Pair Natural Orbital Restricted Open-Shell Configuration Interaction

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A Pair Natural Orbitals Restricted Open Shell Configuration Interaction (PNO-ROCIS) Approach for Calculating Xray Absorption Spectra of Large Chemical Systems Dimitrios Maganas, Serena DeBeer, and Frank Neese J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b10880 • Publication Date (Web): 09 Jan 2018 Downloaded from http://pubs.acs.org on January 13, 2018

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The Journal of Physical Chemistry

A Pair Natural Orbitals Restricted Open Shell Configuration Interaction (PNO-ROCIS) Approach for Calculating X-ray Absorption Spectra of Large Chemical Systems Dimitrios Maganas,[1] Serena DeBeer[1] and Frank Neese[1]* 1

Max Planck Institute for Chemical Energy Conversion, Stiftstr. 34-36, 45470 Mülheim an der Ruhr, Germany

Corresponding Author *[email protected]

ACS Paragon Plus Environment

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ABSTRACT In this work, the efficiency of first principles calculations of X-ray absorption spectra of large chemical systems is drastically improved. The approach is based on the previously developed restricted open shell configuration interaction singles (ROCIS) method and its parameterized version, based on a density functional theory (DFT) ground state determinant ROCIS/DFT. The combination of the ROCIS or DFT/ROCIS methods with the well-known machinery of the pair natural orbitals (PNOs) leads to the new PNO-ROCIS and PNO-ROCIS/DFT variants. The PNO-ROCIS method can deliver calculated metal K-, L- and M-edge XAS spectra orders of magnitude faster than ROCIS, while maintaining an accuracy with calculated spectral parameters better than 1% relative to the original ROCIS method (referred to as canonical ROCIS). The method is of a black box character, as it does not require any user adjustments while it scales quadratically with the system size. It is shown that for large systems the size of the virtual molecular orbital (MO) space is reduced by more than 90% with respect to the canonical ROCIS method. This allows one to compute the X-ray absorption spectra of a variety of large ‘real-life’ chemical systems featuring hundreds of atoms using a first principles wavefunction based approach. Examples chosen from the fields of bioinorganic and solid-state chemistry include the Co K-edge XAS spectrum of the aquacobalamin [H2OCbl]+, the Fe L-edge XAS spectrum of deoxy-myoglobin (DMb), the Ti Ledge XAS spectrum of TiO2 rutile and the Fe M-edge spectrum of α-Fe2O3 hematite. In the largest calculations presented here, molecules with more than 700 atoms and cluster models with more than 50 metal centers were employed. In all the studied cases, very good to excellent agreement with experiment is obtained. It will be shown that PNO-ROCIS method provides an unprecedented performance of wavefunction-based methods in the field of computational X-ray spectroscopy.


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I.

Introduction Over the last decades X-ray absorption spectroscopy (XAS) has been established as a

powerful analytical tool for understanding the local geometric and electronic structure of chemical systems in the fields of bioinorganic and solid-state chemistry. Given the element specific nature of X-ray techniques, highly specific local information can be obtained, even in complicated systems. This is of great utility for obtaining insight into the geometric and electronic structures of catalytic centers in homogeneous and heterogeneous catalysis.1-9 The quantitative interpretation of such spectra from first principles is a challenging task that is perhaps most rigorously approached using wavefunction based methodologies. Wavefucntion techniques allow for an explicit treatment of ligand field, covalency and multiplet effects provided that proper spin- and space-symmetry adapted configuration state functions (CSFs) are employed. Furthermore, wavefunction techniques can explicitly represent all magnetic sublevels (MS components) of all multiplets, which allows for a detailed and physically correct treatment of spin-orbit coupling phenomena. It may be considered a drawback of wavefunction based methods, however, a treatment of electron correlation is necessary to obtain quantitatively accurate results, and this is associated with high technical and computational complexity. Thus, in order to maintain computational efficiency if not feasibility, approximations must be introduced. One way to deal with the complex problems eluded to above is to resort to multireference (MR) ab initio wavefunction methodologies. The most elementary MR method is the complete active space configuration interaction (CASSCF) method, which is the MR counterpart of the Hartree-Fock meanfield method. Various methods exist to account for dynamic correlation on top of a CASSCF wavefunction including second order perturbation theory (CASPT2, NEVPT2) and multireference configuration interaction (MRCI).


