Atomic orbitals wave properties

The wave functions (6.8) are known as atomic orbitals, for / = 0, 1,2, 3, etc., they are referred to as s, p, d, f, respectively, with the value of n as a prefix, i.e. Is, 2s, 2p, 3s, 3p, 3d, etc., From the explicit forms ofthe wave functions we can calculate both the sizes and shapes of the atomic orbitals, important properties when we come to consider molecule formation and structure. It is instructive to examine the angular parts of the hydrogen atom functions (the spherical harmonics) in a polar plot but noting from (6.9) that these are complex functions, we prefer to describe the angular wave functions by real linear combinations of the complex functions, which are also acceptable solutions of the Schrodinger equation. This procedure may be illustrated by considering the 2p orbitals. From equations (6.8) and (6.9) the complex wave functions are... 

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. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2, but in somewhat different ways. Both assume that electron waves behave like more familiar waves, such as sound and light waves. One important property of waves is called interference in physics. Constructive interference occurs when two waves combine so as to reinforce each other (in phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2.2). Recall from Section 1.1 that electron waves in atoms are characterized by then- wave function, which is the same as an orbital. For an electron in the most stable state of a hydrogen atom, for example, this state is defined by the I5 wave function and is often called the I5 orbital. The valence bond model bases the connection between two atoms on the overlap between half-filled orbitals of the two atoms. The molecular orbital model assembles a set of molecular- orbitals by combining the atomic orbitals of all of the atoms in the molecule. 

A convenient orbital method for describing eleetron motion in moleeules is the method of molecular orbitals. Molecular orbitals are defined and calculated in the same way as atomic orbitals and they display similar wave-like properties. The main difference between molecular and atomic orbitals is that molecular orbitals are not confined to a single atom. The crests and troughs in an atomic orbital are confined to a region close to the atomic nucleus (typieally within 1-2 A). The electrons in a molecule, on the other hand, do not stick to a single atom, and are free to move all around the molecule. Consequendy, the crests and troughs in a molecular orbital are usually spread over several atoms. 

The bond orbitals of o, and relate to the other property of waves apart from the phase, that is, the amplitude. The bonding orbitals have large amplitudes on the low-lying atomic orbitals, i.e., on C of o, and on O of (Scheme 8). The antibonding orbitals have large amplitudes on the high-lying atomic orbitals. 

Overlapping Ion Model. The ground-state wave function for an individual electron in an ionic crystal has been discussed by Lowdin (24). To explain the macroscopic properties of the alkali halides, Lowdin has introduced the symmetrical orthogonaliz tion technique. He has shown that an atomic orbital, x//, in an alkali halide can be given by... 

Wolfgang Pauh (1900-1958), an American physicist, was awarded a Nobel Prize in 1945 for developing the exclusion principle. In essence, it states that a particular electron in an atom has only one of fom energy states and that all other electrons are excluded from this electron s energy level or orbital. In other words, no two electrons may occupy the same state of energy (or position in an orbit around the nucleus). This led to the concept that only a certain number of electrons can occupy the same shell or orbit. In addition, the wave properties of electrons are measmed in quantum amounts and are related to the physical and, thus, the chemical properties of atoms. These concepts enable scientists to precisely define important physical properties of the atoms of different elements and to more accmately place elements in the periodic table. 

Equation 1.132 has the property of never giving an energy that is lower than the true energy resonance principle cf Pauling, 1960). This property allows us to assign to the MO wave functions that are obtained by linear combination of the AO functions of the separate atoms (Linear Combination of Atomic Orbitals, or LCAO, method), by progressive adjustment of the combinatory parameters, up to achievement of the lowest energy. 

Because electrons have wave-like properties, atomic orbitals can be described with wave functions and can overlap in different ways ... 

In our QM systems, we have temporarily restricted ourselves to systems of one electron. If, in addition, our system were to have only one nucleus, then we would not need to guess wave functions, but instead we could solve Eq. (4.16) exactly. The eigenfunctions that are determined in that instance are the familiar hydrogenic atomic orbitals. Is, 2s, 2p, 3s, 3p, 3d, etc., whose properties and derivation are discussed in detail in standard texts on quantum mechanics. For the moment, we will not investigate the mathematical representation of these hydrogenic atomic orbitals in any detail, but we will simply posit that, as functions, they may be useful in the construction of more complicated molecular orbitals. In particular, just as in Eq. (4.10) we constructed a guess wave function as a linear combination of exact wave functions, so here we will construct a guess wave function as a linear combination of atomic wave functions (p, i.e.,... 

The reason for proceeding with the ligand field approach is the belief that it is the symmetry properties of the (/-orbitals which are their critical feature. The changes brought about by admixture of donor atom orbitals can be accommodated by changing the appropriate details of the wave function describing the effective (/-orbital set, but the major features of behaviour will remain because the symmetry is maintained. 

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