Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
Supporting Information
for
“Assignment of Near-Edge X-Ray Absorption Fine Structure (NEXAFS) Spectra
of Metalloporphyrins by means of Time-Dependent Density-Functional
Calculations”
by N. Schmidt, R. Fink, and W. Hieringer
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
1. Geometries
The following Figure S1 illustrates the conformations of the tetraphenyl porphyrin molecules
as used in the study.
Figure S1: Optimized molecular geometries of CoTPP (D2d, top row), ZnTPP (D2d,
center row), and 2HTPP (C2v, bottom row) each shown from three different
perspectives; RIBP/TZVP level.
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
The highest (singly) occupied orbital of CoTPP in D2d symmetry is, at the present level,
actually a singly occupied orbital in an E irreducible representation within the majority spin
channel (orbital 43e β), which indicates that the geometry is likely to undergo a spontaneous
symmetry lowering from D2d symmetry. This process is actually observed, but the resulting
geometry deviates only very little from perfect D2d symmetry, and the NEXAFS spectra were
found to be very similar.
Additional information on the geometries, electronic structure, and spectral data for all
molecules can be obtained from the authors upon request (contact W. H.).
2. Influence of basis set on the calculated NEXAFS spectra
As noted in the Section 2 of the main paper, the basis set used in the present work is rather
limited owing to the size of the molecules investigated. To estimate the effects of basis set
inaccuracies on the NEXAFS spectra, we have recalculated the spectra for ZnTPP using three
more basis sets, namely the TZVP (triple-zeta valence plus polarization)1 basis from the
Turbomole library, an augmented version of the SV(P) basis where one set of diffuse
functions2 of s, p, and d type have been added to the standard SV(P) basis for all nonhydrogen atoms (denoted aug-SV(P) in the following), and a mixed basis set where the IGLOIII3 basis set was used for all four N atoms and the SV(P) basis for all other atoms (denoted
IGLO-III(N)+SV(P) in the following). The resulting N 1s and C 1s spectra for ZnTPP are
shown in Figures S2 and S3.
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
Figure S2: Calculated N 1s NEXAFS spectra of ZnTPP using various basis sets (see the
main paper for details on the method used).
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
Figure S3: Calculated C 1s NEXAFS spectra of ZnTPP using various basis sets (see the
main paper for details on the method used).
It can be seen that the improvements due to the TZVP and IGLO-III basis sets do not change
the spectra significantly compared to those calculated with the smaller SV(P) basis set.
However, as expected the addition of diffuse functions has some impact on the spectra in the
high-energy regions, while the low-energy part is again largely unaffected.
3. Comparison of calculated NEXAFS spectra based on single-particle excitations, (timedependent) Hartree-Fock, and a GGA functional
To illustrate the effect of the response corrections to the orbital-energy differences we include
a comparison of NEXAFS spectra calculated using the TD-BHandH-LYP method (see the
main paper) and the Time-dependent Hartree-Fock (TD-HF) method with spectra derived
from the corresponding orbital energy differences and dipole matrix elements (denoted ΔεS5
Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
BHandHLYP for orbitals calculated with the BHandH-LYP hybrid functional, and Δε-HF for
orbitals calculated with the Hartree-Fock method). We also include calculations using the TDBLYP method, which is based on the BLYP GGA functional (the spectra using orbital energy
differences only are very similar to the TDDFT calculations in this case and therefore
omitted). It can be seen that all these methods lead to larger deviations in energy than the TDBHandHLYP method with respect to the experimental resonances, and also show different
spectral signatures, hampering the assignment of the experimental spectra. Similar
observations have been reported for other molecules before (see the Introduction for
references).
Figure S4: Calculated N 1s NEXAFS spectra of ZnTPP using various computational
schemes; see the text and the main paper for details.
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
Figure S5: Calculated C 1s NEXAFS spectra of ZnTPP using various computational
schemes; see the text and the main paper for details.