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However, as the number of final states that one needs to consider in X-ray absorption techniques can easily reach hundreds and the computational effort is a steep function of system size, the applicability of these methodologies is drastically limited. Recently, truncated CI techniques on the basis of restricted active space configuration interaction methods (RASSCF, or RASCI) have proven successful to treat medium size molecules, especially when the missing dynamic correlation is partially recovered perturbatively through the e.g. the RASSCF/RASPT2 protocol. 10-16 An alternative approach is provided by the Restricted Open shell Configuration Interaction Singles (ROCIS) method and its DFT variant DFT/ROCIS (also referred to as ROCIS/DFT).17-18 This method is computationally much more feasible, since it is based on the single open-shell spin restricted determinant and the excited state manifold is spanned by spin- and space-symmetry adapted CSFs. The method can be thought of as a molecular generalization of the familiar LS coupling scheme. ROCIS can treat classes of molecular systems from transition metal compounds up to polymetallic clusters with more than 100 atoms.18-19 However, ROCIS does not incorporate dynamic electron correlation explicitly, which leads to limitations in its accuracy. Hence, a slightly parameterized version, ROCIS/DFT has been developed in order to strike the best balance between accuracy and efficiency. The ROCIS/DFT method employs DFT orbitals and incorporates a downscaling of two-electron repulsion integrals in order to implicitly account for dynamic electron correlation (‘screening’) effects. A limitation of the ROCIS/DFT method is that the ground state is assumed to be representable by a single Slater determinant in which all unpaired electrons are aligned with the same spin. Thus, more complex systems, for example those featuring an antiferromagnetically coupled ground state, cannot be treated. Nevertheless, the choice of ROCIS/DFT has been made to obtain a method of black box character that can be used to calculate spectroscopic properties of open shell systems throughout the entire range


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of spectroscopic techniques ranging from microwaves (electron paramagnetic resonance parameters) to UV/vis absorptions spectra to X-ray absorption spectra. The application of this methodology to treat the metal L-edge problem of various closed- and open-shell compounds, as well as polymetallic clusters, has recently been explored.17, 19-22 In the same spirit, metal M-edge problems can also be treated, while recently ROCIS/DFT has shown to perform equally well with TD-DFT in treating metal K-edge problems.23 In a recent extension of the method, ROCIS/DFT showed excellent performance to treat valence to core resonant X-ray emission (VtC RXES) problems, while it should be readily applicable for the calculation of a wide range of RXES/RIXS processes.23 In this work, we aim at further improving the efficiency of the ROCIS and ROCIS/DFT methods. The concept is based in the idea of pair natural orbitals (PNOs) that have been used with great success in the framework of linear-scaling coupled cluster approaches.24-28, PNO based methods have shown excellent performance in delivering accurate ground state energetics for molecular and surface systems.29-31 Using PNOs in combination with excited state methods is a much less explored field. Hättig and co-workers have shown that excited state PNOs can be generated from correlated methods for excited states that are based in either configuration interaction or couple cluster schemes like CIS(D), CC2, ADC2,32-34 thus leading to state specific PNO-CI and PNO-CC based methods. More recently, we have shown that a ground state PNO scheme can be used in combination with equation of motion based techniques like EOM-CC and STEOM-CC ones to form the back-transformed (bt) bt-PNO-EOM and bt-PNO-STEOM-CCSC methods, which provide a balanced description for charge transfer and Rydberg states in optical spectroscopy.35-36 The genuine DLPNO-STEOM-CCSD method holds great promise for the accurate treatment of larger systems.


