Pauli matrices corrections

Due to the integral approximations used in the MNDO model, closed-shell Pauli exchange repulsions are not represented in the Hamiltonian, but are only included indirectly, e.g., through the effective atom-pair correction terms to the core-core repulsions [12], To account for Pauli repulsions more properly, the NDDO-based OM1 and OM2 methods [23-25] incorporate orthogonalization terms into the one-center or the one- and two-center one-electron matrix elements, respectively. Similar correction terms have also been used at the INDO level [27-31] and probably contribute to the success of methods such as MSINDO [29-31],... 

Even if the g-density is A -representable, this Q-matrix is not A -representable because its largest eigenvalue exceeds the upper bound N /Q N — Q+ ). (That is, this g-matrix violates the Pauli exclusion principle for g-tuples of electrons.) Approximating the correction term, Tp Jpg[ (]], seems difficult, and neglecting this term would give poor results, although the results improve with increasing Q [2, 10]. 

A simple and asymptotically correct [26-28] model is the Ruderman-Kittel-Kasuya-Yosida or RKKY exchange between two localized moments in a Pauli-paramagnetic matrix. For a free-electron gas of wave-vector kF,... 

Unfortunately, this does not give the correct answer, giving instead the state where all the electrons are in the lowest energy Kohn-Sham orbital this violates the Pauli exclusion principle. Satisfying the Pauli exclusion principle requires that every state of the system be occupied by no fewer than zero and no more than two electrons (one with spin a and one with spin ft). This indicates that the eigenvalues of the first-order density matrix [it follows from the defining Eq. (64) that the eigenvectors of y(r,r ) are the Kohn-Sham orbitals]... 

We see then that the G-matrix for finite nuclei is expressed as the sum of two terms the first term is the free G-matrix with no Pauli corrections included, while the second term accounts for medium... 

We have provided a pedagogical derivation of the traditional, nonrelativistic form of multiple scattering theory based on the optical potential formalism. We have also discussed in detail each of the important advances made over the past ten years in the numerical application of the NR formalism. These include the full-folding calculation of the first-order optical potential, off-shell NN t-matrix contributions, relativistic kinematics and Lorentz boost of the NN t-matrix, electromagnetic effects, medium corrections arising from Pauli blocking and binding potentials in intermediate states, nucleon... 

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