4. Verification of the "block-pole" approximation invoked by a rearrangement of the
single-particle spectrum
We have tested the accuracy of the "block-pole approximation" invoked in this paper (i. e.,
the solution of the TDDFT matrix equation based on a modified one-electron spectrum as
outlined in Section 2 of the main paper) for a number of smaller molecules against results
from a full diagonalization of the TDDFT matrix equation based on the unmodified (original)
single-particle spectrum (in the following called "conventional"), which requires the timeconsuming calculation of all excited states up to the desired core excitations. The pyrrole and
benzene molecules have been chosen as benchmarks, because these can be considered as the
basic building blocks of tetraphenyl porphyrins, and a calculation of the full spectrum of these
molecules is still feasible. In addition, the excitation block corresponding to the B2 irreducible
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
representation was fully calculated for ZnTPP (D2d symmetry) and tested against our
approximate calculations. The results are summarized in Table S1, where we list the mean
absolute and maximum deviations per excitation in energy and oscillator strength from the
conventional results observed among the calculated NEXAFS excitations for two choices of
the parameter Δ0 (see main paper).
Table S1: Deviations in excitation energies and oscillator strengths between the present
block-pole approximation and conventional calculations for core excitations of various
molecules; Δ0 = f · (εa,max – εa,min) with f = 1.0 or 100.0 (see main paper for meaning of
Δ0); BHandH-LYP/SV(P) level of theory.
Molecule
Typea
exc.b
MAD (max.)c energy, eV
MAD (max.)c osc. str.
f = 1.0
f = 100.0
f = 1.0
f = 100.0
Benzene
C 1s
100
0.005 (0.027)
0.001(0.003)
0.0036 (0.145,33%)
0.0003 (0.014, 3%)
Pyrrole
C 1s
100
0.004 (0.020)
0.001 (0.003)
0.0021 (0.044,22%)
0.0003 (0.007, 3%)
Pyrrole
N 1s
62d
0.004 (0.032)
<0.001(0.002)
0.0018 (0.025,33%)
0.0001 (0.002, 2%)
ZnTPP
C 1se
120
<0.001(0.002)
<0.001(0.002)
0.0006 (0.016, 4%)
0.0004 (0.012, 3%)
ZnTPP
N 1se
40
<0.001(0.001)
<0.001(0.001)
0.0002 (0.003,3%)
0.0001 (0.002, 2%)
a: Type of NEXAFS excitation; b: Total number of (lowest) core excitations compared in test; c: Mean absolute
deviation per excitation; in brackets: maximum deviation; d: all N 1s excitations within current basis set; e: only
excitations of B2 irreducible representation (D2d point group) have been calculated (full diagonalization of B 2
block in SV(P) basis set using Turbomole 5.9; 12624 total B2 excitations, first C 1s excitation: No. 8813, first N
1s excitation: No. 11970 ).
These results suggest that the errors for core excitation energies introduced by our present
block-pole approximation decrease with increasing Δ0. The deviations are consistently below
0.01 eV for the Δ0 used in this work (see column "f = 100.0" in Table S1) for all molecules
tested. (Note that such errors are below the systematic errors expected from the approximate
hybrid density-functional method and the basis set.) The corresponding errors in oscillator
strengths (intensities) are below 5% for the most intensive excitations. A comparison with the
results for smaller Δ0 (see columns "f = 1.0" in Table S1) shows that the calculated excitation
energies are quite insensitive to the choice of Δ0. The deviations in absolute oscillator
strengths, on the other hand, may be sizeable if a small Δ0 is chosen. To illustrate the
significance of such deviations, Figures S6-S8 show calculated C 1s and N 1s spectra of
benzene, pyrrole, and ZnTPP (B2 part) for various choices of Δ0.
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
Figure S6: Influence of the present block-pole approximation on C 1s spectrum of
benzene; conventional (full) diagonalization of full matrix (black line, "exact") vs. blockpole approximations using various Δ0 = f · (εa,max – εa,min) (colored lines, number after
"block-pole" represents factor f; see main paper for the meaning of Δ 0); spectra are
neither scaled nor shifted.
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
Figure S7: Influence of the present block-pole approximation on C 1s (top part) and N
1s (bottom part) spectra of pyrrole; conventional (full) diagonalization of full matrix
(black line, "exact") vs. block-pole approximation using Δ0 = 1 · (εa,max – εa,min) (red line);
spectra are neither scaled nor shifted.
