Pauli correlation

The physical interpretation of the derivative vx(r) isthat it is the sum of two terms, one representative of Pauli correlations Wx (r), and the other W (r) part of the contribution of the correlation-kinetic effects19 ... 

The physical interpretation of the functional derivative vx(r) shows that it is comprised of a term Wx (r) representative of Pauli correlations, and a term wj (r) that constitutes part of the total correlation-kinetic contribution Wt (r). cThe exact asymptotic structure of these components in the vacuum has been determined and shown to also be image-potential-like. Although the structure of vx(r) about the surface and asymptotically in the vacuum and metal-bulk regions is comprised primarily of its Pauli component, the correlation-kinetic contribution is not insignificant for medium and low density metals. It is only for high density systems (rs < 2) that vx(r) is represented essentially by its Pauli component Wx (r). Thus, we see that the uniform electron gas result of -kF/ir for the functional derivative vx(r), which is the asymptotic metal-bulk value, is not a consequence of Pauli correlations alone as is thought to be the case. There is also a small correlation-kinetic contribution. The Pauli and correlation-kinetic contributions have now been quantified. 

As a result of the Pauli correlation, the effective Coulomb interaction between a 2s electron of spin a, say, and a Isa electron will be different from the interaction with a IsyS electron. Therefore, the Is spin-orbitals will have slightly different spatial parts. This is considered in the so-called unrestricted Hartree—Fock calculations. 

M.K. HARBOLA and V. SAHNI, Theories of electronic structure in the Pauli-correlated approximation. J. Chem. Educ., 70, 920 (1993). 

Independent electron models are only an approximation. Any effect not included within a particular independent electron model is called a correlation. Note that electron correlations are defined with respect to a specific model and therefore depend on the model used. Thus exchange forces appear as a Pauli correlation in Hartree s model. The main effect of Pauli correlations is to reduce the probability of electrons with parallel spins approaching each other. Owing to this reduction, each electron seems to be surrounded by a hole or a space devoid of other electrons. 

Table 3. Total ground-state energies of noble gas and closed s-subshell atoms as determined within Slater theory, the Work-interpretation Pauli-correlated approximation, and Hartree-Fock theory. The negative values of the energies in atomic units are quoted...

Fig. 6. Ground-state energy differences (in ppm) of Slater theory and the Work-interpretation Pauli-correlated approximation with respect to Hartree-Fock theory...

As shown above, the differential equation of Eq. (102) is the Kohn-Sham equation for asymptotic positions of the electron. However, this is also the differential equation when all Coulomb correlations are neglected, i.e. when the fields e(f) and Z, (r) vanish. (Recall that the difference between the work WP(r) and the functional derivative vP(r) arises from Coulomb correlations.) We therefore refer to this equation as being that of the Work-interpretation Pauli-correlated approximation. It is also the differential equation originally proposed by Harbola and Sahni [9] for the case when only Pauli-correlations are considered to be present. 

Table 1. Comparison of the highest occupied eigenvalue of the Work-interpretation Pauli-correlated approximation for atoms with those of exact (fully-correlated) Kohn-Sham theory. The negative values in Rydbergs are quoted...

Comparison of Highest-Occupied Eigenvalues of the Work-Interpretation Pauli-Correlated Ap oximation with Exact Removal Enei es... 

The parameters, and, are given in table 2, where i = Pauli, SRD, or the Pauli-SRD cross term. Effects due to variations in the nudear surface region are estimated by allowing the Pauli correlation range to depend on the local value of the density, where Ro.pauii V5lk (r) and... 

The correlation potential in eq. (3.97) corresponds to that in eqs. (42), (43) and (44) in ref. [Ra79] for the Pauli, SRD, and Pauli-SRD cross term, respectively. I ng range correlations due to center-of-mass constraints and intrinsic, permanent deformations have also been included as in eq. (50) in ref. [Ra79] and eqs. (7) and (8) in ref. [Ra83c], respectively. Following a similar derivation, the local form of the Pauli correlation contribution (largest part) to the spin-orbit potential can be obtained the explidt result is in eq. (56) in ref. [Ra79]. The correlation potentials in these references... 

This may also be called Pauli correlation because it correlates the dynamics of electrons of like spin. Obviously electrons also repel each other because of their charge. 

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