Simulation of Near-Edge X-ray Absorption Fine ... - ACS Publications

Letter pubs.acs.org/JPCL

Simulation of Near-Edge X‑ray Absorption Fine Structure with TimeDependent Equation-of-Motion Coupled-Cluster Theory Daniel R. Nascimento and A. Eugene DePrince, III* Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, United States S Supporting Information *

ABSTRACT: An explicitly time-dependent (TD) approach to equation-of-motion (EOM) coupled-cluster theory with single and double excitations (CCSD) is implemented for simulating near-edge X-ray absorption fine structure in molecular systems. The TD-EOMCCSD absorption line shape function is given by the Fourier transform of the CCSD dipole autocorrelation function. We represent this transform by its Padé approximant, which provides converged spectra in much shorter simulation times than are required by the Fourier form. The result is a powerful framework for the blackbox simulation of broadband absorption spectra. K-edge X-ray absorption spectra for carbon, nitrogen, and oxygen in several small molecules are obtained from the real part of the absorption line shape function and are compared with experiment. The computed and experimentally obtained spectra are in good agreement; the mean unsigned error in the predicted peak positions is only 1.2 eV. We also explore the spectral signatures of protonation in these molecules.

ear-edge X-ray absorption fine structure (NEXAFS) is a powerful spectroscopic technique that probes excitations of core electrons to unoccupied valence orbitals in molecular systems. The resulting spectra contain rich structural information that is simultaneously atom-specific and extremely sensitive to chemical environment. When coupled with theory and computations that guide the interpretation of spectra, NEXAFS can provide detailed information regarding hybridization, charge transfer, or molecular orientation relative to a surface. The development of sophisticated quantum-chemical methodologies that can reliably treat core-level excitations is thus essential to the application of NEXAFS within surface and materials science. A variety of popular electronic structure methods have been applied to the prediction of NEXAFS in many-electron systems, including linear response time-dependent (TD) density functional theory (DFT),1,2 real-time TDDFT,3−5 and orthogonality-constrained DFT,6,7 as well as many-body approaches such as coupled-cluster (CC) response theory,8−14 equation-of-motion (EOM) CC theory,15,16 and the secondorder algebraic diagrammatic construction [ADC(2)].17,18 Among these methods, TDDFT appears to be the most popular due to its low cost relative to that of the many-body approaches. However, TDDFT suffers from several well-known deficiencies that diminish its utility for general applications. It is therefore desirable to address the computational shortcomings of more accurate many-body approaches to the excited-state problem. Some of the most reliable quantum-chemical methods for computing valence excited-state energies and properties are based upon the EOM-CC15,19−28 and linear response CC29−33 hierarchies. Unfortunately, conventional EOM-CC and CC response approaches rely on iterative Davidson-type diagonal-

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© XXXX American Chemical Society

ization techniques34,35 to compute excited-state energies and wave functions that, in their simplest form, converge roots sequentially from lowest to highest in energy. Because the energies involved in NEXAFS can span several hundred or thousand electron volts, such bottom-up approaches may involve the determination of thousands of excited-state wave functions, which obviously represents a significant computational challenge. Fortunately, several recent advances have extended the utility of CC-based descriptions of electronically excited states to this regime. Among these approaches are the CC complex polarization propagator (CC-CPP)8−11 and the energy-specific (ES) EOM-CC methods.16 These strategies, while promising, do not necessarily fully resolve the challenges associated with simulating NEXAFS, though. For example, the CC-CPP still requires the solution of many frequencydependent problems, and both ES-EOM-CC and CC-CPP require some a priori knowledge of the spectral region of interest before their application. A complementary approach that provides a slightly more black box description of NEXAFS can be obtained from the recently developed explicitly timedependent EOM-CC (TD-EOM-CC) formalism36 built upon the time evolution of the CC dipole function. In this Letter, we present a TD-EOM-CC with single and double excitations (TD-EOM-CCSD) approach for simulating broadband absorption spectra in molecular systems. The method is formally equivalent to conventional EOM-CCSD while offering several potential advantages. First, the storage requirements are modest; all spectral information is obtained from the time evolution of a single quantity, the CCSD dipole Received: May 15, 2017 Accepted: June 13, 2017 Published: June 13, 2017 2951

