Overlap Pauli exclusion principle

Although (l)-(4) involve polarization and other terms that are reminiscent of a classical dipole-dipole picture, it must be re-emphasized that the nB-ffAH picture is formulated entirely in the quantal framework (including full consistency with the Pauli exclusion principle). Thus, while vague connections to concepts of classical electrostatics can be drawn, the NBO donor-acceptor picture of H-bonding is essentially based on overlap-type ionic resonance (5.29a), not on ionic forces (or the like) of classical type. 

In addition to the Coulombic forces, there is a repulsive force which operates at short distances between ions as a result of the overlapping of filled orbitals, potentially a violation of the Pauli exclusion principle. This repulsive force may be represented by the equation ... 

The nuclei of the atoms in a solid and the inner electrons form ion cores with energy levels little different from corresponding levels in free atoms. The characteristics of the valence electrons arc modified greatly, however. The stale functions of these outer electrons greatly overlap those of neighboring atoms. Restrictions of the Pauli Exclusion Principle and the Uncertainty Principle force modification of the state functions, and the development of a set of split energy levels becomes a quasi-continuous band of levels of width, which are several electron volts for most solids. Importantly, unoccupied levels of the atoms are also split into bands. The electronic characteristics of solids are determined by the relative position in energy of the occupied and unoccupied levels as well as by die characteristics of the electrons within a band. 

The sharing of three pairs of elections between two atoms can be accomplished by extrapolation of die above considerations. That is, since there can only be one o bond connecting the atoms, then die othei two pairs of shared electrons must be in two diffeient tc bonds, each of which is formed by the parallel overlap of a p orbital. Furthermore die n bonds must be mutually orthogonal so as not to violate the Pauli exclusion principle. Hybridization of one s orbital and one p oibital gives two equivalent sp hybiid AOs which are linearly opposite to one anodier. 

At shorter distances the repulsive forces start to dominate. The repulsive interaction between two molecules can be described by the power-law potential l/rn (n>9) caused by overlapping of electron clouds resulting in a conflict with the Pauli exclusion principle. For a completely rigid tip and sample whose atoms interact as 1/r12, the repulsion would be described by W-l/D7. In practice, both the tip and the sample are deformable (Fig. 3d). The tip-sample attraction is balanced by mechanical stress which arises in the contact area. From the Hertz theory [77,79], the relation between the deformation force Fd and the contact radius a is given by ... 

The upper drawing in Fig. 34 is a schematic, electron-domain representation of the spin-density in a plane through two neighboring comer atoms and the adjacent center atom in Slater s model of the alkali metals. Solid circles represent the atoms kernels (M+ cations). The Pauli Exclusion Principle permits domains occupied by electrons of opposite spin to overlap (comer atoms with the central atom), but prohibits overlap between domains occupied by electrons of the same spin (comer atoms with comer atoms). 

Obviously two particles cannot overlap significantly (the Pauli Exclusion Principle and our experience tells us this). Thus, we have the boundary condition... 

Each atom in a bar of sodium has the same outer 3s orbital containing one electron. The individual orbitals of the atoms in the bar overlap, creating a huge number of molecular orbitals. These groups of closely spaced energy states are called energy bands. Molecular orbitals within those bands, however, must obey the Pauli exclusion principle. So each one of this huge number of... 

If the wave function is the exact wave function, we obtain for E the exact (nonrelativistic) energy. If is an approximate wave function, the variational principle (Levine, 1983) tells us that the lower the E the more closely il> resembles the exact wave function, so long as it satisfies certain conditions, the most important being the Pauli exclusion principle. If i ( contains parameters, then that choice of parameters giving the lowest E will give the best wave function in the sense of maximum overlap with the true wave function. 

It is concluded that total nonadditive effects are dominated by SCF terms, which are directly attributed to electric polarization. However, since the polarization is constrained by the Pauli exclusion principle, classical models which neglect exchange phenomena may incur certain errors, especially in regions of strong overlap between electron clouds of the monomers. Correlation contributions to nonadditivity do appear small enough that they can be safely ignored, with efforts better concentrated on an accurate portrayal of the SCF phenomena. 

As two atoms approach each other, some degree of merging of the particles occurs. As the particles merge, the electron orbits from the electrons of each atom begin to overlap, and a limitation in the occupation of available electron states arises due to the Pauli Exclusion Principle. Under such circumstances, the... 

Van der Waals forces The term van der Waals forces denotes the short-range interactions between closed-shell molecules. Van der Waals forces include attractive forces arising from interactions between the partial electric charges and repulsive forces arising from the Pauli exclusion principle and the exclusion of electrons in overlapping orbitals. A very commonly used potential is the so-called Lennard-Jones (6-12) potential to describe the attractive and repulsive components of van der Waals forces. 

A further possible mechanism is provided by the overlap of the charge clouds for the unpaired electron on the radical with the charge clouds for the filled orbitals of the molecule. The operation of the Pauli exclusion principle tends to keep electrons of parallel spin apart, and there will be a contraction of the orbitals containing electrons with spins parallel to the unpaired spin of the radical and a corresponding expansion of the orbital containing the antiparallel spin (Fig. 16). 

As the crystal grows, every new atom added brings one orbital to the bonding and one to the antibonding orbital set. As the orbitals or electron wave functions overlap, they must broaden as shown in Fig. 2.lie. because of the Pauli exclusion principle. 

In chapter i we discussed the interaction between two widely separated water molecules and concluded that it could conveniently be treated in terms of (i) the interaction between the permanent electric moments of the molecules, (ii) the polarization or delocalization terms due to the distortion of the electron cloud of one molecule by the field of the other, (iii) dispersion forces having their origin in co-ordinated electronic motion in the two molecules, and (iv) the repulsion due to the action of the Pauli exclusion principle when the two electron clouds begin to overlap. Most calculations of the cohesive energy of the ice crystal amount to an evaluation of these contributions for more or less realistic models. Among these it is convenient to distinguish two different types ... 

The second method is based on the continuum representation of the solvent and the two contributions are treated separately. The starting point is the genersd expression derived from the intermolecular forces. In the previous discrete approach, however, this equation is applied to calculated or estimated potentials, available from the literature. We now substitute for a suitable expression taken directly from the theory considering that, as originated from the Pauli exclusion principle, the repulsion forces between two interacting molecules increase with the overlap of the two distributions and they are strictly related to the density of electrons with the same spin. 

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