Coefficients Of Wave Functions Research Articles

Wave functions for pentadiagonal matrices in the weak coupling limit

We consider a pentadiagonal matrix which will be described in this paper. We demonstrate practical methods for obtaining weak coupling expressions for the lowest eigenvector in terms of the parameters in the matrix, v and w. It is found that the expressions simplify if the wave function coefficients are put in the denominator.

Neural network setups for a precise detection of the many-body localization transition: Finite-size scaling and limitations

Determining phase diagrams and phase transitions semi-automatically using machine learning has received a lot of attention recently, with results in good agreement with more conventional approaches in most cases. When it comes to more quantitative predictions, such as the identification of universality class or precise determination of critical points, the task is more challenging. As an exacting test-bed, we study the Heisenberg spin-1/2 chain in a random external field that is known to display a transition from a many-body localized to a thermalizing regime, which nature is not entirely characterized. We introduce different neural network structures and dataset setups to achieve a finite-size scaling analysis with the least possible physical bias (no assumed knowledge on the phase transition and directly inputing wave-function coefficients), using state-of-the-art input data simulating chains of sizes up to L=24. In particular, we use domain adversarial techniques to ensure that the network learns scale-invariant features. We find a variability of the output results with respect to network and training parameters, resulting in relatively large uncertainties on final estimates of critical point and correlation length exponent which tend to be larger than the values obtained from conventional approaches. We put the emphasis on interpretability throughout the paper and discuss what the network appears to learn for the various used architectures. Our findings show that a it quantitative analysis of phase transitions of unknown nature remains a difficult task with neural networks when using the minimally engineered physical input.

Rank reduced coupled cluster theory. II. Equation-of-motion coupled-cluster singles and doubles.

Equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) is a reliable and popular approach to the determination of electronic excitation energies. Recently, we have developed a rank-reduced CCSD (RR-CCSD) method that allows the ground-state coupled-cluster energy to be determined with low-rank cluster amplitudes. Here, we extend this approach to excited-state energies through a RR-EOM-CCSD method. We start from the EOM-CCSD energy functional and insert low-rank approximations to the doubles amplitudes. The result is an approximate EOM-CCSD method with only a quadratic number (in the molecular size) of free parameters in the wavefunction. Importantly, our formulation of RR-EOM-CCSD preserves the size intensivity of the excitation energy and size extensivity of the total energy. Numerical tests of the method suggest that accuracy on the order of 0.05-0.01 eV in the excitation energy is possible with 1% or less of the original number of wavefunction coefficients; accuracy of better than 0.01 eV can be achieved with about 4% or less of the free parameters. The amount of compression at a given accuracy level is expected to increase with the size of the molecule. The RR-EOM-CCSD method is a new path toward the efficient determination of accurate electronic excitation energies.

Branching corrected surface hopping: Resetting wavefunction coefficients based on judgement of wave packet reflection

We present a new interpretation of the decoherence correction in surface hopping by examining the inconsistency of the traditional time-dependent Schrödinger equation and propose an elegant decoherence correction algorithm to deal with wave packet branching. In contrast to the widely used approaches based on decoherence rates, our branching corrected surface hopping (BCSH) resets the wavefunction directly after wave packet branching is identified through prediction of trajectory reflection. The appealing simplicity and reliability of BCSH are demonstrated in a series of widely studied one-dimensional and two-dimensional scattering models using exact quantum solutions and existing surface hopping approaches as references. The BCSH approach exhibits a high performance in all investigated systems, showing good potential for applications in general nonadiabatic dynamics simulations.

Quantum Ising model in transverse and longitudinal fields: chaotic wave functions

The construction of a statistical model for eigenfunctions of the Ising model in transverse and longitudinal fields is discussed in detail for the chaotic case. When the number of spins is large, each wave function coefficient has the Gaussian distribution with zero mean and variance calculated from the first two moments of the Hamiltonian. The main part of the paper is devoted to the discussion of various corrections to the asymptotic result. One type of correction is related to higher order moments of the Hamiltonian, and can be taken into account by Gibbs-like formulae. Other corrections are due to symmetry contributions, which manifest as different numbers of non-zero real and complex coefficients. The statistical model with these corrections included agrees well with numerical calculations of wave function moments.