DOI: 10.1021/acs.jpclett.7b01206 J. Phys. Chem. Lett. 2017, 8, 2951−2957

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The Journal of Physical Chemistry Letters function, which has the same dimension as the ground-state CCSD wave function. Second, because the CCSD dipole autocorrelation function provides spectral information over an arbitrarily broad energy window, TD-EOM-CCSD is naturally suited for simulating the high-energy features relevant to NEXAFS. We demonstrate the utility of TD-EOM-CCSD in the context of NEXAFS by computing K-edge absorption spectra for carbon, nitrogen, and oxygen in several small molecules: carbon monoxide, hydrogen cyanide, and formaldehyde. The predicted principal absorption features at each edge appear within 2 eV of those determined from experiment. We also explore the effect of protonation on these features. In the case of the formyl and isoformyl cations, the carbon K-edge is equally sensitive to protonation at either carbon or oxygen, whereas the oxygen K-edge is only sensitive to protonation at oxygen. Hence, NEXAFS can clearly differentiate between the formyl and isoformyl cations. Within the framework of EOM-CC, it can be shown36 that each Cartesian component of the linear absorption line shape can be expressed in terms of the time evolution of ground-state quantities as

|Mξ(0)⟩ = μξ̂ |Φ0⟩

respectively. These dipole functions are expanded in the same determinant basis that defines the EOM-CCSD excitation manifold as ⎛ ⟨M̃ ξ (t )| = ⟨Φ0|⎜⎜m̃ 0 + ⎝

∫−∞ dt e−iωt ⟨Φ0|(1 + Λ̂)μξ̂ eiH̅ t μξ̂ |Φ0⟩

⎛ |Mξ(t )⟩ = ⎜⎜m0 + ⎝

∑ tiaaa†̂ aî + ia

1 4

Iξ(ω) =

(2)

Iξ(ω) =

∑ λaiaî †aâ + ia

1 4

i

(3)

(4)

(5)

where â† and â represent Fermionic creation and annihilation operators, respectively. Here, the indices i and j (a and b) represent spin orbitals that are occupied (unoccupied) in the reference configuration. The cluster and de-excitation amplitudes are obtained from a conventional ground-state CCSD computation. We introduce TD left and right CCSD dipole functions, which are defined at time t = 0 as ⟨M̃ ξ (0)| = ⟨Φ0|(1 + Λ̂)μξ̂

ijab

⎠

∫−∞ dt e−iωt ⟨M̃ξ(0)|Mξ(−t )⟩

(10)

∫0

∞

dt e iωt ⟨M̃ ξ (0)|Mξ(t )⟩ + c.c.

(11)

∂ |Mξ(t )⟩ = H̅ N|Mξ(t )⟩ ∂t

(12)

Equation 12 can be integrated numerically using standard numerical methods; in this work, we employ the fourth-order Runge−Kutta procedure. In practice, the integration of eq 12 requires repeated application of the normal-ordered similaritytransformed Hamiltonian on the right dipole function. At the TD-EOM-CCSD level of theory, this operation is equivalent to the usual sigma-vector builds that arise within conventional, frequency domain EOM-CCSD methods. With a discretized time coordinate, the absorption line shape given by eq 11 reduces to a discrete Fourier transform. Because the discrete transform has poor numerical convergence properties in general, fully resolving closely spaced spectral features may require that one propagate the dipole function for a prodigious and perhaps computationally infeasible number of time steps. However, this procedure can be significantly accelerated by considering the Padé approximant to the discrete Fourier transform. A similar approach was employed recently to accelerate the Fourier transform of the TD dipole moment in the context of real-time TDDFT.5 The Padé representation of the line shape is essential for the practical application of TD-EOM-CCSD to NEXAFS. Consider a simulation consisting of P total time steps of length Δt; the time at the pth step is t = pΔt. In this case, the discrete Fourier form of the absorption line shape can be expressed as a polynomial in an auxiliary parameter, z = eiωΔt, as

