On the Accurate Description of Photoionization Dynamical Parameters
Calculation of dynamical parameters for photoionization requires an accurate description of both initial and final states of the system, as well as of the outgoing electron.We here show, that using a linear combination of atomic orbitals (LCAO) B-spline density functional (DFT) method to describe the outgoing electron, in combination with correlated equation-of-motion coupled cluster singles and double (EOM-CCSD) Dyson orbitals, gives good agreement with experiment and outperforms other simpler approaches, like plane and Coulomb waves, used to describe the photoelectron. Results are presented for cross sections, angular distributions and dichroic parameters in chiral molecules, as well as for photoionization from excited states. We also present a comparison with the results obtained using Hartree-Fock (HF) and density-functional theory molecular orbitals selected according to Koopmans' theorem for the bound states.
<!>Supporting Information Available
<p>The last two decades have witnessed a remarkable increase in the study of sophisticated light-matter interactions, due to enormous advancements in synchrotron and laser sources. This is dictating high demands for reliable theoretical and computational methodologies. 1,2 Photoelectron spectroscopy (PES) offers detailed information about the electronic structure of the target molecule and the properties of the ejected electrons. Photoionization has recently become important as an ultrafast probe in time-resolved experiments, both in the femto and atto second domains, as well as in the description of many strong-field phenomena. [2][3][4][5] However, the interpretation of such spectroscopic signatures requires that relaxation and correlation effects in the description of bound and continuum states are taken into account. Even though, in general, the distinction between relaxation and correlation is not always well defined, we focus here on correlation effects, which are the dominant effects for valence-shell ionization. Correlation effects in the bound states and on the ionization energies have been studied extensively using several many-body approaches, like configuration interaction (CI), [6][7][8] perturbation theory (PT), 6 coupled cluster (CC) [9][10][11][12][13] and Green's functions (GF). 6,[14][15][16] Additional correlation effects are hidden in the description of the continuum, the interaction between continuum channels or between continuum and bound states.</p><p>The calculation of the photoionization observables is based on the evaluation of the dipole transition moments,</p><p>which connect the initial (Ψ N i ) and final (Φ f ) wave functions. The D if transition elements are linked to the differential cross section, given by (in a.u.) 17 dσ</p><p>where α is the fine structure constant, ω is the photon energy and k is the momentum of the photoelectron in the molecular frame.</p><p>Koopmans' theorem is the simplest way to elucidate PES. The basic assumption is a single determinant (or single particle) description of both the initial and final states, without including orbital relaxation effects. This approximation gives a one-to-one correspondence between a molecular orbital and a spectral band, such that the ionization energy equals minus the corresponding orbital energy, IE= − i . The model fails for ionizations involving multielectron excitations, shake-up or satellite bands. 7,16 To overcome these limitations, correlation effects need to be taken into account. A key quantity in doing so is the Dyson orbital. Considering a single channel approximation of the final state 11,18,19</p><p>the photoelectron matrix element D if can be rewritten as</p><p>where N is the number of electrons in the initial state, µ is the electric dipole moment operator, φ is the wave function of the ejected electron and φ d if is the Dyson orbital. The latter is defined as the overlap of the normalized wave functions for the bound N and (N −1) systems: 14,20</p><p>As indicated by the second equality, the Dyson orbital can be shown to reduce to a linear combination of ground-state molecular orbitals (φ p ), with coefficients γ p . We refer to the Supporting Information for details on the derivation of Eqs. ( 4) and (5).</p><p>Thus, within the single channel approach, correlation effects in the bound states (both initial and final), that are in many cases the dominant ones, can be effectively incorporated via the use of the Dyson orbital approach. The nature of the Dyson orbitals acts as a marker of the effect of correlation upon changing from neutral to ionic system. It can be thought of as a correlated counterpart of a Hartree-Fock orbital describing the initial state of the ionized electron within Koopmans' theorem.</p><p>In this work we use a EOM-CCSD framework, 11 where the left (label L) and right (label R) Dyson orbitals are distinct. 