Multireference electron correlation methods with density matrix renormalisation group reference functions

Recent advances in quantum chemical density matrix renormalisation group (DMRG) theory are presented. The DMRG, originally devised as an alternative to the exact diagonalisation in condensed matter physics, has become a powerful quantum chemical method for molecular systems that exhibit a multireference character, e.g., excited states, π-conjugated systems, transition metal complexes, and in particular for large systems by combining it with conventional multireference electron correlation methods. The capability of the current quantum chemical DMRG is demonstrated for an application involving the potential energy curve of the chromium dimer, which is one of the most demanding multireference systems and thus requires the best electronic structure treatment for non-dynamical and dynamical correlation as well as large basis sets.     

Two-dimensional quantum-spin-1/2 XXZ magnet in zero magnetic field: Global thermodynamics from renormalisation group theory

Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.     

Global correlation matrix spectra of the surface temperature of the oceans from random matrix theory to Poisson fluctuations

In this work we use the random matrix theory (RMT) to correctly describe the behavior of spectral statistical properties of the sea surface temperature of oceans. This oceanographic variable plays an important role in the global climate system. The data were obtained from National Oceanic and Atmospheric Administration (NOAA) and delimited for the period 1982 to 2016. The results show that oceanographic systems presented specific β values that can be used to classify each ocean according to its correlation behavior. The nearest-neighbors spacing of correlation matrix for north, central and south of the three oceans get close to a RMT distribution. However, the regions delimited in the Antarctic pole exhibited the distribution of the nearest-neighbors spacing well described by the Poisson model, which shows a statistical change of RMT to Poisson fluctuations.     

Sum Rule in a Consistent Relativistic Random-Phase Approximation

A fully consistent relativistic random-phase approximation (RRPA) is studied in the sense that the relativistic mean-field (RMF) wavefunction of nucleus and the particlehole residual interactions in the RRPA are calculated from the same effective Lagrangian. A consistent treatment of RRPA within the RMF approximation, i.e., no sea approximation, has to include also the pairs formed from the Dirac states and occupied Fermi states, which is essential for satisfying the current conservation. The inverse energy-weighted sum rule for the isoscalar giant monopole mode is investigated in the constrained RMF. It is found that the sum rule is fulfilled only in the case where the Dirac state contributions are included.     

Random matrix theory filters in portfolio optimisation: A stability and risk assessment

Random matrix theory (RMT) filters, applied to covariance matrices of financial returns, have recently been shown to offer improvements to the optimisation of stock portfolios. This paper studies the effect of three RMT filters on the realised portfolio risk, and on the stability of the filtered covariance matrix, using bootstrap analysis and out-of-sample testing.We propose an extension to an existing RMT filter, (based on Krzanowski stability), which is observed to reduce risk and increase stability, when compared to other RMT filters tested. We also study a scheme for filtering the covariance matrix directly, as opposed to the standard method of filtering correlation, where the latter is found to lower the realised risk, on average, by up to 6.7%.We consider both equally and exponentially weighted covariance matrices in our analysis, and observe that the overall best method out-of-sample was that of the exponentially weighted covariance, with our Krzanowski stability-based filter applied to the correlation matrix. We also find that the optimal out-of-sample decay factors, for both filtered and unfiltered forecasts, were higher than those suggested by Riskmetrics [J.P. Morgan, Reuters, Riskmetrics technical document, Technical Report, 1996. http://www.riskmetrics.com/techdoc.html], with those for the latter approaching a value of α=1.In conclusion, RMT filtering reduced the realised risk, on average, and in the majority of cases when tested out-of-sample, but increased the realised risk on a marked number of individual days–in some cases more than doubling it.     

Random matrix theory analysis of cross-correlations in the US stock market: Evidence from Pearson’s correlation coefficient and detrended cross-correlation coefficient

In this study, we first build two empirical cross-correlation matrices in the US stock market by two different methods, namely the Pearson’s correlation coefficient and the detrended cross-correlation coefficient (DCCA coefficient). Then, combining the two matrices with the method of random matrix theory (RMT), we mainly investigate the statistical properties of cross-correlations in the US stock market. We choose the daily closing prices of 462 constituent stocks of S&P 500 index as the research objects and select the sample data from January 3, 2005 to August 31, 2012. In the empirical analysis, we examine the statistical properties of cross-correlation coefficients, the distribution of eigenvalues, the distribution of eigenvector components, and the inverse participation ratio. From the two methods, we find some new results of the cross-correlations in the US stock market in our study, which are different from the conclusions reached by previous studies. The empirical cross-correlation matrices constructed by the DCCA coefficient show several interesting properties at different time scales in the US stock market, which are useful to the risk management and optimal portfolio selection, especially to the diversity of the asset portfolio. It will be an interesting and meaningful work to find the theoretical eigenvalue distribution of a completely random matrix R for the DCCA coefficient because it does not obey the Mar?enko–Pastur distribution.     

