Coupled Cluster Study of Photoionization and Photodetachment Cross

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Coupled cluster study of photoionization and photodetachment cross sections Bruno Nunes Cabral Tenório, Marco Antonio Chaer Nascimento, Sonia Coriani, and Alexandre Braga Rocha J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00524 • Publication Date (Web): 19 Jul 2016 Downloaded from http://pubs.acs.org on July 23, 2016

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Journal of Chemical Theory and Computation

Coupled Cluster Study of Photoionization and Photodetachment cross sections Bruno Nunes Cabral Tenorio,† Marco Antonio Chaer Nascimento,† Sonia Coriani,‡ and Alexandre Braga Rocha∗,† UFRJ - Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos, 149, Rio de Janeiro - RJ, 21941-909, Brasil, Dipartimento di Scienze Chimiche e Farmaceutiche, Università degli Studi di Trieste, via L. Giorgieri 1, I-34127, Trieste, Italy, and Aarhus Institute of Advanced Studies, Aarhus University, 8000 Aarhus C, Denmark E-mail: [email protected]

Abstract To simulate the photoionization and photodetachment cross sections, as well as the polarizability dispersion profiles below the ionisation threshold, the discretized (pseudo-)spectrum stretching over the entire frequency region (including the continuum) obtained from an asymmetric Lanczos algorithm at the coupled cluster singles and doubles level is used to reconstruct the complex dipole polarizability, on which an analytic continuation procedure is then applied. Through a suitable selection of points in the complex plane, which we have shown that can be quite general, we were able to perform the analytical continuation procedure. Results are reported for the atoms ∗

To whom correspondence should be addressed UFRJ - Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos, 149, Rio de Janeiro - RJ, 21941-909, Brasil ‡ Dipartimento di Scienze Chimiche e Farmaceutiche, Università degli Studi di Trieste, via L. Giorgieri 1, I-34127, Trieste, Italy ¶ Aarhus Institute of Advanced Studies, Aarhus University, 8000 Aarhus C, Denmark †

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He and Ne, the molecules H2 , N2 , CH4 , H2 CO, C2 H2 , CO2 , CO, H2 O, NH3 and SO2 , and the anions H− , F− , OH− and NH− 2 . The method employed has proved to work well with a rather small Lanczos chain length as well as with medium size correlation consistent basis sets supplemented with a limited number of continuum-like Gaussian functions. Such features, suggest the applicability of the method to larger systems.

1

Introduction

The study of light-induced ejection and scattering of electrons in a variety of systems, from atoms, molecules, ions and radicals, to clusters and biomolecules, is a very active area of contemporary research. 1–6 Electron-in-the-continuum processes like photoionization and photodetachment play a central role in many emerging fields of modern atomic, molecular, chemical and optical physics, and can provide information about the instantaneous electronic and vibrational state of many-body systems, in other words their structural and dynamical properties. Modern light sources, like the high intensity femtosecond pulses in the extreme UV to X-ray spectral range produced at the free electron laser facilities, the attosecond X-UV sources obtained in high harmonic generation, the high resolution synchrotron and laser facilities, have, with their unprecedented intensity and/or temporal resolution, further enlarged the scope of the investigation of such processes. At the same time, they call for accurate theoretical models to interpret the data collected from the experimental investigation of molecular photo-absorption, photoionization and ionic photo-fragmentation processes. Yet, the theoretical description of photoionization and photodetachment still poses significant challenges, one of them being a fully correlated treatment of both the initial and final states. While ab initio methodologies rooted on sophisticated wave-function ansatzes like, for instance, the coupled cluster one, are routinely or almost routinely used to interpret spectroscopic data of bound states, 7 their application to the description of molecular photoionization phenomena is still lagging behind. One difficulty is extracting the asymptotic information of the unbound electron common to all correlated methods based on square-integrable (so called L2 ) 2


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finite basis sets. In the seventies, Langhoff and coworkers 8–10 proposed a formally convergent procedure to compute total photoionization cross sections based on the application of a Stieltjes imaging technique to a discretized representation of the continuum part of the spectrum. Stieltjes Imaging was rather popular in the eighties and nineties, when it was applied primarily within the Random Phase Approximation (RPA) for Hartree-Fock and Multiconfigurational Self Consistent Field wavefunctions. 11–19 A few years ago, Averbukh and coworkers proposed to apply the Stieltjes Imaging technique to Lanczos pseudo-spectra obtained within the second-order Algebraic Diagrammatic Construction ADC(2) formalism to describe ionization, 20 autoionization 21 and intermolecular decay phenomena. 22–24 Inspired by these studies, Coriani and coworkers 25,26 implemented a similar approach where the Stieltjes quadrature spectrum was generated from coupled-cluster pseudo-spectra obtained by means of an asymmetric Lanczos algorithm for the hierarchy of coupled cluster approximations CCS (coupled cluster singles), CC2 (coupled cluster singles and approximate doubles) and CCSD (coupled cluster singles and doubles). 27,28 Both photoionization and photodetachment cross-sections of closed shell atoms and small molecules/ions in the VUV-UV region frequency region were studied. In 1974, Broad and Reinhardt 29 suggested a method for calculating photoionization cross-section using L2 basis sets by constructing a discrete representation of the dynamic polarizability, and applied it to atomic hydrogen. This strategy was also used in the RPA approximation to calculate photoionization cross-sections of atoms and molecules. 30 Inspired by the work of Broad and Reinhardt, 29 Nascimento and Goddard proposed to obtain both photoionization cross-sections and dynamic polarizabilities in the normal dispersion region using a discrete basis set to represent both the bound and the continuum states. This discrete-basis-set representation was used to build an approximation to the dynamic polarizability, which, in turn, was used in an analytical continuation procedure for complex values of the frequency, employing Padé approximants. The methodology has been applied

