. (73) and (74), we are calculating... 

Transfer matrix calculations of the adsorbate chemical potential have been done for up to four sites (ontop, bridge, hollow, etc.) or four states per unit cell, and for 2-, 3-, and 4-body interactions up to fifth neighbor on primitive lattices. Here the various states can correspond to quite different physical systems. Thus a 3-state, 1-site system may be a two-component adsorbate, e.g., atoms and their diatomic molecules on the surface, for which the occupations on a site are no particles, an atom, or a molecule. On the other hand, the three states could correspond to a molecular species with two bond orientations, perpendicular and tilted, with respect to the surface. An -state system could also be an ( - 1) layer system with ontop stacking. The construction of the transfer matrices and associated numerical procedures are essentially the same for these systems, and such calculations are done routinely [33]. If there are two or more non-reacting (but interacting) species on the surface then the partial coverages depend on the chemical potentials specified for each species. 

We choose to use the primitive coupled basis, rather than the 712 coupled basis that is often employed, primarily because the primitive coupled basis has a direct product form. This form makes it easy to accomplish the transformation from ji, j2, fci to 0i, 02, - The potential matrix, V, may be written as a numerical quadrature over the angular grid points in the following matrix form ... 

The tacit assumption above is that the monodromy matrix is defined with respect to the primitive unit cell, with sides (5v, 8fe) = (0,1) and (1, 0), because the twist angle that determines the monodromy is given by A9 = — (Sv/Sfe)j.. However, situations can arise where other choices are more convenient. For example, the energy levels within a given Fermi resonance polyad are labeled by a counting number v = 0,1,... and an angular momentum that takes only even or only odd values. Thus the convenient elementary cell has sides (8v, 8L) = (0,2) and (1, 0), and the natural basis, say, y, is related to the primitive basis, x, by... 

Approximate linear dependence of AO-based sets is always a numerical problem, especially in 3D extended systems. Slater functions are no exceptions. We studied and recommended the use of mixed Slater/plane-wave (AO-PW) basis sets [15]. It offers a good compromise of local accuracy (nuclear cusps can be correctly described), global flexibility (nodes in /ik) outside primitive unit cell can be correct) and reduced PW expansion lengths. It seems also beneficial for GW calculations that need low-lying excited bands (not available with AO bases), yet limited numbers of PWs. Computationally the AOs and PWs mix perfectly mixed AO-PW matrix elements are even easier to calculate than those involving AO-AO combinations. 

Almost immediately after the discovery of presolar grains, it was clear that they could only be found in the most primitive chondrites, those that had suffered the least amount of thermal metamorphism. Further work showed that the abundances of presolar grains, when normalized to the content of fine-grained matrix where the grains reside, correlated strongly... 

Cosmochemistry is the study of the chemical compositions of various solar system materials. Chondrites are the most abundant primitive samples. They are essentially sedimentary rocks composed of mechanical mixtures of materials with different origins (chondrules, refractory inclusions, metal, sulfide, matrix), which we will call components. Chondrites formed by the accretion of solid particles within the solar nebula or onto the surfaces of growing planetesimals. They are very old (>4.5 billion years, as measured by radioactive chronometers) and contain some of the earliest formed objects in the solar system. Chondrites have bulk chemical compositions very similar to the solar photosphere, except... 

Write a matrix that converts an (incorrectly) C-centered monoclinic lattice to the proper primitive one. 

In the symmetrization process the primitive period isometric transformations Fp play an outstanding role. For all SRMs with r(3 (Fp) e SO(3) these operators are represented in the representation r NCI St by the unit matrix (cf. Sect. 2.3.3). 

The primitive VB model is defined in terms of overlap and Hamiltonian matrix elements over the basis states of eqn. (2.1.3). For fixed there are 2N possible spin-product functions so that this gives the dimension of the model s space. Indeed (though not originally formulated in this manner) the model may be mathematically represented entirely in spin space, despite the fundamental spin-free nature of the interactions. One may introduce a spin-space overlap operator by integrating out the spin-free coordinates... 

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