Coupled Cluster and Time-Dependent Density Functional Theory

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Spectroscopy and Excited States

Coupled Cluster and Time-Dependent Density Functional Theory Description of Inner Shell Photoabsorption Cross Sections of Molecules Bruno Nunes Cabral Tenorio, Ricardo Rodrigues Oliveira, Marco Antonio Chaer Nascimento, and Alexandre Braga Rocha J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00375 • Publication Date (Web): 06 Sep 2018 Downloaded from http://pubs.acs.org on September 7, 2018

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Journal of Chemical Theory and Computation

Coupled Cluster and Time-Dependent Density Functional Theory Description of Inner Shell Photoabsorption Cross Sections of Molecules Bruno Nunes Cabral Tenorio, Ricardo Rodrigues Oliveira, Marco Antonio Chaer Nascimento, and Alexandre Braga Rocha∗ UFRJ - Universidade Federal do Rio de Janeiro, Instituto de Química, Av. Athos da Silveira Ramos, 149, Rio de Janeiro - RJ, 21941-909, Brasil E-mail: [email protected]

Abstract Near K-edge photoabsorption cross section spectra of a number of molecules, namely, water, ammonia, acetone, acetaldehyde, furan, and pyrrole were obtained at the nitrogen, oxygen and carbon K-edges with the Coupled Cluster ansatz (CC) and with the Time-Dependent Density Functional Theory (TDDFT) by treating the inner shell excitations as individual channels, separated from the valence part of the spectrum. The discretized electronic pseudo-spectrum, obtained with quadratically integrable basis sets (a.k.a. L2 ) at the CC or TDDFT level is used to reconstruct the complex dipole polarizability function, from which the photoabsorption cross section near the K-edge is obtained by a continued fraction based analytic continuation procedure. The CC2 and CCSD results are in good agreement with experimental data while the TDDFT results yield reliable cross sections. Overall, the results obtained in this work indicate that our method can be used for the treatment of the NEXAFS spectra, with the advantage that

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the electronic excitations in the K-edges can be easily obtained at low computational cost using TDDFT.

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Introduction

Near edge X-ray absorption fine structure (NEXAFS) is one important type of X-Ray absorption spectroscopy (XAS) with a large range of recent applications in different systems like gas-phase peptides, 1,2 peptide in water, 3,4 peptide in a zwitterion structure, 5 nitroxide free radicals, 6 organic molecules on surfaces, 7–10 fullerenes isomers, 11–13 corroles compounds 14 and organic films. 15–17 Another important application of XAS is in polycyclic aromatic hydrocarbons (PAH) molecules with astrochemistry implications. 18 PAHs are abundant in the interstellar X-ray photon-rich environments. 19,20 Photo-stability of this class of molecules and their hydrogenated derivatives in X-ray region can be understood by analyzing the inner-shell absorption cross sections. 18,20,21 Processes involving the electron in the continuum, such as photoionization, play a central role in many emerging fields of modern atomic and molecular physical chemistry. With the advent of new synchrotron light sources, more accurate experiments relating the excitation of core electrons have become available and then, theoretical approaches capable of yielding accurate cross-section spectra for those processes are needed to interpret the data collected from the experimental investigations. Many quantum mechanical methods relying on quadratically integrable basis set (a.k.a. L2 ) have been used in order to simulate NEXAFS electronic spectra. Recently, several multiconfigurational and multireference levels have been theoretically studied. Despite the accurate results, only small systems can be studied with these approaches. 22–29 Core-electron excitation spectra have been also obtained with the static exchange method (STEX), 30,31 the algebraic diagrammatic construction approaches 32 ADC(2), 33,34 ADC(3), 35 and ADC within

