Inter nuclear repulsion

If the total attraction energy exceeds the inter-nuclear repulsion, there will be a net bonding effect and the molecule will be stable. If, on the other hand, the electron is off to one side, it will attract both nuclei, but it will attract the closer one much more strongly, owing to the inverse-square nature of Coulomb s law. As a consequence, the electron will now help the electrostatic repulsion to push the two nuclei apart. 

The empirical finding that increased strength of multiple bonds over electron-pair bonds is mainly caused by an increase in single-bond strength at the closer interatomic approach that becomes possible due to screening of inter-nuclear repulsion, can now be examined more closely. Whenever the number of valence-electron pairs on an atom exceeds the number of electron-pair bonds to that atom, the excess density may screen the nucleus. Screening becomes effective when the excess density occurs in atomic s-type states with an appreciable s contribution. The first-period diatomics considered before illustrate this screening condition well. 

T = is the inverse of the overlap matrix. The total energy is obtained by adding the inter-nuclear repulsion energy. 

This expression involves no reference to a wave function, and electronic information is provided by the electronic density p(r), described in the Dirac delta formalism as given in Eq. (303). The computation of the expectation value force operator requires only a simple sum of a classical contribution from the electronic charge density and the inter-nuclear repulsion. 

FIGURE 3.11 The forces between the particles in Hj. (a) The inter-nuclear repulsion always opposes bonding the nuclei together, (b) An electron positioned in a region that will tend to bond the nuclei together, (c) An electron positioned in a region that will tend to pull the nuclei apart. 

Pack and Hirschfelder (133) discuss the energy corrections to the Bom-Oppenheimer approximation, while Bunker (134) discusses the partial breakdown of the Bom-Oppenheimer approximation. The Hell-mann-Feynman theorem is also discussed by Tuan (135) with reference to multiconfiguration SCF-theory, while Loeb and Rasiel (136) discuss constraints upon the LiH molecule with reference to the Hellmann-Fe5mman theorem. King (137) developed a theory of effective Cartesian force constants which relate to the sum of the squares of the normal frequencies these effective force constants are independent of the inter-nuclear repulsion of the nuclei in a molecule, and thus removes one of the indeterminacy of the Hellmann-Feynman method described above. 

In deriving the molecular electronic virial theorem (14.25), we omitted the inter-nuclear repulsion... 

The expression for the Hartree-Fock molecular electronic energy hf is given by the variation theorem as = D H i + FawI-D), where D is the Slater-determinant Hartree-Fock wave function, and the purely electronic Hamiltonian and the inter-nuclear repulsion are given by (13.5) and (13.6). Since doesn t involve electronic coordinates and D is normalized, we have D Vnn D) = Vnn D D) = The operator is the sum of one-electron operators /, and two electron operators gj/, we have gij, where (in atomic units)... 

FIGURE 7.2 A graph of potential energy versus inter-nuclear distance for the H2 molecule. If the hydrogen atoms are too far apart, attractions are weak and no bonding occurs. If the atoms are too close, strong repulsions occur. When the atoms are optimally separated, the energy is at a minimum. 

All the necessary elements can be calculated in terms of integrals over the atomic orbitals x, integrals which involve either the nuclear attraction operators H v) or the inter electronic repulsions 1/r y. 

We have employed the PPP model with independent bond distortions for a fixed electron-phonon coupling constant. The distortions of the bond lead to change in the corresponding transfer integrals. The bond distortions are introduced such that the total bond-order of the chain remains a constant. In the absence of this constraint, the purely tt electron Hamiltonian would lead to a collapse of the chain to a point since the total energy of the system tends to decrease with decrease in chain length equivalently, the tt electron Hamiltonian does not have an in-built repulsive term which keeps the atoms apartsince this term comes from the a framework and the internuclear potentials. Imposition of the constraint of constant total bond-order serves the purpose of the a framework and the inter-nuclear potentials. 

Square well model This is a mathematical simplification of the more realistic model possessing finite repulsions and attractions. The region of attraction is restricted to a range bounded by discontinuities at two inter-nuclear separations at the smaller of these, infinite repulsion occurs, and beyond the larger no interaction exists. 

The nuclear charge increases by one in going from Na to Mg, and the increased electron-nuclear attraction more than offsets the effects of inter-electron repulsion. 

Although not errtirely known the HK fimctiorral has a remarkably property it is imiversally, in a sense that both the kinetic and inter-electrorric repulsion are independent of the corrcemed system. The consequence of such imiversal nature offers the possibility that once it is exactly or approximately knew the HK fimctiorral for a given external potential V r) remain valuable for any other type of potential F (r) applied on the concerned many-electronic system. Let s note the fact that V(r) should be not reduced only to the Coulombic type of potentials but is carrying the role of the generic potential applied, that could beg of either an electric, magnetic, nuclear, or even electronic nature as far it is external to the system fixed by the N electrons in the investigated system. 

Applications of time-dependent DFT to the calculation of polarizabilities have proved promising. has also been successful in deriving non-bonded interaction parameters for molecular force fields. The attractive term in the Lennard-Jones or Lennard-Jbnes-like potential was obtained from the calculated long-range dipole-dipole dispersion term (R R is the inter-nuclear separation). The repulsive term can be obtained by fitting the potential to experimental information about internuclear separations or some other size-dependent parameter such as the van der Waals radius. ... 

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