Diradicaloid

Biradicaloids or diradicaloids are molecules with two radical electrons that have significant interaction with each other. The two unpaired electrons are coupled and can either form a singlet ground state or a triplet ground state .
This is in contrast to "disbiradicals," where the two radical electrons have no significant interaction and act independently as isolated radical species. Diradicals are characterized by their diradical character, commonly quantified using an indicator. In the limit of fully degenerate frontier molecular orbitals, approaches a value of 1, representing 100% diradical character. However, diradicaloids have a small gap between the highest occupied molecular orbital and the lowest occupied molecular orbital and thus can be described as having incomplete diradical character, generally corresponding to a value of between 0.20 and 0.80. Diradicals have historically been characterized as transient species describing the transition state of a bond breaking and/or making process, but recently, the introduction of steric strain to prevent bond formation and substitution of carbon atoms with main-group elements have been found to significantly stabilize diradical species, leading to their isolation and structural characterization. However, these modifications decrease diradical character, leading these species to be more properly designated as diradicaloids. Diradicaloids have found applications in small molecule activation, molecular switching, nonlinear optics, and spintronics.

Theoretical description

Electronic structure

Due to the coupling interaction between the radical electrons in a diradical species, they cannot be simply described as the union of two independent radical centers. Both the open-shell singlet and triplet states must be considered to fully describe the electronic structure of diradical species.
The triplet state wavefunction can be described as a single electronic configuration with a single Slater determinant. However, when the frontier molecular orbitals are degenerate or nearly degenerate, the lowest-energy singlet state wavefunction must account for multiple electronic configurations. Thus, is most accurately represented as a combination of Slater determinants. Here, the configuration interaction coefficients and define the contribution of each determinant to the total wavefunction, where refers to the HOMO and refers to the LUMO:
When , and are degenerate, and the singlet wavefunction describes a perfect diradical. As the HOMO-LUMO gap increases, the wavefunction approaches that of a classical closed-shell species; approaches 1 and approaches 0 so that the lowest-energy singlet state is dominated by the doubly occupied HOMO.
To gain a more intuitive understanding of the diradical nature of the wavefunction, the triplet and singlet wavefunctions can be represented using a localized orbital basis, where and are the two localized orbitals. Assuming and are orthogonal, the overlap integral becomes 0. The HOMO can be decomposed into the in-phase overlap of and, while the LUMO can be decomposed into the out-of-phase overlap of and :
Consequently, the singlet wavefunction can be expressed as the combination of a covalent contribution and an ionic contribution. The covalent component represents the electron configuration in which both localized orbitals are singly occupied; this corresponds to diradical character. The ionic component represents the electron configuration in which one localized orbital is doubly occupied, leaving the other localized orbital empty; this corresponds to zwitterionic character:
where and
When, and ; thus, this situation describes 100% diradical character. As the HOMO-LUMO gap increases, approaches 1 and approaches 0, which results in ; thus, this situation reduces to the complete delocalization of the electrons over the two-orbital system, which is equivalent to the electron configuration of the closed-shell species.

Indicators of diradical character

The CI coefficients and can be used to provide a quantification of diradical character. Some common indicators are listed below:
All of the above indicators effectively describe how much greater the relative weight of the covalent contribution is to the singlet wavefunction compared to the ionic contribution. Thus, the greater the values of these indicators, the greater the diradical character. In the limit of 100% diradical character, these indicators approach a value of 1; in the limit of 100% classical closed-shell character, these indicators approach a value of 0.
Natural orbital occupation numbers are also another theoretical indicator of diradical character. The occupancy of the lowest unoccupied NO is equal to the indicator and ranges from 0 to 1; the closer the calculated occupancy is to 1, the greater the predicted diradical character. On the other hand, the occupancy of the highest occupied NO ranges from 1 to 2; the closer the calculated occupancy is to 1, the greater the predicted diradical character. These natural orbital occupancy numbers can be calculated using almost all computational methods and therefore can often be obtained with less computational cost than calculating using CI methods.
A small singlet-triplet energy gap can also indicate increased diradical character. Lastly, if the calculated A-B distance is elongated compared to the sum of the covalent radii but is shorter than the sum of the van der Waals radii, this may also suggest the presence of a diradicaloid. Incorporating sterically bulky substituents and introducing ring strain in heterocycles can help to prevent bond formation and/or generate elongated bonds.

Synthesis

Cyclobutane-1,3-diyl analogues

Cyclobutane-1,3-diyl

Cyclobutane-1,3-diyl is the planar four-membered carbon ring species with radical character localized at the 1 and 3 positions. The singlet cyclobutane-1,3-diyl is predicted to be the transition state for the ring inversion of bicyclobutane, proceeding via homolytic cleavage of the transannular carbon-carbon bond.
A 1,3-dimethyl substituted derivative in the triplet state was detected by electron paramagnetic resonance spectroscopy; the diradical species was generated via irradiation of the precursor diazo compound below 25 K in a solid matrix. However, the all-carbon cyclobutane-1,3-diyl is very short-lived and quickly reacts to form the bicyclobutane isomer.

