Covalent Bonds: Definition, Properties, and the Quantum Mechanical Perspective

Covalent chemical bonding is a cornerstone concept in chemistry. Covalent bonds occur when pairs of electrons are shared by atoms. Atoms will covalently bond with other atoms in order to gain more stability, which is gained by forming a full electron shell. This article explores the nature of covalent bonds, their properties, and the ongoing debate surrounding their physical origin, delving into the quantum mechanical explanations that underpin this fundamental chemical process.

Introduction: The Covalent Bond and Its Paradoxical Nature

While bond formation is arguably the most fundamental chemical process, its physical origin is still the subject of debate, even today when accurate quantitative molecular electronic structure calculations of ever-increasing accuracy and complexity have become widely available. Seemingly, there is a chasm between numerical and physical resolutions of the covalent bond. The idea of shared electron pairs corresponding to chemical bonds was introduced by Lewis [1] over a hundred years ago in a landmark publication, a decade before Schrödinger [2] developed the method that effectively laid the foundations of quantum chemistry and provided the tools with which Lewis’ ideas could be rigorously tested and interpreted. Lewis’ model built on the earlier work of Abegg [3] who had introduced and explored the Octet Rule. Abegg’s work was also inspirational in Kossel’s development of ionic bonding [4]. The theories of Abegg, Lewis, and Kossel were extended by Langmuir [5,6,7,8] who extended the Octet Rule by developing the 18- and 32-electron rules and introduced the name “Covalent Bond” for a shared pair of electrons. The development of quantum theory provided a theoretical underpinning of covalent bonding, via Burrau’s [9,10] quantum mechanical calculation on the hydrogen molecule ion H2+ and, in the same year, the Heitler-London calculation [11] of the geometry and bond energy of the H2 molecule. Burrau’s work [9,10], in good agreement with experiment [12,13,14], provided theoretical support for the existence of one electron covalent bonds. The Heitler-London work [11] especially made the connection between the Lewis shared pair of electrons [1] and its physical quantitative description. The valence-bond (VB) theory of Pauling [15,16] has provided a natural bridge between these two theories and, especially in its qualitative form, for many years it has been the bonding theory of choice of most chemists. Molecular orbital (MO) theory, promoted initially by Mulliken [17], Hund [18], Hückel [19], and others [20,21,22,23,24,25] has, however, come to be preferred by the chemical community, especially when dealing with multicenter bonds, and more generally for problems of electronic excitation, reactivity, and transition metal chemistry. By the 1960s and 70s, when the development and distribution of quantum chemical computer codes had become widespread, the majority of the methods, such as those in the Gaussian suite of programs [26,27], first released in 1970, were Hartree-Fock Self-Consistent Field MO (HF-SCF) [28,29] based.

Understanding Covalent Bonds

By sharing their outer most (valence) electrons, atoms can fill up their outer electron shell and gain stability. Nonmetals will readily form covalent bonds with other nonmetals in order to obtain stability, and can form anywhere between one to three covalent bonds with other nonmetals depending on how many valence electrons they posses. Covalent bonds are characterized by the sharing of electrons between two or more atoms.

The Octet Rule

The Octet Rule requires all atoms in a molecule to have 8 valence electrons--either by sharing, losing or gaining electrons--to become stable. For Covalent bonds, atoms tend to share their electrons with each other to satisfy the Octet Rule. It requires 8 electrons because that is the amount of electrons needed to fill a s- and p- orbital (electron configuration); also known as a noble gas configuration. Each atom wants to become as stable as the noble gases that have their outer valence shell filled because noble gases have a charge of 0.

Types of Covalent Bonds

The types of covalent bonds can be distinguished by looking at the Lewis dot structure of the molecule. For each molecule, there are different names for pairs of electrons, depending if it is shared or not. A pair of electrons that is shared between two atoms is called a bond pair.

Single Bonds

A single bond is when two electrons--one pair of electrons--are shared between two atoms. It is depicted by a single line between the two atoms. Below is a Lewis dot structure of Hydrogen Chloride demonstrating a single bond. As we can see from the picture below, Hydrogen Chloride has 1 Hydrogen atom and 1 Chlorine atom. Hydrogen has only 1 valence electron whereas Chlorine has 7 valence electrons.

