1. Introduction
Hypervalent iodine compounds (HVI) are important alternatives to transition metal reagents because of their reactivity, synthetic utility, low cost, abundance, and non-toxic nature [
1,
2,
3,
4,
5,
6]. HVIs are involved in a multitude of reactions such as: reductive elimination, ligand exchange, oxidative addition, and ligand coupling [
7,
8]. The three-center four-electron bonds (3c–4e) in HVI are weak and polarizable, which is valuable in synthetic organic chemistry, as they can exchange leaving groups or accept electrophilic/nucleophilic ligands depending on their surroundings [
9]. Despite such utility, there are still unknowns regarding the intrinsic bonding nature of HVIs and hypervalency in general. Though iodine is a halogen, it behaves like a metal; it is the heaviest non-radioactive element of the periodic table and is the most polarizable halogen [
10,
11]. Because of its diffuse electron density (van der Waals (vdW) radius of ca 2 Å), iodine is a good electron donor, but can also serve as an electron acceptor [
12,
13]. Iodine is not known to participate in d-orbital or
-interactions, though this could be further investigated [
14,
15].
HVIs commonly exist in the oxidation states 3, 5, and 7, which support 10, 12, and 14 valence electrons, respectively [
16]. Most common are the oxidation states 3 and 5, which are referred to as
and
-iodanes [
17].
-iodanes form distorted T-shaped molecular geometries, while
-iodanes generally prefer square pyramidal geometries, as confirmed through both X-ray crystallography and computational studies [
18,
19]. These somewhat unusual molecular geometries are the result of the pseudo-Jahn–Teller effect [
20]. The atoms that make up the “T” in
-iodanes form improper dihedrals and non-ideal bond angles. The causes of these angular and dihedral deviations are unknown, but have been related to the anisotropic nature of the electronic density distribution in iodine [
21,
22,
23,
24,
25]. Bader et al. showed that the vdW radius in iodine is larger at the equatorial position than at the axial position [
26,
27,
28]. This supports the observation that electronegative ligands favor the axial positions in iodine [
29].
Hypervalency has been defined in several ways: Musher characterized main group elements in higher oxidation states as hypervalent [
30]. A successful concept employed to explain hypervalency without involving d-orbitals is the formation of multi-centered electron-deficient bonds [
31,
32,
33]. In this context, the 3c–4e bond model of Pimentel–Rundle is especially useful. According to this model, three atoms linearly align, each of which contributes an atomic orbital to form three molecular orbitals; a bonding orbital, a non-bonding orbital, and an antibonding orbital. Since only four electrons are involved, the antibonding orbital is unoccupied. As a result, two bonds share a single bonding electron pair (i.e., they have a fractional bond order of 0.5). The formation of two or more electron-deficient bonds allows hypervalent compounds to have higher oxidation states without necessarily expanding their octets. A direct consequence of this model is that the 3c–4e bond is expected to be substantially weaker than the two-center two-electron (2c–2e) bond in a given hypervalent molecule. Even though there are various works showing that d-orbital contributions to hypervalent bonds (HVB) are minimal, many chemistry text books still make use of the idea of an extended octet and the formation of spd-hybrid orbitals to explain HVB [
34]. There is a strong overlap between the concepts of fractional bond order, the 3c–4e bond, and the halogen bond (XB) in trihalides, which are considered prime examples of 3c–4e bonding, but also strong XB [
35,
36]. A formal definition of XB is given in the following paragraph.
3c–4e HVI bonding (3c–4e HVIB) draws comparisons to the secondary bonding interaction due to the weak bond strength, high reactivity, and long internuclear distances exceeding covalent bond lengths [
37]. 3c–4e HVIB also shares similarities with non-covalent interactions, along with hydrogen bonding [
38,
39,
40], XB [
35,
41,
42,
43], pnicogen bonding [
43,
44,
45], chalcogen bonding [
43,
46], and tetrel bonding [
47]. XB is a non-covalent interaction between an electrophilic halogen (X) and a nucleophile with a lone pair (lp(A)) of donating electrons [
42,
43]. For the remainder of this work, we will express lone pairs as (lp).The nucleophile/halogen acceptor (A), donates electrons to the antibonding (
*(XY)) orbital of the halogen donor (Y) [
48,
49]. XB is also known to have an X–A distance that is shorter than the sum of the vdW radii with Y–X–A angles close to 180 degrees [
35,
41,
50,
51]. Because of the obvious similarities between 3c–4e HVIB and XB, it has been argued that HVB should not be considered as a special bonding class [
52,
53]. On the other hand, the term hypervalency has been widely accepted by the chemistry community, and therefore, its continuous use has been advocated [
54]. Based on this controversy, we decided to delve deeper into the bonding nature of
-iodanes.
