Potential Energy Operator Research Articles

Analytic Advantages of Spherically Symmetric Step-Function Potentials in the Dirac Equation with Scalar and Fourth Component of Vector Potential

The root mean square radii of the particle orbits are calculated (semi)analyticallyfor every bound state, using the Dirac equation with a scalar potentialUs andfourth component of a vector potentialUv in the case of a spherically symmetricstep-function shape with the same radiusR for these potentials. In addition, a(semi)analytic expression of the expectation value of the corresponding potentialenergy operator is derived. For the above quantities, expressions of the energyeigenvalues in terms of the potential parameters are needed and approximateformulas may be used in certain cases. This study emphasizes the analyticadvantages of the relativistic, spherically symmetric step-function potential model.Its applicability is discussed in connection with a problem of physical interest,namely that of the motion of a Λ particle in hypernuclei.

Non-Markovian evolution of the density operator in the presence of strong laser fields

We present an accurate, efficient, and flexible method for propagating spatially distributed density matrices in anharmonic potentials interacting with solvent and strong fields. The method is based on the Nakajima–Zwanzig projection operator formalism with a correlated reference state of the bath that takes memory effects and initial/final correlations to second order in the system–bath interaction into account. A key feature of the method proposed is a special parametrization of the bath spectral density leading to a set of coupled equations for primary and N auxiliary density matrices. These coupled master equations can be solved numerically by representing the density operator in eigenrepresentation or on a coordinate space grid, using the Fourier method to calculate the action of the kinetic and potential energy operators, and a combination of split operator and Cayley implicit method to compute the time evolution. The key advantages of the method are: (1) The system potential may consist of any number of electronic states, either bound or dissociative. (2) The cost for arbitrarily long solvent memories is equal to only N+1 times that of propagating a Markovian density matrix. (3) The method can treat explicitly time-dependent system Hamiltonians nonperturbatively, making the method applicable to strong field spectroscopy, photodissociation, and coherent control in a solvent surrounding. (4) The method is not restricted to special forms of system–bath interactions. Choosing as an illustrative example the asymmetric two-level system, we compare our numerical results with full path-integral results and we show the importance of initial correlations and the effects of strong fields onto the relaxation. Contrary to a Markovian theory, our method incorporates memory effects, correlations in the initial and final state, and effects of strong fields onto the relaxation; and is yet much more effective than path integral calculations. It is thus well-suited to study chemical systems interacting with femtosecond short laser pulses, where the conditions for a Markovian theory are often violated.

A pseudospectral algorithm for the computation of transitional-mode eigenfunctions in loose transition states. II. Optimized primary and grid representations

A highly optimized pseudospectral algorithm is presented for effecting the exact action of a transitional-mode Hamiltonian on a state vector within the context of iterative quantum dynamical calculations (propagation, diagonalization, etc.). The method is implemented for the benchmark case of singlet dissociation of ketene. Following our earlier work [Chem. Phys. Lett. 243, 359 (1995)] the action of the kinetic energy operator is performed in a basis consisting of a direct product of Wigner functions. We show how one can compute an optimized (k,Ω) resolved spectral basis by diagonalizing a reference Hamiltonian (adapted from the potential surface at the given center-of-mass separation) in a basis of Wigner functions. This optimized spectral basis then forms the working basis for all iterative computations. Two independent transformations from the working basis are implemented: the first to the Wigner representation which facilitates the action of the kinetic energy operator and the second to an angular discrete variable representation (DVR) which facilitates the action of the potential energy operator. The angular DVR is optimized in relation to the reference Hamiltonian by standard procedures. In addition, a scheme which exploits the full sparsity of the kinetic energy operator in the Wigner representation has been devised which avoids having to construct full-length vectors in the Wigner representation. As a demonstration of the power and efficiency of this algorithm, all transitional mode eigenstates lying between the potential minimum and 100 cm−1 above threshold have been computed for a center-of-mass separation of 3 Å in the ketene system. The performance attributes of the earlier primitive algorithm and the new optimized algorithm are compared.

A five-dimensional form of the Dirac equation

A Dirac equation in a covariant form with respect to proper orthochronous rotations in (4 + 1)-dimensional pseudo-orthogonal space, i.e. Minkowski space extended by one real dimension is introduced. It contains a five-vector potential with a non-electromagnetic fifth component. The invariance of this equation under the CPT transformation is conditioned by the assumption that the real fifth coordinate changes its sign under charge conjugation, and that it simultaneously changes its sign either under time reversal or under space inversion. The energy levels of an electron under the simultaneous action of Coulomb and central gravitational fields are determined. To this end, (1) new eigenspinors of the total angular momentum operator are derived, with non-zero entries in the first and fourth or in the second and third row of the column matrix and (2) a scalar function is constructed from doubly-periodic Jacobian elliptic functions which, in the limit of the vanishing modulus of the elliptic functions, replaces the function exp(i t) in the stationary-state solutions. The iterated five-dimensional equation contains the ten components of the antisymmetric field tensor. It also contains a term determining the potential energy operator of electron spin density in a non-electromagnetic field. The Pauli equation is derived from the five-dimensional equation, with the transformational characteristics of the original equation. It contains a spin-orbit coupling term depending on the non-electromagnetic potential.