Double Bonds

A Double bond is when two atoms share two pairs of electrons with each other. It is depicted by two horizontal lines between two atoms in a molecule. Below is a Lewis dot structure of Carbon dioxide demonstrating a double bond. As you can see from the picture below, Carbon dioxide has a total of 1 Carbon atom and 2 Oxygen atoms. Each Oxygen atom has 6 valence electrons whereas the Carbon atom only has 4 valence electrons. To satisfy the Octet Rule, Carbon needs 4 more valence electrons.

Triple Bonds

A Triple bond is when three pairs of electrons are shared between two atoms in a molecule. It is the least stable out of the three general types of covalent bonds. Below is a Lewis dot structure of Acetylene demonstrating a triple bond. As you can see from the picture below, Acetylene has a total of 2 Carbon atoms and 2 Hydrogen atoms. Each Hydrogen atom has 1 valence electron whereas each Carbon atom has 4 valence electrons. Each Carbon needs 4 more electrons and each Hydrogen needs 1 more electron. Hydrogen shares its only electron with Carbon to get a full valence shell. Now Carbon has 5 electrons. Because each Carbon atom has 5 electrons--1 single bond and 3 unpaired electrons--the two Carbons can share their unpaired electrons, forming a triple bond.

Polar and Nonpolar Covalent Bonds

Only when two atoms of the same element form a covalent bond are the shared electrons actually shared equally between the atoms. When atoms of different elements share electrons through covalent bonding, the electron will be drawn more toward the atom with the higher electronegativity resulting in a polar covalent bond.

Polar Covalent Bonds

A Polar Covalent Bond is created when the shared electrons between atoms are not equally shared. This occurs when one atom has a higher electronegativity than the atom it is sharing with. The atom with the higher electronegativity will have a stronger pull for electrons (Similiar to a Tug-O-War game, whoever is stronger usually wins). As a result, the shared electrons will be closer to the atom with the higher electronegativity, making it unequally shared. A polar covalent bond will result in the molecule having a slightly positive side (the side containing the atom with a lower electronegativity) and a slightly negative side (containing the atom with the higher electronegativity) because the shared electrons will be displaced toward the atom with the higher electronegativity. As a result of polar covalent bonds, the covalent compound that forms will have an electrostatic potential. This potential will make the resulting molecule slightly polar, allowing it to form weak bonds with other polar molecules. As you can see from the picture above, Oxygen is the big buff creature with the tattoo of "O" on its arm. The little bunny represents a Hydrogen atom. The blue and red bow tied in the middle of the rope, pulled by the two creatures represents--the shared pair of electrons--a single bond.

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Nonpolar Covalent Bonds

A Nonpolar Covalent Bond is created when atoms share their electrons equally. This usually occurs when two atoms have similar or the same electron affinity. The closer the values of their electron affinity, the stronger the attraction. This occurs in gas molecules; also known as diatomic elements. Nonpolar covalent bonds have a similar concept as polar covalent bonds; the atom with the higher electronegativity will draw away the electron from the weaker one. As you can see from the picture above, Hydrogen gas has a total of 2 Hydrogen atoms. Each Hydrogen atom has 1 valence electron. Since Hydrogen can only fit a max of 2 valence electrons in its orbital, each Hydrogen atom only needs 1 electron.

Properties of Covalent Compounds

When compared to ionic compounds, covalent compounds usually have a lower melting and boiling point, and have less of a tendency to dissolve in water. Covalent compounds can be in a gas, liquid, or solid state and do not conduct electricity or heat well.

The Ongoing Debate: Electrostatic vs. Kinetic Energy

With respect to the basic physics of covalent bonding a long-held and still widespread view is that chemical bonding is essentially an electrostatic phenomenon. Namely, the energy lowering that corresponds to bond formation is thought to be the result of the decrease in potential energy due to the attractive interaction between the nuclei and the electronic charge that is accumulated in the bond region. This essentially classical and static picture of interacting charge distributions is appealing in its simplicity, indeed it appears to be a straightforward extension of the Lewis theory. Moreover, the above electrostatic view appears to be consistent with the Virial Theorem [34,35,36], according to which the ratio of total potential to kinetic energy of a molecule at its equilibrium geometry, always equals −2. In other words, the attractive component of the binding energy is therefore due to the (electrostatic) potential energy, whereas the kinetic component is repulsive. This electrostatic view was originally advanced by Slater [36] in 1933, supported by Feynman [37] in 1939, and later by Coulson [24], whose book Valence of 1952 has had a strong influence on the chemical community.