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to calculate photoionization cross-section and dynamic polarizabilities of H− (Ref. 31), Li, Li+ (Ref. 32), He (21S, 23S), 33 He (1 S) and H2 . 34 Here we are presenting a methodology which can be regarded as an alternative to the well know Stieltjes imaging procedure when one wants to calculate photoionization cross sections using L2 basis set. Our methodology employs the coupled-cluster pseudo-spectra obtained by means of the asymmetric Lanczos algorithm, to generate the approximation to the dynamic polarizability, on which the analytical continuation procedure for complex values of the frequency via continued fraction is then performed by means of a suitable selection of points on the complex plane which can be quite general, as we discuss in the Appendix. The converged continued fraction yields a representation for the complex polarizability from which the dynamic polarizability and the photoionization cross-section can be immediately extracted as the real and imaginary components, respectively. The results presented in the following sections show that the reduction of the Lanczos chain-length did not lead to significant loss of accuracy, which, together with an adequate choice of the basis set, opens the possibility of application of the method to larger systems. The methodology is tested in calculations of photoionization and photodetachment crosssections of several atomic, molecular and ionic systems, as well as the dynamic polarizability of the neutral atoms and molecules.

2 2.1

Theory The Coupled-Cluster Lanczos (Pseudo-)Spectrum

The CC wave-function ansatz (for a closed-shell system) is defined by the exponential parametrization |CCi = exp(T )|HFi;

T =

X

t µ τµ

µ

4


(1)

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where |HFi is the reference (Hartree-Fock) wave function, and T is the cluster operator, with tµ being the cluster amplitudes and τµ the corresponding excitation operators. The ground state energy and amplitudes are conventionally determined by projection of the Schrödinger equation onto the reference state, and onto a manifold of excitations out of the reference state, respectively

E = hHF| exp(−T )H exp(T )|HFi Ωµ = hµ| exp(−T )H exp(T )|HFi = 0

(2) (3)

In conventional CC linear response (LR) theory, excitation energies (ωj ) and left (Lj ) and right (Rj ) excitation vectors are usually obtained solving the asymmetric eigenvalue equations

ARj = ωj Rj ;

Lj A = ωj Lj

(4)

under the biorthogonality condition Lj Ri = δij . The Jacobian matrix A is defined as

Aµν =

∂Ωµ = hµ| exp(−T )[H, τν ] exp(T )|HFi ∂tν

(5)

Transition strengths (for operator components X and Y ) are then determined from the single residues of the linear response function, and take the form

XY = S0→j

1 X Y Y X ∗ Tj0 ) T0j Tj0 + (T0j 2

(6)

where the left and right transition moments are given by X ¯ j (ωj )ξ X ; T0j = η X Rj + M

X Tj0 = Lj ξ X

(7)

¯ j ) are obtained from the solution of the linear and the auxiliary Lagrangian multipliers M(ω 5



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equation ¯ j (A + ωj I) = −FRj . M

(8)

See e.g. Ref. 35 for a definition of the remaining building blocks. In most implementations, Eq. (4) is solved iteratively via some generalization of the Davidson algorithm. 36 This yields excitation energies (and transition strengths) in a bottomup fashion, which can be inconvenient when aiming at energies and moments above the ionization threshold. An alternative approach to solve Eq. (4), which produces a discretized spectrum covering the whole frequency range, consists in building a (truncated) tridiagonal representation T of the Jacobian matrix A by application of an asymmetric Lanczos algorithm, followed by its straightforward diagonalization. 27,28 The non-zero elements of the tridiagonal matrix T = PT AQ (where PT Q = 1) are given by

Tll = αl =

pTl Aql ;

Tl+1,l = βl =

q

pTl+l ql+1 ;

Tl,l+1 = γl = sgn{pTl+1 ql+1 }βl

(9) (10)

with

ql+1 = βl−1 (Aql − γl−1 ql−1 − αl ql )

(11)

pTl+1 = γl−1 (pTl A − βl−1 pTl−1 − αl pTl )

(12)

The diagonalization of T, conveniently truncated to dimension k ≪ K, K being the full dimension, generates an effective excitation spectrum. The latter is known to converge from the bottom and from the top towards the exact excitation spectrum. 27,28 The dimension k is referred to as Lanczos chain-length. −1 X X η (where uX = ||ξ X || and and pT1 = vX Choosing as Lanczos seeds q1 = u−1 X ξ X X vX = u−1 X η ξ ) yields an approximate diagonal representation of the (complex) linear re-

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sponse function in terms of the left (Li ) and right (Ri ) eigenvectors of the Lanczos pseudospectrum. 27,28 The residues of such response function give the transition strengths

