C(subscript)S Point-Group Symmetry in Compound(s) I: Role of Overlap of Thiolate 3p(z) AO and Porphyrin a2u MO

These diagrams show the bent angle of the sulfur-iron "bond(s)" [the bond comprises the various molecular orbitals (MOs) that allow for pi-type bonding interactions between the sulfur 3p(x) and 3p(y) AOs with the 3d(xz) and 3d(yz) AOs of iron (or, rather, the MOs of non-protein-bound heme that those AOs of iron contribute to the formation of)] and sigma-type interactions (I'd define a sigma-type interaction, in this context, as being "overlap of quasi-cylindrical symmetry about the S-Fe axis," or something like that) of the 3p(z) AO of sulfur with the 3d(z2) AO of iron (or, rather, the MOs of heme that the 3d(z2) AO of iron has already participated in the formation of) and the a2u MO of heme [see Ogliaro et al., 2001: (http://www.ncbi.nlm.nih.gov/pubmed/11948872); Harris, 2001: (http://www.ncbi.nlm.nih.gov/pubmed/11738185)]. The bent angle of protein-cysteine-liganded heme and also in nitrosylhemes (Porphyrin-Fe-N=O) and in iron(II)-hemes bound to O2 (unless the Fe is bound to both oxygens of O2, in which case it's C2v symmetry, I think) causes those heme species to be assigned to the Cs point group, in which the only possible symmetry operations are the identity operation and the reflection about a sigma(h) plane of symmetry (I think I've drawn it correctly) [for a depiction of the bent angles of the Fe-N and N=O bonds, away from the traditionally-assigned z-axis that is perpendicular to the plane of the porphyrin ring, that characterize Cs point-group symmetry, see Novozhilova et al., 2006, p. 2100, Figs. 6 and 7: (http://www.ncbi.nlm.nih.gov/pubmed/16464112)(http://harker.chem.buffalo.edu/group/publication/370.pdf)]. The only other characteristic of Cs point group symmetry is that it's "abelian," and I forget what that is or maybe don't understand it. I'd need some more math to understand the symmetry stuff fully [for the point-group character tables, see: (http://www.webqc.org/symmetrypointgroup-cs.html)]. One begins with a flow chart and then goes from there. The angle is bent, in part, "because" the angle allows for the constructive overlap of the sulfur's 3p(z) AO with the a2u molecular orbital of heme.

The other main point is that there's potential for confusion in the nature of the interaction of sulfur's 3p(z) orbital with the a2u orbital, on the one hand, and, on the other hand, with the 3d(z2) AO of iron. I don't have time, at the moment, to get into this and refer to various articles, but one might say that one "reason" that the sulfur's "ungerade" 3p(z) AO's can exhibit constructive overlap with the "ungerade" a2u MO of heme [this would be symmetry-forbidden in the planar, isolated porphyrin ring that exhibits D4h point-group symmetry, by virtue of its nitrogen atoms being coplanar with the rest of the porphyrin ring (rather than "domed" out of the plane of the porphyrin ring, as in Cs and C4v point-group symmetries)], for example, is that the a2u MO of heme cease to really be "ungerade" in Cs point-group symmetry (and also in the C4v point group). The 3d(z2) AO and other AOs of the isolated ferryl moiety are "gerade," but those AOs and the a2u MO are neither gerade nor ungerade in Cs point-group symmetry, given that there's no inversion point in Cs point-group symmetry (there's a center of mass of heme, but no inversion point is assigned to it). They're all either a' or a'', in the Cs point group. Anyway, the change in the Fe-S bonding angle changes the symmetry point group that one assigns heme to and, more importantly, changes many of the interactions of the porphyrin-localized MOs of heme with the MOs of heme that are predominantly-localized to the FeO moiety, & it's interesting.

