Oscillators in one and two dimensions and ladder operators for the Morse and Coulomb problems

We exhibit that the radial eigenfunctions of a 2D-harmonic oscillator (2DHO) may be regarded as 1D-harmonic oscillator (1DHO) matrix elements. From this simple fact and using as a starting point the ladder operators â^± for 1DHO, we obtain ladder operators for 2DHO. Furthermore, by using the relationship between the Coulomb and Morse problems with a 2DHO, we are able to obtain the ladder operators for the former problems without explicitly recurring to the factorization method. Some uses of the technique presented are suggested. © 1997 John Wiley & Sons, Inc.     

Method of computer algebraic calculation of the matrix elements in the second quantization language

An automated method by the algebraic programming language REDUCE3 for specifying the matrix elements expressed in second quantization language is presented and then applied to the case of the matrix elements in the TDHF theory. This program works in a very straightforward way by commuting the electron creation and annihilation operators (a^? and a) until these operators have completely vanished from the expression of the matrix element under the appropriate elimination conditions. An improved method using singlet generators of unitary transformations in the place of the electron creation and annihilation operators is also presented. This improvement reduces the time and memory required for the calculation. These methods will make programming in the field of quantum chemistry much easier. © 1995 John Wiley & Sons, Inc.     

Unitary group approach to the many-electron problem. II. Adjoint tensor operators for U(n)

In this paper we present a derivation of the U(n) adjoint coupling coefficients for the representations appropriate to many-electron systems. Since the states of a many-fermion system are to comprise the totally antisymmetric Nth rank tensor representation of U(2n), the work of this paper enables the matrix elements of the U(2n) generators to be evaluated directly in the U(n) × U(2) (i.e., spin orbit) basis using their transformation properties as adjoint tensor operators. A connection between the adjoint coupling coefficients, as derived in this paper, and the matrix elements of certain (spin independent) two-body operators is also presented. This indicates that in CI calculations, one may obtain the matrix elements of spin-dependent operators from the known matrix elements of certain spin-independent two-body operators. In particular this implies a segment-level formula for the matrix elements of the U(2n) generators in the spin-orbit basis.     

Comparison of diatomics-in-molecules and simple valence-bond potential surfaces for FH2

The method of diatomics-in-molecules (DIM) is applied to the FH₂ system. With spin—orbit interaction neglected, all elements of the 24 × 24 hamiltonian matrix are tabulated as analytic functions of the six diatomic fragment potential curves. It is found that neglect of off-diagonal 8 × 8 blocks in the DIM hamiltonian matrix leads to an energy expression for the ground 1 ²A′ level which is identical to the valence-bond formula used by Blais and Truhlar in dynamical studies of the F + D₂ reaction. The ²A″ excited level from DIM theory is identical to the result derived by Blais and Truhlar, without neglect of the 8 × 8 off-diagonal blocks. The DIM and simple valence-bond energies are compared numerically for noncollinear geometries.     

Improved treatment for matrix elements of spin-dependent operators

In this paper a general method for the evaluation of the matrix elements of spin-dependent operators is proposed to improve the treatment primitively suggesteed by Cooper and Musher. This approach is largely based on the recent results which the present authors have achieved in the representation theory for the inner- and outer-product reduction of the symmetric group. It is shown that the so-called outer-product coupling coefficients (OPCC ) can be used to generalize the method for constructing the irreducible tensor operators of group S_n. Together with the use of inner-product coupling coefficients (IPCC ), an expression for the matrix elements of spin-dependent operators is presented as the product of a Racah coefficient for S_n and a reduced matrix element which can be expressed in terms of IPCC, OPCC , and the related integrals. The treatment for one- and two-electron spin-dependent operators is discussed in detail.     

Inequalities for fermion density matrices

A partial trace over the occupation numbers of all but k states in the density matrix of an ensemble with an arbitrary number of single-particle states is defined as the (reduced) k-state density matrix. This matrix is used to obtain a complete, practical solution to the problem of determining the representability of the diagonal elements of the one- and two-particle (reduced) density matrices. This solution is expressed as a series of linear inequalities involving the density-matrix elements; the inequalities are identical with those derived previously by Davidson and McCrae by a different method. In addition, our method is used to obtain nonlinear, matrix inequalities on the off-diagonal elements of the density matrices.     