XX 2 S0→j = uX vX Lj1 R1j − vX

X

Flj

l

Lj1 Ll1 , (ωj + ωl )

(13)

where we refer to Refs. 27 and 28 for further definitions and details. Clearly, when X is a Cartesian component of the electric dipole operator, the discrete dipole oscillator strengths fj for transitions from state 0 to state j (in the length gauge) are obtained as 2 XX YY ZZ + S0→j + S0→j ) fj = ωj (S0→j 3

2.2

(14)

Analytical continuation procedure

The averaged dynamic polarizability is given by 37

α(ω) = with (df /dε) =

P∞

i=1

Z

∞ 0

(df /dε)dε ε2 − ω 2

(15)

fi δ(ωi − ε) + g(ω), and with ωi , fi and g(ω) representing the transition

frequencies, bound and continuum oscillator strengths, respectively. This definition can be extended to complex values of frequency, which leads to

α(z) =

Z

∞ 0

(df /dε)dε ε2 − z 2

(16)

The dynamic polarizability in complex values of frequencies, 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 is the first ionization threshold of the system. If we take the complex frequency as being close enough of the real axis by considering

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z = limη→0 (ω + iη), the dynamic polarizability then takes the form 10

lim α(ω + iη) =

η→0

X i=1

fi +P 2 ωi − ω 2

Z

∞ εI

iπg(ω) g(ε)dε + , ε2 − ω 2 2ω

(17)

where P is the principal value of the Cauchy integral. Separating the real and the imaginary part of limη→0 α(ω + iη)

lim Re[α(ω + iη)] =

η→0

X i=1

fi +P 2 ωi − ω 2

lim Im[α(ω + iη)] =

η→0

Z

∞ εI

g(ε)dε ε2 − ω 2

πg(ω) , 2ω

(18)

(19)

and recalling that the cross section can be written as 37

σ(ω) =

2π 2 g(ω) , c

(20)

we obtain straightaway the following relations

σ(ω) =

4πω lim Im[α(ω + iη)] c η→0

α(ω) = lim Re[α(ω + iη)] η→0

(21) (22)

A crucial point in this approach is to recognize the dynamic polarizability as a Stieltjes series. 38 A Stieltjes series can be written as

χ(z) =

∞ X

χm (−z)m

(23)

m=0

with χm being the moments of a given distribution, θ(u), where

χm =

Z

∞

um dθ(u),

m = 0, 1, 2...

0

8


(24)

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From the Stieltjes theory of moments, we know that a given distribution can be reconstructed once its moments are known. 39 We rewrite Eq. (15) changing the variable ω 2 = −y Z

α(y) =

∞

(df /dε)dε ε2 + y

0

(25)

Its n-fold derivative is dn α(y) = (−1)n n! dy n

Z

∞

(df /dε)dε , (ε2 + y)n+1

0

n = 0, 1, 2...

(26)

Expanding α(y) about the origin, we obtain

α(y) =

∞ X

(27)

αn (−y)n ,

n=0

where αn =

Z

∞ 0

(df /dε)dε , (ε2 )2n+2

n = 0, 1, 2...

(28)

which can be also rewritten as

αn =

∞ X

fi

ωi2n+2 i=1

+

Z

∞ εI

g(ε)dε , ω 2n+2

n = 0, 1, 2...

(29)

The dynamic polarizability is thus a Stieltjes series, where we recognize the terms αn as the moments of the distribution of oscillator strengths, also called sum rules. Such observation enforces our need for a careful choice of the basis set which can reproduce the sum rules of the oscillator strengths in order to construct an appropriate approximate representation of the dynamic polarizability. In the spirit of the work of Broad and Reinhardt, 29 our methodology consists in approximating α(z) by a finite sum α(z) =

k X i=1

9

f˜i ω˜i 2 + z 2


(30)


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where the pseudo spectrum {ω˜i , f˜i }i=1,k stands for the approximate transition frequencies and oscillator strengths obtained in the L2 basis-set calculation, which, in the present case, is the pseudo spectrum obtained from Lanczos-based coupled cluster linear response calculations. 27 We now turn our discussion back to the fact that the dynamic polarizability is a Stieltjes series. It is well known that, for a Stieltjes series, continued fractions, which are equivalent to a [N − 1/N ] Padé approximant, will converge, as N goes to infinity, to the function α(z) in the cut. 39 It was shown that the convergence can be achieved with small values of N . 31,32,34 Observing that the dynamical polarizability is a Stieltjes series gives us a clue on how to perform the analytical continuation, and in this work we use a continued fraction procedure as described in Ref. 40. Having the CC (pseudo-)spectrum in hand, we construct the approximation for α(z) as in Eq. (30) and calculate α(z) at a number of points in the complex plane. These points are fitted by the continued fractions procedure, providing a representation of α(z) in the complex plane. Using this representation, from converged continued fraction, we calculate α(z) on the real axis where it equals α(ω). The imaginary part of α(z) on the real axis provides the cross section by Eq. (21).

