Effective interactions and block diagonalization in quantum-mechanical problems

Many models of condensed-matter systems have interactions with unexpected features: for example, exclusively distant-neighbor spin–orbit interactions. On first inspection these interactions seem physically questionable in view of the basis states used. However, such interactions can be physically reasonable if the model is an effective one, in which the basis states are not exactly as described, but instead include components of states removed from the problem. Mathematically, an effective model results from partitioning the Hamiltonian matrix, which can be accomplished by energy-dependent or energy-independent methods. We examine effective models of both types, with a special emphasis on energy-independent approaches. We show that an appropriate choice of basis makes the partitioning simpler and more accurate. We illustrate the method by calculating the spin–orbit splitting in graphene.     

Effective mass schrödinger equation for exactly solvable class of one-dimensional potentials

We deal with the exact solutions of Schr?dinger equation characterized by position-dependent effective mass via point canonical transformations. The Morse, P?schl-Teller, and Hulthén type potentials are considered, respectively. With the choice of position-dependent mass forms, exactly solvable target potentials are constructed. Their energy of the bound states and corresponding wavefunctions are determined exactly.      

Path integral Monte Carlo simulation of the absorption spectra of an Al atom embedded in helium

We use a multilevel path integral Monte-Carlo (PIMC) method to simulate the arrangement of He atoms around a single Al atom doped in a He cluster. High-level ab initio Al-He pair potentials and a Balling and Wright pairwise Hamiltonian model are used to describe the full potential and the electronic asymmetry arising from the open-shell character of the Al atom in its ground and excited electronic states. Our calculations show that the doping of the Al 3p electron strongly influences the He packing. The results of the PIMC simulation are used to predict the electronic excitation spectrum of an Al atom embedded in He clusters. With inclusion of tail corrections for the ground and excited states potentials, the calculated 3d<--3p spectrum agrees reasonably well with the experimental spectrum. The blueshift of the calculated spectrum associated with the 4s<--3p transition of solvated Al is about 25 nm (2000 cm-1) larger than seen in experiments on Al embedded in bulk liquid He. We predict that the spectrum associated with the 4p<--3p transition will be blueshifted by approximately 7000 cm-1 (nearly 1 eV).     

An accurate and efficient empirical approach for calculating the dielectric self-energy and ion-ion pair potential in continuum models of biological ion channels

This paper presents empirical formulas for calculating the dielectric self-energy and ion-ion pair interactions in cylindrical ion channels. The proposed approach can be extended to more complex channel structures, for example, (i) a "straight" channel with variable radius and (ii) a "curved" channel with constant radius. For calibration purposes, we compare results obtained based on the approximate effective potentials developed herein to exact electrostatic calculations obtained via the algorithm of Graf et al.: the agreement is satisfactory. A dynamic lattice Monte Carlo (DLMC) technique is used to further assess the accuracy and efficiency of the proposed empirical potentials. The concentration profiles and current-voltage curves produced with our simple empirical energy formulas are in excellent agreement with numerical results obtained using the algorithm of Graf et al., which calculates all relevant electrostatic forces exactly. The use of effective ion-ion potentials greatly reduces the computer memory required to perform DLMC ion permeation simulations in dielectrically inhomogeneous environments, thus enabling treatment of larger systems than can be handled by numerically exact techniques.     

Near‐exact supersymmetric partner potentials: Construction and exploitation

Precise supersymmetric (SUSY) partner potentials can be generated only for exactly solvable problems of the stationary Schrödinger equation. This is a severe restriction, as most problems are not amenable to exact solutions. We employ here a linear variational strategy to explicitly construct approximate SUSY partners of a few common, not exactly solvable potentials and subsequently examine their properties to explore the advantages in practical implementation. The efficacy of our proposed scheme is commendable. We demonstrate that, for symmetric potentials, the constructed partners may be so good that the overall recipe has the nicety of generating the whole eigenspectrum by employing only half of the full Hilbert space functions. A similar strategy is shown to work for the odd states too, with proper boundary conditions. Pilot calculations involve a number of low‐lying states of some mixed oscillator and double‐well potentials. Analysis of the results reveals a few interesting features of the problem of construction of approximate SUSY partners and their practical use. Particularly, we identify places where the operator‐level approximations are involved and how far they affect the bounding properties of energies that are obtained as eigenvalues of a matrix diagonalization problem associated with linear variations. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

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.      

