Wave Properties of Electrons in Orbitals

A standing wave. The fundamental frequency of a guitar string is a standing wave with the string alternately displaced upward and downward. 

We like to picture the atom as a miniature solar system, with the electrons orbiting around the nucleus. This solar system picture satisfies our intuition, but it does not accurately reflect today s understanding of the atom. About 1923, Louis de Broglie suggested that the properties of electrons in atoms are better explained by treating the electrons as waves rather than as particles. 

The Is orbital. The Is orbital is similar to the fundamental vibration of a guitar string. The wave function is instantaneously all positive or all negative. The square of the wave function gives the electron density. 

A circle with a nucleus is used to represent the spherically symmetrical s orbital. 

First harmonic of a guitar string. The two halves of the string are separated by a node, a point with zero displacement. The two halves vibrate out of phase with each other. 

The 2p orbital. The 2p orbital has two lobes, separated by a nodal plane. 

The two lobes are out of phase with each other. When one has a plus sign, the other has a minus sign. 

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Here, the orbital phase theory sheds new light on the regioselectivities of reactions [29]. This suggests how widely or deeply important the role of the wave property of electrons in molecules is in chemistry. 

Molecular properties and reactions are controlled by electrons in the molecules. Electrons had been thonght to be particles. Quantum mechanics showed that electrons have properties not only as particles but also as waves. A chemical theory is required to think abont the wave properties of electrons in molecules. These properties are well represented by orbitals, which contain the amplitude and phase characteristics of waves. This volume is a result of our attempt to establish a theory of chemistry in terms of orbitals — A Chemical Orbital Theory. 

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. 

Erwin Schrodinger (1887-1961) and others considered the wave properties of electrons and proposed that electrons were not orbiting around the nucleus in an atom but were in electron-cloud probability areas outside the atomic nucleus. These probability areas were designated as energy levels. 

Figure 17 shows the present-day model of the atom, which takes into account both the particle and wave properties of electrons. According to this model, electrons are located in orbitals, regions around a nucleus that correspond to specific energy levels. Orbitals are regions where electrons are likely to be found. Orbitals are sometimes called electron clouds because they do not have sharp boundaries. When an orbital is drawn, it shows where electrons are most likely to be. Because electrons can be in other places, the orbital has a fuzzy boundary like a cloud. 

In the Bohr-Sommerfeld theories of the atom, the electrons are moving in orbits which are precisely specified, and the velocities are given exactly. Those theories are therefore concerned with properties which cannot be measured precisely. This difficulty is avoided, however, if one develops theories based on the wave properties of electrons we have already seen, with reference to Figure 1.1, that such theories remove some of the arbitrariness inherent in the Bohr-Sommerfeld approach. Modern theories of atoms and molecules are, therefore, wave theories, which have led to a very considerable increase in our understanding. In the remainder of this chapter we will describe aspects of wave mechanics, or quantum mechanics, that will be of help to biologists in appreciating the nature of the molecular structures with which they are concerned. 

The main difference between the two models is that, while Bohr considered the electrons to be traditional particles whose motion could be described by the classical mechanics of Newton, the quantum mechanical model treats the electrons as waves. The wave properties of electrons provide a logical explanation for the existence of allowed orbits in Bohr s atomic model. 

In this section, you saw how the ideas of quantum mechanics led to a new, revolutionary atomic model—the quantum mechanical model of the atom. According to this model, electrons have both matter-like and wave-like properties. Their position and momentum cannot both be determined with certainty, so they must be described in terms of probabilities. An orbital represents a mathematical description of the volume of space in which an electron has a high probability of being found. You learned the first three quantum numbers that describe the size, energy, shape, and orientation of an orbital. In the next section, you will use quantum numbers to describe the total number of electrons in an atom and the energy levels in which they are most likely to be found in their ground state. You will also discover how the ideas of quantum mechanics explain the structure and organization of the periodic table. 

Chapter 14 deals with orbital correlation diagrams following Woodward and Hoffmann [3]. State wave functions and properties of electronic states are deduced from the orbital picture, and rules for state correlation diagrams are reviewed, as a prelude to an introduction to the field of organic photochemistry in Chapter 15. 

Despite these restrictions, the GVB and SC methods generally provide energies that are much closer in quality to CASSCF than to Hartree Fock (19), and wave functions that are close to the CASSCF wave function having the same number of electrons and orbitals in the active space. This property has been used to devise a fast method to get approximate SC wave functions. 

In 1926, Erwin Schrodinger used de Broglie s idea that matter has wavelike properties. Schrodinger proposed what is now known as the quantum mechanical model of the atom. In this new model, he abandoned the notion of the electron as a small particle orbiting the nucleus. Instead, he took into account the particle s wavelike properties, and described the behaviour of electrons in terms of wave functions. 

The Schrodinger equation describes the wave properties of an electron in terms of its position, mass, total energy, and potential energy. The equation is based on the wave function, " I, which describes an electron wave in space in other words, it describes an atomic orbital. In its simplest notation, the equation is... 

Because every matches an atomic orbital, there is no limit to the number of solutions of the Schrodinger equation for an atom. Each P describes the wave properties of a given electron in a particular orbital. The probability of finding an electron at a given point in space is proportional to A number of conditions are required for a physically realistic solution for P ... 

In the bonding molecular orbital the electron density is greatest between the nuclei of the bonding atoms. In the antibonding molecular orbital, on the other hand, the electron density decreases to zero between the nuclei. We can understand this distinction if we recall that electrons in orbitals have wave characteristics. A property unique to waves allows waves of the same type to interact in such a way that the resultant wave has either an enhanced amplitude or a diminished amphtude. In the former case, we call the interaction constructive interference in the latter case, it is destructive interference (Figure 10.21). 

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