Sharon E. Worthington, Christopher J. Cramer*, Frederic J. Dulles,
and Joey W. Storer

Department of Chemistry and Supercomputer Institute
University of Minnesota
207 Pleasant St. SE
Minneapolis, MN 55455-0431

Abstract: Aromatic nitrenium ions have been postulated as carcinogenic species derived from the in vivo catabolism of aromatic amines. Nitrenium ions incorporate a disubstituted, positively charged nitrogen atom, and may thus have either singlet or triplet spin multiplicity. The former spin state is implicated in carcinogenesis. We have calculated singlet-triplet gaps for a number of small to medium-sized nitrenium ions, in particular nitrenium, imenium, methylnitrenium, aziridenium, and phenylnitrenium; similar calculations have also been done for the analogous isoelectronic carbenes and for other valence isoelectronic analogs in the XH2 series. Several levels of theory have been employed, including multi-configuration self consistent field, multi-reference configuration interaction, coupled-cluster theory, and density functional theory. The electronic and steric factors that influence the singlet-triplet gaps for these species are discussed, and the levels of theory are compared. Density functional theory appears to hold promise for larger, biologically relevant nitrenium ions.

Table of Contents

Why nitrenium ions?


Aromatic amines

Theoretical testing ground

Expanded Summary:

Nitrenium ions (R-N-R'+) are reactive intermediates; they act as carcinogens when they derive from aromatic amine catabolism.[1-7] Metabolic activation of the aromatic amine is postulated to proceed via oxidative formation of the corresponding arylhydroxylamine esters.[1-9] The reactive nitrenium ion is produced upon heterolysis of the N-O bond. The divalent nitrenium ion, which is isoelectronic to a carbene, then goes on to covalently modify DNA. Experimental studies on the lifetimes of nitrenium ions in aqueous solution suggest that in certain instances they may be sufficiently long-lived to diffuse to reach genetic material.[10-12]

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Nitrenium ions are isoelectronic to carbenes

- Smaller bond angle at hypovalent center.
- Favored by electronegative and/or pi-donating substituents.
- Paired electrons are in low-energy orbital, but suffer from Coulomb repulsion.
- Challenging because it requires at least a two-configuration wavefunction.
- Larger bond angle at hypovalent center.
- Favored by electropositive and/or pi-accepting substituents.
- Unpaired electrons enjoy exchange and less Coulomb repulsion.

Expanded Summary:

Electronic structure calculations for hypovalent molecules (like nitrenium ions and carbenes) are challenging for theoretical methods; in particular, prediction of the singlet-triplet splitting (S-T gap) requires a subtle balancing of several energetic effects.[13,14] There is inherent difficulty in treating open- and closed-shell systems on an equal footing, in part because they have different numbers of exchange interactions. Additionally, accurately accounting for electron correlation is very important in these molecules that have rather low-lying virtual orbitals, i.e. single-determinant wavefunctions are usually inadequate.

We have been intrigued by the possibility that density functional theory (DFT) might be particularly suited to overcome some of these challenges. DFT includes correlation within an SCF formalism and additionally has the desirable properties that in its spin-polarized form it generally suffers minimally from spin contamination.[15,16] And, of course, it is extremely rapid by comparison to multiconfigurational techniques.[17,18] DFT is not in general applicable to molecular excited states, but has been shown to be formally applicable to the lowest energy electronic state for each symmetry irreducible representation for a given system.[19-21] There is thus no ambiguity in calculating S-T gaps for systems in which the ground state symmetries for the two multiplicities are disparate, like those considered here.

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Nitrenium and methylene

Table 1. 1A1 - 3B1 Singlet-Triplet gaps (kcal/mol) for nitrenium ion and methylene

Level of Theory                Nitrenium   Methylene

MCSCF(6,6)/B1                    29.7        11.9
MRCISD/B1//MCSCF(6,6)/B1         32.0        11.5 
MCSCF(6,6)/B2                    29.2        11.3
MRCISD(Q)/B1//MCSCF(6,6)/B2      30.5        10.0
BVWN5/B1                         29.5         9.8
BLYP/B1                          31.2        10.5
BVWN5/B2                         29.3         9.6
BLYP/B2                          30.6        10.0

B1 = correlation-consistent polarized valence-double-zeta (Dunning)
B2 = correlation-consistent polarized valence-triple-zeta (Dunning)
B3 = correlation-consistent polarized valence-quadruple-zeta (Dunning)

Expanded Summary:

There is good agreement between MCSCF and DFT for the energetic and geometric details of nitrenium and methylene. A more detailed discussion may be found in the next section, where these two molecules are considered in the context of nine valence isoelectronic congeners. As will be noted in later sections as well, the bond angle at the divalent center in the triplets is generally predicted to be roughly 5 degrees wider at the DFT level than is the case at the MCSCF (or other MO theoretical) levels.

