The spin distribution was also calculated at ROHF level, where all but one electron are defined to belong to closed shells. By definition, the gross spin density will then be exactly one. The results compared to UHF do not improve much, however, even if the amount of unpaired electron density is constrained to a more realistic value of exactly one; no spin density is found around iron, all of the unpaired electron is delocalised around the porphyrin ring, as seen in Figure 3.7.
3.5 DFT versus CC2
The properties studied in Papers I and II, the charge and spin distribu-tion, have both been suggested as possible problem childs of DFT. Prob-lems with the definition of spin-DFT was already discussed in Section 2.3. Another source of concern could be the alternative interpretation of ρα and ρβ, proposed by Perdew, Savin and Burke [109]. In approxi-mate DFT, it might apparently sometimes be favourable to use ρα−ρβ
for a better description of the on-top pair density instead of the spin density. Gr¨afenstein et al. went as far as considering the DFT spin den-sity to lack physical meaning [110]. Further, the ability of contemporary functionals to describe charge delocalisation in charged species has been questioned [111–113].
Therefore, it was desirable to corroborate the results in some man-ner. In Paper III, a study of small iron complexes, posing as models for iron porphyrins, was conducted, with the aim of comparing density func-tional results with those of a wave-function based method, the second order coupled-cluster method, CC2 [114] in its density fitting RI-CC2 formulation [115].
Iron complexes of three different classes of spin states were chosen for the study; low spin (S = 0 or 1/2), intermediate spin (S = 1 or 3/2) and high spin (S = 2 or 5/2). The results of the study were reassuring. Both the charge and spin-density distributions were described in a very similar manner with both DFT and CC2 methods. The charge delocalisation is of comparable magnitude and spin polarisation is exhibited in comparable amounts.
42 Results
Figure 3.8: The distribution of the unpaired spin, calculated at a) SVWN b) BLYP and c) CC2 levels, for Fe(R)2(NH3)2, R = NHCHNH.
Excess αdensity in blue, excessβ density in red. An isocontour value of 0.003 e has been used.
Figure 3.8 compares the spin distributions in the low-spin model cal-culated at SVWN, BLYP and CC2 level.
CC2 does not, of course, deliver the final words on the true charge and spin densities. For this, higher-order methods, including multi-reference methods would have to be used. What CC2 does provide, is a corrob-oration of the previous results, originating from a very different start-ing point compared to DFT. That a correlated wave-function method essentially describes the two possibly problematic properties of DFT in the same way is reassuring. Paper III discusses only Fe(II) and Fe(III) species. For general oxidase studies, where also Fe(IV) states are impor-tant, recent reports by Ghosh and Taylor provide further insight into the performance of DFT in comparison with correlated wave function methods [116, 117].
3.6 Point charges for the metal centres of the oxidase
Quantum chemical studies are quite accurate in their description of biomolecular systems, but the computational effort, even for DFT, pre-vents studies of really large protein portions. Especially when performing molecular dynamics runs, which simulate the behaviour of the protein over time, methods based on classical molecular mechanics and force fields have to be employed.
One of the most important interactions affecting the inner workings
3.6. Point charges for the metal centres of the oxidase 43
of cytochromecoxidase is the electrostatics. To get a reliable description of this interaction, the atomic point charges contained in the force field need to be as reliable and sound as possible. With an enzyme like CcO, with several redox active centres and varying ligand states, point charge development is a time consuming task. Paper IV presents a consistent methodology for producing charges for the metal centres, as well as fitted and optimised charges for important states of the enzyme.
The computational level chosen in the work, the hybrid B3LYP func-tional in connection with a triple zeta quality basis set, is suitable for the treatment of the systems considered. Many previous studies have employed an insufficient theoretical level. Previous work has not system-atically presented atomic charges for all four redox active metal centres, CuA, haema, haem a3 and CuB, either.
