The Famous Schrödinger Equation

Yes, yes, I know, it's been a while since I wrote something here. This year I'm devoting myself a bit more to writing in this blog, I want to spread some knowledge and that can't be done if I don't write something on a weekly basis. 


Today I'm going to talk about the Schrödinger equation, this well known equation is the heart of quantum physics and quantum chemistry. As a quantum/theoretical chemist the aim is to solve this powerful equation. First I'll tell you a story...

Schrödinger wrote a paper in 1926, it was called Quantisierung als Eigenwertproblem, it came out on Annalen der Physik (the same magazine that published Albert Einstein's work). In this paper he stated that the old rules of the quantization of the energy could be substituted (in the simplest case: the non-relativistic, unperturbed hydrogen atom) by another postulate that didn't have any whole numbers but it introduces integers that arise in the same natural ways as in the old quantum theory. In a series of paper after this one Schrödinger applied his new wave equation to various cases, such as: 

• The harmonic oscillator

• The rigid rotator

• The diatomic molecule 

• The H atom in an electric field

For the hydrogen atom in an electric field Schrödinger developed a new way, different to that of Newton, Lagrange, and Hamilton, to describe a dynamic system. The Perturbation Theory differs in that one does not seek equations to describe the system at a given point but one finds a function of the coordinates of the system and the time. With these type of functions probable values of the coordinates and other dynamic properties can be predicted. The word probable gives us a hint of a fundamental rule of quantum mechanics, we can't describe in exact detail the behavior of a system. With this arises the famous, Heisenberg's uncertainty principle. 

With the Schrödinger equation we can determine a function ψ of the coordinates of the system and the time. These functions are called the Schrödinger wave functions or probability amplitude function. If you find the square of the absolute value of the Schrödinger wave function you find a probability distribution function for the coordinates of the system represented in the wave function. By similar means, the wave function can also be used to determine the energy of the stationary state. 

What if we include time? 

Well...it gets complicated.

I hope you remember your classical mechanics...

As a start we will consider a Newtonian system with one degree of freedom, so it only consists of a particle of a mass m moving along a fixed straight line, we will take the x-axis in this case. Another assumption is that the system can be described with a potential energy function V(x) that goes from minus infinity to infinity, so

Now, in order to simplify this equation we'll have to do some tricks. 

This equation is closely related to a fundamental equation in classical Newtonian Mechanics, 

this equation states that the total energy W of the system is equal to the sum of the kinetic energy T and the potential energy V, thus this is equal to the Hamiltonian function. Now if we introduce the momentum and the coordinate the equation becomes

Now, we replace the momentum and the energy by their differential operators arbitrarily and introducing the wave function we obtain

And this equation is identical to the classical Newtonian equation and is thus written in the simple form

This definition has only formal significance, remember that!

I think that's enough for this post, to be continued...

References: 

Quantisierung als Eigenwertproblem, Schrödinger, E. 1926 Annalen der Physik (Erste, Zweite, Dritte und Vierte Mittteilung) 

Introduction to Quantum Mechanics, Griffiths.

Classical Dynamics of Particles and Systems, Marion. 

Chemical Bonding in Coordination Compounds

Well hello, again.

Today I'm going to talk about the models used to describe the different type of bondings that occur in coordination compounds. There are various theories that historically have been used to try to describe the interaction that appear:

• Valence Bond Theory

• Crystal Field Theory

• Ligand Field Theory

• Angular Overlap Theory

Before we start I will define some words that will be going to appear along the text: 

Complex: Usually a complex is defined by a number of ligands that are bonded to a central atom (usually a metal) that have a defined geometry. 

Ligand: An atom or molecule bonded to a metal center. 

Coordination number: In complexes the coordination number is simply the number of species that are bonded to the central cation (the metal). For example [Fe(CN)₆]^3-, the iron has a coordination number of 6.

I'm inserting here a table with the common geometries from the VSEPR model, so that you can see them if they are not familiar to you: 

So.

We've already talked about the VB theory, but now I'm going to talk about its applications to coordination compounds. So, Pauling stated the geometry of the coordination complex should depend on the number of hybridization processes that occur. This will depend on the metal's nature, the number of ligands and the coordination number of the metal. For example: 

For tetrahedral geometries, for example a  [CoCl4]2- will (according to Co electronic configuration) 3 unpaired electrons, this unpaired electrons make the complex paramagnetic, that is, susceptible to external magnetic fields. The bonds with the metal will be formed in the 4s and 4p orbitals. This orbitals are, according to Pauling, hybridized. 

The red box shows the hybridized orbitals.

Now for square planar geometries:

You can observe how the hybrid orbitals include one of the 3d orbitals. 

Now octahedral geometries can have two type of compounds according to Pauling. High spin and low spin. You can see the analogy in the next electronic configurations of two different complexes: 

The difference is that the CN complex takes 3d orbitals and the F complex takes the 4d orbitals.