Contribution of the transition moments, form of the absorption band, exciton, and the correlation effects in the linear and nonlinear optical properties of conjugated polymers

This work compares the linear and nonlinear optical properties of polyenes and polyenynes. The simulation was made for finite and infinite conjugation of conjugated polymers, such as polyacetylene, β-carotene, bis (p-toluene sulfonate) (PTS) polyenyne, and a short conjugated polyenyne poly-2,6-decadyin-1,6-ylene azelate (PHDAz). The resonance energy and degree of conjugation are determined by fitting the linear absorption spectra. These parameters are then used for calculating the two photon and third-order nonlinear optical properties. The contribution of the transition moment, the form of the absorption band, the exciton, and phonons in the optical properties are determined. The difference of the experimental values is assigned to correlation effects. We found that the exciton, the correlation effects, and the conduction band are more important in the optical properties of polyenynes than of polyenes. In this way, the dependence of the optical properties of polyenynes on the conduction band makes it possible to increase their nonlinear optical properties by interaction with photons (θ ≈ 0). The dependence of the optical properties on the conduction band also produces that the finiteness of the conjugation strongly decreases the optical properties of polyenynes in relation to polyenes with similar conjugation. On the other hand, the phonons are more important in the optical properties of polyenes than of polyenynes. Consequently, the band is indirect for the studied polyenes and direct for the polyenynes. Therefore, the nonlinear optical properties in the resonance frequency of polyenyne PTS are higher than those for polyacetylene. We also found that asymmetric oscillations of χ(3) in the Brillouin zone increases the χ(3) final value in polyenynes. In addition, we found a change of sign of the wave function coefficients by the Peierls distortion of polyenes to become polyenynes, but this change of sign does not affect the optical properties. As a corollary, relaxation times for these conjugated polymers are estimated and compared to experimental data.

Towards Gauging Time-Reversal Symmetry: A Tensor Network Approach

It is well known that unitary symmetries can be `gauged', i.e. defined to act in a local way, which leads to a corresponding gauge field. Gauging, for example, the charge conservation symmetry leads to electromagnetic gauge fields. It is an open question whether an analogous process is possible for time reversal which is an anti-unitary symmetry. Here we discuss a route to gauging time reversal symmetry which applies to gapped quantum ground states that admit a tensor network representation. The tensor network representation of quantum states provides a notion of locality for the wave function coefficient and hence a notion of locality for the action of complex conjugation in anti-unitary symmetries. Based on that, we show how time reversal can be applied locally and also describe time reversal symmetry twists which act as gauge fluxes through nontrivial loops in the system. As with unitary symmetries, gauging time reversal provides useful access to the physical properties of the system. We show how topological invariants of certain time reversal symmetric topological phases in $D=1,2$ are readily extracted using these ideas.

A coefficient average approximation towards Gutzwiller wavefunction formalism

Gutzwiller wavefunction is a physically well-motivated trial wavefunction for describing correlated electron systems. In this work, a new approximation is introduced to facilitate the evaluation of the expectation value of any operator within the Gutzwiller wavefunction formalism. The basic idea is to make use of a specially designed average over Gutzwiller wavefunction coefficients expanded in the many-body Fock space to approximate the ratio of expectation values between a Gutzwiller wavefunction and its underlying noninteracting wavefunction. To check with the standard Gutzwiller approximation (GA), we test its performance on single band systems and find quite interesting properties. On finite systems, we noticed that it gives superior performance over GA, while on infinite systems it asymptotically approaches GA. Analytic analysis together with numerical tests are provided to support this claimed asymptotical behavior. Finally, possible improvements on the approximation and its generalization towards multiband systems are illustrated and discussed.

An economic prediction of the finer resolution level wavelet coefficients in electronic structure calculations.

