Analytical block-diagonalization of matrices using unit spherical tensors

A new method for block-diagonalizing large Hamiltonian matrices, in closed form, is described. The method is based on (i) a general unitary transformation due to Slichter, and (ii) Fano's unit spherical operatorsÛ _Q ^K (I _i,I _i). The method is illustrated with a simple three spin 1/2 dipolar coupled spin system, characterized by off-block-diagonal unit spherical tensorsÛ ₀ ² (3/2,1/2,) andÛ ₀ ² (3/2,1 /2,). In addition, it is pointed out that any Hamiltonian matrix can be re-labelled in terms of fictitious spin labels, enabling a wide variety of unit spherical tensors to be used in block-diagonalization. For example, a single spin 5/2 matrix can be re-labelled using three spin labels 1/2, 1/2, and 1/2, respectively. Thus the tensor algebra required to block-diagonalize a 6 x 6 matrix is determined solely by the properties of the Pauli spin matrices. Finally, it is shown that re-labelling within the unit spherical tensor framework provides a unifying framework for standard basis operators, fictitious spin 1/2 and 1 operators, and others. The fictitious spin 1 / 2 unit spherical operators discussed in this paper differ from those of Vega and Pines.     

A finite field method for calculating molecular polarizability tensors for arbitrary multipole rank

A finite field method for calculating spherical tensor molecular polarizability tensors α_lm;l′m′ = ?Δ_lm/??_l′m′* by numerical derivatives of induced molecular multipole Δ_lm with respect to gradients of electrostatic potential ?_l′m′* is described for arbitrary multipole ranks l and l′. Interconversion formulae for transforming multipole moments and polarizability tensors between spherical and traceless Cartesian tensor conventions are derived. As an example, molecular polarizability tensors up to the hexadecapole–hexadecapole level are calculated for water using the following ab initio methods: Hartree–Fock (HF), Becke three‐parameter Lee‐Yang‐Parr exchange‐correlation functional (B3LYP), Møller–Plesset perturbation theory up to second order (MP2), and Coupled Cluster theory with single and double excitations (CCSD). In addition, intermolecular electrostatic and polarization energies calculated by molecular multipoles and polarizability tensors are compared with ab initio reference values calculated by the Reduced Variation Space method for several randomly oriented small molecule dimers separated by a large distance. It is discussed how higher order molecular polarizability tensors can be used as a tool for testing and developing new polarization models for future force fields. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Measuring distances between half-integer quadrupolar nuclei and detecting relative orientations of quadrupolar and dipolar tensors by double-quantum homonuclear dipolar recoupling nuclear magnetic resonance experiments

We studied the possibility of using double-quantum homonuclear dipolar recoupling magic angle spinning nuclear magnetic resonance experiments for structural analysis of systems of half-integer quadrupolar nuclei. We investigated symmetry-based recoupling schemes R2(2) (1) and R2(2) (1)R2(2) (-1) and showed that the obtained double-quantum filtered signals depend substantially on magnitudes and relative orientations of dipolar and quadrupolar tensors. Experimental results measured on aluminophosphate molecular sieve AlPO(4)-14, containing dipolar-coupled spin-52 aluminum nuclei, were compared to results of time-consuming numerical simulations. The comparison for short mixing times allowed us to roughly measure internuclear Al-Al distances, if constraints about relative tensor orientations were available. Inspection of relative orientations of dipolar and quadrupolar tensors, using known distances between nuclei, required experimental and simulated data for long mixing times and yielded less accurate results. Two experimental protocols were employed for measuring double-quantum filtered curves, the symmetric protocol, in which excitation and reconversion periods are incremented simultaneously, and the asymmetric protocol, in which only the length of the excitation period is incremented and the length of the reconversion period is kept constant. The former experimental protocol was more convenient for the detection of internuclear distances, and the latter one was more appropriate for the inspection of relative orientations of interaction tensors.     

Separate-local-field NMR spectroscopy on half-integer quadrupolar nuclei

New approaches to the characterization of resonances in the solid-state NMR spectroscopy of half-integer quadrupolar nuclei are explored, on the basis of the acquisition of heteronuclear separate-local-field spectra on rotating solids. In their two-dimensional version, these experiments correlate for each chemical site a second-order quadrupolar MAS powder pattern with the dipolar MAS sideband pattern to nearby heteronuclei. As 3D NMR sequences, such 2D anisotropic correlation spectra become separated for inequivalent chemical sites along a third, isotropic dimension. Extending in such manner separate-local-field NMR approaches to quadrupoles facilitates the assignment of inequivalent resonances to specific structural environments, and provides new tools for the investigation of dynamics in solids. Details about these 2D and 3D NMR experiments are given, and their application is illustrated with 1H-23Na recoupling experiments on mononucleotides possessing multiple bound cations.     

Fifth-rank molecular polarization tensors

Fifth-rank molecular polarization tensors are discussed in terms of their tensor group which includes both the spatial symmetry and the permutation symmetry of the suffixes. This latter may be described by certain four-dimensional point groups, which in the nontotally symmetric cases may be projected into three-dimensional space. The study leads to a determination of the number of independent components of these tensors and a discussion of the types of phenomena which they may be expected to produce.     