• High spin: takes 4d
• Low spin: takes 3d

How can you predict which complexes will be high or low spin? It is according to the nature of the ligand. If you can say that the ligand has a strong field nature, then it is low spin. Thus if the ligand is of weak field nature you can say it will be high spin. 

But how do you know that?

Answer: The spectrochemical series

I− < Br− < S2− < SCN- < Cl− < NO3− < N3− < F− < OH− < C2O42− ≈ H2O < NCS− < CH3CN < py (pyridine) < NH3 < en (ethylenediamine) < bipy (2,2'-bipyridine) < phen (1,10-phenanthroline) < NO2− < PPh3 < CN− ≈ CO

Where do strong ligands start? This is a really good question, it is not really well stablished and basically after ammonia you can consider them strong.

And so, according to the hybridization you can predict its magnetic properties. If there are unpaired electrons the complex is paramagnetic and if there are no unpaired electrons it is called diamagnetic.

Also, something important:

• Tetrahedral geometries are always high spin, its independent of the nature of the ligand.
• Square planar are always low spin (same reason).

So, what have we learned?

We now know the concepts of low and high spin, we've presented the spectrochemical series for the determination of the nature of a ligand and with all this we can predict how the electrons will distribute in a complex giving arise to paramagnetism or diamagnetism.

Well, that's all, folks!
See ya, in the next entries I'm going to talk about the other theories I mentioned.

References:

Atkins, P. . (2014). Química Inorgánica. México: Reverté.

Cotton, F. . (1999). Advanced Inorganic Chemistry. USA: Wiley.

The Chemical Bond

Hello everyone!

Now I'm going to give you a short introduction on what I want to do. I've already told you that I want to dedicate my work to solving the problem of the chemical bond. You may wonder: why?

I'll give you (in parts) a brief summary of what we know of the chemical bond.

So.

First let's define what a chemical bond is: A chemical bond it is say to be formed when two or more atoms are held together strong enough to form a molecule.

So, basically that is a chemical bond. There are various theories to describe it. But before the dawn of quantum mechanics the explanations were rather inaccurate. Since the proposal of the quantum theory and the birth of the Schrödinger equation theoretical chemists and physicists have developed several theories that unite the wave-particle duality of the electron and the interactions with the nuclei. The problem is that physics stumble with a great problem. This problem comes from classical mechanics. The many body problem. We can solve accurately for just two particles.

How many particles are in the simplest atom? The hydrogen atom has one proton, one electron and the nucleus. Too much bodies for the Schrödinger equation, so...now what?

Well before going into that I'm going to present a series of theories that have been proposed to explain the chemical bond.

One of the most remarkable ones, considered a classic in modern chemistry is the work of Linus Pauling, an american chemist, his work: The Nature of the Chemical Bond was a revolution in the theory of what the chemical bond was. He used quantum mechanics to explain the movement of the electrons and defined what are called orbitals. Those are solutions to the Schrödinger equation that define a spatial probability of where the electron may be found. So, every type of atom has its own set of atomic orbitals, they change depending on the energy level. There are multiple types of orbitals. In the next picture you can see a bunch of them!

And how did Pauling describe the chemical bond?

He proposed hybrid orbitals, atomic orbitals that combine each other in order to become hybrid and be able to form a bonding interaction. The bonding interaction forms by the overlap of two or more hybridized orbitals. A characteristic of these hybrid orbitals is that they are degenerate and localized, that is, they have the same energy, all of the atomic orbitals that combined to form the hybridized one. Also a very important thing is that the atom forms the necessary hybridized orbitals to form the bond. That is, if it needs 5 hybridized orbitals to form the bond, only 5 bonds will be created the other atomic orbitals that are left can be used to form new bonds. Pauling's theory is also known as the Valence Bond Theory.

But there is a problem with the VB theory, the idea that the hybridized orbitals that form bonds are degenerate and localized, that is pretty unlikely. There's another theory that helps us explain the chemical bond. The molecular orbital theory or MO for short. This theory is maybe the most used one in modern days, it describes the bond formation by the formation of molecular orbitals that form from the combination of atomic orbitals. The difference with the VB theory is that there molecular orbitals are delocalized over the molecule. Also atomic orbitals tend to combine better if they have similar energies or have the same size. Bigger atoms have bigger orbitals and thus, smaller atoms have smaller orbitals. Big + Big = good, Big + Small = not so good.

Also a very important concept in MO theory is symmetry, if the orbitals are not of the same symmetry the interaction between can be non existent or really really weak. That is the reason that for more complex molecules molecular symmetry and group theory is used in order to be able to classify the type of orbitals that will arise in that complex environment and thus predict the type of interactions that the molecule will have.

So, how is a molecular orbital described in terms of the wave functions?

