Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary function
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RESEARCH PAPER
Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary functions A. Bağcı1 · P. E. Hoggan2 Received: 30 May 2020 / Accepted: 29 August 2020 / Published online: 22 September 2020 © Accademia Nazionale dei Lincei 2020
Abstract This work describes the fully analytical method for the calculation of the molecular integrals over Slater-type orbitals with non-integer principal quantum numbers. These integrals are expressed through relativistic molecular auxiliary functions derived in our previous paper (Bağcı and Hoggan in Phys Rev E 91(2):023303, 2015). The procedure for computation of the molecular auxiliary functions is detailed. It applies both in relativistic and non-relativistic electronic structure theory. It is capable of yielding highly accurate molecular integrals for all ranges of orbital parameters and quantum numbers. Keywords Non-integer principal quantum numbers · Slater-type orbitals · Multi-center integrals
1 Introduction Formulae for interpreting visible (Balmer) and all observed electronic spectra of hydrogen were first characterized by two integers n1 (visible n1 = 2 ) and n2 . The idea of dropping the restriction to integer values of these numbers was first suggested by Rydberg (1890) in 1890. In the Bohr atom model (1913), the integers “n” became known as “quantum numbers” before full identification with the principle quantum number, with the solution of non-relativistic Schrödinger equation for the Coulomb interaction in atomic units. This was much earlier than any attempt to develop a stable method for electronic structure calculation of manyelectron systems or to construct a more flexible basis orbital to be used in this method. The solution of non-relativistic Schrödinger equation for the Coulomb interaction in atomic units (a.u.) leads to an expression for the wavelengths 𝜆 of spectral line emitted in a * A. Bağcı [email protected]; [email protected] P. E. Hoggan [email protected] 1
Department of Physics, Faculty of Arts and Sciences, Pamukkale University, Çamlaraltı, Kınıklı Campus, 20160 Denizli, Turkey
Institute Pascal, UMR 6602 CNRS, University Blaise Pascal, 24 avenue des Landais, BP 80026, 63177 Aubiere Cedex, France
2
transition of the atom from quantum state n2 to state n1 Bethe and Salpeter (1957), ) ( ) Z2 1 1 1 1 ( , − = E − En1 = (1) 𝜆 2𝜋 n2 4𝜋 n21 n22 where Z is the nuclear charge, En1 , En2 are (the lower ) and upper energy levels for hydrogen-like atoms. n1 , n2 are the principal quantum numbers. According to Bohr theory, they have integer values. Rydberg through investigation of alkali metal spectra showed that for many-electron atoms, a similar empirical expression could be used:
) 1 1 ( = En2 − En1 𝜆 2𝜋 ( ) 1 Z2 1 = − . 4𝜋 (n − 𝛿 )2 (n − 𝛿 )2 1 1 2 2
(2)
Here, n∗ = n − 𝛿 is an effective quantum number with noninteger values. The quantity 𝛿 is called the quantum defect Seaton (1983). For atomic orbitals, it depends on the angular momentum quantum number l. Equation (2) was obtained
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