DOI: 10.1039/C9CP03696B
(Paper)
Phys. Chem. Chem. Phys., 2019, Advance Article

Rasmus Faber* and
Sonia Coriani*

DTU Chemistry - Department of Chemistry, Technical University of Denmark, Kemitorvet Building 207, DK-2800 Kongens Lyngby, Denmark. E-mail: rfaber@kemi.dtu.dk; soco@kemi.dtu.dk

Received
30th June 2019
, Accepted 8th August 2019

First published on 8th August 2019

The iterative subspace algorithm to solve the complex linear response equation of damped coupled cluster response theory presented, up to CCSD level, by Kauczor et al., J. Chem. Phys., 2013, 139, 211102, and recently extended to the solution of the complex left response multipliers by Faber and Coriani, J. Chem. Theory Comput., 2019, 15, 520, has been modified to include a core–valence separation projection step in the iterative procedure. This allows one to overcome serious convergence issues that specifically manifest themselves at the CCSD level when addressing core-related spectroscopic effects using large basis sets. The spectra, obtained adopting the new scheme for X-ray absorption and circular dichroism, as well as resonant inelastic-X-ray scattering, are presented and discussed. Core–valence separated results for non-resonant X-ray emission are also reported.

The method relies on the ability to solve linear response equations for a complex, or damped, frequency,^{10} and it has been successfully implemented at various levels of electronic structure theory, including Hartree–Fock and time-dependent density functional theory,^{1–3,12} as well as multiconfigurational self-consistent field,^{1,2} algebraic diagrammatic construction (ADC)^{13} and coupled-cluster (CC) theories.^{7–9,14} Extensions to solvated environments^{15,16} and in the relativistic domain^{17} have also been presented. Applications encompass the calculation of linear absorption spectra in different frequency regions, including X-ray absorption spectra,^{4,5,8,14,17} electronic circular dichroism spectra,^{18} magnetic-field and nuclear-spin induced circular dichroism,^{19–21} magneto-chiral dichroism and birefringence dispersion,^{22} two-photon absorption in both UV-vis and X-ray regimes,^{6,23} Cauchy coefficients at imaginary frequency,^{7,13,24} and, more recently, resonant inelastic X-ray scattering.^{9,25}

At the CC level, two different algorithms have been proposed to obtain complex response functions. In the first one,^{7,8} a diagonal basis of eigenvectors was generated by diagonalization of an approximate Lanczos-based tridiagonal representation of the CC Jacobian, and used to construct the imaginary or real components of the complex linear response function via a sum-over-state-like expression that includes the damping parameter γ. In the second one,^{9,14} the damped linear response equations yielding the complex (real and imaginary components thereof) amplitudes and multipliers were solved via a generalization of the reduced-space algorithm used in the conventional CC response.

The main drawbacks of the first approach are the need to pre-decide the size of the truncated Lanczos basis, which prevents the a priori control of the convergence thresholds, and the need to store a large number of Lanczos vectors on file, resulting in disk and I/O issues for larger systems. Moreover, if the target excited states lie in the X-ray region, large Lanczos chain lengths are required to obtain converged X-ray energies, unless specific core–valence separated (CVS) techniques are adopted.^{26–28}

In the CPP-CC method of ref. 14, on the other hand, the damped response solver manifests severe convergence issues at the CC singles and doubles (CCSD) level in the high-energy frequency region and in particular when larger basis sets are used.

These issues can be rationalized as originating from the high density of doubly excited/ionized valence states that form the continuum in which the X-ray absorption bands are embedded. Thus, even though the essence of the CPP method is to introduce a damping parameter γ in the (linear) response functions to specifically remove the singularities when the external frequency approaches a resonant value, in the CCSD case, this damping is not sufficient to guarantee convergence as the basis set increases, due to the enormous number of closely lying double excited/ionized states that become accessible when a large basis set is used. Too many potential resonances can occur for just one parameter to take care of, and the less and less diagonal the CC Jacobian becomes, which compromises the efficiency of the diagonal preconditioner that is typically used to accelerate convergence. As further proof, in the case of the CC singles and approximate doubles method, CC2, where the double-double block of the Jacobian exactly corresponds to the orbital energy differences matrix, the damped response equations do converge.