A molecular orbital is the result of two atomic wave functions interacting with each other, they have two type of interactions, one is a bonding interactions and one an anti bonding interaction. A molecule is said to have bonds if the number of bonding interactions exceed the anti bonding interactions.

Both interactions between the orbitals are expressed by that plus/min sign. The Sa and the Sb correspond to the wave function of each atomic orbital. And N is a normalization constant.

So, I haven't said anything about the many body problem, guess I'll leave that for the next!

References:

Atkins, P. . (2014). Química Inorgánica. México: Reverté.
Cotton, F. . (1999). Advanced Inorganic Chemistry. USA: Wiley.

My inspirations

Hello everyone!

Well, I told you I was going to talk about my inspirations to venture into theoretical chemistry...Everyone has their heroes, their inspirations and the people who motivate them to go forward. I have a few:

First of all, the most renowned scientist the world has known. Born on December 25th in 1642, his name? Sir Isaac Newton.

I think almost everyone admires Newton, why would someone not admire him? He did, well basically, explained the world with his physics. Gravitation, optics, CALCULUS! Along with Leibniz they developed the most used tool in everyday life. What could we do without calculus? Almost every formula you've encountered in your life has a differential, an integral, a limit.

Why do I admire him? He had the ability to create science that I would compare to art, his mathematical and physical creations are simply beautiful, the ability to explain the natural phenomena that were present in his everyday life. The motion of the planets and the stars, comets and asteroids. The idea of a force that brings things together. Newton was simply magnificent.

Sir Isaac Newton

Next, another great man that made one of the greatest breakthroughs of the history of mankind. Just to say that he derived the equations of the electromagnetic theory and the kinetic gas theory...His name? James Clerk Maxwell, one of the greatest scientists ever to have existed.

So, why him? The importance of his work is monumental, we can compare him to Newton. He made another unifying theory. His admiration to Faraday led him to formulate the equations that describe the behavior of electricity and magnetism. If you are now in college and have taken physics I can bet that you've encountered his name quite a few times. More if you've already taken a course in electromagnetism or physical chemistry. Maxwell's work opened the doors to many new disciplines, it helped many other scientist in their work. Maxwell's explanations of things were like poetry, everything was fluent, everything was like a rhyme.

There are a couple more...

Now a chemist, the one who inspired me to research the chemical bond. The great: Linus Pauling. 

The Nature of the Chemical Bond, by Linus Pauling. The title of one of the greatest pieces of scientific research that has ever been published. I dream on publishing an article called "The New Nature of the Chemical Bond". Linus Pauling is an inspiration to every chemist that want to be called a quantum one. His work on the chemical bond had no precedents. The idea of just a binding force between atoms abandoned the place as soon as Pauling published his finding. With the help of quantum mechanics he was able to create a portrait of the chemical bond. 

What else did he do? Well if that wasn't enough...If you are a chemist or have some notion of it you'll understand this: He conceived electronegativity. 

What else can I say? We now take electronegativity as a basic concept, it is taught in high school, in college...everywhere! It is so simple but imagine the time where there was no electronegativity...

How could you explain the course of a chemical reaction? Electronegativity helps you recognize the active spots of a molecule, why they will react. It is the key to a very big part of chemistry. 

My inspirations are not only scientists. My family is my greatest inspiration of them all. Thanks to them I've been able to achieve the things that I have done. They provided me with the opportunities, with the support to pursue my dreams. I can confidently say that they are proud of what I've done until now. I want to thank them for everything that they have done, they've brought me to this point in life where I can make my own future, where I can make a difference. 

My father, Jorge Gálvez. (in the middle)

My mother, Adriana Vallejo. (it's obvious who she is)

My brother, Gerardo Gálvez. (the one in the blue sweater)

Thanks to them I've been able to succeed in everything I have done and I am in eternal gratitude with them.

And then there's me in the striped sweater.

Thanks for reading! In the next post I'll write something about what I've done until now.

See you!

Square One

Well hello all of you!

This, as the name of the blog implies, shall be the story of my journey into theoretical chemistry. My name is Jorge Luis Gálvez Vallejo, I'm currently an undergraduate chemistry student from Mexico. I study at the Universidad de las Américas Puebla and I'm currently in my fifth semester out of eight. 

My passions in theoretical chemistry are the chemical bond, a problem that has troubled many great minds and has brought enormous advances in the entire field of chemistry, the other one is chemical kinetics and the modeling of dynamic chemical systems. 

I'm basically just a regular guy who really loves maths, physics and of course chemistry. I want to work in something that unites them all. That is why I chose to specialize in theoretical chemistry, in order to be able to use as much maths as I need, give the maths physical explanations and use that to understand the world of chemistry. 

What has brought me to this "square one"? 

Since high school I developed a great love for science and since my second semester of college a love for research started growing in me, everything has led me to the point in which I am right now. I just want to know things, solve problems, have fun doing all of it. 

In the next post I shall talk about my inspirations for doing theoretical chemistry! 

See you soon!