Independent particle theory with electron correlation

We formulate an effective independent particle model where the effective Hamiltonian is composed of the Fock operator and a correlation potential. Within the model the kinetic energy and the exchange energy can be expressed exactly leaving the correlation energy functional as the remaining unknown. Our efforts concentrate on finding a correlation potential such that exact ionization potentials and electron affinities can be reproduced as orbital energies. The equation-of-motion coupled-cluster approach enables us to define an effective Hamiltonian from which a correlation potential can be extracted. We also make the connection to electron propagator theory. The disadvantage of the latter is the inherit energy dependence of the potential resulting in a different Hamiltonian for each orbital. Alternatively, the Fock space coupled-cluster approach employs an effective Hamiltonian which is energy independent and universal for all orbitals. A correlation potential is extracted which yields the exact ionization potentials and electron affinities and a set of associated molecular orbitals. We also describe the close relationship to Brueckner theory.     

Quantum algorithm for obtaining the energy spectrum of molecular systems

Simulating a quantum system is more efficient on a quantum computer than on a classical computer. The time required for solving the Schr?dinger equation to obtain molecular energies has been demonstrated to scale polynomially with system size on a quantum computer, in contrast to the well-known result of exponential scaling on a classical computer. In this paper, we present a quantum algorithm to obtain the energy spectrum of molecular systems based on the multiconfigurational self-consistent field (MCSCF) wave function. By using a MCSCF wave function as the initial guess, the excited states are accessible. Entire potential energy surfaces of molecules can be studied more efficiently than if the simpler Hartree-Fock guess was employed. We show that a small increase of the MCSCF space can dramatically increase the success probability of the quantum algorithm, even in regions of the potential energy surface that are far from the equilibrium geometry. For the treatment of larger systems, a multi-reference configuration interaction approach is suggested. We demonstrate that such an algorithm can be used to obtain the energy spectrum of the water molecule.     

Efficient combination of Wang-Landau and transition matrix Monte Carlo methods for protein simulations

An efficient combination of the Wang-Landau and transition matrix Monte Carlo methods for protein and peptide simulations is described. At the initial stage of simulation the algorithm behaves like the Wang-Landau algorithm, allowing to sample the entire interval of energies, and at the later stages, it behaves like transition matrix Monte Carlo method and has significantly lower statistical errors. This combination allows to achieve fast convergence to the correct values of density of states. We propose that the violation of TTT identities may serve as a qualitative criterion to check the convergence of density of states. The simulation process can be parallelized by cutting the entire interval of simulation into subintervals. The violation of ergodicity in this case is discussed. We test the algorithm on a set of peptides of different lengths and observe good statistical convergent properties for the density of states. We believe that the method is of general nature and can be used for simulations of other systems with either discrete or continuous energy spectrum.     

The supersymmetric quantum mechanics theory and Darboux transformation for the Morse oscillator with an approximate rotational term

Anharmonic potentials with a rotational terms are widely used in quantum chemistry of diatomic systems, since they include the influence of centrifugal force on motions of atomic nuclei. For the first time the Taylor-expanded renormalized Morse oscillator is studied within the framework of supersymmetric quantum mechanics theory. The mathematical formalism of supersymmetric quantum mechanics and the Darboux transformation are used to determine the bound states for the Morse anharmonic oscillator with an approximate rotational term. The factorization method has been applied in order to obtain analytical forms of creation and annihilation operators as well as Witten superpotential and isospectral potentials. Moreover, the radial Schrödinger equation with the Darboux potential has been converted into an exactly solvable form of second-order Sturm–Liouville differential equation. To this aim the Darboux transformation has been used. The efficient algebraic approach proposed can be used to solve the Schrödinger equation for other anharmonic exponential potentials with rotational terms.     

Electronic energy levels with the help of trajectory-guided random grid of coupled wave packets. I. Six-dimensional simulation of H2

As a preliminary to future work on the behavior of atoms and molecules in strong time-dependent fields, we apply the coupled coherent-states (CCS) technique of multidimensional phase-space quantum dynamics to obtain Born-Oppenheimer energy levels of electrons in molecules. Unlike traditional approaches based on atomic and molecular-orbital basis sets and time-independent Schrodinger equation the CCS method exploits the solution of the time-dependent Schrodinger equation in the basis of Monte Carlo-selected trajectory-guided coherent states, which treat classical electron correlations exactly. In addition the CCS trajectories move over averaged potentials, which remove the Coulombic singularities.     