11 For ionisation from the ground state, their expansion coefficients γ L p (defined as</p><p>) and γ R p (defined as</p><p>) are given by:</p><p>whereas for ionisation from an excited state they are found as</p><p>In Eqs. ( 6)-( 9), we have used the standard EOM-CCSD notation, 12,21,22 Λ =</p><p>Having taken care of correlation effects in the bound states via, e.g., the Dyson orbital, the next step is to choose a description of the single particle continuum. We remark here that the development of computational approaches to molecular photoionization has spanned several decades. [23][24][25] Nonetheless, full close-coupling approaches, taking in principle account of interchannel interactions, have progressed slowly, and are still computationally intensive and limited to relatively small systems. In fact, most approaches for polyatomic molecules are still at the static-exchange level, which is the counterpart for the photoelectron of the Koopmans description. Exceptions are the Stieltjes approach, 23,26 and the equivalent Padé approach, 27 that completely avoid the explicit construction of continuum states, but reconstructs cross section from purely bound pseudo-spectra. However, due to the lack of proper asymptotic boundary conditions, the separation of individual channels becomes ambiguous, and angular distributions cannot be obtained. Also, large basis sets are required and the energy resolution is quite limited. Even then, these methods have proven quite successful in combination with highly correlated electronic structure methods such as linear response coupled cluster [28][29][30][31] and algebraic diagrammatic construction (ADC). [32][33][34] The simplest approach to approximate the continuum is with plane waves (PW) or</p><p>Coulomb waves (CW), possibly orthogonalized to the Dyson orbital. 9,35,36 Although this approach becomes accurate at high photoelectron energies, at least several hundred eV, it is generally rather poor at low energies, especially for angular distributions, or description of single particle continuum structures, like shape resonances, Cooper minima, and so on. A more accurate compromise, based on the muffin-tin approximation, is the multiple scattering approach. 23 Another strategy is to explicitly solve the Schrödinger equation for the photoelectron wave function, which can be rather accurately done, for instance, at the static-exchange DFT level, as we do in this work. Actually, even a time-dependent TDDFT continuum can be employed, which is currently under development. An initial result will be presented here for the argon atom.</p><p>To obtain an accurate description of the continuum in large systems, a basis set approach capable of a multicentric (LCAO) description is needed. Again, several possibilities exist. 24,[37][38][39][40] A tricky aspect is that the description of the continuum requires basis sets which obey the particular boundary conditions of the continuum and, at the same time, reproduce the oscillating behaviour of the photoelectron wave function to infinity. Conventional Gaussian-or Slater-type orbitals are unsuitable for this purpose. B-spline functions, on the other hand, are quite useful in this regards. They are very flexible, they accurately describe both the bound and continuum states with minimal numerical dependencies, 41 and they are becoming popular for continuum calculations. 38,40,42 Here, we employ a LCAO B-spline approach where the primitive basis functions χ jlm are expressed as product of radial, B j (r), and angular Y lm (θ, φ) functions,</p><p>A long-range, large angular momentum basis is centred on the origin, to describe the asymptotic part. Much smaller expansions are employed instead at the nuclear positions, to take care of the Coulomb cusps, and to avoid linear dependence. Converged results are easily obtained even for quite large systems and heavy atoms.</p><p>The continuum orbitals are computed by solving the Schrödinger equation in the angular momentum representation</p><p>with real (K-matrix) boundary conditions. In the Hamiltonian operator</p><p>the term V N , V C and V XC are the nuclear attraction, Coulomb and exchange-correlation potentials, respectively.</p><p>Angular momentum analysis of the differential photoionization cross-section in Eq. ( 2)</p><p>gives</p><p>This shows that the angular distribution of the photoelectron in the solid angle dk along the direction of propagation (k) can be completely defined by three parameters: the crosssection σ, the asymmetry parameter β and the dichroism parameter D. Here, P i is a Legendre polynomial of order i, whereas θ is the scattering angle between k and the laboratory frame defined by the polarization or the direction of propagation of the light. The parameter m r specifies the light polarization, with three possible values, 0, +1 and −1, for linearly polarized (LP), left circularly polarized (LCP) and right circularly polarized (RCP) light, respectively.</p><p>For LP, Eq. ( 16) assumes a very simple form,</p><p>The details of the incorporation of EOM-CCSD Dyson orbital coefficients to obtain the differential cross-section in Eq. ( 16) can be found in the Supporting Information.</p><p>In this letter, we present a combined EOM-CCSD Dyson-orbital/DFT continuum approach for the photoionization observables, and compare our results with both experiments and other computational approaches, including PW or CW results. In more detail, we compare the partial cross-sections and asymmetry parameters for atomic (Ar For most of the molecules, experimental geometries from the NIST database were used. 43 The geometry of (S)-methyloxirane was taken from previous work, 44 and so was the geometry used for furan. 45 The EOM-CCSD Dyson orbitals were computed using Q-Chem 11,46 and the aug-cc-pVTZ basis set. PW and CW cross-sections and asymmetry parameters were computed using the ezDyson code. 47 As anticipated, we adapted the LCAO B-spline code 42 in order to take into account the different left and right Dyson orbital coefficients. (See the discussion in the Supporting Information). To compute the continuum functions, ground state electronic densities were obtained from ADF using the LB94 exchange-correlation potential and the DZP basis from the ADF database. 48 The choice of the functional was guided by literature studies showing its good performance for the calculation of photoionization observables. 