arXiv subjects

Wenjian Liu

Publications and source records attributed to Wenjian Liu.

46 records · Page 3Linked to original sources

arXiv2020

Perspective: Essentials of Relativistic Quantum Chemistry

Relativistic quantum chemistry has evolved into a fertile and large field and is now becoming an integrated part of mainstream chemistry. Yet, given the much-involved physics and mathematics (as compared with nonrelativistic quantum chemistry), it is still necessary to clean up the essentials underlying the relativistic electronic structure theories and methodologies (such that uninitiated readers can pick up quickly the right ideas and tools for further development or application) and meanwhile pinpoint future directions of the field. To this end, the three aspects of electronic structure calculations, i.e., relativity, correlation, and QED, will be highlighted.

physics.chem-ph↗

arXiv2020

Iterative Configuration Interaction with Selection

Even when starting with a very poor initial guess, the iterative configuration interaction (iCI) approach can converge from above to full CI very quickly by constructing and diagonalizing a small Hamiltonian matrix at each macro/micro-iteration. However, iCI scales exponentially with respect to the numbers of electrons and orbitals. The problem can be mitigated by observing that a vast number of configurations have little weights in the wave function and do not contribute to the correlation energy. The real questions are then (a) how to identify important configurations in the early stage of the calculation and (b) how to account for the residual contributions of those unimportant configurations. It is generally true that if a high-quality yet compact variational space can be determined for describing the static correlation, a low-order treatment of the residual dynamic correlation would be sufficient. While this is common to all selected CI schemes, the `iCI with selection' scheme presented here has the following distinctive features: (1) Full spin symmetry is maintained. (2) Although the selection is performed on individual CSFs, it is orbital configurations (oCFG) that are used as the organizing units. (3) Given a coefficient pruning-threshold, the selection of important oCFGs/CSFs is performed iteratively until convergence. (4) At each iteration for the growth of wave function, the first-order interacting space is decomposed into disjoint subspaces to reduce memory requirement and facilitate parallelization. (5) Upper bounds for the interactions between oCFG pairs are used to screen each first-order interacting subspace before the first-order coefficients of individual CSFs are evaluated. (6) Upon termination of the selection, dynamic correlation is estimated by using the ENPT2 theory. The efficacy of the iCIPT2 scheme is demonstrated numerically by taking several examples.

physics.chem-ph↗

arXiv2019

The non-tightness of the reconstruction threshold of a 4 states symmetric model with different in-block and out-block mutations

The tree reconstruction problem is to collect and analyze massive data at the $n$th level of the tree, to identify whether there is non-vanishing information of the root, as $n$ goes to infinity. Its connection to the clustering problem in the setting of the stochastic block model, which has wide applications in machine learning and data mining, has been well established. For the stochastic block model, an "information-theoretically-solvable-but-computationally-hard" region, or say "hybrid-hard phase", appears whenever the reconstruction bound is not tight of the corresponding reconstruction on the tree problem. Although it has been studied in numerous contexts, the existing literature with rigorous reconstruction thresholds established are very limited, and it becomes extremely challenging when the model under investigation has $4$ states (the stochastic block model with $4$ communities). In this paper, inspired by the newly proposed $q_1+q_2$ stochastic block model, we study a $4$ states symmetric model with different in-block and out-block transition probabilities, and rigorously give the conditions for the non-tightness of the reconstruction threshold.

stat.ML↗

arXiv2018

Big Data Information Reconstruction on an Infinite Tree for a $4\times 4$-state Asymmetric Model with Community Effects

The information reconstruction problem on an infinite tree, is to collect and analyze massive data samples at the $n$th level of the tree to identify whether there is non-vanishing information of the root, as $n$ goes to infinity. This problem has wide applications in various fields such as biology, information theory and statistical physics, and its close connections to cluster learning, data mining and deep learning have been well established in recent years. Although it has been studied in numerous contexts, the existing literatures with rigorous reconstruction thresholds established are very limited. In this paper, motivated by a classical deoxyribonucleic acid (DNA) evolution model, the F$81$ model, and taking into consideration of the Chargaff's parity rule by allowing the existence of a guanine-cytosine content bias, we study the noise channel in terms of a $4\times 4$-state asymmetric probability transition matrix with community effects, for four nucleobases of DNA. The corresponding information reconstruction problem in molecular phylogenetics is explored, by means of refined analyses of moment recursion, in-depth concentration estimates, and thorough investigations on an asymptotic $4$-dimensional nonlinear second order dynamical system. We rigorously show that the reconstruction bound is not tight when the sum of the base frequencies of adenine and thymine falls in the interval $\left(0,1/2-\sqrt{3}/6\right)\bigcup \left(1/2+\sqrt{3}/6,1\right)$, which is the first rigorous result on asymmetric noisy channels with community effects.