石墨烯能带中的重叠矩阵效应:Tight-Binding方法在模拟中的研究

采用紧束缚方法计算了石墨烯的价带(π)和导带(π*),考虑了非正交基矢下重叠矩阵效应,重叠积分参量s越小,导带越靠近费米面,而价带越远离费米面.在重叠积分参量s≤0.1时,基本保持了原子在实际空间中重叠所引起的能带的改变,太大(s=0.4)则会导致物理上失效.计算了石墨烯的能态密度,在费米面ε=0处(对应Dirac点)的能态密度为零,并且在Dirac点附近呈线性变化.     

Molecular-crystalline approach to evaluation of correlation corrections in the theory of chemical bonding in crystals: Electronic structure of Ti2O3 crystals

The problem of the partial inclusion of electron correlation effects has been considered in the framework of the unrestricted Hartree-Fock method. The calculation of the electronic structure of the [Ti₂O₉]^12? cluster is performed. The results obtained demonstrate that, in some cases, a major part of static correlation effects can be taken into account in the unrestricted Hartree-Fock approximation. The influence of these effects on the local characteristics of crystals is analyzed.     

On the statistical mechanics approach in the random matrix theory: Integrated density of states

We consider the ensemble of random symmetricn×n matrices specified by an orthogonal invariant probability distribution. We treat this distribution as a Gibbs measure of a mean-field-type model. This allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self-consistent equation.     

Unit cell dependence of optical matrix elements in tight-binding theory: The case of zigzag graphene nanoribbons

In the tight-binding theory, momentum matrix elements (MMEs) needed to calculate the optical properties are normally computed using a formulation based on the gradient of the Hamiltonian in the k space. We demonstrate the inadequacy of this formulation by considering the case of zigzag graphene nanoribbons. We show that one obtains wrong values of MMEs, in violation of the well-known selection rules, if the unit cell chosen in the calculations does not incorporate the symmetries of the bulk. This is in spite of the fact that the band structure is insensitive to the choice of the unit cell. We substantiate our results based on group-theoretic arguments. Our observations will open an avenue for proper formulation of MMEs.     

Construction of a Langevin model from time series with a periodical correlation function: Application to wind speed data

A Langevin-type equation for stochastic processes with a periodical correlation function is introduced. A procedure of reconstruction of the equation from time series is proposed and verified on simulated data. The method is applied to geophysical time series–hourly time series of wind speed measured in northern Italy–constructing the macroscopic model of the phenomenon.     

The correlation function in Wilson's theory of phase transitions: Above and below the transition

The correlation function and the correlation length are discussed in the theoretical framework of the Wilson-Feynman diagram expansion for small =4–d. It is shown explicitly that to order ² the scaling relation = (2–) is satisfied and that the correlation function is a homogeneous function ofk and . The explicit form of the scaled correlation function is exhibited.     

Random matrix theory of singular values of rectangular complex matrices I: Exact formula of one-body distribution function in fixed-trace ensemble

The fixed-trace ensemble of random complex matrices is the fundamental model that excellently describes the entanglement in the quantum states realized in a coupled system by its strongly chaotic dynamical evolution [see H. Kubotani, S. Adachi, M. Toda, Phys. Rev. Lett. 100 (2008) 240501]. The fixed-trace ensemble fully takes into account the conservation of probability for quantum states. The present paper derives for the first time the exact analytical formula of the one-body distribution function of singular values of random complex matrices in the fixed-trace ensemble. The distribution function of singular values (i.e. Schmidt eigenvalues) of a quantum state is so important since it describes characteristics of the entanglement in the state. The derivation of the exact analytical formula utilizes two recent achievements in mathematics, which appeared in 1990s. The first is the Kaneko theory that extends the famous Selberg integral by inserting a hypergeometric type weight factor into the integrand to obtain an analytical formula for the extended integral. The second is the Petkovšek–Wilf–Zeilberger theory that calculates definite hypergeometric sums in a closed form.     

Mechanisms of interaction between luteolin and the catalytic zinc ion in matrix metalloproteinases: a computational study

Many experimental studies have found that flavonoids including luteolin can inhibit the activities of matrix metalloproteinases (MMPs), but the related theoretical studies are rather lacking. In this paper, we perform PM6 quantum chemistry calculations together with modeling of ligand‐water exchange reactions to investigate the mechanisms of interaction between luteolin and catalytic zinc ion in MMPs. The calculations show that the electron transfer from the luteolin molecule to the catalytic zinc ion in MMPs occurs when the catalytic zinc ion coordinates with the O atoms of substituent groups at various positions of A, B, and C rings of luteolin molecule. It is found that the more the number of the electron transfer from one coordinating O atom of substituent groups of luteolin molecule to the catalytic zinc ion, the stronger the coordinating ability between them. We further find that comparing with the O atoms of hydroxy groups at 5‐, 7‐, 3′‐, and 4′‐positions of luteolin molecule, the coordinating ability for the O atom of carbonyl group at its 4‐position with the catalytic zinc ion is the strongest, which indicates that when luteolin inhibits MMPs activity, the catalytic zinc ion should coordinate with the carbonyl group at 4‐position of luteolin molecule, rather than the hydroxy groups at its other positions, in agreement with the relevant experimental results reported in previous literature. This paper may be helpful for designing the new MMPs inhibitors having higher biological activities by carrying out the structural modifications of luteolin molecule. Copyright © 2012 John Wiley & Sons, Ltd.     

