K-matrix

Equation (B.IO) stands for the j,k) matrix element of the left-hand side of Eq. (B.7). Next, we consider the (j,k) element of the first term on the right-hand side of Eq. (B.7), namely. 

The force constants k 2 and k2 are the off-diagonal elements of the matrix. If they are zero, the oscillators are uncoupled, but even if they are not zero, the K matrix takes the simple fomi of a symmetrical matrix because ki2 = k2. The matrix is symmetrical even though may not be equal to k22-... 

The m matrix is already diagonalized. Take the masses and the force constant to be 1 arbitrary unit for simplicity and concentrate on the force constant matrix. We can diagonalize the k matrix... 

Diagonalizing the K matrix converts arbitrary systems in generalized coordinate systems q... 

The sensitivity of the analytical system in the case of multicomponent analysis with a square K matrix may be defined as the absolute value of the deterrninant of K. 

So now we see that we can organize each of our 5 pure component spectra into a K matrix. In our case, the matrix will have 100 rows, one for each wavelength, and 5 columns, one for each pure spectrum. We can then generate... 

Classical least-squares (CLS), sometimes known as K-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the classical expression of the Beer-Lambert Law of spectroscopy ... 

Recognizing the difficulty satisfying the requirements for successful CLS, you may wonder why anyone would ever use CLS. There are a number of applications where CLS is particularly appropriate. One of the best examples is the case where a library of quantitative spectra is available, and the application requires the analysis of one or more components that suffer little or no interference other than that caused by the components themselves. In such cases, we do not need to use equation [33] to calculate the pure component spectra if we already have them in a library. We can simply construct a K matrix containing the required library spectra and proceed directly to equation [34] to calculate the calibration matrix K., . 

Multiple Linear Regression (MLR), Classical Least-Squares (CLS, K-matrix), Inverse Least-Squares (ILS, P-matrix)... 

Choosing the continuum inhomogeneity differently yields the alternative K -matrix normalized wave functions ... 

The continuum electron-phase shifts induced by the short-range scattering off the chiral molecular potential are most conveniently introduced by a third choice of continuum function, obtained by diagonalizing the K-matrix by a transformation U, resulting in a set of real eigenchannel functions (apart from normalization) [41] ... 

Magnesium Photoionization a K-Matrix Calculation with GTO Bases... 

The present method is an extension of the K-matrix technique pioneered by Fano (7). Our previous works have discussed thoroughly its general aspects, the discretization procedure (4,8) and the implementation upon the short-range GTO bases (6), so only a concise description will be given here. 

The K-matrix method is essentially a configuration interaction (Cl) performed at a fixed energy lying in the continuum upon a basis of "unperturbed funetions that (at the formal level) includes both diserete and eontinuous subsets. It turns the Schrodinger equation into a system of integral equations for the K-matrix elements, which is then transformed into a linear system by a quadrature upon afinite L basis set. 

The formal basis employed in the K-matrix calculation includes the relevant partial wave channel (pwc) subspaces plus a "localized channel" (/c) of discrete functions. These last are usual Cl states and their inelusion in the basis allows to efficiently reproduce the autoionizing states and the eorrelation effects. 

Using GTO bases, it cannot be expected that the variational representations of the electron waves are snfficiently accnrate far ontside the so-called molecular region , i.e. the rather limited region of space where the potential clearly deviates from the asymptotic Conlomb form. Therefore the phaseshifts of the pwc basis states cannot be obtained from the analysis of their long-range behaviour, as was done in previous works with the STOCOS bases. In the present approach, this analysis may be avoided since the K-matrix techniqne allows to determine, by equation [3] below, the phase-shift difference between the eigenfunctions of Hp and the auxiliary basis functions... 

It should be noted that the integral equations [2] determining the elements Kp B. aE derived as an energy-variational problem, correspond also to the stationary condition of the variational functional proposed by Newton (9). Thus the K-matrix elements obeying equation [2] guarantee a stationary value for the K-matrix on the energy shell. 

This relation allows, as said above, to obtain the phaseshifts of the basis functions by a single-channel K-matrix calculation on the basis whose non-... 

As discussed in (4), the K-matrix has a pole at energies near a resonance and this yields a convenient method for the analysis of the narrow autoionizing states. The matrix representation of equation [2] upon a finite basis may be in fact recast in the form (4)... 

The present method does not involve the analysis of the long-range behaviour of the states, so its application requires only that the narrow wavepackets are accurate inside the molecular region. By equation [3], the phaseshifts of these states may be determined through a K-matrix calculation on the auxiliary basis, so it is assumed that the narrow wavepackets might be continued outside the molecular region as shifted Coulomb waves. 

It may be eoneluded that a method based on the K-matrix teehnique may be conveniently adapted to ealeulate the eontinuum properties using variational basis funetions that are aeeurate only inside the molecular region . This means that the calculations may be carried out upon GTO bases, which allow the extension of the proposed method to moleeular systems, as aheady eheeked for H2 (13). 

Magnesium photoionization a K-matrix calculation with GTO bases R. Moccia and P. Spizzo... 

Instead of the symbol A and the term sensitivity matrix also the symbol K (matrix of calibration coefficients, matrix of linear response constants etc) is used. Because of the direct metrological and analytical meaning of the sensitivities aj - in the A-matrix the term sensitivity matrix is preferred. 

More extensive, multicomponent system are described by the sensitivity matrix (matrix of partial sensitivities according to Kaiser [1972], also called K-matrix according to Jochum et al. [1981]) ... 

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