Assessment of dressed time-dependent density-functional theory for the low-lying valence states of 28 organic chromophores

Almost all time-dependent density-functional theory (TDDFT) calculations of excited states make use of the adiabatic approximation, which implies a frequency-independent exchange-correlation kernel that limits applications to one-hole/one-particle states. To remedy this problem, Maitra et al. [N.T. Maitra, F. Zhang, R.J. Cave, K. Burke, Double excitations within time-dependent density functional theory linear response theory, J. Chem. Phys. 120 (2004) 5932 ] proposed dressed TDDFT (D-TDDFT), which includes explicit two-hole/two-particle states by adding a frequency-dependent term to adiabatic TDDFT. This paper offers the first extensive test of D-TDDFT, and its ability to represent excitation energies in a general fashion. We present D-TDDFT excited states for 28 chromophores and compare them with the benchmark results of Schreiber et al. [M. Schreiber, M.R. Silva-Junior, S.P.A. Sauer, W. Thiel, Benchmarks for electronically excited states: CASPT2, CC2, CCSD, and CC3, J. Chem. Phys. 128 (2008) 134110]. We find the choice of functional used for the A-TDDFT step to be critical for positioning the 1h1p states with respect to the 2h2p states. We observe that D-TDDFT without HF exchange increases the error in excitations already underestimated by A-TDDFT. This problem is largely remedied by implementation of D-TDDFT including Hartree-Fock exchange.     

Application of kernel ridge regression in predicting neutron-capture reaction cross-sections

This article provides the first application of the machine-learning approach in the study of the cross-sections for neutron-capture reactions with the kernel ridge regression (KRR) approach. It is found that the KRR approach can reduce the root-mean-square (rms) deviation of the relative errors between the experimental data of the Maxwellian-averaged $(n,\gamma )$ cross-sections and the corresponding theoretical predictions from 69.8% to 35.4%. By including the data with different temperatures in the training set, the rms deviation can be further significantly reduced to 2.0%. Moreover, the extrapolation performance of the KRR approach along different temperatures is found to be effective and reliable.     

Quantum phase transition of alkaline-earth fermionic atoms confined in an optical superlattice

Using the density matrix renormalization group method, we evaluate the spin and charge gaps of alkaline-earth fermionic atoms in a periodic one-dimensional optical superlattice. The number of delocalized atoms is equal to the lattice size and we consider an antiferromagnetic coupling between delocalized and localized atoms. We found a quantum phase transition from a Kondo insulator spin liquid state without confining potential to a charge-gapped antiferromagnetic state with nonzero potential. For each on-site coupling, there is a critical potential point for which the spin gap vanishes and its value increases linearly with the local interaction.     

The contact in the BCS–BEC crossover for finite range interacting ultracold Fermi gases

Using mean-field theory for the Bardeen–Cooper–Schriefer (BCS) to the Bose–Einstein condensate (BEC) crossover we investigate the ground state thermodynamic properties of an interacting homogeneous Fermi gas. The interatomic interactions modelled through a finite range potential allows us to calculate the thermodynamic behaviour as a function of the potential parameters in the whole crossover region. We concentrate in studying the Contact variable, the thermodynamic conjugate of the inverse of the s-wave scattering length. Our analysis leads to predict a quantum phase transition – like in the case of large potential range. This finding is a direct consequence of the k-dependent energy gap.     

Schrödinger Equations on Damek–Ricci Spaces

In this paper we consider the Laplace–Beltrami operator Δ on Damek–Ricci spaces and derive pointwise estimates for the kernel of e ^τΔ, when τ ∈ ?* with Re τ ≥0. When τ ∈i?*, we obtain in particular pointwise estimates of the Schrödinger kernel associated with Δ. We then prove Strichartz estimates for the Schrödinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice [4 Anker , J.-P. , Pierfelice , V. ( 2009 ). Nonlinear Schrödinger equation on real hyperbolic spaces . Ann. Inst. H. Poincaré (C) Non Linear Analysis 26 : 1853 – 1869 . [Google Scholar]] on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schrödinger equation associated with a distinguished Laplacian on Damek–Ricci spaces, showing that in this case the standard L ¹ → L ^∞ estimate fails while suitable weighted Strichartz estimates hold.     

Semi-Classical Green Kernel Asymptotics for the Dirac Operator

We consider a semi-classical Dirac operator in d ∈ ? spatial dimensions with a smooth potential whose partial derivatives of any order are bounded by suitable constants. We prove that the distribution kernel of the inverse operator evaluated at two distinct points fulfilling a certain hypothesis can be represented as the product of an exponentially decaying factor involving an associated Agmon distance and some amplitude admitting a complete asymptotic expansion in powers of the semi-classical parameter. Moreover, we find an explicit formula for the leading term in that expansion.     

Optimal Bounds for Self-Similar Solutions to Coagulation Equations with Product Kernel

We consider mass-conserving self-similar solutions of Smoluchowski's coagulation equation with product kernel of homogeneity 2λ ∈ (0, 1). We establish rigorously that such solutions exhibit a singular behavior of the form x ^?(1+2λ) as x → 0. This property had been conjectured, but only weaker results had been available up to now.     

Pointwise Green's Function Estimates Toward Stability for Degenerate Viscous Shock Waves

ABSTRACT

We consider degenerate viscous shock waves arising in systems of two conservation laws, where degeneracy describes viscous shock waves for which the asymptotic endstates are sonic to the hyperbolic system (the shock speed is equal to one of the characteristic speeds). In particular, we develop detailed pointwise estimates on the Green's function associated with the linearized perturbation equation, sufficient for establishing that spectral stability implies nonlinear stability. The analysis of degenerate viscous shock waves involves several new features, such as algebraic (nonintegrable) convection coefficients, loss of analyticity of the Evans function at the leading eigenvalue, and asymptotic time decay of perturbations intermediate between that of the Lax case and that of the undercompressive case.     

Uniqueness of ∞-Harmonic Functions and the Eikonal Equation

Comparison results are obtained between infinity subharmonic and infinity superharmonic functions defined on unbounded domains. The primary new tool employed is an approximation of infinity subharmonic functions that allows one to assume that gradients are bounded away from zero. This approximation also demystifies the proof in the case of a bounded domain. A second, quite different, topic is also taken up. This is the uniqueness of absolutely minimizing functions with respect to other norms besides the Euclidean, norms that correspond to comparison results for partial differential equations which are quite discontinuous.