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Register a new userUltra-Compact accurate wave functions for He-like and Li-like iso-electronic sequences and variational calculus. I. Ground state
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Several ultra-compact accurate wave functions in the form of generalized Hylleraas-Kinoshita functions and Guevara-Harris-Turbiner functions, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC), or, equivalently, the Non-Relativistic QED (NRQED), for the ground state energies (4-5 significant digits (s.d.)) of He-like and Li-like iso-electronic sequences in the static approximation with point-like, infinitely heavy nuclei are constructed. It is shown that for both sequences the obtained parameters can be fitted in $Z$ by simple smooth functions: in general, these parameters differ from the ones emerging in variational calculations. For the He-like two-electron sequence the approximate expression for the ground state function, which provides absolute accuracy for the energy $sim 10^{-3}$,a.u. and the same relative accuracies $sim 10^{-2}-10^{-3}$ for both the cusp parameters and the six expectation values, is found. For the Li-like three-electron sequence the most accurate ultra-compact function taken as the variational trial function provides absolute accuracy for energy $sim 10^{-3}$,a.u., 2-3 s.d. for the electron-nuclear cusp parameter for $Z leq 20$ and 3 s.d. for the two expectation values for $Z=3$.
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As a continuation of Part I (Int. Journal of Quantum Chem. 2021; 121: qua.26586), dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges $Z leq 20$, a few ultra-compact wave functions in the form of generalized Hylleraas-Kinoshita functions are constructed, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for energies (4-5 significant digits (s.d.)) of two excited states of He-like ions: the spin-singlet (first) excited state $2^1 S$ and for lowest spin-triplet $1^3 S$ state. For both states it provides absolute accuracy for energy $sim 10^{-3}$,a.u., exact values for cusp parameters and also for 6 expectation values the relative accuracy $sim 10^{-2}$. Bressanini-Reynolds observation about the special form of nodal surface of $2^1 S$ state for Helium is confirmed and extended to ions with $Z > 2$. Critical charges $Z=Z_B$, where ultra-compact trial functions loose their square-integrability, are estimated: $Z_B(1^1 S)approx Z_B(2^1 S)sim 0.905$ and $Z_B(1^3 S)sim 0.902$. For both states the Majorana formula - the energy as the second degree polynomial in $Z$ - provides accurately the 4-5 significant digits for $Z leq 20$.
Schrodingers equation (SE) and the information-optimizing principle based on Fishers information measure (FIM) are intimately linked, which entails the existence of a Legendre transform structure underlying the SE. In this comunication we show that the existence of such an structure allows, via the virial theorem, for the formulation of a parameter-free ground states SE-ansatz for a rather large family of potentials. The parameter-free nature of the ansatz derives from the structural information it incorporates through its Legendre properties.
We study a special case at which the analytical solution of the Lippmann-Schwinger integral equation for the partial wave two-body Coulomb transition matrix for likely charged particles at negative energy is possible. With the use of the Focks method of the stereographic projection of the momentum space onto the four-dimensional unit sphere, the analytical expressions for s-, p- and d-wave partial Coulomb transition matrices for repulsively interacting particles at bound-state energy have been derived.
A general formalism is used to express the long-range potential energies in inverse powers of the separation distance between two like atomic or molecular systems with $P$ symmetries. The long-range molecular interaction coefficients are calculated for the molecular symmetries $Delta$, $Pi$, and $Sigma$, arising from the following interactions: He($2 ^1P$)--He($2 ^1P$), He($2 ^1P$)--He($2 ^3P$), and He($2 ^3P$)--He($2 ^3P$). The electric quadrupole-quadrupole term, $C_{5}$, the van der Waals (dispersion) term $C_{6}$, and higher-order terms, $C_{8}$, and $C_{10}$, are calculated textit{ab initio} using accurate variational wave functions in Hylleraas coordinates with finite nuclear mass effects. A comparison is made with previously published results where available.
Dielectronic recombination has been investigated for Delta-n = 1 resonances of ground-state Li+(1s^2) and for Delta-n = 0 resonances of metastable Li+(1s2s ^3S). The ground-state spectrum shows three prominent transitions between 53 and 64 eV, while the metastable spectrum exhibits many transitions with energies < 3.2 eV. Reasonably good agreement of R-matrix, LS coupling calculations with the measured recombination rate coefficient is obtained. The time dependence of the recombination rate yields a radiative lifetime of 52.2 +- 5.0 s for the 2 ^3S level of Li+.
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