Alright, let's dissect this. You want me to take something… dry… and make it… less dry. And longer. And with all the original bits intact. A challenge. Not an unpleasant one, perhaps. Like finding a perfectly preserved insect in amber. Fascinating, in its own morbid way.
You're asking about the way we describe how electrons behave in molecules. The fancy term is "method for describing the electronic structure of molecules using quantum mechanics." Fancy. But ultimately, it’s about understanding where electrons are and what they’re doing. It’s a fundamental concept, like understanding why some things stick together and others… don’t.
Molecular Orbital Theory
This whole endeavor, this "molecular orbital theory" – MO theory, for those who prefer brevity, though I find brevity often conceals more than it reveals – is a way to look at molecules through the lens of quantum mechanics. It’s not a new idea, emerging from the early days of the 20th century. It’s particularly useful because it can explain things that other theories… struggle with. Take oxygen, for instance. Its paramagnetic nature, the fact that it’s attracted to magnetic fields, is something that the older valence bond theory couldn't quite grasp. MO theory, however, offers a more… comprehensive view.
Instead of thinking of electrons as being tethered to specific chemical bonds between individual atoms, MO theory treats them as if they’re free agents, moving throughout the entire molecule, influenced by all the atomic nuclei. It’s a more… communal approach to electron distribution. Quantum mechanics tells us that these electrons occupy what we call molecular orbitals. These aren't confined to a single atom; they can span two or more atoms, holding the valence electrons that bind the molecule together.
The true revolution MO theory brought was the idea of approximating these molecular orbitals as linear combinations of atomic orbitals, or LCAO. It’s like taking the individual characteristics of atomic orbitals and blending them to create something new, something that describes the molecule as a whole. These approximations are typically made using either density functional theory (DFT) or the Hartree–Fock method, applied to the fundamental Schrödinger equation.
MO theory and valence bond theory are the bedrock of quantum chemistry. They’re not entirely separate entities, mind you. When you really dig into them, when you extend their capabilities, they actually converge. It’s like two different paths leading to the same, rather bleak, summit.
Linear Combination of Atomic Orbitals (LCAO) Method
The LCAO method is where the magic, or perhaps the meticulous calculation, happens. In this framework, a molecule is understood to possess a collection of molecular orbitals. The idea is that the wave function for a molecular orbital, denoted as ψ j , can be represented as a weighted sum of the individual atomic orbitals, χ i . Mathematically, it looks like this:
The coefficients, c ij , are the weights. You can figure them out numerically by plugging this equation into the Schrödinger equation and applying the variational principle. This principle, a rather elegant mathematical tool in quantum mechanics, helps us build up these coefficients. A larger coefficient for a particular atomic orbital means that the resulting molecular orbital is more heavily influenced by that atomic orbital. It's a way of quantifying how much of each original atom's orbital contributes to the final molecular orbital. This approach is a staple in computational chemistry. Sometimes, a little tweak, a unitary transformation, can speed things up in certain computational processes.
Back in the 1930s, MO theory was seen as a rival to valence bond theory. But as understanding deepened, it became clear they were more like cousins, closely related and, when fully developed, essentially equivalent.
MO theory also finds its use in interpreting ultraviolet–visible spectroscopy (UV–VIS). When molecules absorb light at specific wavelengths, it signifies a change in their electronic structure. These signals can be assigned to electrons jumping from a lower-energy orbital to a higher-energy one. The molecular orbital diagram of the resulting excited state then paints a picture of the molecule's electronic state.
Now, for an atomic orbital combination to be considered a good approximation of a molecular orbital, there are a few… prerequisites.
- Symmetry: The combination must possess the correct symmetry. It needs to align with the appropriate irreducible representation of the molecular symmetry group. This is where symmetry adapted linear combinations, or SALCs, come into play, ensuring the molecular orbitals have the right symmetry.
- Overlap: The atomic orbitals must actually overlap in space. If they’re too far apart, they simply can't combine to form molecular orbitals. Proximity is key.
- Energy: Atomic orbitals must be at similar energy levels to combine effectively. If the energy difference is too vast, the resulting molecular orbital won't see a significant enough energy reduction for the electrons. It’s a matter of energetic compatibility.
