Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)
Chapter 10. The Hydrogen Atom: Quantum Theory
IN 1925 SCHRÖDINGER AND HEISENBERG separately developed quan tum theory. Their two formulations are mathematically different, but their theories are rigorous and form the underpinning of modern quantum theory. At about the same time, Dirac made major contributions as well. First, he presented a unified view of quantum theory that showed that the Schrödinger and Heisenberg theories, while mathematically different, were equivalent representations of quantum mechanics. In addition, he developed a quantum theory for the hydrogen atom that is also consistent with Einstein’s Theory of Relativity. The formulation by Schrödinger is the most often used to describe atoms and molecules. Therefore, most of our discussions, starting with the hydrogen atom and then going on to larger atoms and molecules, will be based on the concepts and language that is inherent in the Schrödinger approach.
THE SCHRÖDINGER EQUATION
We used a very simple but correct mathematical method for obtaining the energy levels of the particle in the box and the wavefunctions, but the method we used is not general. For example, it cannot be used to find the energy levels and the wavefunctions for the hydrogen atom. In fact, the language we have been using, that is, wavefunctions and probability amplitude waves, comes from Schrödinger’s formulation of quantum theory. In 1925 Schrödinger presented what has come to be known as the Schrödinger Equation. The Schrödinger Equation is a complicated differential equation in three dimensions. We will not do the mathematics necessary to solve the Schrödinger equation for the hydrogen atom or other atoms or molecules. However, we will use many of the results to learn about atoms and molecules, beginning with the hydrogen atom.
The solution of the hydrogen atom problem using the Schrödinger Equation is particularly important because it can be solved exactly. The hydrogen atom is a “two-body” problem. There are only two interacting particles, the proton and the electron. The next simplest atom is the helium atom, which has a nucleus with a charge of +2 and two negatively charged electrons. This is a three-body problem that cannot be solved exactly. In classical mechanics, it is also not possible to solve a three-body problem. The problem of determining the orbits of the Earth orbiting the Sun with the Moon orbiting the Earth cannot be solved exactly with classical mechanics. However, in both quantum mechanics and classical mechanics, there are very sophisticated approximate methods that permit very accurate solutions to problems that cannot be solved exactly. The fact that a method is approximate does not mean it is inaccurate. Nonetheless, because the hydrogen atom can be solved exactly with quantum theory, it provides an important starting point for understanding more complicated atoms and molecules.
WHAT THE SCHRÖDINGER EQUATION TELLS US ABOUT HYDROGEN
What does the solution to the Schrödinger Equation for the hydrogen atom give? It gives the energy levels of the hydrogen atom, and it gives the wavefunctions associated with each state of the hydrogen atom. The wavefunctions are the three-dimensional probability amplitude waves that describe the regions of space where the electron is likely to be found. Schrödinger’s solution to the hydrogen atom problem gives energy levels consistent with the empirically obtained Rydberg formula. The energy levels are
where n is the principal quantum number. It is an integer that can take on values ≥1, that is, greater than or equal to 1. The difference in energy between any two energy levels is the Rydberg formula. However, in the Schrödinger solution, RH is not an empirical parameter. In solving the problem, Schrödinger found the Rydberg constant is determined by fundamental constants, with
h is Planck’s constant. e is the charge on the electron. εo is a constant called the permittivity of vacuum. εo = 8.54×10-12 C2/J m, with the units Coulombs squared per Joule - meter. μ is the reduced mass of the proton and the electron. It is
where mp and me are the masses of the proton and electron, respectively. The charge on the electron and the proton and their masses were given above.
While Rydberg took experimental data and developed an empirical formula that described the line spectra of the hydrogen atom, the results of Schrödinger’s solution to the hydrogen atom problem using quantum theory are fundamentally different. We have to spend a moment to marvel at the triumph of quantum theory that emerged in 1925. There are no adjustable parameters in Schrödinger’s derivation of the energy levels of the hydrogen atom. All of the necessary constants are fundamental properties of the particles and the electrostatic interaction that attracts the electron’s negative charge to the proton’s positive charge. Schrödinger did not look at the experimental data and then adjust a constant, RH, until it fit the data. He set up a theoretical formalism and applied it to the hydrogen atom. The application of his theory accurately reproduced the experimental observables, the hydrogen atom line spectra, using only fundamental constants. In contrast to Bohr’s theory, the Schrödinger Equation has been successfully applied to a tremendous number of other problems including atoms other than hydrogen and small and large molecules. As mentioned above, for systems larger than the hydrogen atom, that is, atoms and molecules involving more than two particles, the Schrödinger Equation cannot be solved exactly. However, many powerful approximation techniques have been developed that enable accurate solutions to the Schrödinger Equation for atoms, molecules, and other types of quantum mechanical problems. With the advent of computers, and the current tremendous power of computers, it is possible to solve the Schrödinger Equation for very large and complex molecules. As discussed in subsequent chapters, molecules have shapes. The solutions to the Schrödinger Equation for a molecule give its energy levels and its wavefunctions. The wavefunctions provide the necessary information to describe the shapes of molecules.
