Linnett double-quartet theory (LDQ) is a method of describing the bonding in molecules which involves separating the electrons depending on their spin, placing them into separate 'spin tetrahedra' to minimise the Pauli repulsions between electrons of the same spin. Introduced by J. W. Linnett in his 1961 monograph and 1964 book, this method expands on the electron dot structures pioneered by G. N. Lewis. While the theory retains the requirement for fulfilling the octet rule, it dispenses with the need to force electrons into coincident pairs. Instead, the theory stipulates that the four electrons of a given spin should maximise the distances between each other, resulting in a net tetrahedral electronic arrangement that is the fundamental molecular building block of the theory.
By taking cognisance of both the charge and the spin of the electrons, the theory can describe bonding situations beyond those invoking electron pairs, for example two-centre one-electron bonds. This approach thus facilitates the generation of molecular structures which accurately reflect the physical properties of the corresponding molecules, for example molecular oxygen, benzene, nitric oxide or diborane. Additionally, the method has enjoyed some success for generating the molecular structures of excited states, radicals, and reaction intermediates. The theory has also facilitated a more complete understanding of chemical reactivity, hypervalent bonding and three-centre bonding.
Historical background The cornerstone of classical bonding theories is the Lewis structure, published by G. N. Lewis in 1916 and continuing to be widely taught and disseminated to this day. In this theory, the electrons in bonds are believed to pair up, forming electron pairs which result in the binding of nuclei. While Lewis’ model could explain the structures of many molecules, Lewis himself could not rationalise why electrons, negatively-charged particles which should repel, were able to form electron pairs in molecules or even why electrons can form a bond between atoms.
Lewis’ theory has been seminal in the understanding of the chemical bond. Yet despite this, it was formulated before the discovery of electron spin, a key intrinsic property of electrons which manifests itself through inter-electronic interactions. While spin was known about ever since the publication of Stern and Gerlach's results in 1922, with the Pauli exclusion principle being formulated in 1925, the importance of 'spin correlation' for understanding when and why electrons form pairs in molecules was not understood until the work of Lennard-Jones in the 1950s. During the latter decade, J. W. Linnett and his students began to explicitly study the role of spin in determining the electronic structures of various molecules. This resulted in Linnett's landmark 1961 publication, and subsequent 1964 book, in which he outlined what became known as “Linnett double-quartet” theory.
Linnett continued to expand on his theory through a number of publications until his death in 1975. In these writings, Linnett recognised the continued importance of the Lewis model of bonding and the importance of satisfying the octet rule. However, he also argued that this view overemphasises the importance of electron pairing in the formation of chemical bonds. Hence, his theory sought to introduce spin into the conventional model of bonding and hence rectify some of the problems associated with Lewis’ theory. While LDQ theory is a relatively simple extension of Lewis’ bonding theory, the additional freedom of the electrons to separate into two sets, differentiated by their spins, has bestowed upon the theory exquisite agreement with the results of many experiments.
In its nascent years, LDQ theory attracted the interest of many researchers, furnishing greater insights into the structures of many molecules. However, LDQ theory began to fade from the spotlight in the 1970s and was mostly abandoned by researchers in the United States, Great Britain and Europe by the mid-1980s.
Formulation of Linnett double-quartet theory
Basic principles A key trait of LDQ theory that is shared with Lewis theory is the importance of using formal charges to determine the most important electronic structure. LDQ theory produces the spatial distributions of the electrons by considering the two fundamental physical properties of said electrons:
The mutual repulsion of electrons with like spins, in accordance with the Pauli exclusion principle. Hence, electrons with like (parallel) spins tend to keep as far away from each other as possible by refusing to occupy the same spatial region, while electrons with unlike (antiparallel) spins can occupy the same spatial region. This effect is known as ‘spin correlation’. The mutual Coulombic repulsion between electrons. This effect tends to keep electrons as far away from each other as possible, regardless of their relative spins. This is known as ‘charge correlation’.
In Linnett's interpretation, correlation is “the mutual effect the electrons have on one another’s spatial positions”. In the absence of charge correlation, the situation would be as follows:
If an equal number of both spins is present, the electrons will tend to pair up. If an unequal number of spins is present, then the probability distribution of the possible structures is independent of the mutual disposition of the two spin sets.
When one adds the effects of charge correlation, the situation is modified somewhat:
For electrons with the same spin, charge correlation works in tandem with spin correlation to yield a strong repulsion between the electrons. For electrons of opposite spin, charge correlation effects will work against spin correlation effects.
