Published: July 08, 2010 r2010 American Chemical Society 5096 dx.doi.org/10.1021/jp103506g | J. Phys. Chem. B 2011, 115, 5096–5102 ARTICLE pubs.acs.org/JPCA 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 har- monic generation. From a theoretical perspective, a notable difference in treatments of near-resonant or resonant as com- pared to nonresonant spectroscopies is the fact that the electro- nic 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 contribu- tions 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 presented 2,3 and seamlessly encompasses nonresonant as well as resonant 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 documenta- tion 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