Atomic structure
We have now been introduced to a number of empirical properties of atoms and molecules. We have seen the ‘old’ quantum theory, which was the first set of ad hoc explanations of quantum theory that bolted on quantum mechanical elements to classical physics because it had to, rather than because they were derived from first principles. One of the most successful and insightful models in ‘old’ quantum theory is the Bohr atom. Electron orbits are quantized because they have to be, and they do not fall into the nucleus because there must be a minimum value of the angular momentum. These orbits have associated with them a principal quantum number n because we need it. Then Pauli says there must also be a half-integer spin of the electron, which makes no sense, but fits the data and seems to make things work. Another extremely successful idea is that vibrations and photons are quantized with an energy directly proportional to their frequency.
The importance of the old quantum theory is that it allowed us to get used to the strange consequences that result from the mathematics of proper quantum mechanics. We have also seen quantum theory applied from first principles to model systems: free particle, particle in a box, harmonic oscillator, and particle on a sphere. It is time to bring the quantum mechanical machinery to bear on the atom and derive its structure from first principles. Thereby, we will discover what the quantum description of atoms, molecules and bonding really entails. We will also discover even more nuance and richness in the amazing world of light–matter interactions (spectroscopy, lasers and photochemistry) and the insight they reveal on the workings of the universe. This is our task over the next several chapters.
22.1 The hydrogen atom
The H atom corresponds to a radial Coulomb field with a potential proportional to the inverse of distance,
Classically this is unstable. The lowest energy in the system corresponds to the electron spiraling into the nucleus where it crash-lands and stays. Bohr was able to rectify this situation by asserting that the angular momentum of the electron must be quantized and its minimum value is not zero. Doing so allowed him to derive an expression for the lines observed in the H atom emission (or absorption) spectrum. He surmised that the energy level structure of the H atom corresponds to levels with distinct energies. Transitions between levels correspond to quantum jumps that only occur at well-defined energies. The energy difference between levels is made up by the absorption or emission of a photon in order to conserve energy. Now, we seek a more fundamental derivation of the H atom structure that arises from the postulates of quantum mechanics. Such a derivation should contain within the mathematics additional predictions and deeper insight.
Within Bohr’s model, the radius of the electron’s orbit is also quantized. The motion of an electron in a Coulomb field constrained to concentric shells is analogous to the particle on a sphere model that we solved in the last chapter. Thus, much of the heavy lifting in terms of the mathematics has already been done. From this we understand the relevance of spherical harmonics to atomic orbitals. At least, the angular part has been solved. So far we have a collection of concentric spheres but we do not know how they relate to one another as a function of radius. That is, there is a radial dependence to the problem, which we need to understand.
22.1.1 Set up hamiltonian and solve Schrödinger equation
The procedure is to define a complete Hamiltonian, and then to solve the resulting Schrödinger equation for the infinite set of wavefunctions that describe the energy level structure. With the coordinate system defined as in Fig. 22.1 we can write the Hamiltonian in terms of the kinetic energy operator plus the electrostatic potential of the electron–proton attraction,
where μ is the reduced mass
Figure 22.1 Coordinate system for the hydrogen atom in spherical polar coordinates.
Spherical symmetry calls for the Laplacian to be written in spherical polar coordinates as done in Eq. (21.82), which we repeat here
Because of the analogy to the particle on a sphere model, we expect the angular part of the wavefunction to be solved by the spherical harmonics Ylm(θ,φ). Assume a separable solution composed of the product of radial and angular parts,
We do not know a priori that the radial wavefunction Rnl(r) will depend on n and l, but this will indeed be the case.