Spin-spin coupling tensors in fluoromethanes

All spin-spin coupling tensors J of the fluoromethanes CH3F, CH2F2, and CHF3 are obtained theoretically by multiconfiguration self-consistent field linear response (MCSCF LR) ab initio calculations. Furthermore the principal values and the orientation of the principal axis systems of each theoretical J tensor are specified. Experimental liquid crystal NMR (LC NMR) data on the tensorial properties of the CF spin spin coupling in CH3F and CH2F2, and the FF spin-spin coupling in CHF3 are also reported. In the analysis of the experiments, the contributions from molecular vibrations, as well as that of the correlation of vibrational and rotational motion to the experimental anisotropic couplings, D(exp), are taken into account. The information of the anisotropic indirect coupling, 1/2J(aniso), is detected as the difference between D(exp) and the calculated dipolar coupling, D(calc). The extracted indirect contributions, 1/2J(aniso), are in fair agreement with the ab initio results. All relative (experimental and theoretical) CF and FF indirect contributions, 1/2J(aniso)/D(exp), are negative and under 1.7% in magnitude, when the observed molecular orientations are used. Therefore, in the one bond CF couplings and in the two bond FF couplings, the indirect contribution can normally be ignored without introducing serious error to the determination of molecular orientation and/or structure. However, a more accurate method is to partially correct for the indirect contribution by utilising the transferability of the spin-spin coupling tensors in related molecules. This is due to the fact that even small contributions may be significant, if the order parameter of the internuclear direction is negligibly small, leading to dominating indirect contributions. The very good agreement of the experimental values with the calculated coupling constants and the reasonable agreement in the anisotropic properties, which are experimentally much more difficult to define, indicates that the MCSCF LR method is capable of producing reliable J tensors for these systems, contrary to the case of density-functional theory.     

Alternative equivalent icosahedral irreducible tensors

Some sets of icosahedral irreducible tensors can only be defined with respect to one of two seemingly equivalent choices of axes. The problem is explained and the cases when such an alternative choice may cause mixing of sets belonging to different irreducible representations are determined. It is found that the amount of mixing is a property of the spherical basis sets and is independent of the rank and of the permutation symmetry of the tensorial sets. It is further found that certain operators which change the handedness of the coordinate axes can similarly affect these sets, notwithstanding the centrosymmetry of the icosahedron.     

Icosahedral irreducible tensors and their applications

After a discussion of different concepts of anisotropy with reference to the icosahedron, the unusual problems surrounding the construction of icosahedral irreducible tensors are discussed. Tables of these tensors are then presented and applied to orbital problems, molecular polarization phenomena and to the construction of projection operators.     

The symmetry of molecular polarization tensors

Polarization tensors are discussed in terms of their intrinsic symmetry group which is a direct product of the point group and the subgroup of the permutation group relevant to the experiment. The study of these latter groups is simplified by use of the isomorphism with certain point groups and permutations of suffixes can be visualized by rotations and reflections of the vertices of various objects in space. The approach unites the previous treatments and provides a means of constructing the bases for the irreducible tensor components. The difficulties introduced by Laplace's equation are explained and the information obtainable from induced birefringence experiments (Kerr and Cotton–Mouton effects) discussed for various systems.     

Rotational diffusion tensors for non-spherical molecules

The rotational diffusion tensors of several rigid molecules are determined from the rotational friction tensor. The molecular friction tensor is modeled as a sum over the translational Stokes law friction coefficients of the constituent atoms of the molecule. Agreement of the theory with experimental rotational diffusion constants and onentational correlation times is surprisingly good.     

Local Raman tensors in adenosine triphosphoric acid

A polarized laser (488.0 nm) Raman spectroscopic measurement has been made on a single crystal of a disodium salt of adenosine triphosphoric acid (Na₂ATP·3H₂O) by the use of a Raman microscope. The crystal belongs to an orthorhombic system of the space group P2₁2₁2₁, and has dimensions of 10, 100 and 100 μm along the crystallographic axes a, b and c, respectively. For each Raman band in the 300–1800 cm⁻¹ range, the scattering intensity ratio I _bb/I _cc of the bb and cc polarization components has been determined. For a few bands, the relative intensities of the bc components were also estimated. To augment the data, the depolarization ratio of each Raman band of ATP has also been determined for its acidic (pH = 2.42) H₂O and D₂O solutions. From these experimental results, the shapes and orientations of the Raman scattering tensors which are considered to be localized in the adenine-H⁺ portion and in the phosphate portion of the molecule have been derived.     

