Two-electron approaches

Whereas the one-electron exponential form Eq. (5.5) is easily implemented for orbital-based wavefunctions, the explicit inclusion in the wavefunction of the interelectronic distance Eq. (5.6) goes beyond the orbital approximation (the determinant expansion) of standard quantum chemistry since ri2 does not factorize into one-electron functions. Still, the inclusion of a term in the wavefunction containing ri2 linearly has a dramatic impact on the ability of the wavefunction to model the electronic structure as two electrons approach each other closely. 

In this formulation of CTCB the off-diagonal orbital communications have been shown to be proportional to the corresponding Wiberg [52] or related quadratic indices of the chemical bond [53-63]. Several illustrative model applications of OCT have been presented recently [38,46-48], covering both the localized bonds in hydrides and multiple bonds in CO and C02, as well as the conjugated n bonds in simple hydrocarbons (allyl, butadiene, and benzene), for which predictions from the one- and two-electron approaches have been compared in these studies the IT bond descriptors have been generated for both the molecule as whole and its constituent fragments. 

Another more subtle condition is the electron-electron cusp condition. As two electrons approach each other, their Coulomb interaction dominates, and this leads to a cusp in the exchange-correlation hole at zero separation[15]. It is most simply expressed in terms of the pair distribution function. We define its spherically-averaged derivative at zero separation as... 

For nearly a century, the Lewis valence theory [1] and the subsequent development of the effective atomic number (BAN) rule [2, 3] as well as the valence bond theory [4] have constituted the fundamental basis concepts used for rationalizing the structure and bonding in a tremendously large area of covalent chemistry [5]. However, there are families of compounds, which have been, at least in part, reluctant to stick to this conventional two-center/two-electron approach, in particular those in which hypervalency and/or hypercoordination are present. This is the case, of... 

In Equation 1.86, we found that if we try to put two electrons with the same spin at the same point, the wave function is equal to zero. It is quite easy to see in Equation 1.87 that if the two electrons approach each other, the determinant wave function tends to zero and is proportional to the distance between them, 6. (Set 1 = fitt and 2 = 2a = (5 + 5)a in Equation 1.87 and use the Taylor expansion to get Vji( 2) = Vji(h + 5). The result is a sum of two Slater determinants where one has two columns equal and the other one is proportional to 5.) This means that the density of electrons with the same spin, that is, the absolute square of the wave function, tends to zero as 5. If the position of an electron is assumed fixed, the probability density of electrons with the same spin tends to zero near to the fixed electron. The excluded probability density amounts to a full electron, as will be proven for a Slater determinant in Chapter 2. This hole is called the exchange hole. Electrons with the same spin are thus correlated in a Slater determinant. The correlation problem is the problem of accounting for a correlated motion between the electrons. 

One particular problem that massively increases the cost of correlated calculations is the slow convergence of the correlation energy with the size of the basis set that is used for the discretization of the equations. This problem comes about because all commonly used correlation methods try to expand the wavefunction in a linear combination of antisymmetrized orbital products, i.e. Slater determinants. This ansatz, however, cannot correctly describe short-range correlation effects, i.e. the shape of the wavefunction when two electron approach each other closely, and very large expansions augmented with extrapolation techniques are needed to get a sufficiently converged correlation energy. 

Obviously, the main difficulty for obtaining an accurate solution to the corresponding (stationary) electronic Schrodinger equation arises from the Coulomb interaction between the electrons. Not only it couples all electronic motions, it also becomes singular as two electrons approach each... 

Once the requisite one- and two-electron integrals are available in the MO basis, the multiconfigurational wavefunction and energy calculation can begin. Each of these methods has its own approach to describing tlie configurations d),. j included m the calculation and how the C,.] amplitudes and the total energy E are to be... 

The ultimate approach to simulate non-adiabatic effects is tln-ough the use of a fiill Scln-ddinger wavefunction for both the nuclei and the electrons, using the adiabatic-diabatic transfomiation methods discussed above. The whole machinery of approaches to solving the Scln-ddinger wavefiinction for adiabatic problems can be used, except that the size of the wavefiinction is now essentially doubled (for problems involving two-electronic states, to account for both states). The first application of these methods for molecular dynamical problems was for the charge-transfer system... 

MMVB is a hybrid force field, which uses MM to treat the unreactive molecular framework, combined with a valence bond (VB) approach to treat the reactive part. The MM part uses the MM2 force field [58], which is well adapted for organic molecules. The VB part uses a parametrized Heisenberg spin Hamiltonian, which can be illustrated by considering a two orbital, two electron description of a sigma bond described by the VB determinants... 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

In the study of (electronic) curve crossing problems, one distinguishes between a situation where two electronic curves, Ej R), j — 1,2, approach each other at a point R = Rq so that the difference AE[R = Rq) = E iR = Rq) — Fi is relatively small and a situation where the two electronic curves interact so that AE R) Const is relatively large. The first case is usually treated by the Landau-Zener fonnula [87-92] and the second is based on the Demkov approach [93]. It is well known that whereas the Landau-Zener type interactions are... 

The problem with most quantum mechanical methods is that they scale badly. This means that, for instance, a calculation for twice as large a molecule does not require twice as much computer time and resources (this would be linear scaling), but rather 2" times as much, where n varies between about 3 for DFT calculations to 4 for Hartree-Fock and very large numbers for ab-initio techniques with explicit treatment of electron correlation. Thus, the size of the molecules that we can treat with conventional methods is limited. Linear scaling methods have been developed for ab-initio, DFT and semi-empirical methods, but only the latter are currently able to treat complete enzymes. There are two different approaches available. 

Since the first formulation of the MO-LCAO finite basis approach to molecular Ilartree-Pock calculations, computer applications of the method have conventionally been implemented as a two-step process. In the first of these steps a (large) number of integrals — mostly two-electron integrals — arc calculated and stored on external storage. Th e second step then con sists of the iterative solution of the Roothaan equations, where the integrals from the first step arc read once for every iteration. 

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 ... 

There are some boundary conditions which can be used to fix parameters and Ag. For example, when the distance between nucleus A and nucleus B approaches zero, i.e., R g = 0.0, the value of the two-electron two-center integral should approach that of the corresponding monocentric integral. The MNDO nomenclature for these monocentric integrals is. 

An alternative approach is in terms of frontier electron densities. In electrophilic substitution, the frontier electron density is taken as the electron density in the highest filled MO. In nucleophilic substitution the frontier orbital is taken as the lowest vacant MO the frontier electron density at a carbon atom is then the electron density that would be present in this MO if it were occupied by two electrons. Both electrophilic and nucleophilic substitution thus occur at the carbon atom with the greatest appropriate frontier electron density. 

Both HMO calculations and more elaborate MO methods can be applied to the issue of the position of electrophilic substitution in aromatic molecules. The most direct approach is to calculate the localization energy. This is the energy difference between the aromatic molecule and the n-complex intermediate. In simple Hiickel calculations, the localization energy is just the difference between the energy calculated for the initial n system and that remaining after two electrons and the carbon atom at the site of substitution have been removed from the conjugated system ... 

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