Critical Assessment of Time-Dependent Density Functional Theory for Excited States of Open-Shell Systems: II. Doublet-Quartet Transitions

Compared with closed-shell systems, open-shell systems place three additional challenges to time-dependent density functional theory (TD-DFT) for electronically excited states: (a) the spin-contamination problem is a serious issue; (b) the exchange-correlation (XC) kernel may be numerically instable; and (c) the single-determinant description of open-shell ground states readily becomes energetically instable. Confined to flip-up single excitations, the spin-contamination problem can largely be avoided by using the spin-flip TD-DFT (SF-TD-DFT) formalism, provided that a noncollinear XC kernel is employed. As for the numerical instabilities associated with such a kernel, only an ad hoc scheme has been proposed so far, viz., the ALDA0 kernel, which amounts to setting the divergent components (arising from density gradients and kinetic energy density) simply to zero. The ground-state instability problem can effectively be avoided by introducing the Tamm-Dancoff approximation (TDA) to TD-DFT. Therefore, on a general basis, the SF-TDA/ALDA0 Ansatz is so far the only promising means within the TD-DFT framework for flip-up single excitations of open-shell systems. To assess systematically the performance of SF-TDA/ALDA0, in total 61 low-lying quartet excited states of the benchmark set of 11 small radicals [<i>J. Chem. Theory Comput.</i> <b>2016</b>, <i>12</i>, 238] are investigated with various XC functionals. Taking the MRCISD+Q (multireference configuration interaction with singles and doubles plus the Davidson correction) results as benchmark, it is found that the mean absolute errors of SF-TDA/ALDA0 with the SAOP (statistical averaging of model orbital potentials), global hybrid, and range-separated hybrid functionals are in the range of 0.2–0.4 eV. This is in line not only with the typical accuracy of TD-DFT for singlet and triplet excited states of closed-shell systems but also with the gross accuracy of spin-adapted TD-DFT for spin-conserving excited states of open-shell systems

SOiCISCF: Combining SOiCI and iCISCF for Variational Treatment of Spin–Orbit Coupling

It has recently been shown that the SOiCI approach [Zhang, N.; J. Phys.: Condens. Matter 2022, 34, 224007], in conjunction with the spin-separated exact two-component relativistic Hamiltonian, can provide very accurate fine structures of systems containing heavy elements by treating electron correlation and spin–orbit coupling (SOC) on an equal footing. Nonetheless, orbital relaxations/polarizations induced by SOC are not yet fully accounted for due to the use of scalar relativistic orbitals. This issue can be resolved by further optimizing the still real-valued orbitals self-consistently in the presence of SOC, as done in the spin–orbit coupled CASSCF approach [Ganyushin, D.; et al. J. Chem. Phys. 2013, 138, 104113] but with the iCISCF algorithm [Guo, Y.; J. Chem. Theory Comput. 2021, 17, 7545–7561] for large active spaces. The resulting SOiCISCF employs both double group and time reversal symmetries for computational efficiency and the assignment of target states. The fine structures of p-block elements are taken as showcases to reveal the efficacy of SOiCISCF

Performance of TD-DFT for Excited States of Open-Shell Transition Metal Compounds

Time-dependent density functional theory (TD-DFT) has been very successful in accessing low-lying excited states of closed-shell systems. However, it is much less so for excited states of open-shell systems: unrestricted Kohn–Sham based TD-DFT (U-TD-DFT) often produces physically meaningless excited states due to heavy spin contaminations, whereas restricted Kohn–Sham based TD-DFT often misses those states of lower energies. A much better variant is the explicitly spin-adapted TD-DFT (X-TD-DFT) [<i>J. Chem. Phys.</i> <b>2011</b>, <i>135</i>, 194106] that can capture all the spin-adapted singly excited states yet without computational overhead over U-TD-DFT. While the superiority of X-TD-DFT over U-TD-DFT has been demonstrated for open-shell systems of main group elements, it remains to be seen if this is also the case for open-shell transition metal compounds. Taking as benchmark the results by MS-CASPT2 (multistate complete active space second-order perturbation theory) and ic-MRCISD (internally contracted multireference configuration interaction with singles and doubles), it is shown that X-TD-DFT is indeed superior to U-TD-DFT for the vertical excitation energies of ZnH, CdH, ScH₂, YH₂, YO, and NbO₂. Admittedly, there exist a few cases where U-TD-DFT appears to be better than X-TD-DFT. However, this is due to a wrong reason: the underestimation (due to spin contamination) and the overestimation (due to either the exchange-correlation functional itself or the adiabatic approximation to the exchange-correlation kernel) happen to be compensated in the case of U-TD-DFT. As for [Cu­(C₆H₆)₂]²⁺, which goes beyond the capability of both MS-CASPT2 and ic-MRCISD, X-TD-DFT revises the U-TD-DFT assignment of the experimental spectrum

Spin-Multiplet Components and Energy Splittings by Multistate Density Functional Theory

Kohn–Sham density functional theory has been tremendously successful in chemistry and physics. Yet, it is unable to describe the energy degeneracy of spin-multiplet components with any approximate functional. This work features two contributions. (1) We present a multistate density functional theory (MSDFT) to represent spin-multiplet components and to determine multiplet energies. MSDFT is a hybrid approach, taking advantage of both wave function theory and density functional theory. Thus, the wave functions, electron densities and energy density-functionals for ground and excited states and for different components are treated on the same footing. The method is illustrated on valence excitations of atoms and molecules. (2) Importantly, a key result is that for cases in which the high-spin components can be determined separately by Kohn–Sham density functional theory, the transition density functional in MSDFT (which describes electronic coupling) can be defined rigorously. The numerical results may be explored to design and optimize transition density functionals for configuration coupling in multiconfigurational DFT