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Other Valence Isoelectronic XH2 Results


Theoretical Trends:

Singlet-Triplet Gaps: Competition between three trends

  1. Coulombic attraction favors singlet going from anion to cation.
    Due to s character in (lone-pair-like) 3a1 orbital vs. (non-bonding) 1b1 orbital.

  2. Electron-electron repulsion favors the triplet going from anion to cation.
    Due to contraction of orbital size.
    This effect is dominant in the first row.
    Dominates trend going from boron to nitrogen.

  3. Going first to third row, orbitals generally become more diffuse.
    This counters repulsion effect above.
    Thus the first trend becomes evident.
    Note in second and third rows the singlet is uniformly favored.

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Imenium and vinylidene

Table 2. 1A1 - 3B2 Singlet-Triplet gaps (kcal/mol) for imenium ion and vinylidene

Level of Theory                Imenium     Vinylidene

CCSD(T)/B1                      -58.4       -44.3
MCSCF(10,10)/B1                 -54.7       -38.0 
BVWN5/B2                        -61.5       -47.9
BLYP/B2                         -61.5       -48.3
BVWN5/B2//MCSCF(10,10)/B1       -61.1       -47.6
BLYP/B2//MCSCF(10,10)/B1        -61.2       -48.0
BVWN5/B3//MCSCF(10,10)/B1       -61.2       -48.0
BLYP/B3//MCSCF(10,10)/B1        -61.3       -48.3

B1 = correlation-consistent polarized valence-double-zeta (Dunning)
B2 = correlation-consistent polarized valence-triple-zeta (Dunning)
B3 = correlation-consistent polarized valence-quadruple-zeta (Dunning)

Expanded Summary:

For these systems, the S-T gap is quite large. The high s character of the "lone pairs" on the hypovalent atoms make these very low energy orbitals. Indeed, the triplets are formed by promotion out of pi orbitals, not out of sigma lone pairs. That is, the electronic state of the triplet is 3B2, not 3B1.

The high energies of the triplet states may give rise to the larger discrepancies observed for these cases between CCSD(T), MCSCF, and DFT. The DFT gaps are relatively insensitive to the theoretical level at which the geometries were optimized. They also appear to be well converged with respect to basis set. The CCSD(T) gaps are about 3 kcal/mol smaller, and the MCSCF numbers are roughly that much smaller again. In this instance, the MCSCF calculations are full valence complete active space, so it seems unlikely that multirefrence configuration interaction would account for the 6 to 7 kcal/mol difference between MCSCF and DFT. Another possibility is that the MCSCF and DFT numbers would agree more closely were the MCSCF calculations to be carried out at the valence-triple-zeta level. The observed quantitative discrepancies for such a large gap are probably of minimal import.

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Methylnitrenium and ethylidene

Table 3. 1A1 - 3B1 Singlet-Triplet gaps (kcal/mol) for 
         methylnitrenium ion and ethylidene

Level of Theory                Methylitrenium     Ethylidene

CCSD(T)/B1//HF/B1                  13.2             7.3
MCSCF(6,6)/B1//HF/B1               14.9             8.0 
BVWN5/B2//HF/B1                     6.8             4.3
BLYP/B2//HF/B1                      6.9             4.2
BVWN5/B3//HF/B1                     6.9             4.1
BLYP/B3//HF/B1                      6.8             3.9

B1 = correlation-consistent polarized valence-double-zeta (Dunning)
B2 = correlation-consistent polarized valence-triple-zeta (Dunning)
B3 = correlation-consistent polarized valence-quadruple-zeta (Dunning)

Expanded Summary:

The S-T gaps in these molecules are reduced relative to the parent XH2 systems because of the improved pi donor characteristics of a methyl group compared to a hydrogen atom.[13] There are significant quantitative discrepancies between the various levels of theory. The system is rendered somewhat difficult to analyze, however, because the singlets are quite unstable to rearrangement, forming iminium and ethylene, respectively.[13,23-25] In this instance, all geometries were taken from Hartree-Fock optimizations (at which level the singlet is stationary), and it does remain disturbing that there are disagreements on the order of 6 to 7 kcal/mol between MCSCF and CCSD(T) on the one hand, and DFT on the other. Again, the DFT numbers appear well converged with respect to basis set, but more work remains to be done for the other levels. Choice of active space is also an issue in these MCSCF calculations. A full valence space calculation would be (14,14); such a calculation is impractical to carry out. As such, MRCI corrections to the smaller MCSCF calculation might have a moderate effect on the gaps. Work remains to be done in the analysis of this system.