∑ λabijaî †aj†̂ ab̂ aâ ijab

⎞

∑ mijabaa†̂ ab̂ †aĵ aî ⎟⎟|Φ0⟩

The time dependence of the right dipole function is governed by the TD Schrödinger equation

and Λ̂ =

ia

1 4

Note that the line shape could equivalently be expressed in terms of the time evolution of the left dipole function.36 Further, it can be shown that the CC dipole function possesses the time-reversal symmetry, |M(−t)⟩ = |M(t)⟩*, and as a result, the line shape function can be re-expressed as

∑ tijabaa†̂ ab̂ †aĵ aî ijab

∑ miaaa†̂ aî +

∞

(1)

The symbol T̂ represents the CC excitation operator, and ECC represents the energy of the CC ground-state wave function. At the TD-EOM-CCSD level of theory, both the excitation and de-excitation operators are truncated at the level of single and double excitations as T̂ =

⎠

ijab

At time t = 0, the m and m̃ amplitudes can be evaluated from eqs 6 and 7 according to Wick’s theorem;37 these amplitudes then carry the time dependence of the dipole functions. Now, recognizing that the exponential in eq 1 is the complex conjugate of the time evolution operator, the absorption line shape can be expressed compactly as

Here, |Φ0⟩ represents a single (Hartree−Fock) reference determinant, Λ̂ is the CC de-excitation operator, μ̂ ξ is the ξ component of the dipole operator (ξ = x, y, or z), and H̅ N represents the normal-ordered similarity-transformed Hamiltonian ̂ ̂ H̅ N = e−T Ĥ eT − ECC

⎞

∑ mab̃ ij aî †aj†̂ ab̂ aâ ⎟⎟

(9)

N

2 ω ∑ ℜ{Iξ(ω)} 3 ξ

ia

1 4

and

and the oscillator strength is obtained from the real part of this line shape f (ω) =

∑ mã iaî †aâ +

(8)

∞

Iξ(ω) =

(7)

(6)

and 2952


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Figure 1. Convergence of Fourier- and Padé-derived oscillator strengths with the number of time steps (T/Δt) for Π-symmetry states at the (a) carbon and (b) oxygen K-edges in carbon monoxide.

Figure 2. Computed broadband absorption spectrum for carbon monoxide.

⎡ P ⎤ I(z) = ⎢ ∑ cpz p + c.c.⎥ − c0 ⎢⎣ ⎥⎦ p=0

1. Specifically, we consider the convergence of the oscillator strength at the carbon and oxygen K-edges in carbon monoxide. We consider only excitations of Π-symmetry in this analysis; because the molecule is aligned along the z-axis, the Πsymmetry oscillator strength includes contributions from Ix(ω) and Iy(ω). The number of time steps was set by fixing Δt = 0.01 atomic units and varying the total simulation time (T). The principal peaks at the carbon K-edge (∼287 eV) and the oxygen K-edge (∼535 eV) are well described by the Padé (blue lines) form of the absorption line shape after only 2000 time steps. Additional experimentally relevant features in the 285− 300 eV and 530−545 eV ranges (which correspond to bound states) emerge after 5000 time steps and are fully resolved after 10 000 time steps. These results contrast significantly with those from the Fourier (red lines) representation of the absorption line shape. Only after 10 000 time steps is it even clearly apparent that the principal absorption feature emerges at each edge, and some features are not resolved even after 20 000 time steps. In fact, the Fourier form of the line shape function does not yield fully resolved spectra with fewer than 100 000 time steps. Hence, the Padé approximant representation of the absorption line shape function requires far shorter simulation times than the usual Fourier representation of the line shape. In this case, the use of Padé approximants reduces the computational effort by a factor of 10.

