Harmonic oscillator tensors. V. The doubly degenerate harmonic oscillator

The step operators of the two-dimensional isotropic harmonic oscillator are shown to be separable into the basis elements of two disjoint Heisenberg Lie algebras. This separability leads to two sets of irreducible tensors, each of which is based upon its associated underlying Heisenberg Lie algebra. The matrix elements of these tensors are evaluated, along with those of some vibrational operators of physical interest. The possibility of other irreducible tensors are discussed and their usefulness is compared with that of those found here. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 343–357, 1998     

Harmonic oscillator tensors. I. The nondegenerate case

It is shown that the Heisenberg Lie algebra of the nondegenerate harmonic oscillator leads to a basis {J₊, J₀, J_?} of LASU (2). The Hamiltonian of the system is proportional to J₀, and the basis elements give rise to irreducible tensors in the associative enveloping algebra of the Heisenberg Lie algebra. The construction of these irreducible tensors is studied with special attention being paid to the case in which they act upon a single vector space spanned by the harmonic oscillator basis functions. A tensor coupling rule is developed, and useful application is made of it in the calculation of general expressions for vibrational operators and their matrix elements. Throughout, the value of the additional algebraic quantum numbers (l, m) is emphasized.     

Harmonic oscillator tensors. IV. A tensorial approach to internal rotations in molecules

A mapping of 2×2 matrices into the space of single boson operators is shown to lead to the angular momentum operators that give rise to irreducible tensors for the harmonic oscillator. The mapping may also be used to define an axis of quantization. A rotation about this axis induces a wave function and Hamiltonian that may be applied to the study of internal rotations in molecules. The example of a molecule containing two coaxial symmetric tops is presented as a case in point. The case of a potential with a high barrier leads to the approximation of an internal rotation as a torsional oscillator and, consequently, to torsional oscillator tensors whose properties are the same as those of the harmonic oscillator. The possibility of studying more complex potentials is discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 305–315, 1997     

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.     

Harmonic oscillator tensors. III: Efficient algorithms for evaluating matrix elements of vibrational operators

Succinct expressions for the matrix elements of various vibrational operators have been derived in the basis of the nondegenerate harmonic oscillator. Among these are the matrix elements of and , which are found to be dependent upon two quantities and their derivatives. Furthermore, the derivative property of the commutator is used to obtain an explicit expression for the derivatives of an operator in terms of its nested commutator with the conjugate momentum. It may be applied to any of the above cases to obtain the matrix representatives of expressions such as the mixed products , for example. In addition, a simple expression for 1/q is given and its derivatives may be evaluated by this commutator technique. Also the matrix elements of a Gaussian-type operator has been evaluated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Magnetizability tensors from auxiliary density functional theory

The working equations for the calculation of the magnetizability tensor in the framework of auxiliary density functional theory with gauge including atomic orbitals (ADFT-GIAO) are derived. Unlike in the corresponding conventional density functional theory implementations the numerical integration of the GIAOs is avoided in ADFT-GIAO. Our validation shows that this simplification has no effect on the accuracy of the methodology. As a result, a reliable and efficient implementation for the calculation of magnetizabilities of systems with more than 1000 atoms and 14 000 basis functions is presented.     

NMR shielding tensors from auxiliary density functional theory

The working equations for the calculation of NMR shielding tensors in the framework of auxiliary density functional theory are derived. It is shown that in this approach the numerical integration over gauge-including atomic orbitals can be avoided without the loss of accuracy. New integral recurrence relations for the required analytic electric-field-type integrals are derived. The computational performance of the resulting formalism permits shielding tensor calculations of systems with more than 1000 atoms and 15,000 basis functions.     

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.     

Connections between perturbation theory and the Van Vleck transformation: Illustrative calculations on the perturbed harmonic oscillator

We continue to examine the connection between perturbation theory and the Van Vleck unitary transformation. Here we illustrate the formalism derived earlier by applying it to compute the stationary states of the perturbed harmonic oscillator. We find that each solution of the traditional Brillouin-Wigner perturbation theory equations gives rise to a different unitary transformation which, when operating on the unperturbed ground state, produces one or the other of the perturbed eigenstates. With any of the perturbed states able to be reached by a unitary transformation on the unperturbed ground state, we advise caution in using approximate solutions of the perturbation equations in general cases, lest an unexpected stationary state be obtained.     

