Linear Absorption Spectra from Explicitly Time-Dependent Equation-of

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Linear Absorption Spectra from Explicitly Time-Dependent Equationof-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 ABSTRACT: We report an explicitly time-dependent approach to the generation of linear absorption spectra for molecular systems within the framework of equation-of-motion (EOM) coupled-cluster (CC) theory. While most time-dependent CC approaches consider the perturbation and timeevolution of a CC wave function, the present work considers the time-evolution of a CC dipole function. The dipole function formalism introduces no approximations and requires the evolution of only one time-dependent quantity, either the left or right dipole function. This time-dependent framework can be used to compute linear absorption spectra for molecules with a high density of states over a broad spectral range, a case for which conventional frequencydomain computations may become impractical. We validate the approach by comparing absorption spectra for small molecules computed at EOM secondorder approximate CC (CC2) and time-dependent EOM-CC2 (TD-EOM-CC2) levels of theory. TD-EOM-CC2 computations are also used to predict extreme ultraviolet absorption spectra for third-row ions that are in reasonable agreement with experiment.

1. INTRODUCTION Time-domain approaches to the ab initio description of light− matter interactions are increasingly common1−7 as they display a number of potential advantages over their frequency-domain counterparts. First, when modeling the response of a manyelectron system to an intense external electric field, explicitly time-dependent methods provide a more robust description of the electron dynamics than that afforded by perturbative approaches. Second, time-dependent approaches can lead to a drastic reduction in the storage requirements for excited-state computations on large systems. For example, in a frequencydomain computation, obtaining information about broad spectral features in molecules with large densities of states may require the determination of hundreds of excited-state wave functions,8 each of which must be stored either in main computer memory or on disk. On the other hand, explicitly time-dependent approaches yield similar information from the analysis of a single time-domain signal and require the storage of only a handful of wave-function-sized objects. Most explicitly time-dependent approaches to the electronic excited-state problem are built upon single-electron theories, such as time-dependent Hartree−Fock theory,9,10 density functional theory (DFT),11−16 or configuration interaction with single excitations.17−21 Explicit time dependence within a many-body method like coupled-cluster (CC) theory22 is much more rare, most likely due to the complexity of the underlying CC ansatz. Nonetheless, several examples of explicitly timedependent CC have recently emerged.2−4,23,24 The primary scope of these studies has been the response of many-electron molecules to intense external electric fields.2−4 Far fewer studies have explored explicitly time-dependent CC theory as a realistic alternative to conventional frequency-domain ap© XXXX American Chemical Society

proaches to the generation of linear absorption spectra. Huber and Klamroth23 generated excited-state energies at the time-dependent coupled cluster with single and double excitations (CCSD) level of theory, which, for some small molecules, were found to be in good agreement with those from equation-of-motion (EOM) CCSD. However, in that work, the quality of the results in general suffered from both numerical instabilities in the propagation scheme and the use of approximate reconstructed configuration-interaction-like wave functions to evaluate the time-dependent dipole moment. The results of Huber and Klamroth were obtained according to the usual approach to explicitly time-dependent electronic structure theory. When a quantum-mechanical system is exposed to a time-dependent external perturbation, the response of the system carries information regarding the excited states accessed via the perturbation. The Fourier transform of measurables such as the dipole moment are then interpreted as absorption or emission spectra. If the perturbation is weak, then spectra obtained in this way should be qualitatively similar to linear-response spectra generated from oscillator strengths obtained from frequency-domain computations. Alternatively, linear-response absorption spectra can be obtained, without approximation, from the Fourier transform of a dipole autocorrelation function. This approach, while well-known,25−43 has largely been neglected within electronic structure theory. In this article, we extend the dipole autocorrelation function formalism to the framework of explicitly time-dependent (TD) EOM-CC. Somewhat surprisingly, it appears that a dipoleReceived: August 11, 2016 Published: October 25, 2016 A

DOI: 10.1021/acs.jctc.6b00796 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation function-based time-dependent EOM-CC approach requires the time evolution of only a single time-dependent quantity: either the left or right dipole function. On the other hand, a time-dependent CC approach that follows the time evolution of the expectation value of the dipole moment requires the time propagation of two quantities: the left and right CC wave functions. Hence, in principle, the dipole-function-based timedependent CC formalism reduces the computational effort required to evaluate a given component of a linear absorption line shape by a factor of 2. Here, we present the working equations for a time-dependent EOM approximate secondorder CC (CC2)44 method (TD-EOM-CC2). We validate our implementation by computing absorption spectra at both the canonical frequency-domain EOM-CC2 and TD-EOM-CC2 levels of theory. For a set of small closed-shell molecules, EOMCC2 spectra are well-reproduced by TD-EOM-CC2.

