Calculations of CoTPP in the gas-phase were carried out, both with U = 0 applied and with a self consistent U of 6.2 eV as calculated with the self-consistent U linear response method of Cococcioni et al [81], using the optB86b-vdW density functional [74, 75]. This value is higher than values of U typically used in DFT+U calculations of metalloporphyrins, where U = 3 eV is found to give the correct spin state for metalloporphyrins [148, 84, 149], and higher than the U value of 4.9 eV calculated by Perera et al for tetrabromophenylporphyrin [150]. However it should be noted that values forU will be expected to change depending on the projections that were used to self-consistently determineU [151, 152] and also on the underlying exchange correlation functional.
As discussed in section 2.1.4, in spin polarised DFT it is possible to obtain multiple meta-stable spin states, and this happens for CoTPP. When a U value of 6.2 eV is
−4 −2 0 2 4 PDOS
(a) CoTPP (U = 0 eV) dyz
dxz dxy dx2−y2 dz2 −4 −2 0 2 4 PDOS
(b) CoTPP (U = 6.2 eV, low spin)
−10 −8 −6 −4 −2 0 2 4 6 E - Efermi(eV) −4 −2 0 2 4 PDOS
(c) CoTPP (U = 6.2 eV, high spin)
Figure 4.2: PDOS of Cod-states of CoTPP, (a) from a calculation withU = 0 eV, (b) from a meta-stable low-spin state (S = 1/2), with U = 6.2 eV, (c) from a high-spin state (S= 3/2), also withU = 6.2 eV, more stable than the low-spin state by 77 meV. applied, a high-spin state (S = 3/2) is found to be the most stable, with a low-spin state (S = 1/2) meta-stable and higher in energy by 77 meV. The partial density of states (PDOS) of the high- and low-spin systems, along with the PDOS of the system when noU is applied given in Figure 4.2.
The extent to which the addition of the self-consistent U improves the energetics of the Co d-states can be ascertained by comparing these CoTPP results with other calculations and ESR spectra. A calculation by Liao and coworkers [153] looked at gas- phase CoTPP using a Vosko-Wilk-Nusair spin density potential with Burke exchange and Perdew’s correlation, with a molecular orbital basis set. This calculation resulted in an electronic structure where the HOMO was of (dπ)4 character, the SOMO of (dz2)1
agreement with analysis of ESR spectra [154].
da Silva and coworkers [155] did a GW study of TBrPP-Co (5, 10, 15, 20-Tetrakis- (4-bromophenyl)-porphyrin-Co), i.e. CoTPP but with bromophenyls substituted in place of the phenyl groups. For this system the behaviour of the Cod-orbitals would be expected to be very similar. The results of this calculation were very similar to the results of Liao, with the exception that the HOMO-1 was higher in energy than the SOMO.
Both the results of the GGA calculation and the low-spin GGA+U calculation have an orbital ordering that is similar to that calculated by da Silva [155], whereas the high-spin state, with itsdx2−y2 SOMO and unoccupied (dπ)2 orbital has an electronic
configuration that has not been reported in any other work.
By contrasting the PDOS plotted in figures 4.2 (a) and (b) it becomes apparent that one of the main effects of the addition of the U parameter is that it increases the splitting between the occupied and unoccupied d-orbitals, which is a well known consequence of theU parameter [87] (c.f. Section 2.1.5 for a heuristic explanation). In Table 4.1 the splitting of the dz2 SOMO and the HOMO-LUMO gap is compared for
the two low-spin GGA(+U) calculations and to the GW calculation of daSilva et al. of TBrPP-Co. The GW calculation is more suited to the calculation of excited states and so would be expected to give a better value for the splitting between ground and excited states than ground-state DFT, such as GGA [132]. Table 4.1 shows that the addition of the U parameter to the GGA calculation brings both energies into much closer agreement with the GW calculation.
Table 4.1: Comparison of orbital properties between GW, [155] GGA+U (U = 6.2 eV) and GGA calculations.
GW GGA+U GGA
dz2 Splitting (eV) 6.3 6.23 2.12
HOMO-LUMO Gap (eV) 2 1.91 0.49
In Liao [153] a Mulliken analysis of their results for CoTPP was carried out, in Table 4.2 this is compared with a similar analysis of the GGA(+U) calculations, where the charge is decomposed by projection onto the PAW projectors. This shows that the low-spin calculations have a charge density that bears a much closer resemblance to
the work of Liao.
Table 4.2: Comparisons between Mulliken charge analysis between work involving a molecular orbital basis set [153] and GGA and GGA+U (U = 6.2 eV) calculations.
Liao et al. [153] GGA + U (high-spin) GGA + U (low-spin) GGA
d 7.6 7.146 7.327 7.23
s 0.36 0.415 0.377 0.378
p 0.47 0.509 0.489 0.495
Co 0.57 0.93 0.807 0.897
In summary, for the gas phase, the addition of a self-consistent U has two effects. Firstly, it brings the splitting of the Codz2-states into agreement with GW calculations;
without its inclusion the occupied and unoccupied d-states are too close in energy. Secondly, it makes a high-spin S = 3/2 state into the most stable spin state, which runs contrary to what has been observed in previous calculations and in ESR spectra.