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Aziridenium and cyclopropylidene

Table 4. 1A1 - 3B1 Singlet-Triplet gaps (kcal/mol) for aziridenium ion 
         and cyclopropylidene

Level of Theory                Aziridenium     Cyclopropylidene

CCSD(T)/B1//MP2/B1                10.8             -13.3
MCSCF(8,8)/B1                     11.4             -13.0 
BVWN5/B2                           9.4             -13.1
BLYP/B2                            8.4             -12.9
MCSCF(8,8)/B2//MCSCF(8,8)/B1      10.4             -14.1
BVWN5/B2//MCSCF(8,8)/B1           11.0             -13.6
BLYP/B2//MCSCF(8,8)/B1            10.7             -13.5
BVWN5/B3//MCSCF(8,8)/B1           11.1             -13.7
BLYP/B3//MCSCF(8,8)/B1            10.8             -13.6

B1 = correlation-consistent polarized valence-double-zeta (Dunning)
B2 = correlation-consistent polarized valence-triple-zeta (Dunning)
B3 = correlation-consistent polarized valence-quadruple-zeta (Dunning)

Density Plot: The density difference map for the 1A1 and 3B1 states (0.01 au contour intervals, dashed lines are negative, solid lines are positive) obtained by subtraction of the former from the latter for aziridenium (left) and cyclopropylidene (right) taken in the plane of the three membered ring. The hypovalent atom is at the right in each case.

Expanded Summary:

Geometries: Bonds between the ring methylenes and the hypovalent center are predicted to be longer at the MCSCF level of theory than at the DFT level (this trend is maintained for identical basis sets). A similar trend is observed for the triplets. A far more remarkable result is the surprisingly long bond between the two methylene groups: for cyclopropylidene this bond is over 1.6 Å, while for aziridenium it is nearly 1.8 Å! The greater lengths calculated using DFT may be in part because DFT favor bonds angles at the divalent centers in triplets that are about 5 degrees wider than predicted by MCSCF methods, as noted above.[22]

These long bonds are explained beginning with analysis of the electronic structures of the singlets. Most carbene and nitrenium ion singlets are typically regarded as having a dominant configuration where the highest occupied molecular orbital (HOMO) corresponds to a lone-pair-like orbital in the plane of the substituents; the perpendicular p orbital lies empty. In these molecules, however, the "back" of the lone pair orbital has considerable overlap with the bonding orbital connecting the two methylene groups at the base of the three-membered ring. Hybridization thus occurs to create a HOMO with considerable amplitude in the C-C bonding region. Hybridization occurs to a larger degree for the nitrenium cation because of its lower energy lone pair. Since formation of the triplet involves promotion of one electron from the HOMO to the lowest unoccupied molecular orbital (LUMO), some C-C bonding density is removed. Mulliken population analysis of aziridenium at the MCSCF level supports this view; a shift in charge of 0.17 electrons from the two methylene groups to the nitrogen atom occurs on going from the singlet to the triplet. The above density plot illustrates this point with a physical observable, namely density difference maps in the planes of the rings. Both maps illustrate a loss of density from the C-C bonding region.

Energies: CCSD(T), MCSCF, and DFT calculations all agree quite well for both these systems. S-T gaps calculated at the DFT level are sensitive to molecular geometry. There are insufficient experimental data to judge whether DFT or MCSCF (or MP2) geometries are more accurate. The DFT predicted S-T gaps are apparently well converged with respect to basis set. The gap for cyclopropylidene is in good accord with that predicted by Honjou et al. at the SDQCI/DZP//HF/DZP level, -12.4 kcal/mol.[26]