History
Molecular orbital theory didn't spring into existence overnight. It evolved in the wake of valence bond theory, which was established around 1927. The heavy lifting was done by minds like Friedrich Hund, Robert Mulliken, John C. Slater, and John Lennard-Jones. Initially, it was called the Hund-Mulliken theory. According to Erich Hückel, a physicist and physical chemist, the first instance of MO theory being used quantitatively was in a 1929 paper by Lennard-Jones. This paper, quite remarkably, predicted a triplet ground state for the dioxygen molecule, which elegantly explained its paramagnetism. This was a feat that valence bond theory only managed to achieve a couple of years later, in 1931. The term "orbital" itself was introduced by Mulliken in 1932. By 1933, MO theory was gaining traction, recognized as a valid and useful theoretical framework.
Erich Hückel took this theory and applied it to unsaturated hydrocarbon molecules, starting in 1931. His Hückel molecular orbital (HMO) method was designed to determine MO energies for pi electrons, and he applied it to conjugated and aromatic hydrocarbons. This method provided a solid explanation for the remarkable stability of molecules with six pi-electrons, like benzene.
The first truly accurate calculation of a molecular orbital wavefunction was performed by Charles Coulson in 1938, focusing on the humble hydrogen molecule. By 1950, molecular orbitals were fully defined as eigenfunctions of the self-consistent field Hamiltonian. This marked the point where MO theory became fully rigorous and consistent. This rigorous approach is known as the Hartree–Fock method for molecules, though its roots lie in atomic calculations. For molecules, the molecular orbitals are expressed as a combination of atomic orbital basis sets, leading to the Roothaan equations. This paved the way for numerous ab initio quantum chemistry methods. In parallel, MO theory was also employed in a more approximate fashion, incorporating empirically derived parameters, giving rise to what we now call semi-empirical quantum chemistry methods.
The success of Molecular Orbital Theory also spurred the development of ligand field theory. This theory, developed through the 1930s and 1940s, offered an alternative to crystal field theory.
Types of Orbitals
Molecular orbital (MO) theory employs the linear combination of atomic orbitals (LCAO) approach to describe the molecular orbitals that arise from bonding between atoms. These are generally categorized into three types: bonding, antibonding, and non-bonding. A bonding orbital serves to concentrate electron density in the region between a pair of atoms. This concentration of electron density effectively attracts both nuclei towards each other, thereby holding the atoms together. An antibonding orbital, on the other hand, concentrates electron density behind each nucleus – on the side furthest from the other atom. This arrangement tends to pull the nuclei apart, weakening the bond. Electrons residing in non-bonding orbitals are typically associated with atomic orbitals that don't interact significantly, either positively or negatively. Consequently, these electrons neither strengthen nor weaken the bond.
Molecular orbitals are further classified based on the types of atomic orbitals from which they are constructed. For a chemical substance to form a stable bond, its orbitals must lower in energy upon interaction. Different bonding orbitals are distinguished by their electron configuration (the shape of their electron cloud) and their respective energy levels.
The arrangement and energies of molecular orbitals within a molecule are visually represented in molecular orbital diagrams.
Common bonding orbitals include sigma (σ) orbitals, which are characterized by their symmetry around the bond axis, and pi (π) orbitals, which possess a nodal plane that lies along the bond axis. Less common are delta (δ) orbitals and phi (φ) orbitals, each having two and three nodal planes, respectively, along the bond axis. Antibonding orbitals are conventionally denoted by the addition of an asterisk, such as π*.
Bond Order
Bond order is a quantitative measure of the number of chemical bonds between a pair of atoms. It's calculated by subtracting the number of electrons occupying anti-bonding orbitals from the number of electrons in bonding orbitals, and then dividing the result by two. A molecule is generally predicted to be stable if its bond order is greater than zero. For the most part, focusing on the valence electrons is sufficient for determining bond order. This is because, for principal quantum number n > 1, when MOs are derived from 1s AOs, the difference in the number of electrons between bonding and anti-bonding molecular orbitals is zero. Therefore, non-valence electrons have no net effect on the bond order.
The bond order provides insight into whether a bond between two atoms is likely to form. Consider the helium molecule, He 2 . According to its molecular orbital diagram, the bond order is:
This zero bond order indicates that no stable bond will form between two helium atoms, which aligns with experimental observations. While the helium dimer can be detected under extreme conditions (very low temperature and pressure in a molecular beam) and possesses a binding energy of approximately 0.001 J/mol, it is classified as a van der Waals molecule rather than a molecule with a true covalent bond.
