Excited State Solutions Research Articles

Nonlinear effects in the excited states of many-fermion Einstein-Dirac solitons

We present an analysis of excited-state solutions for a gravitationally localized system consisting of a filled shell of high-angular-momentum fermions, using the Einstein-Dirac formalism introduced by Finster, Smoller, and Yau [Phys. Rev. D 59, 104020 (1999)]. We show that, even when the particle number is relatively low ($N_f\ge 6$), the increased nonlinearity in the system causes a significant deviation in behavior from the two-fermion case. Excited-state solutions can no longer be uniquely identified by the value of their central redshift, with this multiplicity producing distortions in the characteristic spiraling forms of the mass-radius relations. We discuss the connection between this effect and the internal structure of solutions in the relativistic regime.

Variational Density Functional Calculations of Excited States via Direct Optimization

The development of variational density functional theory approaches to excited electronic states is impeded by limitations of the commonly used self-consistent field (SCF) procedure. A method based on a direct optimization approach as well as the maximum overlap method is presented, and the performance is compared with previously proposed SCF strategies. Excited-state solutions correspond to saddle points of the energy as a function of the electronic degrees of freedom. The approach presented here makes use of a preconditioner determined with the help of the maximum overlap method to guide the convergence on a target nth-order saddle point. This method is found to be more robust and converge faster than previously proposed SCF approaches for a set of 89 excited states of molecules. A limited-memory formulation of the symmetric rank-one method for updating the inverse Hessian is found to give the best performance. A conical intersection for the carbon monoxide molecule is calculated without resorting to fractional occupation numbers. Calculations on the excited states of the hydrogen atom and a doubly excited state of the dihydrogen molecule using a self-interaction corrected functional are presented. For these systems, the self-interaction correction is found to improve the accuracy of density functional calculations of excited states.

Half-Projected σ Self-Consistent Field For Electronic Excited States.

Fully self-consistent mean-field solutions of electronic excited states have been much less accessible compared to ground state solutions (e.g., Hartree-Fock). The main reason for this is that most excited states are energy saddle points, and hence energy-based optimization methods such as Δ-SCF often collapse to the ground state. Recently, our research group has developed a new method, σ-SCF [ J. Chem. Phys. 2017 , 147 , 214104 ], that successfully solves the "variational collapse" problem of energy-based methods. Despite the success, σ-SCF solutions are often spin-contaminated for open-shell states due to the single-determinant nature; unphysical behaviors such as disappearing solutions and discontinuous first-order energy derivatives are also observed along with the spontaneous breaking of spin or spatial symmetries. In this work, we tackle these problems by partially restoring the broken spin-symmetry of a σ-SCF solution through an approximate spin-projection scheme called half-projection. Orbitals of the projected wave function are optimized in a variation-after-projection (VAP) manner. The resulting theory, which we term half-projected (HP) σ-SCF, brings substantial improvement to the description of singlet and triplet excitations of the original σ-SCF method. Numerical simulations on small molecules suggest that HP σ-SCF delivers high-quality excited-state solutions that exist in a wide range of geometries with smooth potential energy surfaces. We also show that local excitations in HP σ-SCF can be size-intensive.

Effects of Electronic-State-Dependent Solute Polarizability: Application to Solute-Pump/Solvent-Probe Spectra.

Experimental studies of solvation dynamics in liquids invariably ask how changing a solute from its electronic ground state to an electronically excited state affects a solution's dynamics. With traditional time-dependent-fluorescence experiments, that means looking for the dynamical consequences of the concomitant change in solute-solvent potential energy. But if one follows the shift in the dynamics through its effects on the macroscopic polarizability, as recent solute-pump/solvent-probe spectra do, there is another effect of the electronic excitation that should be considered: the jump in the solute's own polarizability. We examine the spectroscopic consequences of this solute polarizability change in the classic example of the solvation dye coumarin 153 dissolved in acetonitrile. After demonstrating that standard quantum chemical methods can be used to construct accurate multisite models for the polarizabilities of ground- and excited-state solvation dyes, we show via simulation that this polarizability change acts as a contrast agent, significantly enhancing the observable differences in optical-Kerr spectra between ground- and excited-state solutions. A comparison of our results with experimental solute-pump/solvent-probe spectra supports our interpretation and modeling of this spectroscopy. We predict, in particular, that solute-pump/solvent-probe spectra should be sensitive to changes in both the solvent dynamics near the solute and the electronic-state-dependence of the solute's own rotational dynamics.

