Complex Polarization Propagator Approach in the Restricted Open

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Complex Polarization Propagator Approach in the Restricted Open-Shell, Self-Consistent Field Approximation: The Near K-Edge X-ray Absorption Fine Structure Spectra of Allyl and Copper Phthalocyanine Mathieu Linares,†,‡ Sven Stafstr€om,† Zilvinas Rinkevicius,‡ Hans Ågren,‡ and Patrick Norman*,† † ‡

Department of Physics, Chemistry and Biology, Link€oping University, SE-581 83 Link€oping, Sweden Laboratory of Theoretical Chemistry, Royal Institute of Technology, SE-106 91 Stockholm, Sweden ABSTRACT: A presentation of the complex polarization propagator in the restricted open-shell self-consistent field approximation is given. It rests on a formulation of a resonant-convergent, first-order polarization propagator approach that makes it possible to directly calculate the X-ray absorption cross section at a particular frequency without explicitly addressing the excited states. The quality of the predicted X-ray spectra relates only to the type of density functional applied without any separate treatment of dynamical relaxation effects. The method is applied to the calculation of the near K-edge X-ray absorption fine structure spectra of allyl and copper phthalocyanine. Comparison is made between the spectra of the radicals and those of the corresponding cations and anions to assess the effect of the increase of electron charge in the frontier orbital. The method offers the possibility for unique assigment of symmetry-independent atoms. The overall excellent spectral agreement motivates the application of the method as a routine precise tool for analyzing X-ray absorption of large systems of technological interest.

I. INTRODUCTION In several spectroscopies, one uses driving electromagnetic fields that are in resonance with one or more of the electronic transition frequencies of the system. Examples are provided not only by the various visible, ultraviolet, and X-ray absorption spectroscopies but also by, for example, electronic circular dichroism and resonant-enhanced Raman scattering and harmonic generation. From a theoretical perspective, a notable difference in treatments of near-resonant or resonant as compared to nonresonant spectroscopies is the fact that the electronic response functions become complex instead of real, as due to increasing significance of the imaginary damping terms that describe the relaxation mechanisms in the system.1 Terms in the response functions that are in resonance with the perturbation will dominate the sum-over-states expressions, and if contributions from nonresonant terms are neglected, computationally tractable expressions may be derived for quantities that are either purely real or imaginary. One example of this technique is provided by the calculation of oscillator strengths that relate to the imaginary part of the polarizability in the limit of zero spectral broadening. More recently, as far as standard time-dependent formulations in quantum chemistry are concerned, developments of the complex polarization propagator (CPP) have been presented2,3 and seamlessly encompasses nonresonant as well as resonant r 2010 American Chemical Society

spectroscopies with a uniform formulation and implementation of response theory. In the present work, we provide the explicit details of the complex linear response function in the restricted open-shell, self-consistent field (SCF) approximation. Since the original work is based on a general multiconfiguration SCF wave function,2,3 it is clear that open-shell systems are, in principle, included in the formalism. But even so, we believe that a presentation focused on how we deal with the special case of a restricted open-shell single determinant reference state is called for, not only for documentation of the implementation but also because, as we shall see shortly, it provides an understanding of spectral intensities of radicals as compared with those of the corresponding ionic species. A formulation of the method in the restricted open-shell approximation is also motivated by making it available to the corresponding open-shell restricted density functional theory, thereby considerably widening its application area to large-size radicals of practical interest. We will apply the code for the determination of X-ray absorption spectra of allyl and copper phthalocyanine (CuPc). Special Issue: Shaul Mukamel Festschrift Received: April 19, 2010 Revised: June 11, 2010 Published: July 08, 2010 5096

dx.doi.org/10.1021/jp103506g | J. Phys. Chem. B 2011, 115, 5096–5102

The Journal of Physical Chemistry B

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The spectra of the radicals will be compared with the spectra of the cationic and anionic forms of the molecules. The allyl radical and ions play an important role in chemical reactions and constitute simple examples of π-resonant systems4 that enables detailed spectral analysis. For CuPc, we benefit from the experimental high-resolution, gas phase experiment performed by Evangelista et al.5 to benchmark the virtues and shortcomings of our approach.