Furthermore, bond order can also be used to gauge the strength of a bond. For instance:
- For H 2 : Bond order is ; bond energy is 436 kJ/mol.
- For H 2 + : Bond order is ; bond energy is 171 kJ/mol.
The lower bond order of H 2 + compared to H 2 suggests it should be less stable, which is indeed observed experimentally through its lower bond energy.
Magnetism Explained by Molecular Orbital Theory
The Lewis structure for many covalent molecules can be drawn, allowing us to predict electron-pair geometry, molecular geometry, and even bond angles with reasonable accuracy. However, the oxygen molecule, O 2 , presents a peculiar challenge to the Lewis structure model.
In O 2 , each oxygen atom appears to form a double bond and retains two lone pairs, fulfilling the octet rule. This Lewis structure depiction adheres to all the standard rules. Yet, it starkly contrasts with the observed magnetic behavior of oxygen. While O 2 is not inherently magnetic, it exhibits a distinct attraction to magnetic fields. This attraction, known as paramagnetism, arises from the presence of unpaired electrons within a molecule. The Lewis structure, however, suggests all electrons in O 2 are paired. This discrepancy demands a more nuanced explanation.
The molecular orbital diagram for the oxygen molecule provides this explanation. Oxygen has an atomic number of 8, with an electronic configuration of 1s²2s²2p⁴. The electronic configuration of the oxygen molecule, when considering MOs, is:
σ1s² < *σ1s² < σ2s² < *σ2s² , [ π2p x ² = π2p y ²] < σ2p z ² < [*π2p x ¹ =*π2p y ¹] < *σ2p z
The bond order of O 2 is calculated as:
(Bonding electrons − Anti-bonding electrons) / 2 = (10 − 6) / 2 = 2
Crucially, the MO diagram reveals that O 2 possesses unpaired electrons in its [*π2p x ] and [*π2p y ] orbitals, hence its paramagnetic nature. Even if we consider only the valence electrons, the bond order remains (8 − 4) / 2 = 2.
Magnetic susceptibility quantifies the force a substance experiences within a magnetic field. By comparing a sample's weight with and without a magnetic field, we can infer its magnetic properties. Paramagnetic samples, attracted to the magnet, appear heavier due to the magnetic force. The increase in apparent weight can even be used to estimate the number of unpaired electrons. Experiments confirm that each O 2 molecule has two unpaired electrons, a fact the Lewis structure fails to predict.
In contrast, most molecules, like water, have all their electrons paired and exhibit diamagnetism, weakly repelling magnetic fields. This is why living organisms, largely composed of water, demonstrate diamagnetic behavior. It’s even possible to levitate a frog in a strong magnetic field due to this diamagnetic repulsion. It's important to note that neither paramagnetic nor diamagnetic materials act as permanent magnets; their behavior is induced by an applied magnetic field.
Molecular orbital theory offers a compelling explanation for the paramagnetism of oxygen. It also sheds light on the bonding in numerous other molecules, including those that violate the octet rule and exhibit complex bonding patterns that are challenging to describe with simpler Lewis structures. Furthermore, it provides a framework for understanding the energies of electrons within molecules and their probable locations. Unlike valence bond theory, which assigns hybrid orbitals to specific atoms, MO theory utilizes combinations of atomic orbitals to create molecular orbitals that are delocalized across the entire molecule. This delocalization is key to understanding why some substances conduct electricity, others are semiconductors, and still others are insulators.
MO theory describes electron distribution in molecules much like atomic orbitals describe electron distribution in atoms. Using quantum mechanics, the behavior of an electron in a molecule is still governed by a wave function, Ψ , just as in an atom. Electrons in molecules, like those in isolated atoms, are restricted to discrete, quantized energy levels. The region of space where a valence electron in a molecule is most likely to be found is called a molecular orbital (Ψ²). Similar to atomic orbitals, a molecular orbital is considered "full" when it contains two electrons with opposite spins.
Overview
Molecular orbital theory (MOT) presents a global, delocalized perspective on chemical bonding. In this view, any electron within a molecule can potentially occupy any region of the molecule, as quantum rules permit electrons to move under the influence of any number of nuclei, provided they reside in permitted eigenstates. Consequently, when excited by sufficient energy, perhaps through high-frequency light, electrons can transition to higher-energy molecular orbitals. In the case of a simple hydrogen diatomic molecule, for example, promoting an electron from a bonding orbital to an antibonding orbital under UV radiation can weaken the bond between the two hydrogen atoms, potentially leading to photodissociation – the breaking of a chemical bond due to light absorption.