(13)

with expansion coefficients, cp = ⟨M̃ (0)|M(pΔt)⟩. We adopt a diagonal Padé approximant scheme similar to that described in ref 5, in which this polynomial is expressed as a ratio of polynomials with new sets of expansion coefficients {ap} and {bp} P /2

P

∑ cpz p=0

p

=

∑q = 0 aqzq P /2

∑q = 0 bqzq

(14)

These new coefficients are obtained from the solution of the set of overdetermined equations defined by equating terms at the same power in z in P

∑ p=0

P /2

cpz p· ∑ bqzq = q=0

P /2

∑ aqzq q=0

(15)

Note that this procedure, which is described in detail in ref 5, is performed as a postprocessing step, and it does not significantly contribute to the total computational effort required to generate a TD-EOM-CCSD spectrum. We explore the relative convergence properties of the Fourier and Padé representations of the absorption line shape in Figure 2953


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(∼0.5% error). We make no comparisons between theory and experiment for higher-energy Rydberg states (ns/p/d... with n ≥ 4) as accurate computations likely require a basis set with additional diffuse or Rydberg functions.42−44 Computed K-edge features from TD-EOM-CCSD are similarly accurate with respect to experiment for other small molecules, such as hydrogen cyanide and formaldehyde. Table 1 provides experimentally obtained core-level excitation

One of the useful features of the TD-EOM-CC framework is that one simulation provides spectral information over an arbitrarily broad energy window. For example, the carbon and oxygen K-edge spectra in Figure 1 that correspond to a given number of time steps were obtained from a single computation. Figure 2 illustrates the full broadband TD-EOM-CCSD absorption spectrum for carbon monoxide. The spectrum spans 600 eV, encompassing regions relevant to both valence and core-level excitations. The highlighted portions correspond to the UV−vis region (red), the carbon K-edge (green), and the oxygen K-edge (blue). The remaining portions in black represent states above the ionization threshold that are not well described within a finite basis set. The large number of unphysical states above the ionization threshold and below the K-edge for carbon highlights the utility of a computational framework that avoids explicitly computing and manipulating wave functions for these uninteresting states. Note that these unphysical features could be filtered out through the use of core−valence separation13,38 or non-Hermitian approaches originally developed within the framework of real-time TDDFT.4,39 Both of these strategies are compatible with the TD-EOM-CC framework. Figure 3 compares the TD-EOM-CCSD and experimentally obtained NEXAFS spectra for carbon and oxygen in carbon

Table 1. Experimentally Determined Core-Level Excitation Energies (eV) and Errors in Computed Excitation Energies (eV) from Linear Response TDDFT (B3LYP) and the CC2 and CCSD Variants of TD-EOM-CC error molecule CO

H2CO

HCN

MUE

transition

exp.

B3LYP

CC2

CCSD

C 1s → π* C 1s → 3s C 1s → 3p O 1s → π* O 1s → 3s O 1s → 3p C 1s → π* C 1s → 3s O 1s → π* O 1s → 3s C 1s → π* C 1s → 3s N 1s → π*

287.4 292.4 293.3 533.6 538.9 539.9 285.6 290.2 530.8 535.4 286.4 289.1 399.7

−11.2 −12.6 −12.7 −13.8 −15.8 −15.8 −10.5 −11.8 −14.1 −15.3 −10.7 −11.9 −12.1 13.0