In contrast to the electrostatic view, Hellmann [40] proposed that covalent bonding should be understood as a quantum mechanical effect, brought about by the lowering of the ground state kinetic energy associated with the delocalization of the motions of valence electrons between atoms in a molecule. For several decades this kinetic view [40] was ignored by most chemists, possibly because it went against the already accepted seemingly simpler electrostatic explanation [36], and because Hellmann had based his reasoning on the statistical Thomas−Fermi model [41,42], that was subsequently shown to be unable to describe covalent bonding [43,44,45]. Other contributing factors could have been the apparent conflict with the Virial Theorem [34,35,36] that Hellmann was unable to resolve [46] and, quite likely, his early tragic death. Interestingly though, the physicists Peierls [47] and Platt [48] expressed general agreement with Hellmann’s view, as early as 1955 and 1961, respectively.

Ruedenberg's Resolution: Delocalization and Kinetic Energy

The contradiction between these two different qualitative models of the covalent bonding mechanism required a rigorous in-depth analysis for its resolution. The analysis was carried through, on the basis of the quantum mechanical Variation Principle, by Ruedenberg and coworkers [49,50,51,52,53,54,55,56,57,58,59] from 1962 on, first for H2+ and H2 and later for other homonuclear diatomic molecules as well. These investigations showed that covalent bonding is a quantum effect as originally suggested by Hellmann [40], since the critical component of bonding is interatomic electron delocalization, which is the quantum mechanical term for electron sharing. Stabilizing electron delocalization in ground states is associated with combination and constructive interference of the atomic orbitals (AOs) of the atoms in a molecule to form molecular orbitals (MOs). This normally results in bonding by a net decrease in the kinetic energy of the molecule, in agreement with Hellmann’s view [40].

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However, as the internuclear distance becomes smaller than about twice the equilibrium separation, an additional, more complex, effect comes into play, namely intra-atomic (orbital) contraction. In a minimal atomic basis set, as the term implies, orbital contraction corresponds to an increase in the orbital exponents of the basis functions, resulting in tighter orbitals, and thus electron densities, around the nuclei. In practice, the orbital exponents, being non-linear parameters, are optimized at each distinct geometry, by numerical minimization of the molecular energy. (Alternatively, the use of extended but geometry independent basis sets, such as those in state-of-the-art HF-SCF or density functional calculations, allows orbital contractions to be resolved via the optimization of the occupied MOs). The net result of orbital contraction is additional stabilization by a decrease in the potential energy while the kinetic energy is increased. These energy shifts ensure that the Virial Theorem [34,35,36] is satisfied. As discussed elsewhere, the orbital contractions actually enhance the degree of interatomic delocalization and thereby lower the interatomic kinetic energy further, even though they increase the antibonding intra-atomic deformation energies. The net result is a decrease in the total energy, as demanded by the Variation Principle.

Feynman's Perspective: Electron Dynamics and Kinetic Energy

Interestingly, after expressing support for Slater’s electrostatic view of covalent bonding in 1939, 26 years later Feynman [88] himself, in his famous Lectures on Physics, explained covalent bond formation in H2+, and by extension in other molecules, as the consequence of a flip-flop motion of electrons between bonded atoms, causing a corresponding drop in the electron’s kinetic energy, as a molecule forms. His simplest argument was based on the Heisenberg uncertainty principle, ΔxΔp ≥ ħ/2, that would imply that the lower energy of the electron in the bonding stationary state of the molecule is a consequence of delocalization, i.e., “spreading out” or increasing Δx, that results in a drop in kinetic energy (proportional to (Δp)2) without a significant increase in its potential energy. The opposite would hold for the repulsive antibonding state. This conclusion is in complete agreement with the views of Hellmann [40] and Ruedenberg [49]. Feynman [88] was, so far as we know, first to emphasize the close connection between electron dynamics and kinetic energy in covalent bonding, i.e., between the flip-flop motion and the corresponding kinetic energy lowering.

