Positive-energy space

M. Byhcki, G. Pestka, J. Karwowski. Rela-tivishc HyUeraas configuration-interaction method projected into positive-energy space. Phys. Rev. A, 77(4) (2008) 044501. 

M, constructed from positive-energy eigenfunctions of the matrix DF SCF equation. The M configuration-state functions form a subspace of the positive-energy space... 

Many-electron wave functions correct to oi may be expanded in a set of CSFs that spans the entire N-electron positive-energy space j (7/J 7r), constructed in terms of Dirac one-electron spinors. Individual CSFs are eigenfimctions of the total angular momentum and parity operators and are linear combinations of antisymmetrized products of positive-energy spinors (g D(+ ). The one-electron spinors are mutually orthogonal so the CSFs / (7/J 7r) are mutually orthogonal. The un-... 

A Lorentz invariant scalar product can be defined in the linear vector space formed by the positive energy solutions which makes this vector space into a Hilbert space. For two positive energy Klein-... 

Here, exc(p(r)) is the exchange-correlation energy per particle of a uniform electron gas of density p( ). This energy per particle is weighted with the probability p(r) that there is in fact an electron at this position in space. Writing Exc in this way defines the local density approximation, LDA for short. The quantity exc(p(r)) can be further split into exchange and correlation contributions,... 

Consider a system composed of n identical, but distinguishable, particles. The distinguishability of the particles may result, for example horn positions in space, e.g. their coordinates. It is useful in this simplified mo to assume, furthermore, that the energy of interaction between the partic... 

Under the assumption that the induced field is much smaller than the external one, and that the latter is still small on the atomic energy scale, the induced field will be proportional to the external field. Both assumptions are valid for experimentally accessible field strenghts, i.e. up to more than 10 Tesla. The proportionality coefficient between them depends on the chosen position in space, and is called chemical shielding, commonly denoted by aafi(R) ... 

In order to achieve this, we should in principle calculate the energy of a given aggregate of atoms as a function of their positions in space. The results can be expressed as a many dimensional potential surface, the minima in which correspond to stable molecules, or aggregates of molecules, while the cols separating the minima correspond to the transition states for reactions leading to their interconveision. If such calculations could be carried out with sufficient accuracy, one could not only... 

The most basic type of DFT calculation is to compute the total energy of a set of atoms at prescribed positions in space. We showed results from many calculations of this type in Chapter 2 but have not said anything about how they actually are performed. The aim of this section is to show that this kind of calculation is in many respects just like the optimization problems we discussed above. 

By representing the operator containing the potential energy in position state space and the one containing the kinetic energy in momentum space, one obtains the following phase space discretized path integral representation ... 

Let us consider the simplest possible case of a system which consists of two orbitals, XA( 1) and / (I), with energies eA and which can interact. It should be emphasized here that these may be orbitals of any kind, atomic orbitals, group orbitals, or complicated MOs. We wish to investigate the results of the interaction between them, that is, what new wave functions are created and what their energies are. Let us also be clear about what the subscripts A and represent. The subscripts denote orbitals belonging to two physically distinct systems the systems, and therefore the orbitals, are in separate positions in space. The two systems may in fact be identical, for example two water molecules or two sp3 hybrid orbitals on the same atom or on different but identical atoms (say, both C atoms). In this case, eA = - Or the two systems may be different in... 

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