Vibronic fine structure in the nitrogen 1s photoelectron spectra from Franck-Condon simulations II: Indoles

The vibronic coupling effect in nitrogen 1s X-ray photoelectron spectra (XPS) was systematically studied for a family of 17 bicyclic indole molecules by combining Franck-Condon simulations (including the Duschinsky rotation effect) and density functional theory. The simulated vibrationally-resolved spectra of 4 molecules agree well with available experiments. Reliable predictions for this family further allowed us to summarize rules for spectral evolution in response to three types of common structural changes (side chain substitution, CH$\leftrightarrow$N replacement, and isomerization). Interestingly, vibronic properties of amine and imine nitrogen are clearly separated: they show negative and positive $\Delta$ZPE (zero-point vibration energy of the core-ionized with respect to the ground state), respectively, indicating flatter and steeper PESs induced by the N 1s ionization; amine N's show stronger mode mixing effects than imine N's; the 1s ionizations on two types of nitrogens led to distinct changes in local bond lengths and angles. The rules are useful for a basic understanding of vibronic coupling in this family, and the precise spectra are useful for future reference and data mining studies

Vibronic fine structure in the nitrogen 1s photoelectron spectra from Franck-Condon simulations. III. Rules for amine/imine N atoms in small N-heterocycles

Vibronic coupling plays a crucial role in X-ray photoelectron spectra (XPS) of molecules. In a series of three papers, we present a comprehensive exploration of the N-heterocycles family, known for their diverse structures, to summarize the general rules of vibronic coupling in high-resolution vibrationally-resolved XPS spectra at the N1s edge. Building upon our previous studies on six-membered monocyclic azines [Phys. Rev. A 106, 022811 (2022)] and fused bicyclic compounds indoles with five and six members [Phys. Rev. A 108, 022816 (2023)], in this study, we focus on investigating a series of 12 five-membered N-heterocycles using Franck-Condon simulations, incorporating Duschinsky rotation effects and density functional theory. Our calculations reveal distinct spectral characteristics of amine and imine within these 12 systems in binding energies, spectral characteristics, structural changes, vibrational coupling strengths, and effects of hydrogenation. Furthermore, we expand our analysis to encompass all 35 N-heterocycles discussed in the three papers and consolidate these findings into the general rules. we find that 1s ionization in amine nitrogen induces more substantial geometrical changes, resulting in larger vibronic coupling strength compared to imine nitrogens. The spectra of imine nitrogens exhibit two distinct characteristic peaks originating from the 0-0 and 0-1 transitions, whereas the spectra of amine nitrogens are characterized by a broad peak with numerous weak fingerprints due to significant mixing of various 0-$n$ transitions. We observe that amine (imine) nitrogens generally cause a negative (positive) change in zero-point vibrational energy. This study provides valuable insights into vibronic coupling in N-heterocycles, shedding light on the distinguishing features and behavior of amine and imine nitrogens in vibrationally-resolved XPS spectra.Comment: 9 figure

Temperature and Tautomeric Effects in High-Resolution Oxygen 1s X-ray Photoelectron Spectroscopy of Purines and Pyrimidines

Purines and pyrimidines, crucial building blocks in biological systems, have attracted significant interest across molecular physics, biochemistry, pharmacology, and chemistry. Extensive spectroscopies have been employed for characterization, while the temperature and potential tautomeric effects can complicate the interpretation of underlying physics and chemistry. Here, we conducted first-principles simulations to analyze the vibrationally-resolved O1s X-ray photoelectron spectra of 6 common biomolecules at different temperatures, comprising 3 purine (xanthine, caffeine, and hypoxanthine) and 3 pyrimidine (thymine, 5F-uracil, and uracil) derivatives, and the tautomeric effect of hypoxanthine at varying temperatures. Using both time-independent (TI) and time-dependent (TD) methods under the Franck-Condon approximation, we obtained theoretical spectra that exhibited excellent agreement with experiments. Our analysis of these systems, all featuring carbonyl oxygens, unveiled distinctive characteristics of oxygen in N-CO-N (O2) compared to that within a N-CO-C structure (O1), showcasing higher O1s binding energy and total vibrational reorganization energy. We observed small differences in the zero-point vibration energies between the core-ionized and ground states, indicating a weak Duschinsky rotation effect. We consistently found that O1s ionization resulted in elongation of the O*=C bond length. The TI method facilitated the assignment of experimental spectra to different atoms or tautomers, where the atom-specific vibronic profiles of all 6 molecules exhibited similarity, with the 0-2 transitions dominating. TD enabled a more comprehensive exploration of the temperature effect, and the tautomeric effect of hypoxanthine by incorporating the Boltzmann population ratios of tautomers. We observed significant temperature dependence in the vibronic features present in these spectra.Comment: 6 figure

