ORTHOGONAL OPERATORS (ОРТОГОНАЛЬНЫЕ ОПЕРАТОРЫ)

ОРТОГОНАЛЬНЫЕ ОПЕРАТОРЫ

Смольякова Анастасия Александровна

магистрант, факультет математики и естественнонаучного образования педагогического института, Белгородский государственный национальный исследовательский университет,

РФ, г. Белгород

Богачев Роман Евгеньевич

научный руководитель, доц., факультет иностранных языков, Белгородский государственный национальный исследовательский университет,

РФ, г. Белгород

ABSTRACT

The relevance of this topic lies in the fact that orthogonal operators play an important role in the Euclidean space. It preserves the dot product, which in turn means that the lengths of the vectors and the angles between them are preserved. Orthogonal operators are widely used in geometry, physics, and medicine.

Keywords: orthogonal operators.

Why use orthogonal operators when the usual Slater-Condon approach such as the Cowan code [4, 5] is so easy to use as a general tool? The Cowan code has been a blessing to the experimental atomic physicist for the past 50 years, and it will no doubt remain so for years to come. The orthogonal operator approach can be seen as an extension and refinement of the usual least squares fitting (LSF) approach, but it also increases the need to find physically acceptable initial estimates for a fairly large number of small parameters, especially those with an incomplete spectrum. On the other hand, the stability of a parameter compared to changing or adding others is a big advantage of the method. Figure 1 gives a clear picture of this aspect [6].

In the first match (from left to right), only the eav parameter was allowed to be changed with all other parameters fixed to zero, while in subsequent matches the parameter indicated by the abscissa was added. (b) Values ​​of the parameter E av using the same procedure as in (a), but using a set of orthogonal equivalents.

Rice. 1. (a) Values of the Eav parameter in the LSF series in the 3d3 configuration in Cr IV.

Both conventional and orthogonal operator approaches are based on the semi-empirical LSF of physical parameters. The mean error can be interpreted as the 'blobsize' used by the artists (Seurat and others) of the 'Pointillism' art movement [3]. They used blobs for their paintings and couldn't represent details smaller than blobs. Similarly, we cannot describe the level structures in detail when the distance between the levels is less than or comparable to the average error.

In many cases, the average error of an orthogonal LSF operator is an order of magnitude smaller than that of a conventional LSF. In principle, this leads to an improvement in native vector compositions and, consequently, to an improvement in transition probabilities. In some cases refinements are appropriate and these will be discussed below. In Table 1, some characteristics of both methods are set side by side.

The term analysis of complex spectra, such as Mn IV [7], Re III [1], and Os III [2], only achieved prediction accuracy of orthogonal operators. Such predictions concern both energy levels and electric dipole intensities.

In addition, reliable calculations of the forbidden magnetic dipole and electric quadrupole intensities are provided by exact egal envectors from the orthogonal operator LSF of the corresponding system.On a more theoretical level, the use of orthogonal operators makes it possible to establish small and hitherto ignored interactions that reflect multibody, relativistic, and correlation effects.

The orthogonal operator approach is briefly reviewed and compared with the conventional LSF approach. The enhanced stability of rectangular Operators Creates room to meaningfully include double-story magnetic Operators and high-order efficient electrostatic Operators. This means that the LSF error is thereby significantly reduced, which should give better eigenvector compositions and improved transition Probabilities. However, 'orthogonal' operators have of course no plug and play method: initial EST and mathes require isoionic/isoelectronic extrapolation, pre-ab initio calculations, or both. Experience with neighboring spectra obviously helps. Little experience has been gained in the open pn and f n shells yet, although we have recently implemented orthogonal operators for both cases; Applications for f n configurations with n 2 ap planned in the near future.

While large-scale calculations with the Cowan code (including many individual configurations) certainly lead to satisfactory results in a fast and reliable manner, important magnetic effects remain on the table, and this can be a problem for close levels. Orthogonal operators (including many effective operators) are inherently more perturbed than variational ones.As far as higher-order electrostatic effects are concerned, it is not easy to compare the effects of a large number of unobservable, scaled configurations (which actually act as effective operators) with effective 3- and 4-body orthogonal operators: they probably only partially represent the same effects. . When a strong configuration interaction comes into play, the orthogonal space of the model is expanded by a limited number of configurations. The average error of the LSF is still clearly smaller, but closer to the average error of the conventional approach in these cases.

In this article, we looked at a more specific example of how orthogonal operators help solve critical problems in other scientific fields, namely in physics.

References:

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2. Azarov, V., Chang-Brie, V.U., Gayasov, R. Analysis of the spectrum of transitions (5d 6 +5d 56s) - (5d56p + 5d46s6p) of double ionized osmium (Os III). U. Data Nucl. Datasheets 2018, - 121, 345–377. [CrossRef]
3. Hautecoeur, L., Impressionist, Georges Seurat; Atrium: Alphen aan den Rijn, The Netherlands, 1972.
4. Cowen, R. Theory of the structure and spectra of atoms; University of California Press: Berkeley, California, USA, 1981.
5. Kramida, A. The Cowan Code: 50 years of growing influence on atomic atomic physics. Atoms 2019, - 7.64. [CrossRef]
6. Hansen, J., Wilings, P., Raassen, A. Parametric fitting with orthogonal operators. Phys. Skr. 1988 - 37, 664–672. [CrossRef]
7. Chang-Brieux, W.W., Artru, M.C., Wyart, J.F. 3d4-3d34p Triple-and-Ionized Manganese Transitions (Mn IV). Phys. Skr. 1986, - 33, 390–400 pp. [CrossRef]