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In this study, we combine the powerful machinery of the PNOs with the ROCIS and ROCIS/DFT methods and we present the core PNO-ROCIS and PNO-ROCIS/DFT methods. The usage of PNOs here is somewhat unconventional since we are not using them to treat electron correlation effects in a state specific manner. Rather, the PNOs are used to identify the relevant part of the virtual space that can be reached by excitation out of local core orbitals. This subspace of the virtual space is local, thus leading to a linear scaling, state universal method. After demonstrating the accuracy of the approximations introduced, the computational efficiency of the method is shown by calculating the Co K-edge XAS spectrum to B12 aqua-cobalamine, the Fe L-edge XAS spectrum of deoxy-myoglobin (DMb), the Ti L-edge XAS spectrum of TiO2 rutile and the Fe M-edge spectrum of α-Fe2O3 hematite.

II.

Theory

A. ROCIS The ROCIS methodology has been described in length elsewhere.17-18 In a nutshell, in the framework of ROCIS or ROCIS/DFT methods upon defining a proper set of ROHF or DFT orbitals one enters the CI process by defining a multideterminantal zeroth order wavefunction, which is written as a linear combination of the 0th order wavefunction and excited spin-adapted configuration state functions (CSFs)

.

Solution of the ROCIS eigenvalue problem: (1)

provides the variational determination of the coefficients with which the CSFs enter into the ROCIS wavefunction with matrix elements:


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and

(2)

In which the Born-Oppenheimer operator is given by:

(3)

Here

refers to matrix elements of the one-electron operator and

are the two-

electron repulsion integrals in Mulliken (charge-cloud) notation. The one-electron operator may be supplemented with matrix elements of an external potential (e.g. modeling a crystal environment), a scalar relativistic potential, for example provided by the second-order Douglass-Kroll-Hess (DKH237-41) or 0th order for relativistic effects (ZORA42-44) methods and also possible continuum solvation effects such as CPCM.45 The excited configurations are constructed by performing single excitations relative to each CSF in reference space. Thus, the ROCIS wavefunction with total spin S' = S , S' = S − 1 and S' = S +1 are given by equations 4-6:

(4)

(5)

(6) where throughout uppercase labels refer to many electron quantities, lowercase labels to one-electron quantities. Indices i,j,k,l refer to doubly occupied orbitals in the reference


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determinant, t,u,v,w to singly occupied ones and a,b,c,d to virtual orbitals while p,q,r,s refer to general orbitals.

B. Quasi Degenerate Perturbation Theory (QDPT) On top of the three ROCIS calculations for the three different multiplicities, SOC is introduced on the basis of the quasi degenerate perturbation theory by solving the quasidegenerate eigenvalue problem with the matrix elements:

(7)

Here,

is a Clebsch-Gordon coefficient and

is a

reduced matrix element. In this approach, the SOC appears as an effective one-electron operator which contains one- and two-electron SOC integrals and also incorporates the spinother orbit interaction. It takes the form:

(8)

where

and

refer to the coordinates and spin-operators of electron i respectively.

Further details on the ROCIS/DFT method, working equations and further implementation details describing how ROCIS and ROCIS/DFT are implemented in the ORCA computational package are given elsewhere.46

C. Construction of approximate Singles PNOs Since we are dealing with single excitations in the ROCIS method, the ‘PNOs’ that need to be constructed are ‘singles PNOs’. Thus, as in the case of the singles excitations in the


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coupled cluster method, the term pair natural orbital is used in a rather liberal sense, since no electron pairs are involved in the treatment. It would have been possible refer to the treatment as “natural virtual space” treatment. However, since we are using the established machinery of the PNO approach, we prefer to keep the acronym PNO with the understanding of the discussion above. As in the original LPNO-CCSD24, DLPNO-CCSD27-28 and DLPNO-MP247 methods the singles PNOs are constructed using the quasi-doubles amplitudes

(9)