R̂ = r0 +

ia

∑ tiaaa†̂ aî + ia

1 4

Iξ(ω) =

where â† (â) represents a creation (annihilation) operator, and, throughout, indices i, j, k, and l (a, b, c, and d) represent spin orbitals that are occupied (unoccupied) in the reference determinant. The cluster amplitudes for the ground state are determined by solving the system of coupled nonlinear ̂ ̂ −T̂ ̂ T̂ equations, ⟨Φai |e−TĤ eT|Φ0⟩ = 0, ⟨Φab ij |e He |Φ0⟩ = 0, etc., ̂ ̂ and the ground-state energy (ECC) is given by ⟨Φ0|e−TĤ eT|Φ0⟩ a ab = E CC . Here, |Φi ⟩ and |Φ ij ⟩ represent, respectively, determinants that are singly or doubly excited, relative to the reference determinant. While explicit time-dependence can be incorporated directly into the t-amplitudes,23,24,45−47 it will be more convenient to consider time dependence in the context of the EOM-CC formalism.48−50 In EOM-CC theory, excited-state wave functions are determined as eigenfunctions of the similaritŷ ̂ transformed Hamiltonian, H̅ = e−TĤ eT, or a shifted similaritytransformed Hamiltonian, H̃ = H̅ − ECC. Note that H̃ is not Hermitian, and excitation energies can be determined from the solution of either the right eigenvalue equation

I(ω) =

I(ω) =

ia

∑ ⟨Φ0|L̂0μR̂ ̂F|Φ0⟩⟨Φ0|L̂ FμR̂ ̂ 0|Φ0⟩δ(ωF − ω)

The Dirac δ-function can be replaced by its Fourier integral iω′t δ(ω′) = ∫ ∞ −∞ dt e , giving ∞

I(ω) =

∫−∞ dt e−iωt ∑ ⟨Φ0|L̂0μR̂ ̂F|Φ0⟩ F

̂ ̂ 0|Φ0⟩ × ⟨Φ0|L̂ F eiωFt μR

(9)

Now, ⟨Φ0|L̂ F is an eigenvector of H̃ , so e can be replaced by the complex conjugate of the time evolution operator, and eq 9 becomes iωFt

∞

I(ω) =

∫−∞ dt e−iωt ∑ ⟨Φ0|L̂0μR̂ ̂F|Φ0⟩ F

iHt̃

̂ ̂ 0|Φ0⟩ × ⟨Φ0|L̂ F e μR

(10)

Recall that ∑FR̂ F|Φ0⟩⟨Φ0|L̂ F = 1̂, so eq 10 simplifies to ∞

I(ω) =

∑ labijaî †aj†̂ ab̂ aâ + ... ijab

(7)

(8)

(3)

1 4

∑ |⟨Ψ0|μ|̂ ΨF⟩|2 δ(EF − E0 − ω)

F

Here, the left and right eigenvectors for state P are given by ⟨Φ0|L̂ P and R̂ P|Φ0⟩, respectively, and ωP represents the difference in energy between state P and the ground state. The symbols L̂ P and R̂ P represent linear excitation operators of the form

∑ laiaî †aâ +

(6)

The ground and excited states are taken to be those obtained from EOM-CC theory, and we recognize that EF − E0 = ωF:

or the left eigenvalue equation

L̂ = l0 +

∑ ρI |⟨Ψ|I μξ̂ |ΨF⟩|2 δ(EF − EI − ω)

F

(2)

⟨Φ0|L̂ PH̃ = ⟨Φ0|L̂ PωP

(5)

ijab

Here, EI and EF are the energies of the initial and final electronic states, respectively, ω is the frequency of the incident radiation, ρI is the Boltzmann factor for the initial states (assuming that the system is in thermal equilibrium), and μ̂ξ is the component of the dipole operator parallel to the ξ axis, defined in second quantization as μ̂ξ = ∑pqμξpqâ†pâq. Indices p and q run over all spin orbitals. For the sake of clarity, we ignore the label ξ for the remainder of this derivation. At zero temperature, the Boltzmann factor is zero for all states except the ground state (|Ψ0⟩), and eq 6 becomes

(1)

̃ ̂P|Φ0⟩ = ωPR̂P|Φ0⟩ HR

∑ rijabaa†̂ ab̂ †aĵ aî + ...