II. THEORY In this section, we will present the explicit and computationally tractable expressions for the complex polarization propagator in the restricted open-shell, self-consistent field approximation, or in other words, the Hartree-Fock and Kohn-Sham density functional theory (KS-DFT) approximations. The applications we have in mind involve radicals of doublet spin symmetry with a single R-spin electron occupying a spatial molecular orbital denoted by upper case I (also referred as the SOMO) and with two electrons occupying each of the inactive spatial orbitals denoted by lower case i. The running index, s, is used for unoccupied, secondary, spatial orbitals. With use of this notation, the time-dependent state may be parametrized according to jψðtÞæ ¼ ei^k ðtÞ j0æ

ð1Þ

where the Hermitian operator, κ̂, equals i XXh ^si þ k/si ðtÞ E îs ^ ðtÞ ¼ ksi ðtÞ E k s

þ

Xh

i

ksI ðtÞ

â†sRâIR

þ k/sI ðtÞ â†IRâsR

i Xh kiI ðtÞ â†Iβâiβ þ k/iI ðtÞ â†iβâIβ

i

ð2Þ

and in turn, the singlet electron excitation operators are defined in terms of electron creation and annihilation operators according to ^si ¼ â†sRâiR þ a† âiβ , îs ¼ E ^†si E E ð3Þ sβ

The summations in eq 2 are thus carried out over spatial orbitals with spin degrees of freedom explicitly accounted for by the introduction of the singlet excitation operators given in eq 3 and the separate consideration of the SOMO. It is clear that this situation is a special case of the more general treatment provided by eqs 40 and 41 in the original work,3 where summations run over complete sets of inactive and secondary spin orbitals. Therefore, by collecting our first-order response parameters and excitation (and deexcitation) operators according to 0 1 ðωÞ kð1Þ n i/ A @h kð1Þ ðωÞ -n h i/ ð1Þ ð1Þ ð1Þ ð1Þ ¼ ksi ðωÞ, ksI ðωÞ, kiI ðωÞ, ksi ð-ωÞ ,

^qn ^q

!

i/ h i/ T ð1Þ ð1Þ ksI ð-ωÞ , kiI ð- ωÞ

T îs , â†IRâsR , â† âIβ ^si , â†sRâIR , â† âiβ , E ¼ E Iβ iβ

ð4Þ

ð6Þ

~ [2]) are The overlap and relaxation matrices (S[2] and R diagonal without a direct dependence on the electronic struc~ [2] are introduced in a ture—the relaxation parameters in R phenomenological manner—whereas the elements of the nondiagonal, but diagonal-dominant, Hessian matrix, E[2], on the other hand, do depend on the electron density. The expressions of the elements of the respective matrices read as ½2 ^ 0 j0æ; Enl ¼ - Æ0j½^q†n , ½^ql , H ½2

Snl ¼ Æ0j½^q†n , ^ql j0æ; ~nl½2 ¼ γn Æ0j½^q†n , ^ql j0æ R

ð7Þ

and for the study of the induced polarization along molecular axis R in the electric-dipole approximation, the expressions of the property gradients read as †

i

h

½2 ½2 ~ ½2 -1 ½1 ^; B ^ææω ¼ - A½1 ÆÆA n ½E - pωS - ipR nl Bl

^R j0æ; B½1 ^β j0æ A½1 qn , μ qn ,- μ n ¼ - Æ0j½^ n ¼ Æ0j½^

s

þ

the linear response equation will be of the form presented in eq 72 of ref 3 (we have here used n as a running index for the pairs of orbital indices with a separation into positive and negative index values):

ð8Þ

where, for the A[1] vector, the minus sign in front of the commutator stem from the expansion of the exponential operator in the parametrization (eq 1) and, for the B[1] vector, the minus sign on the operator is associated with the coupling of the dipole moment and the external electric field. With this choice of operators, the linear response function in eq 6 will equal the a tensor element of the electric-dipole polarizability, R Rβ(ω). We are in the present work concerned with calculations of linear absorption spectra that are proportional to the imaginary part of isotropic average of R(ω) according to1 σðωÞ ¼