Exact calculations of Franck–Condon overlaps and of matrix elements of some potentials by means of the coherent state representation

Franck–Condon overlaps are described as the matrix elements of unitary operators related to the spatial displacement and the frequency shift. They are calculated exactly by means of the coherent state representation. Furthermore, the generalized matrix elements of x^j, e^?αx, and e^?βx₂ between two states with different equilibrium coordinates and frequency are evaluated in the same way.     

Harmonic oscillator tensors. II. angular momentum expressions of matrix elements of vibrational operators

The spatial symmetries of the harmonic oscillator and the recently found irreducible tensors constructed from the associated annihilation and creation operators are exploited to obtain new expressions for the elements of the matrix representatives of several examples of vibrational operators. Since all vibrational operators are expressible in terms of the irreducible tensors, their matrix elements reflect the angular momentum symmetry inherent in them, for the results derived here are in terms of the Clebsch–Gordan coefficients and the isoscalar factors that arise from the couplinig rule of the irreducible tensors. Familiarity with the mathematical properties of these quantities derived from the elementary theory of angular momentum facilitates the evaluation of many vibrational operators that may be of importance in the study of potentials in this basis. In particular, it is shown that the nonvanishing of matrix elements is governed by a law of conservation of angular momentum along the axis of quantization of the nondegenerate harmonic oscillator. © 1993 John Wiley & Sons, Inc.     

Geometry of density matrices. VI. Superoperators and unitary invariance

We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one-particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group-theoretic sense, for the group of unitary transformations of the set of one-electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle-number-conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non-particle-number-conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.     

Configuration interaction matrix elements for atoms using permutation group algebra

A procedure is described for the efficient evaluation of the energy matrix elements necessary for atomic configuration-interaction calculations. With the orbital configurations of an N electron system in spin state S written as the irreducible representations [2^1/2N?S, 1^2S] of the permutation group S( N ), it is possible to evaluate readily the energy matrix elements of a spin-free Hamiltonian expressed in terms of the generators of the unitary group. We show how the use of angular momentum ladder operators permits the effective generation of a basis of eigenstates of ??², ??_z as well as ??² and ??_z, for which the energy matrix elements may be evaluated with ease.     

Analogy between real irreducible tensor operator and molecular two-particle operator

. Molecular matrix elements of a physical operator are expanded in terms of polycentric matrix elements in the atomic basis by multiplying each by a geometrical factor. The number of terms in the expansion can be minimized by using molecular symmetry. We have shown that irreducible tensor operators can be used to imitate the actual physical operators. The matrix elements of irreducible tensor operators are easily computed by choosing rational irreducible tensor operators and irreducible bases. A set of geometrical factors generated from the expansion of the matrix elements of irreducible tensor operator can be transferred to the expansion of the matrix elements of the physical operator to compute the molecular matrix elements of the physical operator. Two scalar product operators are employed to simulate molecular two-particle operators. Thus two equivalent approaches to generating the geometrical factors are provided, where real irreducible tensor sets with real bases are used. Received: 3 September 1996 / Accepted: 19 December 1996     

A general expression for matrix elements of reciprocal powers of the displacement coordinateq in the harmonic oscillator basis

A formula is derived that allows one to determine the matrix elements of an arbitrary integral reciprocal power of the dimensionless displacement coordinate q of the harmonic oscillator from those ofq ^–1 in an exact manner. This relation is obtained from the use of the chain rule and irreducible tensors expressed in terms of the creation and annihilation operators of the harmonic oscillator.     

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

This 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.     