1.6 2.2 2.2 1.5 −0.9 1.4 2.5 2.2 1.4 −0.6 2.1 1.8 1.8 1.7

0.3 1.1 1.3 1.9 1.4 1.6 0.9 1.0 1.4 2.1 0.6 0.8 0.9 1.2

energies for these molecules45,46 and carbon monoxide,40,41 as well as errors in the computed excitation energies at several levels of theory. Results are provided for linear response TDDFT (using the B3LYP functional), TD-EOM-CCSD, and approximate second-order TD-EOM-CC (TD-EOM-CC2).36 Again, TD-EOM-CCSD core excitation energies are in reasonable agreement with experiment, displaying a mean unsigned error (MUE) of 1.2 eV. TD-EOM-CC2 provides slightly less accurate excitation energies with a MUE of 1.7 eV. Linear response TDDFT with the B3LYP functional yields a considerably larger MUE of 13.0 eV, but we note that other DFT-based approaches, such as the orthogonality-constrained DFT,6,7 display significantly better agreement with experiment. The maximum deviations between TD-EOM-CCSD and experimentally obtained excitation energies are roughly 2 eV, which is in reasonable agreement with previous CC response theory studies in slightly larger basis sets.8 These errors can be attributed to both the incompleteness of the one-electron basis set employed as well as an incomplete description of dynamical electron correlation; Coriani et al. have demonstrated that the inclusion of iterative triple excitations reduces the error in the oxygen 1s → π* transition in carbon monoxide by more than 1 eV.8 We now consider the effect of protonation on core-level excitations in carbon monoxide, hydrogen cyanide, and formaldehyde. The protonated species are known to exist in the interstellar medium,47−49 and they can be generated and studied in a laboratory by irradiating interstellar ice analogues at low temperatures and pressures.50−58 The formyl and isoformyl cations have also been studied spectroscopically in the condensed phase at room temperature.59 To our knowledge, there exist no detailed computational or experimental studies

Figure 3. Computed carbon and oxygen K-edge spectra for carbon monoxide. Experimental excitation energies (gray bars) and peak assignments were obtained from refs 40 and 41.

monoxide. The carbon K-edge is characterized by an intense peak at 287.4 eV assigned to the C 1s → π* transition followed by two low-intensity Rydberg peaks at 292.4 and 293.3 eV assigned to the C 1s → 3s(σ) and C 1s → 3p(π) transitions, respectively.40 The oxygen K-edge is characterized by an intense peak at 533.6 eV assigned to the O 1s → π* transition followed by two low-intensity peaks at 538.9 and 539.9 eV assigned to the O 1s → 3s(σ) and O 1s → 3p(π) transitions, respectively.41 The positions of these transitions, as determined by experiment, are indicated by vertical gray lines. The computed spectra are in good agreement with experiment. Carbon K-edge features (Figure 3a) are all predicted to lie within 1.3 eV of the experimentally observed features (∼0.4% error); the computed excitation energies for the π* and 3s(σ) states are slightly overestimated, while that for the 3p(π) state is slightly underestimated. Computed oxygen Kedge features (Figure 3b) are overestimated by 1.4−1.9 eV 2954


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H2COH+, where protonation quenches much of the rich structure present in the spectra of the unprotonated species, particularly at the carbon K-edge. In summary, we have developed an implementation of TDEOM-CCSD that, when coupled with a Padé-approximantbased representation of the absorption line shape function, yields high-quality absorption spectra over arbitrarily broad energy windows. We demonstrated the utility of the approach for simulating NEXAFS in molecular systems. One particularly useful feature in this context is that a single simulation provides spectral information at the K-edge for multiple atoms in the molecule, despite the fact that these features may be separated by hundreds of electron volts. At present, the greatest barriers to the application of TDEOM-CCSD to core-level spectra in larger systems or molecules containing heavier elements are the storage of the four-index quantities associated with the similarity-transformed Hamiltonian and the large number of time steps required to fully resolve the spectra. We are currently exploring several strategies to overcome these limitations. First, the storage of many four-index tensors can be avoided within the densityfitting approximation by re-expressing the equations for the sigma-vector construction in terms of three-index electron repulsion integrals; such an approach will result in a large increase in the number of floating point operations for a single sigma-vector build, though. Second, both the core−valence separation13 and multilevel CC14 schemes are compatible with TD-EOM-CCSD and would result in a large decrease in both the storage and floating point requirements of the approach. Last, we are currently exploring strategies to parallelize the time propagation of the dipole function itself. Our preliminary efforts in this area suggest that this strategy is indeed a viable approach to accelerate a TD-EOM-CCSD computation, and this topic is a subject of ongoing work.

illustrating how protonation modifies the near-edge features for carbon, nitrogen, and oxygen in these molecules. Figure 4 illustrates computed carbon, nitrogen, and oxygen K-edge spectra for carbon monoxide, hydrogen cyanide, and