Quantum Dynamical Description of Covalent Bonding

While our own work over a period of 25 years [68,69,72,75,77,78,79,80,81,82,83], has agreed with Hellmann’s [40], Ruedenberg’s [49,50,51,52,53,54,55,56,57,58,59], and Feynman’s [88] views, it has also expanded on them by exploring the quantum dynamical description of covalent bonding. Noting that Thomas-Fermi (TF) theory [41,42], the original and simplest type of density functional theory (DFT) [45,89], is unable to describe covalent bonding [43,44,45], we analyzed the reasons for this failure in an effort to better understand the basic physics of bonding [68,78,79,80]. We have found that, at least in simple systems, the source of the problem is the simplified semi-classical form of the TF kinetic energy functional, resulting in a theory that is unable to account for dynamical constraints (non-ergodicity) and slow electron transfer and thus for any hindered internal electron dynamics in atoms and molecules.

The first suggestion that covalent bonding was best seen as a quantum dynamical phenomenon was, as noted above, published by Feynman [88] in 1965, in his wide ranging lectures on physics, three years after Ruedenberg’s groundbreaking paper in Reviews of Modern Physics [49]. Feynman [88], unlikely to have read Ruedenberg’s paper, treats H2+ as a simple two-state system of bonding and antibonding MOs and uses time-dependent quantum theory to show that the bonding electron in H2+, must, if localized, oscillate between the two nuclei with a frequency that is directly related to the energy difference between the delocalized MOs. These MOs are stationary states and their energy difference is approximately twice the bond energy. As noted above, this energy difference, part of the time-independent quantum description of the molecule and due to the orthogonality of ground and first excited states, is essentially kinetic in character. However, at the time-dependent level, the degree of bonding could be seen to be proportional to the rate of interatomic electron oscillation, which in turn is dependent on delocalization of the electron. The dynamical picture of interatomic electron oscillation Feynman [88] referred to as the “flip-flop mechanism” of covalent bonding.

Ruedenberg and co-workers [49,50,51,52,53,54,55,56,57,58,59,83] reached an equivalent energy analysis of bonding utilizing the results of time-independent variational calculations, their main conclusion being that the physical basis of bonding is interatomic electron delocalization. They went on to analyze the importance of kinetic energy to bonding, the role of orbital contraction and the Virial Theorem [34,35,36]. In addition to his treatment of H2+, Feynman [88] discussed bonding in H2 as well as in benzene and (conjugated) dye molecules, as other examples of two-state systems. He demonstrated the generality of his idea of attraction by particle exchange between attractive centers by the analogous treatment of the umbrella inversion in ammonia and the interaction of nucleons. As noted above, the quantum dynamical view of covalent bonding is not unique. Nor is at odds with the Hellmann-Ruedenberg theory [40,49,50,51,52,53,54,55,56,57,58,59,83] or the Virial Theorem [34,35,36]. Instead it provides a fully consistent alternative interpretation which sheds light on covalent bonding while avoiding, or in a deeper analysis helping to resolve, the apparent contradiction between the theory and the theorem.

The Strength of Covalent Bonds

Covalent bonding occurs when two atomic orbitals come together in close proximity and their electron densities overlap. The strongest type of covalent bonds are sigma bonds, which are formed by the direct overlap of orbitals from each of the two bonded atoms. bond strength: Directly related to the amount of energy required to break the bond between two atoms. bond length: The distance between the nuclei of two bonded atoms. Double and triple covalent bonds occur when four or six electrons are shared between two atoms, and they are indicated in Lewis structures by drawing two or three lines connecting one atom to another.

Covalent bonds between atoms are quite strong, but attractions between molecules/compounds, or intermolecular forces, can be relatively weak. However, the Lewis theory of covalent bonding does not account for some observations of compounds in nature. The theory predicts that with more shared electrons, the bond between the two atoms should be stronger. According to the theory, triple bonds are stronger than double bonds, and double bonds are stronger than single bonds. This is true. However, the theory implies that the bond strength of double bonds is twice that of single bonds, which is not true.

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