Algebraic Hamiltonian Research Articles

Analyzing cyclobutane: A vibrational hamiltonian approach using dynamical U(2) Lie Algebras

This study uses a one-dimensional Lie algebraic framework, precisely adapted to the symmetry of molecules, to investigate the vibrational frequencies of cyclobutane. A vibrational Hamiltonian is also constructed using one-dimensional Morse oscillators, preserving the D2d point group symmetry for each molecule. The analysis compares the calculated fundamental vibrational frequencies with the observed experimental frequencies, resulting in a root mean square deviation of 0.886 cm-1. The result highlights the exceptional precision of the U(2) Lie algebraic Hamiltonian in accurately predicting the vibrational frequencies and their combination bands at the sub-cm-1 level of precision. Specifically, this computational approach has the potential to achieve these results with lower computational costs compared to traditional theoretical approaches. The broader implications of our study suggest that the U(2) Lie algebraic framework can be effectively applied to a wide range of molecular systems, offering significant advantages in fields such as material science, drug design, and environmental monitoring by providing precise and efficient vibrational spectral analyses. KEY WORDS: Vibrational Hamiltonian, Vibrational frequencies, Morse oscillator, Cyclobutane, Lie algebraic method Bull. Chem. Soc. Ethiop. 2024, 38(6), 1887-1896. DOI: https://dx.doi.org/10.4314/bcse.v38i6.29

Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras

We discuss a procedure to determine finite sets M within the commutant of an algebraic Hamiltonian in the enveloping algebra of a Lie algebra g such that their generators define a quadratic algebra. Although independent from any realization of Lie algebras by differential operators, the method is partially based on an analytical approach, and uses the coadjoint representation of the Lie algebra g. The procedure, valid for non-semisimple algebras, is tested for the centrally extended Schrödinger algebras Ŝ(n) for various different choices of algebraic Hamiltonian. For the so-called extended Cartan solvable case, it is shown how the existence of minimal quadratic algebras can be inferred without explicitly manipulating the enveloping algebra.

Vibrational Hamiltonian of tetrachloro-, tetrafluoro-, and mono- silanes using U(2) Lie algebras

In this paper, we have applied the symmetry adapted one-dimensional framework of the U(2) Lie algebras to estimate the vibrational frequencies of tetrachloro-, tetrafluoro-, and mono- silanes in the gas phase having the spectroscopic interest of terrestrial volcanic plumes and other planetary atmospheres. A vibrational Hamiltonian that preserves the Td point group symmetry of each of these silane molecules is devised using ten interacting Morse oscillator bound state spectra. The calculated vibron numbers and locality parameters indicate that the vibrational motion is highly anharmonic in SiH4 (nearest to local mode), moderately anharmonic in SiF4 (mixed mode) and the least anharmonic in SiCl4 (near to local mode). RMS deviations of the derived vibrational frequencies [0.41 cm−1 (SiH4), 0.83 cm−1 (SiCl4), and 0.63 cm−1 (SiF4)], with reference to their experimental counterparts, assert that the U(2) Lie algebraic Hamiltonian is successful in deriving all the fundamental vibrations, their higher overtones and combination bands up to the fifth excitation, of each of the three silane molecules at the sub-cm−1 level of accuracy, possibly at a much lower computational cost as compared to other theoretical methods.

Schematic SO(4) model for the description of hadrons

An algebraic model is proposed which simulates the strong interaction between quarks, for the description of hadron states. The model is based on the SO(4)-group. The effects of the long-range quark–quark interactions, like zero-color confined states, symmetries and energy dependence, required for a correct description of hadrons at low energy, are considered through the assignment of SO(4)-quantum numbers to the hadron states. A simpler version of this model was used earlier to illustrate the relevance of many-body methods implemented for pseudo-scalar and vector meson-like states, particularly the low energy portion of the pion spectrum. In this work an algebraic Hamiltonian is proposed, its spectrum is calculated for several meson quantum numbers J P = 0±, 1±, 2±, as well as baryon quantum numbers , and compared to data. Our purpose is to investigate how adequate is this model when applied to the description of several characteristics of the hadron spectrum, like the observed masses, density of states and sequences of spin and parity.