Finally, placing the isoelectronic carbene or nitrenium centers in a 3-membered ring has similar effects on the calculated S-T gaps relative to the parent XH2 systems. Relative to CH2 and NH2+, with experimentally determined S-T gaps of 9.00 +/- 0.01[27] and 30.1 +/- 0.2[28] kcal/mol, respectively, in each case the singlet is stabilized by roughly 20 kcal/mol. Most of this stabilization probably arises from the small bond angles enforced by the ring at the divalent center. Disubstituted carbene and nitrenium triplets generally have much wider bond angles than their respective singlets since they allocate more s character to their bonding orbitals. As shown in the density difference maps, the deficiency of bonding density in the region between the two methylene groups coincides with the triplets being able to open to the calculated MCSCF bond angles at the hypovalent centers.[29]

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Phenylnitrenium and phenylcarbene

Table 5. 1A' - 3A" Singlet-Triplet gaps (kcal/mol) for phenylnitrenium  
         and phenylcarbene

Level of Theory                Phenylnitrenium     Phenylcarbene

PMP4/B1                             -26.8             2.4
BVWN5/B2                            -21.2             4.3
(Expt. XH2)                          30.1             9.0
(Effects of Ph substitution)        -51.3            -4.7

B1 = correlation-consistent polarized valence-double-zeta (Dunning)
B2 = correlation-consistent polarized valence-triple-zeta (Dunning)
B3 = correlation-consistent polarized valence-quadruple-zeta (Dunning)

The results underline the dissimilarities found in these isoelectronic systems.

MCSCF calculations were prohibitively expensive for the size of the active space deemed necessary to fully describe the systems.

In particular, orbitals with significant contributions from the nitrogen in phenylnitrenium were low enough in energy to interact with the sigma framework. Perturbation theory is at a disadvantage because spin contamination is so high (S**2 = 2.7) in the UHF triplet.

DFT generally has much lower levels of spin contamination in open shell systems relative to UHF.

Expanded Summary:

Calculations at the BVWN5/B2//BVWN5/B1 level of theory (294 basis functions) were carried out for these two aromatic systems.[16] Key results are (i) the predicted S-T gaps (including zero-point corrections) are -21.2 and 4.3 kcal/mol for phenylnitrenium ion and phenylcarbene, respectively. (ii) Triplet phenylcarbene is a planar species and has a predicted barrier to rotation about the exocyclic bond of 2.7 kcal/mol; the planar phenylnitrenium triplet, on the other hand, is a rotational transition state (TS) structure. The local minimum has instead the N-H bond perpendicular to the aromatic ring with a rotational barrier of 1.8 kcal/mol. Although the DFT Hessian for the planar phenylnitrenium structure has one negative force constant, high spin contamination[30] causes the UHF Hessian matrix to have none, a qualitatively incorrect result. (iii) Singlet phenylcarbene is a planar species with a predicted barrier to rotation about the exocyclic bond of 11.0 kcal/mol--singlet phenylnitrenium does not rotate about this bond, but instead inverts through nitrogen with a barrier of 26.0 kcal/mol.

The large singlet stabilization by phenyl substitution for nitrenium derives from the considerably enhanced pi-acceptor characteristics of a positively charged nitrogen compared to a neutral carbon--this differential pi-accepting ability is more manifest in the singlets, where an empty p orbital serves as acceptor, than it is in the triplets. Singlet stabilization for the weakly electron-deficient carbene is by comparison smaller in magnitude[13]--one measure of this is that while singlet phenylnitrenium prefers to isomerize by inverting (like an imine), singlet phenylcarbene breaks pi conjugation by rotating out of plane. The orbitals displayed in the picture accompanying the title illustrate these points exactly. In particular, the coefficient of the p orbital on the nitrenium nitrogen is quite large for the lowest energy pi orbital of phenylnitrenium. In phenylcarbene, on the other hand, there is very little contribution from the carbene p orbital to this MO. The next higher energy MO is similarly differentiated.

The singlet is the lower energy spin state for phenylnitrenium. However, substitution of the aromatic ring with pi-acids and/or sigma-donors can decreases the gap between the pi LUMO and the sigma HOMO, and can thereby make the triplet spin state accessible.[16,31,32,33,34] DFT should be a promising method for modeling this effect.

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Specific molecules

* Greater orbital contraction at the cationic hypovalent center increases triplet stabilization compared to uncharged or anionic centers.

* The increased s-character of sigma-orbitals at sp hypovalent centers increases singlet stability; this effect is larger for the nitrenium ions than for carbenes.

* Changes in the S-T gap arising from placing isoelectronic hypovalent centers into 3-membered rings are energetically similar (about 20 kcal/mol differential destabilization of the triplet)--there is less energetic similarity between carbenes and nitrenium ions in other substituted systems.