THERE ARE FOUR QUANTUM NUMBERS
The energies of the different states of the hydrogen atom only depend on a single quantum number, n. However, there are actually four quantum numbers associated with electrons in atoms. These come out of solving the hydrogen atom with quantum theory. One of these only comes into play for atoms and molecules that have more than one electron. In that sense, the hydrogen atom is a special case because it has only one electron. In the hydrogen atom, in addition to the principal quantum number n, the two other quantum numbers are l and m. l is called the orbital angular momentum quantum number and m is called the magnetic quantum number. These two quantum numbers, when combined with n, determine how many different states are associated with a particular energy, and they determine the shapes of the wavefunctions. The fourth quantum number is s. It is called the spin quantum number. When Bohr solved the hydrogen atom problem with old quantum theory, the electron moved in orbits that had different energies and shapes. Schrödinger’s correct quantum solution to the hydrogen atom gave the energies and the wavefunctions, which, in correspondence to Bohr’s orbits, are called “orbitals.” In discussing atoms and molecules, we often use the term wavefunction and orbital interchangeably. The orbitals are probability amplitude waves that obey Heisenberg’s Uncertainty Principle, in contrast to Bohr’s orbits.
As stated above, the principal quantum number, n, can take on values, n ≥ 1, that is, 1, 2, 3, 4, etc. l can have values from 0 to n-1 in integer steps. m can take on values from l to -l in integer steps. s can only take on two values, +½ or -½. These values are summarized in Table 10.1.
For historical reasons, the states with different quantum numbers l are given different names. An s orbital has l = 0. A p orbital has l = 1. A d orbital has l = 2. An f orbital has l = 3. For our discussions of all atoms, we will only need to go to f orbitals, that is, l = 3. As shown below, these different orbital types have different shapes.
Because the energies of the states (orbitals) of the hydrogen atom only depend on the quantum number n, there will be more than one state with the same energy for n > 1. For n = 1, l = 0, and m = 0 (see the table). Therefore, there is a single orbital with n = 1. It has l = 0, so it is referred to as the 1s orbital. The 1 is the n value, and s means that l = 0. For n = 2, l can equal 0, giving rise to the 2s orbital. However, for n = 2, l can also equal 1. For l = 1, m can equal 1, 0, or -1 (see the table). l = 1 is a p orbital, and there are three different p orbitals that can be called 2p1, 2p0, and 2p-1. The 2 is the principal quantum number n. The p means l = 1, and the three subscripts are the three possible m values. So for n = 2, there are four different states.
TABLE 10.1. Quantum Numbers.
If n = 3, then l can equal 0, to give the 3s orbital. l can also equal 1 with m = 1, 0, and -1, to give 3p1, 3p0, and 3p-1. In addition, l can equal 2. For l = 2, m can equal 2, 1, 0, -1, and -2. These are the d orbitals, 3d2, 3d1, 3d0, 3d-1, and 3d-2. There are five different d orbitals. Therefore, for n = 3, there are nine different states: an s orbital, three p orbitals, and five d orbitals. When n = 4, there is the 4s orbital, the three different 4p orbitals, 4p1, 4p0, and 4p-1, the five different 4d orbitals, 4d2, 4d1, 4d0, 4d-1, and 4d-2. In addition, there are seven f orbitals, 4f3, 4f2, 4f1, 4f0, 4f-1, 4f-2, and 4f-3. Therefore, for n = 4, there are a total of 16 states: an s orbital, three p orbitals, five d orbitals, and seven f orbitals.