Relativistic spin-orbit effects on hyperfine coupling tensors by density-functional theory

A second-order perturbation theory treatment of spin-orbit corrections to hyperfine coupling tensors has been implemented within a density-functional framework. The method uses the all-electron atomic mean-field approximation and/or spin-orbit pseudopotentials in incorporating one- and two-electron spin-orbit interaction within a first-principles framework. Validation of the approach on a set of main-group radicals and transition metal complexes indicates good agreement between all-electron and pseudopotential results for hyperfine coupling constants of the lighter nuclei in the system, except for cases in which scalar relativistic effects become important. The nonrelativistic Fermi contact part of the isotropic hyperfine coupling constants is not always accurately reproduced by the exchange-correlation functionals employed, particularly for the triplet and pi-type doublet radicals in the present work. For this reason, ab initio coupled-cluster singles and doubles with perturbative triples results for the first-order contributions have been combined in the validation calculations with the density-functional results for the second-order spin-orbit contributions. In the cases where spin-orbit corrections are of significant magnitude relative to the nonrelativistic first-order terms, they improve the agreement with experiment. Antisymmetric contributions to the hyperfine tensor arise from the spin-orbit contributions and are discussed for the IO2 radical, whereas rovibrational effects have been evaluated for RhC, NBr, and NI.     

Vibrational circular dichroism theory: formulation defining magnetic dipole atomic polar tensors and vibrational nuclear magnetic shielding tensors

The theory of vibrational circular dichroism is formulated in terms of magnetic dipole atomic polar tensors, an extension of infrared atomic polar tensors to magnetic dipole transitions. Relations are provided to define a vibrational nuclear magnetic shielding tensor that describes electronic shielding of an external magnetic field for a nucleus.     

Harmonic theory of thermal two-photon absorption in benzene

A correlation function formalism is applied to compute the two-photon absorption spectrum of benzene. Using harmonic Hamiltonians for the ground and excited electronic states, we find that the theory agrees qualitatively with the experimentally observed sparsity of the thermal two-photon absorption spectrum as compared with the single-photon absorption spectrum. An expression for the average vibrational energy in the excited state is derived. We find that cooling of the nascent vibrational energy in the electronically excited state is not as extensive in the two-photon absorption process as compared to the single-photon case.     

Sequence-adapted molecular tensors: Algebraic methods and application to crystal field theory

Tensorial sets adapted to sequences of finite subgroups are applied to the crystal field problem, and a general method for generating sequence-adapted molecular tensors using finite group algebra is formulated. All subgroup sequences of the abstract finite group G(24), isomorphic to the octahedral, O, tetrahedral, T_d, and symmetric, S(4), groups are tabulated with explicit isomorphisms provided. The sequences fall into eight equivalence classes. A catalog of irreducible representations of G(24) adapted to a member of each of the eight sequence classes is given together with the transformations which generate representations adapted to all other sequences. With this data it is possible to systematically generate tensorial sets adapted to any sequence of a realization of G(24). Unitary transformations which adapt conventional forms of first- and second-rank irreducible tensorial sets of the rotation group to the eight sequences of the octahedral group are provided. Forms suitable for use with magnetic fields are included. The problem of a d¹ ion in a trigonal crystal field is treated with sequence-adapted molecular tensors, and the utility of different sequences for descent in symmetry is discussed.     

Accounting for translational invariance in analyzing contributions of perturbation theory to force-constant tensors of molecules

It has been shown that there are an infinite number of modes of representation of force constants in the form of a sum of the contributions of first-order and second-order perturbation theory, depending on the selection of the admixture of translation of the nuclear skeleton as a whole to the displacement of the nuclei. In particular, any force constant can be reprsented in the form of a specially constructed second-order relaxation term (or first-order with zero contribution); a representation has been found in which the first-order contributions for the displacements of all nuclei in the molecule are identical; rules of the sums for the relaxation terms have been derived, relating the changes in the Gel-Mann-Feynman forces on the nuclei in the molecule with displacements of various nuclei. For the electronic ground state of the molecule, the optimal representation of the force constants has been found, in which the minimum in the first-order contribution is combined with the minimum in the absolute magnitude of the second-order relaxation contribution. It has been proven that in the case of an unperturbed electronic state, the first-order contribution to any diagonal force constant is positive.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 21, No. 3, pp. 280–287, May–June, 1985.     