2.3

Multipoint Padé Approximants

In this section we briefly describe our methodology to obtain the analytical representation for Eq. (30) interpolating it by a continued fraction procedure at a number of points N in the complex plane. These points are commonly arbitrarily chosen. For many choices, the representation Eq. (30) converges to the function α(z). Nevertheless, we can find a “better” choice for the points, which we describe in detail in Appendix. The multipoint Padé approximants are the ratios of two polynomials and can be obtained from a continued fraction function. The details of the algorithm are as follows: 40 given a number N of complex points {zi }i=1,..,N , we insert them in Eq. (30) which yields N approximate values [α(zi ) ≡ αi ]i=1,...,N . The multipoint Padé approximant with these knots can be 10


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written as the continued fraction a1

α(z) =

(31)

a2 (z − z1 )

1+

a3 (z − z2 )

1+

1 + ...

aN (z − zN −1 ) 1

where the coefficients ai are given by the recursion formula

g1 (zi ) = αi ,

i = 1, ..., N

(32)

gp−1 (zp−1 ) − gp−1 (z) , (z − zp−1 )gp−1 (z)

p≥2

(33)

ai = g(zi )i , and gp =

The complex function Eq. (31) is the analytical representation for Eq. (30) and its real and imaginary parts can be separated to give the dynamic polarizability and the photoionization cross section via (21), respectively.

3

Computational details

The discretized spectrum covering the entire frequency region was obtained from the asymmetric Lanczos-based version of CCSD linear response theory, 27,28 as implemented in the Dalton program package. 41 The construction and analytical continuation procedure using the Lanczos pseudospectrum were subsequently used in the Padé approximant procedure, which was implemented as a stand-alone Python code. The code is available from the author upon request. 42 For comparison with our theoretical curve of the (averaged) dynamic polarizability, we also calculated the frequency dependent polarizability at the CCSD level using standard

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CCSD linear response at a number of chosen frequency values. Correlation consistent valence and core-valence basis sets of Dunning and co-workers 43 have been employed for all systems, namely y-cc-p(C)VYZ, with Y = T or Q, and either singly (y=aug), doubly (y=d-aug) or triply (y=t-aug) augmented, depending on the case at hand. As the continuum information about the system is extremely important in our L2 procedure, additional Kaufmann’s continuum-like Gaussian basis functions 44 were added at the center of mass of all studied systems with quantum number n ranging from 2 to 10 or 20, depending on the system. Polarization and diffuse functions are also needed for a reasonable description of the region of discrete excitations. Experimental equilibrium geometries were used for all molecular species, taken from the NIST web page 45 . All electrons were correlated in the calculations, unless otherwise specified.

4

Discussion of results

The methodology illustrated in the previous sections has been applied to calculate the total photoabsorption cross sections and dynamic polarizabilities of the (neutral) systems Ne, N2 , He, H2 , CH4 , C2 H2 , H2 CO, CO2 , NH3 , CO, H2 O and SO2 , and the photodetachment cross sections of the closed shell anions H− , F− , OH− and NH− 2 . The study of the photoionization of neutral species and the photodetachment of closed shell anions will be analysed separately in the following sections.

4.1

Photoionization cross section of neutral species

To investigate how the selection of dimension k of the Lanczos chain affects the quality of the approximation in Eq. (30) and, subsequently, the photoionization cross sections obtained from the analytical continuation procedure via Eqs. (31) and (21) we carried out a specific analysis on the Neon atom and the N2 molecule. 12


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In Fig. 1, we present the CCSD photoionization cross sections of Neon obtained using the aug-cc-pVQZ basis set supplemented with (9s,9p,9d) continuum-like functions for different values of the Lanczos chain length k. Decreasing the dimension of the chain length k from 2000 to 1500, 1000, 500, 200 and 100, as shown on Fig. 1, no considerable difference can be noted on the photoionization cross section calculations. The oscillator strength sum rules S(−1) to S(−6) of Neon, collected on Table 1, were within 5% of absolute deviation according to the experimental values from Olney et al. 46 As described on Section 2.2, the dynamic polarizability is a Stieltjes series, and the moments of its distribution are the sum rules of the oscillator strength. Therefore, the quality of the sum rules represented in Table 1 is an indication of both the adequacy of the applied basis set and of the approximation Eq. 30, that complements the good agreement between the theoretical and experimental cross section depicted on Fig. 1. Turning to N2 , we considered three values of the chain length, namely k =1000, 500 and 300, respectively. For each resulting Lanczos pseudo spectrum, we performed the analytical continuation procedure as shown in Fig. 2. As our results show, good convergence was achieved for all the chosen chain lengths. Good agreement was also obtained between the computed sum rules of the oscillator strength and the experimental values of Olney et al. 46 The S(−n) results obtained from the Lanczos pseudo spectra, collected in Table 2, were within 9% of absolute deviation. Once again, the satisfactory results rely on the good adequacy of the applied basis set, aug-cc-pVTZ supplemented with (9s,9p,9d) Kaufmann et al. 44 continuum-like Gaussian basis functions placed at the center of mass of the molecule. Inspecting the values of sum rules collected in Tables 1 and 2, we note, as common trend, an almost negligible dependence on the value of the chain length. Even when increasing the order of the spectral moments up to 16, for example, the numerical deviations do not exceed 0.02% in relative values. This rapid convergence of the Lanczos (truncated) chain approach is believed 25 to be due to the fact that we are only addressing the region of the valence photoionization which is expected to be dominated by the fast convergence of the