* Unusually long bonds between the two methylene groups in the 3-membered rings are attributed to cationic one-electron bonds.

* Hyperconjugative stabilization of singlets is much more important for (cationic) nitrenium ions than for carbenes; this is dramatically illustrated in the case of phenyl substitution.

Implications for modeling aromatic amine carcinogenesis:

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We follow throughout the standard convention that x/y//z/w denotes a calculation with method x and basis y at a geometry optimized with method z and basis set w. Singlet and triplet carbenes and nitrenium ions were optimized at a variety of theoretical levels, employing Hartree-Fock (HF, restricted for singlets, unrestricted for triplets), second-order perturbation theory (MP2), and multiconfiguration self-consistent-field (MCSCF) techniques. MCSCF active spaces were constructed from the two "lone-pair" orbitals on the hypovalent atoms and the heavy-atom-heavy-atom bonding/antibonding orbitals. When practical, the active space was expanded to include the entire valence space. All MCSCF calculations are of the complete active space (CAS) variety.[35,36] Employed basis sets include the correlation-consistent polarized valence double-zeta (cc-pVDZ, hereafter referred to as basis set B1) and the correlation-consistent polarized valence triple-zeta (cc-pVTZ, hereafter B2) basis sets of Dunning and Woon [37]. For Ga, Ge, and As, the relativistic effective core potential basis of Stevens et al.[38] was used with either 10-electron (Ga) or 28-electron cores (Ge, As). This basis set has a double-zeta -41G contraction for its valence orbitals, to which was added a d function (exponent 0.207) and additional s (exponent 0.0205) and p (exponent 0.011) functions; the 6-31G** (GeH2 and AsH2+) and 6-31++G** (GaH2-) bases[39,40] were used for hydrogen atoms in these third-row dihydrides.

Geometry optimizations were also carried out using spin-polarized density functional theory (DFT) with both local and gradient-corrected functionals. Several combinations of functionals were employed. BVWN5 combines Becke's non-local exchange functional[41] with a local correlation functional of Vosko, Wilks, and Nusair.[42] BLYP, replaces the correlation functional with a local/non-local alternative developed by Lee, Yang, and Parr.[43] In the XH2 series, BP replaces the correlation functional with a non-local alternative developed by Perdew.[44] Carbene and nitrenium ion optimizations at the DFT level were carried out using the B1 and B2 basis sets. However, for the isoelectronic XH2 series, DFT calculations employed the uncontracted Slater-type-orbital triple-zeta basis sets of Snijders et al.[45] with one p and one d function added to hydrogen and one d and one f function added to the heavy atoms. Effective core potentials for the third row atoms were used only for the innermost 18 electrons; 28-electron core potentials (i.e., including the 3d block) correlated about 1 kcal/mol less well with the MRCI results. Auxiliary s, p, d, f, and g functions were used for all nuclei to fit the molecular density during the SCF procedure. These levels of theory, with and without the Perdew non-local correction, respectively, are referred to as NL(x)DA/TZ2P and NL(xc)DA/TZ2P in the figure above.

Stationary points were characterized by analytic frequency calculations where appropriate.

Spin contamination ranged from slight to severe in UHF triplet calculations. The worst case was phenylcarbene, for which = 2.7. With DFT, spin contamination was minimal, never exceeding 2.02.

Single point calculations were performed at various geometries using both larger basis sets (for DFT up to the valence quadruple-zeta cc-pVQZ[37] basis set with g functions removed, hereafter B3) and selected post-HF levels of theory. In particular, coupled-cluster calculations including all single, double, and perturbative triple excitations (CCSD(T))[46-49] out of the HF reference configuration were performed. Spin-projected fourth-order perturbation theory (PMP4) was also employed in the phenylnitrenium/phenylcarbene case. Finally, in some cases single point multireference configuration interaction calculations were performed considering all single and double excitations from the CASSCF reference space (core orbitals were frozen) into the space of all virtual orbitals (MRCISD). Corrections for quadruple excitations were sometimes estimated using the multireference analog of the Langhoff-Davidson correction [50] suggested by Bruna et al. [51]. This level of theory is referred to as MRCISD(Q).

HF, MCSCF, and MRCI calculations were carried out using the GAMESS program suite; [52] HF, MP2, CCSD(T), and DFT calculations were carried out using the GAUSSIAN 92/DFT program suite.[53] Some DFT calculations employed the Amsterdam Density Functional Program.[54]

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