Orbitals and the 4th Quantum Number, (M7Q6) – UW-Madison Chemistry 103/104 Resource Book (2024)

Introduction

Orbital Shapes

An orbital is the quantum mechanical generalization of Bohr’s orbit. In contrast to his concept of a simple circular orbit with a fixed radius, orbitals are mathematically derived regions of space with differentprobabilitiesof having an electron in them.

s-Orbitals

One way of representing electron probability distributions is illustrated in Figure 1 for the 1s orbital of hydrogen. Recall that orbitals are described by wavefunctions, which are represented with ψ. Recall from the M7Q6, that Max Born proposed an interpretation of the wavefunction,ψ, that is still accepted today: the probability density of finding an electron at a given point in space is [ψ(r)]². In Figure 1a, a dot density diagram is represented. This dot density diagram displays black dots where there is a probability of finding an electron—the higher density of dots, the higher the probability of finding the electron in that region. Taking the information from the dot density diagram, a plot of [ψ(r)]²versus distance from the nucleus (r) can be constructed. This type of plot (shown in Figure 1b) is a plot of the probability density. This plot shows the probability of finding the electron near a particular point in space that lies a distance r from the nucleus. The 1sorbital is spherically symmetrical, so the probability of finding a 1selectron at any given point dependsonlyon its distance from the nucleus. The probability density is greatest atr= 0 (at the nucleus) and decreases steadily with increasing distance. At very large values ofr, the electron probability density is tiny butnotexactly zero.

In contrast to the probability density, we can calculate theradial probability(the probability of finding a 1selectron at a distancerfrom the nucleus) by adding together the probabilities of an electron being at all points on a series ofxspherical shells of radiusr₁,r₂,r₃,…,r_{x− 1},r_x. In effect, we are dividing the atom into very thin concentric shells, much like the layers of an onion (Figure 1a), and calculating the probability of finding an electron on each spherical shell. Recall that the electron probability density is greatest atr = 0 (Figure 1b), so the density of dots is greatest for the smallest spherical shells in Figure 1a.

The surface area of each spherical shell is equal to 4πr², which increases rapidly with increasingr (Figure 1c). Because the surface area of the spherical shells increases at first more rapidly with increasingr than the electron probability density decreases, the plot of radial probability has a maximum at a particular distance (Figure 1d). Importantly, when ris very small, the surface area of a spherical shell is so small that thetotal probability of finding an electron close to the nucleus is very low; at the nucleus, the electron probability vanishes because the surface area of the shell is zero (Figure 1d).

Another representation of the 1sorbital is given in Figure 2. Again, this dot density diagram illustrates the regions where an electron is likely to be found in the “electron cloud”.

Not all s orbitals are the same, however. While the 1sorbital looks like that shown in Figure 2, recall that there are alsosorbitals inn= 2, 3, etc. (called 2s, 3s, and ns orbitals). Figure 4 compares the electron probability densities for the hydrogen 1s, 2s, and 3sorbitals. Note that all three are spherically symmetrical. For the 2sand 3sorbitals, however (and for all othersorbitals as well), the electron probability density does not fall off smoothly with increasingr. Instead, a series of minima and maxima are observed in the radial probability plots (Figure 4c). The minima correspond to radial nodes (regions of zero electron probability), which alternate with spherical regions of nonzero electron probability. The orbitals depicted in Figure 4 are of the s type, thus l = 0 for all of them.

Three things happen tosorbitals asn increases (Figure 4):

1. They become larger, extending farther from the nucleus.
2. They contain more nodes. This is similar to a standing wave that has regions of significant amplitude separated by nodes (points with zero amplitude). The number of radial nodes in an orbital is n – l – 1.
3. For a given atom, thesorbitals also become higher in energy asn increases because of the increased distance from the nucleus.

Orbitals are generally drawn as three-dimensional surfaces that enclose 90% of theelectron density.Although such drawings show the relative sizes of the orbitals, they do not normally show the spherical nodes in the 2sand 3sorbitals because the spherical nodes lie inside the 90% surface. Fortunately, the positions of the spherical nodes are not important for chemical bonding. This makes sense because bonding is an interaction of electrons from two atoms which will be most sensitive to forces at the edges of the orbitals.

See Also

SpinEvaluating Spin MultiplicityIF THERE ARE THREE POSSIBLE VALUES FOR SPIN QUANTUM NUMBER, THEN POSSIBLE NUMBER OF ELEMENTS IN THE 5TH PERIOD OF PERIODIC TABLE WOULD BE- HOW??Quantum Numbers for Atoms

p-Orbitals

Onlysorbitals are spherically symmetrical. As the value oflincreases, the number of orbitals in a given subshell increases, and the shapes of the orbitals become more complex. Because the 2psubshell hasl= 1, with three values ofm_l(−1, 0, and +1), there are three 2p orbitals.

The dot density diagram for one of the hydrogen 2porbitals is shown inFigure 5. This orbital has two lobes of electron density arranged along the z axis with an electron density of zero in thexy plane. This type of node is called a nodal plane (instead of a radial node which was seen in thes orbitals). Since the xy plane is the nodal plane in Figure 5, it is a 2p_z orbital (the orbital is oriented along thez axis).

As shown in Figure 6, the other two 2porbitals have identical shapes, but they lie along thexaxis (2p_x) andyaxis (2p_y), respectively. Note that each 2p orbital has just one nodal plane.