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
Figure S8: Influence of the present block-pole approximation on C 1s (top part) and N
1s (bottom part) spectra of ZnTPP, considering only excitations belonging to the B2
irreducible representation of the D2d molecular point group; conventional (full)
diagonalization of full matrix (black line, "exact") vs. block-pole approximation using Δ0
= 1 · (εa,max – εa,min) (red line); spectra are neither scaled nor shifted.
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
Figure S6 shows that the intensities obtained using the block-pole approximation approach the
intensities from the exact calculations with increasing Δ0, and the deviations obtained for Δ0 =
100 · (εa,max – εa,min) are already negligible. Figures S7 and S8 show that the intensities
obtained using the present block-pole approximation are close to the "exact" intensities (from
an conventional TDDFT calculation) for pyrrole and ZnTPP (both N 1s and C 1s) even for
small Δ0 = 1 · (εa,max – εa,min) (spectral lines for larger Δ0 have not been included in Figures S7,
S8 for better clarity). It can therefore be concluded that a choice of Δ0 = 100 · (εa,max – εa,min)
(see main paper) leads to a negligible error due to the present block-pole approximation for
these molecules, and all spectral features are fully recovered.
Note that the accuracy in energies shown above is indeed only observed for core excitations
(which are sufficiently separated in energy from other excitations). A modification of the
orbital energy difference spectrum in regions of valence excitations leads to large deviations
(not shown), which is in line with expectations. The approximation invoked here is thus only
accurate for the calculation of excitation blocks which are well separated from other blocks as
in the present case of N 1s and C 1s core excitations.
5. An example illustrating the present shifting method leading to a "block-pole"
approximation
We finally give a numerical example on the orbital shifting procedure used to invoke the
present block-pole approximation. For ZnTPP, the C 1s core orbital energies at the chosen
level are computed to lie between -10.631 a.u. (-289.28 eV) and -10.589 a.u. (-289.15 eV).
The LUMO energy is computed to -0.0608 a.u. (-1.65 eV). By user input, we now specify that
the energies of the C 1s core orbitals from which the desired NEXAFS excitations shall be
computed are, for example, below -10.3 a.u. (no undesired, higher-lying orbitals should be
below this number). In our implementation, this means that orbital energy differences Δεia
which are lower than -10.2392 a.u. (= -10.3 a.u. + 0.0608 a.u.) are shifted up in energy by Δ0
= 35314 a.u. which is simply (and somewhat arbitrarily) taken to be the difference of the
highest virtual orbital energy and the lowest occupied orbital energy times 100. In effect, all
low-lying orbital energy differences, including the valence transitions, are thus shifted to very
high energies. The result is a modified orbital energy difference spectrum where the desired
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Supporting Information for „Assignment of NEXAFS of …” by N. Schmidt et al.
core excitations end up as the lowest-energy orbital excitations. This modified spectrum
enters the standard eigensolver routines implemented in Turbomole4 to yield the TDDFT
solutions for the desired core excitation energies.
In passing, we note that we have also tested a scheme which in addition to the lower orbital
energy differences also shifts orbital energy differences above a certain threshold. This avoids
the computation of excitations to unoccupied orbitals of very high energies, which may not
bear any physical significance in finite basis sets. So far we have found no clear advantage in
using this scheme, however, and skip any further discussion of this option here.
Please contact W. H. for any further details on the implementation.
1.
2.
3.
4.
F. Weigend and R. Ahlrichs, Phys. Chem. Chem. Phys. 7, 3297-3305 (2005).
R. A. Kendall, T. H. Dunning and R. J. Harrison, J. Chem. Phys. 96, 6796 (1992).
W. Kutzelnigg, U. Fleischer and M. Schindler, The IGLO-Method: Ab Initio
Calculation and Interpretation of NMR Chemical Shifts and Magnetic Susceptibilities.
(Springer, Heidelberg, 1990).
F. Furche and D. Rappoport, in Computational Photochemistry, edited by M. Olivucci
(Elsevier, Amsterdam, 2005), pp. 93-128.
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