As we demonstrated in Chapter 21 in Eq. (21.86), the spherical harmonics Ylm(θ,φ) are eigenfunctions of the Legendrian operator (the angular part of the Laplacian operator) with eigenvalue l(l + 1), thus
Now we need to solve the radial equation, which from Eq. (21.85) we know to be
While perhaps not immediately obvious unless you sit around looking at differential equations all the time, the solution to this second-order differential equation is made easier by the substitution
which yields
Equation (22.9) is a one-dimensional Schrödinger equation in which the Coulomb potential has been replaced by an effective potential energy
The first term is the Coulomb potential. The second is a repulsive centrifugal angular momentum dependent term. Substituting
and dividing by E, which is negative for bound states, we can rewrite Eq. (22.9) as
To make this even more compact, we introduce
which allows us to write
The form of the Rnl(r) that provides the solution to Eq. (22.14) is given by
The v(ρ) term is related to the associated Laguerre polynomials, which are polynomials in ρ of degree (n − l − 1) . For an excellent review of the method of solving for these functions, see the book by Griffiths in Further Reading. The form of Rnl(r) is (exponential) × (polynomial). Examples of the Rnl(r) are given in Table 22.1. The exponentially decaying term constrains the wavefunction to be localized in the region of space close to the nucleus. The polynomial of order (n − l − 1) introduces (n − l − 1) radial nodes into the wavefunction. A general rule will be that the greater the number of nodes, the higher the energy of the associated state. More nodes correlate with greater curvature in the wavefunction and, therefore, higher kinetic energy.
Table 22.1 Hydrogenic radial wavefunctions derived from solution of the Schrödinger equation.
| Orbital | n | l | Rn,l |
|---|---|---|---|
| 1s | 1 | 0 | R10 = 2a− 3/2 exp( − r/a) |
| 2s | 2 | 0 |  |
| 2p | 2 | 1 |  |
| 3s | 3 | 0 |  |
| 3p | 3 | 1 |  |
| 3d | 3 | 2 |  |
| 4s | 4 | 0 |  |
| 4p | 4 | 1 |  |
| 4d | 4 | 2 |  |
| 4f | 4 | 3 |  |
 [footnote 1]1 |
The radial wavefunction is found to depend on both the principal quantum number n,
and the orbital angular momentum (azimuthal) quantum number l. The value of l is constrained by n such that
For each l value there are (2l + 1) possible values of m. This means that the degeneracy of each level is
However, the energy of the H atom levels depends only on n according to
There is no l dependence to the energy of the H atom. Therefore, only Rnl is required to derive its energy level structure. If the reduced mass is calculated specifically for the H atom using μ = μH, then the value of the constants out front is equal to RH = 109 677.581 cm–1. If an infinite nuclear mass is used such that μ = me, then we obtain R∞ = 109 737.31534 cm–1. The number of digits may seem extreme and, indeed it is. However, the Rydberg constant is arguably the most accurately measured constant in the pantheon of physical constants. It is known with a relative uncertainty of about 8 × 10–13 from the work of Theodor W. Hänsch and coworkers.2 Hänsch was awarded the Nobel Prize in Physics 2005 for his work in high-resolution laser spectroscopy. At this level of accuracy, spectroscopy on the H atom and its exotic cousin muonic hydrogen (a muon bound by a proton) is able to probe the structure of the proton and provide a stringent test to the Standard Model of particle physics.
The H atom spatial wavefunction is represented by
These wavefunctions form an orthonormal set that obeys the condition
In the next section we will see that this wavefunction is not yet complete because it does not include the electron spin.
22.2 How do you make it better? The Dirac equation
Great excitement accompanied the verification that the Schrödinger equation returned energies that correctly predicted the structure of the H atom spectrum. It won Erwin the Nobel Prize in Physics, which he shared with Dirac in 1933. However, as experimental resolution and sensitivity increased, systematic deviations from the predictions of the Schrödinger equation became apparent almost immediately. These deviations indicated that the gross structure of the H atom must be as predicted but that an underlying fine and even hyperfine structure needed further explanation.
