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Quantum mechanics employs a wave function, y, to describe the physical state of an electron in an atom or molecule. The value of the wave function (which may be complex) depends upon the position of the electron. Cartesian coordinates ( x, y, z ) may be used, but it is frequently more convenient to use spherical coordinates (r, q, f ). Imagine a line segment connecting the origin (which is the position of the nucleus in an atom) and the point r,q,f. The variable r is the length of the line segment, q is the angle between the z axis and the line segment, and f is the angle between the x axis and the projection of the line segment onto the xy plane.
An orbital is the region of space where an electron exists and is described by the wave function. The quantity y2 (or y*y for complex wave functions) describes the probability of interacting with the electron at the point r,q,f. For this reason the wave function can be used to predict where an electron is likely to be found in an atom.
Each orbital is characterized by three quantum numbers.
Principal Quantum Number, n
The principal quantum number describes the size of the orbital and may have the value of any positive (nonzero) integer. Each orbital has an energy, E, and for a hydrogen-like atom (an atom with only one electron) the energy is dictated solely by the principal quantum number and the charge, Z, on the nucleus.
En = - 13.61 eV ( Z/n )2
Angular Momentum Quantum Number, l
The electron possesses angular momentum by virtual of its motion around the atom. It is tempting to envision this angular momentum in the same terms as that arising from the motion of a planet around the sun, but this view is incorrect. One example of the invalidity of this view is the fact that for l = 0, there is no orbital angular momentum, a behavior that does not exist for a planet in orbit. l may have the value of zero and of any positive integer that is less than the value of n.
Magnetic Quantum Number, m
The magnetic quantum number is any integer (positive, negative, or zero) whose absolute value does not exceed the value of l.
Orbitals are designated by the notation: nSg. As indicated above, n is the principal quantum number. The symbol S indicates the orbital angular momentum. The subscript g provides information on the angular geometry of the orbital. The s orbitals are spherically symmetric, thus no extra description is required. For p, d, and f orbitals, however, the geometry of the orbital (which depends upon the value of m) is described in the subscript.
|1||p||x, y, z|
|2||d||z2, xy, xz, yz, x2-y2|
|3||f||5z3-3zr2, 5xz2-xr2, 5yz2-yr2, zx2-zy2, xyz, x3-3xy2, y3-3yx2|
The lowest energy orbital in the hydrogen atom is the 1s orbital, which corresponds with n = 1, l = 0, and m = 0.
The interactions between atoms is an important issue in chemistry, because such interactions form the basis of chemical bonding and intermolecular forces, which govern the behavior of substances. Interactions between atoms are the result of interactions between orbitals on the atoms. For this reason it is important to understand the properties and geometries of the various orbitals.
Geometrically, orbitals are three dimensional structures with complicated features, which makes visualization difficult. Chemists employ a variety of graphical representations to depict the shape and structure of an orbital. Each representation provides a different perspective on the orbital.
The wave functions for an atom (but not a molecule) can be separated into two functions: Rnl(r) and Ylm(q,f). Rnl(r) depends only upon the distance from the nucleus and is called the radial function. Ylm(q,f) depends only upon the angular orientation about the nucleus and is called the angular function.
The plot at the lower left shows the variation of Rnl(r) with the distance from the nucleus for the 1s orbital. As is evident from the plot, the wave function is largest at the nucleus and decreases exponentially as the distance from the nucleus increases.
On average, how far away from the nucleus is the electron? This seems like a simple question. The Rnl(r) vs r plot suggests the electron spends most of its time very close to the nucleus, but this perception is misleading. The function y*y describes the probability of finding an electron in a small box of volume dx dy dz at a position x, y, z. At the nucleus (r = 0), there is only one infinitesimal box dx dy dz in which the electron can exist. As large values of r, however, there are many different infinitesimal boxes and thus more chances to find the electron. In fact, the volume of an infinitesimally thin spherical shell at a distance r from the nucleus has a volume proportional to r2 dr.
The radial distribution plot, shown at the lower right for the 1s orbital, shows the variation of r2Rnl(r) with r. The function r2Rnl(r) is the product of Rnl(r), which describes the probability per unit volume of finding the electron, and r2, which describes the volume. The plot shows that the electron is never found at the nucleus (because the volume there is zero) and the electron is unlikely to be found far from the electron (because the wave function is miniscule far from the nucleus). The electron is most likely to be found between 0.5 ao and 2 ao where r2Rnl(r) is large; the most likely distance is ao. The constant ao is called the Bohr radius and has a value of 0.529 angstroms. Thus the electron in the hydrogen 1s orbital spends most of its time 0.26 to 1.0 angstroms from the nucleus.
|Radial Function ||Radial Distribution Plot |
Ordinary two-dimensional plots show the variation of a function with a single variable, which limits the amount of information that can be conveyed. This limitation is particularly problematic when angular properties of an orbital are of interest. An electron density plot depicts the electron density (y*y) on a particular plane. The electron density is represented by the intensity of the color. If the color at a particular point is very bright, there is a high probability of finding the electron at that point. If the color is dim, the probability is low.
|Electron Density Plot || |
The electron density plot at the left depicts the hydrogen 1s orbital. The button below can be used to display the axes.
Notice that the color is most intense at the origin (the nucleus) and diminishes as the distance from the nucleus increases.
In addition, observe that the 1s orbital is symmetric; there is no angular variation in the electron density.
An orbital isosurface is a surface on which all points have the same y*y value. The isosurface encloses a region of high electron density. An isosurface plot typically encloses 90% of the electron density (though other values are possible). That is, the electron has a 90% probability of being found inside the isosurface.
A virtual reality representation of the hydrogen 1s orbital isosurface is shown at the left. The virtual reality representation is interactive. You can rotate the image, and you may use the controls below the image to navigate through the virtual world.
Notice that the 1s orbital is spherically symmetric; the value of the wave function is independent of the angular position around the nucleus.
This material is based upon work supported by the National Science Foundation under Grant Number 1246120