Additionally, the method has enjoyed some success for generating the molecular structures of excited states, radicals, and reaction intermediates.
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.
[5] 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.
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.
Thus, it has previously[20] been argued that the following should also be included in the basic postulates of LDQ theory: The electron pairing can result in a greater net binding between the nuclei, but this is not necessarily the case in all molecules.
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.
Much like Lewis’ bonding model, LDQ theory assumes that the dominant contributions result from electron-electron and electron-nuclear interactions.
[2][18] This also highlights that the additional degree of freedom afforded by having two distinct spin sets in the LDQ approach allows a single electron in a bond to be shared equally between two atoms, which produces the above structure for benzene.
Thus, a comparison of the magnitude of the inter-electronic repulsions in a series of possible molecular structures can be used to assess their relative energies and hence determine the ground and excited states.
Linnett rationalises this three-electron redistribution by arguing that it is required by the need to both form the two carbon-hydrogen bonds and retain the tetrahedral disposition of the electrons of a given spin.
[2][25] Interestingly, the excited state does not obey the octet rule as the carbon atoms have an average 6.5 valence electrons surrounding them.
[25] Thus, this example illustrates that LDQ theory can be a powerful tool for understanding the geometric rearrangements that occur when excited states are formed.
A major drawback of Lewis’ bonding theory is its inability to predict and understand the structures of radicals due to the presence of unpaired single electrons.
LDQ theory has seen great success in explaining the structures of open shell systems such as nitric oxide or ozone due to the additional degree of freedom associated with having two independent spin sets.
In the cases of nitric oxide and ozone, the maxima of the electron density of the localised orbitals result in distributions which closely mirror the dot-and-cross diagrams produced using LDQ theory.
[21][22][28][29][30][31][32][33] Firestone has previously used the concept of L-strain (see above) to analyse the activation energies in SN2, SH2 and E2 reactions, since the movement of electron density out of the internuclear region is commonly associated with the formation of transition states.
LDQ structures, in particular the coincidence of electron pairs, can be used to rationalise and explain the stability and reactivity of certain families of molecules such as hydrocarbons.
The result is that the energy required to overcome charge correlation and pair the electrons up is compensated to a lesser extent by the bonding in ethylene as compared with ethane.
In the traditional Lewis view, this violates the octet rule as the five phosphorus-chlorine bonds would result in a net ten electrons around the phosphorus atom.
The LDQ approach thus enables each electron to localise in one of the boron-hydrogen internuclear bond regions, rather than being delocalised over the entire three-centre boron-hydrogen-boron moiety.
This approach has previously been shown to produce lower energies as compared to valence bond or molecular orbital wave functions derived from Lewis structures for molecules such as benzene, diborane or ozone.
[4] A recent report[42] on the disilyne and digermyne molecules has shown that their ELFs also result in a toroidal basin surrounding the internuclear axis.
The LDQ structure is in excellent agreement with these computational results: the toroid is angled in comparison with the case in acetylene due to the perturbation caused by the off-axis hydrogen atoms.
The ELF analysis of ClF3 indicates that there is a single toroidal-shaped basin at the 'back' of each fluorine atom, corresponding analogously to the three lone pairs arranged in a ring as generated for the HF molecule (see above).
[49] Linnett's vision of double-quartet theory was limited to elements which did not expand their valence beyond the octet: this produced the familiar spin tetrahedra.
While these results are interesting, they have been contested in the scientific literature due to Luder's abandonment of the octet rule and the author's controversial views on spin correlation.
For example, a recent study found that there is a qualitative correspondence between the molecular structures produced using LDQ theory and those suggested by dynamic Voronoi metropolis sampling.
[52] Another recent example is the correspondence of the results obtained using LDQ theory to those produced using the Fermi-Löwdin orbital self-interaction correction[53] (FLO-SIC) method.
Further, the electronic geometries for many ground state molecules, such as carbon dioxide, produced via FLO-SIC methods were found to generally agree with those derived from LDQ theory.
In a subsequent publication, the authors posited that the Fermi orbital descriptors[55] utilised in their work can be correlated to the electron spins generated in LDQ analyses.
Left: The dot-and-cross diagram of the LDQ structure of ozone (O
3
). The nuclei are as indicated and the electrons are denoted by either dots or crosses, depending on their relative spins. Right: Simplified diagram of the LDQ structure of O
3
, showing electrons in non-coincident pairs using thin lines and a coincident electron pair using a thick line.
Diagram showing (a) the most probable and (b) the least probable disposition of two spin sets, each containing three electrons, around a ring. The electrons are denoted by either dots or crosses, depending on their relative spins. The least probable case has a probability 56% of that for the most probable case,
[
2
]
illustrating that the correlation between the spin sets is weak.
The LDQ structure of the fluoride anion. The central fluorine nucleus is coloured gray while the electrons are coloured either purple or green to distinguish between the spin sets.
The LDQ structure of hydrogen fluoride. The fluorine nucleus is coloured gray and the proton is coloured pink, while the electrons are coloured either purple or green to distinguish between the spin sets.
The LDQ structure of molecular oxygen in the ground state (
3
Σ
g
−
state). The oxygen nuclei are coloured red while the electrons are coloured either purple or green to distinguish between the spin sets.
The LDQ structure of methane. The carbon nucleus is coloured brown and the protons are coloured pink, while the electrons are coloured either purple or green to distinguish between the spin sets.
The dot-and-cross diagram for molecular oxygen in the ground state. The oxygen nuclei are as indicated and the electrons are denoted by either dots or crosses, depending on their relative spins.