Prognostic Value of MET Gene Copy Number and Protein Expression in Patients with Surgically Resected Non-Small Cell Lung Cancer: A Meta-Analysis of Published Literatures

<div><p>Background</p><p>The prognostic value of the copy number (GCN) and protein expression of the mesenchymal-epithelial transition (MET) gene for survival of patients with non-small cell lung cancer (NSCLC) remains controversial. This study aims to comprehensively and quantitatively asses the suitability of MET GCN and protein expression to predict patients' survival.</p><p>Methods</p><p>PubMed, Embase, Web of Science and Google Scholar were searched for articles comparing overall survival in patients with high MET GCN or protein expression with those with low level. Pooled hazard ratio (HR) and 95% confidence intervals (CIs) were calculated using the random and the fixed-effects models. Subgroup and sensitivity analyses were also performed.</p><p>Results</p><p>Eighteen eligible studies enrolling 5,516 patients were identified. Pooled analyses revealed that high MET GCN or protein expression was associated with poor overall survival (OS) (GCN: HR = 1.90, 95% CI 1.35–2.68, <i>p</i><0.001; protein expression: HR = 1.52, 95% CI 1.08–2.15, <i>p</i> = 0.017). In Asian populations (GCN: HR = 2.22, 95% CI 1.46–3.38, <i>p</i><0.001; protein expression: HR = 1.89, 95% CI 1.34–2.68, <i>p</i><0.001), but not in the non-Asian subset. For adenocarcinoma, high MET GCN or protein expression indicated decreased OS (GCN: HR = 1.49, 95% CI 1.05–2.10, <i>p</i> = 0.025; protein expression: HR = 1.69, 95% CI 1.31–2.19, <i>p</i><0.001). Results were similar for multivariate analysis (GCN: HR = 1.61, 95% CI 1.15–2.25, <i>p</i> = 0.005; protein expression: HR = 2.18, 95% CI 1.60–2.97, <i>p</i><0.001). The results of the sensitivity analysis were not materially altered and did not draw different conclusions.</p><p>Conclusions</p><p>Increased MET GCN or protein expression was significantly associated with poorer survival in patients with surgically resected NSCLC; this information could potentially further stratify patients in clinical treatment.</p></div

Relativistic GVVPT2 Multireference Perturbation Theory Description of the Electronic States of Y₂ and Tc₂

The multireference generalized Van Vleck second-order perturbation theory (GVVPT2) method is used to describe full potential energy curves (PECs) of low-lying states of second-row transition metal dimers Y₂ and Tc₂, with scalar relativity included via the spin-free exact two-component (sf-X2C) Hamiltonian. Chemically motivated incomplete model spaces, of the style previously shown to describe complicated first-row transition metal diatoms well, were used and again shown to be effective. The studied states include the previously uncharacterized 2¹Σ_g⁺ and 3¹Σ_g⁺ PECs of Y₂. These states, together with 1¹Σ_g⁺, are relevant to discussion of controversial results in the literature that suggest dissociation asymptotes that violate the noncrossing rule. The ground state of Y₂ was found to be X⁵Σ_u^– (similar to Sc₂) with bond length <i>R</i>_e = 2.80 Å, binding energy <i>D</i>_e = 3.12 eV, and harmonic frequency ω_e = 287.2 cm^–1, whereas the lowest 1¹Σ_g⁺ state of Y₂ was found to lie 0.67 eV above the quintet ground state and had spectroscopic constants <i>R</i>_e = 3.21 Å, <i>D</i>_e = 0.91 eV, and ω_e = 140.0 cm^–1. Calculations performed on Tc₂ include study of the previously uncharacterized relatively low-lying 1⁵Σ_g⁺ and 1⁹Σ_g⁺ states (i.e., 0.70 and 1.84 eV above 1¹Σ_g⁺, respectively). The ground state of Tc₂ was found to be X³Σ_g^– with <i>R</i>_e = 2.13 Å, <i>D</i>_e = 3.50 eV, and ω_e = 336.6 cm^–1 (for the most stable isotope, Tc-98) whereas the lowest ¹Σ_g⁺ state, generally accepted to be the ground state symmetry for isovalent Mn₂ and Re₂, was found to lie 0.47 eV above the X³Σ_g^– state of Tc₂. The results broaden the range of demonstrated applicability of the GVVPT2 method

Evaluation of human mesenchymal-epithelial transition (MET) by immunohistochemistry (IHC) in the selected studies in the selected studies.

<p>NA: not available; NSCLC, non-small cell lung cancer; ADC, adenocarcinoma; IHC, immunohistochemistry; HR: hazard ratio, obtained by estimated (E) or reported in text (R). “M” means the HR come from multivariate analysis, and “U” means HR come from univariate analysis; EGFR, epidermal growth factor receptor; HGF, hepatocyte growth factor.</p

Forest plot (A) assessing MET protein expression in NSCLC stratified by histological subtypes; Forest plot (B) assessing MET protein expression in NSCLC stratified by ethnic source.

<p>Forest plot (A) assessing MET protein expression in NSCLC stratified by histological subtypes; Forest plot (B) assessing MET protein expression in NSCLC stratified by ethnic source.</p

Forest plot (A) assessing MET gene copy number in NSCLC stratified by histological subtypes; Forest plot (B) assessing MET gene copy number in NSCLC stratified by ethnic source.

<p>Forest plot (A) assessing MET gene copy number in NSCLC stratified by histological subtypes; Forest plot (B) assessing MET gene copy number in NSCLC stratified by ethnic source.</p