The Bohr model of the atom, in which electrons orbit about the nucleus, is a convenient representation. Unfortunately, it is not accurate. The results of scientific experiments suggest that electrons act more like electromagnetic waves than orbiting particles. According to a basic principle of quantum mechanics, it is impossible to know simultaneously both the exact position and momentum of an electron; this means that the trajectory of the electron cannot be precisely determined.
Waves. What can be determined is a region of space around the nucleus where there is a high probability of finding the negative charge of an atom. Mathematically, this probability distribution is similar to an equation that describes a wave. In other words, the electron distribution in an atom can be described by the mathematical formulas and physical concepts of a standing wave. A standing wave is basically a stationary, bound vibration—the vibration of a guitar string, for example. Viewed in slow motion, the plucking of a guitar string first displaces it a certain distance from its original position. The string rebounds to its origin and is then displaced in the opposite direction by the same distance. Figure 1 illustrates such a wave motion. The upward displacement is assigned a plus sign, while the downward one is assigned a negative sign. These signs are called phase signs. The point where the wave crosses the original position is called a node, a point of zero amplitude in the wave.
Quantum numbers. In 1926, Erwin Schrödinger derived a wave equation that incorporated both the particle and wave characteristics of the electron. This equation allowed the calculation of a probability function whose solution generated four quantum numbers. The first quantum number, the principal quantum number ( n ), tells the location of the probability region relative to the nucleus. It corresponds to the orbit, or shell, designation. The second quantum number, the angular momentum quantum number ( l ), tells the shape of the probability region. It corresponds to the orbital designations. The third quantum number, the magnetic quantum number ( m ), describes the orientation of the probability region in space. The fourth quantum number, the spin quantum number ( ms ), describes the direction of spin of the electron. The first three quantum numbers define the region in space about the nucleus of an atom where there is the highest probability of finding the electron density. This region is referred to as an atomic orbital.
The shapes of the two most common orbitals found in organic compounds are spherical in atomic s orbitals and hourglass shaped in atomic p orbitals. The atomic p orbitals can be oriented along the x, y, and z axes in three-dimensional space.
Linear combination of atomic orbitals. Just as waves can interact to reinforce or diminish themselves, atomic orbitals can combine to form new orbitals. This combining process is referred to as the linear combination of atomic orbitals. In this process, the total number of orbitals remains constant; that is, the number of new orbitals always equals the number of orbitals combined to form them. Linear combination can occur between orbitals of two different atoms, creating molecular orbitals, or between two orbitals of the same atom, creating hybrid atomic orbitals.
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