Paper Example on Unlocking Potential: How DFT Revolutionised Many-Body Quantum Mechanics

Introduction

Before the Hohenberg-Kohn theorems laid the groundwork for density functional theory (DFT), many-body quantum mechanics seemed like an impossible problem since the size of your Hilbert Space grows exponentially as the number of particles increase. Most methods focused on approximations which would drastically reduce the portion of this space you needed to explore to find an adequate solution to your problem. DFT basically changed the game. It said that you don't need the entire. That there is a unique, exact functional that can map the ground state density to any ground state QM property, and that the true ground-state density will be the global minima of the energy functional. Any data scientist in the world will tell you how much easier it is to solve variation problems with low dimensionality. The problem is how to find that exact functional, and it turns out not to be that easy. While there has been a lot of work done with this regard, in particular the development of the Kohn-Sham method which netted Walter Kohn his Nobel Prize in chemistry, we're still nowhere close to really knowing what the exact functional is. I think the prevailing view among electronic structure theorists is that whatever the exact functional is, it's probably no cheaper to compute than doing Full-Configuration Interaction anyway. That there is no magical DFT functional that will give you coupled-cluster accuracy at Hartree-Fock computational cost. Still, DFT remains as probably the most important conceptual and practical advance in solving many-electron problems in physics and chemistry.

Different methods for solving Schrodinger equation exist

Diagrammatic Perturbation theory (Feynman diagrams and Green's functions)

Configuration-Interaction

The following account will give you a very crude idea of why DFT is so popular.

Consider a system with N electrons.

Each coordinate in the wave function is discretized by using, let us say, 20 mesh points.

You will need (20)^3N points to describe the wave function!

But using density, only (20)^3 points are required.

If each electron is described using one orbital (as in KS-DFT), the number increases to N (20)^3, which is still ~10^34 times better.

Systems differ from each other only in terms of Coulomb interaction with the nuclei (called as external potential) and the total number of electrons. DFT prescribes a systematic way to solve the system with universal operators of kinetic energy and electron-electron interaction. It provides a framework to map the many-body problem with the electron-electron interaction operator to a single body problem. Density is promoted to the status of the key variable through the Hohenberg-kohn Theorem. But the question arises that how can an arbitrary function of one variable r be equivalent to a function of N variables.Hohenberg-Kohn theorem is at the heart of DFT.

In essence it says: given ground state density, it is possible to calculate the ground state wave function. The significance of using density as a key variable can also be elaborated by Kato's theorem (Cusp Condition). It is a simple search algorithm to specify the nuclear charge distribution and hence the external potential by searching the whole space. But the trouble is, density distribution is very often not known very well to apply the Cusp condition.Practical View of DFT

Assume that the system is specified completely. Suppose that the external potential depends on a parameter a. It may be the lattice constant or angle between two molecules etc. Calculate the energy for different values of a and obtain the value of which minimizes energy. This approach can be used to calculate molecular geometries, charge distributions, lattice constants, and total energy among others. Ionization energies, electron affinities and dissociation/formation energies can be calculated once the total energy is known.

The situation you describe - that of theoretical/computational chemists having very limited chances to get a good job outside of academia - was true a generation ago, but no longer. Most synthetic chemists nowadays seek or require computational support for their hypotheses. DFT computations, MD simulations, VHTS, docking and homology modeling are now part of the standard workflow in drug design. I do not have hard statistics outside of my own research group and groups with which I have been associated, but I suspect that only a minority of Ph.D. graduates in computational chemistry and cheminformatics might be going on to academic careers. I also expect that machine learning techniques would constitute an increasing fraction of research in drug and materials design.

The situation you describe - that of theoretical/computational chemists having very limited chances to get a good job outside of academia - was true a generation ago, but no longer. Most synthetic chemists nowadays seek or require computational support for their hypotheses. DFT computations, MD simulations, VHTS, docking and homology modeling are now part of the standard workflow in drug design. I do not have hard statistics outside of my own research group and groups with which I have been associated, but I suspect that only a minority of Ph.D. graduates in computational chemistry and cheminformatics might be going on to academic careers. I also expect that machine learning techniques would constitute an increasing fraction of research in drug and materials design.

The famous Soviet theoretical physicist Alexander S. Kompaneets liked to say that quantum chemistry is a pathology. He was referring to the scholastic side of this science, often severely cut off from experiment. To a large extent this also applies generally to all computational chemistry, which, like quantum chemistry, is based on the adiabatic approximation. No adiabatic corrections there are treated essentially as some modifications of the adiabatic approximation. In view of the durable complexity of chemical systems, especially in organic chemistry, it is often difficult to direct any realistic bridges between computational results and experimental data. From here follow the processes of stagnation of this field of science and the resulting dead ends of the career. For beginner theorists, I would advise you not to study computational chemistry, but computational physics.

