In December 1926, Schroedinger published his seminal paper titled "An Undulatory Theory of the Mechanics of Atoms and Molecules" in Physical Review Letters. A few years later, Paul Dirac wrote in his paper (published in Proceedings of the Royal Society) as follow:
"The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation." P. A. M. Dirac (1929)
To get an idea, consider a small molecule $C_{6}H_{6}$) which contains 6 carbon atoms (6 $\times$ 6 electrons) and 6 hydrogen atoms (6 $\times$ 1 electrons), totaling 36+6=42 electrons.
You may know the Schrodinger's equation for a single particle in a potential $V(x,y,z)$ can be written as
$-\frac{{\hbar}^2}{2m}\dot {{\bigtriangledown}^2}{\psi(x,y,z)}+V(x,y,z)\psi(x,y,z)=E\psi(x,y,z)$
Here, the equation is written in its time-independent form. For time-dependent form see here in the page. Since Kohn-Sham equations were originally developed for ground state systems, now I move forward with time-independent Schrodinger's equation. Later, I will discuss more advanced topics.
(Later P W Anderson wrote a paper opposing Dirac's view. See that paper titled "More is different" published in Science Magazine.
To get an idea, consider a small molecule $C_{6}H_{6}$) which contains 6 carbon atoms (6 $\times$ 6 electrons) and 6 hydrogen atoms (6 $\times$ 1 electrons), totaling 36+6=42 electrons.
You may know the Schrodinger's equation for a single particle in a potential $V(x,y,z)$ can be written as
$-\frac{{\hbar}^2}{2m}\dot {{\bigtriangledown}^2}{\psi(x,y,z)}+V(x,y,z)\psi(x,y,z)=E\psi(x,y,z)$
Here, the equation is written in its time-independent form. For time-dependent form see here in the page. Since Kohn-Sham equations were originally developed for ground state systems, now I move forward with time-independent Schrodinger's equation. Later, I will discuss more advanced topics.
(Later P W Anderson wrote a paper opposing Dirac's view. See that paper titled "More is different" published in Science Magazine.
OK. Let us take the words of Dirac. Approximate practical methods of applying quantum mechanics have to be developed.
What are the approaches? Firs of all what we should know? Total energy. What is total energy? The total energy (kinetic energy + potential energy+ any other energy) associated with the total system that include nucleus and electrons.
As a simple approach, consider that there is only kinetic energy and potential energy contribution and set all other contribution to zero. Then,
Total energy = kinetic energy + potential energy.
Let us assume $C_{6}H_{6}$) molecule. For this molecule, as we saw above, there are 42 electrons. So, considering 3 co-ordinates for each electrons, $42 \times 3$ becomes 126 coordinates (see this description from Prof. Hardy Gross's lecture to understand this).
(for electron) simply kinetic energy = $\frac{p^{2}}{2m}$, where $p$ is the moment of the electron and $m$ is the mass of the electron. What is the potential energy? We can calculate the potential energy of a single electron on positive or negative charges. But, our system is a interacting system. right? Lets come to that interacting system later. Now let us focus on the non-interacting system. Consider that the electrons are non-interacting with each other and they interact only with the nucleus (positive ions). Then, we need to account only the nucleus-electron attraction and the kinetic energy of the electron. This is exactly what Thomas-Fermi model does in electronic structure calculation (look for post on this). This omit the electron-electron interaction completely and considers that the electrons are like gases.
Now, what we need is to calculate the Hamiltonian for a given system (let us construct for C6H6). How to construct the kinetic energy functional (in terms of density) and electron-neucleus potential functional? Let us see further in this.
Kinetic enregy = $\frac{p^{2}}{2m}$.
Now, this is in momentum space. We need to convert this kinetic energy in to a functional of density. Let us do that.
The exchange correlation functional is an important correction to the the total energy which gives about 3% of total energy. Eventhough it is small percentage, it is crucial in dermining (....)
(this page will be updated regularly).
(for electron) simply kinetic energy = $\frac{p^{2}}{2m}$, where $p$ is the moment of the electron and $m$ is the mass of the electron. What is the potential energy? We can calculate the potential energy of a single electron on positive or negative charges. But, our system is a interacting system. right? Lets come to that interacting system later. Now let us focus on the non-interacting system. Consider that the electrons are non-interacting with each other and they interact only with the nucleus (positive ions). Then, we need to account only the nucleus-electron attraction and the kinetic energy of the electron. This is exactly what Thomas-Fermi model does in electronic structure calculation (look for post on this). This omit the electron-electron interaction completely and considers that the electrons are like gases.
Now, what we need is to calculate the Hamiltonian for a given system (let us construct for C6H6). How to construct the kinetic energy functional (in terms of density) and electron-neucleus potential functional? Let us see further in this.
Kinetic enregy = $\frac{p^{2}}{2m}$.
Now, this is in momentum space. We need to convert this kinetic energy in to a functional of density. Let us do that.
The exchange correlation functional is an important correction to the the total energy which gives about 3% of total energy. Eventhough it is small percentage, it is crucial in dermining (....)
(this page will be updated regularly).