In wavelet based electronic structure calculations, introducing a new, finer resolution level is usually an expensive task, this is why often a two-level approximation is used with very fine starting resolution level. This process results in large matrices to calculate with and a large number of coefficients to be stored. In our previous work we have developed an adaptively refined solution scheme that determines the indices, where the refined basis functions are to be included, and later a method for predicting the next, finer resolution coefficients in a very economic way. In the present contribution, we would like to determine whether the method can be applied for predicting not only the first, but also the other, higher resolution level coefficients. Also the energy expectation values of the predicted wave functions are studied, as well as the scaling behaviour of the coefficients in the fine resolution limit.

Dynamic Stress Concentration of a Circular Cavity Subjected by Steady Plane SH Waves in an Elastic Quarter with a Semi-Circular Canyon

Steady state responses of a circular cavity and a semi-circular canyon subjected by plane SH wave in an elastic quarter are presented by using Fourier-Hankel wave function expansion method and image method with Fourier series expansion on the boundary conditions to determine linear algebraic equations of unknown wave function coefficients. Especially, displacement and stress component expressions are formulated for incident, reflected, scattering waves, respectively. This method can provide an analytical ideas and methods for further studies of elastodynamic interface problems.

Ergodicity breaking in a model showing many-body localization

We study the breaking of ergodicity measured in terms of return probability in the evolution of a quantum state of a spin chain. In the non-ergodic phase a quantum state evolves in a much smaller fraction of the Hilbert space than would be allowed by the conservation of extensive observables. By the anomalous scaling of the participation ratios with system size we are led to consider the distribution of the wave function coefficients, a standard observable in modern studies of Anderson localization. We finally present a criterion for the identification of the ergodicity-breaking (many-body localization) transition based on these distributions which is quite robust and well suited for numerical investigations of a broad class of problems.

Exact solutions of the Manning–Rosen potential plus a ring-shaped like potential for the Dirac equation: spin and pseudospin symmetry

The analytical bound state solutions of the Dirac equation for the Manning–Rosen potential plus a ring-shaped like potential in spin and pseudospin symmetries are investigated by a proper approximation to the centrifugal term. Using the Nikiforov–Uvarov method, an explicit form of the energy eigenvalue and the corresponding energy eigenstates, expressed in terms of the Jacobi polynomials, are derived. The normalization coefficients of wave functions for spin and pseudospin symmetry solutions are found explicitly.

Communication: Tensor hypercontraction. III. Least-squares tensor hypercontraction for the determination of correlated wavefunctions

The manipulation of the rank-four tensor of double excitation amplitudes represents a challenge to the efficient implementation of many electronic structure methods. We present a proof of concept for the approximation of doubles amplitudes in the tensor hypercontraction (THC) representation. In particular, we show how THC can be used to both reduce the scaling with respect to molecular size of coupled cluster singles and doubles (CCSD) (and related methods) by two orders [from O(N(6)) to O(N(4))] and remove the memory bottleneck associated with storage of the doubles amplitudes. The accuracy of correlated methods as integral and amplitude approximations are introduced is examined. For a set of 20 small molecules, single and double-excitation configuration interaction (CISD), quadratic CISD (QCISD), and CCSD correlation energies could be reproduced with millihartree accuracy after the introduction of these approximations. Our approach exploits otherwise hidden factorizable tensor structure in both the electron repulsion integrals and the wavefunction coefficients and should be applicable with suitable modifications to many methods in electronic structure theory.

Studying the general properties of potentials by means of dimensionless scaling variables

The general properties of the three-parametric Woods-Saxon potential and the two-parametric Hulthen and square-well potentials are studied by converting them to dimensionless scaling variables. The relations between the parameters of the potentials and such characteristics of a system as binding energies and the asymptotic normalization coefficients of wave functions are investigated.