To improve the predictions of the Schrödinger equation as solved above, we need to fix the Hamiltonian by making it consistent with relativity. Schrödinger had attempted this but abandoned the effort because of what appeared to be internal inconsistencies. Wolfgang Pauli delivered an essential clue about how to proceed when he proposed3 the necessity for a new quantum number to describe the intrinsic magnetic moment of the electron. For historical reasons, he called this quantum number ‘spin,’ even though this is conceptually misleading. Importantly, he showed that the spin of an electron is half-integer, a concept that had previously been thought to be untenable. This insight allowed Paul Dirac to formulate the Dirac equation in 1930 – a fully relativistic form of the Schrödinger equation.4 The Dirac equation combines quantum theory with Maxwell’s theory of the electromagnetic field to produce quantum field theory, from which the dual nature (wave/particle as well as matter/antimatter) of subatomic particles can be deduced directly. The Dirac equation is much more complicated than the Schrödinger equation and we will not show it here. However, the resulting first-order corrections can be simply explained.
The Dirac equation shows us how to fix the Hamiltonian to account for relativistic effects. In so doing it will also show us a basic principle of how to treat systems quantum mechanically.
- Work out the Hamiltonian that contains the most important interactions (the minimum set of interactions that give the system its fundamental structure).
- Then add smaller perturbations to this minimal set of interactions by adding additional terms to the Hamiltonian.
To improve the accuracy of the Hamiltonian for the H atom this is done by adding three terms to the zeroth-order kinetic + potential energy term 
derived from the Schrödinger equation,
From the 
term, the energy only depends on n. The three corrections are the mass-velocity term 
, which accounts for relativistic variation of the inertia of the electron with velocity. Written with the fine structure constant α = e2/ℏc
E – V is the kinetic energy of the electron. Thus, this term is the relativistic correction in the form of − Ek(Ek/2mc2). The Darwin term is physically obscure, though some state that it corrects for the relativistic electric moment of the electron and nonlocalizability5
The third term is the spin–orbit term,
It represents the energy associated with the interaction of the intrinsic magnetic moment of the electron and the magnetic field generated by the orbital motion of the electron as it moves in the electric field of the nucleus. To simplify Eqs (22.24–22.25), we use atomic units in which distance is measured in terms of the Bohr radius a0 and energy is in Rydberg units.
Quantization within a four-dimensional space-time leads to the natural introduction of a fourth quantum number, the spin s, just as quantization within a three-dimensional space leads to the need for the three quantum numbers n, l and m.6 As expected, the energies of the states become dependent on the spin when a Hamiltonian that takes into account relativistic effects is solved in four-dimensional space-time. In fact, the spin–orbit term makes the energy dependent on the coupling of l and s, represented by j = l + s. The total energy of each level, now specified by the quantum numbers n, l and j is is the atomic units unit of energy known as the Rydberg, Ry. The spin–orbit term increases in size with increasing atomic number Z. For heavier atoms, this term results in large shifts that will be dealt with in more detail in Chapter 23.
The term
The simple Schrödinger equation leads to a single ground-state level with n = 1 and l = 0. There are two degenerate excited states, n = 2, l = 0, 1. The Dirac equation leads to a ground state with n = 1, l = 0, j = 1/2. The two excited states with n = 1 and l = 0, 1 are now split into two different energies in which the j = 1/2 state lies 0.365 cm–1 below the j = 3/2 state. These splitting are shown in Fig. 22.2.
Figure 22.2 The progressive development of the H atom spectrum as higher levels of theory from the simple Schrödinger equation to the Dirac equation to quantum electrodynamics (QED) to the coupling of QED corrections with nuclear hyperfine splitting match higher levels of experimental resolution and sensitivity.
Quantum electrodynamics and accounting for the finite size of the nucleus provides the next correction. The ground state 2S1/2 shifts up by 0.2722 cm–1. There are now three excited states as the j = 1/2 state splits by 0.0354 cm–1 between the state derived from l = 1 (lower E, 2P1/2) and the state from l = 0 (2S1/2). This is the Lamb shift, which is related to vacuum fluctuations (i.e., the fabric of the universe or the quantization of the electromagnetic field). These fluctuations cause deviations in electron trajectories from spherical symmetry. They are also responsible for spontaneous emission, that is, fluorescence. The two j = 1/2 states both couple to the continuum of states resulting from vacuum fluctuations. The difference in coupling of theses states via a photon to the ground state leads to a slightly different lifetime for each state. Because there is an uncertainty relationship between lifetime and energy, the slight difference in lifetime leads to a slight shift in energy. The l = 1, j = 3/2 state (2P3/2) shifts upward.