4πω Im½R h ðωÞ c

ð9Þ

In the frequency region of an electronic resonance, corresponding to a particular excitation n = n0 > 0 in the operator manifold of eq 5, we have E½2 n0 n0 - pωSn0 n0 0

ð10Þ

whereas other diagonal elements will be nonzero. The corre[2] sponding matrix elements for n = -n0 are given by E[2] -n0 -n0 = En0 n0 [2] [2] 0 0 0 0 and S-n -n = -Sn n , so there occurs no cancellation in this case. Under the assumption of a diagonal-dominant Hessian, the matrix inverse in eq 6 will therefore be dominated by element n = l = n0 , and the value of this element will roughly equal i ~ ½2 pR n0 n0 and the value of the response function becomes equal to ^; B ^ææω ¼ - A½1 ÆÆA n0

ð5Þ

-n

5097

i B½1 n0 ~ ½2 pR 0 0 n n

ð11Þ

dx.doi.org/10.1021/jp103506g |J. Phys. Chem. B 2011, 115, 5096–5102


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The value of the response is purely imaginary, corresponding to a peak in the absorption spectrum, whereas the real part of R(ω) (corresponding to the refractive index) has a zero crossing. If one compares the lowest transition in the absorption spectrum of a radical to that of the corresponding cation—the SOMO of the radical corresponds to the lowest unoccupied molecular orbital (LUMO) of the cation—we note that in the former case, the relevant excitation operator is â†Iβâiβ, and in the latter case, it is, ^si. From eqs 7 and 8, it is straightforward to determine instead, E the matrix elements needed to evaluate eq 11 in these two cases. For the diagonal elements of the polarizability tensor, we obtain the results ( 2jμisR j2 , n0 ¼ ðs, iÞ ðcationÞ i ð12Þ RRR ðωÞ ¼ iI 2 0 j , n ¼ ðIβ, iβÞ ðradicalÞ jμ pγn0 R where μif denotes the matrix element of the dipole moment operator between the initial and the final orbital in the excitation process. If the final electronic state (orbital) of the radical closely resembles that of the cation, then the transition moments are equal for the two systems, and one thus expects the intensity in the cation spectrum to be twice of that in the radical spectrum.

III. COMPUTATIONAL DETAILS The structures of isolated copper phthalocyanine and allyl were optimized at the unrestricted Kohn-Sham DFT level of theory using the hybrid B3LYP6 exchange-correlation functional in conjunction with Dunning’s correlation consistent basis sets (cc-pVDZ)7 for light elements and the Stuttgart effective core potential (SDD)8 for copper. Structure optimizations of CuPc and allyl where performed in the D2h and the C2v point groups, respectively. An illustration of the structure of CuPc is given in Figure 1, where also the four nonequivalent carbons have been labeled. The z axis has been chosen as the out-of-plane axis for both CuPc and allyl. Calculations on the cationic and anionic systems were performed using the molecular structures of the corresponding radicals. In the calculations of the polarizability, we employ the Coulomb attenuated method B3LYP (CAM-B3LYP) functional9 with a set of parameters that guarantees a correct asymptotic limit of the Coulomb hole-electron interaction10 (R = 0.19, β = 0.81, and μ = 0.33). In the property calculations of allyl and CuPc, we employed for light elements the taug-cc-pVDZ and aug-cc-pVDZ basis sets,7 respectively, and for copper, we employed the SDD.8 The chosen basis set for CuPc allows for a description of valence but not Rydberg transitions. The relaxation parameters in eq 6 that govern the broadening of the absorption spectra were chosen as γn = 1000 cm-1. Structure optimizations were performed with the Gaussian program,11 and property calculations were performed with a version of the Dalton program12 that includes an implementation of the CAM-B3LYP functional by Peach et al.13 IV. RESULTS AND DISCUSSION A. Allyl. The core and frontier orbitals of allyl are depicted in Figure 2 together with a collection of the canonical Kohn-Sham core orbital energies, and with shadings indicating signs of orbitals. One of the core orbitals (referred to as φ1 and being of A1 symmetry) is localized to the center carbon whereas the

Figure 1. Molecular structure and atomic labeling of copper phthalocyanine.