IF

∑ tijabaa†̂ ab̂ †aĵ aî + ... ijab

1 4

The eigenfunctions ⟨Φ0|L̂ P and R̂ P|Φ0⟩ form a biorthogonal basis, so ⟨Φ0|L̂ PR̂ Q|Φ0⟩ = δPQ, and ∑PR̂ P|Φ0⟩⟨Φ0|L̂ P = 1̂, where the sum runs over all states. This latter property will allow us to develop a time-domain approach to EOM-CC theory that requires the explicit determination of the right and left eigenvectors for the ground state only. 2.2. Linear Absorption Line Shape Function. A detailed derivation relating the Schrö dinger representation of an absorption line shape to the Heisenberg representation of the time evolution of the dipole operator can be found in ref 26. Here, we provide a similar derivation specific to EOM-CC theory entirely within the Schrödinger picture. Consider first the Fermi’s Golden Rule expression for the line shape function for incident radiation polarized in the ξ direction (ξ ∈ {x, y, z}):

2. THEORY 2.1. EOM-CC Theory. The ground-state CC wave function ̂ can be written as |ΨCC⟩ = eT|Φ0⟩, where T̂ is the cluster operator, and in this work, |Φ0⟩ represents a single closed-shell (restricted Hartree−Fock) determinant. The cluster operator is defined in second-quantized notation as T̂ =

∑ riaaa†̂ aî +

∫−∞ dt e−iωt ⟨Φ0|L̂0μê iHt̃ μR̂ ̂ 0|Φ0⟩

(11)

Left and right dipole functions can be defined as ⟨M̃ (0)| = ⟨Φ0|L̂ 0μ̂ and |M(0)⟩ = μ̂ R̂ 0|Φ0⟩, respectively, where |M(0)⟩ indicates that the dipole function is evaluated at time t =

(4)

and B


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Journal of Chemical Theory and Computation

The elements of the left CC2 dipole function at time t = 0 are obtained by applying Wick’s theorem53 to eq 15

0. The complex conjugate of the time-evolution operator can be used to propagate the left dipole function forward in time ∞

I(ω) =

∫−∞ dt e−iωt ⟨M̃ (t )|M(0)⟩

m̃ 0 =

(12)

i

or the right dipole function backward in time ∞

I(ω) =

∫−∞ dt e

−iωt

⟨M̃ (0)|M( −t )⟩

∑ μab λbi − ∑ (μij λaj − μjj λai) + ∑ μbj λabij b

j

bj

(13)

(20) ij mab ̃ = μbj λai +

∑ μkk λabij + 7(ab) ∑ μac λcbij c

k

− 7(ij) ∑ (14)

μik λabkj

(21)

k

where in eq 14 we have reintroduced the label ξ. Quite surprisingly, it appears that the present time-dependent CC formalism requires the propagation of either the right or left dipole function but not both. In previous explicitly timedependent CC investigations, optical response properties were obtained by perturbing the system with a time-dependent external electric field and following the response of the system in time. Spectroscopic information was then extracted from the time evolution of the dipole moment, calculated as ⟨Ψ̃(t)|μ̂|Ψ(t)⟩. Because the similarity-transformed Hamiltonian is not Hermitian, knowledge of such an expectation value requires that both left and right wave functions be evolved.2,4,23,51,52 Hence, the numerical effort required to evaluate the x-, y-, or z-component of the absorption line shape is reduced by a factor of 2 in the present formalism. 2.3. CC Dipole Functions. In EOM-CC theory, the ground-state right and left eigenfunctions are given by R̂ 0 = 1 and L̂ 0 = 1 + Λ̂, respectively, where Λ̂ is a de-excitation operator. At the approximate second-order CC (CC2)44 level of theory, the de-excitation operator is defined as 1 Λ̂ = ∑ia λaiaî †aâ + 4 ∑ijab λabijaî †aj†̂ ab̂ aâ , and the cluster operator, T̂ , is similarly truncated at the level of single and double excitations. The λ-amplitudes can be determined with knowledge of the t-amplitudes from a standard ground-state CC2 computation. The left and right CC2 dipole functions at time t = 0 are then ̃ | = ⟨Φ0|(1 + Λ̂)μ ̂ ⟨M(0)

