User:Francisle1992/sandbox

The Early Days of Quantum Chemistry

The age of Quantum Chemistry began in the 1910s under two traditions: physical chemistry and molecular spectroscopy. Gilbert Newton Lewis, and Niels Bohr were among the first scientists in this field of study.

On April 5th, 1913, Bohr published “On the Constitution of Atoms and Molecules”, which was considered to be one of the first article in Quantum Chemistry. In this article, he used a system of several nuclei with electrons surrounding it in a way as if there weren’t any other nuclei. The electrons orbit in the circle connected to the nuclei and this ring was the only thing that connected the system together. However, there was a small number of electrons in the outer levels were arranged differently. Along with that, Bohr also suggested that the hydrogen molecule offered the heat of formation, which was double compared to the experimental data. For this reason, the calculations were too complicated for more complex molecules.

In 1926, Lewis suggested the model in which the atom shared electron pairs satisfied even complicated molecules. For the simplest molecules, such as hydrogen, the electrons were classified into two different groups. The first group suggested that the electron orbited around both nuclei, while the other one suggested was similar to that of Bohr’s model, in which the electron move either in a plane either perpendicular to the axis of the two nuclei or in the cross orbits.

Beside Bohr and Lewis, Niels Bjerrum, a friend of Bohr, whom followed his classical electro dynamical identity between the radiation frequency and motion one. This model was called the hybrid model, which quantified the rational energy. With this model, Bjerrum suggested a model of the infrared molecular spectra for simple diatomic molecule and established the base for other quantum theory.

Then in 1919 – 1920, three scientists Torsten Herrlinger, Adolf Kratzer, and Karl Schwarzschild began to expand Bohr’s work. According to their work, the motion of electron could be applied into the interpretation of the rational and vibrational motions of molecule under a theoretical umbrella. Edwin Crawford Kemble, an American physicist indicated the interpretation of band spectra that spectroscopic frequencies are the energy differences and are not the same as the ones of motion like they were mentioned in Bjerrum’s theory.

Heitler and London

Walter Heitler and Fritz London were the first scientists to be successful in providing a proper structure for the hydrogen molecule. In the early 1927, they both traveled to the University of Zurich to study with Erwin Schrodinger, whom developed one of the most important equations in quantum mechanic under his name.

In the beginning, Heitler and London’s goal was to determine the interaction of the charges of two atoms. However, the Coulomb integral, which represent electron’s energy in an atomic orbital, was not supportive to their calculation. Even with help from Heisenberg’s work regarding the quantum mechanical resonance phenomenon, Heitler and London were not able to achieve their goal.

Years later, Heitler and London were still trying to figure the problem out and began to perform their calculation by considering the two hydrogen atom approaching closely to one another. They made an assumption that the two electron 1 and 2 belonged to two different atoms a and b respectively. Because the two electrons were identical, the wave function was written as:

$$\Psi=c_1\Psi_a(1)\Psi_b(2)+c_2\Psi_a(2)\Psi_b(1)$$

And they also minimized the energy in other to calculate for  and :

$$E=\frac{\int \Psi H\Psi d\Tau}{\int \Psi^2d\Tau}$$

They ended up with the two results for energy:

$$E_1=2E_0+\frac{C+A}{1+S_{1,2}} $$ and $$E_2=2E_0+\frac{C-A}{1-S_{1,s}}$$

Where S is the overlap of two atomic wave functions ; C is the Coulomb integral, and A is the exchange integral, where both C and A are negatives. This indicates that $$\frac{c_1}{c_2}=1$$ for $$E_1$$ and $$\frac{c_1}{c_2}=-1$$ for $$E_2$$,

The wave functions were rewritten as:

$$\Psi_1=\Psi_a(1)\Psi_b(2)+\Psi_a(2)\Psi_b(1)$$

$$\Psi_2=\Psi_a(1)\Psi_b(2)-\Psi_a(2)\Psi_b(1)$$