Figure 2. Canonical Kohn-Sham orbitals of allyl. Carbon core orbital energies are given in a.u.

other two (referred to as φ2 and φ3 of A1 and B2 symmetry, respectively) are delocalized over the two end carbons. The orbital energies of φ2 and φ3 are almost degenerate, whereas that of φ1 differs from the other two. For the radical, φ1 is the most stable of the core electronic states (separated by 0.2 eV from φ2 and φ3), but for the cation, the reversed ordering is observed, and the separation amounts in this case to 1.6 eV. The reason for this destabilization of the φ2 and φ3 orbitals for the radical (as compared to the cation) is the increased electronic screening of the end carbon nuclei by the population of the SOMO. This trend is continued for the anion, in which further enhanced screening of the end carbon nuclei yields an orbital energy gap of 1.4 eV (φ1 being the most stable state). For convenience in the discussion below, we will refer to the three valence orbitals in Figure 2 as HOMO, SOMO, and LUMO for all three systems, although this is an orbital reference that is strictly appropriate only for the radical. The 1s f π* transitions are induced by an electric field along the molecular z direction, and since the z dipole operator spans 5098



Figure 3. Theoretical carbon NEXAFS spectra of isolated allyl—the individual panels show the spectra for the cation (upper), radical (mid), and anion (lower), respectively. The absorption for a randomly oriented molecule (circles) and the absorption as due to out-of-plane polarized radiation (solid line) are shown.

the irreducible representation B1, we expect the NEXAFS spectrum to be dominated by transitions from either φ1 or φ2 to the LUMO as well as from φ3 to the SOMO. The spectrum of the cation, which is shown in the upper panel of Figure 3, provides a direct reflection of these expected valence transitions. The lowest band is found around 282.8 eV, and it is due to electron excitations from orbital φ3 to the SOMO (which in this case is unoccupied). There are no contributions from absorption of in-plane polarized radiation, which is reflected by the complete overlap of the two spectral curves (denoted by “zz” and “ave”, respectively) in this frequency region. The second band of the cation has a peak at 286.2 eV, and it is connected with electronic transitions from orbital φ1 to the LUMO. The intensity of this second band is lower than that of the first band, which is explained by the differences in the overlap of orbital densities between the initial and final electronic states in the excitation processes; for the second band, the electron density associated with the initial state is localized to the center carbon, whereas the final state density (the density of the LUMO) is delocalized over all three carbon atoms, but for the first band, both the initial and final state densities are localized to the end carbons. The intensity of the third band with a peak at 289.3 eV is smaller, which, in view of the argument of density overlaps, is somewhat surprising. The excitation process is in this case taking place from orbital φ2 (which is localized on the end carbons) to