(19)

ia

mã i = μia +

An absorption spectrum or oscillator strength can then be extracted from the real part of either line shape 2 f (ω) = ω ∑ ℜ{Iξ(ω)} 3 ξ

∑ μii + ∑ μia λai

Here, the symbol 7 represents an antisymmetric permutation operator defined as 7(ij)A(i)B(j) = A(i)B(j) − A(j)B(i). 2.4. Time Evolution. The time evolution of the right or left dipole function is governed by the time-dependent Schrödinger equation → i ∂t |M(t )⟩ = H̃ |M(t )⟩ (22) or its left-hand variant ← −i⟨M̃ (t )| ∂t = ⟨M̃ (t )|H̃

(23)

→ ← where the symbols ∂t and ∂t indicate that the time-derivative operator acts to the right or left, respectively. Here, it is assumed that H̃ , and hence the underlying orbitals and groundstate t-amplitudes, are independent of time. The time derivatives of the m- and m̃ -amplitudes are given by

ṁ 0 =

∑ mia -ia + ia

ṁ ia =

1 4

∑ mijab⟨ij

ab⟩ (24)

ijab

∑ mib -ab − ∑ mja -ij + ∑ mjb >jabi + ∑ mijab - jb b

j

1 − 2

∑

jb

mjkab >jkib

jkb

1 + 2

∑

jb

mijbc >ajbc (25)

jbc

ṁ ijab = 7(ab) ∑ mkb >kaij + 7(ij) ∑ mic >acbj − mijabΔijab

(15)

k

c

and

(26)

|M(0)⟩ = μ|̂ Φ0⟩

and

(16)

m̃̇ 0 = 0

We express the dipole functions in the determinant basis that defines the EOM-CC2 excitation manifold, so ⎛ ⟨M̃ (t )| = ⟨Φ0|⎜⎜m̃ 0 + ⎝

mã̇ i =

⎞ 1 ∑ mã iaî †aâ + ∑ mab̃ ij aî †aj†̂ ab̂ aâ ⎟⎟ 4 ijab ⎠ ia

∑ m̃ bi -ba − ∑ mã j - ji + ∑ m̃ bj >ibaj b

j

1 − 2

(17)

and ⎛ |M(t )⟩ = ⎜⎜m0 + ⎝

(27)

∑

jk mab ̃ >ibjk

jkb

bj

1 + 2

∑ m̃ bcij >bcaj (28)

jbc

ij mab ̃̇ = 7(ab) ∑ m̃ bk >ijka + 7(ij) ∑ mc̃ i >cjab

⎞ 1 ∑ miaaa†̂ aî + ∑ mijabaa†̂ ab̂ †aĵ aî ⎟⎟|Φ0⟩ 4 ijab ⎠ ia

k

+

7(ab)7(ij)mã i - jb

c

−

ij mab ̃ Δijab

(29)

Δab ij

Here, = ϵi + ϵj − ϵa − ϵb, where ϵi is the energy of the ith molecular orbital, and - and > are, respectively, one- and two-body components of the similarity-transformed Hamiltonian, the elements of which are given in Table 1. Equations

(18)

and the m- and m̃ -amplitudes carry the time dependence of the dipole functions. At time t = 0, the elements of the right CC2 dipole function are simply m0 = ∑iμii, mai = μai, and mab ij = 0. C


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Journal of Chemical Theory and Computation 22 and 23 can be integrated using standard numerical techniques. Table 1. One- and Two-Body Elements of the CC2 Similarity-Transformed Hamiltoniana 1

