Natural general spin orbitals

An appealing way to apply the constraint expressed in Eq. (3.14) is to make connection with Natural Orbitals (31), in particular, to express p as a functional of the occupation numbers, n, and Natural General Spin Ckbitals (NGSO s), yr,, of the First Order Reduced Density Operator (FORDO) associated with the N-particle state appearing in the energy expression Eq. (3.8). In order to introduce the variables n and yr, in a well-defined manner, the... 

This arbitrariness most clearly manifests itself in going beyond the scope of the HF approximation, as evidenced by a wide variety of definitions for molecular systems available in the literature for valences and bond orders in the case of post-Hartree-Fock methods for molecular systems [570,578-580]. In post-HF methods local characteristics of molecular electronic structure are usually defined in terms of the first-order density matrix and in this sense there is no conceptual difference between HF and post-HF approaches [577]. It is convenient to introduce natural (molecular) spin orbitals (NSOs), i.e. those that diagonalise the one-particle density matrix. The first-order density matrix in the most general case represents some ensemble of one-electron states described by NSOs... 

Before proving this theorem, we will make some general remarks about the nature of the one-electron functions ipk(x) or spin orbitals. For the two values of the spin coordinate f — 1, such a function y)k(r, f) has two space components... 

This is the most general expression obtained from a set of natural spin orbitals written in spinor form as... 

General relationships between AOM and crystal field parameters are shown in Table 23. Using the AOM one can easily compute the electronic energy levels, inclusive of spin-orbit coupling, without any symmetry assumption or perturbation procedure, and it is also easy to account for the different chemical natures of the ligands and for differences in bond distances. It is also possible to handle anisotropic n interactions, which can be expected to occur with pyridine or pyridine iV-oxide ligands.366,367 General review articles on the AOM and its applications have already appeared.364,368-371... 

The general setting of the electronic structure description given above refers to a complete (and thus infinite) basis set of one-electron functions (spin-orbitals) (f>nwave functions, an additional assumption is made, which is that the orbitals entering eq. (1.136) are taken from a finite set of functions somehow related to the molecular problem under consideration. The most widespread approximation of that sort is to use the atomic orbitals (AO).17 This approximation states that with every problem of molecular electronic structure one can naturally relate a set of functions y/((r). // = M > N -atomic orbitals (AOs) centered at the nuclei forming the system. The orthogonality in general does not take place for these functions and the set y/ is characterized... 

Calculations on the ground state of Ha, at the SCF and SCF-CI levels, but using a basis of generalized diatomic orbitals, have been reported by Helfrich.43 The resulting wavefunction was then subjected to a natural spin orbital (NSO) analysis. 

The nature of the CFI in compounds of samarium, such as SmAl2 (151,152), SmPt2 (153,154), SmAl3 (155), SmSn3 (153,156-158) and SmX (X = P, As, Sb, Bi) (159,160), have been examined in detail and reviewed critically in recent years (2,3). In general, it has been observed that the spin-orbit interaction term plays a significant role, and in cubic compounds the effects of both fourth and... 

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