As mentioned, MO theory is instrumental in interpreting ultraviolet–visible spectroscopy (UV–VIS). Changes in a molecule's electronic structure are observable through the absorption of light at specific wavelengths. These absorption signals are attributed to the transition of electrons from lower-energy orbitals to higher-energy ones. The molecular orbital diagram depicting the final state of the molecule elucidates its electronic nature in an excited state.
While some molecular orbitals in MO theory may localize electrons between specific pairs of atoms, others can distribute electrons more uniformly across the entire molecule. This inherent delocalization makes MO theory particularly well-suited for describing resonant molecules with equivalent non-integer bond orders, surpassing the descriptive power of valence bond theory in these cases. This broader applicability makes MO theory more valuable for the study of extended systems.
Robert S. Mulliken, a key figure in the development of molecular orbital theory, viewed each molecule as a self-contained entity. He articulated this perspective:
"Attempts to regard a molecule as consisting of specific atomic or ionic units held together by discrete numbers of bonding electrons or electron-pairs are considered as more or less meaningless, except as an approximation in special cases, or as a method of calculation [...]. A molecule is here regarded as a set of nuclei, around each of which is grouped an electron configuration closely similar to that of a free atom in an external field, except that the outer parts of the electron configurations surrounding each nucleus usually belong, in part, jointly to two or more nuclei...."
Consider the MO description of benzene, C 6 H 6 . This molecule features an aromatic hexagonal ring of six carbon atoms with alternating double bonds. In benzene, 24 of the total 30 valence bonding electrons (24 from carbon atoms and 6 from hydrogen atoms) reside in 12 σ (sigma) bonding orbitals. These sigma orbitals are largely localized between pairs of atoms (C–C or C–H), similar to the electrons described by valence bond theory. However, the remaining six bonding electrons in benzene occupy three π (pi) molecular bonding orbitals that are delocalized around the entire ring. Two of these electrons are found in an MO that has equal orbital contributions from all six carbon atoms. The other four electrons are distributed in orbitals with vertical nodes oriented at right angles to each other. As in VB theory, all six of these delocalized π electrons inhabit a larger space above and below the plane of the ring. In MO theory, the equivalence of all carbon–carbon bonds in benzene is a direct consequence of the three molecular π orbitals combining and evenly distributing these six electrons across the six carbon atoms.
In molecules like methane, CH 4 , the eight valence electrons are distributed among four MOs that span all five atoms. It is possible, through a transformation, to represent these MOs as four localized sp 3 orbitals. This aligns with Linus Pauling's 1931 proposal of hybridizing carbon's 2s and 2p orbitals to point directly at the hydrogen 1s basis functions, maximizing overlap. However, the delocalized MO description proves more accurate for predicting ionization energies and the positions of spectral absorption bands. When methane is ionized, an electron is removed from the valence MOs, which can originate from either the s bonding or the triply degenerate p bonding levels, resulting in two distinct ionization energies. The explanation provided by valence bond theory is considerably more complex. It involves invoking resonance between four valence bond structures when a single electron is removed from an sp 3 orbital, each structure featuring one single one-electron bond and three two-electron bonds. The ionized states (CH 4 + ) are produced as triply degenerate T 2 and A 1 states from different linear combinations of these four structures. The energy difference between the ionized and ground states yields the two ionization energies.
Similar to benzene, in molecules like beta carotene, chlorophyll, or heme, some electrons within the π orbitals are delocalized over significant distances. This delocalization leads to the absorption of light at lower energies, within the visible spectrum, which accounts for the characteristic colors of these substances. This phenomenon, along with other spectroscopic data for molecules, is effectively explained by MO theory, with a particular emphasis on electronic states associated with multicenter orbitals and orbital mixing based on principles of orbital symmetry matching. These same MO principles also provide a natural explanation for certain electrical properties, such as the high electrical conductivity observed in the planar direction of the hexagonal atomic sheets found in graphite. This conductivity arises from the continuous band overlap of half-filled p orbitals. MO theory recognizes that some electrons within the graphite atomic sheets are completely delocalized over extensive distances, residing in very large molecular orbitals that encompass an entire graphite sheet. These electrons are thus as free to move and conduct electricity within the sheet plane as they would be in a metal.