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individual states at the lower and upper edge of the eigenvalue spectrum, a characteristic attribute of the Lanczos method. We also see from Tables 1 and 2 that the spectral moments S(−1) to S(−3) are in good agreement with the experimental values, 46 whereas the spectral moments S(−4) and higher tend to deviate more noticeably from experiment. We should expect the quality of the calculated oscillator strength sum rules of higher order to increase by drastically augmenting the basis set expansion with continuum-like Gaussian basis functions, which would then reflect also on the numerical agreement with experiment. One should not forget, however, that there are also other factors that influence the latter, as for instance the intrinsic limitations of the chosen wave function ansatz (CCSD) and the neglect of vibrational effects (for molecules). Such calculations could be easily handled for small systems, however we refrained from doing so as our purpose is to present a methodology that is sufficiently reliable and still relatively cheap computationally to be applicable to larger systems. It is appropriate to mention that the even coupled-cluster spectral moments (also known as Cauchy moments) S(−2m) can also be obtained by the analytic procedure of Ref. 47, which computes the dispersion coefficients of the dipole polarizability – and hence the Cauchy moments – as frequency derivatives of the coupled cluster linear response function. This is however not sufficient in our context, as we request both odd and even Cauchy moments. Based on the results for Ne and N2 , we found that a suitable choice for the chain length k could be k =300. All the following results were therefore obtained with Lanczos chain length set to 300. More details about the analytical continuation procedure can be found in the Appendix. In 1985, Nascimento 34 published theoretical results of photoionization cross section and dynamic polarizability from He and H2 using the same analytical continuation procedure applied in our present work. In his work, Nascimento used configuration interaction (CI) wave functions to evaluate oscillator strength and transition frequencies from the ground state which, in turn, were used to compute the sum rules and the complex dynamic polarizability. In that case, good agreement between theory and experiment data available at the time was

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achieved. Our CCSD results are compared to newer experiment data from Samson et al. 48,49 Note that for two electron systems the CCSD ground state wavefunction is equivalent to Full CI. The computed photoionization cross sections are shown in Fig. 3 and Fig. 4 for He and H2 , respectively, whereas the sum rules are given in Table 3. Details about the basis set used can be found in the figure captions. Also for these two systems, the agreement between experiment and theory is rather satisfactory. In a recent work of Ruberti et al., 24 the ADC-Lanczos-Stieltjes method at the ADC(1), ADC(2), and ADC(2)x levels of theory was applied for the series of eight molecules of first row elements HF, NH3 , H2 O, CO2 , H2 CO, CH3 , C2 H2 , and C2 H4 , showing a good accuracy for total molecular photo-ionization cross-sections in the valence region. Cukras et al. 25 also published pilot results for the atoms He, Ne, and Ar and the molecules H2 , H2 O, NH3 , HF, CO and CO2 at the CCS, CC2 and CCSD levels using the asymmetric Lanczos algorithm and the Stieltjes Imaging technique to obtain photoionization cross section. Their work was followed by a CC study of photodetachment cross sections of closed-shell anions H− , Li− , Na− , F− , Cl− , and OH− (Ref. 26) where the applied technique yielded reasonably accurate results for the studied systems. The results of our calculations for H2 , He, Ne, N2 , C2 H2 , CH4 , CO, CO2 , H2 CO, H2 O, NH3 and SO2 are presented from Fig. 1 to Fig. 12. Once again, details of the basis set used here for the various atomic or molecular systems are included in their respective figure captions. In Figs. 5 to 9 we show the total photoabsorption cross sections of CH4 , H2 CO, C2 H2 , CO2 and CO, respectively. The theoretical results are plotted together with the experimental points, and a reasonably good agreement between theory and experiment is observed. C2 H2 and CO are worth particular mention since even the sharp increase in the absorption region of the spectrum near 10 eV is well reproduced by our theoretical curves. The cross section profiles of the molecules H2 O, NH3 and SO2 , shown from Fig. 10 to Fig. 12, have some common features that we briefly comment upon here. All three systems show a small absorption feature before the principal absorption region – below 10 eV –

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that is reproduced by our theoretical curves. On the other hand, the experimental results also indicate the presence of 50–52 a sharp peak near 13 eV (see inset in the Figures 10, 11, 12), before the principal peak (i.e., the maximum cross section), that is not present in our theoretical curves, but the main peak of all three spectra is well reproduced and in reasonable good agreement with the experiments. The quality of our profiles could be enhanced by improving the basis set and enlarging the chain length, but this is beyond the purpose of this study. In Table 3 we collect the sum rules, S(0) to S(−6), for all neutral species, together with the corresponding experimental values. As seen already for Ne and N2 , the agreement with experiment is best for the lower-order sum rules. Since we are applying L2 basis set to reproduce properties from the continuum, we should expect that the numerical agreement with experiment of the calculated oscillator strength sum rules of higher orders could be increased by drastically augmenting the basis set expansion with the continuum-like Gaussian 44 basis functions as well as considering better correlation of the wave functions (e.g. including triple excitations in the cluster operator), but such an option would increase the computational demand, which is not our goal. Moreover, the Lanczos approach has not yet been extended to wave functions ansatzes going beyond CCSD.