To overcome these problems, we here propose to apply a core–valence separation projector^{27} during the solution of the CCSD complex response equations, to remove the continuum of valence ionized states. The performance of the resulting CVS-CPP algorithm is illustrated by calculations, within the CCSD linear response (LR-CCSD) framework, of near-edge-absorption fine structure (NEXAFS) and X-ray circular dichroism (XCD) and, within CCSD quadratic response (QR-CCSD), of resonant inelastic X-ray scattering (RIXS). CVS-CCSD results for non-resonant X-ray emission (XES), computed, according to our previously proposed recipe,^{9} as transitions between valence- and core-ionized states, are also reported for completeness. Equation-of-motion (CVS-CPP-EOM-CCSD) variants of the same properties could also be derived by modifications to the property Jacobian matrix and final property expressions.^{9,29,30} An alternative derivation of RIXS and XES exploiting a conceptually analogous CVS-DIIS damped solver within the frozen-core core–valence separated fc-CVS-EOM-CCSD framework^{31} is presented in ref. 32.

σ_{XAS}(ω) ∝ − ωℑ〈〈μ_{α};μ_{α}〉〉^{γ}_{ω}
| (1) |

σ_{XCD}(ω) ∝ ωℜ〈〈μ_{α};m_{α}〉〉^{γ}_{ω}
| (2) |

(3) |

In the CC damped linear response, the complex polarizability is computed according to:

(4) |

For the NEXAFS cross section, the imaginary part of the complex dipole–dipole polarizability in eqn (4) is needed, which is obtained according to

(5) |

t^{X}(ω + iγ) = t^{X}_{ℜ}(ω + iγ) + it^{X}_{ℑ}(ω + iγ),
| (6) |

−t^{X}_{ℑ}(−ω + iγ) = t^{X}_{ℑ}(−ω − iγ),
| (7) |

−t^{χ}_{ℜ}(−ω + iγ) = t^{χ}_{ℜ}(−ω − iγ).
| (8) |

An imaginary operator is needed to compute the XCD cross section in eqn (2), according to

(9) |

In the RIXS amplitudes, the operators X and Y are always real, and both real and imaginary components are needed. The general complex expressions of the KHD amplitudes within damped CC response theory (as well as EOM-CCSD) were presented in ref. 9, and can be split into real and imaginary parts as given below for ^{CC}^{f0}_{XY}(ω):

(10) |

While referring to, e.g., ref. 9 and 35, for a definition of the remaining CC building blocks in the expressions above, we return to the solution of the complex response equations needed to obtain the real and imaginary components of the complex amplitudes t^{X}(ω + iγ) and multipliers ^{X}(ω − iγ):

(A − (ω + iγ)I)t^{X}(ω + iγ) = −ξ^{X}
| (11) |

^{X}(ω′ − iγ)(A + (ω′ − iγ)I) = −η^{X} − Ft^{X}(ω′ − iγ)
| (12) |

They can be recast in (pseudo-symmetric) matrix form as, e.g.,

(13) |

An important step to ensure convergence in the iterative algorithm is the generation of new trial vectors as

(14) |

(15) |

Despite the preconditioner, and as anticipated in the introduction, the iterative subspace algorithm of ref. 9 and 14 does not converge at the CCSD level (in larger basis sets) when solving for the complex response amplitudes at positive values of ω falling in the X-region. The same occurs when solving for the response multipliers at negative frequencies in the X-region.