As mentioned above, each of the orbitals has a different shape. It is common to rename the orbital with an indication of its shape. For example, the three different 2p orbitals, rather than being called 2p1, 2p0, and 2p-1, are usually called 2px, 2pz, and 2py. The relation between the subscript and the shape will become clear when the shapes are presented.
HYDROGEN ATOM ENERGY LEVELS
Figure 10.1 shows an energy level diagram for the hydrogen atom. Levels are shown for n = 1 to 5. The spacings between the levels are not scaled properly for clarity of presentation, but as shown, the spacing between levels gets smaller as n increases. Also as n in-creases, the number of different states (orbitals) associated with the particular n increases. Hydrogen is a special case because it only has one electron. For hydrogen, all orbitals with the same n have the same energy. As discussed in the next chapter, for atoms with more than one electron, for a given n, orbitals with different l values have different energies.
FIGURE 10.1. Hydrogen energy level diagram. The spacings between the levels are not to scale. The first five energy levels are shown. The energy only depends on the principal quantum number, n. The orbitals and the number of each type are also shown. For n = 4, there is a single s orbital, three different p orbitals, five different d orbitals, and seven different f orbitals. The diagram would continue with the n = 6 level. The different levels are sometimes referred to as shells.
HYDROGEN ATOM s ORBITALS
Although the hydrogen energies only depend on the principal quantum number n, l and m still play an important role. These quantum numbers determine the shapes of the orbitals, and they determine other aspects of the hydrogen atom’s properties. For example, the m quantum number is called the magnetic quantum number. The three 2p orbitals, 2p1, 2p0, and 2p-1, differ by their m quantum number. When the hydrogen atom is put in a magnetic field, the energies of these three orbitals are no longer the same.
From the energy levels calculated with the Schrödinger equation (see Figure 10.1) it is clear how the empirical diagram in Figure 9.3 arises. The optical transitions seen in the line spectrum of the hydrogen atom and described by the Rydberg formula are transitions between the energy levels of the hydrogen atom, energy levels that are calculated using quantum theory with no adjustable parameters.
As mentioned above, the n, l, and m quantum numbers all go into determining the shapes of the wavefunctions. The s orbitals have l = 0. l = 0 means that the electron has no angular momentum in its motion relative to the nucleus of the atom. All directions look the same, so s orbitals are spherically symmetric three-dimensional probability amplitude waves. Figure 10.2 shows schematic representations of the 1s, 2s, and 3s orbitals (probability amplitude waves). The darker shading indicates a greater probability of finding the electron that distance from the center. The distances at which the probabilities have maxima are shown by the solid circles. The centers of the white regions in the 2s and 3s orbitals (dotted circles) are nodes, that is, regions where the probability of finding the electron goes to zero. In going from the 1s to the 2s to the 3s, the size of the orbital becomes much larger. The electron has a greater probability of being found further away from the nucleus as the n quantum number increases.
FIGURE 10.2. The 1s, 2s, and 3s orbitals shown in two-dimensional representations. These are actually spherical. Darker represents a greater probability of finding the electron. Solid circles are distances with peaks in the probability. Dashed circles are nodes where the probability goes to zero. The way the orbitals are represented, they have a fairly sharp outer edge. The orbitals are waves that become very small at large distances, but only decay to zero as the distance from the center goes to infinity.
The increased size of the orbitals is the reason that the energy increases as the quantum number n increases. The formula for the energy levels of the hydrogen atom has a negative sign in front of it, En = -RH/n2. We use a sign convention that a lower energy is a more negative energy. The hydrogen atom is composed of a proton and an electron attracted to each other through a Coulomb interaction, that is, an electrostatic interaction. Opposite charges attract. The proton is positively charged and the electron is negatively charged. When a proton and electron are infinitely far apart, they do not feel each other. There is no attraction because they are so far apart. The system has its zero of energy when the particles are separated at infinity. The electron and proton attraction increases as they get closer to each other. The energy of the system decreases, becoming increasingly negative. The 2s orbital has the electron further away from the proton on average than the 1s, and the 3s orbital has the electron still further away from the proton on average. This is clear from Figure 10.2. As the quantum number increases, the energy is a smaller negative number. For larger values of n, it takes less energy to separate the electron and proton, that is, to ionize the atom. Ionization is the process of pulling the electron out of an atom so that they are no longer bound together. For n = 1, it takes an energy of RH to ionize the atom. This is the amount of energy that needs to be put into the atom to overcome the binding of -RH. When n = 2, it only takes RH/4 to ionize a hydrogen atom. When n = 3, even less energy, RH/9, is needed to ionize the atom.
