Electron repulsion approximations

The HF equations are approximate mainly because they treat electron-electron repulsion approximately (other approximations are mentioned in the answer suggested for Chapter 5, Harder Question 1). This repulsion is approximated as resulting from interaction between two charge clouds rather than correctly, as the force between each pair of point-charge electrons. The equations become more exact as one increases the number of determinants representing the wavefunctions (as well as the size of the basis set), but this takes us into post-Hartree-Fock equations. Solutions to the HF equations are exact because the mathematics of the solution method is rigorous successive iterations (the SCF method) approach an exact solution (within the limits of the finite basis set) to the equations, i.e. an exact value of the (approximate ) wavefunction l m.. 

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. 

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. 

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 ... 

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. 

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. 

Many problems with MNDO involve cases where the NDO approximation electron-electron repulsion is most important. AMI is an improvement over MNDO, even though it uses the same basic approximation. It is generally the most accurate semi-empirical method in HyperChem and is the method of choice for most problems. Altering part of the theoretical framework (the function describing repulsion between atomic cores) and assigning new parameters improves the performance of AMI. It deals with hydrogen bonds properly, produces accurate predictions of activation barriers for many reactions, and predicts heats of formation of molecules with an error that is about 40 percent smaller than with MNDO. 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

As the distance between the two particles varies, they are subject to these long-range r " attractive forces (which some authors refer to collectively as van der Waals forces). Upon very close approach they will experience a repulsive force due to electron-electron repulsion. This repulsive interaction is not theoretically well characterized, and it is usually approximated by an empirical reciprocal power of distance of separation. The net potential energy is then a balance of the attractive and repulsive components, often described by Eq. (8-16), the Lennard-Jones 6-12 potential. 

You will see shortly that an exact solution of the electronic Schrodinger equation is impossible, because of the electron-electron repulsion term g(ri, r2). What we have to do is investigate approximate solutions based on chemical intuition, and then refine these models, typically using the variation principle, until we attain the required accuracy. This means in particular that any approximate solution will not satisfy the electronic Schrodinger equation, and we will not be able to calculate the energy from an eigenvalue equation. First of all, let s see why the problem is so difficult. 

The necessity for going beyond the HF approximation is the fact that electrons are further apart than described by the product of their orbital densities, i.e. their motions are con-elated. This arises from the electron-electron repulsion operator, which is a sum of ten-ns of the type... 

As well as being attracted to the nucleus, each electron in a many-electron atom is repelled by the other electrons present. As a result, it is less tightly bound to the nucleus than it would be if those other electrons were absent. We say that each electron is shielded from the full attraction of the nucleus by the other electrons in the atom. The shielding effectively reduces the pull of the nucleus on an electron. The effective nuclear charge, Z lle, experienced by the electron is always less than the actual nuclear charge, Ze, because the electron-electron repulsions work against the pull of the nucleus. A very approximate form of the energy of an electron in a many-electron atom is a version of Eq. 14b in which the true atomic number is replaced by the effective atomic number ... 

Valence The highest-energy electrons in an atom, which an atom loses, gains, or shares in forming a chemical bond. Valence shell electron-pair repulsion (VSEPR) A procedure based on electron repulsion in molecules that enables chemists to predict approximate bond angles. 

Consider a crude approximation to the ground state of the lithium atom in which the electron-electron repulsions are neglected. Construct the ground-state wave function in terms of the hydrogen-like atomic orbitals. 

An important consequence of the only approximate treatment of the electron-electron repulsion is that the true wave function of a many electron system is never a single Slater determinant We may ask now if SD is not the exact wave function of N interacting electrons, is there any other (necessarily artificial model) system of which it is the correct wave function The answer is Yes it can easily be shown that a Slater determinant is indeed an eigenfunction of a Hamilton operator defined as the sum of the Fock operators of equation (1-25)... 

Other approximate, more empirical methods are the extended Huckel 31> and hybrid-based Hiickel 32. 3> approaches. In these methods the electron repulsion is not taken into account explicitly. These are extensions of the early Huckel molecular orbitals 4> which have successfully been used in the n electron system of planar molecules. On account of the simplest feature of calculation, the Hiickel method has made possible the first quantum mechanical interpretation of the classical electronic theory of organic chemistry and has given a reasonable explanation for the chemical reactivity of sizable conjugated molecules. 

As noted above, many integrals describing electron repulsion are very small in magnitude, especially those such as 4> (1)< jv(1) when p v. The simplest semi-empirical approach, termed the zero-differential overlap (ZDO) approximation, is therefore to assume that these integrals can be ignored. Mathematically expressed, this is equivalent to the following ... 

The purpose of most quantum chemical methods is to solve the time-independent Schrodinger equation. Given that the nuclei are much more heavier than the electrons, the nuclear and electronic motions can generally be treated separately (Born-Oppenheimer approximation). Within this approximation, one has to solve the electronic Schrodinger equation. Because of the presence of electron repulsion terms, this equation cannot be solved exactly for molecules with more than one electron. 

Two different correlation effects can be distinguished. The first one, called dynamical electron correlation, comes from the fact that in the Hartree-Fock approximation the instantaneous electron repulsion is not taken into account. The nondynamical electron correlation arises when several electron configurations are nearly degenerate and are strongly mixed in the wave function. 

While the enthalpy of formation is the property of interest in chemical thermodynamics of materials, many books focus on the lattice enthalpy when considering trends in stability. The static non-vibrational part of the lattice enthalpy can be deconvoluted into contributions of electrostatic nature, due to electron-electron repulsion, dispersion or van der Waals attraction, polarization and crystal field effects. The lattice enthalpy is in the 0 K approximation given as a sum of the potential energies of the different contributions ... 

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