Here

is a scaling parameter that is introduced to guarantee numerical stability. It takes

the default value

,

are orbital energies of virtual orbitals and

is the

diagonal element of the Fock operator for the core orbital i that acts as donor orbital in the core level excitation process. The integral

is an exchange integral. Since the donor

orbital i is strongly localized, it is obvious that the orbitals ‘a’ and ‘b’ must be located in the same region of space in order to provide a non-negligible exchange integral. This is where the locality enters the treatment. We refer to these amplitudes as ‘quasi’ amplitudes since they technically resemble MP2 double excitation amplitudes, but since we are treating single excitations here the amplitudes are not physical but rather mathematical objects. From the quasi-amplitudes, quasi-densities are constructed: (10) Where the amplitudes without tilde are not scaled. We emphasize that scaling of the amplitudes is crucial in order to ensure that the occupations of the surviving PNOs are not suffering from numerical noise effects. In general, the densities are averaged over all core


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orbitals of the chosen XAS absorber atom to ensure that we have identical expansions for all edges:

Diagonalization of this density forms the PNOs expanded in terms of virtual MOs (11) Here,

denotes the eigenvectors of

and

is a diagonal matrix that contains the

occupation numbers of the natural orbitals as diagonal entries. The PNOs to be considered in the subsequent CI calculation are selected according to their occupation numbers ni > TCutPNO , where TCutPNO is a threshold with a default value that will be discussed below. In a final step, the truncated set of PNOs are recanonicalized with respect to the Fock operator. Subsequently, the original virtual orbitals are replaced by the recanonicalized PNOs and the upper limit for the excitation is set to the last surviving PNO. After this, the ROCIS calculation proceeds exactly like in the canonical case but with the truncated PNO set spanning the virtual space. This leads to large savings. Given, the locality and the nature of the XAS process, the number of donor orbitals is not increasing with system size. By construction, the size of the PNO space is also not increasing with system size. Hence, the number of ROCIS amplitudes to be solved for does not increase with system size either and neither does the size of the integral list required to perform the calculation. However, the algorithm is not genuinly linear scaling, since we have used the canonical virtual space to expand the PNOs and the machinery for the subsequent integral transformation into the PNO basis is also not linear scaling.


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D. Core orbitals for PNO construction As discussed in the previous section, the densities for the PNOs construction are averaged over the entire number of the available core orbitals in the studied molecular system for the centers that core electron spectra will be calculated. For this reason in the calculation preparation step the necessary core orbitals are identified automatically and are localized before the program enters the PNOs construction and selection step. For a given element across the periodic table the relevant core orbitals that need to be included in the PNOs construction process are provided in Table 1. Table 1. Core orbitals that are used to construct the PNOs across the elements of the periodic table. Atoms Li-Be Na-Ar K-Kr Rb-Xe Cs-Ba Fr-Uuo La-Lu Ac-Lr

Core Orbitals 1s 1s, 2s, 2p 1s, 2s, 2p, 3s, 3p 1s, 2s, 2p, 3s, 3p, 4s, 4p 1s, 2s, 2p, 3s, 3p, 4s, 4p, 5s, 5p 1s, 2s, 2p, 3s, 3p, 4s, 4p, 5s, 5p, 6s, 6p 1s, 2s, 2p, 3s, 3p, 4s, 4p, 5s, 5p 1s, 2s, 2p, 3s, 3p, 4s, 4p, 5s, 5p, 6s, 6p

For example, in the case of a first-row transition metal complex, the densities will be averaged over the 1s, 2s, 2p, 3s and 3p based MOs. In this way, it is possible to 1) use a global set of PNOs for all kinds of core electron excitation processes and 2) produce PNOs with larger population numbers. An example is provided in Figure 1 for the [MnVN(CN)4]-2 (25) molecule of the molecular test-set presented in Figure 2. At TCutPNO=10-11 threshold the populations of the generated PNOs by considering only the 1s core orbitals range between 106

and 10-11, while only 10 PNOs are selected. As it is seen in Figure 1, the populations of the

generated PNOs by considering the 2s-2p or the 3s-3p core orbitals increase by 4 and 6