Parity effects in eigenvalue correlators, parametric and crossover correlators in random matrix models: Application to mesoscopic systems

This paper summarizes some work that I have been doing on eigenvalue correlators of random matrix models which show some interesting behavior. First we consider matrix models with gaps in their spectrum or density of eigenvalues. The density-density correlators of these models depend on whether N, where N is the size of the matrix, takes even or odd values. The fact that this dependence persists in the large N thermodynamic limit is an unusual property and may have consequences in the study of one electron effects in mesoscopic systems. Secondly, we study the parametric and cross correlators of the Harish Chandra-Itzykson-Zuber matrix model. The analytic expressions determine how the correlators change as a parameter (e.g. the strength of a perturbation in the Hamiltonian of the chaotic system or external magnetic field on a sample of material) is varied. The results are relevant for the conductance fluctuations in disordered mesoscopic systems.     

Comparative analysis of energy-level splitting of Pr3＋ doped in LiYF4 and LiBiF4 crystals： a complete energy matrix calculation

Based on the combination of Racah's group-theoretical consideration with Slater's wavefunction, a 91 × 91 complete energy matrix is established in tetragonal ligand field D_2d for Pr³⁺ ion. Thus, the Stark energy-levels of Pr³⁺ ions doped separately in LiYF₄ and LiBiF₄ crystals are calculated, and our calculations imply that the complete energy matrix method can be used as an effective tool to calculate the energy-levels of the systems doped by rare earth ions. Besides, the influence of Pr³⁺ on energy-level splitting is investigated, and the similarities and the differences between the two doped crystals are demonstrated in detail by comparing their several pairs of curves and crystal field strength quantities. We see that the energy splitting patterns are similar and the crystal field interaction of LiYF₄:Pr³⁺ is stronger than that of LiBiF₄:Pr³⁺.     

The scattering matrix for size distributions of irregular particles: An application to an olivine sample

We have compared simulated and measured scattering matrices of a size distribution of olivine particles at wavelengths of 633 and 442 nm. The computations were carried out for size distributions of irregularly-shaped compact particles with different average projected areas using the discrete-dipole approximation (DDA). The results of the comparison show that the model of irregularly-shaped particles mimic the observations far better than the results given by spheres, spheroids or rectangular prisms having a wide range in aspect ratios. The computed scattering matrices for size distributions of irregularly-shaped particles do not depend very strongly on the precise particle shape assumed, providing a method to infer certain physical properties of an ensemble of natural dust particles, such as the refractive index, when some information on the sample, as the size distribution, is known a priori.     

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to a polarized laser beam: spatially varying reflection matrix

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to a polarized, Gaussian laser beam directed perpendicular to the surface is studied. The focus of this investigation is the 4×4, spatially varying reflection matrix that can be used to determine the normally backscattered radiation when the polarization of the incident radiation is specified. An inverse integral transform is used to construct the spatially varying reflection matrix from the generalized reflection matrix found in a previous study. The elements of this matrix depend on location specified by optical radius and azimuthal angle. The azimuthal variation is found by performing part of the inverse transform analytically, while the radial variation is described by five functions that are calculated numerically via an inverse Hankel transform. Benchmark numerical results for these five functions are presented, and the effects of beam radius and particle concentration are discussed. Expressions that describe the behavior of the reflection functions at small and large optical radii are developed, and comparisons are made to the one-dimensional and scalar situations. The scalar approximation fails to predict the three-dimensional effects produced by the polarized beam, and even when the incident radiation is unpolarized, the error in the scalar reflection function can be as high as 20%.     

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to spatially varying polarized radiation: generalized reflection matrix

Three-dimensional vector radiative transfer in a semi-infinite medium exposed to spatially varying, polarized radiation is studied. The problem is to determine the generalized reflection matrix for a multiple scattering medium characterized by a 4×4 scattering matrix. A double integral transform is used to convert the three-dimensional vector radiative transfer equation to a one-dimensional form, and a modified Ambarzumian's method is then applied to derive a nonlinear integral equation for the generalized reflection matrix. The spatially varying backscattered radiation for an arbitrarily polarized incident beam can be found from the generalized reflection matrix. For Rayleigh scattering and normal incidence and emergence, the generalized reflection matrix is shown to have five non-zero elements. Benchmark results for these five elements are presented and compared to asymptotic results. When the incident radiation is polarized, the vector approach used in this study correctly predicts three-dimensional behavior, while the scalar approach does not. When the incident radiation is unpolarized, both the vector and scalar approaches predict a two-dimensional distribution of the intensity, but the error in the scalar prediction can be as high as 20%.