In this work, we investigated the intrinsic nature of HVIB in
-iodanes and its relation to XB, 3c–4e bonding in trihalides, and covalent bonding to determine if there is a smooth transition between these interactions. Additionally, we studied the role equatorial ligands play in strengthening the 3c–4e bond in
-iodanes, as well as substituent effects in the axial ligands. We utilized density functional theory (DFT), vibrational spectroscopy, quantum theory of atoms in molecules (QTAIM) combined with the Cremer–Kraka criterion for covalent bonding [
55,
56,
57], and the natural bond orbital (NBO) analysis to characterize the nature and the intrinsic strength of HVIB. This investigation was rationalized by studying a diverse group of 34 HVI compounds shown in
Figure 1, including known chemical compounds complemented by some model compounds. The remainder of this work is presented as follows: data, results, and discussion are presented in
Section 2;
Section 3 gives a description of the computational methods utilized; and
Section 4 gives conclusions, the outlook, and future goals.
3. Computational Methods
DFT was utilized to optimize molecular geometries and to calculate for each stationary point molecular vibrational frequencies including the L-modes of Konkoli and Cremer [
60,
61,
62] and the determination of local mode force constants (
), NBO charges, electron densities
, and energy densities
; where
is a bond critical point. Each stationary point was confirmed as a minimum by absence of imaginary normal mode frequencies. Available experimental geometries for the ICl
dimer, IF
, IF
, dichloroiodobenzene (PhICl
), and diacetoxyiodobenzene (PhI(OAc)
) [
18,
63,
64,
65,
66] were used to gauge the accuracy of the DFT calculations. Experimental and calculated geometries using different model chemistries for this set of compounds are compared in
Table A1 and
Table A2 (See
Appendix A). We initially employed Grimme’s Rung 5 double hybrid density functional B2PLYP [
67] and Dunning’s cc-pVDZ basis set [
68,
69,
70,
71] with a tight convergence criterion and an ultra-fine integration grid. The B2PLYP functional combines the generalized gradient approximation exchange functional of Becke [
72,
73] and the Lee–Yang–Parr correlation functional [
74] with exact Hartree–Fock exchange and Møller–Plesset perturbation theory [
75,
76,
77,
78] of second order (MP2) [
79,
80,
81]. This functional has shown close agreement between calculated and experimental geometries and vibrational frequencies for heavy atoms [
82,
83]. However, for our set of molecules, the cc-pVDZ basis set did not produce the desired accuracy (
Table A1 and
Table A2), and the B2PLYP/aug-cc-pVTZ level of theory became computationally expensive. The combination of MP2 and a relatively small double-zeta basis set is known to provide a fortuitous cancellation of error [
84,
85]. MP2 overestimates correlation energy, but this is compensated by the cc-pVDZ basis set [
86]. Therefore, we tested MP2/cc-pVDZ for reducing the computer time. However, results calculated at this level of theory gave less accurate results than
B97X-D/aug-cc-pVTZ [
87,
88], while calculations at the B2PLYP/Def2TZP level of theory led to inaccurate results in several cases. For Br and I, scalar relativistic effects were assessed by using effective core potentials (ECPs) in combination with the Dunning basis sets [
89,
90].
Although geometries are first order properties and therefore less sensitive to the level of theory, B2PLYP/aug-cc-pVTZ and
B97X-D/aug-cc-pVTZ calculations turned out to be in closest agreement with experimental data, while for a small subset of compounds, close agreement between
B97X-D/aug-cc-pVTZ and CCSD(T)/aug-cc-pVTZ was obtained (
Table A3). To further rationalize these results, gauge-independant atomic orbital (GIAO) magnetic shielding tensors [
91,
92,
93,
94,
95] were calculated and isotropic shielding constants were converted into chemical shifts utilizing the linear regression method of Tantillo et al. for PhICl
, PhI(OAc)
, and 1-Hydroxy-1,2-benziodoxol-3(1H)-one [
96,
97,
98,
99,
100,
101]. This method requires the calculation of isotropic magnetic shielding tensors for a test set of molecules at a given level of theory (in our case,
B97X-D/aug-cc-pVTZ and B3LYP/aug-cc-pVTZ), plotting the raw calculated isotropic value against experimental NMR chemical shifts, and using the following relationship to develop an equation for calculating chemical shifts (
Figure A1):
where
is the derived chemical shift and
is the calculated isotropic magnetic shielding tensor. The margin of error for proton-NMR chemical shifts turned out to be 0.24–6.91% for the B3LYP functional and 0.19–5.81% for the
B97X-D functional (
Table A4) [
18,
63,
102]. Although both
B97X-D and B3LYP gave satisfactory and similar calculated chemical shifts,
B97X-D gave more accurate geometries and frequencies.