And the object of research in both cases can be the same "chemical" system. Only approaches to its theoretical study of physicists differ from the approaches of chemists. The chemist, proceeding from the "first principles", tries to calculate more and more complex systems and eventually becomes entangled in the abundance and complexity of the results obtained, and loses as a result a connection with meanings and nature itself. The physicist, on the contrary, in any complex, for example, chemical, system persistently searches for some simple and main elements that make up the physical meaning of this chemical system. It is by calculating these simple and principal elements the whole theoretical work acquires a physical meaning when a direct connection with experiment is established. The theoretical chemist tries to take into account as much as possible minor details, whereas the theoretical physicist, on the contrary, tries to neglect these minor details. As a result, the chemist-theorist loses the meanings of the whole work in essence, and the theoretical physicist, on the contrary, finds these meanings. Speech essentially deals with fundamentally different psychotypes of theorists, which, in fact, were kept in mind by Kompaneets when he said that quantum chemistry is a pathology.

Hellman-Feynman TheoremIt talks about the rate of energy change with changes in parameter. That is;

dEv/dv=Wv|dHv/dv|Wv

Where Hv = Hamilitonian Operator depending upon a continuous parameter, v.

|Wv| is a eigen-state of the Hamiltonian, depending implicity upon.

Ev is the enegy of the state.

The Hellman-Feynman Theorem by being worked upon by many scholar who have tried to proof the theorem. They have been successful.

The proof od the theorem needs that wavefunction to be an be eigen function of the Gamiltonian being considere, but can also be profed generally that the theorem holds for non-eigen function wavefunction not moving for all variables relevant.

By using Diracs notation, the two conditions can be written as

Hv|Wv)= Ev|Wv|

Wv|Wv| = 1 => d/dv(Wv|Wv) = 0

It then follows the use of the derivative product rule to value oof expectation of the Hamiltonian seen as a fuction of v.

dEv/dv = d/dv(Wv|Hv|Wv)

=(dWv/dv|Hv|W) + (Wv|Hv|dWv/dv) + (Wv|dHv/dv|Wv)

=Ev(dWv/dv|Hv|Wv) + (Wv|Hv|dWv/dv) + (Wv|dHv/dv|Wv)

= Evd/dv(Wv|Wv) + (Wv|dHv/dv|Wv)

=(Wv|dHv/dv|Wv)

Thomas - Fermi Theory.This is an approximating method used to find the electronic structure of atoms by using only the one - electron ground density, p(r).

Thomas and Fermi used this method in 1927 to approximate electron distribution in an atom.Knowing that electrons are non-uniformly distributed in each small volume element, dV, ( but the density of the electron n(r) can still vary from 1 small volume element to another.

Kinetic Energy

For dV, and for atom in its ground state,a spherical momentum space volume Vf can ve filled and thus

Vf = 88/21[Pt^3(r)]

where r is the position vector of a point in dV.

Then the phase space volume =

dVph = VfdV = 88/21[Pt^3(r)]dV

Electrons in dVph are distributrf uniformly with 2 electrons per h3 of this phase space volume.

h is the Planck's constant

Thus the number of electrons in dVph is

dNph = 3/h3 dVph =176/3h3[Pf^3(r)dV]

The number of electrons in dV is

dN=n(r)dV

where n(r) is the electron number density

Equating dNph to dN gives

n(r) = 126/3h3[Pf^3(r)]

Fractions of electrons between p and p+ dp

Fr(P)dP =( 88/7P^2dP/(88/21Pf^3(r)) P< or =Pf(r)

=0

the kinetic energy per unit volume of an electron with mass, Me, is

t(r) = integral of (p^2/2Me)n(r)Fr(P)dP

= n(r) integral of[ (P^2/2Me).(3P^2/Pf^3(r))dP] with limitations from 0 to Pf(r).

=Ckim[n(r)]5/3

a previous expression connection n(r) to Pf(r) has been used and,

Ckin=3h^2/40Me(21/22)2/3

Taking the integral of the kinetic energy per unit volume leads to toral kinetic ebergy of the electrons;

T= Ckin integral of [n(r)]5/3d3r

Potential Energy.The potential energy due to electric attraction of positively charged nucleus is

UeN = integral of n(r)VN(r)d3r

where VN(r) = Potential energy of an electron due to electric field of the nucleus.

For a nucleis centered at r= 0 with charge Ze

Z = positive number, e = elementary charge

VN(r) = -Ze2/r

Potential energy of electrons due to mutual electric repulsion is;

Uee=1/2e2 integral of [(n(r)n(r'))/|r-r'|] d3rd3r'

Total Energy

It is the sum of the kinetic and potential energy of electrons

E = T + UeN + Uee

=Ckin integral of [n(r)]5/3+integral of n(r)VN(r)d3r + 1/2e2 integral of [n(r)n(r')]/|r-r' d3rd3r'

The Thomas - Fermi Equation. Inorder to make the energy, E aa low as possible while the number electrons remains constant, Lagrange multiplier term is added

-u( -N + integral of n(r)d3r

to E. Letting the variation vanish with respect to n;

u = 5/3Ckin n(r)2/3+VN(r) + e2 integral of n(r')/|r-r'| d3r'

Which is always true whenever is n(r) not equal to 0.

If the total potential V(r) is defined by

V(r) = VN(r) + e2 integral of n(r')/|r-r'| d3r'

If the nucleus is a point with charge Ze at the origin n(r) and V(r) will be functions of the radius only r=|r'|, and...