Quantum Mechanical Model

It is observed that when an electric spark is released into a container of gas, the result is a glow. If this light is run through a prism, it's seen that the light does not span the optical spectrum. Instead, it consists of only a very few discrete wavelengths of light. Further, gases of different elements will have a different characteristic set of frequencies, which allows astronomers to calculate the composition of bodies in distant galaxies.

If we reason along classical lines, the discrete nature of these spectra is something of a mystery. Coming off the wire, the electrons have a distribution of energies that is essentially random. Therefore, if the electrons are losing energy to the gas, which then releases the energy as light, the emission should be random as well. These spectra, discovered before the idea of atoms (and therefore subatomic particles), were perhaps the earliest sign that something was wrong with classical mechanics.

The spectra suggest the electrons are only able to accept energy at specific values that correspond to gaps between different states the electron can occupy. Peering backward in time, aided by our knowledge of atoms, we can imagine the negatively charged electron as orbiting the positively charged nucleus. If the electron were allowed to orbit only at a discrete set of radii, or equivalently, at a discrete set of angular momenta, we'd have the discrete energy states we're in search of.

If we take seriously the intuition that we gained in analyzing the experimental spectra, we can formulate a model of the hydrogen atom first popularized by Niels Bohr, before the maturation of quantum theory. The hydrogen atom is especially simple, with a single electron (of charge \(-e\)) orbiting a single proton (of charge \(+e\)).

For simplicity, and to ease the comparison with quantum mechanics, we'll restrict bound electrons to a given set of angular momenta, rather than radii. In particular, we'll insist that the angular momentum of the electron can only come in integer multiples of Planck's constant \(\hbar\), thus \(L \in \{\hbar, 2\hbar, \ldots\}\). As a result, the orbital velocity of the electrons is \(v = n\hbar/m_er\).

We'll also assume that the electron is a particle that orbits the nucleus (just a proton) in a circle, just like a satellite around Earth.

Thus we have

\[k \frac{e^2}{r^2} = \frac{m_ev^2}{r}.\]

Solving for the angular momentum, \(m_evr\), we have

\[L = \sqrt{k m_e e^2 r}.\]

As \(L\) is some integer multiple of \(\hbar\), the allowable radii of the electron are given by \(r = \dfrac{n^2\hbar^2}{k m_e e^2}\).

Now, we're ultimately interested in the relative energy of these orbital states, which is just the sum of the kinetic and potential energies of the electron:

\[\begin{align} E^\textrm{tot} &= K + V \\ &= \frac12 m_ev^2 - k\frac{e^2}{r}. \end{align}\]

Substituting our result for \(r\) and our result for \(v\), we find

\[\begin{align} E^\textrm{tot}_n &= \frac12 \frac{m_en^2\hbar^2}{m_e^2r^2} - k\frac{e^2}{r} \\ &= -\frac12\frac{k^2m_ee^4}{n^2\hbar^2}, \end{align}\]

which we can re-write very simply as \(\displaystyle E_n = -\frac{E_1}{n^2},\) where \(E_1 = \dfrac{k^2m_ee^4}{2\hbar^2}\).

If our electron transitions between some excited state \(E_n\) and the ground state \(E_1\), the energy of the emitted light will simply be

\[\begin{align} E_\textrm{photon} &= E_n - E_1 \\ &= E_1\left(1-\frac{1}{n^2}\right), \end{align}\]

and we can find its wavelength using the Planck-Einstein equation \(E = hc/\lambda:\)

\[\lambda_\textrm{photon}^{-1} = \frac{E_1}{hc}\left(1-\frac{1}{n^2}\right).\]

Miraculously, this extremely simple model matches the emission spectrum of hydrogen exactly for all transitions to the ground state \(n \rightarrow 1\).

When the electron moves from a higher state to a lower state, we call this transition a "relaxation" of the hydrogen atom. Similarly, we call the move from a lower state to a higher state an "excitation". Our model predicts that hydrogen atoms will only "glow" when one of the electrons in the spark chamber has an energy exactly equal to the gap between the ground state and some excited state \(E_n\).

