Spin orbital pair energies

We now investigate this invariance problem with our minimal basis model of two noninteracting H2 molecules. The following example also provides us with an opportunity to introduce the concept of spin-adapted pair correlation energies. Up to this point we have considered spin-orbital pair energies. For example, in a 4-electron system with and 2 doubly occupied, we must calculate six spin-orbital pair energies, i.e., e, 12, 12, T2> i2> 22- However, not all of the corresponding spin-orbital pair... 

Koopmans theorem. According to the concept of molecular orbitals developed from atomic orbitals (L.C.A.O.— M.O.) the electrons are allotted one at a time to spin orbitals of energy which occur in pairs... 

The Cl(2p) and Mo(3d) x-ray photoelectron spectra (XPS) of 15 were kindly recorded by Prof. R.E. McCarley and Mr. M. Luly. The chlorine XPS, displayed in Fig. 12, clearly shows two types of Cl atoms (each peak is a doublet composed of the 2P3/2, 2Pi/ spin-orbit pair) separated by 1.2 eV. The higher energy peak (198.0 eV referred to C(1s) at 285.0 eV) may be attributed to a bridging Cl-atom and the lower energy peak (197.6 eV) to a terminal chlorine. Structure XII is assigned to 15, even though only one Mo(3d) peak could be resolved in the Mo XPS (the energy separation may be too small to resolve due to the broad peaks). 

We call jEcorr the correlation energy and the contribution of the spin orbital pair y>t, y> ) to the correlation energy. With this definition the correlation energy can be expressed exactly as a sum of pair contributions. So far we have only been concerned with the definition of the Sij and we have not learned anything about how to calculate the sij. Still in the same philosophy, we introduce some other convenient definitions. 

With the STO-3G minimal basis set, the above correlation energy is -0.0258 a.u. as compared to —0.0275 a.u. obtained using spin-orbital pairs. Thus pair theory is not only variant to unitary transformations of degenerate HF... 

An impressive success of the lEPA is its prediction of the ground state energy of the beryllium atom. The spin-orbital pair correlation energies obtained by... 

Table 5.1 Spin-orbital pair correlation energies (a.u.) of Be ...

Spin-orbit splittings as well as binding energies of a particular electron level increase with increasing atomic number. The intensity ratio of the peaks from a spin-orbit doublet is determined by the multiplicity of the corresponding levels, equal to 2j + 1. Hence, the intensity ratio of the j = and j = components of the Rh 3d doublet is 6 4 or 3 2. Thus, photoelectron peaks from core levels come in pairs -doublets - except for s levels, which normally give a single peak. 

Electrons must occupy all the orbitals of a given energy level singly with the same spin before pairing begins (Hund s Rule). 

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