Just as with thesorbitals, the size and complexity of theporbitals for any atom increase as the principal quantum numbernincreases. The shapes of the 90% probability surfaces of the 3p, 4p, and higher-energyp orbitals are, however, essentially the same as those shown in Figure 6.The shapes of porbitals for n > 2 look like strings of beads oriented along the x, y, and z axis with two additional nodes per unit increases in n.

d-Orbitals

Subshells withl= 2 have fived orbitals; the first shell to have adsubshell corresponds ton = 3. The five dorbitals havem_lvalues of −2, −1, 0, +1, and +2.

The hydrogen 3d orbitals, shown in Figure 7, have more complex shapes than the 2porbitals. All five 3d orbitals contain two nodal planes, as compared to one for each p orbital and zero for eachsorbital. In three of thedorbitals, the lobes of electron density are oriented between thexandy,xandz, andyandzplanes; these orbitals are referred to as the 3d_xy, 3d_xz, and 3d_yzorbitals, respectively. A fourthdorbital has lobes lying along thexandy axes; this is the 3d_x²-y²orbital. The fifth 3d orbital, called the 3d_z²orbital, has a unique shape: it looks like a 2p_zorbital combined with an additional doughnut of electron probability lying in thexy plane. Despite its peculiar shape, the 3d_z² orbital is mathematically equivalent to the other four and has the same energy. Like thesandporbitals, asnincreases, the size of thed orbitals increases, but the overall shapes remain similar to those depicted in Figure 7.

f-Orbitals

Principal shells with n = 4 can have subshells with l = 3 and ml values of −3, −2, −1, 0, +1, +2, and +3. These subshells consist of seven f orbitals. Although these orbitals play an important role in the heavier elements, their shapes are complicated and we will not consider these shapes in this course

Orbital Energies

Although we have discussed the shapes of orbitals, we have said little about their comparative energies. We begin our discussion oforbital energies by considering atoms or ions with only a single electron (such as H or He⁺). This is the simplest case.

The relative energies of the atomic orbitals withn ≤ 4 for a hydrogen atom are plotted in Figure 8. Note that the orbital energies depend ononlythe principal quantum numbern, not on l. Consequently, the energies of the 2sand 2porbitals of hydrogen are the same; the energies of the 3s, 3p, and 3dorbitals are the same; and so forth. The orbital energies obtained for hydrogen using quantum mechanics are exactly the same as the allowed energies calculated by Bohr. In contrast to Bohr’s model, however, which allowed only one orbit for each energy level, quantum mechanics predicts that there are4orbitals with different electron density distributions in then= 2 principal shell (one 2sand three 2p orbitals),9in then= 3 principal shell, and16in then = 4 principal shell. As we have just seen, quantum mechanics predicts that in the hydrogen atom, all orbitals with the same value of n(e.g., the three 2porbitals) aredegenerate, meaning that they have the same energy. Figure 8 shows that the energy levels become closer and closer together as the value ofnincreases, as expected because of the 1/n²dependence of orbital energies.

It was demonstrated in the 1920s that when hydrogen line spectra (which we saw previously) are examined at extremely high resolution, some lines are actually not single peaks but rather pairs of closely spaced lines. This is the so-called fine structure of the spectrum, and it implies that there are additional small differences in energies of electrons even when they are located in the same orbital. These observations led Samuel Goudsmit and George Uhlenbeck to propose that electrons have a fourth quantum number. They called this the spin quantum number, or m_s.

The other three quantum numbers—n, l, and m_l—are properties of specific atomic orbitals that define the most likely location in space of an electron. These orbitals are a result of solving the Schrödinger equation for electrons in atoms. The electron spin is a different kind of property. It is a completely quantum phenomenon with no analogues in the classical realm. In addition, it cannot be derived from solving the Schrödinger equation and is not related to the normal spatial coordinates (such as the Cartesian x, y, and z). Electron spin describes an intrinsic electron “rotation” or “spinning.” Each electron acts as a tiny magnet, just as if it were a tiny rotating object charged with an angular momentum, even though this rotation cannot be observed in terms of the spatial coordinates.

The magnitude of the overall electron spin can only have one value, and an electron can only “spin” in one of two quantized states. One is termed the α state, with the z component of the spin being in the positive direction of the z axis. This corresponds to the spin quantum number m_s = +½. The other is called the β state, with the z component of the spin being negative and m_s = -½. Any electron, regardless of the atomic orbital it is located in, can only have one of those two values of the spin quantum number. The energies of electrons having m_s = -½ and m_s = ½ are different if an external magnetic field is applied.

Figure 9illustrates this phenomenon. An electron acts like a tiny magnet. Its moment is directed up (in the positive direction of the z axis) for the +½ spin quantum number and down (in the negative z direction) for the -½ spin quantum number. A magnet has a lower energy if its magnetic moment is aligned with the external magnetic field (the left electron in Figure 19) and a higher energy for the magnetic moment being opposite to the applied field. This is why an electron withm_s = +½ has a slightly lower energy in an external field in the positive z direction, and an electron with m_s = -½ has a slightly higher energy in the same field. This is true even for an electron occupying the same orbital in an atom. A spectral line corresponding to a transition for electrons between the same orbitals but with different spin quantum numbers has two possible values of energy; thus, the line in the spectrum will show a fine structure splitting.