Linear Combination of Atomic Orbitals LCAO Approximation

4 Linear Combination of Atomic Orbitals (LCAO) Approximation 

The solutions to the HF model, are known as the molecular orbitals (MOs). These orbitals generally span the entire molecule, just as the atomic orbitals (AOs) span the space about an atom. Since organic chemists consider the atomic properties of atoms (or collection of atoms as functional groups) to persist to some extent when embedded within a molecule, it seems reasonable to construct the MOs as an expansion of the AOs, 

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The VB and the MO methods are rooted in very different philosophies of describing molecules. Although at the outset each method leads to different approximate wave functions, when successive improvements are made the two ultimately converge to the same wave function. In both the VB and MO methods, an approximate molecular wave function is obtained by combining appropriate hydrogen-like orbitals on each of the atoms in the molecule. This is called the linear combination of atomic orbitals (LCAO) approximation. 

These one-electron basis functions, 4>, constitute the basis set. When the basis functions represent the atomic orbitals for the atoms in the molecule, eq. 3.4 corresponds to a linear combination of atomic orbitals (LCAO) approximation. 

By far the commonest approximation employed to reduce the notion of an MO to an explicit, practical form is the linear combination of atomic orbitals (LCAO) approximation. Each MO is written as a linear combination of atomic orbitals on the various atoms. Denoting the /th atomic orbital , and the A th molecular orbital y/k, we write... 

Molecular orbitals are characterized by energies and amplitudes expressing the distribution of electron density over the nuclear framework (1-3). In the linear combination of atomic orbital (LCAO) approximation, the latter are expressed in terms of AO coefficients which in turn can be processed using the Mulliken approach into atomic and overlap populations. These in turn are related to relative charge distribution and atom-atom bonding interactions. Although in principle all occupied MOs are required to describe an observable molecular property, in fact certain aspects of structure and reactivity correlate rather well with the nature of selected filled and unfilled MOs. In particular, the properties of the highest occupied MO (HOMO) and lowest unoccupied MO (LUMO) permit the rationalization of trends in structural and reaction properties (28). A qualitative predictor of stability or, alternatively, a predictor of electron... 

The procedure is called the linear combination of atomic orbitals (LCAO) approximation and can be used for molecules of any size. H2+ is a special case in that a wavefimction can be found that will solve the Schrodinger equation exactly, yet the MO approach will be used so that molecular orbitals can be derived. The simplest trial function for the H2+ system is written ... 

A molecular orbital is assumed to be represented by a linear combination of atomic orbitals. This assumption is called the linear combination of atomic orbitals (LCAO) approximation. When the atomic orbitals and developing coefficients are denoted by xrand Crj, respectively, the molecular orbital (,) can be written by... 

Equation 6 is merely a statement of the linear combination of atomic orbitals (LCAO) approximation. 

Each of these molecular orbitals, Fi, in turn is described as a linear combination of basis functions, i. This is the linear combination of atom orbitals (LCAO) approximation ... 

The dimension of this basis set, m, sets the dimension of the quantum chemical problem. Normally, the basis functions are centered on the atoms of the molecular system (the Linear Combination of Atomic Orbitals (LCAO) approximation). Thus, the size is approximately proportional to the number of atoms in the system. 

Chapter 2 presents the principles of the Linear Combination of Atomic Orbitals (LCAO) approximation. This is the dominant approximation of qualitative (and 99 % of quantitative) MO-theory. It has become so established that we very often lose sight of the fact that it is an approximation and there have been many controversial discussions about methods for analysing the properties of individual atoms within molecules over the last 25 years. Chapter 2 also describes the effects of the electronegativities of the individual elements and introduces the Walsh diagram approach to analysing the structures of small molecules and fragments. 

The Linear Combination of Atomic Orbitals (LCAO) approximation is fundamental to many of our current models of chemistry. Both the vast majority of the calculational programs that we use, be they ab initioy density functional, semiempirical molecular orbital, or even some sophisticated force-fields, and our qualitative understanding of chemistry are based on the concept that the orbitals of a given molecule can be built from the orbitals of the constituent atoms. We feel comfortable with the Ji-HOMO (Highest Occupied Molecular Orbital) of ethylene depicted as a combination of two carbon p-orbitals, as shown in Fig. 2.1, although this is not a very accurate description of the electron density of this Molecular Orbital (MO). The use of the Jt-Atomic Orbitals (AOs), however, makes it easier to understand both the characteristics of the MO itself and the transformations that it can undergo during reactions. 

In the practical procedure, the one particle orbital is expanded as a linear combination of proper functions (atomic orbital) whose center is located at the nuclei. (Linear Combination of Atomic Orbital LCAO approximation)... 

Similar concerns apply to molecular orbitals. One constructs molecular orbitals and populates them with electrons is a manner analogous to an individual atom by adopting the linear combination of atomic orbitals (LCAO) approximation. While this might lend the impression that molecular orbitals are merely an extension of atomic orbitals, they are conceptually distinct. An atomic orbital is a description of the state of motion of an electron subject to the influence of a single nucleus plus other electrons. But molecular orbitals describe electron motions in the field of two or more nuclei plus the other electrons and the use of the LCAO method is merely a matter of mathematical convenience (Gavroglu and Simoes 2012, p. 83). The delocalized character of molecular orbitals is conceptually quite distinct from the idea of atomic orbitals, and Mulliken - one of the originators of the molecular orbital approach - was at pains to distinguish his conceptual scheme from the methods employed to compute them (ibid, pp. 84—85). 

The coefficients indicate the contribution of each atomic orbital to the molecular orbital. This method of representing molecular orbital wave functions in terms of combinations of atomic orbital wave functions is known as the linear combination of atomic orbitals (LCAO) approximation. The combination of atomic orbitals chosen is called the basis set. A minimum basis set for molecules containing C, H, O, and N would consist of 2s, Ip, 2py, and 2p orbitals for each C, N, and O and a Is orbital for each hydrogen. The basis sets are mathematical expressions describing the properties of the atomic orbitals. 

Linear Combination of Atomic Orbital (LCAO) approximation. 

We consider a three-dimensional periodic polymer or molecular crystal containing m orbitals in the elementary cell of one or more atoms. For the sake of simplicity the number of elementary cells in the direction of each crystal axis is taken equal to an odd number Ni = N2 = Ni = 27V 4-1. We assume further that there is an interaction between orbitals belonging to different elementary cells. In that case we can describe, in the one-electron approximation, the delocalized crystal orbitals of the polymer with the aid of the linear-combination-of-atomic-orbitals (LCAO) approximation in the form... 

The term basis set refers to the set of atom-centered mathematical functions chosen to describe atomic orbitals. These atomic orbitals are subsequently combined into molecular orbitals in the LCAO (see Linear Combination of Atomic Orbitals (LCAO)) approximation. Minimal basis sets, split-valence basis sets, and split-valence sets augmented with diffuse and polarization functions have all been used to examine the properties of hydrogen-bonded complexes. The weight of the evidence suggests that basis set which are not at least augmented split-valence basis sets are inadequate. 

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