Vibrational Hamiltonian of Naphthalene (C10H8) Using Dynamical U(2) Lie Algebras

This article is the first to demonstrate the application of dynamical U(2) Lie algebras to model a vibrational Hamiltonian of naphthalene (C10H8), a prototype polycyclic aromatic hydrocarbon molecule containing a fused pair of benzene rings, in the gas phase having considerable astrophysical interest. A vibrational Hamiltonian that preserves the D2h symmetry of naphthalene is modeled using 19 coupled one-dimensional Morse oscillators to describe the eight C-H single bonds, the eleven C-C aromatic bonds, and their dynamic interactions, making use of the Casimir and Majorana operators. Results indicate that the dynamical U(2) Lie algebraic Hamiltonian has successfully reproduced the vibrational frequencies at the 48 fundamental excitations near to the level of spectroscopic accuracy, as well as predicted their first three overtones. The observed degeneracy of the stretching, bending, and combination vibrations at the fundamental and lower overtone excitations are expected to modify the shape of the two major bands (centered at 782 cm−1 and 3065 cm−1), and several minor bands in the infrared absorption spectrum of naphthalene in the gas phase.

Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems

The notion of hidden symmetry algebra used in the context of exactly solvable systems (typically a non semisimple Lie algebra) is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By construction, these algebras do not depend on the choice of realizations by vector fields of the underlying Lie algebra, allowing to propose a new approach to analyze polynomial algebras as those subspaces in an enveloping algebra that commute with a given algebraic Hamiltonian. These polynomial algebras play an important role in context of superintegrability, but are still poorly understood from an algebraic point of view. Among the main results, we present finitely generated quadratic algebras of dimensions 4, 5 and 6, as well as cubic algebras of dimensions 3 and 5, and various Abelian algebras, all of dimension 3. Based on the observation how superintegrability is associated with exact solvability, we propose a procedure that connects the underlying Lie algebra with algebraic integrals of motion. As the integrals constructed in such way are now independent on the realization, alternative choices of realizations can provide new explicit models with the same symmetry algebra. In this paper, we consider examples of such equivalent Hamiltonians in terms of differential operators for the three cases and connected to the underlying Lie algebra gl(2,R)⋉R2⊕T1 as well as to the maximal parabolic subalgebra of gl(3,R). We also point out differences between the enveloping algebra of Lie algebras and the enveloping algebra of the related differential operators realization.

Vibrational Hamiltonian of methylene chloride using U(2) Lie algebra

ABSTRACT This paper is the first to demonstrate the application of U(2) Lie algebraic Hamiltonian in reproducing and predicting the vibrational energy levels of an asymmetric penta-atomic molecule – methylene chloride (CH2Cl2) – having odorant properties similar to ether in the gas phase. The results indicated that the U(2) Lie algebraic Hamiltonian is successful in realising seven of the nine fundamental (stretching and bending) vibrational modes of CH2Cl2 observed experimentally. The lowest, highest, and RMS deviations of the derived vibrational fundamental frequencies stand, respectively, within 0.006%, 0.23%, and 0.7075 cm−1 of the experimental values suggested that the proposed model is successful in predicting the experimental results at the realised fundamental mode vibrations. This study also suggests that further research is needed to make the U(2) Lie algebraic Hamiltonian more robust so as to accommodate for the unrealised out-of-the-plane complex bending vibrational modes of CH2Cl2 molecule.

Equations of motion for noncompact symplectic dynamical symmetry

When the high dimension of the quantum Hilbert space creates a serious challenge for conventional numerical diagonalisation of the Hamiltonian matrix, the equations of motion (EOM) method may solve efficiently the eigenvalue problem in some instances. EOM assumes the existence of a spectrum generating Lie algebra (SGA) for which the Hamiltonian operator is a polynomial (typically a quadratic) in the algebra’s basis operators. The method computes values of low-energy excitation energies and corresponding algebra matrix elements by solving a coupled set of commutation relations. To test EOM’s effectiveness, the paper applies it to SU(1, 1) Sp(1, ), whose unitary discrete series representation spaces are infinite dimensional. The paper investigates two (1, 1) Hamiltonians. The first Hamiltonian has a phase transition that challenges brute force diagonalisation. The second Hamiltonian is the one-dimensional quartic potential. For the quartic case, the algebraic mean field Hamiltonian, an approximation to the SGA Hamiltonian, provides a qualitative solution and a useful starting point for an EOM computation. EOM yields fast and accurate results on a laptop computer for both Hamiltonians.