Unfortunately, this intuitive modeling approach breaks down when we try to apply it to more complicated atoms. This is because the orbits of electrons are not circles, or even spheres (except for the ground state), but more complicated shapes that defy the simple picture we present here. Incredibly though, the Bohr model still nails the emission spectrum for photon radiation from hydrogen.

In general, we need to resort to full blown quantum mechanics, solving the Schrödinger wave equation to find the orbital states of electrons about nuclei:

\[\begin{align} E \lvert \psi \rangle &= \mathbf{H} \lvert \psi\rangle \\ &= \left(\frac{p^2}{2m} - k\frac{e^2}{r}\right)\lvert \psi \rangle \\ &= - \left( \frac{\hbar^2\nabla^2}{2m} + k\frac{e^2}{r}\right)\lvert\psi\rangle. \end{align}\]

Mathematically, this pursuit becomes impractical after the hydrogen atom, and instead, the wave equation is solved numerically with computers.

Main Article: orbitals and quantum numbers

When Heisenberg put forward his uncertainty principle, which said that, at any given instant, it is impossible to calculate both the momentum and the location of an electron in an atom; it is only possible to calculate the probability of finding an electron within a given space.

And thus the quantum mechanical model redefined the way electrons travel according to this approach, and we cannot simply say that the electron exists at a particular point in space. Instead of defining a particular path, it proposed some region in space around the nucleus, called an orbital, where the probability of finding an electron is maximum. Thus the electron doesn't always remain at a definite distance from the nucleus.

Energy Level and Sub-Energy Levels

The energy levels classify the orbitals based on their proximity towards the nucleus; these are represented as

\[\text{K }(n=1), \text{L }(n=2), \text{M }(n=3), \text{N }(n=4), \text{O }(n=5), \ldots.\]

The lowest energy level is \(\text{K}\) or \(1,\) the next being \(\text{L}\) or \(2,\) and so on. Thus for an electron the energy level describes the path of the electron and the energy of the electron given by the equation

\[\begin{align} E_n = - \hbar cR_\infty\dfrac{Z^2}{n^2} &= -\dfrac{2\pi^2mZ^2e^4}{n^2h^2}\text{kJ/mol}\\ &\approx -\dfrac{2.18\times 10^{-19}Z^2}{n^2}\text{J/atom}\\ &\approx -\dfrac{13.6Z^2}{n^2}\text{eV/atom}, \end{align}\]

where \(\hbar\) is the Plank's constant, \(c\) is the speed of light, \(R_\infty\) is the Rydberg's constant, \(Z\) is the atomic number of the element, and the electron is present in the \(n^\text{th}\) energy level.

The energy levels are sub-divided into sub-shells, which are designated as \(s\), \(p\), \(d\) and \(f\), and the number of sub-shells in each energy level is given by the number itself. For example, the \(\text{K}\)-energy level has only \(1\) sub-shell \(s\), and the \(\text{L}\)-energy level has \(2\) sub-shells \(s\) and \(p\).

The sub-shells are precisely defined with the help of quantum numbers, which govern the number of orbitals in each sub-shell. The magnetic quantum number \((m_l)\) visualizes the behavior of an electron under the influence of a magnetic field (like Earth). We know that the movement of electric charge can generate a magnetic field, and under the influence of an external magnetic field the electrons tend to orient themselves in certain regions around the nucleus (called orbitals), which is why this quantum number gives the number of orbitals in a particular sub-shell.

The values of the magnetic quantum number depend upon the azimuthal quantum number \((l)\). For example, if the azimuthal quantum number of an atom is \(l\), then the magnetic quantum numbers range as follows:

\[m_l= -l,( -l+1), . . . , 0 , . . ., (l-1), l.\]

So, there are \(2l+1\) values of \(m_l\) for a given value of \(l\), i.e. there will be \(2l+1\) orbitals. According to the magnetic quantum number, the number orbitals in

• \(s\) sub-shell is \(1\) because the value of \(l=0 \implies m_l = 0;\)
• \(p\) sub-shell is \(3\) because the value of \(l=1 \implies m_l = -1, 0, 1;\)
• \(d\) sub-shell is \(5\) because the value of \(l=2 \implies m_l = -2, -1, 0, 1, 2;\)
• \(f\) sub-shell is \(7\) because the value of \(l=3 \implies m_l = -3, -2, -1, 0, 1, 2, 3.\)

Note: We see that each sub-shell can have many orbitals, but all orbitals are assumed to have equal energy as their energies differ by very small, negligible values.