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the core-valence separation scheme (CVS-ADC). 36,37 Besides, Ekström et al. 38 obtained absorption cross sections with the random phase approximation (RPA) and density-functional based electronic structure methods directly from scanning the imaginary part of the complex dipole polarizability over the relevant frequency region. Other methodologies based on density functional theory can be found elsewhere. 15,16,39–46 X-ray spectroscopy studies based on coupled cluster ansatz have been investigated via equation-of-motion coupled cluster method (EOMCC) by Nooijen and co-workers 47 and by Coriani and co-workers 48 with damped CC linear response (LRCC) theory based on an asymmetric Lanczos-chain algorithm 49 which further extended the approach to a core-valence separation scheme (CVS-CC). 50,51 Core excited states lies in a manifold of high energy valence-excited states which makes their calculation a difficult task. However, the well-known locality of these states makes the calculation ideal for local correlation treatments such as CC and TDDFT. Moreover, L2 based methods are largely used for simulating NEXAFS spectra with the help of a simple approach consisting of taking the individual excitation and oscillator strength, each broadened by a Gaussian or a Lorentzian function. Despite of being very simple and appealing, this approach does not permit taking any information about the continuum region of the spectra, which contains valuable information about the electronic structure of the system. Obtaining information for both the discrete and continuum regions of the spectrum using electronic structure methods based on L2 basis sets represent a challenge since the asymptotic behavior of continuum wave functions is not met by L2 wave functions. To overcome this implication, without using methods based on obtaining the explicit electron wave function of the continuum, we apply a methodology first reported by Nascimento et al. 52,53 based on an analytical continuation procedure where the electronic pseudo-spectrum, obtained with any electronic structure method based on L2 basis sets wave functions, is used to construct an approximated representation for the complex dipole dynamic polarizability function 52–58 which is then calculated at a number of points in the complex plane and subsequently fitted by a continued fraction procedure. With the analytic function of the complex dynamic

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dipole polarizability in hand, we obtain the photoabsorption/photoionization cross section directly from its imaginary part, and from its real part we obtain the real dynamic dipole polarizability. On recent papers, we demonstrated that the vacuum ultraviolet (VUV) spectra of a number of molecules can be accurately obtained at coupled cluster (CC) level for small molecules and also at TDDFT level for larger systems. 57,58 Herein, we show for the first time, using water and ammonia as model systems, that the K-edge photoionization cross section can be also simulated using the hierarchy of CC with singles (CCS), doubles (CC2), and singles and doubles (CCSD) excitations as well as TDDFT calculations, with good agreement with the experimental results. In addition, photoionization cross section of a number of organic molecules, namely, acetone, acetaldehyde, furan, and pyrrole at the nitrogen, oxygen and carbon K-edges were obtained yielding good agreement with experimental data, demonstrating the reliability of this level (TDDFT) of calculation for inner shell molecular electronic states in agreement with our previous work for VUV region of the spectra. 58 We agree that the present method, as well as the Stieltjes imaging technique, 59,60 are not the most suitable approaches to treat the ionization pre-edge. The strength of both methods is in the description of the photoionization phenomena. Thus, one of the goals of this work is to provide a useful tool for interpreting NEXAFS data as well as to propose an accessible method capable of predicting photoionization cross section data to the general community and to research groups that may be interested in information above the inner-shell thresholds or where experimental cross-section data may not be available.

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Theoretical Approach

We have applied a different methodology from the common approach largely used in connection to linear response based methods 39,45,48 where the photoabsorption spectra is obtained directly from scanning the imaginary part of the complex dipole polarizability over the

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frequency region. Our methodology is based on an analytic continuation procedure to a electronic pseudo-spectra 52,53,55–58 from which an analytic complex valued expression for the dynamic dipole polarizability is obtained and the photoabsorption cross section is directly extracted from the imaginary part of the analytic dynamic dipole polarizability expression. The methodology is outlined in the following. The averaged dynamic dipole polarizability is given by Z

∞

α(ω) = 0

where (df /dε) =

P∞

i=1

(df /dε)dε ε2 − ω 2

(1)

fi δ(ωi − ε) + g(ω), ωi , fi and g(ω) being the transition frequencies,

bound and continuum oscillator strengths, respectively. Extending this definition to complex values of frequency, α(z) becomes Z

∞

α(z) = 0

(df /dε)dε ε2 − z 2

(2)

The complex dynamic dipole polarizability, α(z), is analytical throughout the complex plane, except for an infinite number of poles along the real axis, and a branch cut in the photoionization interval εI ≤ Re(z) < ∞, where εI stands for the ionization threshold of the system. Taking the complex frequency as being close enough to the real axis and considering z = limη→0 (ω + iη), the dynamic polarizability takes the form 59

lim α(ω + iη) =

η→0

X i=1

fi +P 2 ωi − ω 2

Z

∞

εI

g(ε)dε iπg(ω) + , 2 2 ε −ω 2ω

(3)

with P representing the Cauchy principal value of the integral. Separating the real and the imaginary parts in equation 3 we obtain

lim Re[α(ω + iη)] =

η→0

X i=1

fi +P 2 ωi − ω 2

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Z


∞

εI

g(ε)dε ε2 − ω 2

(4)


lim Im[α(ω + iη)] =

η→0

πg(ω) , 2ω

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(5)

since the cross section can be written as 61 2π 2 g(ω) , c

(6)

4πω lim Im[α(ω + iη)] c η→0

(7)