Two-dimensional excitons and hydrogenlike impurities on a curved surface

The shifts and splittings are found for a two-dimensional donor and a Wannier-Mott exciton on a curved surface. The appearance of fine structure in the excitonic luminescence line as a result of the motion of the system as a whole in a potential of geometric origin is estimated. The relative contributions of the Coulomb potential and kinetic energy operator to the corrections for curvature depend strongly on the geometric form of the surface.

The Mechanism of the Anomalous Energy Shift between s-States in Mirror Nuclei with a Halo Structure

A mechanism of the anomalous energy shift between s-states in mirror nuclei is investi­ gated by using the complex scaling method (CSM). The s-states are described by a resonant core+p system and by a loosely bound core+n system, where the latter reproduces a neu­ tron halo structure. To understand the mechanism of the anomalous energy shift appearing in the resonant s-state of the core+p system, we pick out the complex expectation values of the Coulomb potential and kinetic energy operators from the Hamiltonian for resonant states, using the procedure recently developed in the framework of the CSM. We find that the kinetic energy reduction is large, in addition to the Coulomb energy reduction in the ordinary Thomas-Ehrman shift. As a result, a large energy shift is reasonably explained by the combination of two mechanisms of the kinetic energy reduction and the Coulomb energy reduction.

A heuristic screening approach for atoms includingl-splitting

TheZ-expansion theory of Atomic Structure is supplemented with the inclusion of the external screening concept, calculated from a simplified version of the works of M. Kregar, where the two-body potential energy operators are split into the sum of effective one-body operators. The use ofZ- andN-dependent screening parameters instead of screening constants, gives a simplified Screened Hydrogenic Model which compares favourably with more sophisticated methods, specially for ionized atoms.

Application of the quantum mechanical hypervirial theorems to even-power series potentials

The class of the even-power series potentials,V(r)=-D+∑ k-0 ∞ Vkλkr2k+2,V 0=ω2>0is studied with the aim of obtaining approximate analytic expressions for the nonrelativistic energy eigenvalues, the expectation values for the potential and kinetic energy operators, and the mean square radii of the orbits of a particle in its ground and excited states. We use the hypervirial theorems (HVT) in conjunction with the Hellmann-Feynman theorem (HFT), which provide a very powerful scheme for the treatment of the above and other types of potentials, as previous studies have shown. The formalism is reviewed and the expressions of the above-mentioned quantities are subsequently given in a convenient way in terms of the potential parameters, the mass of the particle, and the corresponding quantum numbers, and are then applied to the case of the Gaussian potential and to the potentialV(r)=−D/cosh2(r/R). These expressions are given in the form of series expansions, the first terms of which yield, in quite a number of cases, values of very satisfactory accuracy.

Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theoretically and computationally insightful. © 1996 John Wiley & Sons, Inc.

Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theoretically and computationally insightful. © 1996 John Wiley & Sons, Inc.

On the use of a Hamiltonian with projected potential for the calculation of scattering wave functions: Method and general properties

The theoretical framework of a method that utilizes a projected potential operator to construct scattering wave functions is presented. Theorems and spectral properties of a Hamiltonian with the potential energy operator represented in terms ofL2(ℛ3) are derived. The computational advantages offered by the method for calculating spectroscopic quantities, like resonance energies, decay probabilities and photoionization cross-sections, are discussed.

The vibrational structure of H+4 and D+4

A potential energy surface (PES) for the H+4 system which is a fit to local, high quality ab initio multiple reference single and double excitations configuration interaction (CI) calculations (142 data points) is reported. The potential energy surface obtained here has been calculated by maintaining H+3 as a core in its equilibrium geometry and moving the remaining hydrogen atom around it (three-dimensional potential energy surface). The new surface supposes an improvement on the preceding potential energy surfaces considering both the basis set size and the ab initio method used here. The energy minimum of the potential presented here has been found to be about 2.5 kcal/mol lower than those obtained in previous studies, indicating that H+4 and D+4 are more stable ions than previously believed. A three-dimensional vibrational kinetic energy operator in internal coordinates without singularities has been derived. Energy and wave functions of the vibrational levels of the dissociating hydrogen in H+4 and D+4 systems have been calculated by using the derived potential and kinetic energy operators and integrating the vibrational Hamiltonian with the normal coordinates finite elements method. The vibrational states with energies below the new dissociation limit are reported and characterized, giving a more complete description of the vibrational structure. The number of bound vibrational levels obtained here is 7 for H+4 and 24 for D+4.