Vibrationally-resolved X-ray spectra of diatomic systems: Time-independent and time-dependent simulations

We systematically investigated vibronic coupling effects in X-ray spectra of diatomic systems using time-independent (TI) and time-dependent (TD) methods. Under the TI framework, we studied 5 systems (N$_2$, N$_2^+$, NO$^+$, CO, CO$^+$) in their lowest C/N/O 1s excited or ionized states, generating 10 X-ray absorption (XAS) or photoelectron (XPS) spectra using density functional theory (DFT) with two pure (BLYP, BP86) and two hybrid (B3LYP, M06-2X) functionals. Excellent agreement between theoretical and experimental spectra was found in most systems, except that in O1s XAS of CO and NO$^+$, intensities of higher-energy peaks were underestimated. We established a connection between their complex vibronic structures and the significant geometrical changes induced by the O1s hole. Functional dependence in diatomic systems is generally more pronounced than in polyatomic ones. In all examined cases, pure functionals exhibit better or similar spectral accuracy to hybrid functionals, attributed to superior prediction accuracy in bond lengths and vibrational frequencies. With the TD wavepacket method, we simulated vibrationally-resolved XAS of CO$^+$, NO$^+$, and CO using potential energy curves (PECs) generated at both DFT and multiconfigurational levels. Both TD and TI generate similar C/O 1s XAS spectra of CO$^+$. For O1s XAS of NO$^+$ and CO, TD calculations significantly improved the corresponding TI results, demonstrating sensitivity to the anharmonic effect and the PEC quality. TI and TD approaches are complementary, with practical applications depending on the ease and accuracy of excited-state geometry optimization or PEC scanning, and the significance of anharmonicity. DFT with pure functionals is recommended for diatomic calculations due to its easy execution and reliable accuracy. TI is optimal for most scenarios, but TD is needed for problems with strong anharmonic effects.Comment: 11 figure

On the Convergence of Deep Learning with Differential Privacy

In deep learning with differential privacy (DP), the neural network achieves the privacy usually at the cost of slower convergence (and thus lower performance) than its non-private counterpart. This work gives the first convergence analysis of the DP deep learning, through the lens of training dynamics and the neural tangent kernel (NTK). Our convergence theory successfully characterizes the effects of two key components in the DP training: the per-sample clipping (flat or layerwise) and the noise addition. Our analysis not only initiates a general principled framework to understand the DP deep learning with any network architecture and loss function, but also motivates a new clipping method -- the global clipping, that significantly improves the convergence while preserving the same privacy guarantee as the existing local clipping. In terms of theoretical results, we establish the precise connection between the per-sample clipping and NTK matrix. We show that in the gradient flow, i.e., with infinitesimal learning rate, the noise level of DP optimizers does not affect the convergence. We prove that DP gradient descent (GD) with global clipping guarantees the monotone convergence to zero loss, which can be violated by the existing DP-GD with local clipping. Notably, our analysis framework easily extends to other optimizers, e.g., DP-Adam. Empirically speaking, DP optimizers equipped with global clipping perform strongly on a wide range of classification and regression tasks. In particular, our global clipping is surprisingly effective at learning calibrated classifiers, in contrast to the existing DP classifiers which are oftentimes over-confident and unreliable. Implementation-wise, the new clipping can be realized by adding one line of code into the Opacus library