Orbitals

FIELDS OF STUDY: Inorganic Chemistry, Organic Chemistry

ABSTRACT

Atomic and molecular orbitals are theoretical constructs of the modern theory of atomic structure, based on the mathematical description of quantum mechanics. They are ascribed specific geometries about the atomic nucleus according to the probability that an electron with the appropriate energy level will be in that region of space. Orbitals describe the physical and chemical behavior of electrons in atoms and molecules.

The Dual Nature of Electrons

The modern theory of atomic structure, based on quantum mechanics, dates back to the 1920s and is still being developed and refined by researchers. The foundations of the theory were laid in 1897, when J. J. Thomson (1856–1940) identified electrons as charged particles with negligible mass, and were reinforced in 1909, when Ernest Rutherford (1871–1937), Hans Geiger (1882–1945), and Ernest Marsden (1889–1970) identified protons as much heavier particles bearing the opposite charge. Subatomic particles had not been identified as discrete entities before that time, though a number of theories about the structure of the atom had already been proposed. A more complete description of the internal structure of atoms required a third particle that was electrically neutral, termed the "neutron." Because neutrons bear no electrical charge, their direct observation by electrical means was not possible, and it was not until James Chadwick (1891–1974) demonstrated their existence by indirect means in 1932 that the basic theoretical principles of atomic structure were resolved.

A large pool of experimental observations and measurement data relating to the interaction of atoms and molecules with light also existed. This information was crucial to ascribing energy values to electrons in atoms and molecules, with perhaps the most important discovery being the relationship between mass and energy, defined by Albert Einstein (1879–1955) in 1905 as

E = mc²

where E is energy, m is mass, and c is the speed of light. Earlier the same year, Einstein had proposed that electromagnetic energy is transferred to electrons not at a constant rate but in minute, individual units—that is, as particles. These particles, known as "photons," have no mass and exhibit wave-particle duality, meaning that they display properties characteristic of both waves and particles. In 1924, Louis de Broglie (1892–1987) combined Einstein’s mass-energy equation with his observations about photons to propose that all particles, including electrons, exhibit the same wave-particle duality. This duality makes the concept of electron orbitals central to the understanding of chemical behavior.

Quantum Numbers

The quantum-mechanical model of the atom describes the electrons in each atom in terms of quantum numbers. A "quantum" (plural "quanta") is the smallest possible unit of some physical property, such as energy or magnetism. In atoms, each electron is described by four quantum numbers: the principal, orbital, magnetic, and spin quantum numbers, each of which can take only certain specific values.

The principal quantum number (n) defines the energy level of the electron and its likely distance from the nucleus, effectively identifying which electron shell it inhabits. Each electron shell can be thought of as containing one or more subshells. The orbital or azimuthal quantum number (l) determines the electron’s angular momentum and therefore the shape of its orbital; it is related to the probability density function, which describes the probability that an electron of a specific energy level will be found in a given location within the atom. The magnetic quantum number (m) relates to the electron’s energy within a magnetic field and determines the number and orientation of the orbitals within each subshell. Depending on the orbital number, a subshell may contain one, three, five, or seven orbitals. These first three quantum numbers are all mathematically related and can only take integer values.

The fourth number is the spin quantum number (s), which describes the orientation of the electron’s intrinsic angular momentum—in other words, the direction of its spin—and can take one of only two values, +1/2 (up) or −1/2 (down). Because of this, only two electrons can occupy any individual orbital at any one time, and they must spin in opposite directions. This is due to the Pauli exclusion principle, which states that two electrons in the same atom cannot have the same values for all four quantum numbers. Because two electrons in the same orbital must have the same principal, orbital, and magnetic quantum numbers, their spin quantum numbers must be different, and because there are only two possible spin quantum numbers, there cannot be more than two electrons in one orbital.

Orbital Shapes

Atomic orbitals are most easily visualized as three-dimensional shapes. Because the regions of space around a nucleus are defined by wave functions, there are specific locations in which an electron cannot exist. These locations are the "nodes" of the particular wave function and are best thought of as boundary surfaces, where the electron can be on one side or the other of the boundary surface but not at any point actually on the surface.

