Nuclear repulsion

There can be subtle but important non-adiabatic effects [14, ll], due to the non-exactness of the separability of the nuclei and electrons. These are treated elsewhere in this Encyclopedia.) The potential fiinction V(R) is detennined by repeatedly solving the quantum mechanical electronic problem at different values of R. Physically, the variation of V(R) is due to the fact that the electronic cloud adjusts to different values of the intemuclear separation in a subtle interplay of mutual particle attractions and repulsions electron-electron repulsions, nuclear-nuclear repulsions and electron-nuclear attractions. 

Note the stnicPiral similarity between equation (A1.6.72) and equation (Al.6.41). witii and E being replaced by and the BO Hamiltonians governing the quanPim mechanical evolution in electronic states a and b, respectively. These Hamiltonians consist of a nuclear kinetic energy part and a potential energy part which derives from nuclear-electron attraction and nuclear-nuclear repulsion, which differs in the two electronic states. 

Here, f/ei is the electionic Hamiltonian including the nuclear-nuclear repulsion terms, Pji is a Caitesian component of the momentum, and Mi the mass of nucleus /. One should note that the bra depends on z while the ket depends on z and that the primed R and P equal their unprimed counterparts and the prime simply denotes that they belong to the bra. 

The iotal energy of a system is equal to the sum of the electronic energy and the Coulombic nuclear repulsion energy ... 

Since the nuclear-nuclear repulsion is constant for a fixed configuration ofatoms, this term also drops out. The Hamiltonian isnow purely electronic. 

After solving the electronic Schrodinger equation (equation 4), to calculate a potential energy surface, you must add back nuclear-nuclear repulsions (equation 5). 

The potential energy of vibration is a function of the coordinates, xj,. .., z hence it is a function of the mass-weighted coordinates, qj,. .., q3N. For a molecule, the vibrational potential energy, U, is given by the sum of the electronic energy and the nuclear repulsion energy ... 

Physically, the strength of the spring representing the bond is affected by a subtle balance of nuclear repulsions, electron repulsions and electron-nuclear attractions. None of these is affected by nuclear mass and, therefore, k is not affected by isotopic substitution. 

That is, the sum of the electronic energy and nuclear repulsion energy of the molecule at the specified nuclear configuration. This quantity is commonly referred to as the total energy. However, more complete and accurate energy predictions require a thermal or zero-point energy correction (see Chapter 4, p. 68). 

The first term corresponds to electron-nuclear attraction, the second to electron-electron repulsion, and the third to nuclear-nuclear repulsion. 

All terms except the nuclear-nuclear repulsion are functions of p, the electron density E is given by the following expression ... 

Finally we have to remember to add on the nuclear repulsion and, if we repeat the calculation for a range of values of the internuclear separation, we arrive at the potential energy curves shown in Figure 4.3 for the ground-state (singlet)... 

This potential is referred to in electromagnetism texts as the retarded potential. It gives a clue as to why a complete relativistic treatment of the many-body problem has never been given. A theory due to Darwin and Breit suggests that the Hamiltonian can indeed be written as a sum of nuclear-nuclear repulsions, electron-nuclear attractions and electron-electron repulsions. But these terms are only the leading terms in an infinite expansion. 

Consider now the Hamilton operator. The nuclear-nuclear repulsion does not depend on electron coordinates and is a constant for a given nuclear geometry. The nuclear-electron attraction is a sum of terms, each depending only on one electron coordinate. The same holds for the electron kinetic energy. The electron-electron repulsion, however, depends on two electron coordinates. 

The nuclear repulsion operator does not depend on electron coordinates and can immediately be integrated to yield a constant. 

The derivative in eq. (10.88) of the nuclear repulsion (third term) is trivial since it does not involve electron coordinates. The one-electron derivatives are given as... 

The individual terms in (5.2) and (5.3) represent the nuclear-nuclear repulsion, the electronic kinetic energy, the electron-nuclear attraction, and the electron-electron repulsion, respectively. Thus, the BO Hamiltonian is of treacherous simplicity it merely contains the pairwise electrostatic interactions between the charged particles together with the kinetic energy of the electrons. Yet, the BO Hamiltonian provides a highly accurate description of molecules. Unless very heavy elements are involved, the exact solutions of the BO Hamiltonian allows for the prediction of molecular phenomena with spectroscopic accuracy that is... 

The solution of the Schrodinger equation with Helec is the electronic wave function xTelec and the electronic energy Eelec. depends on the electron coordinates, while the nuclear coordinates enter only parametrically and do not explicitly appear in Pelec. The total energy Etot is then the sum of Eelec and the constant nuclear repulsion term, M M... 

Still, the reasons for this success were not quite clear for instance, the good performance of the method in reproducing the equilibrium C—H bond distance in methane was thought to depend on fortuitous cancellation of the nuclear repulsion and of the change in... 

Molecular mechanics force fields rest on four fundamental principles. The first principle is derived from the Bom-Oppenheimer approximation. Electrons have much lower mass than nuclei and move at much greater velocity. The velocity is sufficiently different that the nuclei can be considered stationary on a relative scale. In effect, the electronic and nuclear motions are uncoupled, and they can be treated separately. Unlike quantum mechanics, which is involved in determining the probability of electron distribution, molecular mechanics focuses instead on the location of the nuclei. Based on both theory and experiment, a set of equations are used to account for the electronic-nuclear attraction, nuclear-nuclear repulsion, and covalent bonding. Electrons are not directly taken into account, but they are considered indirectly or implicitly through the use of potential energy equations. This approach creates a mathematical model of molecular structures which is intuitively clear and readily available for fast computations. The set of equations and constants is defined as the force... 

The discussion thus far has focused on the forces between an array of atoms connected together through covalent bonds and their angles. Important interactions occur between atoms not directly bonded together. The theoretical explanation for attractive and repulsive forces for nonbonded atoms i and j is based on electron distributions. The motion of electrons about a nucleus creates instantaneous dipoles. The instantaneous dipoles on atom i induce dipoles of opposite polarity on atom j. The interactions between the instantaneous dipole on atom i with the induced instantaneous dipole on atom j of the two electron clouds of nonbonded atoms are responsible for attractive interactions. The attractive interactions are know as London Dispersion forces,70 which are related to r 6, where r is the distance between nonbonded atoms i and j. As the two electron clouds of nonbonded atoms i and j approach one another, they start to overlap. There is a point where electron-electron and nuclear-nuclear repulsion of like charges overwhelms the London Dispersion forces.33 The repulsive... 

The hydrogen molecule is the simplest molecule that contains an electron-pair bond. The electronic Hamiltonian for H2, with a term for nuclear repulsion, is... 

The potential energy function U(R) that appears in the nuclear Schrodinger equation is the sum of the electronic energy and the nuclear repulsion. The simplest case is that of a diatomic molecule, which has one internal nuclear coordinate, the separation R of the two nuclei. A typical shape for U(R) is shown in Fig. 19.1. For small separations the nuclear repulsion, which goes like 1 /R, dominates, and liniR >o U(R) = oo. For large separations the molecule dissociates, and U(R) tends towards the sum of the energies of the two separated atoms. For a stable molecule in its electronic ground state U(R) has a minimum at a position Re, the equilibrium separation. 

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