While a finite value of γ ensures that eqn (13) will not be exactly singular for any value of ω, the solution of this equation becomes nonetheless unfeasible if many excitation energies are very close to ω. The X-ray region is beyond the (valence)ionization limit of common molecules and there is therefore a continuum of ionized states near any given core-excitation energy. In calculations using finite basis sets, a discrete set of ionization energies is found and, for methods that only include single excitations, this set might be sparse enough not to prevent convergence of the damped response equations in the X-ray region. This explains why methods like CC2 and ADC(2) in general do not manifest (according to our experience) converge problem. For models, such as CCSD, that explicitly include double excitations, however, the set of ionized states is so dense that the normal damped response equations will not converge except for the smallest molecules and basis sets.

The problem of separating bound states from the continuum of ionized states can be solved using the CVS approximation,^{26} which has recently been introduced in the context of CC theory to describe core-excited and core-ionized states.^{27,31} In the present work, we propose to use a CVS projector in eqn (13), whenever the frequency ω is positive and falls in the X-ray region.

This projector allows only the part of the CC amplitudes in eqn (11) that involve core electrons to respond to the external electric field, effectively removing the valence ionization contribution. Similarly, a CVS projector is applied in the multiplier equation, eqn (12), for negative ω′. In the opposite case, negative ω in eqn (11) and positive ω′ in eqn (12), the equations are strongly diagonally dominant for absolute values of ω in the X-ray region and can easily be solved without applying projectors of any kind.

Comparing to the formulation of CVS-CCLR of Coriani and Koch,^{27} projecting out the valence excitation space from the right and left response equations only for the above-mentioned frequency values corresponds to projecting out exclusively from the eigenvalue equations. However, in ref. 27, it was suggested that one could apply CVS also to the left multiplier _{f} equations, even though they are convergent. To obtain a damped-response CVS scheme equivalent to a CVS scheme where the _{f} equations are also projected out, we would need to apply the CVS projector to all equations where the magnitude of the frequency is in the X-ray region, no matter its sign. Even though the effect of projecting out from the _{f} vectors is negligible, our recommendation is however not to project them.

The XCD of methyloxirane has been experimentally measured by Turchini et al.^{38} at the carbon K-edge. The study of Turchini et al. is the first (and only) CD measurement on a randomly oriented system and it was performed in the vapour phase. Computational investigations have been presented at the random phase approximation level by Alagna et al.,^{40} and using STEX by Carravetta et al.^{41}

In Fig. 3, we compare the XCD spectrum of methyloxirane calculated using CVS-CPP-CCSD with the one obtained from a CVS-CCSD-LR calculation. The CPP-CVS-CCSD calculations were performed on a grid with a spacing of 5 × 10^{−4} Hartree between points and with a broadening factor of 1000 cm^{−1}. Our calculated spectra present bands and sticks somewhat consistent with the computed spectral sticks of ref. 41, whereas the agreement with the experimental spectrum is far from satisfactory. Further investigations are required to firmly assess the origin of the observed discrepancies.

Fig. 4 Full-space CPP-CCSD (black) versus CVS-CPP-CCSD (red) results for the RIXS (at the indicated incident frequency) and XES spectra of water in the 6-311++G**+Ryd basis set. |

As another example of the applicability of the CVS-CPP approach to RIXS, we considered the case of acetone at the oxygen K-edge. Despite the still moderate size of the molecule, its RIXS spectra could not be computed using the full-space CPP-CCSD approach, whereas they are easily obtained with the CVS-CPP solver. Table 1 collects the excitation energies and strengths of the valence excited states that have been considered in the spectral simulation. The resonant pump frequency was chosen at the value of the first core excitation at the oxygen K-edge, see also Table 2. The computed RIXS and XES spectral slices are shown in Fig. 6. Experimental resonant (RIXS) and non-resonant (XES) spectral slices at the oxygen K-edge in solution are available from ref. 42, see also ref. 43. The re-digitized experimental curves have been overlapped with the computed spectra without any shift on the energy axis. Despite the misalignment, the main spectral features at the lower core excitation in the resonant RIXS spectrum, as well as the XES spectrum, are reproduced. As can be appreciated from Fig. 6, the overall shift of the XES spectrum is slightly larger than for the RIXS one, possibly an indication that CCSD describes relaxation effects on XES (or, at least, on XES as computed here^{9} at the CCSD level) less accurately than those on RIXS.