math.PR↗

arXiv2018

Classifications and canonical forms of tensor product expressions in the presence of permutation symmetries

Complicated mathematical equations involving products of tensors with permutation symmetries, frequently encountered in fields such as general relativity and quantum chemistry (e.g., equations in high-order coupled cluster theories), require computer-based automatic derivations and manipulations. In these processes, a key step is the collection of tensor product terms that can be found identical by utilizing permutation symmetries of tensors or relabeling dummy indices, which is usually achieved by defining a canonical form for tensor product expressions. However, the problem of finding a canonical form is nontrivial, and can be potentially of exponential cost in the number of indices. In this work, we provided a general solution to this tensor canonicalization problem.

physics.chem-ph↗

arXiv2018

Large Degree Asymptotics and the Reconstruction Threshold of the Asymmetric Binary Channels

In this paper, we consider a broadcasting process in which information is propagated from a given root node on a noisy tree network, and answer the question that whether the symbols at the nth level of the tree contain non-vanishing information of the root as n goes to infinity. Although the reconstruction problem on the tree has been studied in numerous contexts including information theory, mathematical genetics and statistical physics, the existing literatures with rigorous reconstruction thresholds established are very limited. In the remarkable work of Borgs, Chayes, Mossel and Roch (The Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels), the exact threshold for the reconstruction problem for a binary asymmetric channel on the d-ary tree is establish, provided that the asymmetry is sufficiently small, which is the first exact reconstruction threshold obtained in roughly a decade. In this paper, by means of refined analyses of moment recursion on a weighted version of the magnetization, and concentration investigations, we rigorously give a complete answer to the question of how small it needs to be to establish the tightness of the reconstruction threshold and further determine its asymptotics of large degrees.

math.PR↗

arXiv2018

Evolution of Regional Innovation with Spatial Knowledge Spillovers: Convergence or Divergence?

This paper extends endogenous economic growth models to incorporate knowledge externality. We explores whether spatial knowledge spillovers among regions exist, whether spatial knowledge spillovers promote regional innovative activities, and whether external knowledge spillovers affect the evolution of regional innovations in the long run. We empirically verify the theoretical results through applying spatial statistics and econometric model in the analysis of panel data of 31 regions in China. An accurate estimate of the range of knowledge spillovers is achieved and the convergence of regional knowledge growth rate is found, with clear evidences that developing regions benefit more from external knowledge spillovers than developed regions.

econ.EM↗

arXiv2017

Reconstruction for the Asymmetric Ising Model on Regular Trees

It is known that the Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels. In this paper, we will adopt a refined analysis of moment recursion on a weighted version of the magnetization, which is engaged in Sly [2011] to handle the symmetric Potts model, and establish the critical condition of the asymmetric Ising model to make Kesten-Stigum bound the reconstruction threshold on regular $d$-ary trees.

math.PR↗

arXiv2017

The Tightness of the Kesten-Stigum Reconstruction Bound of Symmetric Model with Multiple Mutations

It is well known that reconstruction problems, as the interdisciplinary subject, have been studied in numerous contexts including statistical physics, information theory and computational biology, to name a few. We consider a $2q$-state symmetric model, with two categories of $q$ states in each category, and 3 transition probabilities: the probability to remain in the same state, the probability to change states but remain in the same category, and the probability to change categories. We construct a nonlinear second order dynamical system based on this model and show that the Kesten-Stigum reconstruction bound is not tight when $q \geq 4$.

math.PR↗

arXiv1997

Orbital Localization and Delocalization Effects in the U 5f^2 Configuration: Impurity Problem

Anderson models, based on quantum chemical studies of the molecule of U(C_8H_8)_2, are applied to investigate the problem of an U impurity in a metal. The special point here is that the U 5f-orbitals are divided into two subsets: an almost completely localized set and a considerably delocalized one. Due to the crystal field, both localized and delocalized U 5f-orbitals affect the low-energy physics. A numerical renormalization group study shows that every fixed point is characterized by a residual local spin and a phase shift. The latter changes between 0 and π/2, which indicates the competition between two different fixed points. Such a competition between the different local spins at the fixed points reflects itself in the impurity magnetic susceptibility at high temperatures. These different features cannot be obtained if the special characters of U 5f-orbitals are neglected.

cond-mat.str-el↗