Molecular Orbital Theory
Atomic and Molecular Orbital Theory
By Michelle
Image source:
Note: not what an atom actually looks like, as this article will later explore!
Introduction
At school, we routinely learn that neon has the electronic configuration ‘1s22s22p6’ while sodium is ‘1s22s22p63s1’—but what does this jumble of letters really mean? We learn that hydrogen is diatomic but helium isn’t, and that covalent bonding happens because atoms ‘want a full outer shell’—immediately I questioned: why on Earth would atoms ‘want’ a full outer shell if they aren’t human beings?
The truth is, a lot of the chemistry curriculum at school uses a simplified model of what actually happens, because the most accurate explanation we have found so far is extremely complex. But after attending a random chemistry club at lunch and finding that the intricate explanations behind atoms and covalent bonding, I quite literally fell in love. Amidst finding out that oxygen is actually magnetic and discovering the reason why atoms actually ‘want’ a full outer shell, studying atomic and molecular orbital theory gave me a greater insight into the link between quantum physics and atomic theory (and this is why chemistry is just applied physics).
The Wave Model of the Electron
Electrons exhibit wave-particle duality: in J.J. Thomson’s cathode ray tube experiment (figure 1), it was found that electrons possess both mass and charge because they can be deflected by electric and magnetic fields, behaving like particles; later on, in diffraction experiments, it was also found that electrons can behave like a wave and electron intensity would spread out through a diffraction grating, creating an interference pattern (figure 2). Therefore it is impossible to know accurately both the exact position and velocity of the electron at the same time because they behave like waves at the subatomic level. [1]
Figure 1: cathode ray tube. Image source: [2]
Figure 2: interference pattern from the diffraction of electrons. Image source: [3]
Thus in the quantum model of the atom, the exact position of an electron is not defined. Electrons are not orbiting around the nucleus like planets around a sun. However, you can describe an electron and its wave behaviour using something called the wave function, denoted by Ψ. This can be obtained by solving the Schrodinger equation (a differential equation). [4] The amplitude or ‘size’ of the wave squared, Ψ2 is the probability of finding an electron at a given point in time once normalised. Normalisation means adjusting the wave function so that the sum of Ψ2 over all positions in space (r) and time (t) equals to 1, because the total probability of finding an electron at any point in space and time is 1. [5]
An electron can be described using the ‘particle in a box’ model, which models a particle that is free to move within a box. The wave function must equal to 0 at the boundaries of the box as the electron cannot be found outside of the box (the probability = 0). So the electron behaves like a standing wave, with the ends of the function having 0 amplitude—they are the nodes of a standing wave. [5] This is shown by Figure 3 below.
Figure 3: The particle in a box model—the particle can be described by a standing wave. [5]
Atomic Orbitals
Separating the wave function gives information about the wave properties of the electron in terms of 4 discrete quantum numbers: n (principal quantum number), l (orbital quantum number), ml (magnetic quantum number), and ms (spin) [5]. In order for solutions to exist,
n can take values from n = 1, 2, 3, etc.
l can take values from 0, 1, 2, ... , n-1 for a given n
ml can take values from -l, -l+1, …. 0, l+1, …, l for a given l
ms can take values +½ or -½ for an electron (spin up and spin down)
[5]
The state of an electron can be described by these 4 quantum numbers, and the 4 quantum numbers combined gives information about the energy of an electron. Since the quantum numbers can only take certain values, the electron’s energy is quantised and can only take discrete values. The lowest energy level is when n = 1, and energy increases with n. Different values of l and ml will give slightly different energies. A set of particular values of n, l and ml defines an orbital and corresponds to a particular energy state [6]. This can be displayed on an energy level diagram in figure 4. An orbital also describes a particular spatial distribution: there is a 95% chance of finding the electron inside the orbital.
if an electron has n = 1, then the electron’s l can only be 0. ml can only be 0, and ms can be +½ or -½. The values n=1, l=0 and ml=0 define what is known as the 1s orbital
for an electron with n = 2, l can be 0 or 1. When l = 0, ml can only be 0 as above, but when l = 1, ml can be -1, 0 or 1. There is 1 orbital for when n = 2 and l = 0 (only 1 possible value of ml): the 2s orbital. There are 3 orbitals for when n=2 and l=1 (3 possible values of ml): the 2px, 2py and 2pz orbitals.
for an electron with n = 3, l can be 0, 1, or 2. When l = 0 or 1, the situation is the same as above, giving rise to the 3s, 3px, 3py and 3pz orbitals. When l = 2, ml can be -2, -1, 0, 1 or 2, giving rise to 5 3d orbitals.
