A basis set of a one-particle Hilbert space, as the name indicates, consists of a complete set of linearly independent one-particle functions. Any arbitrary one-particle function in that Hilbert space can be expanded in terms of these basis functions. All complete basis sets are equivalent, however they are not equally efficient to use in practical calculations. Therefore they have to be chosen judiciously for any given problem.

The spin orbitals used in a Slater determinant corresponding to N-particles are to be linearly independent, or the Slater determinant will be identically vanishing. They have to be also properly normalised in order for the Slater determinant being both normalised and correspond to N particles (in the treatments to be discussed in the following paragraphs, these properties are enforced by using the Lagrange method, through introducing appropriate Lagrange multipliers). 

In the framework of the conventional Hartree-Fock theory, one considers an arbitrary Slater determinant from the class of the above-mentioned Slater determinants and on the basis of the variational principle arrives at an eignevalue problem (the one-particle Hartree-Fock equation that is to be solved self-cosistently) that makes the expectation value of the Hamiltonian with respect to the adopted N-particle Slater determinant stationary. In the self-consistent solution of the Hartree-Fock equation, one endeavours to obtain the set of N one-particle eigenfunctions that minimise the just-mentioned expectation value (at any local minimum the energy expectation value is stationary, but an arbitrary stationary point is not necessarily a minimum, and if a minimum, it is not necessarily the absolute minimum).

At any given step of iterative solution of the Hartree-Fock equation, this equation can be solved by a variety of methods. One method consists of choosing an appropriate set of one-particle basis functions and formally expanding the to be calculated eigenfunctions in terms of these basis functions. In this way, the to be solved Hartree-Fock differo-integral equation is transformed into a matrix eigenvalue problem. If the adopted set of basis functions form an orthonormal set, then the matrix eigenvalue problem is an ordinary one, otherwise it is a generalised eigenvalue matrix equation.

Ps. For the derivation of the Hartree-Fock equation on the basis of the variational principle, consult pp. 332-337 of the book  Solid State Physics, by NW Ashcroft and ND Mermin.

Behnam's answer is very thorough and mathematically correct. I'd like to offer an analogy in order to try and make it a bit more understandable before you dive into the deep mathematical wording.

You can think about it a little bit like computer graphics, where an image is typically made up of pixels. Imagine if you want to draw some shape on your screen, for example a circle. If you have a low resolution, i.e. a small number of pixels to use, your circle will look choppy. But with an increasing number of pixels, the "blocky" approximation to that circle gets better and better. In the best case, you would use an infinetly large number of pixels and have a perfectly round circle; but infinitely large computer screens are a bit hard to produce, obviously, so we typically go with finite resolutions that are large enough to not disturb the picture anymore.

That's a bit like what the basis functions do. They are your "pixels" that you use to find a good approximation to the "shape" (your orbitals); the more basis functions you have, the better you can approximate the orbitals. And in the end, their number is hopefully large enough that you don't have noticeable artifacts from the necessarily finite number of basis functions anymore. Unfortunately, this is often not the case, and the effects from using finite basis sets are still very much visible in most of the things that people do today.

"What is the difference between one-electron spin orbitals we use in Slater determinant and the basis set functions?"

The spin-orbitals that are used in Slater determinants are expressed in terms of the basis functions as

 |i> = \sum_u C_{ui} |u>

where |i> is a spin-orbital, and C_{ui} is the coefficient for the basis set function |u> for spin-orbital i.

The way you get the spin-orbitals is by solving the Hartree-Fock (or Kohn-Sham) equations. That is, you minimize the energy with respect to the spin-orbitals, yielding the best possible single Slater determinant wave function. Because we've assumed a basis set, this means you just solve for the coefficients C_{ui}.

So, spin-orbitals (a.k.a. molecular orbitals) are obtained by solving the Hartree-Fock equations in the basis set, which is composed of the basis functions on the atoms (atomic orbitals). Spin-orbitals are by definition the solutions to the Hartree-Fock equations.

"What is then the difference between wavefunctions we get from these two approaches?"

I'm not sure what you mean by the "difference between wave functions we get from these two approaches". After all, you haven't defined what is the wave function you're interested in.

