Slope matrix

In the multivariate case, one slope vector is not enough a square slope matrix must be generated with dimensions equal to the number of independent unknowns Xi one wishes to detemiine. The slope matrix is... 

The n-fold procedure (n > 2) produces an n-dimensional hyperplane in n -b 1 space. Lest this seem unnecessarily abstract, we may regard the n x n slope matrix as the matrix establishing a calibration srrrface from which we may determine n unknowns Xi by making n independent measurements y . As a final generalization, it should be noted that the calibration surface need not be planar. It might, for example, be a curwed sruface that can be represented by a family of quadratic equations. 

Elements in the slope matrix A are proportional to absorptivities and concentrations are in parts per million. We shall take this as the true slope matrix. 

Subtracting the slope matrix obtained by the multivariate least squares tieatment from that obtained by univariate least squares slope matiix yields the error mahix... 

The reader is asked to find the standard deviations of the slopes matrix in Problem 9 below. 

Not until the so called third generation of LMTOs [58], was there a way to properly include the interstitial part of the muffin-tin potential and perform calculations without the ASA, in fact it was possible to perform calculations for exact muffin-tins using the Exact Muffin-Tin Orbitals (EMTO) method. Since the structural dependent part is called the slope matrix in the EMTO method, this is the name I will use for the rest of this thesis when discussing the EMTO method. 

In actual calculations one does not solve this equation, but instead calculates the poles of the scattering path operator to the kink matrix. Nevertheless, it is clear that we need to find an expression for the slope matrix. This can be derived from the bare (or canonical) KKR structure constant matrix Sr,l,rl(k2), and this will be shown below, as well as how to compute the first energy derivative of the slope matrix. 

The EMTO slope matrix can be obtained from the bare KKR structure constant matrix Sr,l,rl(k2) as [58]... 

Note that the screening parameters, and likewise the slope matrix, depend on energy through lo. Applying the product rule, from Eq.(4.21) we obtain the n-th energy derivative of the slope matrix as... 

In a generic EMTO calculation, the slope matrix is calculated at an expansion center and later Taylor expanded to the energy points on the integration contour ... 

Figure 4.2. The relative error of the expansion of (a) the slope matrix using a one-center Taylor expansion and (b) using a two-center expansion.The error is defined in Eq. (4.35) and the expansion center is (0,0) for the one center expansion, and (0,0) and (-10,0) for the two-center expansion.

These conditions lead to a system of linear equations for an+i,an+2,. Solving these equations, we obtain a (n + m+ l)th order expansion for Sa. In Paper I, this expansion is examined and compared with the old one-center Taylor expansion. The results show a marked improvement in the expansion of the fee slope matrix, see Fig.(4.2), and we also find some cases where it is necessary to use the two-center expansion in order to produce correct results. The error in Fig.(4.2) is defined as... 

Another idea we had, was to try to set the expansion center somewhere out in the complex plane instead for on the real energy axis. In Fig.(4.3) is shown the real (a) and imaginary (b) part of the ss element of the fee slope matrix. 

Figure 4.3. The real part (a) and imaginary part (b) of the ss-block of the EMTO fee slope matrix.

A short introduction to electronic structure theory is included as well as a chapter on Density Functional Theory, which is the underlying method behind all calculations presented in the accompanying papers. Multiple Scattering Theory is also discussed, both in more general terms as well as how it is used in the methods employed to solve the electronic structure problem. One of the methods, the Exact Muffin-Tin Orbital method, is described extensively, with special emphasis on the slope matrix, which energy dependence is investigated together with possible ways to parameterize this dependence. 

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