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COMPUTATIONAL DETAILS The TD-EOM-CCSD approach was implemented as a plugin to the PSI4 electronic structure package.61 All computations employed the Cholesky decomposition approximation to the electron repulsion integral tensor with an error threshold of 10−4 Eh. The time propagation was carried out using a fourthorder Runge−Kutta numerical integrator with a time step of 0.01 atomic units (∼0.24 as). The spectra in Figures 2−4 were generated using a total simulation time of 500 atomic units. In all simulations, the dipole autocorrelation function was artificially damped using the Lorentz distribution function, w(t) = e−t/τ, with τ = 0.005 atomic units. All molecular geometries were optimized within the aug-cc-pVTZ basis set at the CCSD with a perturbative treatment of triple excitations [CCSD(T)]62 level of theory. Linear response TDDFT computations with the B3LYP functional were performed using the Q-Chem electronic structure package.63 The aug-ccpVTZ basis set was employed within all computations on neutral molecules. TD-EOM-CCSD computations on the cation species employed the cc-pVDZ basis set for hydrogen and the aug-cc-pVTZ basis set for all other atoms. We note that, in this case, the core-level spectra are insensitive to the choice of basis set for hydrogen. We repeated the computations presented in Figure 4 using TD-EOM-CC2 and the aug-ccpVTZ basis set on all atoms, and in this case, the qualitative changes to the spectra upon protonation were essentially the same as those discussed above. These data are provided in the Supporting Information. Also included in the Supporting

Figure 4. Computed K-edge spectra for small neutral and protonated molecules.

formaldehyde, as well as those of four protonated species: HCNH+, H2COH+, HCO+, and HOC+. Some spectral signatures of protonation are consistent across all four species. In each cation, the 1s → π* transition for the protonated atom (e.g., nitrogen in HCNH+) is slightly quenched, whereas the opposite trend is observed for the 1s → π* transition in the atom that is not protonated; these intensity changes may arise from changes in the delocalization of the π* orbital60 upon protonation. The only exception is in the formyl cation, where the 1s → π* transition in oxygen is slightly less intense upon protonation at carbon. We also observe a consistent hypsochromic shift in the 1s → π* transition for the protonated atom in all four species. Changes to the 1s → π* transition in the atom that is not protonated are less consistent. For HCNH+ and HOC+, we observe a similar hypsochromic shift in the carbon 1s → π* transition. However, in the case of carbon in H2COH+ and oxygen in HCO+, we observe a slight bathochromic shift in the 1s → π* transitions. The lack of any significant change to the principal peak at the oxygen K-edge in the formyl cation provides a clear way of differentiating the isoformyl and formyl cations within the context of NEXAFS. Changes in the character of the Rydberg excitations are more complex. The most significant changes occur in HCNH+ and 2955


