Position-dependent effective mass Schrödinger equations for PT-symmetric potentials

We use the method of point canonical transformations and choose the Rosen-Morse-type potential as the reference potential to study exact solutions of the position-dependent effective mass Schr?dinger equations. Choosing three position-dependent mass distributions, we construct seven exactly solvable target potentials with PT symmetry. The energy spectra of the bound states and corresponding wavefunctions for the PT-symmetric potentials are given in the exact closed forms. We also discuss the isospectrality of different Schr?dinger equations with the same mass distribution or different mass distributions for different PT-symmetric potentials.      

Irreducible Brillouin conditions and contracted Schrödinger equations for n-electron systems. III. Systems of noninteracting electrons

We analyze the structure and the solutions of the irreducible k-particle Brillouin conditions (IBCk) and the irreducible contracted Schr?dinger equations (ICSEk) for an n-electron system without electron interaction. This exercise is very instructive in that it gives one both the perspective and the strategies to be followed in applying the IBC and ICSE to physically realistic systems with electron interaction. The IBC1 leads to a Liouville equation for the one-particle density matrix gamma1=gamma, consistent with our earlier analysis that the IBC1 holds both for a pure and an ensemble state. The IBC1 or the ICSE1 must be solved subject to the constraints imposed by the n-representability condition, which is particularly simple for gamma. For a closed-shell state gamma is idempotent, i.e., all natural spin orbitals (NSO's) have occupation numbers 0 or 1, and all cumulants lambdak with k> or =2 vanish. For open-shell states there are NSO's with fractional occupation number, and at the same time nonvanishing elements of lambda2, which are related to spin and symmetry coupling. It is often useful to describe an open-shell state by a totally symmetric ensemble state. If one wants to treat a one-particle perturbation by means of perturbation theory, this mainly as a run-up for the study of a two-particle perturbation, one is faced with the problem that the perturbation expansion of the Liouville equation gives information only on the nondiagonal elements (in a basis of the unperturbed states) of gamma. There are essentially three possibilities to construct the diagonal elements of gamma: (i) to consider the perturbation expansion of the characteristic polynomial of gamma, especially the idempotency for closed-shell states, (ii) to rely on the ICSE1, which (at variance with the IBC1) also gives information on the diagonal elements, though not in a very efficient manner, and (iii) to formulate the perturbation theory in terms of a unitary transformation in Fock space. The latter is particularly powerful, especially, when one wishes to study realistic Hamiltonians with a two-body interaction.     

Non-Markovian stochastic Schrödinger equations in different temperature regimes: a study of the spin-boson model

Stochastic Schrodinger equations are used to describe the dynamics of a quantum open system in contact with a large environment, as an alternative to the commonly used master equations. We present a study of the two main types of non-Markovian stochastic Schrodinger equations, linear and nonlinear ones. We compare them both analytically and numerically, the latter for the case of a spin-boson model. We show in this paper that two linear stochastic Schrodinger equations, derived from different perspectives by Diosi, Gisin, and Strunz [Phys. Rev. A 58, 1699 (1998)], and Gaspard and Nagaoka [J. Chem. Phys. 13, 5676 (1999)], respectively, are equivalent in the relevant order of perturbation theory. Nonlinear stochastic Schrodinger equations are in principle more efficient than linear ones, as they determine solutions with a higher weight in the ensemble average which recovers the reduced density matrix of the quantum open system. However, it will be shown in this paper that for the case of a spin-boson system and weak coupling, this improvement does only occur in the case of a bath at high temperature. For low temperatures, the sampling of realizations of the nonlinear equation is practically equivalent to the sampling of the linear ones. We study further this result by analyzing, for both temperature regimes, the driving noise of the linear equations in comparison to that of the nonlinear equations.     