Based on these findings, the
B97X-D/aug-cc-pVTZ level of theory was chosen for this study due to its displayed ability to predict accurate first and second order experimental properties in HVI molecules in addition to the previous findings of Oliveira et al. that this level of theory is suitable for the detailed analysis of XB [
41].
Vibrational spectroscopy was applied to quantify the intrinsic strength of HVIBs. Chemists have utilized vibrational spectroscopy to obtain information about the electronic structure of molecules and their framework of bonds. However, normal vibrational modes cannot be used as a direct bond strength measure because they are delocalized due to electronic and mass coupling, a fact that often has been overlooked [
103,
104]. The electronic coupling is eliminated by solving the Wilson equation of spectroscopy [
105] and transforming to normal coordinates. Konkoli and Cremer showed that the remaining mass (kinematic) coupling can be eliminated by solving a mass-decoupled equivalent of the Wilson equation, leading to local vibrational modes, which are associated with internal coordinates
such as bond lengths, bond angles, and dihedral angles [
60,
106]. Zou and Cremer verified that there is a one-to-one relationship between local and normal vibrational modes through an adiabatic connection scheme (ACS) [
107,
108,
109], allowing a normal mode decomposition into local mode contributions [
44,
110,
111] and, as such, the detailed analysis of a vibrational spectrum. This is of particular value, given the fact that L-modes can be applied to both calculated and measured spectra [
61,
112].
Another important feature of L-modes is the direct relationship between the local stretching force constant (
) of a chemical bond and its intrinsic strength [
113]. This has enhanced our knowledge about chemical bonding and the often overlooked, but highly important weak intermolecular interactions, providing a wealth of new insight into: (i) covalent bonding [
113], stretching from peculiar cases of reversed bond length-bond strength relationships [
114,
115], to a new design recipe for fluorinating agents [
116]; (ii) weak chemical interactions including hydrogen bonding [
117,
118], XB [
35,
41,
42], pnicogen bonding [
43], chalcogen bonding [
50], weak interactions in gold clusters [
119], as well as non-classical hydrogen bonds in boron–hydrogen
interactions [
120,
121]. In addition, new electronic parameters and rules were derived [
122,
123,
124].
When comparing a larger set of
, the use of a relative bond strength order (BSO
) is convenient [
103,
104]. The BSO
of a bond is obtained by utilizing the extended Badger rule [
103,
104,
125] according to which BSO
n is related to
by a power relationship, which is fully determined by two reference values and the requirement that for a zero-force constant, the BSO
value becomes zero:
The constants
a and
b are calculated from
values of two reference compounds with known BSO
values
n and
n via:
and,
In this work, we chose as reference compounds FI and IF
representing BSO
values of one and 0.5, respectively, guided by the corresponding Mayer bond orders [
126] of 0.940 and 0.543 evaluated at the
B97X-D/cc-pVTZ level of theory. More than 50% of iodine bonds in this work include an atom from the second period. This renders the FI/IF
reference system ideal (a second period atom bound to iodine), in addition to providing a spectrum with a full 3c–4e bond from a trihalide (IF
) on the one end and a full covalent bond (FI) on the other end.
Using the
of 1.913 mdyn/Å for IF
and 3.953 mdyn/Å for FI (
B97X-D/cc-pVTZ level of theory), the constants
a and
b in the power relationship Equation (
3) were determined to be
a = 0.269 and
b = 0.955, leading to:
Because the chosen reference system was designed for 3c–4e interactions particular to this study, a scaling procedure was used to obtain appropriate BSO
values for covalent I–C interactions between the equatorial ligands and the central iodine. The equatorial bonds are fully-formed single bonds, but the C–I bond is much less polar and weaker than the I–F bond used as a reference. We calculated
= 2.557 mdyn/Å for the I–C bond in iodobenzene. From Equation (
6), we calculated BSO
= 0.659. The scaling factor was obtained by setting
n = 1 for this I–C bond. The scaling factor is 1/0.659 = 1.517, which was applied to BSO
of all equatorial I–C bonds. Multiplying the scaling factor through Equation (
6) provided a new BSO
equation for assessing the strength of the equatorial I–C bonds in this study:
The Cremer–Kraka criterion was applied to assess the covalent nature of HVIB [
42,
55,
56,
77,
127]. According to this criterion, a covalent bond between two atoms A and B is defined by (1) the existence of a zero-flux surface and bond critical point
between atoms A and B (
necessary condition) and (2) a negative and thereby stabilizing local energy density
(
sufficient condition).
will be close to zero or positive if the interaction between A and B is non-covalent, that is electrostatic or of the dispersion type.
is defined as:
where
is the kinetic energy density (always positive, destabilizing) and
is the potential energy density (always negative, stabilizing). In addition to the established Cremer–Kraka criterion, a molecular fragmentation scheme for estimating electron density shifts has recently emerged as a potential tool for the qualitative investigation of non-covalent interactions at low computational cost [
128].