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coupled cluster, density functional, and static-exchange levels of theory. J. Chem. Phys. 2013, 138, 124311. (12) List, N. H.; Coriani, S.; Kongsted, J.; Christiansen, O. Lanczosdriven coupled-cluster damped linear response theory for molecules in polarizable environments. J. Chem. Phys. 2014, 141, 244107. (13) Coriani, S.; Koch, H. Communication: X-ray absorption spectra and core-ionization potentials within a core-valence separated coupled cluster framework. J. Chem. Phys. 2015, 143, 181103. (14) Myhre, R. H.; Coriani, S.; Koch, H. Near-Edge X-ray Absorption Fine Structure within Multilevel Coupled Cluster Theory. J. Chem. Theory Comput. 2016, 12, 2633−2643. (15) Nooijen, M.; Bartlett, R. J. Description of core-excitation spectra by the open-shell electron-attachment equation-of-motion coupled cluster method. J. Chem. Phys. 1995, 102, 6735−6756. (16) Peng, B.; Lestrange, P. J.; Goings, J. J.; Caricato, M.; Li, X. Energy-specific equation-of-motion coupled-cluster methods for highenergy excited states: Application to K-edge X-ray absorption spectroscopy. J. Chem. Theory Comput. 2015, 11, 4146−4153. (17) Plekan, O.; Feyer, V.; Richter, R.; Coreno, M.; de Simone, M.; Prince, K.; Trofimov, A.; Gromov, E.; Zaytseva, I.; Schirmer, J. A theoretical and experimental study of the near edge X-ray absorption fine structure (NEXAFS) and X-ray photoelectron spectra (XPS) of nucleobases: Thymine and adenine. Chem. Phys. 2008, 347, 360−375. (18) Wenzel, J.; Wormit, M.; Dreuw, A. Calculating X-ray Absorption Spectra of Open-Shell Molecules with the Unrestricted AlgebraicDiagrammatic Construction Scheme for the Polarization Propagator. J. Chem. Theory Comput. 2014, 10, 4583−4598. (19) Geertsen, J.; Rittby, M.; Bartlett, R. J. The equation-of-motion coupled-cluster method: Excitation energies of Be and CO. Chem. Phys. Lett. 1989, 164, 57−62. (20) Comeau, D. C.; Bartlett, R. J. The equation-of-motion coupledcluster method. Applications to open- and closed-shell reference states. Chem. Phys. Lett. 1993, 207, 414−423. (21) Stanton, J. F.; Bartlett, R. J. The equation of motion coupledcluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties. J. Chem. Phys. 1993, 98, 7029−7039. (22) Stanton, J. F. Many-body methods for excited state potential energy surfaces. I. General theory of energy gradients for the equationof-motion coupled-cluster method. J. Chem. Phys. 1993, 99, 8840− 8847. (23) Stanton, J. F.; Gauss, J. Analytic energy derivatives for ionized states described by the equation-of-motion coupled cluster method. J. Chem. Phys. 1994, 101, 8938−8944. (24) Stanton, J. F.; Gauss, J. Analytic energy derivatives for the equation-of-motion coupled-cluster method: Algebraic expressions, implementation and application to the S1 state of HFCO. Theor. Chim. Acta 1995, 91, 267−289. (25) Krylov, A. I. Size-consistent wave functions for bond-breaking: the equation-of-motion spin-flip model. Chem. Phys. Lett. 2001, 338, 375−384. (26) Levchenko, S. V.; Krylov, A. I. Equation-of-motion spin-flip coupled-cluster model with single and double substitutions: Theory and application to cyclobutadiene. J. Chem. Phys. 2004, 120, 175−185. (27) Krylov, A. I. Spin-flip equation-of-motion coupled-cluster electronic structure method for a description of excited states, bond breaking, diradicals, and triradicals. Acc. Chem. Res. 2006, 39, 83−91. (28) Piecuch, P.; Gour, J. R.; Włoch, M. Left-eigenstate completely renormalized equation-of-motion coupled-cluster methods: Review of key concepts, extension to excited states of open-shell systems, and comparison with electron-attached and ionized approaches. Int. J. Quantum Chem. 2009, 109, 3268−3304. (29) Monkhorst, H. J. Calculation of properties with the coupledcluster method. Int. J. Quantum Chem. 1977, 12, 421−432. (30) Dalgaard, E.; Monkhorst, H. J. Some aspects of the timedependent coupled-cluster approach to dynamic response functions. Phys. Rev. A: At., Mol., Opt. Phys. 1983, 28, 1217−1222. (31) Sekino, H.; Bartlett, R. J. A linear response, coupled-cluster theory for excitation energy. Int. J. Quantum Chem. 1984, 26, 255−265.

Information is a brief study justifying the choice of the aug-ccpVTZ basis set itself at the TD-EOM-CC2 level of theory.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b01206. Simulations of K-edge spectra for protonated species and a study of basis set effects at the TD-EOM-CC2 level of theory (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

A. Eugene DePrince III: 0000-0003-1061-2521 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant No. CHE-1554354. Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund for partial support of this research (54668-DNI6).

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REFERENCES

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