Irreducible Brillouin conditions and contracted Schrödinger equations for n-electron systems. IV. Perturbative analysis

The k-particle irreducible Brillouin conditions IBCk and the k-particle irreducible contracted Schr?dinger equations ICSEk for a closed-shell state are analyzed in terms of a M?ller-Plesset-type perturbation expansion. The zeroth order is Hartree-Fock. From the IBC2(1), i.e., from the two-particle IBC to first order in the perturbation parameter mu, one gets the leading correction lambda2(1) to the two-particle cumulant lambda2 correctly. However, in order to construct the second-order energy E(2), one also needs the second-order diagonal correction gammaD(2) to the one-particle density matrix gamma. This can be obtained: (i) from the idempotency of the n-particle density matrix, i.e., essentially from the requirement of n-representability; (ii) from the ICSE1(2); or (iii) by means of perturbation theory via a unitary transformation in Fock space. Method (ii) is very unsatisfactory, because one must first solve the ICSE3(2) to get lambda3(2), which is needed in the ICSE2(2) to get lambda2(2), which, in turn, is needed in the ICSE1(2) to get gamma(2). Generally the (k+1)-particle approximation is needed to obtain Ek correctly. One gains something, if one replaces the standard hierarchy, in which one solves the ICSEk, ignoring lambda(k+1) and lambda(k+2), by a renormalized hierarchy, in which only lambda(k+2) is ignored, and lambda(k+1) is expressed in terms of the lambdap of lower particle rank via the partial trace relation for lambda(k+2). Then the k-particle approximation is needed to obtain E(k) correctly. This is still poorer than coupled-cluster theory, where the k-particle approximation yields E(k+1). We also study the possibility to use some simple necessary n-representability conditions, based on the non-negativity of gamma(2) and two related matrices, in order to get estimates for gammaD(2) in terms of lambda2(1). In general these estimates are rather weak, but they can become close to the best possible bounds in special situations characterized by a very sparse structure of lambda2 in terms of a localized representation. The perturbative analysis does not encourage the use of a k-particle hierarchy based on the ICSEk (or on their reducible counterparts, the CSEk), it rather favors the approach in terms of the unitary transformation, where the k-particle approximation yields the energy correct up to E(2k-1). The problems that arise are related to the unavoidable appearance of exclusion-principle violating cumulants. The good experience with perturbation theory in terms of a unitary transformation suggests that one should abandon a linearly convergent iteration scheme based on the ICSEk hierarchy, in favor of a quadratically convergent one based on successive unitary transformations.     

Nonadiabatic effects in the nuclear probability and flux densities through the fractional Schrödinger equation

Nonadiabatic effects in the nuclear dynamics of the H₂⁺ molecular ion, initiated by ionization of the H₂ molecule, is studied by means of the probability and flux distribution functions arising from the space fractional Schrödinger equation. In order to solve the fractional Schrödinger eigenvalue equation, it is shown that the quantum Riesz fractional derivative operator fulfills the usual properties of the quantum momentum operator acting on the bra and ket vectors of the abstract Hilbert space. Then, the fractional Fourier grid Hamiltonian method is implemented and applied to molecular vibrations. The eigenenergies and eigenfunctions of the fractional Schrödinger equation describing the vibrational motion of the H₂⁺ and D₂⁺ molecules are analyzed. In particular, it is shown that the position-momentum Heisenberg's uncertainty relationship holds independently of the fractional Schrödinger equation. Finally, the probability and flux distributions are presented, demonstrating the applicability of the fractional Schrödinger equation for taking into account nonadiabatic effects.     

Pseudospectral methods of solution of the Schrödinger equation

Several different pseudospectral methods of solution of the Schrödinger equation are applied to the calculation of the eigenvalues of the Morse potential for I₂ and the Cahill–Parsegian potential for Ar₂ [Cahill, Parsegian, J. Chem. Phys. 121, 10839 (2004)]. The calculation of the eigenvalues for the Woods–Saxon potential are also considered. The convergence of the eigenvalues with a quadrature discretization method is found to be very fast owing to the judicious choice for the weight function, basis set and quadrature points. The weight function used is either related to the exact ground state wavefunction, if known, or an approximation to it from some reference potential. We compare several different pseudospectral methods.     

Exact solution of Schrödinger equation for Pseudoharmonic potential

Exact solution of Schr?dinger equation for the pseudoharmonic potential is obtained for an arbitrary angular momentum. The energy eigenvalues and corresponding eigenfunctions are calculated by Nikiforov–Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The energy eigenvalues are calculated numerically for some values of ℓ and n with n ≤ 5 for some diatomic molecules.      