-ca = fca + ∑kd tkd⟨ak ∥ cd⟩ − 2 (∑k tka -kc + ∑k fkc tka + ∑kld τklad⟨kl ∥ cd⟩) 1

-ik = fki + ∑lc tlc⟨kl ∥ ic⟩ + 2 (∑c tic -kc + ∑c fkc tic + ∑lcd τilcd⟨kl ∥ cd⟩)

-kc = fkc + ∑ld tld⟨kl ∥ cd⟩ >kbcj = ⟨kb ∥ cj⟩ + ∑d t jd⟨kb ∥ cd⟩ − ∑l tlb⟨kl ∥ cj⟩ − ∑ld (t jldb + t jdtlb)⟨kl ∥ cd⟩

Figure 1. Molecules used to validate the TD-EOM-CC2 method against canonical EOM-CC2.

>kbij = ⟨kb ∥ ij⟩ − ∑c fkc tijbc + ∑cd tict jd⟨kb ∥ cd⟩ + 7(ij) ∑c tic⟨kb ∥ cj⟩ − ∑l tlb⟨kl ∥ ij⟩ − 7(ij) ∑lc tlbt jc⟨kl ∥ ic⟩ − ∑lcd tict jdtlb⟨kl ∥ cd⟩

were used within the Hartree−Fock and CC2 portions of the TD-EOM-CC2 procedure, respectively. For the smallest molecules considered (Figure 2a,b), the TDEOM-CC2 spectra are nearly indistinguishable from those obtained from canonical EOM-CC2. Small differences in oscillator strengths begin to emerge for butadiene and malonaldehyde, but in general, the canonical EOM-CC2 spectra are well-reproduced by TD-EOM-CC2. In Figure 2d, significant deviations between the spectra can be found at energies exceeding 18 eV. This difference emerges because the canonical EOM-CC2 computation involved only 100 total roots (50 of A′ symmetry and 50 of A″ symmetry), and this number is insufficient to span the energy window from 2−20 eV. Hence, this discrepancy reveals one of the benefits of a time-domain approach: one need not consider how many eigenfunctions must be determined to span an arbitrary energy window. Similar deviations can be seen at high energies for the spectra corresponding to larger molecules in panels e−h of Figure 2. The canonical EOM-CC2 absorption spectrum for hexatriene accounts for the 50 lowest Au- and 50 lowest Busymmetry excited states (transitions to the Ag and Bg states are dipole forbidden). The spectrum for methylsalicylate accounts for 10 states each of A′ and A″ symmetry, and the spectra for the photoactive yellow protein (PYP) and green fluorescent protein (GFP) chromophores include 25 states of A′ symmetry and 25 states of A″ symmetry. In each case, the low-energy portions of the spectra are well-reproduced by TD-EOM-CC2, and the spectra obviously differ in the higher-energy regions for the reasons given above. The TD-EOM-CC2 spectra presented in Figure 2 were generated from the time evolution of the right dipole function (see eq 13). However, the formalism presented in Section 2.2 suggests that equivalent line shapes can be obtained from the time evolution of the left dipole function (see eq 12). Figure 3 demonstrates the equivalence of these line shapes for carbon monoxide, methane, butadiene, and malonaldehyde; spectra from both propagation schemes are totally indistinguishable. Figure 3 also includes the difference between the spectra as a percentage, relative to the largest feature in each spectrum in the 4−20 eV range. The maximum deviations between the spectra are only on the order of 1%. The shape of the difference curves suggests small, possibly systematic disparities between excitation energies determined by the propagation of the left or right dipole function. These differences are quite small, though; for butadiene and malonaldehyde, the left and right propagation schemes yield maxima in the lowest-energy features that differ by only 1.6 and 4.3 meV, respectively. The excellent agreement between spectra lead us to conclude

>akcd = ⟨ak ∥ cd⟩ − ∑l tla⟨lk ∥ cd⟩

>klic = ⟨kl ∥ ic⟩ + ∑d tid⟨kl ∥ dc⟩ 1

Here, τijab = tijab + 4 7(ij)7(ab)tiat jb . The elements of these tensors were taken from the Supporting Information of ref 54.