The next correction is nuclear hyperfine splitting, which results from the interaction of the nuclear spin I with the electron. F = 0 and F = 1 states in ground state split by 0.0475 cm–1.
The perturbations introduced as corrections to the Hamiltonian also slightly change the wavefunctions of the H atom. These changes are extremely small in comparison to the structure imposed by the zeroth-order Hamiltonian. We need not consider the changes in any of our discussions below. However, it will be worth keeping them in mind, as well as the ideas of perturbations and the emergence of more levels at high resolution when we explore atomic spectroscopy in the next chapter. While the splittings and corrections introduced are quite small, there are three very important concepts to take away from this section:
- A fully relativistic treatment of the H atom demands the inclusion of spin in the complete wavefunction, which makes the Pauli exclusion principle a natural consequence of the electron being a fermion.
- An extremely effective method of modeling the behavior of complex systems is to establish a Hamiltonian that contains the essence of the system, and then bolt on to this smaller perturbations that improve the description of the system.
- Radiative relaxation of an excited state (called in different contexts spontaneous emission, fluorescence or phosphorescence) is caused by the interaction of excited states with fluctuations in the vacuum field that arise from the quantization of the electromagnetic field that fills all space.
22.2.1 Directed practice
Why are the corrections imposed by the Dirac equation much larger for the hydrogenic Cs ion than for the H atom?
22.3 Atomic orbitals
The Dirac equation tells us that the solution to the Schrödinger equation is incomplete. The complete form of the wavefunction is a product not only of the radial and angular parts but also a spin eigenvector
That spin needed to be included was already known from the work of Pauli. Indeed, his proposal of a half-integer quantum number was an essential clue that aided Dirac in formulating and solving his equation. However, the spin part of the overall wavefunction is distinct from the radial and angular parts in that it does not depend on spatial coordinates. It only depends on whether the spin of the electron is up (ms = 1/2) or down (ms = –1/2). It is this portion of the total wavefunction that is responsible for only two electrons being able to occupy the same orbital.
22.3.1 Comparing representations
Solutions to the Schrödinger equation are given different names to emphasize different aspects depending on context, see Table 22.2. Energy levels, orbitals and wavefunctions are closely related. They emphasize different aspects of the same thing. Quantum numbers are used to describe all of these. The values of the quantum numbers determine the values of quantities such as energy, angular momentum, magnetic moment, and so forth.
Table 22.2 A comparison of different terms of reference to solutions to the Schrödinger equation.
For the H atom En = −hcRH/n2 and the explanation of the Rydberg formula, is the same obvious explanation given by the Bohr model. The energy levels are quantized and the energy depends on n. There are an infinite number of bound states, n = 1 → ∞. The energy of the n = ∞ level is 0. Transitions correspond to jumps between these quantized energy levels and are called bound-bound transitions. Bound-bound transitions correspond to sharp lines in spectra because they have well-defined energies. The photon that connects these levels through absorption or emission must match the energy difference between the states to ensure energy conservation. Small corrections to the Rydberg formula reflect corrections introduced by higher-level theories.
Above the ionization limit the electron is no longer bound to the nucleus. Because it is not confined to the region of the nucleus, the electron’s energy is no longer quantized. Excitation above n = ∞ corresponds to ionization of the atom
Ionization
Ionization is a bound-free transition that removes the electron from the atom into a continuum of states that can have any value of kinetic energy. The electron has been taken out of its spherical box. Such a transition is still subject to the limits of energy conservation, just as in the photoelectric effect. However, these transitions do not give rise to sharp lines; instead, bound-free transition gives rise to continuous spectra. They may still exhibit structure because the strength of the transition into the continuum will depend on the frequency of the light that excites it; but this structure resembles broad overlapping features rather than sharp peaks.
