Harmonium

Finally, an Addendum (p. 70) has been added during the final processing of the manuscript and presents some most recent applications concerning charmonium and harmonium systems in the IV-dimensional case. This section was written by F. Vinette. 

Addendum. The Application of so(2,1) to the Study of Charmonium and Harmonium in iY-Dimension with the Use of Symbolic Computation6... 

The hydrogen atom in a linear external field (charmonium) and in a quadratic external field (harmonium) has already been studied by Vrscay (1983, 1984, 1985). In both cases, Lie algebraic methods provided a considerable number of coefficients in the perturbation series expansion for small values of the coupling constant. However, these results were obtained for a... 

Consider the Schrodinger equation for the TV-dimensional hydrogen atom in a radial external field Xra, where a = 1 corresponds to charmonium and a = 2 corresponds to harmonium,... 

The readers together with the author admired an exact solution of the Schrodinger equation for the harmonium. It is truly a wonder two electrons, but still we get an exact (and simple) solution. Is it the end of such wonders ... 

So here is something really interesting. Indeed, we can manipulate the masses arbitrarily and therefore create links among some apparently unrelated systems. Note that if V is chosen as the Coulombic repulsion, Eq. (4.50) describes either two-electron Hooke atom (i.e., harmonium ), and then we put mi = m2 [Pg.215]

Harmonium represents the two-electron Hooke atom. A Hodce diaromic molecule means two heavy particles (nuclei) interacting by Coulomb forces. The same is true with electrons, but the heavy particle-light particle interactions are harmonic. 

Nicolaus Copernicus University (Torun, Poland). The solutions already known in the literature turn out to be the special cases of this general one the already-described harmonium N = 3), Hj Hooke s molecule. and H2 Hooke s molecule. " ... 

The Hohenberg-Kohn theorem ean be proved for an arbitrary external potential-this property of the density is ealled the v-representability. The arbitrariness mentioned above is necessary in order to define in future the functionals for more general densities (than for isolated molecules). We will need that generality when introducing the functional derivatives (p. 584) in which p(r) has to result from any external potential (or to be a v-representable density). Also, we will be interested in a non-Coulombic potential corresponding to the haniionic helium atom (cf. harmonium, p. 589) to see how exact the DFT method is. We may imagine p, which is not u-representable e.g., discontinuous (in one, two, or even in every point like the Dirichlet function). The density distributions that are not u-representable are out of our field of interest. 

We Have a Beacon—Exact Electron Density Distribution of Harmonium... 

Table 11.1. Harmonium (harmonic helium atom). Comparison of the components (a.u.) of the total energy E[po calculated by the HF, BLYP, and BP methods with the exact values (row KS exact Kohn-Sham solution). ...

Fig. 11.10. Efficiency analysis of various DFT methods and comparison with the exact theory for the harmonium (with force constant k = ) according to Kais et al. Panel (a) shows one-electron effective potential up = a -f Ueoui -I- Vxc, with external...

Fig. 11.11. Exchange potential. Efficiency analysis of various DFT methods and comparison with the exact theory for the harmonium (with the force constant k = ) according to Kais et al. Panel (al shows exchange potential as a function of the radius r, and Panel (b) uses a function of the density distribution p. The notation of Fig. 11.10 is used. It is seen that both DFT potentials produce plots that differ by nearly a constant from the exact potential (it is, therefore, an almost exact potential). The two DFT methods exhibit some non-physical oscillations for small r.

The DFT models can be tested when applied to exactly solvable problems with electronic correlation (like the harmonium, as discussed in Chapter 4). It turns out that despite the exchange and correlation DFT potentials deviating from the exact ones, the total energy is quite accurate. 

Biconfluent Heun equation in quantum chemistry Harmonium and related systems... 

Abstract Schrodinger equation for harmonium and related models may be transformed to the biconfluent Heun equation. The solubility of this equation and its applications in quantum chemistry are briefly discussed. 

Keywords Schrodinger equation Biconfluent Heun equation Harmonium Exactly solvable models... 

Harmonium may be defined as a quantum three-body problem described by the Schrodinger equation with harmonic interactions between particles 1—3 and 2 — 3 and the Coulombic interaction between particles 1—2. The... 

In some analyses, this recurrence relation may be very useful. In terms of the parameters describing harmonium, it reads... 

Cioslowski J, Pernal K (2000) The ground state of harmonium. J Chem Phys 113 8434-8443... 

Matito E, Cioslowski J, Vyboishchikov SF (2010) Properties of harmonium atoms from PCI calculations calibration and benchmarks for the ground state of the two-electron species. Phys Chem Chem Phys 12 6712-6716... 

Mtiller-Herold U (2006) On the emergence of molecular strucmre from atomic shape in the 1/r harmonium model. J Chem Phys 124 014105... 

Intracule-Extracule. - The properties of the electron intracule and one-electron densities of the ground state of harmonium were determined and expressed in terms of asymptotic expansions and Pade approximants. Topologies of both densities were investigated in detail. An attractor composed of a cage critical point and a (1,-1) critical sphere. 

We have a beacon - exact electron density distribution of harmonium... 

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