Lan Cheng Department of Chemistry Johns Hopkins University Scalar Relativistic Equation-of-Motion Coupled-Cluster Calculations of Core Ionized/Excited States Lan Cheng Department of Chemistry Johns Hopkins University

Motivation With rapid advancement of synchrotron radiation resources, core- level spectroscopy has become useful tools for studying a wide range of phenomena Aiming at development of generally applicable methods for accurate calculations of core-excited and core-ionized states Excitation/ionization energies Intensities Dynamics

Summary of computational developments in the literature Density functional theory (DFT) techniques Applicable to large molecules Moderate accuracy (e.g., Norman, Li, Besley, Evangelista …) Green’s function theory Can be systematically improved Perturbative in nature (Cederbaum, Schirmer, Dreuw …) Equation-of-motion coupled-cluster (EOMCC) methods Accurate and can be systematically improved Computationally challenging (Coriani, Koch, Li …)

Computational challenges for core ionized/excited states

Computational challenges for core ionized/excited states Convergence of the EOM-CC equation for core excited/ionized states The target state is (energetically) embedded in continuum (Many other states energetically close to the target state) Standard Davidson’s algorithm is best for targeting the lowest roots

Computational challenges for core ionized/excited states Convergence of the EOM-CC equation for core excited/ionized states The target state is embedded in continuum (Many other states energetically close to the target state) Standard Davidson’s algorithm is best for targeting the lowest roots Treatment of relativistic effects scalar-relativistic effects spin-orbit coupling

Computational challenges for core ionized/excited states Convergence of the EOM-CC equation for core excited/ionized states Core-valence separation (CVS) Asymmetric Lanczos algorithm Filter diagonalization Treatment of relativistic effects scalar-relativistic effects spin-orbit coupling Cederbaum, Domcke, Schirmer, Phys. Rev. A 22, 206 (1980); Coriani, Koch, J. Chem. Phys. 143, 181103 (2015); Coriani, Fransson, Christiansen, Norman, J. Chem. Theor. Comp. 8, 1616 (2012); Santra, Breidbach, Zobeley, Cederbaum, J. Chem. Phys. 112, 9243 (2000).

Computational challenges for core ionized/excited states Convergence of the EOM-CC equation for core excited/ionized states Core-valence separation (CVS) Asymmetric Lanczos algorithm Filter diagonalization Treatment of relativistic effects spin-free exact two-component theory spin-orbit coupling by means of degenerate perturbation theory via EOMCC Cederbaum, Domcke, Schirmer, Phys. Rev. A 22, 206 (1980); Coriani, Koch, J. Chem. Phys. 143, 181103 (2015); Coriani, Fransson, Christiansen, Norman, J. Chem. Theor. Comp. 8, 1616 (2012); Santra, Breidbach, Zobeley, Cederbaum, J. Chem. Phys. 112, 9243 (2000).

Implementation in CFOUR program package The implementation is built upon the existing efficient implementation of EOMIP-CCSDT (Matthews and Stanton) and EOM-CCSD in CFOUR (Stanton and Gauss).

Implementation in CFOUR program package Arnoldi algorithm Arnoldi iteration works with Krylov matrix: Diagonalization within this subspace Convergence can be achieved, but may be very expensive. The implementation is built upon the existing efficient implementation of EOMIP-CCSDT (Matthews and Stanton) and EOM-CCSD in CFOUR (Stanton and Gauss).

Implementation in CFOUR program package Arnoldi algorithm Core-valence separation (CVS) Arnoldi iteration works with Krylov matrix: Diagonalization within this subspace Convergence can be achieved, but may be very expensive. Valence multiply excited states can get energetically close to core ionized/excited states. However, core ionized/excited states are spatially isolated from valence excited states. Hence the coupling with pure valence excited states is small and can be neglected. The implementation is built upon the existing efficient implementation of EOMIP-CCSDT (Matthews and Stanton) and EOM-CCSD in CFOUR (Stanton and Gauss).

