A Note on the Symmetry Properties of Löwdin's Orthogonalization Schemes

2008 ◽  
Vol 73 (6-7) ◽  
pp. 937-944 ◽  
Author(s):  
András T. Rokob ◽  
Ágnes Szabados ◽  
Peter R. Surján
Keyword(s):  
Point Group ◽  
Symmetry Operator ◽  
Basis Set ◽  
Symmetry Relation ◽  
Symmetry Properties ◽  
The Matrix

We point out that the well-known symmetry properties of the symmetrically and canonically orthogonalized vectors hold only under certain conditions on the overlapping vectors. In particular, the matrix of the transformation induced by the symmetry operator must be unitary. This requirement is not fulfilled if Cartesian d or f functions are used in the basis set. If such functions are present, canonically orthogonalized orbitals do not transform according to representations of the molecular point group; nor do Löwdin orthogonalized vectors preserve symmetry relation of the original vectors.

scholarly journals The Symmetry of Linear Molecules

2018 ◽  
Author(s):  
Katy L. Chubb ◽  
Per Jensen ◽  
Sergei N. Yurchenko
Keyword(s):  
Eigenvalue Problems ◽  
Point Group ◽  
Basis Set ◽  
Group Symmetry ◽  
Linear Molecule ◽  
Numerical Application ◽  
Set Functions ◽  
Symmetry Properties ◽  
Linear Molecules ◽  
Representation Transformation

A numerical application of linear-molecule symmetry properties, described by the D∞h point group, is formulated in terms of lower-order symmetry groups Dnh with finite n. Character tables and irreducible representation transformation matrices are presented for Dnh groups with arbitrary n-values. These groups are subsequently used in the construction of symmetry-adapted ro-vibrational basis functions for solving the Schrödinger equations of linear molecules as part of the variational nuclear motion program TROVE. The TROVE symmetrisation procedure is based on a set of "reduced" vibrational eigenvalue problems with simplified Hamiltonians. The solutions of these eigenvalue problems have now been extended to include the classification of basis-set functions using ℓ, the eigenvalue (in units of ℏ) of the vibrational angular momentum operator L ^ z . This facilitates the symmetry adaptation of the basis set functions in terms of the irreducible representations of Dnh. 12C2H2 is used as an example of a linear molecule of D∞h point group symmetry to illustrate the symmetrisation procedure.

Microdiffraction analysis of precipitate crystallography and morphology in Al alloys

1990 ◽  
Vol 48 (4) ◽  
pp. 954-955
Author(s):  
B.C. Muddle ◽  
G.R. Hugo
Keyword(s):  
Point Group ◽  
Hexagonal Lattice ◽  
Intersection Point ◽  
Saed Pattern ◽  
Point Symmetry ◽  
Hexagonal Symmetry ◽  
Precipitate Crystal ◽  
The Matrix ◽  
The Subject ◽  
Electron Microdiffraction

Electron microdiffraction has been used to determine the crystallography of precipitation in Al-Cu-Mg-Ag and Al-Ge alloys for individual precipitates with dimensions down to 10 nm. The crystallography has been related to the morphology of the precipitates using an analysis based on the intersection point symmetry. This analysis requires that the precipitate form be consistent with the intersection point group, defined as those point symmetry elements common to precipitate and matrix crystals when the precipitate crystal is in its observed orientation relationship with the matrix.In Al-Cu-Mg-Ag alloys with high Cu:Mg ratios and containing trace amounts of silver, a phase designated Ω readily precipitates as thin, hexagonal-shaped plates on matrix {111}α planes. Examples of these precipitates are shown in Fig. 1. The structure of this phase has been the subject of some controversy. An SAED pattern, Fig. 2, recorded from matrix and precipitates parallel to a <11l>α axis is suggestive of hexagonal symmetry and a hexagonal lattice has been proposed on the basis of such patterns.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

scholarly journals DEVELOPMENT OF MOLECULAR IMPRINTED POLYMER SOLID PHASE EXTRACTION (MISPE) FOR SEPARATION NITROFURANTOIN RESIDUE IN CHICKEN EGGS

2017 ◽  
Vol 10 (6) ◽  
pp. 108
Author(s):  
Sophi Damayanti ◽  
Untung Gunawan ◽  
Slamet Ibrahim
Keyword(s):  
Solid Phase Extraction ◽  
Analytical Methods ◽  
Density Functional ◽  
Solid Phase ◽  
Basis Set ◽  
Phase Extraction ◽  
Molecular Imprinted Polymer ◽  
Imprinted Polymer ◽  
Dft Methods ◽  
The Matrix

