In the previous parts, we learned how to set up the Hückel determinant for conjugated linear and cyclic molecules and calculate the energy and coefficients of molecular orbitals. In this new article, we will use the SHM to calculate different molecular properties. These properties help understand the structure and reactivity of organic molecules.

CALCULATION OF THE pi-ELECTRON DENSITY

The electron density quantifies the amount of electronic charge localized around each atom in the molecule. This information gives a valuable estimation of the electrostatic (partial) charge on each atom of the aromatic system. The concept of fractional charge seems to undermine the idea of an electron as a particle. Still, it is a consequence of the probabilistic interpretation of the quantum realm. The Copenaghen interpretation of quantum mechanics theory states that the square of the wave function, associated with the description of the quantum particle’s wave nature, provides a probability density function (PDF). Whose multiplication with the infinitesimal volume of space produces a probability of localizing the particle in that volume. Therefore, the squared orbitals functions obtained by the SHM can be used to get the amount of negative charge on each atom of the aromatic system. For this purpose, we can sum the square of wave function coefficients of all the occupied orbitals

(1)

where is the number of electrons in the orbital.

The Figure above shows the molecular orbitals for the allyl anion. Let’s now calculate the electron densities on the atoms of these molecules using the formula given before.

CALCULATION OF THE BOND ORDER

The bond order is another valuable property that can easily be derived from the HMO coefficients. It gives tha $#\pi-$$electron distribution in the bonds instead of on the atoms. It is defined as

In the following Figure, the calculation of of the allyl radical is reported. The result shows an equal contribution of 0.707 for both bonds.

The total bond order is obtained by adding the sigma bond (1) to as shown in the examples reported in the figure below.

For hydrocarbons, there is a correlation between bond order and bond lenght that can be used to make an approximate prediction of bond lenght variations. C.A.Coulson in article of 1939 has proposed the following formula that can be used to estimate the effect of the bond order on the length variation of a single bond

$$R=s-\frac{s-d}{1+\frac{k(1-p)}{p}}$$

where s is the average single bond length taken equal to 1.54 A for C-C, d, the double bond length equla to 1.337 A, p the bond order, and k an adjustable parameter. For the example given before, gor p=0.707 and k=0.795, the value of R is

$$R=1.54-\frac{1.54-1.337-d}{1+\frac{0.795(1-0.707)}{0.707}}=1.54-0.153=1.387 \AA$$

CHEMICAL REACTIVITY INDECES

CALCULATION OF THE FREE VALENCE

Finally, the free valence is defined as the difference between the maximum possible bond order and the actual tital bond order. It is defined as

UNDER CONSTRUCTION

If you have found interesting and useful my article, do not forget to press “Like it” and subscribe for updates on new ones!

REFERENCES

1. J. P. Lowe. Quantum Chemistry. 1993, Academic Press. 
2. F.A. Carroll. Structure and Mechanism,1998, BROOKS/COLE Publishing Company.

In the previous article, we have learned how to set up the Hückel determinant for an aromatic molecule based on the topology of the pi-bonds. In this second part, we are going to learn how to calculate from the determinantal equation both the eigenvalues and the eigenvectors, corresponding to the orbital energy and orbital functions of the molecular system.

Continue reading →

Cristallo e fiamma, due forme di bellezza perfetta da cui lo sguardo non sa staccarsi

Italo Calvino, Lezioni americane. Sei proposte per il prossimo millennio.

The successful reception of my previous posts on calculating the lattice energy, the calculation of the Born-Haber cycle and the Madelung constant motivated me to write more about the crystal structures of inorganic and organic compounds. In this series of articles, I will briefly introduce the crystallography. It does not pretend to be comprehensive, but it is meant to give just a flavor of this fascinating topic.

What is a Crystal?

At the atomic level, a crystal is solid with regular arrangements of atoms in space. Depending on the material, these arrangements have different symmetry properties that generate beautiful symmetric shapes at the macroscopic level.A regular and infinite arrangement of atoms can be visualized as a vast and complex network of interconnected particles. Each atom, with its unique properties, contributes to the overall complexity and variety of the system. The repeating patterns of the atoms create an endless and fascinating display. This assembly of atoms forms the basis of the world of matter, showcasing the intricate and beautiful nature of the smallest units of our universe.
We start introducing the Bravais lattice’s fundamental concept to describe and classify these atomic arrangements..The term “Bravais lattice” is a name given to a mathematical concept used to describe how points are arranged in a crystal. It is named after Auguste Bravais, a physicist from France who studied crystals in the 19th century. Bravais introduced the idea of different types of arrangements and symmetry in crystal structures. His work was important in understanding how crystals are classified based on their symmetry. The Bravais lattice is a fundamental concept in crystallography that describes the periodic arrangement of points in a crystal lattice. It provides a framework for classifying and categorizing different types of crystal structures based on their symmetry and arrangement of lattice points.

Continue reading →

Indipendentemente dai motivi del culto, le antiche cattedrali invitano ad un’ammirata contemplazione, ispirano rispetto e quiete. Anche il visitatore più disinvolto non si esimere dal moderare la voce, non insiste in argomenti futili: delle navate, l’eco delle sue stesse parole sembra destare insolite suggestioni. L’impegno di generazioni di architetti e di artigiani e’ stato dimenticato, le loro impalcature sono state rimosse ormai da lungo tempo, i loro stessi errori sono stati cancellati dai secoli. Il monumento che essi crearono, ora compiuto e perfetto, ci appare come la testimonianza di un disegno sopraordinario. Se evochiamo in noi il ricordo di un cantiere in attività, con il rumore ritmato dei martelli, le voci ed i gesti degli operai, l’odore stantio del legno e di tabacco, alle splendide structure che ora ammiriamo non possiamo attribuire altro significato che quello di essere il frutto di un ordine imposto alla mera fatica umana.

Anche la scienza ha i suoi templi, costruiti con gli sforzi di pochi architetti e di molto operai, e di fronte ad essi proviamo lo stesso sentimento. Anche in questi templi l’atmosfera e’ solenne, e forse lo e’ a tal punto da condizionare l’espressione stessa del pensiero scientifico, che una lunga tradizione vuole assai severo e formale.

G.N. Lewis, M. Randall -“Thermodinamica”, Leonardo Edizioni Scientifiche, Roma (1971).

INTRODUZIONE

Gilbert Newton Lewis e Merle Randall nella introduzione alla prima edizione del loro autorevole testo di termodinamica chimica descrivono  la termodinamica come la cattedrale della scienza. La loro non è solo una concessione poetica di un’epoca ancora permeata dal romanticismo scientifico, ma una meravigliosa analogia per questa fondamentale disciplina della scienza che più di ogni altra contiene  le leggi arcane che goverano il nostro Universo e il suo destino.

In questa serie di articoli riporto alcuni appunti su argomenti vari di termodinamica chimica che possono essere utili come riferimento o come materiale didattico.

Continue reading →