49,50 All parameters used to calculate the observables are listed in Table S1 and S2. Notice that we present the results according to conventional symmetry labelling, even though the QChem calculations adopted non-Mulliken symmetry convention. 46 The performance of the various combinations of methods here employed to describe bound and continuum states was assessed for the three lowest-energy ionizations of CO, H In general, the B-spline DFT method for the continuum in conjunction with the EOM-CCSD Dyson orbitals offers the best description of photoionization processes, amongst the various combinations here considered. For CO and H 2 O, there is very little difference between the results obtained from the EOM-CCSD correlated Dyson orbitals and those from the Koopmans theorem-based HF and DFT MOs. This is quite reasonable, given that the major contributing MO in the Dyson orbital has a coefficient greater than 0.9 for the corresponding HF or DFT MO. A slight vertical deviation is seen for our method with respect to experiment for the third lowest-lying ionization (the ionization to B 2 Σ + ) of CO. Even though the PW and CW approximations often reproduce the spectral profile, it shows non-uniform large vertical shifts. The discrepancy is larger near the ionization threshold. This is justifiable as PW and CW are two extreme approximations and the actual charge on the ionized core is photoelectron wave function fail to reproduce the Cooper minimum (see also Fig. S1). The spectral profile of the total cross-section is reproduced using the B-spline DFT technique for the continuum, even though the minimum is shifted by about 15 eV. It is seen to be a feature arising due to ionization from the 3p orbital of Ar. As the magnitude of the partial crosssection of ionization from 3p orbital is much larger than that of 3s, the total cross-section is very similar to the partial cross-section from 3p orbital (as shown in Fig. S1). Electronelectron interaction mixes the 3s and 3p channels. As the 3p cross-section is much larger, the Cooper minimum is also felt in the 3s ionization. However, the DFT B-spline continuum approach lacks correlation in the treatment of the continuum and thus does not yield the minimum in the partial cross-section of ionization from the 3s orbital. The time-dependent (TD)DFT description of the continuum overcomes this problem and provides a remarkable agreement with experiment, as it can be appreciated from Fig. 2. Notable improvement in the asymmetry parameter corresponding to ionization from the 3p orbital is also seen upon consideration of TD-DFT B-spline continuum. The minimum is obtained at the correct photon energy, whereas the dip in the spectral feature is overestimated by about 1 β unit.</p><p>The photoelectron dichroism parameter (D) is governed by the description of the nature of the ionized state. From now onwards, we do not report on the PW and CW descriptions of the continuum, having assessed their inadequacy to properly calculate photoionization dynamical variables.</p><p>Fig. 3 illustrates our approach on the D of (S)-methyloxirane. As immediately apparent, correlation effects must be accounted for while describing the bound state of the target.  ionized state ( 2 A 2 in both cases). The dark state is characterized by excitation from occupied 1a 2 to virtual 10a 1 . However, the bright state is a combination of transition from 1a 2 to 3b 1 and 5b 1 MO's. The square of the Dyson norm (here computed as |φ L • φ R |) corresponding to ionization from both excited states is about 0.48. This value, close to 0.5, reflects ionization from a singly occupied orbital in the initial state.</p><p>For both ground and excited state ionization processes, β is more sensitive than partial σ and shows an initial decrease followed by gradual increase, finally being constant at around 1.5 β units. A steep descent in σ is observed in all cases, where the steepness is a measure of the diffuseness of the orbital from which ionization occurs.</p><p>To summarize, our methodology-using EOM-CCSD Dyson orbital coefficients to de-scribe the bound initial and final states of the target system and B-spline DFT functions to describe the continuum-gives an accurate description of photoionization dynamic properties. On the other hand, simple PW or CW continuum results appear too erratic to be of much use at low electron kinetic energy.</p><p>The present approach is completely general, allowing the calculation of all photoionization observables, including, although not considered in this initial study, the photoionization from molecules aligned or fixed in space. This contrasts Stieltjes or Padé pure bound state approaches, which, although useful in limited contexts, lack sufficient generality. With the DFT B-spline continuum, only interchannel coupling effects, which are less widespread, lie outside our formulation. However, as our preliminary results illustrate in the case of argon, the use of the TD-DFT continuum may alleviate this deficiency. Further theoretical improvements include application to core ionizations by coupling the bound state calculation with the core-valence-separation (CVS) approach, [63][64][65] and the description of multielectron excitations, i.e. shake-up states, which will require the inclusion of higher-order (beyond CCSD) correlation effects for an accurate description of the bound states.</p><p>Finally, the approach is computationally quite efficient and could be used for the descrip-</p><!><p>The Supporting Information is available free of charge.</p>