Background: The use of nitrofurantoin and other nitrofuran antibiotics in food which produced from animals is prohibited by European Union because of potentially carcinogenic and mutagenic. Various methods for analysis of residues of nitrofurantoin has been developed, but because of the interference of the matrix, it is necessary to separate the matrix therefore, the matrix effect will not interfere the analysis. Nowadays, molecular imprinted polymer (MIP) is a well-developed tool in the analytical field, mainly for separating substances in relatively complex matrices.Objective: The purpose of this study is to obtain MISPE that is selective for the separation of nitrofurantoin residues in chicken eggs.Methods: Analytical methods development of nitrofurantoin were optimization of HPLC system and validation of analytical methods performed to obtain the suitable system for nitrofurantoin detection. In silico study used for MIP design by observing the difference Gibbs free energy using Gaussview 5.08 software with Density Functional Theory (DFT) methods using 6-311G as basis set. MIP synthesis was done using bulk method use nitrofurantoin as template, acrylamide as functional monomer, ethyleneglycoldimethacrylate (EGDMA) as crosslinker, and azobisisobutyronitrile (AIBN) as an initiator reaction inside dimethylformamide (DMF) as solvent. Non imprinted polymer (NIP) was synthesized as comparison. MIP and NIP which has been synthesized was inserted into SPE cartridge and characterized using Infrared spectroscopy and HPLC.Result: MISPE that has been synthesized was characterized and compared to non-imprinted polymer solid phase extraction (NISPE) and marketed Solid Phase Extraction (SPE) C18. Sensitivity of MIP, NIP, and SPE C-18 to nitrofurantoin was 84.54 %, 37.73 %, and 33.95 % respectively, based on recovery of nitrofurantoin.Conclusion: Based on the result it was obtained MISPE has high selectivity toward nitrofurantoin compared to NISPE and either marketed SPE.  

scholarly journals Conformational isomerism of 1-butene secondary ozonide as studied by means of matrix isolation infrared absorption spectroscopy

2012 ◽  
Vol 10 (5) ◽  
pp. 1647-1656
Author(s):  
Simona Strazdaite ◽  
Ruta Bariseviciute ◽  
Justinas Ceponkus ◽  
Valdas Sablinskas
Keyword(s):  
Absorption Spectroscopy ◽  
Infrared Absorption ◽  
Theoretical Calculations ◽  
Matrix Isolation ◽  
Dft Method ◽  
Basis Set ◽  
Infrared Absorption Spectroscopy ◽  
Spectral Bands ◽  
Secondary Ozonide ◽  
The Matrix

AbstractTheoretical calculations of structures, stability and vibrational spectra of 1-butene secondary ozonide (SOZ) conformers were performed using DFT method B3LYP with a 6-311++G(3df, 3pd) basis set. The calculations predict six staggered structures of 1-butene SOZ. The FTIR spectra of 1-butene SOZ isolated in Ar, N2 and Xe matrices were recorded. It was found that nitrogen is the best suited for the matrix isolation of 1-butene SOZ. The bandwidth of the spectral bands of the ozonide isolated in nitrogen was as narrow as 2 cm−1. For the first time the existence of five conformers of 1-butene SOZ were confirmed experimentally by means of matrix isolation infrared absorption spectroscopy. The equatorial gauche (∠OCCC=−66.1°) conformer was proved theoretically and experimentally to be the most stable. It was found that due to high potential barriers of the conformational transitions annealing of the matrix is useless for the assignment of spectral bands to various conformers of 1-butene SOZ. Using the hot nozzle technique the van’t Hoff experimental plots were made for three additional conformers of 1-butene SOZ and experimental ΔH values for these additional conformers were established. The crystallization problems of 1-butene SOZ are discussed which accounts for the rich conformational diversity of the ozonide as well as high conformational barriers for axial-equatorial transitions.