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orders of magnitude, respectively. At the same time the number of the selected PNOs also increases in the sequence. In fact, 100 PNOs are selected when considering the 3p core orbitals or when considering the entire sequence of the core orbitals. This implies that the minimum size of the selected virtual MO space depends strongly on the type core electron excitations under consideration. Although a smaller virtual MO space is required for 1s electron excitations, the size of the required MO virtual space necessarily increases in the case of 3p electron excitations. Hence a virtual MO space selection upon considering the entire sequence of the core orbitals is clearly preferable since it will allow us to treat all edges of a given element consistently.

Figure 1. Dependence of the number of PNOs versus the TCutPNO population threshold for the case of [MnVN(CN)4]-2 (25) molecule. The PNOs are generated from densities considering


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only the 1s (orange dots), 2s (green dots), 2p (purple dots), 3s (cyan dots), 3p (blue dots) or all the above (red dots) core orbitals respectively. For the VIVO(acac)2 (24) molecule representative PNOs are visualized in Figure 2 at the TCutPNO populations 1, 10-3, 10-7, 10-11, 10-16. As is evident from this plot, the PNOs with the largest occupation numbers are the empty metal 3d- and 4p-based orbitals (occupation numbers above 10-3), which is in accord with chemical intuition. At lower occupation numbers one finds orbitals with a larger number of radial nodes that are still centered at the vanadium centered. They form a kind of Rydberg series that describes the onset of the continuum (ranging between 10-7 and 10-11). The second type of lower occupation number orbitals are more and more delocalized onto the ligands. Excitation into these orbitals describe low-probability metal-to-ligand charge transfer processes. Most of them have very small occupation numbers and are rejected by the selection process.

Figure 2. Visualization of selected PNOs for the case of VIVO(acac)2 (24) molecule at TCutPNO populations 1, 10-3, 10-7, 10-11 and 10-16


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III.

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Computational Details All calculations were performed using the ORCA suite of programs.48 The BP8649-50

and B3LYP49, 51-52 functionals were used together with the Grimme’s dispersion correction5354

for geometries/frequencies and electronic properties, respectively. The def2−TZVP basis

set of Weigend et al.66,67 is of tripleζ quality55 and was used for all the atoms in combination with the matching Coulomb fitting basis for the resolution of identity56-57 (RI, in BP86 calculations). The ROCIS/DFT and PNO-ROCIS/DFT calculations were performed using the converged restricted RKS or unrestricted UKS Kohn−Sham wavefunctions. For these calculations, the B3LYP density functionals were employed together with the def2−TZVP(−f) basis sets. In addition, the def2-SVP basis set was used in the calculations performed on biomolecules and solid clusters. It should be noted that the accuracy of the ROCIS/DFT is comparable for the def2-TZVP and def2-SVP basis sets.17,

19-22

Scalar

relativistic effects were treated on the basis of the second−order Douglas−Kroll−Hess (DKH)37-39 and ZORA58 methods. In a typical ROCIS calculation, 100 states are included per metal center and per spin multiplicity to ensure saturation of the calculated spectra. In the case of multimetallic clusters, the final spectrum is obtained as the sum of all individual subspectra obtained separately for each ion. Empirical energy shifts were applied to the calculated spectra to match the K- pre-edge, the L3-edge and M3-edge experimental features in the K-, L- and M-edge XAS spectra, respectively. All spectra were generated with the orca_mapspc utility program of the ORCA package. A constant Gaussian broadenings of 0.5, 0.5 and 1.0 eV was applied to the calculated transitions for the metal M, L- and K-edge XAS spectra respectively. As a comparison measure of the shape of two different spectra the mean absolute error of the calculated areas MAEarea 1,2 is used, which reads according to the relation:

MAEarea = % 1,2

area1 − area2 area1

.


