Orbit, concept variational principle

An early example implementing the general approach to take into account first the intrabond correlation, is presented by the PCILO - perturbational configuration interaction of localized orbitals method [121,122], As one of its authors, J.-P. Malrieu mentions in [122], the PCILO method opposes the majority of the QC methods in all the fundamental concepts. In contrast to the majority of the methods based on the variational principle, the PCILO method is based on estimating the energy by perturbation theory. Also, the majority of the QC methods use one-electron HFR approximation, at least as an intermediate construct, whereas the PCILO is claimed to addresses directly the V-electron wave function and takes into account all surviving matrix elements of the electron-electron interactions. In contrast with other QC... 

The student is then shown that a plot of E vs. gives a minimum for 4 = 1 for the H atom. In other words, we get them to appreciate the variation principle from this simple optimization of 4. The virial ratio (—V/T) also is introduced as an important concept of electronic behavior. We do not discuss the excited states of the H atom, but restrict ourselves solely to the ground state. The only connection with experiment is the ionization energy of the H atom and other one-electron atoms. The student is told that orbital energy and total energy mean one and the same thing for the H atom, but that as soon as we turn to He or more complicated atoms, the situation will be dramatically different. Constant contours of the wavefunction are introduced. 

However, the periodicity condition (4.518) for paths is to be maintained and properly implemented in approximating the effective-classical partition function (4.525) being, nevertheless, closely and powerfully related with the quantum beloved concept of stationary orbits defined/described by periodic quantum waves/paths. This way, the effective-classical path integral approach appears as the true quantum justification of the quantum atom and of the quantum stabilization of matter in general, providing reliable results without involving observables or operators relaying on special quantum postulates other than the variational principles - with universal (classical or quantum) value. 

Looking back at the pi-electron calculations, we motivated the linear combination of atomic orbitals (LCAO) concept for electron orbitals, which spread over a whole molecule. We also have in hand the polynomial functions for the radial functions of the H-atom. In the previous example for pi-electrons, we used the variational principle to optimize the coefficients of the atomic orbitals in the linear combinations. We have also observed that the H-atom model can be modified by varying the value of the effective nuclear charge, Z, to model heavier elements. In 1930, Slater [4] proposed deleting the radial nodes and small parts of the H atom orbitals to use functions of the... 

Tethers. Orbital momentum can be exchanged using a tether between two spacecraft. This principle has been proposed to efficiently transfer payloads from Earth orbit to lunar or Martian orbits and even to exchange payloads with the lunar surfece. An extreme version is a stationary tether linking a point on Earth s equator to a craft in geostationary Earth orbit, the tether running far beyond to a countermass. The electrostatic tether concept uses variations in the electric potential with orbital height to induce... 

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