Given these rules, it is found that:
The four electrons in the same spin set will always keep apart as they experience a negative charge correlation and a negative spin correlation. Electrons in different spin sets can pair up (occupy the same spatial region) as they experience a negative charge correlation (which tends to keep them apart) but a positive spin correlation (which favours the spatial proximity of electrons with unlike spins).
Consequences of electron correlation effects An octet is any arrangement which results in a given nucleus having a total of eight valence electrons around it. In Lewis' bonding model, the electrons tend to pair up in bonds such that an atom has a total of four chemical bonds and lone pairs associated with it: thus, the atom can satisfy its octet. LDQ theory also acknowledges that the elements in the ‘first short period’ of the periodic table tend to attain an octet of electrons surrounding them. However, in contrast with Lewis' view, Linnett argued that due to the combined effects of charge correlation and spin correlation, it is physically more meaningful to consider the octet as the sum of two tetrahedral quartets of electrons. Each quartet consists of electrons of one spin only, and these electrons can act and orient themselves independently. One can then obtain molecular structures by arranging the electrons in such a way as to maximise the separations between the electrons, hence minimising the mutual inter-electronic repulsions, while simultaneously ensuring that the basic geometry of the spin sets is not altered. Additionally, Linnett stressed that due to the Pauli exclusion principle, one should prioritise separating electrons of the same spin when considering the overall electronic structure.
Influence of nearby nuclei
In chemical bonding, the presence of additional nuclei causes the electrons to seek to maximise their attractive electrostatic interactions with all nearby nuclei. This can result in the formation of coincident or ‘close-paired’ electron pairs, in accordance with Lewis’ bonding model. Thus, it has previously been argued that the following should also be included in the basic postulates of LDQ theory:
The attraction between the nuclei and the electrons tends to distort the electronic geometry. This distortion acts to force a maximum number of electrons into the internuclear (bond axis) region, helping to efficiently bind the nuclei together. The presence of any additional nearby nuclei can partially relax the influence of correlation effects on the electronic geometry. Therefore, it is possible for two electrons of opposite spin to come together and occupy the same spatial region, effectively forming the classical Lewis electron pair. This can serve to strengthen the binding between the nuclei by increasing the net electron density in the internuclear region. The exact disposition of the electrons is determined by the relative electronegativities of the constituent elements. The electron pairing can result in a greater net binding between the nuclei, but this is not necessarily the case in all molecules. In his discussions, Linnett notes that due to the opposing effects of charge and spin, the correlation between the two spin quartets should be small and so the individual spin tetrahedra can be treated as being partly independent from each other. This then facilitates electron pairing since nearby nuclei can easily force the two electrons together. Linnett also argues that a relatively small deviation from the strictly regular tetrahedra of the rigorous LDQ theory approach could be energetically favourable in some cases.
Balancing the intramolecular interactions The structure obtained from applying LDQ theory balances the three principal interactions in the molecule: electron-electron, electron-nuclear and nuclear-nuclear. Much like Lewis’ bonding model, LDQ theory assumes that the dominant contributions result from electron-electron and electron-nuclear interactions. However, it has previously been shown that the introduction of nuclear-nuclear interactions into LDQ theory can explain some trends in bond angles and bond lengths. In particular, Firestone produced an extensive discussion of the effects of moving bonding electron density out of the internuclear region and highlighted that sometimes such a distortion is necessary to produce a more satisfactory arrangement of the spin sets. Due to the decreased shielding of the nuclear-nuclear interactions and the decreased electron-nuclear interactions associated with this change, the net energy of the molecule tends to increase: this is known as “L-strain” (see section on reactivity later).
Examples of the application of the theory
Understanding structures using LDQ As an example of the application of LDQ theory to molecular bonding, take the case of the fluoride ion. By using LDQ theory, the electronic structure shown below is obtained. The two spin sets are under the action of only one nucleus and so there is no net interaction which will cause the electrons to pair up. Hence, unlike the Lewis model which predicts four lone pairs, all electrons in the fluoride ion are spatially separated. Therefore, the following statement by Luder is found to be true for all mononuclear species:“In an isolated atom, no valence electron is close-paired with another”. If a proton then approaches the fluoride ion, the proton's attractive potential can distort the electronic geometry. Two electrons of opposite spin (necessary to complete the duplet of the hydrogen atom) are attracted to the proton and this attractive potential pulls them together to yield an electron pair localised to the internuclear region. This is illustrated in the LDQ structure of hydrogen fluoride shown below.
Again, while the Lewis picture would predict four coincident electron pairs, the LDQ theory treatment yields only one close pair and two staggered spin tetrahedra that share a vertex. This makes sense as the other six electrons, unlike the two bonding electrons, do not significantly experience the attractive influence of the proton and hence their inter-electronic repulsions keep them separated.
Example: molecular oxygen