Quantum Chemistry: From classical physics to quantum world

From the very beginning of my intermediate college life I was really fascinated by the quantum world due to the “Biggan Chinta” (one of the Bangladeshi science magazines). Almost every month I used to buy a magazine and go through the full book in a very short time. Every magazine was outstanding from different aspects. But what is more amusing for me is  the biography of scientists such as Einstein, Maxwell, Schrodinger, De Broglie, Richard Finemann and many more. You know every person I mentioned above is an eminent figure in the quantum world. They sowed the first seeds of quantum theory. Another thing that mesmerized me was the cosmology. I liked to hear and talk about cosmology. Theory of relativity and its interpretation was something like I am not in this world. However, this article is a little endeavor for me to talk about the fundamental concept of quantum theory in the light of physics and chemistry. I would especially like to discuss how Schrodinger first brought the concept of wave function and how it changed the world of science.

In Newtonian physics waves and particles are completely different things. Law, postulates and equations are also completely different. But when we discuss quantum physics then according to de Broglie, particles could act as a wave and vice versa. It is sometimes weird to hear but it is true. 

Let's look at the wave equation which we used to memorize in college - simple harmonic wave equation. Common wave equations is (equation 1):

Here, A represents the amplitude of the wave. If you imagine a water wave, A corresponds to the height of the water, while in the case of a spring, it describes the vertical displacement of the spring. As we can see, these equations depend on both position and time. Thus, the amplitude A is a function of position (x) and time (t). Equation (1) illustrates this type of wave with various parameters. By differentiating Equation (1) twice — first with respect to x and then with respect to t — we obtain the following results:

From equation (2) and (3) we can have our ultimate classical equation of wave:

The above equation is a second order partial differential equation containing two variables x, and t. These variables can be separated by the following procedure:

After putting this equation into equation (4) and divide both sides by  Y(x). T(t) we get,

Look at the equation. The left hand side only depends on x whereas the right hand side depends on t. Thus we get two separate and ordinary differential equations for Y and T.

The spatial part equation (6) is called “time independent wave equation” or "amplitude equation”. Our interest is electron waves in atoms and molecules are stationary type waves which do not change with time. So, in this writing when I refer to the wave equation it will be only position dependent. 

The general solution of spatial equations is the solution of ordinary differential equation solution and can be represented as the followings:

We now suppose a guitar string clamped at two points. That means its oscillation is not allowed at x=0 and x=L. So, this is the boundary condition. At x =0,

Now apply the second boundary condition at the other fixed end, x=L:

which occurs when:

Hence, 

So the allowed standing wave patterns on a string fixed at both ends are:

This is the solution of the classical standing wave equation. But the question is where is quantum chemistry? Here is the thinking of famous mind E. Schrodinger. He thought, if particles like an electron behaves like waves, the classical equation of wave must fit to them. Only tweaking is instead of normal wavelength; he assumes that wavelength is de Broglie wavelength. Thus, for a particle of wavelength λ moving along x-coordinate,  the spatial part equation becomes:

After putting the value of λ form de Broglie wave-particle duality equation we get the following equation:

This is now in the eigenvalue form:

This is the time-independent Schrodinger equation for one dimension. If we want to extend this to three dimension then it will be something like,

This is what we called the three dimensional Schrodinger equation. Later on, we will discuss how this equation could be extended to complex theory including Hartree-Fock approximation, Density Functional Theory etc. Best of luck. Thank you for reading. If you enjoyed it, I would appreciate your support!

References:

1. Prasad, R. K. (2014). Quantum chemistry (4th rev. ed.). New Age International Publishers. ISBN 978-81-224-2408-9

2. https://www.youtube.com/watch?v=9WZM68aVnGk