Wavefunction and Fourier coefficients of excitons in quantum wells: computation and application

Excitonic dynamic equations, which are derived from the quasi-Boson approach, are useful tools in investigating the ultrafast optical responses of semiconductor nanostructures. To apply these equations to the exciton dynamics in semiconductor quantum wells, we need exciton wavefunctions and their representations in momentum space to obtain the coefficients in the excitonic dynamic equations. By discussing in detail the exciton wavefunctions and their momentum-space representations, we present a method of obtaining the essential coefficients in the excitonic dynamic equations. We finally use these coefficients to understand the nonlinear effects in the terahertz-pulse-induced intraexcitonic transitions caused by high exciton densities. The obtained theoretical results are in good agreement with recent experimental results.

High-Throughput Computational Screening of Chromophores for Dye-Sensitized Solar Cells

Electron injection from a photoexcited chromophore into semiconductor (TiO(2)) nanoparticles is one of the key electron transfer processes in dye-sensitized solar cells. We describe our model for calculations of electron injection times, which is based on partitioning the semiconductor chromophore system into fragments (TiO(2) slab with adsorbed chromophore's anchoring group, and an isolated chromophore), and calculating the imaginary part of the self-energy from the electronic properties of the fragments: density of states of TiO(2) slab, TiO(2)-anchoring group coupling, and chromophore's wave function coefficients. The electronic properties of the semiconductor and its interface with the chromophore's anchoring group are reused for all chromophores with the same anchoring group (carboxylic acid in this study), and only a calculation of the isolated chromophore's lowest unoccupied molecular orbital is required for each chromophore. We use this model to calculate electron injection times for a large set of organic chromophores, including, e.g., perylene dyes and biisonicotinic acid, on TiO(2) rutile (110) and anatase (101) surfaces. The calculated injection times are in good agreement with reported experimental injection times or light conversion efficiencies of solar cells based on these dyes. Our model is computationally efficient and allows us to make reliable predictions of electron injection times for families of chromophores sharing the same adsorption chemistry.

Quasiclassical theory of the Dirac equation with a scalar-vector interaction and its applications in the physics of heavy-light mesons

We construct a relativistic potential quark model of $D$, ${D}_{s}$, $B$, and ${B}_{s}$ mesons in which the light quark motion is described by the Dirac equation with a scalar-vector interaction and the heavy quark is considered a local source of the gluon field. The effective interquark interaction is described by a combination of the perturbative one-gluon exchange potential ${V}_{\mathrm{Coul}}(r)=\ensuremath{-}\ensuremath{\xi}/r$ and the long-range Lorentz-scalar and Lorentz-vector linear potentials ${S}_{\mathrm{l}.\mathrm{r}.}(r)=(1\ensuremath{-}\ensuremath{\lambda})(\ensuremath{\sigma}r+{V}_{0})$ and ${V}_{\mathrm{l}.\mathrm{r}.}(r)=\ensuremath{\lambda}(\ensuremath{\sigma}r+{V}_{0})$, where $0\ensuremath{\le}\ensuremath{\lambda}<1/2$. Within the quasiclassical approximation, we obtain simple asymptotic formulas for the energy and mass spectra and for the mean radii of $D$, ${D}_{s}$, $B$, and ${B}_{s}$ mesons, which ensure a high accuracy of calculations even for states with the radial quantum number ${n}_{r}\ensuremath{\sim}1$. We show that the fine structure of $P$-wave states in heavy-light mesons is primarily sensitive to the choice of two parameters: the strong-coupling constant ${\ensuremath{\alpha}}_{s}$ and the coefficient $\ensuremath{\lambda}$ of mixing of the long-range scalar and vector potentials ${S}_{\mathrm{l}.\mathrm{r}.}(r)$ and ${V}_{\mathrm{l}.\mathrm{r}.}(r)$. The quasiclassical formulas for asymptotic coefficients of wave functions at zero and infinity are obtained.

Method to map one-dimensional electronic wave function by using multiple Brillouin zone angle resolved photoemission

Angle resolved photoemission spectroscopy (ARPES) is a powerful tool to investigate electronic structures of solids and has been widely used in studying various materials. Electronic structure information by ARPES is obtained in the momentum space. However, in the case of one-dimensional systems, we show that real space information can be extracted from ARPES data taken over multiple Brillouin zones (BZs). Intensities in multiple BZs are proportional to the photoemission matrix element which contains information on the coefficient of Bloch wave functions. It is shown that the Bloch wave function coefficients can be indeed extracted from ARPES data, which allows us to construct the real space wave function. As a test, we use ARPES data from proto-typical onedimensional system SrCuO2 and construct the real space wave function.