The evaluation of matrix elements in the analysis of anharmonic molecular vibrations: Functional taylor series expansions

AbstractThis article presents methods for computing Cartesian Gaussian matrix elements using a Taylor series for general potential energy operators that admit well‐behaved radial derivatives. These operators arise in the analyses of anharmonic vibrations in molecules. Application to the evaluation of matrix elements for hydrogen associated two wells illustrates the method. © 1995 John Wiley & Sons, Inc.

Self‐consistent methods for the treatment of low‐lying molecular vibrations

AbstractThe subject of this article is the self‐consistent‐field (SCF) treatment of low‐lying molecular vibrations in molecules subject to solvent effects and light atom migration. The analyses use a Cartesian Gaussian basis and Gaussian functional expansions of potential energy operators. The objective of the work was to establish approximate and practical methods of analysis of vibrational degrees of freedom in molecules that build on and compare well with the highly accurate treatments of vibrations in small molecular systems of the past decade. An application to a system in which hydrogen bonding contributes the major anharmonic effect illustrates the method. © 1995 John Wiley & Sons, Inc.

The evaluation of matrix elements in the analysis of anharmonic molecular vibrations: Optimized expansions and quadratures

AbstractThis article presents methods for computing matrix elements with Cartesian Gaussian wave functions of potential energy operators that depend on functions of the form (r−r0)n exp[−a(r − r0)] as well as matrix elements of the class of polynomial many‐body potentials developed by Murrell et al. The matrix elements arise in the analyses of anharmonic vibrations in molecules. © 1995 John Wiley & Sons, Inc.

The Fourier-grid formalism: philosophy and application to scattering problems using R-matrix theory

The Fourier-grid (FG) method is a recent L2 variational treatment of the quantum mechanical eigenvalue problem that does not require the use of a set of basis functions: it is rather a discrete variable representation approach. The author restates the FG philosophy in more general terms, examine and compare this method with other approaches to the eigenvalue problem, and begin the development of an FG R-matrix method for scattering. The philosophy of the FG method is to use the simplest representation for each of the kinetic and potential energy operators of the Hamiltonian, and use a generalized Fourier transform to put the matrix elements of one of the above operators in the same representation as the other, so the Hamiltonian has a single representation. Thus, the Hamiltonian is represented at discrete points in either configuration or its reciprocal space.

Pseudospectral method for solving the time-dependent Schrödinger equation in spherical coordinates

In this paper we describe a numerically efficient pseudospectral method for solving the time-dependent Schrödinger equation in spherical coordinates. In this method the translational kinetic energy operator is evaluated with a Fourier transform. The angular dependence of the wave function is expanded on a two-dimensional grid in coordinate space and the angular part of the Laplacian is evaluated by a Gauss–Legendre–Fourier transform between the coordinate and conjugate angular momentum representations. The potential energy operator is diagonal. Calculations performed for a model system representing H2 scattering from a static corrugated surface yield transition probabilities identical to those obtained with the close coupled wave packet (CCWP) method. The new algorithm will be more efficient than the CCWP method for problems in which a large number of rotational states are coupled.

Intramolecular dynamics. I. Curvilinear normal modes, local modes, molecular anharmonic Hamiltonian, and application to benzene

The Hamiltonian based on curvilinear normal modes and local modes (CNLM) is discussed using Wilson’s exact vibrational Hamiltonian as basis, the CNLM representation diagonalizing only the normal mode block of FG matrix in curvilinear internal coordinates. Using CNLM the kinetic and potential energy operators for benzene are given, including cubic and quartic anharmonicity in the potential energy and cubic and quartic terms in the kinetic energy expansion in curvilinear coordinates. Using symmetrized coordinates and cubic and higher force constants the number and identity of the independent symmetry allowed (A1g) such force constants are obtained. The relation to conventional anharmonic force constants is then given and the allowed contributions of the latter are obtained. The results are applied to CH overtone spectra and intramolecular vibrational dynamics in Part III of this series.

Position moments linearly averaged over Hartree-Fock orbitals.

Linearly averaged position moments (LAPM's) are introduced for the Hartree-Fock orbitals. When all the LAPM's are well defined, the Hartree-Fock equation is shown to be equivalent to a set of relations between the LAPM's involving the potential-energy operator. The true Hartree-Fock orbitals must satisfy all the relations, and hence the LAPM equations can be used as a sensitive criterion to assess the accuracy of approximate Hartree-Fock orbitals. Such an application is presented for He, Be, and Ne atoms.

Anisotropic effective core potentials

We present a convenient method for accounting for the anisotropy of partially filled d shells that are incorporated in effective core potentials by including the leading anisotropy term of the d-type electron density, which is the quadrupole moment, as an electrostatic potential energy operator in the model Hamiltonian. We present sample calculations on the cobalt hydride and dicobalt systems. We find the quadrupole anisotropy to have a very large effect in the distance regime of CoH. In the dicobalt system, which has a relatively long internuclear distance, the quadrupole anisotropy shifts the equilibrium bond length by nearly 0.02 bohr.