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the LUMO (which is delocalized over all three carbons), so one therefore anticipates a larger overlap of densities as compared to the situation for the second band. We also note that the third absorption band has significant contributions from absorption of in-plane polarization radiation, which is associated with 1s f σ* transitions. We have not performed more detailed characterization of transition of this type in the present work, but instead kept focus at the low-lying 1s f π* transitions. The absorption spectrum of the radical is shown in the middle panel of Figure 3. The first two bands in this spectrum have their direct correspondences in the first two bands of the cation, although they are slightly shifted toward lower energies. We interpret this shift as due to the destabilization of the core orbitals, which was discussed at the beginning of this section. In addition, the intensity of the second band is preserved in the comparison of spectra for the radical and cation, whereas the intensity of the first band for the radical is about one-half of that of the cation. So what we see in the first band of the radical and the cation is eq 12 in action with i denoting φ3 and I and s denoting the SOMO in the cases of the radical and cation, respectively. The third band of the radical is found around 287.0 eV, which thus is a shift in energy by as much as 2.3 eV compared to the third band of the cation. Finally, the absorption spectrum of the anion is shown in the lower panel of Figure 3. The most apparent difference as compared to the spectra of the other two species is the lack of the first band, which, of course, is a direct consequence of the SOMO’s being doubly occupied and inaccessible. Bands two and three are no longer separated, and low-lying Rydberg transitions also appear and become mixed with the valence transitions. B. Copper Phthalocyanine. There is an obvious upside as well as a downside associated with the approach taken in the present work; namely, the absence of spin contamination and the neglect of spin polarization, respectively. To qualify the usefulness of the restricted open-shell CPP in the context of X-ray absorption spectra, benchmark calculations are called for. An important class of molecular compounds is that which uses phthalocyanine as basic building unit, since such compounds are used in a wide variety of applications, including light-emitting diodes,14 solar cells,15 field effect transistor devices,16 and sensor applications.17 The main reasons for this diversity in applications are the ability of the Pc molecule to coordinate metal atoms at its center position, which enables tuning of electronic and optical properties, and the efficiency of intermolecular charge transport. We wish here to address the recent XAS experiment carried out by Evangelista et al.5 on the copper Pc radical in the gas phase. Apart from providing high-resolution experimental spectra, this work also contains a state-of-the-art theoretical analysis based on the static-exchange (STEX) approximation,18-20 and it will serve as an excellent benchmark for the evaluation of the restricted open-shell CPP approach. The frontier molecular orbitals of the CuPc radical are ordered in accordance with the schematic figure made for allyl (Figure 2); that is, the SOMO appears in the band gap region between the doubly occupied and the virtual orbitals. The main character of the SOMO is that of a copper 3d orbital, and due to this local atomic character (and therefore small orbital density overlap), it is for good reasons assumed that it will not contribute to carbon K-edge spectra so that electron excitation operators from the core orbitals to the SOMO can be ignored in the calculations.5 In the CPP approach, on the other hand, the manifold of excitation operators is that of eq 5, and radicals are from the perspective of including a complete set of electron transfer operators therefore 5099



Figure 4. Theoretical carbon NEXAFS spectra of isolated copper phthalocyanine as due to absorption of out-of-plane (z component) polarized radiation; the individual panels show the spectra for the cation (upper), radical (mid), and anion (lower), respectively. The spectra are shifted by þ10.4 eV to align peak B in the radical spectrum with the corresponding peak in the experiment given in ref 5 (which also is included here as an inset).

treated on equal footing with closed-shell species. In addition, concerning the ordering of the frontier molecular orbitals is the fact that, at the Hartree-Fock level of theory, the discussed copper 3d orbital is actually predicted to be lower in energy than the highest occupied π orbital and, thus, giving rise to only a single electron occupancy of the latter orbital. Since the Hartree-Fock orbitals are used as a basis for the formation of the STEX Hamiltonian, it is fortunate that this interchange of orbital levels does not strongly affect the virtual space.5 But it is clear that, for metal organic system in general, it is advantageous to base response calculations on an electron density arising from DFT rather than Hartree-Fock, calculations. The experimental NEXAFS spectrum of CuPc from ref 5 is included as an inset of the mid panel of Figure 4. Six electronic transitions (marked A-F) were identified and characterized in the original work.5 The characterization work using the STEX technique involves the summation of individual spectra obtained from separate calculations with core holes localized to each of the, in this case, four symmetry independent carbons. A compensation for the well-known effect of “over-screening” was also