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IV.

Test set of molecules In an effort to evaluate the performance of the PNO-ROCIS method and extract the

relevant thresholds for truncating the virtual MO space, we choose two sets of molecular systems. The first set consists of 25 mononuclear molecular complexes with variable metal centers, coordination environments, oxidation and spin states (see Figure 3). More specifically this set consist of complexes: TiIVCl4 (1)59, Cs2[CuIICl4] (D2d) (2)60 (2)59, TiIVCpCl3 (3)59, TiIVCp2Cl2 (4)59, V(IV)O(HRL2), Me (5)61, Et (6)61, Cl (7)61, [V(V)O2(R,R'L1)] , R=Me, R'=Ph (8)61, R=H, R'=Me (9)61, R=H, R'=4−Ph (10)61, R=Me, R'=O2NPh (11)61, K4[CrII(CN)6] (12)62, K3[CrIII(CN)6] (13)62, K3[CoIII(CN)6] (14)63,

[CoIII(NH3)6]I3 (15)63,

[CoIII(phen)3](ClO4)3 (16)63, CoIII(dtc)3, (17)63, V(III)(acac)3 (19)64, Cr(III)(acac)3 (20)64, Mn(III)(acac)3 (21)64,

Fe(III)(acac)3 (22), Co(III)(acac)3 (23)64, V(IV)O(acac)2 (24)64 and

(PPh4)2[MnVN(CN)4] (25)65. The following chemical abbreviations have been used for the ligands: Cp, cyclopentadienyl; acac, acetylacetonate; phen,phenathroline; L1, oxyoxime; salicylaldoxime61. In addition, a validation study of PNO-ROCIS was performed on molecular systems extracted from the B12 aqua-cobalamine [H2OCbl]+66 and deoxymyoglobin (DMb)67 biomolecules as well as from the bulk structures of TiO2_rutile68 and hematite α-Fe2O3 69 solid surfaces, as it is shown in Figure 4.


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Figure 3. Graphical representation of the structure of the model molecular complexes consisting the molecular training set used to calibrate the PNO-ROCIS method.


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Figure 4. Graphical representation of the structures of the aqua-cobalamin [H2OCbl]+, deoxy-myoglobin (DMb), TiO2 rutile and Fe2O3 hematite. The structure of DMb is represented by molecular cuts that are performed at 1, 2 and 10 Å, around the metal center. For the TiO2 rutile and Fe2O3 hematite clusters up to 52 Ti and 7 Fe atoms were considered.

V.

PNO-ROCIS Calibration

A. TCutPNO Threshold The optimum TCutPNO thresholds across the edges were estimated by first performing a series of PNO-ROCIS calculations on the transition metal complexes set by varying the TCutPNO thresholds in the range (10-1-10-16). The question arises of how to best


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judge the convergence of the PNO-ROCIS results to the canonical ROCIS method. Clearly, looking at individual transitions and their intensities is highly impractical. Hence, a more global measure of convergence needs to be established. To this end, the calculated PNO-ROCIS spectrum, together with the respective canonical ROCIS spectrum, using the same linewidth parameters are compared. We consider the PNOROCIS results to be sufficiently converged if the mean-absolute error of the calculated spectral area (the integral under the absorption envelope) averaged for all molecules in the test set at a given TCutPNO threshold is within 1% of the area predicted by the canonical ROCIS results. The resulting convergence pattern is shown in Figure 5.

Figure 5. Convergence of the calculated PNO-ROCIS M, L- and K-edge spectra to the respective canonical–ROCIS spectra at variable TCutPNO thresholds. The mean absolute errors MAE(%) correspond to the PNO-ROCIS versus canonical ROCIS calculated spectral areas averaged for all molecules in the test set at a given TCutPNO threshold. The red dot lines


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indicate the convergence criterion (MAE(%)

Pair Natural Orbital Restricted Open-Shell Configuration Interaction

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