Photoexcitation of Light-Harvesting C–P–C60 Triads: A FLMO-TD-DFT Study

The recently proposed linear-scaling time-dependent density functional theory (TD-DFT) [ J. Chem. Comput. 2011 , 7 , 3643 ] is employed to capture more than 300 low-lying excited states for three light-harvesting C-P-C60 triads composed of β-carotenoid polyene (C), diaryl-based porphyrin (P), and pyrrole-fullerene (C60). The simulated optical absorption spectra are grossly in good agreement with experimental observations. To gain insights on the structure-property relations, both top-down and bottom-up analyses of the excited states are made in terms of the underlying fragment localized molecular orbitals (FLMO). A maximum occupation method is further proposed for finding excited-state solutions of self-consistent-field equations and is applied to long-range charge-transfer states that cannot be described by TD-DFT with pure density functionals.

A two-parameter continuation method for computing numerical solutions of spin-1 Bose–Einstein condensates

We describe a novel two-parameter continuation method combined with a spectral-collocation method (SCM) for computing the ground state and excited-state solutions of spin-1 Bose–Einstein condensates (BEC), where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. To compute the ground state solution of spin-1 BEC, we implement the single parameter continuation algorithm with the chemical potential μ as the continuation parameter, and trace the first solution branch of the Gross–Pitaevskii equations (GPEs). When the curve-tracing is close enough to the target point, where the normalization condition of the wave function is going to be satisfied, we add the magnetic potential λ as the second continuation parameter with the magnetization M as the additional constraint condition. Then we implement the two-parameter continuation algorithm until the target point is reached, and the ground state solution of the GPEs is obtained. The excited state solutions of the GPEs can be treated in a similar way. Some numerical experiments on Na23 and Rb87 are reported. The numerical results on the spin-1 BEC are the same as those reported in [10]. Further numerical experiments on excited-state solutions of spin-1 BEC suffice to show the robustness and efficiency of the proposed two-parameter continuation algorithm.

Pseudogap phase in strongly disordered conventional superconductors

Excited states of Bogoliubov--de Gennes equations are examined for a two-dimensional negative-$U$ Hubbard Hamiltonian with on-site disorder. It is shown explicitly that the temperature (pseudogap temperature) when the superconducting gap opens is different from that (the superconducting transition temperature) where the long-range order appears. In the excited-state solutions the system is self-organized into blocks, which behave like superspins and are coupled with their neighbors. Only if the couplings between these blocks become strong enough can the true long-range order be realized.

A spectral-Galerkin continuation method using Chebyshev polynomials for the numerical solutions of the Gross–Pitaevskii equation

We study an efficient spectral-Galerkin continuation method (SGCM) and two-grid centered difference approximations for the numerical solutions of the Gross–Pitaevskii equation (GPE), where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. Some basic formulae for the SGCM are derived so that the eigenvalues of the associated linear eigenvalue problems can be easily computed. The SGCM is implemented to investigate the ground and first excited-state solutions of the GPE. Both the parabolic and quadruple-well trapping potentials are considered. We also study Bose–Einstein condensates (BEC) in optical lattices, where the periodic potential described by the sine or cosine functions is imposed on the GPE. Of particular interest here is the investigation of symmetry-breaking solutions. Sample numerical results are reported.

Excited states of the saddle point equation of the Landau-Ginzburg-Wilson Hamiltonian with random temperature

The phase transition in quenched disordered systems is studied on the level of the saddle point solution. Two-dimensional saddle point equation of the Landau-Ginzburg-Wilson Hamiltonian with random temperature is numerically solved for the excited states. It is shown that the excited-state solutions can be described by the domain walls. The length of domain wall and the free energy increase due to the domain wall are calculated. On the level of the saddle point solution the partition function can be mapped to an Ising model approximately. The coupling between Ising spin is estimated. The phase transition is discussed according to this Ising model. It is found that there are two classes of phase transition: inhomogeneous and homogeneous. Different from the pure system, for the phase transition in disordered systems there are fluctuations on the level of saddle point solution.