These are the conventions, evidently [conventions that few researchers actually follow, evidently, given that it seems like it would become problematic in computer models of MOs (http://vitalii.chemicalblogs.com/2_computational_chemistry/archive/40_conventions_for_symmetry_notations.html)] for the assignment of the principal axis (a C1 axis that makes the molecule nonaxial, given that one ends up back where one started after rotating the molecule 360 degrees, or 2pi radians, about the z-axis). The z-axis that I've drawn intersects 2 atoms (the maximum number), and the x-axis is perpendicular to the plane of the isolated, nonsubstituted porphyrin ring. The sigma(h) plane is a horizontal reflection plane, and the "lines" or axes of the Fe-S and Fe=O bonds are in the same plane (those lines are coplanar, and their plane is coplanar with the sigma(h) plane). The molecule is symmetric upon reflection through the sigma(h) plane but has no other axes of symmetry or planes of reflection. The assignment of a molecule to a point group is not determined by x, y, or z-axis assignments, but one has to assign x, y, and z axes (and, ultimately, a spherical coordinate system) in order to superimpose 3-dimensional plots of MO probability density functions (the orbital "shapes" are dictated by the probability that an electron will be within the region that is defined by the surface of the orbital at any given instant) onto the "coordinate-system-independent" C2, C3, etc., symmetry axes and planes of reflection. Evidently, that's part of the reason there can be confusion, as far as orbital symmetry assignments go. Researchers assign different coordinate systems, and that can, apparently, alter the MO symmetry assignments, across different articles, even when different groups of researchers are referring to precisely the same molecule, in precisely the same electronic state.



The reverse face of heme is shown in the second diagram:

Occupancies of "Frontier Orbitals" of Compound I; Nature of Nonbonding Electrons in the Fe=O Oxygen

These diagrams show the orbital occupations of a commonly-encountered electronic state of perferryl heme (the quartet A2u state of compound I), and many of the "orbital occupancies" are, in other iron(IV) or iron(III) hemes, the same as these or similar to these. The nature of the "lone pair" of oxygen has been a source of confusion in articles, and it's almost certainly a sigma-type bonding interaction of the 2p(z) atomic orbital of oxygen and the 3d(z2) atomic orbital of iron, with minimal 3d(z2) character and somewhat minimal bonding character (i.e., a "nonbonding MO"). [Also, I forgot to put it on the diagram, but the formal charge on oxygen, in this species, would be 6 - 2 (lone pair electrons) - 1 [nonbonding radical in pi*(xz or yz) MO] - 1/2(4 electrons in pi(xz) and pi(yz) bonding MO's) = +1. I think the formal charge can be +2 or something else, in some of the transition states, but that's not the actual charge. For example, in one of the major iron(II)-heme species, porphyrin-Fe(II)-O-O(-), the actual charge is about -0.2 and not the formal charge of -1, apparently [found experimentally, as reported in the abstract of Jensen et al., 2005: (http://www.ncbi.nlm.nih.gov/pubmed/15598490)(http://www.cicum.cup.uni-muenchen.de/ac/kluefers/homepage/L/BAC/heme1.pdf)]. With regard to the oxygen's lone-pair electrons, though, there may be some 2s character, too, and part of the confusion stems from the fact that, in dioxygen, the "nonbonding MO's" have predominantly 2s character and mix to only a minimal extent with the 2p(z) atomic orbitals of oxygen (they're not "sp3" orbitals) [this article discusses the absence of the supposed "rabbit ears," produced by sp3-hybridized, "lone-pair" MO's in the water molecule: Laing, 1987: (http://pubs.acs.org/doi/abs/10.1021/ed064p124), and this article includes a discussion of nonbonding MO's in general, I think: Hurst et al., 1999: (http://pubs.acs.org/doi/abs/10.1021/jp984565h)]. But other articles refer to dioxygen's nonbonding MO's as being the pi*(yz) and pi*(xz) antibonding MO's, which are singly-occupied in ground-state triplet dioxygen (there's also at least one excited triplet state of dioxygen). There's no single way to determine, without looking at experimental data [data from the use of different types of photoelectron spectroscopy (PES): (http://scholar.google.com/scholar?hl=en&q=%22bonding+character%22+%22lone+pair%22+nonbonding+molecular+orbital+%22photoemission+spectroscopy%22+OR+%22photoelectron+spectroscopy%22&btnG=Search&as_sdt=100000001&as_ylo=&as_vis=0)], the extent to which each of the various MO's exhibits "bonding character", within a given molecule. In general, the highest-occupied molecular orbitals (HOMO's, meaning the HOMO, the HOMO-1, the HOMO-2, etc.), which may be either bonding or antibonding MO's, are the nonbonding ("lone-pair") MO's [the highest-energy MO's, in which there is more separation of electron density throughout different parts of a molecule (more "charge separation," for example, in resonance structures within a given MO--not as much delocalization of electrons)]. But in a lot of nitrogen-containing compounds, there may be essentially no MO's that exhibit nonbonding character, as revealed by PES. Anyway, these supplementary data show some of the strange pi- and sigma-type MO's that describe the bonding interactions in the Fe(II)-O-O moiety of iron(II)-heme bound to dioxygen [the two MO's on the right (the 2nd and 3rd) of p.7 (the 2nd one is probably most similar to the "lone-pair" sigma-type interaction in the FeO moiety of iron(IV)-heme, as sketched in the second-to-last diagram below, but those MO's on p.7 of that pdf are pi-type interactions, not sigma-type interactions, as I've shown): (http://www.rsc.org/suppdata/CC/b7/b704871h/b704871h.pdf)]. Anyway, transition metals will tend ("prefer") to "obtain," via overlap with and donation from ligands, 18 electrons in organometallic complexes [the "18-electron rule" (http://scholar.google.com/scholar?hl=en&q=%2218+electron+rule%22+transition+metal&btnG=Search&as_sdt=100000001&as_ylo=&as_vis=0)], and the rest of the electrons are in other MO's of the O-Fe-S moiety and MO's of the porphyrin ring that are shared with the heme iron(IV) (or iron in another oxidation state). In one article that describes orbital occupancies of an iron(II)-heme, the iron(II) shares a total of 20 electrons with the porphyrin ring and the distal (i.e., O2) and proximal (i.e., histidine or cysteine) ligands. I've only shown the electrons that are in the HOMO's, or "frontier orbitals," whose occupancies are most likely to shift during the courses of Cytochrome P450-enzyme-catalyzed reactions, etc. Note that the diagrams should say "3pi(xz)" or "3pi(yz)" constructive overlap of the sulfur's 3px or 3py atomic orbitals (with the porphyrin's a2u molecular orbital).