22.3.1.1 Example
Determine the ionization energy Ei (also known as the ionization potential I) of the first three levels of the H atom. Koopmans’ theorem states that the ionization energy is equal to the negative of the orbital energy of the orbital from which the electron originated. For a one-electron system, this theorem is exact. We will discuss deviations from this in the next chapter. Numerical results are given in Table 22.3.
Table 22.3 The ionization energies of the first three H atom levels.
| n | n2 | I/cm–1 | I/eV |
|---|---|---|---|
| 1 | 1 | 109 679 | 13.599 |
| 2 | 4 | 27 420 | 3.400 |
| 3 | 9 | 12 187 | 1.511 |
22.3.2 Shells and subshells
The solutions of the Schrödinger equation impose a structure on the electronic states that is summarized in Fig. 22.3. This is, of course, the basis of the structure of the Periodic Table. Orbitals with the same nl are called equivalent orbitals, while electrons having the same nl are equivalent electrons. The value of n determines the shell to which an orbital belongs. For each value of nl there is a set of 2l + 1 equivalent orbitals that forms a subshell that can hold a maximum of 2(2l + 1) electrons. The ‘extra’ factor of 2 arises from the Pauli exclusion principle, which dictates that one electron has ms = 1/2, the other ms = –1/2, and that there are no more than two in an orbital because no two electrons can have the same set of quantum numbers.
Figure 22.3 The structure of orbitals categorized by shell (determined by the value of n) and subshell (determined by the value of l).
A subshell with the maximum number of electrons is a closed subshell. For example, in the 2p subshell, there are three p orbitals – commonly referred to as px, py and pz in chemistry – that can hold a maximum of six electrons to form the closed subshell denoted 2p6. Note that 2l + 1 corresponds to the number of different ml values for that value of l. There are 2l + 1 orbitals for each value of l because there are 2l + 1 allowed projections of l.
For a given value of n, the allowed values of l are l = 0, 1, 2, …, n − 1; thus, there are n subshells with that value of n. Taken together, the subshells of a given n form a shell of the atom. Shells are referred to by upper-case letters starting with K for n = 1 and proceeding alphabetically beyond that. Half-filled and fully filled (closed) subshells are particularly important as points of stability and in angular momentum coupling. Tables 22.4 and 22.5 summarize the details described here.
Table 22.4 Allowed values of orbital angular momentum l, subshells and shells as function of the principle quantum number n.
| n | l | Subshells | Shell |
|---|---|---|---|
| 1 | 0 | s | K |
| 2 | 0, 1 | s, p | L |
| 3 | 0, 1, 2 | s, p, d | M |
| 4 | 0, 1, 2, 3 | s, p, d, f | N |
| 5 | 0, 1, 2, 3, 4 | s, p, d, f, g | O |
| etc. |
Table 22.5 Number of orbitals and designations of half-filled and closed subshells for the most commonly encounters subshells. The value of orbital angular momentum l is synonymous with a letter that designates the subshell.
| Subshell | l | # of orbitals = 2l + 1 | Half-filled subshell | Closed subshell |
|---|---|---|---|---|
| s | 0 | 1 | s | s2 |
| p | 1 | 3 | p3 | p6 |
| d | 2 | 5 | d5 | d10 |
| f | 3 | 7 | f7 | f14 |
| g | 4 | 9 | g9 | g18 |
22.3.3 Shapes of orbitals
Wavefunctions of atomic states describe probability amplitudes of finding the electron that occupies that state. Their absolute square is associated with orbitals, the regions of space in which electrons reside. Their energy and shape determine how they interact with one another and, as we shall see, these will also determine how the orbitals interact regarding chemical bonding. The shapes of s, p, and d orbitals are shown in Fig. 22.4. It is the number of nodal planes, equal to the value of l, that distinguishes these characteristic shapes. Any given orbital of the same l will have the same symmetry about the nucleus regardless of the value of n. However, as n increases both the size of the orbital and the number of nodes, equal to n – l – 1, increases. The remaining nodes are radial nodes.
