### Extending the MAP basis sets to the elements H-Lr with use of Frobenius angles

__A.L. Tchougréeff__^{1}, P. Reinhardt^{2}

^{1}A.N. Frumkin Institute of Physical Chemistry and Electrochemistry of RAS, Moscow, Russia.

^{2}Laboratoire de Chimie Théorique, Sorbonne Université et CNRS UMR7616, Paris, France

**Abstract:**

In [1] we revisited the problem of atomic orbital bases from the positions of semi-empirical quantum chemistry: getting ones: (i) having proper nodal structure; (ii) simple functional form; (iii) representing atomic properties with reasonable accuracy. Employing the Ansatz [2, 3] which reduces the the number of parameters to a single one per nℓ subshell: its specific orbital exponent ξ_{nℓ}, and assures the orthogonality of the radial parts on account of polynomial multipliers *P*_{n-ℓ-1}^{2ℓ+1}(2ξ_{nℓ} r) a generalization of hydrogen-like atomic orbitals has been proposed and the values of ξnℓ obtained for the 2nd row elements either by minimization their ground state energy or by extracting from atomic spectra. In [4] the *Ansatz* has been extended to elements H-Xe of the Periodic Table, by minimizing the total energy of respective spectroscopic states. The name of MAP (minimal atomic parameter/Moscow–Aachen–Paris) has been coined for them. The fundamental properties (total energy, radial expectation values, node positions, etc.) of the generated MAP orbital sets are astonishingly close to those obtained with much larger basis sets known in the literature, without numerical inconsistencies. The obtained exponents follow simple relations with respect to the nuclear charge *Z*. In [5] the spatial features as orbital shapes of the MAP sets derived in [4] are analyzed, possible fits to alternative orbital sets, respect of Kato's condition and radial distribution of energy components are considered. For comparing basis sets the Frobenius angle between the orbital subspaces they span is introduced as numerical tool. It is shown that the electronic density of the MAP states is depleted around the nucleus with respect to the other orbital sets. Despite this, the similarity between the respective subspaces in all cases (except a unique case of the Pd atom) as measured by the cosine of the Frobenius angle amounts above 0.96 for all atoms. In the present contribution we describe the results of fitting the MAP orbital exponents to the Koga basis sets [6] by minimizing respective Frobenius angles, show their piecewise linear dependence on *Z* and interpret the latter in terms of known Slater rules [7].

- Popov, I. & Tchougréeff, A. Theor. Chem. Acc., 2019, 138, 9.
- Â.À. Ôîê, Ì.È. Ïåòðàøåíü. ÆÝÒÔ 1934, ò. 4, ¹4 ñ. 295-325.
- Â.À. Ôîê, Ì.È. Ïåòðàøåíü. ÆÝÒÔ 1936, ò. 6, ¹ ñ. 1-8.
- Reinhardt, P.; Popov, I. & Tchougréeff, A. Int. J. Quant. Chem., 2021, 121, e26687.
- Reinhardt, P.; Popov, I. & Tchougréeff, A. Int. J. Quant. Chem., 2021, 121, e26690.
- Koga, T. & Thakkar, A. 1996 J. Phys. B: At. Mol. Opt. Phys. 29 2973.
- J.C. Slater. 1930 Phys.Rev. 36 57.