Padé spectrum decompositions of quantum distribution functions and optimal hierarchical equations of motion construction for quantum open systems

Pade? spectrum decomposition is an optimal sum-over-poles expansion scheme of Fermi function and Bose function [J. Hu, R. X. Xu, and Y. J. Yan, J. Chem. Phys. 133, 101106 (2010)]. In this work, we report two additional members to this family, from which the best among all sum-over-poles methods could be chosen for different cases of application. Methods are developed for determining these three Pade? spectrum decomposition expansions at machine precision via simple algorithms. We exemplify the applications of present development with optimal construction of hierarchical equations-of-motion formulations for nonperturbative quantum dissipation and quantum transport dynamics. Numerical demonstrations are given for two systems. One is the transient transport current to an interacting quantum-dots system, together with the involved high-order co-tunneling dynamics. Another is the non-Markovian dynamics of a spin-boson system.     

Exact solution of Schrödinger equation with deformed ring-shaped potential

Exact solution of the Schrödinger equation with deformed ring-shaped potential is obtained in the parabolic and spherical coordinates. The Nikiforov–Uvarov method is used in the solution. Eigenfunctions and corresponding energy eigenvalues are calculated analytically. The agreement of our results is good.AMS Subject Classification: 03.65.–w, 12.39.Jh, 21.10.–k     

A new method of solving the many-body Schrödinger equation

A new method of solving the many-body Schrödinger equation is proposed. It is based on the use of constant particle-particle interaction potential surfaces (IPSs) and the representation of the many-body wave function in a configuration interaction form with coefficients depending on the total interaction potential. For these coefficients the corresponding set of linear ordinary differential equations is obtained. A hierarchy of approximations is developed for IPSs. The solution of a simple exactly solvable model and He-like ions proves that this method is more accurate than the conventional configuration interaction method and demonstrates a better convergence with increasing basis set.     

Variational solution of the three-dimensional Schrödinger equation using plane waves in adaptive coordinates

A series of improvements for the solution of the three-dimensional Schr?dinger equation over a method introduced by Gygi [F. Gygi, Europhys. Lett. 19, 617 (1992); F. Gygi, Phys. Rev. B 48, 11692 (1993)] are presented. As in the original Gygi's method, the solution (orbital) is expressed by means of plane waves in adaptive coordinates u, where u is mapped from Cartesian coordinates, u=f(r). The improvements implemented are threefold. First, maps are introduced that allow the application of the method to atoms and molecules without the assistance of the supercell approximation. Second, the electron-nucleus singularities are exactly removed, so that pseudo-potentials are no longer required. Third, the sampling error during integral evaluation is made negligible, which results in a true variational, second-order energy error procedure. The method is tested on the hydrogen atom (ground and excited states) and the H(2)(+) molecule, resulting in milli-Hartree accuracy with a moderate number of plane waves.     

Fluorescence properties of systems with multiple Förster transfer pairs

A mathematical model has been developed to compute the spectroscopic properties of fluorescence systems with multiple F?rster transfer pairs in a homogeneous 3-dimensional matrix. This model is based on F?rster energy transfer theory and needs only a limited number of parameters which depend only on the properties of the individual dyes and their pair-wise interactions. Yet, the model allows the accurate prediction on the fluorescence properties of systems comprising mutual F?rster transfer between three dyes. The model is compared to an experimental system composed of reverse micelles and water soluble dyes. Although the experimental system might include additional effects that may influence the fluorescence properties (e.g. adsorption to the micelle walls, aggregation of the dyes) the agreement between the mathematical model and the experimental system is reasonably good.     

Particular and homogeneous solutions of time-independent wavepacket Schrödinger equations: calculations using a subset of eigenstates of undamped or damped Hamiltonians

A variety of causal, particular and homogeneous solutions to the time-independent wavepacket Schr?dinger equation have been considered as the basis for calculations using Chebychev expansions, finite-τ expansions obtained from a partial Fourier transform of the time-dependent Schr?dinger equation, and the distributed approximating functional (DAF) representation for the spectral density operator (SDO). All the approximations are made computationally robust and reliable by damping the discrete Hamiltonian matrix along the edges of the finite grid to facilitate the use of compact grids. The approximations are found to be completely well behaved at all values of the (continuous) scattering energy. It is found that the DAF–SDO provides a suitable alternative to Chebychev propagation. Received: 29 February 2000 / Accepted: 5 April 2000 / Published online: 18 August 2000     

Solving the Schrödinger and Dirac equations of hydrogen molecular ion accurately by the free iterative complement interaction method