Quantum properties of complete solutions for a new noncentral ring‐shaped potential

We propose a new exactly solvable ring‐shaped potential V(r,θ) = ?(α/r) + (σ/r²) + β cos²θ/(r²sin²θ). The exact bound‐state solutions are presented explicitly. The creation and annihilation operators are established directly from the normalized radial wave functions. We present two important recurrence relations among the diagonal matrix elements and obtain some explicit expressions of mean values of r^k (8 ≥ k ≥ ?11). The exact form of continuum states is also obtained analytically. Comments are made on the calculation formula of phase shifts and the analytical properties of the scattering amplitude. It is interesting to find that the exact form of continuum states will reduce to that of the bound states at the poles of the scattering amplitude. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Generalized spherical functions in the theory of many-electron atoms

A new method for finding non-relativistic and relativistic wave-functions of an electron moving in the field of a nuclear charge in the jj coupling scheme is proposed. It is based on the usage of generalized spherical functions. The mathematical apparatus necessary to find the expressions for matrix elements of the non-relativistic and relativistic energy or electron transition operators is developed. The formulas obtained for these matrix elements are more convenient than those usually used in jj coupling scheme; only their radial integrals and some phase multipliers depend on orbital quantum numbers.     

Determination of diabatic states through enforcement of configurational uniformity

. An electronic structure-based construction of diabatic states from adiabatic states is formulated that is applicable when individual diabatic states contain several dominant configurations. It is accomplished by maximizing the electronic uniformity of the diabatic states with respect to their dominant configurations throughout the entire nuclear coordinate region. The configurations are generated from unambiguously defined diabatization-adapted molecular orbitals. The orthogonal transformation from adiabatic to diabatic states is deduced by an intrinsic analysis of the adiabatic CI coefficients, without calculating matrix elements of additional, derivative or non-derivative operators. The practicality of the method is demonstrated by applying it to the conical intersection region of the 1¹ A ₁ and 2¹ A ₁ states of ozone.     

Matrix elements in configuration interaction calculations

A method is proposed for the calculation of matrix elements among various states of atoms. A set of tensor operators is the only entity in the formalism, and all formulas involve merely the vacuum expectation values of these tensor operators and the recoupling transformation coefficients. Some numerical examples are given for the Coulomb interaction matrix elements.     

Transferable integrals in a deformation-density approach to crystal orbital calculations. I

In the usual ab initio method of calculating molecular orbitals, the number of integrals to be evaluated increases as M⁴, where M is the number of basis functions. In this paper, an alternative method is discussed, where the computation time increases much less violently with the number of basis functions. Matrix elements of the deformation potential are evaluated by Fourier transform methods, while matrix elements of the neutral-atom potential are evaluated by means of transferable integrals. The transferable integrals (moments of the neutral-atom potentials) can be evaluated once and for all and incorporated as input data in computer programs. In an appendix to the paper, a general expansion theorem is discussed. This theorem allows an arbitrary spherically symmetric function to be expanded about another center.     

Detection limits in XPS for more than 6000 binary systems using Al and Mg Kα X‐rays

A simple approach to estimating the detection limits of X‐ray photoelectron spectroscopy (XPS) for any element in any elemental matrix is presented, using the intensity of the background at the expected position for the photoelectron peak to be detected. The approach has been extended to estimate the detection limit for all elements from lithium to bismuth in a similar range of elemental matrices. Using a number of assumptions, it is possible to obtain a reasonable estimate of the background intensity at any electron kinetic energy in the XPS spectrum of an element. Therefore, a detection limit for an arbitrary element homogeneously distributed in that matrix can be estimated. The results show that, although most elements are detectable at about the 1 at.% to 0.1 at.% level, for heavy elements in a light element matrix, the detection limit can be better than 0.01 at.%, whereas for light elements in a heavy element matrix, detection limits above 10 at.% are not uncommon. Two charts detailing the detection limits for all combinations of trace and matrix elements from lithium (Z = 3) to bismuth (Z = 83) are provided for Al Kα and Mg Kα X‐ray sources using a typical hemispherical analyser instrument which provides 10⁶ counts eV for the Ag 3d_5/2 peak from pure silver. These detection limits can be scaled to estimate the detection limits for any given instrument and operating conditions if the intensity of the Ag 3d_5/2 peak from pure silver under those conditions is known. © 2014 Crown Copyright. Surface and Interface Analysis © 2014 John Wiley & Sons Ltd.