The coupled-cluster formalism – a mathematical perspective

ABSTRACT

The Coupled-Cluster (CC) theory is one of the most successful high precision methods used to solve the stationary Schrödinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions – Gårding type inequalities – for the local uniqueness of the CC solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. One of the criteria assumes the relative smallness of the total cluster amplitudes (after possibly removing the single amplitudes) compared to the Gårding constants. In the extended CC theory the Lagrange multipliers are wave function parameters and, by means of the bivariational principle, we here derive a connection between the exact cluster amplitudes and the Lagrange multipliers. This relation might prove useful when determining the quality of a CC solution. Furthermore, the use of an Aubin–Nitsche duality type method in different CC approaches is discussed and contrasted with the bivariational principle.     

Disentangling stochastic signals superposed on short localized oscillations

We introduce a procedure for separating periodic oscillations superposed on a stochastic signal. The procedure combines empirical mode decomposition (EMD) of a signal with tools of data analysis based on stochastic differential equations, namely nonlinear Langevin equations. Taking the set of modes retrieved from the EMD of the signal, our procedure is able to separate them into two groups, one composing the periodic signal and another composing the stochastic signal. The framework is robust for a broad family of localized oscillations, in the range of large frequencies. In particular, we show that, in this context, the EMD method outperforms a low-pass filter and is robust for a wide interval of different frequency ranges and amplitudes of the periodic oscillation, as well as for a broad family of different non-linear Langevin processes.     

Deposition pattern of drying droplets

The drying of liquid droplets is a common daily life phenomenon that has long held a special interest in scientific research. When the droplet includes nonvolatile solutes, the evaporation of the solvent induces rich deposition patterns of solutes on the substrate. Understanding the formation mechanism of these patterns has important ramifications for technical applications,ranging from coating to inkjet printing to disease detection. This topical review addresses the development of physical understanding of tailoring the specific ring-like deposition patterns of drying droplets. We start with a brief introduction of the experimental techniques that are developed to control these patterns of sessile droplets. We then summarize the development of the corresponding theory. Particular attention herein is focused on advances and issues related to applying the Onsager variational principle (OVP) theory to the study of the deposition patterns of drying droplets. The main obstacle to conventional theory is the requirement of complex numerical solutions, but fortunately there has been recent groundbreaking progress due to the OVP theory. The advantage of the OVP theory is that it can be used as an approximation tool to reduce the high-order conventional hydrodynamic equations to first-order evolution equations,facilitating the analysis of soft matter dynamic problems. As such, OVP theory is now well poised to become a theory of choice for predicting deposition patterns of drying droplets.     

SnSxSe2-x单晶纳米片的可控制备与表征

锡二硫族化合物可以通过改变硫和硒的含量来连续调控三元合金材料的带隙、载流子浓度等物理化学性质,在电子和光电子器件应用上具有巨大的潜力。本文采用化学气相沉积(CVD)技术可控地制备了不同元素组分的SnS_xSe_2-x(x=0,0.2,0.5,0.8,1.0,1.2,1.5,1.8,2.0)单晶纳米片。采用扫描电子显微镜(SEM)、原子力显微镜(AFM)、能量色散X射线光谱(EDS)、透射电子显微镜(TEM)以及拉曼光谱等手段对SnS_xSe_2-x纳米片进行了综合表征。结果表明本方法成功实现了元素百分比可调的SnS_xSe_2-x单晶纳米片的可控制备。重点研究了依赖于元素百分比的SnS_xSe_2-x的拉曼特征谱,实验结果与基于密度泛函理论(DFT)的第一性原理计算得到的SnS_xSe_2-x的拉曼仿真谱高度吻合,理论计算结果较好地诠释了实验拉曼光谱发生变化的原因。本研究提供了一种元素百分比可调的三元SnS_xSe_2-x单晶纳米片的可控制备方法,同时对锡二硫族化合物的明确、无损识别提供了方案。