Atomic nature of the Si-B4 paramagnetic center assessed by multifrequency electron spin resonance

An electron spin resonance (ESR) study has been carried out in the range 4.2--120 K on the trigonal Si-B4 center in annealed $(200--350\text{ }\ifmmode^\circ\else\textdegree\fi{}\text{C})$ neutron-irradiated $p$-type Si. Detailed observations reveal a rich $^{29}\text{S}\text{i}$ hyperfine (hf) structure, indicating interaction of the unpaired electron with up to five shells of equivalent Si sites. Optimized consistent computer-assisted fitting of spectra observed at three ESR frequencies point toward a set of 1-2-2-3-3 equivalent Si sites for the five observed shells in decreasing hf splitting, respectively, with the 1-3-3-3-3 set, conforming with the trigonal symmetry of the defect, giving satisfactory results as well. The strongest interaction of the unpaired electron is with one Si atom. Full angular mapping of all five hf doublets enabled inference of the hybrid coefficients, i.e., localization of the unpaired electron and $s\text{\ensuremath{-}}p$ ratio over the different Si sites. Stepping from previous theoretical work, two tri-interstitial models are discussed from where, although lack of theoretical results on the molecular wave-function coefficients prevents definite assignment, the tri-interstitial ${\text{I}}_{3}\text{-I}$ model is advanced as a plausible candidate. An uncommon drastic temperature dependence is observed in which all resolved $^{29}\text{S}\text{i}$ hf doublet splittings collectively and evenly narrow with increasing temperature toward 1/4 of the low-temperature values along a bounded exponential (Boltzmann factor) decay, with activation energy ${E}_{\text{a}}\ensuremath{\approx}0.0041\text{ }\text{eV}$. There is an attendant upward shift in ${g}_{\ensuremath{\parallel}}$. An effect due to thermal dilatation as possible origin is excluded. Instead, it is ascribed to even redistribution (delocalization) of the low-temperature unpaired hybrid by 75% over next-neighboring sites, possibly within a four-defect cluster arrangement.

Semiempirical study of perturbations of the landé g factors of electronic-vibrational-rotational levels of hydrogen: IV. The I 1Π−g, R 1Π−g, J 1Δ−g, and S 1Δ−g states of the H2 and D2 molecules

Values of the Lande g factors are obtained for the first time for rovibronic levels of the (3d)J1Δg− state of the D2 molecule and of the (4d)S1Δg− state of the H2 and D2 molecules. Semiempirical values of these quantities are obtained for the first time for the (4d)R1Πg− state of the H2 molecule and the (3d)I1Πg− and (4d)R1Πg− states of the D2 molecule. These values are determined in terms of the nonadiabatic model, which takes into account the interaction between the ndπ1Πg and ndδ1Δg (n = 3 and 4) states in the approximation of pure precession, with the use of the semiempirical values obtained by us for the expansion coefficients of wave functions in the adiabatic basis functions and the results of our numerical calculation of overlap integrals of the vibrational wave functions of these states. The values thus obtained are in good agreement with the data available in the literature. It is found that the interference effects of coupling of the ndπ1Πg and ndδ1Δg states lead to significant perturbations of the dependences of the g factors of vibronic levels on the rotational quantum number as compared to the adiabatic results. These perturbations are regular (reaching a factor of nine) for the I1Πg− and R1Πg− states and irregular (reaching a factor of 50) for the J1Δg− and S1Δg− states. The nonadiabatic values of g factors obtained in our previous studies for the H2, HD, and D2 molecules in the i3Πg− and j3Δg− states are improved by an additional inclusion of adiabatic effects. It is established that the effect of the electron spin on the g factor values for rovibronic levels of the 3dΔg− states is significant and reaches 25% for H2 and 130% for D2.