These diagrams show the higher-energy steric (i.e., repulsion by electron clouds), as the porphyrin molecular orbitals are interacting, via the nitrogens, with the 3d(xy) atomic orbital of iron, in Coordinate System 1 [shown below and, in terms of the relative energy levels of the d(xy) and d(x2-y2) MO's of transitional electronic states of hydrogen abstraction reactions catalyzed by compound I, depicted in Ogliaro et al., 2000, p. 8984, Scheme 4: (http://pubs.acs.org/doi/abs/10.1021/ja991878x)(http://theochem.chem.rug.nl/~filatov/Pubs/JACS_122_8977.pdf)], or, in Coordinate System 2 [not shown below but shown above, in the diagrams of the "frontier-orbital" occupations of commonly-encountered iron(IV)-heme species], with the 3d(x2-y2) atomic orbital of iron. In coordinate System 2, the relative energy levels of the d(x2-y2) and d(xy) atomic orbitals [and the corresponding delta(d(x2-y2)) and delta(d(xy)) MO's of heme] are reversed, in comparison to the situation in Coordinate System 1. Another potential source of confusion is that the delta(3d(x2-y2)) MO of heme is sometimes referred to, in diagrams, as d(x2-y2) (as if it's an atomic orbital of iron), and the pi*(xz) and pi*(yz) MO's of heme are often depicted as being "d(xz)" and "d(yz)" (parentheses mean subscripts in the blogger software that I'm using to type this) orbitals (as if they're atomic orbitals of iron). The pi(xz) and pi(yz) bonding MO's are not-infrequently referred to as being "pi orbitals" or the like [see Lehnert et al., 2001, p. 8289, Scheme 4: (http://www.ncbi.nlm.nih.gov/pubmed/11516278)]. It's always clear that these are MO's, but it's potentially a source of confusion, in my opinion. I have an article in which the authors specifically address this issue of the two different coordinate systems, but I can't find it on my computer. It's potentially a serious source of confusion, partly because there is sometimes no mention of the assignment, of either one or the other coordinate system, that the authors of various articles might be making, in their computer modeling of electron distributions in MO's ("probability density functions," as in "density functional theory," or DFT) or the like.