a

3. COMPUTATIONAL DETAILS The overall time-dependent EOM-CC2 (TD-EOM-CC2) procedure involves a ground-state CC2 computation, the solution of the CC2 λ-amplitude equations, and the time evolution of either the m- or m̃ -amplitudes. Each of these components was implemented within a single plugin to the PSI4 electronic structure package.55 Canonical EOM-CC2 computations were also performed, using the native implementation in PSI4. All TD-EOM-CC2 computations employ either density fitting (DF)56−59 or Cholesky decomposition (CD)60−63 approximations to the electron repulsion integral (ERI) tensor in both the Hartree−Fock and CC2 procedures, whereas canonical EOM-CC2 computations use only conventional fourindex ERIs. In the TD-EOM-CC2 computations, the time propagation was carried out with the fourth-order Runge− Kutta numerical integrator with a time step of 0.05 au (0.0012 fs) and a total simulation time of 1000 au (≈24 fs), unless otherwise noted. Geometries for all molecules, except carbon monoxide and methane, were taken from the Supporting Information of ref 54. Geometries for carbon monoxide and methane were optimized at the CC2/cc-pVTZ level of theory. 4. RESULTS To validate the present TD-EOM-CC2 approach, we consider absorption spectra for the eight molecules shown in Figure 1 generated from both time-dependent and canonical EOM-CC2 computations. Stick spectra (black lines) and artificially broadened spectra (blue dashed lines) from canonical EOMCC2 computations are given in Figure 2. The features of the broadened spectra have Lorentzian line shapes and 0.272 eV full width at half-maximum (fwhm) . Also provided in Figure 2 are broadened TD-EOM-CC2 spectra (solid red lines) extracted from the time evolution of the right dipole autocorrelation function defined by eq 13. TD-EOM-CC2 spectra were broadened by damping the dipole autocorrelation function by e−t/τ, with τ = 4.84 fs; this choice of damping function yields features with line shapes and fwhm that are consistent with those of the broadened canonical EOM-CC2 spectra. Here, all computations employed the cc-pVDZ basis set, and the cc-pVDZ-JK and cc-pVDZ-RI auxiliary basis sets D


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Figure 2. Normalized and artificially broadened absorption spectra computed with the TD-EOM-CC2 and EOM-CC2 methods.

derived from the semiempirical configuration-interaction (CI)based model of ref 64 (labeled “model”). These spectra were artificially broadened, as described above. Features due to oxidation states not considered in our computations are indicated by shaded red areas. We observe reasonable agreement between TD-EOM-CC2 and experimentally observed peak positions for the assigned peaks. Agreement between experimentally observed relative intensities and those from TD-EOM-CC2 is considerably worse, but we observe good agreement between TD-EOM-CC2 and the model of ref 64, particularly for Al3+ and Si4+. Additional 2s22p6 → 2s22p5ns1 and 2s22p6 → 2s22p5(n − 1)d1 transitions with n > 5 were assigned in the spectra of ref 64, but the present computations cannot describe these excitations with any reliability. In particular, we have found that the higher-energy portions of the Na+ and Mg2+ spectra are extremely sensitive to the choice of basis set. For example, the position of the peak corresponding to 2s22p6 → 2s22p55d1 for Na+ (not shown here) changes by roughly 2 eV when employing either the augcc-pVTZ or aug-cc-pVQZ basis set. Hence, we doubt that the aug-cc-pVQZ basis is flexible enough to describe these higherenergy portions of the spectra. We also note that the splittings in the experimentally observed and model spectra are not reproduced by the present computations, which do not account for spin−orbit interactions.

Figure 3. Normalized time-dependent EOM-CC2 absorption spectra computed by propagating the right dipole function (solid red lines), left dipole function (dashed blue lines), and the percent difference between both spectra for (a) carbon monoxide, (b) methane, (c) butadiene, and (d) malonaldehyde.