Shape of Atomic Orbitals

The shape of these orbitals are defined by the wave equation, and they get pretty weird as we reach higher orbitals.

The shape of first five atomic orbitals \(1s, 2s, 2p_x, 2p_y, 2p_z\)[1]

The \(s\) orbitals are spherical in shape and are designated as \(1s, 2s, 3s, 4s, ...,\) and so on with the increase in the principal quantum number.

The \(1s\) orbital[2]

The \(p\) orbitals are somewhat in the shape of a dumbbell with two lobes projecting outwards in the opposite planes; the lobes of the three \(p\) orbitals are mutually perpendicular and have the same energy.

The \(p\)-orbitals in each of the three plane; \(p_x, p_y, p_z\)[2]

The shapes of \(d\) and \(f\) orbitals are pretty complicated, and they are oriented in a very complex fashion. The \(d\) orbitals look like this:

Orientation of \(d\)-orbitals in the \(xyz\)-plane: \(d_{xy}, d_{yz}, d_{zx}, d_{x^2-y^2}, d_{z^2}\)[2]

Main Article: electron configurations.

The concept of atomic orbitals gives us the probability distribution graphs which tell us the most probable position of an electron around the atomic nucleus. Apart from the probability of finding an electron, we also need to understand how the electrons are arranged. This is given by the electronic configuration, and thus electrons follow some rules while taking their places in an atom.

So far we have understood the area where an electron may be present around an atom and we have come to the conclusion that they are found in some kind of clouds called orbitals. Now, we shall see how the electrons are arranged within these orbitals. The spatial arrangement of the electrons in an orbital is governed by many rules. The most important of those are as follows:

1. Aufbau principle
2. Pauli's exclusion principle
3. Hund's rule.

Aufbau Principle: This rule says that the electrons occupy the lowest possible energy level before proceeding to the higher level. The order of increasing energy level is given by

\[ n+l=(\text{Principal quantum number})+(\text{Azimuthal quantum number}). \]

The rule is based on the total number of nodes in the atomic orbital, \(n + ℓ\), which is related to the energy. In the case of equal \(n + ℓ\) values, the orbital with a lower \(n\) value is filled first. The fact that most of the ground state configurations of neutral atoms fill orbitals following this \(n + ℓ\), \(n\) pattern was obtained experimentally by reference to the spectroscopic characteristics of the elements.[4]

Pictorial representation of the Aufbau Principle[3]

Pauli's Exclusion Principle: This rule states that two particles with half-integer spins cannot occupy the same quantum state at the same time. In other words, two electrons moving in an orbital cannot have the same spin quantum number or simply spin.

Half-integer spin: A spin is an intrinsic property of all elementary particles. Fermions, the particles that constitute ordinary matter, have half-integer spin.

For example, in a helium atom, we have two electrons, and they are not present in the same spin, i.e. a helium atom:

\[\begin{align} \text{would be like this: }\ &\ce{_2He}: \boxed{\uparrow\downarrow}\\ \text{and not like this: }\ &\ce{_2He}: \boxed{\uparrow\uparrow}. \end{align}\]

Note: The spin quantum number has two values \(\frac12\) and \(-\frac12,\) which means that the spin of an electron can be clockwise, represented as \(\uparrow,\) or anti-clockwise, represented as \(\downarrow.\)

Hund's Rule: This lays the cornerstone for the internal arrangement of electrons in an atom, and it deals with the order of filling the electrons into the orbitals of the same sub-shell. When more than one orbital of equal energy are available \((\)e.g. \(p_x, p_y, p_z),\) the electrons first occupy the orbitals with parallel spins or same spin. Only then will the pairing of electrons take place.

This happens because if we place two electrons in the same orbital with opposite spins, the repulsive nature increases; however, if the electrons are placed in separate orbitals with parallel spins, the repulsion is greatly minimized. Also, the electrons occupy the orbitals with the same spin, thus reducing the repulsion.