These are approximations for the "lone-pair" MO (sigma(d(z2)), etc.) of oxygen, a MO of heme that is likely to exhibit minimal 3d(z2) character, and the frequently-unoccupied sigma*(d(z2)) (antibonding) MO of heme that becomes occupied during charge-transfer states (ferryl-to-ring and thiolate ligand-to-ring and substrate-to-oxo transitions during high-energy, intermediate electronic states, such as in the electronic states of transition states of overall Cytochrome P450-dependent reactions, or during the multitude of "configuration interaction orbitals" that form during the absorption of visible or UV wavelengths by heme, etc.):

"Sub-states" of Singlet Delta Dioxygen; Changes in the Occupancies and Symmetries of the Orbitals of Singlet States of Dioxygen During Reactions

This article [Kearns, 1969: (http://pubs.acs.org/doi/abs/10.1021/ja01052a003)] helps me understand the difference between the two "sub-states" of singlet delta dioxygen [as discussed here: (http://hardcorephysiologyfun.blogspot.com/2010/05/three-major-electronic-states-of.html)]. This article (along with other articles by Yamaguchi and colleagues) [Yamaguchi et al., 2009: (http://linkinghub.elsevier.com/retrieve/pii/S0277538709000515)] shows the singlet delta state as comprising two "sub-states," also, except Yamaguchi et al. (2009) discuss the 2pi* diradical states of dioxygen (and Fe(IV)=O hemes) in terms of broken symmetry molecular orbitals. I guess that those are eigenfunctions, in which the 2pi* molecular orbitals are either not equivalent at all times or cease to be equivalent, in terms of their spin-density distributions (orbital "shapes"), as the charge-transfer complexes or transition states begin to form, during reactions of dioxygen with other molecules. Another point that Yamaguchi et al. (2009) make is that, as the O2 3sigma(g) molecular orbital (the "single, sigma" bond, formed by constructive overlap of two 2pz atomic orbitals) [Ochiai, 1996: (http://pubs.acs.org/doi/abs/10.1021/ed073p130)] begins to break, that 3sigma(g) molecular orbital begins to exhibit diradical character, albeit to a lesser extent than the true diradical state that the triplet ground state of dioxygen exhibits. Yamaguchi et al. (2009) (p. 2047, Fig. 5A) depicted the 3sigma(u) type (neither of the orbitals, shown as the bracketed pair at the bottom right of Fig. 5A, is symmetric with respect to the center of mass of dioxygen) of "broken-symmetry eigenfunction" molecular orbitals, formed by the configuration of the 3sigma(g) molecular orbitals (configuration interaction means like rehybridization of existing molecular orbitals to produce a new pair of molecular orbitals, with more complex patterns of "mixing" of the original molecular orbitals). I've shown 2pi*(u) "broken-symmetry orbitals" and not the 3sigma(u) type of orbitals shown in Fig. 5A [Yamaguchi et al. (2009) also depicted the pair of 4sigma(u) broken-symmetry molecular orbitals, which are unoccupied (LUMOs), at the top right of Fig. 5A, in brackets], but the concept is similar (for the types of broken-symmetry orbitals I've drawn, see Yamaguchi et al., 1983, p. 104, Fig. 1B: (http://linkinghub.elsevier.com/retrieve/pii/016612808385012X)].


This shows the way in which the lower-energy sub-state (just singlet delta) of singlet delta dioxygen can be distinguished from the higher-energy, singlet delta* sub-state of dioxygen (see Kearns, 1969, p. 6556, column 1).



In contrast to the "situations" shown above, the 2pi* orbitals in the higher-energy, singlet delta* sub-state of dioxygen, shown below, remain singly-occupied by electrons of opposite "spins" (see Kearns, 1969, p. 6556, column 1), and the net spin multiplicity for the overall dioxygen molecule, in each of the two singlet delta sub-states and in the singlet sigma state, is 1 (a singlet state, in terms of the spin multiplicity). This is because the net "spin" is zero [S = 0, and the spin multiplicity is abs.val.(2S) + 1, or 1] if the two electrons of opposite spin are in a single orbital (as in the "molecule-interacting" situations for singlet delta, shown above, or for the singlet sigma state, shown in the last diagram of this posting) or in two singly-occupied orbitals that exhibit singlet coupling (as in the "non-interacting" situation of singlet delta and singlet delta* and in the "molecule-interacting" situation for singlet delta*, as shown below):




This diagram shows the changes that occur in the relative energies of the two sub-states of singlet delta dioxygen (they're degenerate before dioxygen approaches the alkene), upon the approach of dioxygen, in either of the two sub-states, to an alkene. The diagram also shows the change that occurs, in response to the approach of dioxygen to the alkene, in the relative energies of the 2pi* orbitals, such that the 2pi* orbitals cease to be degenerate (at the same energy level):


This diagram shows the orbital occupancies for the singlet sigma state, in which the higher-energy 2pi* molecular orbital is doubly-occupied (this state, overall, is also higher in energy than the two singlet delta sub-states are and is the highest of the commonly-encountered electronic states of dioxygen):