Electronic repulsion

Racah parameters The parameters used to express quantitatively the inter-electronic repulsion between the various energy levels of an atom. Generally expressed as B and C. The ratios between B in a compound and B in the free ion give a measure of the nephelauxetic effect. ... 

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. 

Because the electron-electron repulsion is less effective in the triplet state, it will nonnally be lower in energy than the corresponding singlet state. 

Parallel molecular dynamics codes are distinguished by their methods of dividing the force evaluation workload among the processors (or nodes). The force evaluation is naturally divided into bonded terms, approximating the effects of covalent bonds and involving up to four nearby atoms, and pairwise nonbonded terms, which account for the electrostatic, dispersive, and electronic repulsion interactions between atoms that are not covalently bonded. The nonbonded forces involve interactions between all pairs of particles in the system and hence require time proportional to the square of the number of atoms. Even when neglected outside of a cutoff, nonbonded force evaluations represent the vast majority of work involved in a molecular dynamics simulation. 

The Extended Iliickel method, for example, does not explicitly consider the elTects of electron-electron repulsions but incorporates rep 11 Ision s into a sin gle-clectron poten tial. Th is simplifies th c solution of the Schrbdinger equation and allows IlyperChem to compute the poten tial energy as the sum of the energies for each electron. 

Th e calcn lation of the two-electron repulsion mtegraism ah iniiio method is inevitable and time-consuming. The computational iim e is main ly dom in alcd by th e performance of Ih e two-electron integral calcii lalion. The following item s can con trol the performance of the two-electron integrals. 

IlyperChem uses 16 bytes (two double-precision words) of storage for each electron repulsion integral. The first 8 bytes save thecom-pressed four indices and the second S bytes store the value of the integral. Each index lakes 16 bits. Thus the maximum number of basis fiinctions is 65,535. This should satisfy all users of IlyperChem for the foreseeable future. 

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

Ihc complete neglect of differential overlap (CNDO) approach of Pople, Santry and Segal u as the first method to implement the zero-differential overlap approximation in a practical fashion [Pople et al. 1965]. To overcome the problems of rotational invariance, the two-clectron integrals (/c/c AA), where fi and A are on different atoms A and B, were set equal to. 1 parameter which depends only on the nature of the atoms A and B and the ii ilcniuclear distance, and not on the type of orbital. The parameter can be considered 1.0 be the average electrostatic repulsion between an electron on atom A and an electron on atom B. When both atomic orbitals are on the same atom the parameter is written , A tiiid represents the average electron-electron repulsion between two electrons on an aiom A. 

III fact, while this correction gives the desired behaviour at relatively long separations, it doLS not account for the fact that as two nuclei approach each other the screening by the core electrons decreases. As the separation approaches zero the core-core repulsion iimild be described by Coulomb s law. In MINDO/3 this is achieved by making the cure-core interaction a function of the electron-electron repulsion integrals as follows ... 

In order to calculate higher-order wavefunctions we need to establish the form of the perturbation, f. This is the difference between the real Hamiltonian and the zeroth-order Hamiltonian, Remember that the Slater determinant description, based on an orbital picture of the molecule, is only an approximation. The true Hamiltonian is equal to the sum of the nuclear attraction terms and electron repulsion terms ... 

HMO method. This is because electron repulsion is taken into account in the SCF calculation whereas it is not taken into account in the Huckel calculation. 

Using Program SCF for ethylene and 1,3,5-hexatriene, list the electron repulsion integrals in the foiiii Yjj, Yj2, and so on. Take the coordinates from Figure 8-6. Try small variations in the atomic coordinates to see what their influence is on Yy. 

To see how and under what conditions stability is enhanced or diminished, we need to consider the symmetry of the orbital (9-32), Flectrons in the antisymmetric orbital r r have a 7ero probability of occurring at the node in u where U] = rj. Electron mutual avoidance of the node due to spin correlation reduces the total energy of the system because it reduces electron repulsion energy due to charge... 

One of the limitations of HF calculations is that they do not include electron correlation. This means that HF takes into account the average affect of electron repulsion, but not the explicit electron-electron interaction. Within HF theory the probability of finding an electron at some location around an atom is determined by the distance from the nucleus but not the distance to the other electrons as shown in Figure 3.1. This is not physically true, but it is the consequence of the central field approximation, which defines the HF method. 

Nonbonded interactions are the forces be tween atoms that aren t bonded to one another they may be either attractive or repulsive It often happens that the shape of a molecule may cause two atoms to be close in space even though they are sep arated from each other by many bonds Induced dipole/induced dipole interactions make van der Waals forces in alkanes weakly attractive at most distances but when two atoms are closer to each other than the sum of their van der Waals radii nuclear-nuclear and electron-electron repulsive forces between them dominate the fvan derwaais term The resulting destabilization is called van der Waals strain... 

Set this threshold to a small positive constant (the default value is 10" ° Hartree). This threshold is used by HyperChem to ignore all two-electron repulsion integrals with an absolute value less than this value. This option controls the performance of the SCF iterations and the accuracy of the wave function and energies since it can decrease the number of calculated two-electron integrals. 

The NDDO (Neglect of Diatomic Differential Overlap) approximation is the basis for the MNDO, AMI, and PM3 methods. In addition to the integralsused in the INDO methods, they have an additional class of electron repulsion integrals. This class includes the overlap density between two orbitals centered on the same atom interacting with the overlap density between two orbitals also centered on a single (but possibly different) atom. This is a significant step toward calculatin g th e effects of electron -electron in teraction s on different atoms. 

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