4.2

Dynamic polarizabilities of neutral species

As discussed in Section 2, the analytical continuation provides a way to obtain an analytic complex function approximating the dynamic polarizability, via Eq. (30), from a multipoint Padé approximant interpolation, Eq. (31), on a set of points in the complex plane, selected as discussed in the Appendix. According to Eqs. (21) and (22) we get simultaneously the photoionization cross section as the imaginary part of Eq. (31) plotted on the real axis and the (averaged) dynamic polarizability as the real part of Eq.(31). In the previous section we discussed the results for the photoionization cross sections of the neutral species. Here we are considering the real part of the complex (analytic) functions obtained from the Lanczos 16


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CC calculations of the same neutral species that were shown in the previous section. The quality of the (averaged) dynamic polarizability compared to experiment 53–55 and to theoretical results evaluated with standard CCSD-LR in the same basis sets is another criterion to monitor the convergence of the Padé approximants to the function α(z) in the cut. The complex dynamic polarizability contains both absorptive and dispersive information about the system, thus photoionization cross section and dynamic polarizabilities should be obtained simultaneously once a good representation is found for the complex polarizability. The quality of our representation of the dynamic polarizability is directly related to the flexibility of the basis set, which must contain some of the physical characteristics of the continuum wave function, along with polarization and diffuse functions for a reasonable description of the bound discrete excitations, and the ability of the electronic structure method employed, in the present case CCSD, to account for correlation effects. Our computed dispersion curves of the (averaged) dynamic polarizability are shown in Figs. 1 to 12. For Ne, He, H2 , and CO (Figs. 1, 3, 4 and 9, respectively) high accuracy of the theoretical results compared to the experiments 53,55 is observed. For N2 , CH4 , H2 O, NH3 , CO2 and C2 H2 (Figs. 2, 5, 10, 11, 8 and 7, respectively) all results fall below 3% of relative deviations from the experimental values. For the molecules H2 CO and SO2 we compare our results only to experimental values of static polarizabilities from refraction index, 46 where we find 4% and 7% of relative deviations respectively.

4.3

Photodetachment cross section of closed shell anions

Anionic photodetachment processes are more closely related to electron scattering process since the ejected electron leaves a neutral charged target with the same continuum structure. However, the absence of the static Coulomb contribution to the long range potential provides that the weaker polarization and dispersion forces are made prominent in this case. Looking at these basic differences from the photoionization process, we are tempted to consider that the conditions obtained from the photoionizations tests should be reconsidered when going 17



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to photodetachment processes. We start from hydride anion, H− , and vary the length of the Lanczos chain from 300 to 1000 to generate the pseudo spectrum. The basis set we used before for hydrogen, daug-cc-pVQZ + (9s,9p,9d), had to be enlarged to provide more flexibility in representing the excitations in the discrete region of the spectrum. In this sense we applied the aug-cc-pCVQZ basis set plus addition of (15s,15p,15d) continuum-like functions. 44 Recent works of Cukras et al. 26 and of Oana et al. 56 reported new theoretical results for the hydride anion in a good agreement with older theoretical results as those of Nascimento et al. 31 and of Venuti and Decleva, 57 as well as with experimental results of Smith and Burch. 58 Our results are plotted in Fig. 13 along with the experimental cross section from Smith and Burch, 58 Popp and Kruse 59 and Génévriez and Urbain. 60 Considering that the experimental deviation is about 4%, 60 we conclude that, for both values of the chain length, we can satisfactorily compare with the experimental curve and reproduce the main feature of the photodetachment cross section, the large non-resonant peak at around 1.5 eV, with a maximum height of approximately 40Mb, as well as a sharp peak around 10 eV where there is a series of auto-detaching resonances. 61 The results for the fluorine anion are plotted in Fig. 14 together with the experimental results from Vacquié et al. 62 The basis set details are given in the figure caption. A smaller range of energy was considered here than in the H− for better comparison with the experimental points. 62 The photodetachment curves of OH− and NH− 2 are shown in Figs. 15 and 16, respectively, together with reference experimental value. 63,64 The basis set details can be found in the figure captions. Also in these two cases the set of experimental points 63,64 only covers a small region of low energy, ranging from 1.8 to 3 eV for the hydroxide and from 0.7 to 1.0 eV for the amide anion. In the figures 15 and 16 we show our theoretical result in a smaller range of energy for better comparison to the experimental points 63,64 where we observe a good agreement between our result and the experiments. As the amide anion experimental

18


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data 64 were reported in relative units, we have scaled the experimental cross section points by 5.0 as to match the maximum with the plateau of the theoretical curve. In the theoretical calculations of OH− and NH− 2 , the agreement with experiment is less satisfactory than in the case of H− , and a better combination of chain length value and basis set is probably needed for these two anions.