This is why at school we learn that in the ‘1st quantum shell there is a 1s subshell; in the 2nd quantum shell there is a 2s subshell and 3 2p subshells, etc.’, i.e. the principal quantum number n gives the ‘quantum shell’, n and l give the ‘subshell’, and n, l and ml together give the orbital
Figure 4: orbitals and their energies. Image source: [11]
The Pauli-Exclusion principle for fermions (matter particles in the standard model of fundamental particles) states that no 2 electrons in an atom can have the same state—2 electrons in one atom cannot have the same 4 quantum numbers. Therefore, there can only be a maximum of 2 electrons with the same n, l, ml, because there can only be 2 values of ms (spin up and spin down). (Otherwise, there would be electrons with the same ms as well as the same n,l and ml, which is not allowed.) This means that one orbital can only hold a maximum of 2 electrons with opposite spins. In general, electrons fill up the lowest energy orbitals first before filling up higher energy orbitals:
Figure 5: the filling up of orbitals by electrons. the direction of the arrows represent the spin states of the electrons. The 1s orbital has to be filled before the electrons start occupying the 2s orbital. [12]
In a hydrogen atom, there is 1 electron, and the wave equation gives a probability distribution that is spherical around the nucleus (hence the idea that the 1s orbital has a spherical shape). [6] The shapes of the probability distribution for some other orbitals are shown in figure 6. The p subshell
Figure 6: orbital ‘shapes’. [6]
Covalent Bonding
Valence Bond Theory and the Lewis structure of molecules explains covalent bonding as the sharing of pair(s) of electrons between 2 atoms in order to achieve a ‘full outer shell’. [7] This theory is a good approximation and simplification in many situations, but it does not explain everything: did you know that liquid oxygen can be attracted by magnetic fields? In quantum theory, covalent bonds are explained through molecular orbital theory.
Molecular Orbitals
Molecular orbitals form when atoms interact. Their atomic orbitals ‘overlap’; this is when the standing waves corresponding to each orbital interfere with each other [7] (interference is when 2 waves meet at a point and the resultant wave is given by the vector sum of the individual waves). If the waves are in phase (‘in step with each other’), constructive interference occurs and the wave has maximum amplitude; if the waves are out of phase, destructive interference occurs and the wave has 0 amplitude, i.e. they cancel out. This is shown in figure 7 below.
Figure 7: left: constructive interference; right: destructive interference [7]
If there is constructive interference, there is increased probability of finding an electron between the two nuclei, but if there is destructive interference, then there is minimum probability of finding an electron between the two nuclei. Hence, the constructive interference case forms a bonding orbital and the destructive interference case forms an anti-bonding orbital.
For the bonding orbital, the electrons have a high probability of being found between the nuclei, resulting in the force of attraction between the electrons and the 2 nuclei—this configuration is at a lower energy (because the electric potential energy is more negative due to stronger forces of attraction) and hence is more stable. The antibonding orbital is less stable and is at a higher energy. The Pauli-exclusion principle still applies and each molecular orbital can only hold up to 2 electrons. When 2 atoms interact, the electrons from the atomic orbitals will fill up the lower energy bonding orbitals first before filling up the anti-bonding orbitals.
Figure 8: the energy levels of bonding and anti-bonding orbitals. [8]
The bonding orbital formed when 2 s-orbitals overlap is called a σ orbital whilst the anti-bonding orbital formed is a σ* orbital. There is only 1 s orbital per atom for a particular principal quantum number, so only 1 pair of σ/σ* orbitals form. Similarly when 2 p-orbitals overlap, a π bonding orbital forms and a π* anti-bonding orbital forms. (Hence, ‘σ bond’ and ‘π bond’ in Lewis theory.) Since there are 3 p-orbitals per atom (px, py and pz), 3 pairs of π/π* orbitals will be formed. The overlap of pz orbitals is at a lower energy because this corresponds to a head-on overlap; the force of attraction is stronger as the electrons are closer to the nuclei. The overlap of px and py orbitals is at a higher energy because this corresponds to a side-on overlap.