SPATIAL DISTRIBUTION OF s ORBITALS
To get a better feel for the spatial distribution of the probability of finding the electron in some position, it is useful to make two types of plots of the wavefunctions. One is just to plot the wavefunction as a function of distance from the nucleus. This type of plot is useful but somewhat misleading. The second type of plot is called a radial distribution function, which will be described shortly. Figure 10.3 is a plot of the wavefunction Ψ(r) as a function of the distance from the proton, which is at the center of the atom. This type of plot is the probability amplitude of finding the electron along a single line radially outward from the center. In Figure 10.2, r is along a horizontal line starting at the center of the shaded electron distribution and moving outward to the right. Figure 10.3 shows that the probability of finding the electron along a single line decreases rapidly, and is close to zero by a distance from the nucleus of 3 Å.
The problem with the type of plot shown in Figure 10.3 is it does not account for the three-dimensional nature of the atom. Looking at the 1s orbital in Figure 10.2, you can see that you can find the electron at some distance from the center by moving along a line to the right, but also along a line to the left, or up, or down. You can also move in any diagonal direction a distance r and have the same probability of finding the electron. Since the atom is three dimensional, you can also move in or out of the page and find the electron. If you want to know the probability of finding the electron a certain distance r from the proton, you need to sum all of these different radial directions.
FIGURE 10.3. A plot of the 1s wavefunction Ψ(r) as a function of r, the distance from the proton. Ψ(r) is proportional to the probability of finding the electron along a line radially outward from the center of the atom. The distance r is in Å, which is 10-10m.
What is really being asked is what is the probability of finding the electron a certain distance from the nucleus when I add together all possible directions? The way to state this question is, what is the probability of finding the electron in a thin spherical shell with the spherical shell having a radius of r? As r gets bigger, the volume of the thin spherical shell increases, which for some distance, offsets the fact that the wavefunction is decreasing. To understand the roll of the thin spherical shell, consider a number of hollow rubber balls each with the same thickness rubber wall. A ball with a small radius (small r) will have less rubber in the wall than a ball with a large radius. If you just went in a single straight line from the center of the ball to the wall, and where you hit the wall you asked what is the thickness of the rubber, it would be independent of the radius of the ball. But it is clear that a large hollow ball has more rubber in its wall then a small ball.
The surface area of a sphere is 4πr2, where r is the sphere’s radius. If you multiply this by the wall thickness, you have the volume of the rubber in the ball. Now it is clear that a larger ball has a lot more rubber in the wall than a small ball. If you double the radius, the amount of rubber increases by a factor of 4. Another important fact is that as r goes to zero, the amount of rubber in the ball goes to zero because the surface area, 4πr2, goes to zero. Asking if an electron is a distance r from the nucleus is like asking how much rubber is in the wall of a ball of radius r. It is necessary to account for the increasing surface area as the radius increases.
THE RADIAL DISTRIBUTION FUNCTION
The radial distribution function is exactly what we need to take into account the three-dimensional nature of an atom. As r is increased and we look in all directions to find the electron, we must include a factor of 4πr2. The radial distribution function is a plot of the probability of finding the electron a distance r from the nucleus for all directions. As discussed in Chapter 5, the Born interpretation of the wavefunction says that the probability of finding a particle in some region of space is proportional to the absolute value squared of the wavefunction. Here we want the probability of finding the electron in a thin spherical shell that has radius r. This is the radial distribution function, which is given by 4πr2|Ψ|2. The vertical lines mean absolute value. For the functions we are dealing with, we just need to square the wavefunction.
