Natural orbital function characterized

D-functional theory, explicitly given in terms of natural orbitals and their occupation numbers, has emerged as an alternative method to conventional approaches for considering the electronic correlation. This chapter has introduced important basic concepts for understanding the NOF formalism. We have also offered a brief characterization of almost all references concerning this theory published hitherto. 

This shows that p is similar to the well-known one-particle density of electronic structure theory. Diagonalizing the operator p yields the natural populations and the natural orbitals" defined as the eigenvalues and eigenvectors of p(p). Since we are dealing with distinguishable particles we have a separate density matrix for each degree of freedom. The natural populations characterize the contribution of the related natural orbital to the MCTDH wave function. Small natural populations therefore indicate that the MCTDH expansion converges. The natural populations thus provide us with... 

Secondly, we note that the density matrix may become singular. is a Hermitian, positive semidefinite matrix and - as mentioned above - its eigenvalues, called natural weights, characterize the importance of the corresponding natural orbital. A zero eigenvalue occurs if there is a natural orbital (i.e., a linear combination of the single-particle functions) that does not contribute to the MCTDH wave function. Its time evolution may thus be modified by replacing with p( )... 

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... 

Dimeric 7r-arene-bridged samarium anilide [Sm(NHArl Pr,H)3]2 was employed in the isolation and characterization of a dimeric amido-imido compound. Addition of 4 equivalents of TMA led to (1) cleavage of one of the Ln-NHPh bonds, (2) deprotonation, and (3) adduct formation. Density functional theory calculations, carried out in order better to understand the nature of the imido bonding, revealed the presence of Sm - N 7r-bonding interactions involving the samarium 5d orbitals [218]. 

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