Energy exact atomic step

The resulting functional has been evaluated for the initial orbitals of the lithium and beryllium atoms given in Section 4.6. Using the same process of intra- and inter-orbit optimization carried out in Section 4.6, but substituting, in each step, the value for /(r) given by the Fade approximant obtained from Eq. (189), energy values were obtained for the Li and Be atoms that are indistingishable from those previously calculated with the exact values of/(r), namely, when Eq. (40) is solved. 

We conclude this section by providing a comparison in Figures 1—5of the kinetic-energy potentials of the CGE and several better GGA OF-KEDF s, using accurate densities for H, He, Be, Ne, and Ar atoms. For many-electron atoms, highly accurate densities (from atomic configuration interaction calcu-lations) are fed into the OF-KEDF s. Accurate potentials are obtained via a two-step procedure the exact [p]) is obtained for a given accurate... 

Such a separation is exact for atoms. For molecules, only the translational motion of the whole system can be rigorously separated, while their kinetic energy includes all kinds of motion, vibration and rotation as well as translation. First, as in the case of atoms, the translational motion of the molecule is isolated. Then a two-step approximation can be introduced. The first is the separation of the rotation of the molecule as a whole, and thus the remaining equation describes only the internal motion of the system. The second step is the application of the Born-Oppenheimer approximation, in order to separate the electronic and the nuclear motion. Since the relatively heavy nuclei move much more slowly than the electrons, the latter can be assumed to move about a fixed nuclear arrangement. Accordingly, not only the translation and rotation of the whole molecular system but also the internal motion of the nuclei is ignored. The molecular wave function is written as a product of the nuclear and electronic wave functions. The electronic wave function depends on the positions of both nuclei and electrons but it is solved for the motion of the electrons only. 

The independent-electron approximation was discussed in the previous chapter. The molecular wave functions, ifi, are solutions of the Hartree-Fock equation, where the Fock operator operates on tfi, but the exact form of the operator is determined by the wave-function itself. This kind of problem is solved by an iterative procedure, where convergence is taken to occur at the step in which the wave function and energy do not differ appreciably from the prior step. The effective independent-electron Hamiltonian (the Fock operator) is denoted here simply as H. The wave functions are expressed as linear combinations of atomic functions, x-... 

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