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Table 1: CCSD oscillator strength sum rules for Ne for different values of chain length k in the Lanczos algorithm, alongside with the experimental 46 ones. Note that the “experimental” S(0) value is the exact number of electrons. S(0) S(−1) S(−2) S(−3) S(−4) S(−5) S(−6)

Experiment 46 10 3.60 2.66 2.57 2.94 3.75 5.14

k = 2000 9.53 3.78 2.68 2.54 2.88 3.66 5.02

k = 1500 9.53 3.78 2.68 2.54 2.88 3.66 5.02

k = 1000 9.53 3.78 2.68 2.54 2.88 3.66 5.02

k = 500 9.54 3.78 2.68 2.54 2.88 3.66 5.02

k = 200 9.54 3.78 2.68 2.54 2.88 3.66 5.02

k = 100 9.54 3.78 2.68 2.54 2.88 3.66 5.02

Table 2: CCSD oscillator strength sum rules for N2 at different values of the chain length k in the Lanczos algorithm, alongside with the experimental 46 ones. Note that the “experimental” S(0) value is the exact number of electrons. S(0) S(−1) S(−2) S(−3) S(−4) S(−5) S(−6)

5

Experiment 46 14 9.26 11.75 17.82 30.02 53.92 100.8

k = 1000 12.411 9.50 11.67 17.32 28.54 50.10 91.50

k = 500 12.403 9.50 11.67 17.32 28.54 50.10 91.50

k = 300 12.407 9.50 11.67 17.32 28.54 50.10 91.50

Conclusions

An analytic continuation procedure has been applied to the (averaged) complex polarizability built from the excitation energies and transition strengths obtained from a Lanczos-based implementation of Coupled Cluster Linear Response Theory and performed at a number of values on the complex plane to generate the photoionization cross section profiles and dynamic polarizability dispersion curves of a series of closed shell atoms and molecules, and well as to obtain the photodetachment cross section of four closed shell anions. In particular, we have examined the ability of the approach to sufficiently well reproduce the main features of the cross section profiles when adopting a rather small Lanczos chain length (of the order 20


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Table 3: CCSD oscillator strength sum rules for all investigated neutral species calculated with chain length k = 300. For details on the basis sets, see respective molecular system’s figure. Experimental results from a Ref. 46 and b Ref. 52 shown in parentheses. Note that the “experimental” S(0) value is the exact number of electrons. He H2 CH4 C2 H2 H2 CO CO2 NH3 CO H2 O SO2

S(0) 1.999 (2) 2.001 (2) 9.260 (10) 12.565 (14) 14.379 (16) 19.821 (22) 9.213 (10) 12.366 (14) 9.137 (10) 23.152 (32)

S(−1) 1.506 (1.489a ) 3.042 (3.076a ) 10.59 (10.76a ) 13.45 (13.04a ) 12.52 (12.55a ) 14.28 (13.74a ) 9.232 (9.253a ) 9.647 (9.086a ) 7.358 (7.244a ) 18.37 (17.82b )

S(−2) 1.385 (1.383a ) 5.201 (5.433a ) 16.35 (17.26a ) 22.50 (22.95a ) 17.85 (18.69a ) 17.83 (17.51a ) 14.16 (14.57a ) 13.14 (13.06a ) 9.659 (9.839a ) 26.09 (25.45b )

S(−3) 1.418 (1.423a ) 9.500 (10.22a ) 28.59 (31.36a ) 45.26 (48.08a ) 32.16 (34.87a ) 28.13 (27.89a ) 27.78 (29.00a ) 23.10 (23.81a ) 16.47 (17.09a ) 48.88 (48.14b )

S(−4) 1.547 (1.556a ) 18.07 (20.04a ) 54.26 (62.22a ) 102.4 (113.3a ) 68.13 (76.97a ) 50.28 (50.63a ) 66.83 (70.48a ) 47.51 (50.86a ) 33.96 (35.70a ) 113.4 (114.0b )

S(−5) 1.758 (1.771a ) 35.30 (40.43a ) 109.5 (132.0a ) 251.3 (289.7a ) 163.4 (193.7a ) 96.64 (99.68a ) 190.4 (202.7a ) 109.7 (121.8a ) 81.13 (86.17a ) 321.6 (333.1b )

S(−6) 2.052 (2.069a ) 70.24 (83.18a ) 231.6 (295.0a ) 654.3 (786.3a ) 431.0 (541.1a ) 194.5 (207.4a ) 616.5 (664.0a ) 275.9 (318.1a ) 217.1 (233.1a ) 1101.8 (1182.6b )

of a few hundreds) and a medium size correlation consistent basis set supplemented with a limited number of continuum-like Gaussian functions. The approach proved to perform reasonably well for the photoionization cross sections, but to be less satisfactory for the detachment cross sections.

Acknowledgement M.A.C.N. and A.B.R. acknowledges FAPERJ and CNPq for financial support. B.N.C.T. acknowledges CNPq for financial support S.C. acknowledges financial support from the AIAS-COFUND program (Grant Agreement No. 609033). The COST Actions MP1306 “Modern Tools for Spectroscopy on Advanced Materials (EUSPEC)” and CM1204 “XUV/X-ray light and fast ions for ultrafast chemistry (XLIC)” are also acknowledged.