Calculation of vibrational spectra of some tetraphenyl porphyrins

ABSTRACTIn this paper, we report vibrational energy levels of some tetraphenylporphyrins and its cation radicals, i.e. Cu[TPP], Cu[TPP]+, Ni(TPP), Ni(TPP)+, ClFe(TPP) and ClFe(TPP)+ using U(2) algebraic model Hamiltonian. In our study, we have used five fitting parameters which provide better comparisons between the experimental and theoretical calculations throughout the study. RMS deviation has been improved in this paper due to the inclusion of intermode couplings in algebraic model.Structure of metalloporphyrinsIn our work, we study the spectroscopic features of some biomolecules using Lie algebraic methods. This will be sought to be accomplished by studying the different vibrational energy levels as function of vibrational quantum numbers for certain biomolecules. Possibilities of local-to-normal transition in viable cases will be taken into consideration. In addition, new vibrational parameters for certain biomolecules will be sought to be identified for cases where Lie algebraic approach has not been initiated. The Lie algebraic study may as well be useful in interpreting the vibrational spectra of proteins and other biological systems.

Theoretical calculation of vibrational spectra of metalloporphyrins

In this paper we have reported the theoretical calculation of the vibrational frequencies of various metallotetraphenylporphyrins and its cation radicals for 60 vibrational bands each using U(2) algebraic model Hamiltonian.

A study of vibrational spectra of metallotetraphenyl porphryins: an algebraic approach

Using Lie algebraic method, we report a study of the vibrational frequencies of various metallotetraphenyl porphyrins and its cation radicals, i.e. Cu[TPP], Cu[TPP]+, Ni(TPP), Ni(TPP)+, ClFe(TPP) and ClFe(TPP)+ for 60 vibrational bands each using U(2) algebraic model Hamiltonian. Intermode couplings in algebraic models have been included to achieve improved RMS deviations as well as to get deep insight into detailed spectroscopy for these tetraphenyl porphyrin molecules.

Phase diagram of coupled benders within a [formula omitted] algebraic approach

The phase diagram associated with an algebraic model Hamiltonian that encompasses the main physical ingredients of coupled two-dimensional benders is presented. The model is based on a U(3)⊗U(3) dynamical algebra and can be applied, for instance, to the study of the bending dynamics of tetratomic molecules. A Hamiltonian with three control parameters, encompassing four dynamical symmetries as limiting cases, is defined and its classical limit is obtained using the intrinsic state formalism. The four possible geometric phases can be associated with linear, cis bent, trans bent, and non-planar configurations. The approach, which is inspired by the dynamics of coupled benders in tetratomic molecular species, is completely general.

A study of vibrational excitations of HCN in the framework of an algebraic model

The vibrational description of the hydrogen cyanide molecule is presented in the framework of the U(2) × U(3) × U(2) algebraic model. In our approach the Hamiltonian is first written in configuration space, and thereafter expressed in an algebraic form by introducing a realization of the coordinates and momenta in terms of the familiar bosonic operators for the harmonic oscillator. An anharmonization procedure is then applied to the algebraic Hamiltonian, which consists in replacing the bosonic operators by a new set of operators belonging to the su(2) algebra for the stretches and the su(3) algebra for the bends. This procedure is equivalent to starting the description in terms of interacting Morse oscillators for the stretching degrees of freedom, without any analog with the bends. This method of obtaining the algebraic representation of the Hamiltonian provides in natural form the connection between the spectroscopic parameters and the structure and force constants. The analysis of the vibrational excitations of HCN is considered, where a description of 73 vibrational levels is carried with a standard deviation of rms = 0.91 cm−1, up to now the best description using an algebraic approach. This fit is used to estimate the potential energy surface.

The classical nonlinear properties of molecular highly excited vibration

The concepts of classical nonlinear dynamics are employed to interpret the spectroscopic properties of the molecular highly excited vibration. These concepts include the Morse oscillator, the pendulum dynamics, chaos and the overlapping of resonances leading to chaos. The relations of resonance, constant of motion and the basic dynamical unit to the pendulum dynamics are stressed. An algebraic Hamiltonian in the coset space is employed for the dynamical analysis from which the dynamical potential can be easily obtained. The dynamical potential is closely related to the classical fixed points in which the quantized levels are embedded in various quantal environments. Localized modes are easily identified in various systems which share similar dynamical potentials. The dissociation of DCO radical is finally interpreted by these concepts from the classical nonlinear dynamics.