Maximum feasibility guideline in the design and analysis of protein folding potentials

Protein folding potentials are expected to have the lowest energy for the native shape. The Linear Programming (LP) approach achieves exactly that goal for a training set, or indicates that this goal is impossible to obtain. If a solution cannot be found (i.e., the problem is infeasible) two possible routes are possible: (a) choosing a new functional form for the potential, (b) finding the best potential with a feasible subset of the data, and (or) detecting inconsistent subset of the data in the training set. Here, we explore option (b). A simple heuristic for finding an approximate solution to an infeasible set of linear inequalities is outlined. An approximately feasible solution is obtained iteratively, starting from a certain initial guess, by computing a series of analytic centers of the polyhedra defined by all the inequalities satisfied at the subsequent iterations. Standard interior point algorithms for Linear Programming can be used to compute efficiently the analytic center of a polyhedron. We demonstrate how this procedure can be used for the design of folding potentials that are linear in their parameters. The procedure shows an improvement in the quality of the potentials and sometimes points to flaws in the original data.     

FTIR microscopy of biological cells and tissue: data analysis using resonant Mie scattering (RMieS) EMSC algorithm

Transmission and transflection infrared microscopy of biological cells and tissue suffer from significant baseline distortions due to scattering effects, predominantly resonant Mie scattering (RMieS). This scattering can also distort peak shapes and apparent peak positions making interpretation difficult and often unreliable. A correction algorithm, the resonant Mie scattering extended multiplicative signal correction (RMieS-EMSC), has been developed that can be used to remove these distortions. The correction algorithm has two key user defined parameters that influence the accuracy of the correction. The first is the number of iterations used to obtain the best outcome. The second is the choice of the initial reference spectrum required for the fitting procedure. The choice of these parameters influences computational time. This is not a major concern when correcting individual spectra or small data sets of a few hundred spectra but becomes much more significant when correcting spectra from infrared images obtained using large focal plane array detectors which may contain tens of thousands of spectra. In this paper we show that, classification of images from tissue can be achieved easily with a few (<10) iterations but a reliable interpretation of the biochemical differences between classes could require more iterations. Regarding the choice of reference spectrum, it is apparent that the more similar it is to the pure absorption spectrum of the sample, the fewer iterations required to obtain an accurate corrected spectrum. Importantly however, we show that using three different non-ideal reference spectra, the same unique correction solution can be obtained.     

On orthogonality constrained multiple core‐hole states and optimized effective potential method

An attempt to construct a multiple core‐hole state within the optimized effective potential (OEP) methodology is presented. In contrast to the conventional Δ‐self‐consistent field method for hole states, the effects of removing an electron is achieved using some orthogonality constraints imposed on the orbitals so that a Slater determinant describing a hole state is constrained to be orthogonal to that of a neutral system. It is shown that single, double, and multiple core‐hole states can be treated within a unified framework and can be easily implemented for atoms and molecules. For this purpose, a constrained OEP method proposed earlier for excited states (Glushkov and Levy, J. Chem. Phys. 2007, 126, 174106) is further developed to calculate single and double core ionization energies using a local effective potential expressed as a direct mapping of the external potential. The corresponding equations, determining core‐hole orbitals from a one‐particle Schrödinger equation with a local potential as well as correlation corrections derived from the second‐order many‐body perturbation theory are given. One of the advantages of the present direct mapping formulation is that the effective potential, which plays the role of the Kohn–Sham potential, has the symmetry of the external potential. Single and double core ionization potentials computed with the presented scheme were found to be in agreement with data available from experiment and other calculations. We also discuss core‐hole state local potentials for the systems studied. © 2012 Wiley Periodicals, Inc.     

基于稀疏模型的气相色谱-质谱数据解析及其在严重重叠峰解析中的应用

提出一种气相色谱-质谱（GC-MS）数据解析算法。以色谱峰顶点处的质谱作为待测谱,在谱库中检索一定量相关参考谱,求解关于各纯组分色谱响应值的方程。质谱检索采取分步策略,先利用质谱碎片规律建立高速索引进行粗选,然后使用强峰高概率出峰准则和耐挤压性准则排除无关质谱。为求解色谱响应值方程,提出基于稀疏模型的回归算法,相比传统算法,稀疏算法利于提取待测谱的主要结构,避免"过拟合"。实验结果表明,该质谱检索算法具有较高的精度和规模较小的剩余参考谱集,而所提稀疏模型算法可有效解析严重重叠峰。该算法可作为GC-MS重叠峰解析,特别是严重重叠峰解析的一种有效解决方案。     

A simple and effective technique to variationally interpret the structure of SUSY partners of mirror image potentials

Precise supersymmetric partner potentials can be generated for exactly solvable problems of the stationary Schrödinger equation. Construction of isospectral potential is not always possible for exactly solvable systems. This is a restriction, as most problems are not exactly solvable. Employment of mirror-image property can help to construct an exact isospectral partner of that potential. These potentials have chemical relevance to enantiomers. In this paper, we present a formulation as modelling to explore the form of SUSY pair of these potentials. Through polynomial fit, we correlate all possible basic SUSY partners and optimise it to best fit polynomial to present a typical energy value of N = 50.     