L-modes was carried out with the program COLOGNE2018 [
129], and Mayer bond orders were determined with the program ORCA [
130]. NBO populations were computed using NBO 6 [
131,
132,
133,
134]. The electron density analysis, in particular the calculation of electron density at the bond critical point (
) and
, was performed with the program AIMAll [
135,
136]. All DFT calculations were carried out with GAUSSIAN16 [
137].
4. Conclusions
In this work, we quantified the intrinsic bond strength and bonding nature of a series of HVI compounds through vibrational spectroscopy. Use of DFT in this work was rationalized by testing several levels of theory against first and second order experimental properties of a small set of known HVI reagents. The computed set of 34 HVI molecules was then compared to XB, 3c–4e bonding, and covalent bonding in terms of BSO
,
,
,
, and NBO charges. Recently, Politzer and coworkers [
138] showed that by substituting a ligand in trihalides with a negative point charge, the positive electrostatic potential at the polarized
-hole collinear to the point charge correlates qualitatively well with the interaction energy; substantiating the key role played by electrostatics, which is also reflected in the atomic charge distribution (see
Figure 1) and can be rationalized in terms of the 3c–4e model. The more negative charge at the ligands Y and A compared to the central iodine is due to the presence of a node at the center of the occupied non-bonding orbital [
139]. This charge separation is responsible for the lower covalent character of 3c–4e bonds compared to a classical 2c–2e bond. Politzer and coworkers proposed the existence of a continuum between non-covalent and covalent bonds, the latter being a result of an increased degree of polarization [
138]. Our results do also suggest the existence of such a continuum, but whether covalency can be seen as a degree of polarization is still disputable, especially in view of Ruedenberg’s description of covalent bonding, where energy lowering is a result of the complex interplay of kinetic and potential energy contributions [
140,
141]. The 3c–4e bonds in HVI share properties with XB, but are more closely related to the 3c–4e bonds in trihalides or covalent bonding in extreme cases. The equatorial 2c–2e HVI bond is stronger than comparable 3c–4e bonds (bonds involving the same ligands like in IF
) and is more closely related to a covalent bond. Our results support the following transition: XB < 3c–4e bond in trihalides < 3c–4e bond in HVI < 2c–2e bond in PhIF
< covalent bond. When comparing the difference (equatorial ligands) between trihalides and
-iodanes, we found that the 3c–4e HVIB is strengthened by the equatorial ligand by comparing IF
, PhIF
, and IF
. The equatorial ligand contributes significantly in pulling electron density from the central I, allowing for more polar interactions. Thus, highly electronegative ligands at the equatorial position will form strong interactions, as will axial ligands in such a case. We also found that axial ligands in HVIs have a minimal direct effect on one another in terms of NBO charge analysis, but do play a role in altering charge on the central I. Substituent effects in HVI can alter bond strength in both axial ligands and the equatorial ligand, particularly when F atoms are involved as ligands. The five functional groups studied here play a bond-strengthening and -polarizing role in the following order: OCO > OC(CH
)
> OH > CN > NH
, with OH and OC(CH
)
being partially interchangeable. In terms of
, we found a strong linear correlation with BSO
n.
becoming more negative correlates to an increase in bond strength. Furthermore, large
stabilization in the bonding region correlates to the increased covalent character of a bond. Finally, we found the 3c–4e bond concept to be a valuable descriptor in terms of the linear portion of
-iodanes.
Future goals are to utilize L-modes and the analysis of the electrostatic potential to explain why the T-shaped molecular geometry in
-iodanes contains improper dihedrals and non-ideal bond angles. We also plan to investigate 3c–4e bonding and intramolecular HB in a series of HVI reagents utilizing L-modes and to explore the chemical reactivity of HVI compounds utilizing the unified reaction valley approach developed in our group [
103,
142,
143,
144]. In addition, we will perform a conformational and geometrical study of a series of novel HVI monomeric materials with a strong potential of forming useful polymers [
145,
146].