1. 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

2. 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

3. 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 …)

4. Computational challenges for core ionized/excited states

5. 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

6. 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

7. 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, (2015);
Coriani, Fransson, Christiansen, Norman, J. Chem. Theor. Comp. 8, 1616 (2012);
Santra, Breidbach, Zobeley, Cederbaum, J. Chem. Phys. 112, 9243 (2000).

8. 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, (2015);
Coriani, Fransson, Christiansen, Norman, J. Chem. Theor. Comp. 8, 1616 (2012);
Santra, Breidbach, Zobeley, Cederbaum, J. Chem. Phys. 112, 9243 (2000).

9. 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).

10. 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).

11. 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).

12. 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

13. 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

14. 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

15. 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.

16. 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.

17. 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.

18. 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

19. 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

20. 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

21. 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).

22. 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).

23. 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

24. 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

25. 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
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

26. 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

27. 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

28. 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

29. 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

30. 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

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

32. 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