Proton shielding tensors in potassium hydrogen maleate

The proton NMR in single crystals of potassium hydrogen maleate has been sttudied by means of multiple-pulse line-narrowing techniques. The magnetic shielding tensors of all magnetically inequivalent protons in the unit cell could be determined independently. Two of these protons are carboxylic, forming hydrogen bonds. The orientations of the shift tensors are consistent with the position of the hydrogens at the midpoints of the 0–0 intervals. The range of anisotropy of 32 ppm, found for the shift tensor of the caboxylic hydrogen, is larger than that found for hydrogen bonds in acids and seems to be characteristics of acidic salts.The other protons in the unit cell are olefinic. Two features distinguish this type of protons from those studied so far: (1) The magnetic shielding tensor is not even approximately axially symmetric, the principal values being ?2.4, ?5.1, ?7.3 ± 05 ppm (from adamantane); and (2) the principal directions reflect all characteristic directions of the carboncarbon double bond (while the CH direction is of no importance). The principal value in the direction perpendicular to the sp² system is the least shielded one.     

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.     

Multiple alignment tensors from a denatured protein

The structural content of the denatured state has yet to be fully characterized. In recent years, large residual dipolar couplings (RDCs) from denatured proteins have been observed under alignment conditions produced by bicelles and strained polyacrylamide gels. In this report, we describe efforts to extend our picture of the residual structure in denatured nuclease by measuring RDCs with multiple alignment tensors. Backbone amide 15N-1H RDCs were collected from 4 M urea for a total of eight RDC data sets. The RDCs were analyzed by singular value decomposition (SVD) to determine the number of independent alignment tensors present in the data. On the basis of the resultant singular values and propagated error estimates, it is clear that there are at least three independent alignment tensors. These three independent RDC datasets can be reconstituted as orthogonal linear combinations, (OLC)-RDC datasets, of the eight actually recorded. The first, second, and third OLC-RDC datasets are highly robust to the removal of any single experimental RDC dataset, establishing the presence of three independent alignment tensors, sampled well above the level of experimental uncertainty. The observation that the RDC data span three or more dimensions of the five-dimensional parameter space demonstrates that the ensemble average structure of denatured nuclease must be asymmetric with respect to these three orthogonal principal axes, which is not inconsistent with earlier work demonstrating that it has a nativelike topology.     

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.     

Irreducible tensor operator methods in intermediate-field coupling

The algebra of irreducible tensor operators is developed in the intermediate-field coupling case. The Wigner-Eckart theorem is formulated for a simple irreducible tensor operator as well as for the Kronecker and scalar products of these operators. The expressions required for the calculation of Coulomb repulsion, crystal field splitting, spin-orbit interaction, and Zeeman effect are given in detail. Recent applications to various problems in spectroscopy and magnetism of transition metal compounds are referred to.     

The isotropic invariants of fifth-rank cartesian tensors

An orthonormal set of irreducible fifth-rank tensors having the required permutation symmetry is constructed. Various problems not encountered in the analogous problem for tensors of ranks two, three, four and six are discussed.     

Harmonic oscillator tensors in contact transformation theory

The application of contact transformation theory to the perturbed harmonic oscillator is reexamined in the light of the harmonic oscillator tensors previously presented. It is found that the recasting of the formalism of this problem in terms of harmonic oscillator tensors results in great simplifications, most of which stem from the introduction of the additional algebraic quantum numbers (l, m). The order of magnitude of each fragment of the Hamiltonian is easily recognizable, and the diagonal and nondiagonal parts contained therein are readily identifiable. The determination of the contact transformation operator is reduced to a simple formula. First, an analysis is made for a single mode of vibration, and it is subsequently extended to a multimode case. The perturbed diatomic vibrator is presented as an example.     

Reintroducing anisotropic interactions in magic-angle-spinning NMR of half-integer quadrupolar nuclei: 3D MQMAS

Selective reintroduction of anisotropic interactions such as the chemical shift anisotropy (CSA) and homonucler dipolar (HMD) coupling were implemented in a high-resolution NMR spectroscopy for half-integer quadrupolar nuclei. Rotary resonance recoupling (R(3)) combined with the multiple-quantum magic-angle spinning (MQMAS) in a three-dimensional (3D) experiment provides not only site-specific high-resolution spectra to yield the quadrupolar interaction parameters but also the CSA or HMD interaction parameters. This 3D experiment provides an avenue for the complete local structural information of half-integer quadrupolar nuclei. Three-dimensional MQMAS experiments incorporating R(3) of HMD and CSA interactions were demonstrated on model compounds containing (11)B, (23)Na, and (87)Rb nuclei.     

String defects connecting pairs of half-integer disclinations and tilting transition of a langmuir monolayer

The static and dynamic string defect textures connecting pairs of half-integer disclinations have been observed by Brewster angle microscopy in the solid phase of pentacosadiynoic acid Langmuir monolayers. The static string defect structures have appeared coexisting with two kinds of point disclinations that have four and two black brushes. The use of local laser heating has allowed one to observe kinetics of creation and annihilation of string defects connecting the two-half-integer disclinations in the splitting process of an s = 1 point disclination into fractional disclinations. These kinetics have been analyzed by studying the competition between the orientational elasticity of the molecules and the line tension of the string and the drag force of the disclinations.