There are four known orbital shapes, labeled s, p, d, and f, corresponding to l = 0, 1, 2, and 3, respectively. S orbitals are spherically symmetrical about the nucleus, and the s orbital corresponding to the principal quantum number n = 1 (that is, the 1s orbital) can be thought of as a ball with the nucleus at its center. The 2s orbital can be thought of as a ball surrounding another ball, the 3s orbital can be thought of as a ball surrounding a ball surrounding a third ball, and so on. The surface of each inner ball is the boundary-surface node of the wave function. However, orbitals are merely mathematical representations of the probability of an electron with specific energy being in that particular space; electrons do not actually follow the paths of their orbitals as planets do in a solar system. The most difficult concept to grasp in visualizing the shapes of the orbitals is that all of them exist in the same space at the same time.

The s orbitals are the simplest to visualize, especially for higher principal quantum numbers, as each s subshell only contains one orbital. A p subshell contains three p orbitals, oriented at right angles to each other; they are shaped like figure eights and have the same relationship to each other as the s orbitals as the principal quantum number increases. There are five d orbitals, four of which are shaped like four-leaf clovers and oriented in the spaces between the p orbitals; the fifth d orbital has the shape of a figure eight with a "donut" around its middle and is oriented in the same direction and space as one of the p orbitals. There are seven f orbitals, again shaped like four-leaf clovers, oriented in as many different directions about the nucleus. For the sake of simplicity, although technically a p orbital is one of three orbitals within a p subshell (and a d orbital is one of five, and an f orbital is one of seven), each containing only two electrons, the term "orbital" is sometimes used to refer to the entire subshell instead, and it is said that a p orbital can contain up to six electrons, a d orbital can contain up to ten electrons, and an f orbital can contain up to fourteen electrons.

Electrons in Orbitals

The electrons in atoms occupy the different orbitals in pairs within each electron subshell, and each atom accordingly has an electron configuration that reflects this principle. Electrons fill orbitals in order of increasing energy, from the 1s orbital outward. While the order of the atomic orbitals according to their respective quantum numbers is 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, and so on, with each electron shell after n = 4 containing all four orbital types, the order in which electrons fill these orbitals is slightly different, as some orbitals of different quantum numbers have similar energy requirements. The number of electrons contained in a particular subshell is given as a superscript number following the orbital designation, so that a completely full n = 2 electron shell, for example, would be represented as 2s²2p⁶, while an n = 3 electron shell missing two electrons would be represented as 3s²3p⁶3d⁸ (as the d subshell can hold up to ten electrons).

When all of the electrons in an atom have occupied their proper orbitals, the outermost electron shell of almost all elements contains fewer electrons than the number needed to fill its orbitals. This outermost shell is the valence shell, and the electrons in it are the valence electrons. The number of valence electrons corresponds to the atom’s valence, which is essentially the number of bonds it can form with other atoms.

Hybrid Orbitals

In molecules, the atomic orbitals can combine in different ways to produce hybrid atomic and molecular orbitals. These have their own theoretical shapes, determined by the manner in which they "overlap" mathematically to form chemical bonds. The most common hybrid orbitals are sp, sp², and sp³, formed by the combination of an s orbital with one, two, or three p orbitals, respectively; the superscript number indicates the proportion of each type of orbital involved. Other possible hybrid orbitals include dsp³, d²sp³, and dsp².

PRINCIPAL TERMS

• electron configuration: the order and arrangement of electrons within the orbitals of an atom or molecule.
• electron shell: a region surrounding the nucleus of an atom that contains one or more orbitals capable of holding a specific maximum number of electrons.
• probability density: in reference to electrons, the probability of finding a particular electron in a given region of space within an atom or molecule; also called electron density.
• quantum number: one of four numbers that describe the energy level, orbital shape, orbital orientation, and spin of an electron within an atom.
• valence: the maximum number of bonds an atom can form with other atoms based on its electron configuration; also called valency.

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