Relation between the E(5) symmetry and the interacting boson model beyond the mean-field approximation

In a unified algebraic scheme, we investigate the relation between the E(5) symmetry and the interacting boson model beyond the mean-field level. The results indicate that the E(5) symmetry is actually in between the critical point of the U(5)-O(6) transition and the O(6) limit but it is fairly close to the former based on the phase diagram of the interacting boson model at the large boson number limit. In addition, an algebraic Hamiltonian of the E(5)-β 2n model is proposed.

Vibrational spectroscopy of bio-molecules: an algebraic approach

Porphyrins have received numerous considerable attentions during recent years because of their great biological importance. The most interesting areas of current research in molecular spectroscopy are the study of the vibrational ground and excited states of polyatomic molecules especially for the Metalloporphyrins. In this paper, we have calculated the fundamental and extrapolated vibrational energy levels of Magnesium, Nickel, Copper, Zinc, Metalloporphyrin molecules using U (2) algebraic model Hamiltonian. The results obtained by this method are in good agreement with the experimental data. This study gives a general approach for solving the vibrational spectra of Metalloporphyrin molecules.

Demixing and Cleaning of Wave Functions by Projection, Application to the Assignment of Molecular Vibrations

It is shown how the projection of a quantum mechanical state into an appropriate subspace of the Hilbert space can be useful for the classification and the assignment of states. The projection cleans the wave function and decomposes mixtures caused by near-accidental degeneracy. The idea is applied to vibrational states of the molecule CF(3)CHFI obtained from an algebraic Hamiltonian.

Overtone spectra of porphyrins and its substituted forms: an algebraic approach

We introduce an algebraic model to vibrations of polyatomic Bio-molecules and present, as an example, the vibrational analysis of Cm-H, Cm-C, Cm-D, Cb-Cb, pyrrol breathing and Cb-C, stretching modes of Metalloporphyrins and its substituted forms. The excited energy levels of Cb-C, pyrrol breathing stretching modes of Ni(OEP) and Ni(OEP)-d4 are calculated by using U(2) algebraic mode Hamiltonian. The higher excited energy levels of Cm-H, Cm-C, Cm-D and Cb-Cb vibrational modes of Porphyrin and its substituted forms are predicted upto second overtone. It shows that the energy levels are clustering at the higher overtones. The results obtained by this method are accuracy with experimental data.

Dynamical potential approach to dissociation of H–C bond in HCO highly excited vibration

The highly excited vibrational levels of HCO in the electronic ground state, X̄1 A′, are employed to determine the coefficients of an algebraic Hamiltonian, by which the dynamical potential is derived and shown to be very useful for interpreting the intramolecular vibrational relaxation (IVR) which operates via the HCO bending motion. The IVR inhibits the dissociation of H atom and enhances the stochastic degree of dynamical character. This approach is from a global viewpoint on a series of levels classified by the polyad number which is a constant of motion in a certain dynamical domain. In this way, the seemingly complicated level structure shows very regular picture, dynamically.

Formalism of the displaced squeezed Fock states for variational calculations of highly excited ro-vibrational levels: Diatomic molecules

A set of displaced squeezed number states is proposed as trial wave functions in variational calculations of ro-vibrational energy levels of diatomic molecules. By employing the ladder-operator formalism, we construct such states as well as an algebraic Hamiltonian expressed in terms of normal-ordered boson operators. We also show that this algebraic Hamiltonian can be expanded in terms of pseudoladder operators $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{a}(\ensuremath{\lambda},\ensuremath{\kappa})$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}(\ensuremath{\lambda},\ensuremath{\kappa})$ obtained via a generalized Bogoliubov transformation. In this case, the Hamiltonian matrix is built using the usual Fock basis set and the coherence and squeezing parameters $\ensuremath{\lambda}$ and $\ensuremath{\kappa}$ are optimized variationally. The convergence of the variational calculations is largely improved when using the displaced squeezed number states instead of the usual Fock ones. A class of generalized displaced squeezed number states is also considered, and some numerical applications are given.