Effective local potentials for excited states

The constrained variational Hartree-Fock method for excited states of the same symmetry as the ground state [Chem. Phys. Lett. 287, 189 (1998)] is combined with the effective local potential (ELP) method [J. Chem. Phys. 125, 081104 (2006)] to generate Kohn-Sham-type exact-exchange potentials for singly excited states of many-electron systems. Illustrative examples include the three lowest (2)S states of the Li and Na atoms and the three lowest (3)S states of He and Be. For the systems studied, excited-state ELPs differ from the corresponding ground-state potentials in two respects: They are less negative and have small additional "bumps" in the outer electron region. The technique is general and can be used to approximate excited-state exchange-correlation potentials for other orbital-dependent functionals.     

Theory of photoinduced heterogeneous electron transfer

We consider electron injection into the conduction band of a semiconductor, from an electronically excited state of a dye molecule, adsorbed on its surface. For arbitrary width of the conduction band, the survival probability of the excited state can be calculated using a Green's-function approach. We show that the existence of a split-off state can play an important role in the total injection probability. In the wide band limit, the survival probability decays exponentially, but for finite band widths it does not. We further investigate the effect of vibrations on the process. A Green's operator technique may be used to solve this too exactly. We show that the problem may be reduced to a non-Hermitian eigenvalue problem for the vibrational states alone. Exact results can be obtained for arbitrary bandwidth and for a few vibrational degrees of freedom. In the wide band limit, the dynamics is particularly simple and we find that (1) the survival probability of the excited state is unchanged by the inclusion of vibrational motion, but (2) each vibrational state now has a finite lifetime. Numerical results are presented for the effects of reorganization energy, energy of the injecting level, and the variation of the matrix element for the electron injection, on the survival probability of the electron in the excited state. As an illustration of the approach, we also present results of numerical calculation of the absorption spectrum of perylene adsorbed on TiO(2) and compare it with experimental results.     

Increasing the applicability of density functional theory. II. Correlation potentials from the random phase approximation and beyond

Density functional theory (DFT) results are mistrusted at times due to the presence of an unknown exchange correlation functional, with no practical way to guarantee convergence to the right answer. The use of a known exchange correlation functional based on wave-function theory helps to alleviate such mistrust. The exchange correlation functionals can be written exactly in terms of the density-density response function using the adiabatic-connection and fluctuation-dissipation framework. The random phase approximation (RPA) is the simplest approximation for the density-density response function. Since the correlation functional obtained from RPA is equivalent to the direct ring coupled cluster doubles (ring-CCD) correlation functional, meaning only Coulomb interactions are included, one can bracket RPA between many body perturbation theory (MBPT)-2 and CCD with the latter having all ring, ladder, and exchange contributions. Using an optimized effective potential strategy, we obtain correlation potentials corresponding to MBPT-2, RPA (ring-CCD), linear-CCD, and CCD. Using the suitable choice of the unperturbed Hamiltonian, Kohn-Sham self-consistent calculations are performed. The spatial behavior of the resulting potentials, total energies, and the HOMO eigenvalues are compared with the exact values for spherical atoms. Further, we demonstrate that the self-consistent eigenvalues obtained from these consistent potentials used in ab initio dft approximate all principal ionization potentials as demanded by ionization potential theorem.     

Ab initio MRD-CI study of the spectrum of the TeO molecule employing relativistic effective core potentials

Potential energy curves and properties of the low-lying electronic states of tellurium oxide have been computed using a configuration interaction treatment that includes the spin-orbit coupling interaction. Relativistic effective core potentials (RECPs) are used to describe the inner shells of both the Te and O atoms. Good agreement is obtained for the spectroscopic constants of the X1-X2(3)sigma-, a1delta, and b1sigma+ states for which experimental data are available. The ratio of the parallel and perpendicular b-X transition moments, as well as the radiative lifetime of the b state, was computed, and both results were also found to be in good agreement with measurement. The energetic order of the electronic states in TeO appears to be very similar to that observed for the isovalent O2 molecule, but the Rydberg valence-mixing effects that are so prominent in the latter's spectrum (e.g., for the Schumann-Runge bands) are totally absent in TeO.