Multinuclear solid-state nuclear magnetic resonance and density functional theory characterization of interaction tensors in taurine

A variety of experimental solid-state nuclear magnetic resonance (NMR) techniques has been used to characterize each of the elements in 2-aminoethane sulfonic acid (taurine). A combination of (15)N cross-polarization magic angle spinning (CPMAS), (14)N ultrawideline, and (14)N overtone experiments enabled a determination of the relative orientation of the nitrogen electric field gradient and chemical shift tensors. (17)O spectra recorded from an isotopically enriched taurine sample at multiple magnetic fields allowed the three nonequivalent oxygen sites to be distinguished, and NMR parameters calculated from a neutron diffraction structure using density functional theory allowed the assignment of the (17)O parameters to the correct crystallographic sites. This is the first time that a complete set of (17)O NMR tensors are reported for a sulfonate group. In combination with (1)H and (13)C MAS spectra, as well as a previously reported (33)S NMR study, this provides a very broad set of NMR data for this relatively simple organic molecule, making it a potentially useful structure on which to test DFT calculation methods (particularly for the quadrupolar nuclei (14)N, (17)O, and (33)S) or NMR crystallography approaches.     

Time-dependent density functional theory for calculating origin-independent optical rotation and rotatory strength tensors

An approach to calculate origin-independent electronic chiroptical property tensors using time-dependent density functional theory (TDDFT) and gauge-including atomic orbital (GIAO) basis sets is evaluated. Computations of origin-dependent optical rotation tensors and of rotatory strengths needed to simulate circular dichroism spectra are presented. The optical rotation tensor computations employ solutions of coupled perturbed Kohn-Sham equations for a dynamic electric field and a static magnetic field. Because the magnetic field is time independent, the GIAO treatment is somewhat simplified compared to a previously reported method, at some added computational cost if hybrid functionals are employed. GIAO rotatory strengths are also calculated, using transition density matrices from a standard TDDFT excitation energy module. A new implementation in the NWChem quantum chemistry package is employed for representative computations of origin-invariant chiroptical response tensors for methyloxirane, norbornenone, and the ketosteroid androstadienone. For the steroid molecule the vibrational structure of the CD spectrum is modeled explicitly by using calculated Franck-Condon factors. The agreement with experiment is favorable.     

Predicting 9Be nuclear magnetic resonance chemical shielding tensors utilizing density functional theory

The structures of a series of beryllium containing complexes have been optimized at the B3LYP/6-31G(d) level and their (9)Be magnetic shielding values have been determined using B3LYP/6-311G+g(2d,p) and the gauge-including atomic orbital (GIAO) method. The calculated chemical shifts are in excellent agreement with experimental values. The performance of a variety of NMR methods (SGO, IGAIM, CSGT) were also examined but were found to be inferior to the GIAO method at the chosen level of theory employed. The theoretical method has been utilized to predict the beryllium chemical shifts of structurally characterized complexes for which no measured (9)Be NMR spectrum exists, and to investigate a literature complex with an unusual (9)Be NMR chemical shift. A new standard for beryllium NMR in nonaqueous solvents has been suggested.     

Effective utilization of off-diagonal hypervirial relations considering diagonal hypervirial relation: Harmonic oscillator case

The effective utilization of hypervirial relations is scrutinized to improve the approximate excited-state functions in the harmonic oscillator system. A new method is presented which simultaneously employs the off-diagonal hypervirial relations with the diagonal hypervirial relation. In order to use these relations effectively, the following points are pointed out: (i) the presented method is useful to get better reasonable results for the excitation energies and the state functions; (ii) the ground state given must satisfy the virial theorem; (iii) in the hypervirial operator used here as x^m, the smaller integers of m's present better results; and (iv) the employment of the comparatively small number of trial basis functions of the type exp (?γ|x|) is sufficient for reproducing the exact excited state. Especially among them, condition (ii) plays an important role. Applying all the proposals to the first and the second excited states, one gets a highly improved excitation energy, state function, and other physical quantities (e.g., transition moment and oscillator strength). The presented method is also found to be more effective than the employment of only the off-diagonal hypervirial relations or the method of the scaling operation.     

Local-scaling transformation version of density functional theory

The local-scaling transformation version of density functional theory (LS-DFT ) is reviewed. It is shown that in the context of LS-DFT it is possible to construct N-representable energy density functionals and that the theory provides systematic ways for calculating strict upper bounds to the exact energies. The importance of the concept of “orbit” in LS-DFT is indicated and several approaches leading to intraorbit and interorbit optimization are discussed. Results of the application of these optimization procedures to the determination of upper bounds for the ground-state energy of the beryllium atom are given. Also, numerical results are reported on the use of local scaling transformations for the direct solution of the Kohn-Sham equations via the density-constrained minimization of the kinetic energy of a noninteracting system. © 1995 John Wiley & Sons, Inc.