The nonrelativistic Schr?dinger equation and the relativistic four-component Dirac equation of H(2) (+) were solved accurately in an analytical expansion form by the free iterative complement interaction (ICI) method combined with the variational principle. In the nonrelativistic case, we compared the free ICI wave function with the so-called "exact" wave function as two different expansions converging to the unique exact wave function and found that the free ICI method is much more efficient than the exact method. In the relativistic case, we first used the inverse Hamiltonian to guarantee Ritz-type variational principle and obtained accurate result. We also showed that the ordinary variational calculation also gives a nice convergence when the g function is appropriately chosen, since then the free ICI calculation guarantees a correct relationship between the large and small components of each adjacent order, which we call ICI balance. This is the first application of the relativistic free ICI method to molecule. We calculated both ground and excited states in good convergence, and not only the upper bound but also the lower bound of the ground-state energy. The error bound analysis has assured that the present result is highly accurate.     

Cumulant reconstruction of the three-electron reduced density matrix in the anti-Hermitian contracted Schrödinger equation

Differing perspectives on the accuracy of three-electron reduced-density-matrix (3-RDM) reconstruction in nonminimal basis sets exist in the literature. This paper demonstrates the accuracy of cumulant-based reconstructions, developed by Valdemoro (V) [F. Colmenero et al., Phys. Rev. A 47, 971 (1993)], Nakatsuji and Yasuda (NY) [Phys. Rev. Lett. 76, 1039 (1996)], Mazziotti (M) [Phys. Rev. A 60, 3618 (1999)], and Valdemoro-Tel-Perez-Romero (VTP) [Many-electron Densities and Density Matrices, edited by J. Cioslowski (Kluwer, Boston, 2000)]. Computationally, we extend previous investigations to study a variety of molecules, including LiH, HF, NH(3), H(2)O, and N(2), in Slater-type, double-zeta, and polarized double-zeta basis sets at both equilibrium and nonequilibrium geometries. The reconstructed 3-RDMs, compared with 3-RDMs from full configuration interaction, demonstrate in nonminimal basis sets the accuracy of the first-order expansion (V) as well as the important role of the second-order corrections (NY, M, and VTP). Calculations at nonequilibrium geometries further show that cumulant functionals can reconstruct the 3-RDM from a multireferenced 2-RDM with reasonable accuracy, which is relevant to recent multireferenced formulations of the anti-Hermitian contracted Schrodinger equation (ACSE) and canonical diagonalization. Theoretically, we perform a detailed perturbative analysis of the M functional to identify its second-order components. With these second-order components we connect the M, NY, and VTP reconstructions for the first time by deriving both the NY and VTP functionals from the M functional. Finally, these 3-RDM reconstructions are employed within the ACSE [D. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)] to compute ground-state energies which are compared with the energies from the contracted Schrodinger equation and several wave function methods.     

Bound state solution of the Schrödinger equation for Mie potential

Exact solution of Schr?dinger equation for the Mie potential is obtained for an arbitrary angular momentum. The energy eigenvalues and the corresponding wavefunctions are calculated by the use of the Nikiforov–Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The bound states are calculated numerically for some values of ℓ and n with n ≤ 5. They are applied to several diatomic molecules.      

The Quadrature Discretization Method (QDM) in the solution of the Schrödinger equation

The Quadrature Discretization Method (QDM) is employed in the solution of several onedimensional Schrödinger equations that have received considerable attention in the literature. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to a weight function. For a certain class of problems with potentials of the form that occur in supersymmetric quantum mechanics, the ground state wavefunction is known. In the present paper, the weight functions that are used are related to the ground state wavefunctions if known, or some approximate form. The eigenvalues and eigenfunctions of four different potential functions discussed extensively in the literature are calculated and the results are compared with published values.     

Solutions of N-dimensional Schrödinger equation with Morse potential via Laplace transforms

A study is undertaken to investigate an analytical solution for the N-dimensional Schrödinger equation with the Morse potential based on the Laplace transformation method. The results show that in the Pekeris approximation, the radial part of the Schrödinger equation reduces to the corresponding equation in one dimension. In addition, a comparison is made between the energy spectrum resulted from this method and the spectra that are obtained from the two-point quasi-rational approximation method and the Nikiforov–Uvarov approach.