1,3-diphospha-cyclobutane-2,4-diyl

In 1995, Niecke and coworkers reported the first synthesis of a phosphorus analog of cyclobutane-1,3-diyl, ₂. This species consists of a -four-membered heterocycle with radical character centered on the two carbon atoms. The heterocycle was synthesized from the reaction of arylphosphene with n-butyllithium in a 2:1 ratio, followed by elimination of LiCl. X-ray diffraction revealed that the unit exists in the planar four-membered ring form, rather than as the bicyclic isomer. MCSCF calculations predicted a singlet ground state. In addition, the calculated CI wavefunction has contributions from both the doubly occupied HOMO state and the doubly occupied LUMO state; this corresponded to occupation of the HOMO with 1.6 electrons, indicating considerable diradical character. The diphosphacyclobutane heterocycle is thermally stable, and transannular C-C bond formation is thermally forbidden according to the Woodward-Hoffmann rules. Heating at 100 °C in toluene led to the cleavage of the P-C bond, likely generating a ring-opened carbene intermediate that subsequently performed intramolecular C-H activation.
File:Niecke synthetic route towards diradicaloid.jpg|thumb|Figure 5. Synthetic route towards 2 developed by Niecke et al.
Another synthetic route was developed by Yoshifuji and Ito to access a wider variety of substituents at phosphorus. 2 equivalents of Mes*-substituted phosphaalkyne can be reacted with the lithiated compound of the first substituent on phosphorus, forming the anionic four-membered ring. This intermediate can then be alkylated to attach the second phosphorus substituent. This two-step synthetic pathway allows for the synthesis of unsymmetrically substituted 1,3-diphospha-cyclobutane-2,4-diyls. The substituents on carbon are limited to Mes*, however, due to the limitation of the phosphaalkyne starting material. Most diradicaloids of this type can be handled in air and display high kinetic stability due to the steric protection provided by the Mes* substituents on the carbon radical centers.

1,3-diaza-2,4-dipnicta-cyclobutane-2,4-diyl

These diradical species consist of a heterocyclic core where the radical sites are centered on the pnictogen atoms. The presence of a nitrogen atom in the heterocycle is thought to stabilize the planar form relative to the bicyclic isomer. This is believed to result from the inability of Pn-Pn bond formation in the bicyclobutane form to energetically compensate for the increase in Pn-N-Pn angle strain; consequently, the planar form, which allows for larger Pn-N-Pn angles, is more stable. The lack of electron delocalization found in calculations suggests that aromaticity from the presence of 6π electrons does not play a significant role in stabilization of the planar isomers.
In 2011, Schulz and coworkers synthesized the first example of a four-membered ring diradicaloid with meta-terphenyl and hypersilyl substituents on the nitrogen atoms. The synthetic route begins with the chlorinated P₂N₂ heterocycle, which is then reduced to the diradicaloid with relatively mild titanium or titanium reducing agents. The bulky terphenyl and hypersilyl groups provide kinetic stabilization, preventing dimerization. The terphenyl-substituted diradicaloid is almost indefinitely stable under argon atmosphere at ambient temperatures as a solid and in solvent. The crystal structure reveals a planar four-membered ring and a long distance between the two phosphorus atoms, indicating no significant transannular interactions. Computations also support the diradical character of this species and predict a singlet ground state. The calculated CI wavefunction has contributions from both the doubly occupied HOMO state and the doubly occupied LUMO state; this corresponds to occupation of the HOMO with 1.7 electrons, indicating considerable diradical character.
File:Schulz synthetic route towards diradicaloid.jpg|thumb|Figure 8. Synthetic route towards a diradicaloid developed by Schulz et al.
Using a similar synthetic route, the arsenic analogue was also synthesized from the chlorinated precursor; reduction using magnesium metal generated the arsenic centered diradicaloid. The crystal structure confirmed a long As-As distance, and EPR spectroscopy indicated a singlet ground state. A mixed phosphorus-arsenic diradicaloid was also reported in 2015, the first with different radical centers. The crystal structure revealed a kite-shaped planar four membered ring with a transannular As-P distance of 2.790 Å, which is shorter than the sum of van der Waals radii but longer than the sum of covalent radii.
Heavier derivatives were observed in situ but could not be isolated due to rapid decomposition to the allyl analogues in the presence of magnesium; however, the corresponding diradicaloids could be trapped through cycloadditions with alkynes, thereby providing evidence for their existence. Calculations suggest that the antimony and bismuth-centered diradicaloids have higher diradical character than the lighter pnictogen analogues due to the singlet-triplet energy gap decreasing with heavier, larger pnictogens.