Benchmark calculations: Relativistic effects H2O IP for 1s of O CO IP for 1s of C NH3 IP for 1s of N non-relativistic 539.6 542.3 296.4 405.5 EOMIP-CCSDT/aug-cc-pCVTZ results

Benchmark calculations: Relativistic effects H2O IP for 1s of O CO IP for 1s of C NH3 IP for 1s of N non-relativistic 539.6 542.3 296.4 405.5 SFX2C-1e 540.0 542.6 296.5 405.7 EOMIP-CCSDT/aug-cc-pCVTZ results

Benchmark calculations: Relativistic effects H2O IP for 1s of O CO IP for 1s of C NH3 IP for 1s of N non-relativistic 539.6 542.3 296.4 405.5 SFX2C-1e 540.0 542.6 296.5 405.7 Δ(rel) 0.4 0.1 0.2 EOMIP-CCSDT/aug-cc-pCVTZ results

Benchmark calculations: Higher excitations H2O IP for 1s of O CO IP for 1s of C NH3 IP for 1s of N EOMIP-CCSD 541.5 544.3 297.6 407.0 The aug-cc-pCVTZ basis sets were used.

Benchmark calculations: Higher excitations H2O IP for 1s of O CO IP for 1s of C NH3 IP for 1s of N EOMIP-CCSD 541.5 544.3 297.6 407.0 EOMIP-CCSDT 539.6 542.3 296.4 405.5 The aug-cc-pCVTZ basis sets were used.

Benchmark calculations: Higher excitations H2O IP for 1s of O CO IP for 1s of C NH3 IP for 1s of N EOMIP-CCSD 541.5 544.3 297.6 407.0 EOMIP-CCSDT 539.6 542.3 296.4 405.5 Δ(T) -1.9 -2.0 -1.2 -1.6 The aug-cc-pCVTZ basis sets were used.

Benchmark calculations: Comparison with Experiments H2O IP for 1s of O CO IP for 1s of C NH3 IP for 1s of N EOMCC 539.83 542.48 296.47 405.54 SFX2C-1e-EOMIP-CCSD/aug-cc-pCV5Z results augmented with SFX2C-1e/aug-cc-pCVQZ triples correction

Benchmark calculations: Comparison with Experiments H2O IP for 1s of O CO IP for 1s of C NH3 IP for 1s of N EOMCC 539.83 542.48 296.47 405.54 Experiment 539.78 542.5 296.2 405.6 SFX2C-1e-EOMIP-CCSD/aug-cc-pCV5Z results augmented with SFX2C-1e/aug-cc-pCVQZ triples correction

Benchmark calculations: Comparison with Experiments H2O IP for 1s of O CO IP for 1s of C NH3 IP for 1s of N EOMCC 539.8 542.5 296.5 405.5 Experiment 296.2 405.6 Δ(EOMCC-Experiment) 0.1 0.0 0.3 -0.1 SFX2C-1e-EOMIP-CCSD/aug-cc-pCV5Z results augmented with SFX2C-1e/aug-cc-pCVQZ triples correction

Application to transition-metal complexes Excitations from the inner s-type orbitals of ligands are used to probe “covalency” of metal ligand bonds Mixing of d orbitals of metal and p-type orbitals of the ligands Glaser, Hedman, Hodgson, Solomon, Acc. Chem. Res., 33, 859 (2000).

Application to transition-metal complexes Excitations from the inner s-type orbitals of ligands are used to probe “covalency” of metal ligand bonds Mixing of d orbitals of metal and p-type orbitals of the ligands Charge-transfer excitations Glaser, Hedman, Hodgson, Solomon, Acc. Chem. Res., 33, 859 (2000).

Application to transition-metal complexes D4h [CuCl4]2- Cl 1s-> Cu 3d D2d [CuCl4]2- Excitation energy Oscillator Strength cc-pVDZ (NR) 2832.4 0.0037 2832.3 0.0022 Glaser, Hedman, Hodgson, Solomon, Acc. Chem. Res., 33, 859 (2000). EOMEE-CCSD results obtained at SFX2C-1e-CCSD(T)/ANO1 geometries

Application to transition-metal complexes D4h [CuCl4]2- Cl 1s-> Cu 3d D2d [CuCl4]2- Excitation energy Oscillator Strength cc-pVDZ (NR) 2832.4 0.0037 2832.3 0.0022 cc-pVDZ （SFX2C-1e) 2842.6 0.0036 Glaser, Hedman, Hodgson, Solomon, Acc. Chem. Res., 33, 859 (2000). EOMEE-CCSD results obtained at SFX2C-1e-CCSD(T)/ANO1 geometries