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made, which, considering the different nature of the carbons, means that different energy scaling factors were used depending on whether a given core hole was localized to a carbon of the pyrrole (C1 in Figure 1) or benzene (C2-C4 in Figure 1) rings. All of these matters considered, peaks A-F in the experimental spectrum were interpreted as to arise from excitations from carbons C2-C4 (A), C1 (B), C2-C4 (C), C2-C4 (D), — (E), and C1-C4 (F).5 The theoretical NEXAFS spectrum of the radical CuPc, as obtained with use of the CPP approach in conjunction with DFT and the CAM-B3LYP functional, is shown in the mid panel of Figure 4. The spectrum has been overall shifted by þ10.4 eV to align peak B in the theoretical spectrum with the corresponding peak in the experiment (the corresponding spectra for the anion and cation have been shifted by the same amount). The need for such shifts is associated with the self-interaction error in KohnSham DFT with use of standard exchange-correlation functionals, and if unknown, it can be accounted for by separate calculations of self-interaction correction energies.21 The energy region covered by peaks A-F is compressed by some 1.3 eV as compared to the experiment, so the theoretical band separations are not in perfect agreement with those of the experiment, but there is otherwise a striking and convincing agreement concerning intensities and spectral shape. The theoretical separation of bands D and F is 1.5 eV, which is in good agreement with the experimental value of 1.8 eV,5 but on the other hand, the theoretical separation of bands B and D is underestimating the experimental results by as much as 0.8 eV. In the CPP approach, the characterization of the NEXAFS spectrum is carried out by a study of the imaginary part of the response vector: the atomic localizations of the inactive orbitals are identified for the largest elements of the vector.10,22 When such an analysis is carried out for the spectrum of the CuPc radical, one obtains peak assignments that stand in some contrast to those obtained in the STEX approach (and which were reviewed above). The predominant difference in assignments arises due to the fact that in the CPP approach taken here, the absorption due to the three benzene carbons (C2-C4), gives rise to peaks that are well separated. In other words, the spectral characterization made in the present work suggests that the benzene carbons are quite nonequivalent in terms of their near-edge absorption characteristics. In terms of their ionization characteristics, on the other hand, the X-ray photoelectron spectroscopy results (experimental and theoretical) in ref 5 demonstrated a strong similarity of the benzene carbons. We assign peak A not collectively to the benzene carbons as in ref 5, but rather, as due only to carbon C2. Peak B, which in the original work was assigned to arise solely from the pyrrole carbon,5 is here assigned to carbons C3 and C4 (in about equal proportions). Considering the strength of peak B, an assignment of peak B to C1 is not reflecting the stoichiometric ratio of carbons as pointed out in ref 5, and from this perspective, our assignment appears more reasonable. We assign the remaining peaks that are labeled C, D, E, and F to carbons C3, C1, C2-C3, and C2, respectively. The theoretical NEXAFS spectra of the CuPc cation and anion are depicted in the upper and lower panels of Figure 4, respectively. We have in these cases employed an identical molecular structure as in the calculations for the radical, so spectral differences are thus entirely due to changes in the electronic structure associated with the different electron occupations of the SOMO. In comparing the spectra of the cation and anion, we first note 5100


The Journal of Physical Chemistry B that the number of absorption peaks is the same; that is, there are no additional low-lying peaks in the former spectrum, as due to the transitions to the SOMO (and as was the case for allyl). This observation is a mere confirmation of the conclusion drawn in ref 5 about the SOMO being invisible in the NEXAFS spectrum. But the more striking fact about the anion spectra as compared with that of the radical is the splitting of peak B into two peaks that we have labeled B1 and B2. The reason for these splittings is that peak B has, as discussed before in connection with the spectrum of the radical, contributions from electronic transitions both from the benzene C3 and C4 carbons and that these two are screened differently by the electron density in the SOMO. Peaks B1 and B2 are associated with carbons C3 and C4, respectively. We can address the effects of the coordinated copper atom on the X-ray absorption spectrum of phthalocyanine by comparing the spectra in Figure 4 of the present work with the spectrum of free base phthalocyanine reported in Figure 5 of ref 23. The methodological approach as well as computational details regarding exchange-correlation functional and basis sets are identical in the present work and in ref 23, and the phthalocyanine spectra in the present work are, from that perspective, comparable to that in ref 23. In free base Pc, there are eight nonequivalent carbons (due to the two nitrogen-bonded hydrogens in para positions), and the absorption spectrum is characterized by split peaks that can be assigned to the four pairs of carbon atoms (the splitting being due to the two hydrogens). The splittings of the C1 and C2 pairs in free base Pc were estimated to be 0.5 and 0.4 eV, respectively, whereas the splittings of the C3 and C4 pairs were not resolved.23 The most relevant comparison to be made here is to focus on the peaks of free base Pc that stem from the four carbons in the branches without nitrogen-bonded hydrogens. For these four carbons in free base Pc, the absorption peak found at lowest energy is assigned to the benzene carbon that is labeled by C2 in Figure 1 (it is labeled by C6 in ref 23), and which thus agrees with the assignments made in the present work for CuPc. Ordered by the transition energy, the next two (degenerate) absorption peaks in free base Pc are assigned to the two benzene carbons labeled by C3 and C4 Figure 1 (they are labeled by C7 and C8 in ref 23), and the peak assigned to the pyrrole carbon was found to be highest in energy.