(a) Traditional Lewis structure for hydrogen fluoride, showing eight valence electrons in four coincident pairs. (b) Simplified 2D LDQ structure for hydrogen fluoride, showing the electrons in non-coincident pairs using thin lines and a coincident electron pair using a thick line.
Left: The dot-and-cross diagram of the LDQ structure of NO. The nuclei are as indicated and the electrons are denoted by either dots or crosses, depending on their relative spins. Right: Simplified diagram of the LDQ structure of NO, showing electrons in non-coincident pairs using thin lines.
The LDQ structure of benzene. The carbon nuclei are coloured brown and the hydrogen nuclei are coloured pink, while the electrons are coloured either purple or green to distinguish between the spin sets.
Left: The dot-and-cross diagram of the LDQ structure of benzene. The nuclei are indicated and the electrons are denoted by either dots or crosses, depending on their relative spins. Right: Simplified diagram of the LDQ structure of benzene, showing electrons in non-coincident pairs using thin lines.
The LDQ structure of molecular oxygen in its first excited state (
1
Δ
g
state). The oxygen nuclei are coloured red while the electrons are coloured either purple or green to distinguish between the spin sets.
The LDQ structure of molecular oxygen in its second excited state (
1
Σ
g
+
state). The oxygen nuclei are coloured red while the electrons are coloured either purple or green to distinguish between the spin sets.
The dot-and-cross diagram of the LDQ structure of the ground state of acetylene is shown on the left and that of the first excited state of acetylene is shown on the right. The nuclei are as indicated and the electrons are denoted by either dots or crosses, depending on their relative spins.
The LDQ structure of nitric oxide. The oxygen nucleus is coloured red and the nitrogen nucleus is coloured blue, while the electrons are coloured either purple or green to distinguish between the spin sets.
(a) The top shows both the dot-and-cross diagram and the simplified diagram of the LDQ structure of the CN radical. The nuclei are as indicated and the electrons are denoted by either dots or crosses, depending on their relative spins. Below is shown the dimerisation reaction of the CN monomer into the cyanogen molecule. The electrons in non-coincident pairs are shown using thin lines and the coincident electron pairs are shown using thick lines. (b) The top shows both the dot-and-cross diagram and the simplified diagram of the LDQ structure of the NO radical. Below is shown the dimerisation reaction of the NO monomer into the N
2
O
2
dimer.
The LDQ structures of (a) ethane, (b) ethylene and (c) acetylene. The carbon nuclei are coloured brown and the hydrogen nuclei are coloured pink, while the electrons are coloured either purple or green to distinguish between the spin sets.
The dot-and-cross diagram of the LDQ structure of PCl
5
. The nuclei are as indicated and the electrons are denoted by either dots or crosses, depending on their relative spins. The electrons in non-coincident pairs are shown using thin lines.
(a) The LDQ structure of the B
2
H
7
−
molecule. The nuclei are as indicated and the electrons are denoted by either dots or crosses, depending on their relative spins. The thick lines denote coincident electron pairs. (b) The traditional valence bond theory structure for the B
2
H
7
−
molecule. The horizontal bar stretching across the boron-hydrogen-boron moiety indicates that the two bonding electrons are delocalised across these three centres. (c) The resonance structures for the B
2
H
7
−
molecule.
(a) The LDQ structure of the B
2
H
6
molecule. The nuclei are as indicated and the single electrons are denoted by dots. The thick lines denote coincident electron pairs. (b) The traditional valence bond theory structure for the B
2
H
6
molecule. The thin curved lines stretching across the boron-hydrogen-boron moiety indicate that the two bonding electrons are delocalised across these three centres. (c) The resonance structures for the B
2
H
6
molecule.
(a) The dot-and-cross diagram of the simplified LDQ structure of acetylene. The nuclei are as indicated and the electrons are denoted by either dots or crosses, depending on their relative spins. The ellipse in the centre indicates the relative disposition of the electrons around the carbon-carbon internuclear axis. (b) The ELF for the acetylene molecule generated using an η=0.865 value for the
isosurfaces
. The electron basins are coloured in blue (hydrogen basins) and green (carbon bonding basin), and the carbon core basins are coloured light blue.
(a) The dot-and-cross diagram of the simplified LDQ structure of digermyne. The nuclei are as indicated and the electrons are denoted by either dots or crosses, depending on their relative spins. The ellipse in the centre indicates the relative disposition of the electrons around the germanium-germanium internuclear axis. (b) The ELF for the digermyne molecule generated using an η=0.700 value for the isosurfaces. The electron basins are coloured in blue (hydrogen basins) and green (germanium bonding basin), and the germanium core basins are coloured purple.
(a) The dot-and-cross diagram of the simplified LDQ structure of ClF
3
. The nuclei are as indicated and the electrons are denoted by either dots or crosses, depending on their relative spins. The electrons in non-coincident pairs are shown using thin lines and the coincident electron pairs are shown using thick lines. (b) The ELF for the ClF
3
molecule generated using an η=0.700 value for the isosurfaces. The electron basins are coloured in yellow (fluorine basins) and red (chlorine lone pair basins), and the chlorine core basin is coloured purple. (c) The ELF for the ClF
3
molecule generated using an η=0.835 value for the isosurfaces.
The Luder electron-repulsion theory structures of (a) the zinc atom and (b) the ytterbium atom. The zinc nucleus is coloured gray and the ytterbium nucleus is coloured cyan, while the electrons are coloured either purple or green to distinguish between the spin sets.
