Pauli approximation

The Hartree approximation is usefid as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefiinction is that it does not reflect the anti-synnnetric nature of the electrons as required by the Pauli principle [7], Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the siumnation in equation Al.3.11 does not include the th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation A1.3.11 solved for each orbital. 

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

Because single-electron wave functions are approximate solutions to the Schroe-dinger equation, one would expect that a linear combination of them would be an approximate solution also. For more than a few basis functions, the number of possible lineal combinations can be very large. Fortunately, spin and the Pauli exclusion principle reduce this complexity. 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

It is now known that the view of electrons in individual well-defined quantum states represents an approximation. The new quantum mechanics formulated in 1926 shows unambiguously that this model is strictly incorrect. The field of chemistry continues to adhere to the model, however. Pauli s scheme and the view that each electron is in a stationary state are the basis of the current approach to chemistry teaching and the electronic account of the periodic table. The fact that Pauli unwittingly contributed to the retention of the orbital model, albeit in modified form, is somewhat paradoxical in view of his frequent criticism of the older Bohr orbits model. For example Pauli writes,... 

The correlation error can, of course, be defined with reference to the Hartree scheme but, in modem literature on electronic systems, one usually starts out from the Hartree-Fock approximation. This means that the main error is due to the neglect of the Coulomb correlation between electrons with opposite spins and, unfor-tunetely, we can expect this correlation error to be fairly large, since we force pairs of electrons with antiparallel spins together in the same orbital in space. The background for this pairing of the electrons is partly the classical formulation of the Pauli principle, partly the mathematical fact that a single determinant in such a case can... 

Importantly, the value of the results gained in the present section is not limited to the application to actual systems. Eq. (4.2.11) for the GF in the Markov approximation and the development of the perturbation theory for the Pauli equation which describes many physical systems satisfactorily have a rather general character. An effective use of the approaches proposed could be exemplified by tackling the problem on the rates of transitions of a particle between locally bound subsystems. The description of the spectrum of the latter considered in Ref. 135 by means of quantum-mechanical GF can easily be reformulated in terms of the GF of the Pauli equation. 

So, high gets a contribution from fermion modes with Dirac eigenvalues from the interval Mi to the Pauli-Villars mass M, and Detiow considers eigenvalues less than Mi. The product of these determinants is independent on the scale2 Mi. But, we may calculate both of them only approximately. In (D. Diakonov et.al., 1986) it was demonstrated a weak dependence of the product on M in the wide range of Mi which serves as a check of the approximations. 

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