V. CONCLUSIONS The extension of the complex polarization propagator approach to the restricted open-shell SCF approximation has been presented, and it is thus a formulation free of spin-contamination but it is also a formulation that neglects spin polarization. The formulation and implementation is general in the sense that applies to arbitrary frequencies of the perturbing fields as well as to arbitrary oneelectron property operators. Our motivation for this work has been to be able to address X-ray absorption spectra of radicals, and we illustrate the characteristics of the spectroscopy by the study of the cationic, radical, and anionic forms of the simple allyl system. The resulting spectra are discussed in view of the gradual filling of the π orbital of A2 symmetry, which is empty for the cation, half-filled for the radical, and filled for the anion. The intensity of the corresponding peak in the spectra is reduced by a factor of 2 in going from the cation to the radical, and it of course vanishes for the anion. We benchmark the method against the experimental highresolution gas phase carbon NEXAFS spectrum for the copper phthalocyanine radical found in the work of Evangelista and coworkers.5 Our theoretical results display a good spectral agreement

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in comparison with the experiment although several of the band separations are underestimated, but the agreement concerning relative band intensities enables a clear assignment of peaks in the theoretical spectrum to those in the experiment. From our theoretical calculations, we can characterize absorption peaks as being due to particular symmetry-independent atoms in the molecule. In this respect, our analysis differs from that made in the original work.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT P.N. acknowledges financial support from the Swedish Research Council (Grant No. 621-2007-5269). The authors acknowledge a grant for computing time from the National Supercomputer Centre (NSC), Sweden. ’ REFERENCES (1) Boyd, R. W. Nonlinear Optics, 2nd ed.; Academic Press, San Diego, 2003. (2) Norman, P.; Bishop, D. M.; Jensen, H. J. Aa.; Oddershede, J. J. Chem. Phys. 2001, 115, 10323. (3) Norman, P.; Bishop, D. M.; Jensen, H. J. Aa.; Oddershede, J. J. Chem. Phys. 2005, 123, 194103. (4) Linares, M.; Humbel, S.; Braïda, B. J. Phys. Chem. A 2008 112, 13249. (5) Evangelista, F.; Carravetta, V.; Stefani, G.; Jansik, B.; Alagia, M.; Stranges, S.; Ruocco, A. J. Chem. Phys. 2007, 126, 124709. (6) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (7) Dunning, T. H., Jr. J. Chem. Phys. 1989, 90, 1007. (8) Dolg, M.; Wedig, U.; Stoll, H.; Preuss, H. J. Chem. Phys. 1987 86, 866. (9) Yanai, T.; Tew, D. P.; Handy, N. C. Chem. Phys. Lett. 2004, 393, 51. (10) Ekstr€om, U.; Norman, P. Phys. Rev. A 2006, 74, No. 042722. (11) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu ,G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W. ; Gonzalez, C. ; Pople, J. A. Gaussian 03, revision B05; Gaussian, Inc.: Pittsburgh, PA, 2003. (12) Dalton, a molecular electronic structure program, release 2.0, 2005; see http://www.kjemi.uio.no/software/dalton/dalton.html. (13) Peach, M. J. G.; Helgaker, T.; Salek, P.; Keal, T. W.; Lutnæs, O. B.; Tozer, D. J.; Handy, N. C. Phys. Chem. Chem. Phys. 2006, 8, 558. (14) Blochwitz, J.; Pfeiffer, M.; Fritz, T.; Leo, K. Appl. Phys. Lett. 1998, 73, 729. (15) Peumans, P.; Forrest, S. R. Appl. Phys. Lett. 2001, 79, 126. (16) Di, C. A.; Yu, G.; Liu, Y. Q.; Xu, X. J.; Song, Y. B.; Wang, Y.; Sun, Y. M.; Zhu, D. B.; Liu, H. M.; Liu, X. Y.; Wu, D. X. Appl. Phys. Lett. 2006, 88, 121907. (17) McKeown, N. B. Phthalocyanine Materials: Synthesis, Structure, and Function; Cambridge University Press: Cambridge, New York, 1998. 5101



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