The main difference between the fitting procedure presented in Paper IV and previous studies, is, however, the inclusion of an outer region surrounding the actual structure for which new point charges are to be derived. The purpose of this region is to stabilise the inside region, both structurally and electronically. Several of the haem subsystems have, formally, a highly negative charge; The haema3(III)=O state, for exam-ple, has a charge of −3. This highly anionic state would not be stable if isolated, but with the charge-neutralising effect of the environment, all electrons are bound.
Here, it is appropriate to discuss a refinement to the methodology presented in Paper IV, suggested by one of the anonymous referees. The suggestion was to describe the outer region not with actual atoms, but with point charges instead. Near neutrality would be achieved with a lighter computational effort. The buried atom problem that can appear in charges fitted to the surrounding electrostatic potential could also be alleviated, at least to a degree. Initial test calculations, where the outer region atoms were replaced by standard point charges, showed that problems with unrealistic charges at the periphery appear. Thus, the simpler methodology cannot be employed as is. It would, however, be interesting and important to thread this path further. Perhaps decreasing the size of the atomic outer region from the one used in Paper IV, and
44 Results
then surrounding the real atoms by a layer of point charges could lead to an improved point-charge derivation model, with higher accuracy and lower computational cost.
3.7 Vibrational electron transfer between the haems
Paper V presents the most time-consuming study of this thesis, measured in both wall and CPU time. The project began in 2002 and is still in draft stage, awaiting the conclusions of the final calculations. It will hopefully be submitted for publication early April 2007.
The mechanism of the final electron transfer step in the respiratory chain, that from haem a to a3, is not known. It is quite difficult to measure or study it reliably, both experimentally and theoretically. The studies performed to date have considered the transfer to be due to elec-tron tunnelling, which most likely does play the largest role in the process, even if the details are obscured.
The idea of this work was to look for a possible, more direct mecha-nism of electron transfer. Perhaps the vibrational modes of the enzyme could be involved. Previous studies have shown that vibrational motion does affect the electron transfer rate, even significantly [118–122]. But also these studies have considered the process from a tunnelling point of view; the vibrations move the redox centres relative to each other, affect-ing parameters involved in electron tunnellaffect-ing, e.g., distance between the centres.
By calculating the full vibrational IR spectrum of a combined 323-atom model of haem a and a3, with connecting amino acid residues, the changes in electron density induced by each vibrational mode could be studied. Only few vibrations give rise to even a small shift in electron density between the haems. The character of these vibrations are anal-ysed in detail. It is concluded that some vibrations do shift electron density between the haems, and that this could perhaps be an
assist-3.7. Vibrational electron transfer between the haems 45
Figure 3.9: The electron density difference between the two localised states of the combined system,a[II]/a3[III] anda[III]/a3[II]. Haemato the left. Upon electron transfer, electron density is decreased in the areas coloured light red, and increased in the areas coloured dark blue.
ing mechanism on top of electron tunnelling in the transfer process. The charge shifting, although only temporary, might catalyse the full electron transfer.
Figure 3.9 shows the difference between the two localised spin states of the combined system, that is, between a[II]/a3[III] and a[III]/a3[II].
The very delocalised nature of the transferred electron is clearly seen also in the combined system, further corroborating the results of Papers I–III;
the electron gets shifted from the entire haem a to the entire haem a3. One should note the possible caveat present, however. As the localised states are not the ground states of the electronic structure, tricks have had to be employed to avoid the ground state of the specific level of calculation, a mixed valence state.
General references
In addition to the original literature, the following three books have served as good reference material during my thesis work:
• B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Wat-son. Molecular Biology of the Cell, 3rd ed. (Garland Publishing, New York, 1994).
• R. G. Parr and W. Yang. Density Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989).
• F. Jensen. Introduction to Computational Chemistry (John Wiley
& Sons, Chichester, 1999).
47
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Appendix
In the printed version, Papers I–V follow. Legalities prevent their appear-ance in the electronic version.
57