Figure 10.4 displays the radial distribution function for the 1s state of the hydrogen atom. The distance that has the maximum probability is not the center of the atom because the volume of the spherical shell goes to zero as r goes to zero. The vertical line shows the location of the maximum in the probability distribution. It is r = 0.529 Å. This is an important and interesting number. In Bohr’s old quantum theory of the hydrogen atom, the 1s state had the elec tron going in a circular orbit with a radius of 0.529 . This distance is called the Bohr radius and is given the symbol a0. What we see from the correct quantum mechanical treatment of the hydrogen atom is that the electron is a probability amplitude wave with the distance for the maximum probability equal to the Bohr radius a0. This is not a coincidence. The Bohr radius is actually a fundamental constant. It is given by
FIGURE 10.4. A plot of the radial distribution function for the 1s orbital as a function of r, the distance from the proton. The radial distribution function is the probability of finding the electron in a thin spherical shell a distance r from the proton. The radial distribution function takes into account that the electron can be found in any direction radially outward from the proton. The distance r is in Å, which is 10-10m.
where all of the parameters were given above when the Rydberg constant was defined in terms of fundamental constants. In fact, the energy levels of the hydrogen atom can be written in terms of the Bohr radius as
Figures 10.5 and 10.6 show plots of the wavefunctions (top panels) and the radial distribution functions (bottom panels) for the 2s and 3s orbitals. The wavefunction of the 2s orbital has a node, that is, a place where the wavefunction is zero. Nodes were discussed in connection with the particle in a box wavefunctions (see Figure 8.4). At a node the probability of finding a particle, in this case the electron, is zero. The 2s wavefunction begins positive, crosses zero at the node located at twice the Bohr radius, 2a0, and then is nega tive. The wavefunction then decays to zero. By 8 Å the value of the wavefunction is very small. As we have discussed in detail, the wavefunctions are probability amplitude waves. Like other waves, they can be positive or negative. The bottom panel of Figure 10.5 displays the 2s radial distribution function. This is the probability of finding the electron a distance r from the nucleus. Probabilities are always positive because they are the square of the wavefunction, which is always positive. A wave can be positive or negative, but it makes sense that a probability is always a positive number or zero. The radial distribution function shows that most of the probability is between about 2 and 4 Å, which also can be seen in Figure 10.2 but not as quantitatively. The peak of the probability is located at ∼2.8 Å.
FIGURE 10.5. A plot of the 2s hydrogen atom wavefunction (top panel) and the radial distribution function (bottom panel), as functions of r, the distance from the proton. The wavefunction begins positive, goes through a node at slightly more than 1 Å (2a0), and then decays to zero. The radial distribution function shows that the maximum probability of finding the electron peaks at about 2.8 Å, with most of the probability between 2 and 4 Å (see Figure 10.2). The distance r is in Å, which is 10-10m.
FIGURE 10.6. A plot of the 3s hydrogen atom wavefunction (top panel) and the radial distribution function (bottom panel), as functions of r, the distance from the proton. The wavefunction begins positive, goes through a node, becomes negative, goes through a second node, and becomes positive again. It then decays to zero. The radial distribution function shows that the maximum probability of finding the electron peaks at about 7 Å, with most of the probability between 5 and 11 Å (see Figure 10.2). The distance r is in Å, which is 10-10m.
In Figure 10.6, it can be seen that the 3s wavefunction has two nodes, that is, the wavefunction crosses zero twice. The hydrogen atom wavefunctions have this in common with the particle in a box wavefunctions (see Figure 8.4). For n = 1, there is no node. For n = 2, there is one node. For n = 3, there are two nodes. The number of nodes for the s orbitals is n-1. The 3s wavefunction begins positive, goes negative, and then becomes positive again. It finally de cays to zero, and is very small by about 16 Å. The 3s radial distribution function shows that most of the probability is relatively far from the nucleus. The peak probability is at ∼7 Å. Most of the probability is between 5 and 11 Å. The three radial distribution functions shown in Figures 10.4, 10.5, and 10.6 are quantitative plots of the information shown schematically in Figure 10.2. As the principal quantum number, n, gets larger, the s orbitals become larger and have more nodes.
THE SHAPES OF THE p ORBITALS
For the 2s orbital, n = 2, l = 0, and m = 0. However, for n = 2, l can also equal 1 with the associated three values of m, m = 1, 0, -1. The three different m values give rise to the three different 2p orbitals. These are shown in the energy level diagram, Figure 10.1. The three different 2p orbitals are represented schematically in Figure 10.7. As mentioned above, because of their shapes, the 2p orbitals are usually referred to as 2pz, 2py, and 2px. Each orbital has two lobes, a positive lobe and a negative lobe. Which lobe is assigned positive and negative is arbitrary, but the sign must change because there is an angular nodal plane. The 2pz orbital has its lobes pointed along the z axis. The nodal plane (shaded in the figure) is the xy plane. This plane (z = 0) is a plane where the probability of finding the electron is zero. The sign of a wavefunction changes when it passes through a node. In the 2s orbital, there is a radial node. A radial node is a distance from the center at which there is a node. Each p orbital has an angular node, that is, directions (a plane) where there is a node. The 2p orbitals do not have a radial node, but the 3p orbitals have a radial node in addition to the angular nodal plane, and the 4p orbitals have two radial nodes, etc.
