In brief:

-When electrons matter: quantum mechanical-base methods

They can wave function (WF) based of Density Functional Theory based (DFT).

WF from simplest to most accuarte

- semi-empirical, a lot of integrals are disregarded and empirical parameters included

-Hartre-Fock

-Post-HF: MPn, CI, CCSD...

DFT, from simplest to most accurate functionals

-LDA

-GGA

-meta-GGA

....

-in parallel: hybrid functionals, including a % of HF exchange

-Without explicitly describing electrons (only atoms or groups of atoms and springs), no QM,: molecular mechanics methods (MM), with a lot of different force fields

QM/MM methods combine both kind of methods for the description of different parts of the same system. ONIOM is one of these QM/MM methods.

Well, actually one should write a whole book to answer this question. To make it short: either you are accurate, but wait ages for an answer, or you are quick, but your answer is inaccurate, or sometimes even wrong (and sometimes you do not know if it is just inaccurate, but at least qualitatively acceptable, or completely wrong) :-)

Mr. Pandey,

The formalisms are differenet. Different formalisms means different model equations designed to account for a selected set of properties.

As Professor Tatiana Korona said, this is very large question ! I give here a summary:

(Ab initio methods)

Methods that do not include any empirical or semi-empirical parameters in their equations – being derived directly from theoretical principles, with no inclusion of experimental data – are called ab initio methods. The simplest type of ab initio electronic structure calculation is the Hartree–Fock method (HF), an extension of molecular orbital theory, in which the correlated electron-electron repulsion is not specifically taken into account; only its average effect is included in the calculation.

(Density functional theory (DFT))

In DFT, the total energy is expressed in terms of the total one-electron density rather than the wave function. In this type of calculation, there is an approximate Hamiltonian and an approximate expression for the total electron density. DFT methods can be very accurate for little computational cost. Some methods combine the density functional exchange functional with the Hartree–Fock exchange term and are termed hybrid functional methods.

(Semi-empirical methods)

Semi-empirical quantum chemistry methods are based on the Hartree–Fock method formalism, but make many approximations and obtain some parameters from empirical data. The use of empirical parameters appears to allow some inclusion of correlation effects into the methods.

(Molecular mechanics)

These methods can be applied to proteins and other large biological molecules, and allow studies of the approach and interaction (docking) of potential drug molecules. Large molecular systems can be modeled successfully while avoiding quantum mechanical calculations entirely. Molecular mechanics simulations, for example, use one classical expression for the energy of a compound, for instance the harmonic oscillator. All constants appearing in the equations must be obtained beforehand from experimental data or ab initio calculations.

(Molecular dynamics)

Molecular dynamics (MD) use either quantum mechanics, molecular mechanics or a mixture of both to calculate forces which are then used to solve Newton's laws of motion to examine the time-dependent behaviour of systems. The result of a molecular dynamics simulation is a trajectory that describes how the position and velocity of particles varies with time.

(Quantum mechanics/Molecular mechanics (QM/MM))

QM/MM is a hybrid method that attempts to combine the accuracy of quantum mechanics with the speed of molecular mechanics. It is useful for simulating very large molecules such as enzymes.

(https://en.wikipedia.org/wiki/Computational_chemistry)

In brief:

-When electrons matter: quantum mechanical-base methods

They can wave function (WF) based of Density Functional Theory based (DFT).

WF from simplest to most accuarte

- semi-empirical, a lot of integrals are disregarded and empirical parameters included

-Hartre-Fock

-Post-HF: MPn, CI, CCSD...

DFT, from simplest to most accurate functionals

-LDA

-GGA

-meta-GGA

....

-in parallel: hybrid functionals, including a % of HF exchange

-Without explicitly describing electrons (only atoms or groups of atoms and springs), no QM,: molecular mechanics methods (MM), with a lot of different force fields

QM/MM methods combine both kind of methods for the description of different parts of the same system. ONIOM is one of these QM/MM methods.

From my point of view (I do vibrational spectroscopy),

First of all, IN FULL CI WE TRUST. ;-)

Ab initio methods are the most reliable. However, please note that the reliability is built on large basis sets, and of course, endless waiting. Besides, `reliable' does not mean accurate. They just do some approximation without human modified parameters. So Hartree-Fork is ab initio but not accurate. For electronic ground state single reference calculation, it is believed CCSD(T) with some large basis set is good enough, but it is super slow. When it comes to excited state, use MRCI. But, very very very very very slow.

Density Functional Theory (DFT) is good, much quicker and in chemical accuracy (1kcal/mol). But from my point of view, it is not good enough for spectroscopic research, because we want spectroscopic accuracy (1cm-1).

Semi-empirical method is also good. But again, not accurate enough.

Note that some guys insist that DFT is better than MP2, for example. It is true when you are lucky. However, DFT and semi-empirical methods are all semi-empirical. For hybrid functional, parameters are got from fitting data bases, too.

If your system is very similar with some published work, it can be a good choice to use that functional. However, no one can guarantee that. It is always suggested to test the method you use before massively use it.

You can find some guys use DFT to deal with vibrational spectra. They are NOT wrong. However, the result is not quantitative, only helpful to assign and understand the spectra.

All above are based on Quantum mechanics (QM), more or less.

Molecular mechanics (MM) is the fast, but based on empirical force field parameters. It is useful when dealing with large system, and that is it.

QM/MM is best when parts of your large system have to be deal with QM. Onion is one of QM/MM.

However, as Prof. Korona said, it is a book. Luckily, we have some. The name can be `Essential/Foundamental/Introduction to Theoretical/Computational Chemistry'. But I am writing here because most of current method focus on chemical accuracy ;-( Those who begin with spectroscopy will find my discussion helpful.