The hydrogen wave functions.

The solutions of the Schrödinger equation for the hydrogen atom can be written as

Because the subscripts n, l, and m can take on more than one value, Y_nlm stands really for many different functions. It can be shown that n is always a positive integer; i.e., n = 1,2,3,.... It's called the principal quantum number . The quantum number l is always a non-negative integer. It's called the angular quantum number . For a given n it is restricted to l = 0,1,2,...,n-1. The number m is also always an integer, but its value range from -l to l; i.e., m = -l,-l+1,-l+2,...,l-2,l-1,l. It is called the magnetic quantum number . Combining n, l, and m gives the wave functions Y_1,0,0, Y_2,0,0, Y_2,1,-1, Y_2,1,0, Y_2,1,1, Y_3,0,0, Y_3,1,-1, Y_3,1,0, Y_3,1,1, Y_3,2,-2, Y_3,2,-1, Y_3,2,0, Y_3,2,1, Y_3,2,2, etc. The functions are always grouped. The functions with n = 1 are called the first shell, those with n = 2 are called the second shell, those with n = 3 are called third shell, etc. The functions with l = 0 are called s functions, those with l = 1 are called p functions, those with l = 2 are called d functions, and those with l = 3 are called f function. After f we follow the alphabet; i.e., we get g, h, etc. Combining the shells with the s, p, d, etc. gives subshells. These are called 1s, 2s, 2p, 3s, 3p, 3d, etc. The number is the principal quantum number n. The letter indicates the angular quantum number l. This notation for subshells is also used for other atoms than hydrogen, and it is very widely used when discussing the electronic structures of atoms.

It is very important to have a good idea of how the value of the hydrogen wave functions vary in space. Other atoms have very similar wave functions, and the shape of them determines to a very large extend how and what kind of chemical bonds are formed. In principle, one can get all information of the hydrogen wave function by looking at radial and angular functions. This is indeed necessary when one wants to do actual calculations of chemical bonds. To get some qualitative idea of the wave functions, the mathematical expressions are for most people not very useful. Instead we try to give some plots of the wave functions. This, however, is not trivial.

When we plot a function of one variable, we need all two dimensions of a piece of paper or a computer screen. For a (wave) function of three variable we cannot make a plot without leaving out some information of the function. We need to make sure, that the information that we leave out is either not relevant, or can be added mentally by using other information that we have on the function. Let's start with the simplest wave function; i.e., 1s or

Because 1s is so simple, it is a good candidate to illustrate other ways of plotting wave functions. Sometimes one is mainly interested in how a wave function varies in a particular plane. One then makes a cross section of the wave function, and uses a contour plot to show the variation of the wave function in the cross section. As an example, let's suppose we want to know how 1s varies in the xz-plane. We first need to have an expression that tells us how the wave function depends on x and z. We do this by substituting for the spherical coordinates expressions in terms of Cartesian coordinates x, y, and z. As r = Ö[(x²+y²+z²)] we get

Remember that contour lines indicate lines along which the function is constant. This explains why the contour lines for 1s are all circles. There all two problems with such contour plots. The first is that we do not know the values of the function on the different contours. We can label each contour with the function value, but that is often not done. The second problem is that we have in general no information of the wave functions away from the cross section. For the 1s function this is not really a problem, because we know that for other cross sections we get the same plot, provided the cross section contains the nucleus. In general however this second problem cannot be resolved so easily.

The most common way to give some idea of the shape of the wave function is by showing a drawing of an isosurface . An isosurface is a three-dimensional analog of a contour line. For 1s an isosurface can be defined by

So the question is which sphere will we draw, or what value for the constant c will we take? In many textbooks it is suggested that one should take c so that

All isosurfaces of s functions (i.e., wave function with l = 0) are spherical, and drawings of them all look the same as the one for 1s. The radial dependence is, however, not the same for different s functions. The following figure shows a contour plot of 2s (i.e., Y_2,0,0).

The different colors are used to indicate that the wave function changes sign. Contours of one color are used for regions where the wave functions is positive, and the other color is used for regions where the wave function is negative. Instead of colors the line type may be varied (e.g., solid lines where the wave function is positive and dashed or dotted lines where it is negative). Changes in the sign of a wave function are very important, because they are related to the energy of the wave function. The isosurfaces on which a wave function is zero, are called nodal planes . In general the energy increases when the number of nodal planes increases. The drawback of a figure showing isosurfaces now should be clear. The number of nodal planes can in general not be determined from such figure. Because the nodal plane of 2s is defined by the distance r it is called a radial nodal plane.

The next figure shows isosurfaces of Y_2,1,0 and a contour plot of a cross section in the xz-plane.

This wave function is a 2p function. Because it is oriented along the z-axis, it is called the 2p_z function. There is also a 2p_x and a 2p_y function. These functions are Y_2,1,1 and Y_2,1,-1, respectively. Their isosurfaces look exactly the same as those of 2p_z, except that they are rotated so that they are oriented along the x- and y-axis, respectively. The reason why the figure shows two isosurfaces is that the wave function changes sign. The xy-plane is a nodal plane. One of the isosurfaces is defined by 2p_z(r) = c and the other by 2p_z(r) = -c. To indicate the difference in sign, the isosurfaces have been given a different color. The nodal plane is not defined by the distance r, but by q = p/2. So it is not a radial nodal plane. The regions that the isosurfaces enclose, are often called the lobes of the wave function. We see that 2p functions have two lobes. The probability to find the electron at a certain position is highest in these lobes as can be seen from the contour plot. Also other p functions have two lobes and a flat nodal plane that separates them. However, p function with n > 2 have also radial nodal planes which cut up the lobes in two or more pieces. A np function has n-2 radial nodal planes.

The isosurfaces of the 3s function are again spheres, and the isosurfaces of the 3p functions resemble those of 2p functions. We get something new when we look at 3d functions. The next figure shows isosurfaces of 3d_xy (i.e., Y_3,2,-2), and a cross section in the xy-plane.

We see four lobes. Opposite lobes have the same sign, neighboring lobes have opposite sign. There are two nodal planes. One is the xz- and the other the yz-plane. The functions 3d_xz (Y_3,2,-1), 3d_yz (Y_3,2,1), and 3d_x²-y² (Y_3,2,2) have the same shape, but they are oriented differently. Their nodal planes are the xy- and yz-plane, the xy- and xz-plane, and the planes x = y and x = -y, respectively. The shape of 3d_z² (Y_3,2,0) is very different. The isosurfaces are shown in the next figure, as well as a cross section of the wave function in the xz-plane.

The upper and lower lobe resemble the lobes of a p function, except both lobes have the same sign. The middle lobe is a torus with an opposite sign to that of the upper and lower lobe. The nodal planes for 3d_z² are two cones defined by 3cos²q = 1. The cross section of the wave function in the xz-plane shows these nodal planes better.

Finally a word of warning. The wave functions above are those of the hydrogen atom. The orbitals of other atoms, resemble these wave functions very much. The designation 1s, 2s, 2p, 3s, 3p, 3d, etc. is also used for other atoms. The angular wave functions in fact are even the same, because these functions are a consequence of the spherical symmetry of atoms. The radial parts of orbitals of other atoms are different. The form is more or less the same, but the positions of the radial nodal planes depends very much on the atom number.

• The s, p, and d functions.
• The radial part of the hydrogen wave functions.
• The hydrogen wave functions (this page).
• The energies of the hydrogen atom.