The 2py orbital has its lobes pointed along the y axis, and its nodal plane is the xz plane. The 2px orbital has its lobes pointed along the x axis, and its nodal plane is the yz plane. The schematic illustrations of the 2p orbitals in Figure 10.7 are like the representations of the s orbitals in Figure 10.2. Figure 10.7 gives a feel for the regions that have a large amount of electron probability amplitude. However, it is important to recognize that these are probability amplitude waves that die out smoothly away from the nucleus. In the figure, the lobes terminate abruptly, but the wavefunctions decay at long distances in a manner similar to that shown in Figure 10.3 for the 1s orbital. Nonetheless, Figure 10.7 is useful to get a feel for the shapes of the 2p orbitals. These shapes will be very important when we discuss molecular bonding and the shapes of molecules.
THE SHAPES OF THE d ORBITALS
When n = 3, l can equal 0 to give the 3s orbital. l can equal 1 with m = 1, 0, -1 to give the three different 3p orbitals. In addition, l can equal 2 with m = 2, 1, 0, -1, -2 to give five different 3d orbitals. These are shown in the energy level diagram, Figure 10.1. Figure 10.8 shows the five different 3d orbitals. Like the p orbitals, the d orbitals are often given names that reflect their shapes rather than labeling them with the quantum number, m. Four of the orbitals have the same basic shape. Each has four lobes and two angular nodal planes. Two of the lobes are positive and two are negative. When a nodal plane is crossed the wavefunction changes sign. The fifth orbital, the dz2, has a different shape, but it still has two angular nodal surfaces. One is the xy plane, and the other is the conical surface shown in the diagram. Like the p orbitals, the shaded regions in Figure 10.8 indicate where most of the electron probability amplitude is found. These probability amplitude waves go smoothly to zero as the distance from the nucleus increases.
FIGURE 10.7. Schematic of the three hydrogen atom 2p orbitals, 2pz, 2py, and 2px. Each orbital has two lobes, one positive and one negative. Each has an angular nodal plane, that is, a plane where the probability of finding the electron is zero. The 2pzorbital has its lobes along the z axis, and the nodal plane is the xy plane, which is shaded. The 2pyorbital has its lobes along the y axis, and the nodal plane is the xz plane. The 2pxorbital has its lobes along the x axis, and the nodal plane is the yz plane. The lobes in each diagram show where most of the electron probability amplitude is located. These probability amplitude waves delay smoothly to zero away from the nucleus (the proton) and do not stop abruptly as in the diagrams.
When n = 4, in addition to s, p, and d orbitals, l can equal 3, which gives rise to the seven possible m values. These are the seven f orbitals. The f orbitals have three angular nodes and very complicated shapes. As will be discussed in the next chapter on atoms bigger than hydrogen, only very heavy elements have f orbitals containing electrons, and the f electrons do not usually participate in making chemical bonds. Many molecules, particularly those in which the prime element is carbon called organic molecules, involve mainly 2s and 2p orbitals. However, molecules that contain heavier elements, such as metals, may involve d orbitals as well.
In Chapter 11, we will build on our discussion of the hydrogen atom to understand the properties of all atoms. Because these larger atoms contain more than one electron, the fourth quantum number, s, will come into play. By applying some simple rules, we will be able to understand many of the properties of atoms and how they form molecules.
FIGURE 10.8. Schematics of the five hydrogen atom 3d orbitals, which are named in relation to their shapes. Each orbital has two angular nodes, as well as positive and negative lobes. The angular nodes are the planes in four of the diagrams and the cones and disk in the fifth diagram. When a nodal plane is crossed, the wavefunction changes sign. The lobes in each diagram show where most of the electron probability amplitude is located. Four of the orbitals consist of four lobes. The dz2 orbital has a different shape. It sill has two nodal surfaces, the xy plane and the conical surfaces. These probability amplitude waves decay smoothly to zero away from the nucleus (the proton) and do not stop abruptly as in the diagrams.