σ(ω) =

we finally obtain: σ(ω) =

α(ω) = lim Re[α(ω + iη)] η→0

(8)

To use the function 7 the dynamic dipole polarizability, α(z), is approximated by a finite sum α(z) =

k X i=1

fi ωi2 + z 2

(9)

where z is the complex valued frequency and the pseudo-spectrum i.e., {ωi , fi }i=1,Λ with Λ being the length of the ab initio calculated spectrum, stands for the transition frequencies and oscillator strengths obtained with a L2 basis set. Once the pseudo-spectrum is obtained in a L2 basis set calculation, the approximation for α(z) as in Eq. (9) is constructed from whence we calculate α(z) at a number of (arbitrary) points in the complex plane. Next, these points are fitted by a continued fractions procedure, yielding a representation of α(z) in the complex plane as outlined in Ref. 57. Using this converged continued fraction representation we calculate α(z) on the real axis where it equals α(ω). The imaginary part of α(z) on the real axis yields the cross section by Eq. (7). Simulating photoabsorption cross section spectra for inner-shell states represents a more complicated task when compared to the valence region of the spectrum. RPA and TDDFT methods suffer from the incomplete handling of relaxation effects for the calculation of coreelectron excitation energies. The relaxation effect we refer can be interpreted as a correlation effect since it relates the simultaneous excitation of a core-electron with the relaxation of

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the remaining ones due to the core-hole left behind. The core hole left when an inner-shell electron is excited or removed is a much more significant perturbation to the molecular electronic structure than the excitation/removal of a valence electron. If no treatment is performed to take into account the relaxation of the core hole density, one finds considerable errors in the dipole inner-shell transition frequencies. The multielectron excited character present in the hierarchy of CC approaches enables us to recover this relaxation effect in part, with discrepancies in transition energies about 2 eV with singles and doubles excitations, and below 1 eV when triple excitations are included, capturing up to 98% of the relaxation correlation energies. 51 Nonetheless, it is a common practice to the users of RPA/TDDFT methods to consider that the error induced by the lack of relaxation of the core hole is systematic and equally distributed to all inner-shell electronic transitions which, indeed, can be shifted with some empirical factor, allowing direct comparison of the calculated spectrum with experimental data. 15,16,46 Using the transition frequencies and oscillator strengths coming from inner shell states in equation 9 is equivalent to taking the partial complex dynamic dipole polarizability from individual channels. Considering that the coupling between the channels from the valence excitation with the inner shell excitations is small enough, we can take the contribution from each channel separately. A similar procedure has been already used by Nascimento in Ref. 55 for the H2 photoionization cross section. Converging core-electron photoionization profiles with L2 basis set represents a particular challenge. NEXAFS spectra oscillates quickly within a small energy range until it decreases asymptotically to zero far beyond the ionization potential of the measured edge. The simple selection of the complex points used in our previous works 57,58 for VUV spectra yields strongly oscillating cross section profiles. We found that an efficient manner to smooth out the theoretical inner-shell photoionization cross section is to use the principal pseudo-spectra, (¯ ωip , f¯ip )i=1,p as obtained in the Stieltjes imaging procedure, 59,62 in the construction of the complex points where the α(z) representation on Eq. (9) is evaluated. The Stieltjes principal

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pseudo-spectra in an L2 basis set to calculate inner-shell photoionization cross section has been previously used 30 in the Static Exchange (STEX) approximation and in reference 60 the authors show how the Stieltjes imaging technique can be generalized to include high energy electronic continua and also other types of inner shell ionization phenomena. Herein, we apply the principal pseudo-spectra in a different way, where we used it as the points in the complex plane used to fit the equation 9 via an analytic continuation procedure based on continued fraction functions. In the seventies, Langhoff and co-workers 59,62 proposed an elegant way to extract the principal pseudo-spectra of the Hamiltonian from the converged moments S(-k)

S(−k) =

r X

ωi−k fi ,

k = 0, ..., 2r − 1.

(10)

i=1

The moments of the distribution of oscillator strengths S(−k), k = 0, ..., 2r − 1 define a sequence of orthogonal polynomials of degree 0 to r. The roots and residues of these orthogonal polynomials are used to obtain the principal pseudo-spectrum (¯ ωip , f¯ip )i=1,p which represents the oscillator strength distribution at the quadrature points ωip . The oscillator strength distribution as obtained from the Stieltjes principal pseudo-spectra is a smoothened version of the original electronic pseudo-spectra {ωi , fi }i=1,k (with p

Coupled Cluster and Time-Dependent Density Functional Theory

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