Figure 22.4 (a) An s orbital has l = 0 and ml = 0. The origin lies at the center of the sphere. It has no nodal planes and n – 1 radial nodes. The angular shape shown here of an s orbital is determined by the absolute square of the corresponding spherical harmonic, |Y00(θ, φ)|2. Orbitals have infinite extent but are predominantly localized to a region of space near the origin. What is plotted here is a figure that contains most of the probability distribution. See Fig. 22.5 for a dimensionally correct depiction of the radial probability distribution of the H atom. (b) The p orbitals have l = 1 and ml = 0, +1, and –1. They are dumbbell-shaped, with one nodal plane and n – 2 radial nodes. The angular shape of a p orbital shown here is determined by the absolute square of the corresponding spherical harmonic, |Y01(θ, φ)|2 for a pz orbitals but 
for px and 
for py. The px and py orbitals have the same shape as pz but they are directed along the x and y axes, respectively. The phase of the wavefunction (arbitrarily shown as + and – in the figure) changes across a node. (c) The d orbitals have l = 2 and ml = 0, ±1, and ±2. They have two nodal planes and n – 3 radial nodes. (a) dz2. (b) dxz. (c) dyz. (d) dxy. (e) dx2 − y2 is similar to dxy but rotated by 90°. The phase of the wavefunction (arbitrarily shown as + and – in the figure) changes across a node.
Figure 22.5 Radial distribution function [Pnl(r)] for the lowest energy states of the H atom. (a) Comparing the first four of the s orbitals, we see how the maximum in the probability distribution shifts to longer radial distances with increases in the principal quantum number n. The number of radial nodes (points at which the probability distribution goes to zero) also increases as n – 1 because there are no angular nodes in the s function. (b) The L shell with n = 2 is composed of both an s and a p functions. The total number of nodes is n – 1 thus, the p function has an angular node rather than a radial node. In panels (c) and (d) these patterns continue. It can also be seen that the maximum occurs at smaller r as the azimuthal quantum number l increases, but that this is compensated by smaller maxima at even shorter r.
Most of the charge density from the electron distribution is centered about the outermost antinode, and therefore, within a spherical shell of moderate thickness. As a function of n, the lower the value of n, the closer the electron density is centered about the nucleus. In other words, orbital size for a given orbital class increases with increasing n. For a given value of n, the maximum in charge density is further from the nucleus the smaller the value of l. However, the probability of an electron lying close to the nucleus is greatest for small l. Indeed, only for l = 0 (s electron) is there any probability of finding the electron at r = 0. This means that only s electrons exhibit appreciable interactions with the nucleus, which also means that they are the electrons most susceptible to isotopic shifts. In hydrogenic atoms, these two effects counterbalance each other and the orbital energy depends only on n not l.
This is not the case for multi-electron atoms. For multi-electron atoms, the radial distribution function is similar to but not identical to those shown for hydrogenic atoms in Fig. 22.4. It is still true that only an s electron has a nonvanishing probability at the nucleus. However, the trend in the peak of electron density with l is reversed. Higher l orbitals (for the same value of n) have a peak in the electron density further from the nucleus than lower l orbitals. The result is that the degeneracy of the orbitals is lifted. The energy depends on both n and l and it increases with increases in both n and l.
22.3.4 Directed practice
Why do not most of the d orbitals in Fig. 22.4 look like the l = 2 solutions of the spherical harmonics in Table 21.3?
22.4 Many-electron atoms
22.4.1 Set up the hamiltonian
A multi-electron atom (or ion) is a system with N electrons about a nucleus with charge +Z, where Z is the atomic number. To obtain the energies of all allowed electronic states, we begin by figuring out the appropriate Hamiltonian within a coordinate system such as that shown in Fig. 22.6. The electrons each have kinetic and potential energies given by terms analogous to the one-electron terms derived previously in Eq. (22.2). To this we must add terms to account for the electron–electron repulsions (like charged particles repel) and for the spin–orbit interaction (the interaction of spin and orbital angular momenta of each individual electron introduced by the Dirac equation).
Stay updated, free articles. Join our Telegram channel
Full access? Get Clinical Tree