Symmetry | ω_{n} (a.u.) |
ω_{n} (eV) |
S_{n0} (a.u.) |
f_{n0} |
---|---|---|---|---|

A_{2} |
0.166849 | 4.5402 | 0.0000000 | 0.00000 |

B_{2} |
0.238143 | 6.4802 | 0.1991480 | 0.03162 |

A_{1} |
0.278881 | 7.5887 | 0.0000052 | 0.00000 |

A_{2} |
0.279308 | 7.6003 | 0.0000000 | 0.00000 |

B_{2} |
0.286427 | 7.7941 | 0.0479960 | 0.00916 |

B_{2} |
0.300078 | 8.1655 | 0.1921846 | 0.03845 |

A_{1} |
0.309681 | 8.4269 | 0.3608999 | 0.07451 |

B_{1} |
0.321666 | 8.7530 | 0.0891817 | 0.01912 |

B_{2} |
0.325418 | 8.8551 | 0.0069247 | 0.00150 |

A_{2} |
0.329077 | 8.9546 | 0.0000000 | 0.00000 |

B_{1} |
0.345325 | 9.3968 | 0.0219041 | 0.00504 |

B_{1} |
0.346340 | 9.4244 | 0.0315136 | 0.00728 |

A_{1} |
0.346448 | 9.4273 | 1.3455875 | 0.31078 |

B_{2} |
0.346749 | 9.4355 | 0.0759524 | 0.01756 |

A_{2} |
0.354554 | 9.6479 | 0.0000000 | 0.00000 |

B_{1} |
0.364546 | 9.9198 | 0.0042734 | 0.00104 |

A_{1} |
0.365335 | 9.9413 | 0.0096702 | 0.00235 |

B_{2} |
0.375621 | 10.221 | 0.1865032 | 0.04670 |

B_{2} |
0.378524 | 10.300 | 0.1435405 | 0.03622 |

A_{2} |
0.381769 | 10.388 | 0.0000000 | 0.00000 |

A_{1} |
0.387199 | 10.536 | 0.0279431 | 0.00721 |

A_{2} |
0.388462 | 10.571 | 0.0000000 | 0.00000 |

A_{1} |
0.389988 | 10.612 | 0.1629636 | 0.04237 |

B_{1} |
0.397173 | 10.808 | 0.2212859 | 0.05859 |

A_{1} |
0.398122 | 10.833 | 0.4069430 | 0.10801 |

A_{2} |
0.404636 | 11.012 | 0.0000000 | 0.00000 |

B_{2} |
0.405736 | 11.041 | 0.3600519 | 0.09739 |

B_{1} |
0.408028 | 11.103 | 0.0198221 | 0.00539 |

A_{1} |
0.412197 | 11.216 | 0.1073709 | 0.02950 |

B_{2} |
0.415604 | 11.309 | 0.2804371 | 0.07770 |

ω_{n} (a.u.) |
ω_{n} (eV) |
S_{n0} (a.u.) |
f_{n0} |
---|---|---|---|

19.5892 | 533.05 | 0.0033221 | 0.04338 |

19.7441 | 537.26 | 0.0000000 | 0.00000 |

19.7855 | 538.39 | 0.0001939 | 0.00256 |

19.7880 | 538.46 | 0.0000809 | 0.00107 |

19.7910 | 538.54 | 0.0001268 | 0.00167 |

19.8098 | 539.05 | 0.0000418 | 0.00055 |

Fig. 6 CVS-CPP-CCSD results for the RIXS (bottom panel) and XES (upper panel, label “non-resonant”) spectra at the oxygen K-edge of acetone in the 6-311++G** basis set. The experimental spectral slices in solution from ref. 42 are also shown. Note that the computed spectra have not been shifted to align to the experimental ones. |

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