Molecular orbital LCAO approximation

ITyperChem uses the Linear Combination of Atomic Orhilals-Molecular Orbital (LCAO-MO) approximation for all of itsah initio sem i-empirical melh ods. If rg, represen is a molecular orbital and... 

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. 

Ah, the crux of the problem, is it not Up until now, we ve just assumed we have some set of molecular orbitals i or Vu which we can manipulate at will. But how does one come up with even approximate solutions to the many body Schrodinger equation without having to solve it Start with the celebrated linear combination of atomic orbitals to get molecular orbitals (LCAO-MO) approximation. This allows us to use some set of (approximate) atomic orbitals, the basis functions which we know and love, to expand the MOs in. In the most general terms,... 

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... 

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. 

Thus, the valence-bond method may be regarded as a special case of the molecular orbital (MO) approximation in a linear combination of atomic orbitals (LCAO) modification which omits ionic terms. It should be emphasized, however, that although we have derived valence bonds from molecular orbitals, this is not necessary and has been done merely to show the relationship between the two methods. We could just as easily have written ab initio the particular hnear combination of atomic orbitals which excludes ionic terms. 

This chapter revises basic atomic orbital theory. The chapter begins with the exact results for the case of the hydrogen atom and the orbital concept for many-electron atoms. It is very important to understand these details about atomic orbitals, since the orbital concept is essential in the approximation to chemical bonding known as the Linear Combination of Atomic Orbitals — Molecular Orbital [LCAO-MO] theory. Only in the case of the hydrogen atom are these atomic orbitals, as exact solutions to Schrddinger s equation, available as functions. 

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. 

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). 

This method of representing the molecular orbital wave function in terms of combinations of atomic orbital wave functions is known as the linear combination of atomic orbitals-molecular orbital (LCAO-MO) approximation. The combination of atomic orbitals chosen is called the basis set. A minimal basis set for molecules containing C, H, O, and N would consist of Is-, 2s-, Ip -, 2py-, and 2p -orbitals for C, O, and N, and a Is-orbital for hydrogen, inclusion of additional orbitals in the basis set leads to an extended basis set. Economy dictates use of minimal basis sets whenever possible. 

It is traditional for quantmn theory of molecular systems (molecular quantum chemistry) to describe the properties of a many-atom system on the grounds of interatomic interactions applying the hnear combination of atomic orbitals (LCAO) approximation in the electronic-structure calculations. The basis of the theory of the electronic structure of solids is the periodicity of the crystalline potential and Bloch-type one-electron states, in the majority of cases approximated by a linear combination of plane waves (LCPW). In a quantmn chemistry of solids the LCAO approach is extended to periodic systems and modified in such a way that the periodicity of the potential is correctly taken into account, but the language traditional for chemistry is used when the interatomic interaction is analyzed to explain the properties of the crystalhne sohds. At first, the quantum chemistry of solids was considered simply as the energy-band theory [2] or the theory of the chemical bond in tetrahedral semiconductors [3]. From the beginning of the 1970s the use of powerful computer codes has become a common practice in molecular quantum chemistry to predict many properties of molecules in the first-principles LCAO calculations. In the condensed-matter studies the accurate description of the system at an atomic scale was much less advanced [4]. 

An approximate treatment of tt electron systems was introduced in 1931 by Erich Huckel (Figure 15.17) and is called the Huckel approximation of tt orbitals. The first step in a Huckel approximation is to treat the sigma bonds separately from the pi bonds. Therefore, in a Huckel approximation of a molecule, only the tt bonds are considered. The usual assumption is that the <7 bonds are understood in terms of regular molecular orbital theory. The <7 bonds form the overall structure of the molecule, and the tt bonds spread out over, or span, the available carbon atoms. Such 77 bonds are formed from the side-on overlap of the carbon 2p orbitals. If we are assuming that the tt bonds are independent of the cr bonds, then we can assume that the 77 molecular orbitals are linear combinations of only the 2p orbitals of the various carbon atoms. [This is a natural consequence of our earlier linear combination of atomic orbitals—molecular orbitals (LCAO-MO) discussion.] Consider the molecule 1,3-butadiene (Figure 15.18). The tt orbitals are assumed to be combinations of the 2p atomic orbitals of the four carbon atoms involved in the conjugated double bonds ... 

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. 

In the linear combination of atomic orbitals (LCAO) approximation of the molecular orbital, the energy of the overlap electron density between the atomic orbitals Xu and Xv due to attraction by the core. See Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Relativistic Effective Core Potential Techniques for Molecules Containing Very Heavy Atoms Transition Metal Chemistry and Transition Metals Applications. 

The spatial one-electron wavefunctions, f/n), are represented as a linear combination of atom-centered functions (i.e. atomic orbitals), called the linear combination of atomic orbitals (LCAO) approximation. The functions (pik constitute a basis set. This is the same approach used for multi-electron atoms and for the molecule. The index k refers to the specific atomic orbital wavefunction, and the index / refers to its contribution to a specific molecular orbital. 

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