Figure 9: energy levels of the molecular orbitals formed by 1s, 2s and 2p orbitals [9]
The number of molecular orbitals produced will always be equal to the total number of atomic orbitals brought by the original atoms. [9]
Why some molecules are possible, whilst others are not?
If only the bonding orbital is filled up and the anti-bonding orbital is empty, that means a covalent bond is formed between the atoms. Filling up the anti-bonding orbital ‘cancels out’ the effect of filling the bonding orbital.
Consider the case of 2 hydrogen atoms: each H atom has 1 electron in the 1s subshell. When the atomic orbitals overlap, both of the electrons will occupy the σ bonding orbital, forming a ‘sigma bond’ between the 2 H atoms, hence hydrogen is diatomic.
[9]
When we consider 2 helium atoms however, each He atom has 2 electrons in the 1s subshell. When the 4 electrons occupy the molecular orbitals, 2 of them fill up the bonding orbital but the other 2 have to go into the anti-bonding orbital, hence cancelling out the effect of the bonding orbital. This means that the 2 He atoms cannot form a covalent bond so He2 does not exist.
For oxygen, the following figure illustrates how the electrons fill up the molecular orbitals. As you can see, there are unpaired electrons in 2 of the molecular orbitals—these unpaired electrons are what give it paramagnetism: oxygen molecules are weakly attracted by an external magnetic field.
Figure 10: oxygen molecular orbitals [10]
Bond Order
More accurately, rather than saying that the antibonding orbital ‘cancels out’ the bonding orbital, we can calculate the bond order for 2 atoms. The bond order also corresponds to the ‘number of bonds’ given by valence bond theory.
Bond order = ½ (number of bonding electrons - number of antibonding electrons)
For the 2 helium atoms, the bond order is equal to 0, thus there is no bond between them. For the 2 hydrogen atoms, the bond order = 1, so there is a single bond between them. For the oxygen atoms, bond order = 2, hence a ‘double bond’ forms.
Bibliography
[1] Clugston, Michael, and Rosalind Flemming. Advanced Chemisty. Oxford University Press, 2013.
[2] "Cathode-Ray Tube - Wikipedia". En.Wikipedia.Org, 2021, https://en.wikipedia.org/wiki/Cathode-ray_tube.
[3] Courses.Washington.Edu, 2021, http://courses.washington.edu/phys431/electron_diffraction/Electron_Diffraction.pdf.
[4] "Wave Function | Definition & Facts". Encyclopedia Britannica, 2021, https://www.britannica.com/science/wave-function.
[5] Hyperphysics.Phy-Astr.Gsu.Edu, 2021, http://hyperphysics.phy-astr.gsu.edu/.
[6] "Introduction To Molecular Orbital Theory". Ch.Ic.Ac.Uk, 2021, https://www.ch.ic.ac.uk/vchemlib/course/mo_theory/main.html.
[7] "8.4 Molecular Orbital Theory". Opentextbc.Ca, 2021, https://opentextbc.ca/chemistry/chapter/8-4-molecular-orbital-theory/.
[8] "Bonding And Antibonding Pi Orbitals – Master Organic Chemistry". Master Organic Chemistry, 2021, https://www.masterorganicchemistry.com/2017/02/14/molecular-orbital-pi-bond/.
[9] "Mo Theory". Chem.Fsu.Edu, 2021, https://www.chem.fsu.edu/chemlab/chm1046course/motheory.html.
[10] "9.10: Molecular Orbital Theory Predicts That Molecular Oxygen Is Paramagnetic". Chemistry Libretexts, 2021, https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/09%3A_Chemical_Bonding_in_Diatomic_Molecules/9.10%3A_Molecular_Orbital_Theory_Predicts_that_Molecular_Oxygen_is_Paramagnetic.
[11] "Organizing Atoms And Electrons: The Periodic Table - Annenberg Learner". Annenberg Learner, 2021, https://www.learner.org/series/chemistry-challenges-and-solutions/organizing-atoms-and-electrons-the-periodic-table/.
[12] "Hund’S Rule And Orbital Filling Diagrams | Chemistry For Non-Majors". Courses.Lumenlearning.Com, 2021, https://courses.lumenlearning.com/cheminter/chapter/hunds-rule-and-orbital-filling-diagrams/.
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