Molecular orbitals in momentum space

1965 ◽  
Vol 286 (1406) ◽  
pp. 376-389 ◽  
Keyword(s):  
Momentum Space ◽  
Wave Functions ◽  
Basis Set ◽  
Equilibrium Distance ◽  
Numerical Work ◽  
Algebraic Expressions ◽  
Unitary Transformations ◽  
The Matrix ◽  
Internuclear Distances ◽  
Ground State Energies

The troublesome problem of developing cusps in ordinary molecular wave functions can be avoided by working with momentum-space wavefunctions for these have no cusps. The need for continuum wavefunctions can be eliminated if one works with a hydrogenic basis set in Fock’s projective momentmn space. This basis set is the set of R 4 spherical harmonics and as a consequence one may obtain, solely by the ordinary angular momentum calculus, algebraic expressions for all the integrals required in the solution of the momentum space Schrödinger equation. A number of these integrals and a number of R 4 transformation coefficients are tabulated. The method is then applied to several simple united-atom and l.c.a.o. wavefunctions for H + 2 and ground state energies and corrected wavefunctions are obtained. It is found in this numerical work that the method is most appropriate at internuclear distances somewhat less than the equilibrium distance. In Fock’s representation both l.c.a.o. and unitedatom approximations become exact as the internuclear distance approaches zero. The united-atom expansion can be viewed as an eigenvalue equation for the root-mean-square momentum, p 0 = √( — 2 E ). In the molecule, the matrix operator corresponding to p 0 is related to the operator for the united-atom by a sum of unitary transformations, one for each nucleus in the molecule.

Polar structures in the theory of conjugated molecules. II. Symmetry properties and matrix elements for polar structures

1950 ◽  
Vol 200 (1062) ◽  
pp. 390-401 ◽  
Keyword(s):  
Valence Bond ◽  
Matrix Elements ◽  
Conjugated Molecules ◽  
Energy Calculations ◽  
Valence Bond Theory ◽  
Energy Parameters ◽  
Symmetry Properties ◽  
The Matrix ◽  
Bond Theory ◽  
Usual Formalism

The inclusion of polar structures in the valence-bond theory of π-electrons entails some additions to the usual formalism, and these are given in this part. The symmetry properties of sets of structures, both non-polar and polar, and the matrix elements that come into energy calculations, are dealt with. Using the work of part I, and the conclusions of this part, the energy parameters for work with polar structures are evaluated.

scholarly journals Utilization of the Spin Symmetry in Fitting the Magnetic Data lor Large Exchange Clusters

2013 ◽  
Vol 12 (1) ◽  
pp. 56-61
Author(s):  
Roman Boča
Keyword(s):  
Energy Levels ◽  
Spin Symmetry ◽  
Basis Set ◽  
Magnetic Data ◽  
Coupled Clusters ◽  
Spin Quantum ◽  
Desktop Computers ◽  
Spin Quantum Number ◽  
The Matrix ◽  
Lower Size

Abstract The Heisenberg Hamiltonian appropriate to exchange clusters commutes with the square of the total spin ant its third component. Therefore in studying the exchange coupled clusters of medium/high nuclearity the spin quantum number S can be utilized in factoring of large interaction matrices (dimension of which is 104 - 105). Then the blocks of much lower size can be diagonalized using the desktop computers. To this end, the eigenvalues form the partition function Z(T,B) from which all thermodynamic properties, including the magnetization M(B,T0) and the magnetic susceptibility χ(T,B0), can be reconstructed. The matrix elements of the interaction operators in the coupled basis set of spin kets have been generated with the help of the irreducible tensor operators for a loop for S = Smin until S = Smax. In addition to the modelling of energy levels for different topologies, a fitting of magnetic data is exemplified by a number of examples like [Fe6] and [Mn3Cr4] systems

scholarly journals Duality Identities for Moduli Functions of Generalized Melvin Solutions Related to Classical Lie Algebras of Rank 4

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
S. V. Bolokhov ◽  
V. D. Ivashchuk
Keyword(s):  
Lie Algebras ◽  
Toda Chain ◽  
Scalar Fields ◽  
Cartan Matrix ◽  
Radial Coordinate ◽  
Cylindrically Symmetric ◽  
Symmetry Properties ◽  
The Matrix ◽  
Chain Type ◽  
Internal Symmetries

We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank 4 (namely, A4, B4, C4, and D4) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically symmetric gravitational configuration in D dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions Hs(z) (s=1,…,4) of squared radial coordinate z=ρ2 obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers (n1,n2,n3,n4)=(4,6,6,4),(8,14,18,10),(7,12,15,16),(6,10,6,6) for Lie algebras A4, B4, C4, and D4, respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued 4×4 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A4 case) the matrix representing a generator of the Z2-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances. We also calculate 2-form flux integrals over 2-dimensional discs and corresponding Wilson loop factors over their boundaries.

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