Superconvergence of high order FEMs for eigenvalue problems with periodic boundary conditions

We study Adini’s elements for nonlinear Schrödinger equations (NLS) defined in a square box with periodic boundary conditions. First we transform the time-dependent NLS to a time-independent stationary state equation, which is a nonlinear eigenvalue problem (NEP). A predictor–corrector continuation method is exploited to trace solution curves of the NEP. We are concerned with energy levels and superfluid densities of the NLS. We analyze superconvergence of the Adini elements for the linear Schrödinger equation defined in the unit square. The optimal convergence rate O ( h 6 ) is obtained for quasiuniform elements. For uniform rectangular elements, the superconvergence O ( h 6 + p ) is obtained for the minimal eigenvalue, where p = 1 or p = 2 . The theoretical analysis is confirmed by the numerical experiments. Other kinds of high order finite element methods (FEMs) and the superconvergence property are also investigated for the linear Schrödinger equation. Finally, the Adini elements-continuation method is exploited to compute energy levels and superfluid densities of a 2D Bose–Einstein condensates (BEC) in a periodic potential. Numerical results on the ground state as well as the first few excited-state solutions are reported.

Self-Consistent Field Calculations of Excited States Using the Maximum Overlap Method (MOM)

We present a simple algorithm, which we call the maximum overlap method (MOM), for finding excited-state solutions to self-consistent field (SCF) equations. Instead of using the aufbau principle, the algorithm maximizes the overlap between the occupied orbitals on successive SCF iterations. This prevents variational collapse to the ground state and guides the SCF process toward the nearest, rather than the lowest energy, solution. The resulting excited-state solutions can be treated in the same way as the ground-state solution and, in particular, derivatives of excited-state energies can be computed using ground-state code. We assess the performance of our method by applying it to a variety of excited-state problems including the calculation of excitation energies, charge-transfer states, and excited-state properties.

Computing wave functions of nonlinear Schrödinger equations: A time-independent approach

We present a novel algorithm for computing the ground-state and excited-state solutions of M-coupled nonlinear Schrödinger equations (MCNLS). First we transform the MCNLS to the stationary state ones by using separation of variables. The energy level of a quantum particle governed by the Schrödinger eigenvalue problem (SEP) is used as an initial guess to computing their counterpart of a nonlinear Schrödinger equation (NLS). We discretize the system via centered difference approximations. A predictor–corrector continuation method is exploited as an iterative method to trace solution curves and surfaces of the MCNLS, where the chemical potentials are treated as continuation parameters. The wave functions can be easily obtained whenever the solution manifolds are numerically traced. The proposed algorithm has the advantage that it is unnecessary to discretize or integrate the partial derivatives of wave functions. Moreover, the wave functions can be computed for any time scale. Numerical results on the ground-state and excited-state solutions are reported, where the physical properties of the system such as isotropic and nonisotropic trapping potentials, mass conservation constraints, and strong and weak repulsive interactions are considered in our numerical experiments.

Schrödinger link between nonequilibrium thermodynamics and Fisher information.

It is known that equilibrium thermodynamics can be deduced from a constrained Fisher information extemizing process. We show here that, more generally, both nonequilibrium and equilibrium thermodynamics can be obtained from such a Fisher treatment. Equilibrium thermodynamics corresponds to the ground-state solution, and nonequilibrium thermodynamics corresponds to excited-state solutions, of a Schrödinger wave equation (SWE). That equation appears as an output of the constrained variational process that extremizes Fisher information. Both equilibrium and nonequilibrium situations can thereby be tackled by one formalism that clearly exhibits the fact that thermodynamics and quantum mechanics can both be expressed in terms of a formal SWE, out of a common informational basis. As an application, we discuss viscosity in dilute gases.

Excited-state solutions for an infinite family of one-dimensional potentials

A method of generating analytically and numerically exact ground and excited-state wave functions and energies of an infinite family of one-dimensional potentials is described. Explicit application to a sixth-order polynomial potential is presented.

Excited States in a Model Polaron Hamiltonian

The energy-spectrum problem is treated for a bound stationary polaron. The Pekar Hamiltonian is used to describe the polaron and its polar lattice. This model is based on two main assumptions. First, the carrier is assumed to move much faster than the lattice ions. Second, the polaron is assumed to be spread out over a distance that is much larger than the lattice constant. These assumptions restrict us to a strong-coupling regime. A variational principle especially well suited for finding excited states has been used in a search for solutions of the polaron Hamiltonian. Using trial wave functions having $s\ensuremath{-}$, $p\ensuremath{-}$, and $d$-like symmetry, some well-separated approximate eigenstates of this model polaron have been found. The excited states represent internal and self-consistent excited-state solutions of the polaron Hamiltonian. It is found that the energy of each state is proportional to the square of the polaron coupling constant. Also, the qualitative features of the spectrum, i.e., the number of energy levels and their relative positions, are independent of the coupling constant.