6 6.1

Appendix Selection of the complex points zi

We mentioned in Sec. 2 that the set of complex frequency points zj = xj + iyj where the function Eq. 30 is interpolated is arbitrary, and that the function (31) will converge to α(z) 21



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for several different sets. To avoid confusion with the excitation energies ωi , we use here the symbol xj for the real part of the complex frequency point zj . The approximation in Eq. (30) is very basis set dependent as we are applying L2 basis sets to reproduce properties of the continuum region of the spectrum. Beside, the length of the Lanczos chain also plays an important role, as the quality of the pseudospectrum improves when the length of the chain is increased, with the Lanczos eigenvalues and eigenvectors converging from the extremes. All this calls for care in the selection of those complex points zi . To construct the approximation Eq. (30) we used the full set of transition energies and oscillator strengths, {ωi , fi }i=1,k , from the Lanczos calculations, which, for k = 300, means between 300 and 900 terms, depending on the symmetry of the system. One might speculate about the need to include values for the complex points zj that cover the entire frequency region spanned by the {ωi }i=1,k transition energies in the pseudospectrum. Actually, as we want to cover the frequency domain of the photoionization spectrum, we can take only a few points (let us say, about 30) within the frequency domain of {ωi }i=1,k to construct our multipoint Padé approximant Eq. (31). As an example, we take the water molecule calculation with chain length k = 300, where the Lanczos pseudospectrum, {ωi , fi }i=1,k (for each cartesian component of the polarizability) was obtained. We start by choosing the set of real values xj of zj = xj + iyj . We begin with two points to be used as pure real and better describe the absorption region, specifically x1 = 0 and a small quantity before the first transition frequency, for instance half the first transition frequency, x2 =

ω1 . 2

By noticing that the Lanczos algorithm approaches

convergence from the extremes, we then add the first ten transitions energies from the Lanczos pseudospectrum taken in steps of two, followed by the rest of the transitions in steps of twenty, up to ωi , i = 400. This yields a set of 27 xj values that will represent our real part of the set zj , {0, ω21 , {ωj }j,1,10,2 , {ωj }j,11,400,20 } (where the digits in the subscript indicates the start, end and step, respectively). The imaginary part is quite arbitrary, and in Fig.17 we show two cross sections obtained for two selections of the imaginary part of zj .

22


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The curve labelled “plot-1” was obtained using the value ωj /2 for imaginary part, and on “plot-2” we used the corresponding oscillator strength value fj multiplied by 5.0. Both of them are capable of reproducing the experimental cross section although the convergence is not as good for the selection of the plot-2. In fig. 18 we show different plots for the same selection of values xj for the real part as before, but using Fj × η as imaginary part, i.e. z = {0, ω21 , {ωj + iFj η}j,1,10,2 , {ωj + P iFj η}j,11,400,20 }, where Fj is the cumulative oscillator strength, Fj = jl=1 fl , and η is a real

multiplicative factor, whose value is specified in the legend of the figure. We note that the three plots reproduce the experimental cross section, but while the sets of points with η equal to 1.0 and 2.0 reproduce the spectrum rather well, for η equal to 0.5 we start to lose

convergence. We took the selection in Fig. 18 with η equal to 1.0 as our best choice and applying the same selection of zj for the other systems studied we obtained the plots shown in Fig. 2 and in Figs. 4 to 12. The atomic neutral species Ne and He have a series of auto-ionization resonances which complicates the convergence of the Padé procedure. In those cases, to achieve convergence, we took the first five transition frequencies and oscillator strengths of the Lanczos procedure. For He and Ne atoms we chose the set of zi points as {0, ω21 , {ωj +iFj η}j,1,5,1 , {ωj +iFj η}j,6,400,20 }. For He we used the η = 2 and for the Ne we used η = 3. The plots for Ne and He are shown in Figs. 1 and 3. The anions also need particular mention. In those systems we observe that the selection for the set of zi points as {0, 0.005, {ωj +iFj η}j,1,5,1 , {ωj +iFj η}j,6,20,4 }, which gives 17 points, was sufficient to reproduce the results shown in Figs. 13, 14, 15 and 16. For F− , OH− and − NH− 2 we used the η = 1 and for the H we used η = 0.2.

To summarize, we note that, except for Ne and He, we were able to find one selection suitable for the neutral species which was equally applicable for N2 , CH4 , C2 H2 , H2 CO, CO2 , NH3 , CO, H2 O and SO2 (Fig. 2 and from Fig. 5 to 12) and another suitable selection which

23



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was equally applicable to the anions H− , OH− , F− and NH− 2 (Fig. 13 to 16). In Fig. 19 we used H2 , He, Ne and H2 O as examples, to show that the Stieltjes quadrature points 10,25 (or, in other words, the frequencies and oscillator strengths of the principal pseudospectrum) (ωip , fip )i=1,n of order n, could as well be used as the complex points zi by setting the quadrature frequency ωip as the real part, and the quadrature oscillator strength, fip , as the imaginary part. Thus, when available, the Stieltjes quadrature points also represent a reliable choice for the zi points.

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(40) Radzimirski,

B.

Demonstrations

"Multipoint Project.

Padé 2016;

Approximants"

from

the

Wolfram

http://demonstrations.wolfram.com/

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Coupled Cluster Study of Photoionization and Photodetachment Cross

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