Let's have a look at the examples of nitrogen and oxygen:

If we follow Hund's rule for the nitrogen atom, its electronic configuration would be like this:

\[\ce{_7N}:\underbrace{\boxed{\uparrow\downarrow}}_{\text{1s}}\quad\underbrace{\boxed{\uparrow\downarrow}}_{ \text{2s}}\quad\underbrace{\boxed{\uparrow\color{white}{\downarrow}}\boxed{\uparrow\color{white}{\downarrow}}\boxed{\uparrow\color{white}{\downarrow}}}_{\text{2p}} \ \ \ \text{total spin of unpaired electrons =}\dfrac 12+\dfrac 12+\dfrac12=1\dfrac12.\]

Note that the electrons are placed in orbitals with parallel spins and are not paired up before at least one electron occupies each orbital.

But we can't place the electrons with opposite spin (as in the case below) or start pairing up electrons before filling up each orbital with at least one electron (as in the next case):

\[\begin{align} \ce{_7N}:\underbrace{\boxed{\uparrow\downarrow}}_{\text{1s}}\quad\underbrace{\boxed{\uparrow\downarrow}}_{\text{2s}}\quad\underbrace{\boxed{\uparrow\color{white}{\downarrow}}\boxed{\color{white}{\uparrow}\downarrow}\boxed{\uparrow\color{white}{\downarrow}}}_{\text{2p}}& \ \ \ \text{total spin of unpaired electrons =}\dfrac 12-\dfrac 12+\dfrac12=\dfrac12\\\\ &\text{or}\\\\ \ce{_7N}:\underbrace{\boxed{\uparrow\downarrow}}_{\text{1s}}\quad\underbrace{\boxed{\uparrow\downarrow}}_{\text{2s}}\quad\underbrace{\boxed{\uparrow\downarrow}\boxed{\uparrow\color{white}{\downarrow}}\boxed{\color{white}{\uparrow}\color{white}{\downarrow}}}_{\text{2p}}& \ \ \ \text{total spin of unpaired electrons =}\dfrac12. \end{align}\]

Observe that the total spin of the unpaired electrons is maximum in the correct configuration, which is why this rule is also called Hund's rule of maximum multiplicity. The multiple being the spin of the electron, i.e. \(\frac12\).

The electron in the oxygen atom are arranged in the following way which obeys Hund's rule. The electrons in the \(p\) orbital are first filled up with atleast one electron (in parallel spin) and only then the last electron is paired up:

\[\ce{_8O}:\underbrace{\boxed{\uparrow\downarrow}}_{\text{1s}}\quad\underbrace{\boxed{\uparrow\downarrow}}_{\text{2s}} \quad\underbrace{\boxed{\uparrow\downarrow}\boxed{\uparrow\color{white}{\downarrow}}\boxed{\uparrow\color{white}{\downarrow}}}_{\text{2p}}\ \ \ \text{total spin of unpaired electrons =}\dfrac12+\dfrac 12 = 1.\]

As opposed to this one where the pairing up has taken place in the wrong order, disobeying Hund's rule gives

\[\ce{_8O}:\underbrace{\boxed{\uparrow\downarrow}}_{\text{1s}}\quad\underbrace{\boxed{\uparrow\downarrow}}_{\text{2s}} \quad\underbrace{\boxed{\uparrow\downarrow}\boxed{\uparrow\downarrow}\boxed{\color{white}{\downarrow}\color{white}{\downarrow}}}_{\text{2p}}\ \ \ \text{total spin of unpaired electrons =}0.\]

Again, observe that the total spin of the unpaired electrons is maximum in the correct configuration.

[1] Image from https://en.wikipedia.org/wiki/Atomicorbital#/media/File:Neonorbitals.JPG released into the public domain.

[2] Image from http://www.sparknotes.com/chemistry/fundamentals/atomicstructure/section1.rhtml.

[3] Image from https://commons.m.wikimedia.org/wiki/File:Klechkovski_rule.svg#mw-jump-to-license under the GNU free documentation license.

[4] Aufbau Principle. Wikipedia.org. Retrieved 14:33, April 7, 2016, from https://en.m.wikipedia.org/wiki/Aufbau_principle.