that propagation schemes based on eqs 12 and 13 are indeed equivalent. We have confirmed that TD-EOM-CC2 reproduces canonical EOM-CC2 spectra in the 2−20 eV range for a series of small molecules. We now consider the ability of TD-EOM-CC2 to predict extreme ultraviolet (XUV) spectra for four isoelectronic cations: Na+, Mg2+, Al3+, and Si4+. The XUV spectra provided in Figure 4 illustrate the 2s22p6 → 2s22p5ns1 and 2s22p6 → 2s22p5(n − 1)d1 series for these cations with n ∈ {4, 5}. Experimentally obtained spectra were digitized from ref 64; the photoabsorption measurements reported therein were carried out using the dual laser plasma (DLP) technique.65 The TD-EOM-CC2 simulations employ the aug-cc-pVQZ basis set, and the ERIs within the Hartree−Fock and CC2 portions of the computation were approximated by CD methods, with a CD error threshold of 10−4Eh. In the time propagation, the time step was 0.01 au. The peak assignments were taken from ref 64. Direct comparison of experimentally obtained spectra and the present TD-EOM-CC2 computations is complicated by (i) the sensitivity of the DLP measurements to experimental conditions and (ii) the presence of additional oxidation states of aluminum and silicon.64 To address the former issue, Figure 4 also depicts spectra generated from oscillator strengths

5. CONCLUSIONS We have presented an implementation of explicitly timedependent EOM-CC2 based on the time-evolution of the CC2 dipole function. Like other explicitly time-dependent approaches to the excited-state problem, the present method offers a potential practical advantage over frequency-domain computations; spectral information over an arbitrarily wide energy range can be obtained from a single time-domain simulation, with the storage of only a handful of wave -functionsized objects. Unlike previous implementations of timedependent CC theory, which require the time evolution of both the right and left CC wave functions, the present approach requires the time evolution of only one quantity: the dipole function. Either the left or right dipole function can be evolved; the resulting spectra are nearly indistinguishable. The present procedure can be generalized beyond CC2 to other more sophisticated CC expansions. We have validated our implementation by computing linear absorption spectra for several small molecules using convenE


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Figure 4. Theoretical (red and blue lines) and experimental (black lines) XUV spectra illustrating the 2s22p6 → 2s22p5ns1 and 2s22p6 → 2s22p5(n − 1)d1 transitions for the (a) Na+, (b) Mg2+, (c) Al3+, and (d) Si4+ cations. The shaded red areas in panels (c) and (d) highlight signal from additional oxidation states that were not considered theoretically.64 Theoretical spectra (red lines): TD-EOM-CC2 computations employ the aug-cc-pVQZ basis set; (blue lines) oscillator strengths were taken from the configuration interaction model of ref 64. Experimental spectra were digitized from ref 64.

storage of many uninteresting low-energy eigenstates. Such strategies include a core−valence separation framework,66 energy-specific eigensolvers,67 or complex polarization propagator/damped CC response formalisms.68−70 The latter approach has the desirable property that it can be used to determine spectral features within small arbitrary energy windows. However, a fine-grain description of a large window may nonetheless require that the damped CC response equations be solved for thousands of frequencies. It is in this regime that a time-domain approach could potentially be of use.

tional EOM-CC2 and TD-EOM-CC2 computations; EOMCC2 spectra are well-reproduced by the TD-EOM-CC2 approach. We also numerically demonstrated the equivalence of line shapes generated by the evolution of either the right or left dipole function. Lastly, we demonstrated that TD-EOMCC2 predicts XUV absorption spectra for several third-row ions in reasonable agreement with experimentally obtained spectra. These data demonstrate how time-domain approaches can be used to obtain spectral information over a broad range of frequencies. While the treatment of these bare ions at the conventional EOM-CC2 level of theory does not represent a particularly challenging case, these proof-of-principle computations suggest that TD-EOM-CC methods could potentially be of great use in describing XUV spectra in complexes involving such ions. A comparable study of magnesoscene, for example, using conventional EOM-CC2 approaches and the same quality basis set would likely be impossible due to the presence of an extremely large number of low-energy eigenstates. Similar arguments can be made for the utility of time-domain approaches to X-ray absorption near edge structure (XANES) prediction. Accordingly, we are currently assessing the efficacy of TD-EOM-CC approaches for modeling XANES in small molecules. In this regime, TD-EOM-CC could potentially complement or serve as an alternative to other CC-based approaches specifically designed to avoid the determination and

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Funding

Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund for support of this research (grant 54668-DNI6). Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Schlegel, H. B.; Smith, S. M.; Li, X. J. Chem. Phys. 2007, 126, 244110.

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