Application to transition-metal complexes D4h [CuCl4]2- Cl 1s-> Cu 3d D2d [CuCl4]2- Excitation energy Oscillator Strength cc-pVDZ (NR) 2832.4 0.0037 2832.3 0.0022 cc-pVDZ （SFX2C-1e) 2842.6 0.0036 Δ(rel) 10.2 -0.0001 10.3 0.0000 Glaser, Hedman, Hodgson, Solomon, Acc. Chem. Res., 33, 859 (2000). EOMEE-CCSD results obtained at SFX2C-1e-CCSD(T)/ANO1 geometries

Application to transition-metal complexes D4h [CuCl4]2- Cl 1s-> Cu 3d D2d [CuCl4]2- Excitation energy Oscillator Strength cc-pVDZ-unc 2827.8 0.0036 2827.7 0.0022 Exp. 2820.6 2820.2 Glaser, Hedman, Hodgson, Solomon, Acc. Chem. Res., 33, 859 (2000). SFX2C-1e-EOMEE-CCSD results obtained at SFX2C-1e-CCSD(T)/ANO1 geometries

Application to transition-metal complexes D4h [CuCl4]2- Cl 1s-> Cu 3d D2d [CuCl4]2- Excitation energy Oscillator Strength cc-pVDZ 2842.6 0.0036 0.0022 cc-pVTZ 2836.3 0.0040 2836.4 0.0024 Exp. 2820.7 2820.2 Glaser, Hedman, Hodgson, Solomon, Acc. Chem. Res., 33, 859 (2000). SFX2C-1e-EOMEE-CCSD results obtained at SFX2C-1e-CCSD(T)/ANO1 geometries

Application to transition-metal complexes D4h [CuCl4]2- Cl 1s-> Cu 3d D2d [CuCl4]2- Excitation energy Oscillator Strength cc-pVDZ 2842.6 0.0036 0.0022 cc-pVTZ 2836.3 0.0040 2836.4 0.0024 aug-cc-pVDZ 2843.5 \ Exp. 2820.7 2820.2 Glaser, Hedman, Hodgson, Solomon, Acc. Chem. Res., 33, 859 (2000). SFX2C-1e-EOMEE-CCSD results obtained at SFX2C-1e-CCSD(T)/ANO1 geometries

Application to transition-metal complexes D4h [CuCl4]2- Cl 1s-> Cu 3d D2d [CuCl4]2- Excitation energy Oscillator Strength cc-pVDZ 2842.6 0.0036 0.0022 cc-pVTZ 2836.3 0.0040 2836.4 0.0024 aug-cc-pVDZ 2843.5 \ cc-pCVDZ 2828.9 2829.1 Exp. 2820.7 2820.2 Glaser, Hedman, Hodgson, Solomon, Acc. Chem. Res., 33, 859 (2000). SFX2C-1e-EOMEE-CCSD results obtained at SFX2C-1e-CCSD(T)/ANO1 geometries

Application to transition-metal complexes D4h [CuCl4]2- Cl 1s-> Cu 3d D2d [CuCl4]2- Excitation energy Oscillator Strength cc-pVDZ 2842.6 0.0036 0.0022 cc-pVTZ 2836.3 0.0040 2836.4 0.0024 aug-cc-pVDZ 2843.5 \ cc-pCVDZ 2828.9 2829.1 cc-pCVTZ ? aug-cc-pCVTZ Exp. 2820.6 2820.2 Glaser, Hedman, Hodgson, Solomon, Acc. Chem. Res., 33, 859 (2000). SFX2C-1e-EOMEE-CCSD results obtained at SFX2C-1e-CCSD(T)/ANO1 geometries

Outlook An efficient implementation for the CVS scheme Asymmetric Lanczos algorithm Spin-orbit coupling for L-edge excitations

Acknowledgement Sonia Coriani Technical University of Denmark Henrik Koch Norwegian University of Science and Technology Devin Matthews University of Texas at Austin John Stanton University of Florida Jaime Combariza Maryland Advanced Research Computing Center