A conversation on the quartic equation of the secular determinant of methylenecyclopropene

The Hückel method (HM) is based on quantum mechanics and it is used for calculating the energies of molecular orbitals of π electrons in conjugated systems. The HM involves the setting up of the secular determinant which is expanded to obtain a polynomial which is to be solved. In general, the polynomial is one which may be factorized. However, in May 2020, students brought to my attention that the secular determinant of methylenecyclopropene could not be factorized completely. As a result of this, we used a combination of online tools, technology and visualization to calculate the roots of the secular determinant. This write-up, in a playwriting format, describes the conversation between the facilitator and the students.

Lagrange-multiplier regularization of eigenproblem for Jx

We solve the eigenproblem of the angular momentum $$J_x$$ J x by directly dealing with the non-diagonal matrix unlike the conventional approach rotating the trivial eigenstates of $$J_z$$ J z . Characteristic matrix is reduced into a tri-diagonal form following Narducci–Orszag rescaling of the eigenvectors. A systematic reduction formalism with recurrence relations for determinants of any dimension greatly simplifies the computation of tri-diagonal matrices. Thus the secular determinant is intrinsically factorized to find the eigenvalues immediately. The reduction formalism is employed to find the adjugate of the characteristic matrix. Improving the recently introduced Lagrange-multiplier regularization, we identify that every column of that adjugate matrix is indeed the eigenvector. It is remarkable that the approach presented in this work is completely new and unique in that any information of $$J_z$$ J z is not required and only algebraic operations are involved. Collapsing of the large amount of determinant calculation with the recurrence relation has a wide variety of applications to other tri-diagonal matrices appearing in various fields. This new formalism should be pedagogically useful for treating the angular momentum problem that is central to quantum mechanics course.

Cyclopentadienyl System: Solving the Secular Determinant, π Energy, Delocalization Energy, Wave Functions, Electron Density and Charge Density

In this study, characteristics of Hückel strategy, were abused so as to acquire some significant outcomes, through a theoretical technique with which it is conceivable to get secular equations, π energy, wave functions, electron density and charge density, as an account of cyclopentadienyl system i.e. C5H5+ (cation), C5H5- (anion), and C5H5. (radical) and permitting the expression of delocalization energy of conjugated cyclopentadienyl ring framework. Here, it was presented the secular determinant of the Hückel technique and applied to cyclopentadienyl system framework so as to communicate their orbital energies of cyclopentadienyl system, also to communicate its electron and charge density in terms of stable configuration of a system. It is settled by the Hückel strategy and applied by the assumptions for nearby comparability such as coulomb integrals, exchange integrals and overlap integrals. This simple way hypothetical strategy will allow to graduate and post graduate understudies to understanding the investigation of stable configuration, electron and charge density and also other parameters.

Application of Huckel Molecular Orbital Theory (HMO) on Hetero-Conjugated Molecule, 3-Aminopropenal

Very often the application of quantum mechanics into chemistry represents a challenging task for chemistry students. However, this can be very usual exercise, and we have shown the easiness on the molecule of 3-aminopropenal, which is an interesting example because of the existence of the conjugation system consisting of the carbonyl group, alkenyl system and the lone electronic pair on nitrogen without any symmetry. Coefficients obtained using the Huckel secular determinant were -2.0484, -1.7328, -0.7827, +0.4892 and +1.5747.