The sets of molecular orbitals and atomic orbitals[*] differ only by a unitary rotation. If you calculate the exact solution to the Schrödinger equation (a.k.a. full configuration interaction in chemistry, and exact diagonalization in physics), then it doesn't matter which representation you use.

However, if you're doing anything else, then you're imposing restrictions on the wave function, and then the orbital basis you use matters a lot. Spin-orbitals, i.e. the Hartree-Fock reference, are usually the best starting point [**], because they are the best single Slater determinant for the system. Electron correlation - the description of the instantaneous interactions when electrons are close by in space - is the smallest for this reference, and you can efficiently capture the correlation effects with e.g. perturbation theory (Møller-Plesset), or coupled-cluster theory.

[*] As has been pointed out by others above, the set of atomic orbitals is often linearly dependent, because diffuse functions on neighboring atoms look alike. Thus, the atomic basis set that is actually used is one where the redundant combinations are dropped out. This is done by calculating the eigendecomposition of the overlap matrix, and removing its lowest eigenvalues. This reducible set is the one that's actually used to describe the molecular orbitals.

[**] This only applies when the system is "single-reference", i.e. in the exact wave function the HF determinant has a large weight. There are many systems where this is not true, and the exact wave function has many important configurations. These are called multi-reference systems, and the weight of the HF determinant in the expansion may be very small. For instance, in Cr2, the HF determinant carries only something like 1e-3 weight.

Basis set functions are some functions (usually gaussian) that simulate the real atomic orbitals. A common example is the 6-31G common basis set .It means that each core atomic orbital is simulated by a linear combination of 6 gaussian functions, the more gaussians used the more realistic the result, but remember the computational cost, and to the right of the dash is the valence term  where it is classified into inner and outer parts. The inner part is simulated by 3 gaussian functions and the outer by 1.

The shape and energy of these basis set functions differ significantly after interactions with each others forming new orbitals. These new orbitals are the one-electron spin-orbital functions we use in Slater Determinant SD and in the case of molecule they're called Molecular Orbitals MOs while in case of atom they're still (called) Atomic Orbitals AOs.

In other words, Basis set is a part of the input of a calculation and one-electron spin-orbitals are part of the output. 

Dear Moustafa,

Schreodinger equation gives not gaussian, but exponential exp(-kx) asymptotic for bound states. No number of gaussians can approximate it.

Dear Eugene, 

As far as I know, slater and gaussian orbitals are approximations to the exact hydrogen-like orbitals which are solutions of the schrodinger equation ?

If this is wrong, please clarify it to me in simpler language as I'm not a mathematician.

Regards, 

https://en.wikipedia.org/wiki/Atomic_orbital

https://en.wikipedia.org/wiki/Gaussian_orbital

It is functional analysis, Moustafa. Gaussians is not a complete set of functions. Not any function can be aproximated by them. Evidently, on the whole axis aproximation of exp(-kx) by gaussians can not be made because of different asymptotics. But from Schreodinger equation for bound states wave function asymptotic is exp(-kx).

Actually, you are wrong. Gaussians form a L^2 complete set of functions. Slater type orbitals (STOs) can be expressed exactly using Gaussian type orbitals (GTOs), as the latter can be viewed as a Laplace transform of the former [1].

The discussion has also strayed quite far from the original question, which was on the difference of molecular orbitals and basis functions. The basis functions, often also called atomic orbitals, are not really solutions to the hydrogenic Schrödinger equation. Yes, the form of STOs can be motivated by the hydrogenic Schrödinger equation, and the connection [1] was one reason why GTOs functions gained favor and eventually won over STOs in practical calculations, but the hydrogenic Schrödinger equation is not really relevant for general atoms and molecules.

The reason is that the hydrogenic equation completely dismisses inter-electronic interactions, i.e. Coulomb and exchange interactions. So, basis functions are *not* solutions to the hydrogenic Schrödinger equation. They're typically not solutions to the Hartree-Fock (or Kohn-Sham) equations either. If you look at a general Gaussian basis set, it is composed of a set of contracted functions, and free primitives. The contracted functions *may* be a true atomic orbital - the contraction coefficients being taken from a HF or KS orbitals of a free atom - but these are not fit to any a priori form for the orbital. Instead, the orbitals are determined by running a self-consistent field calculation for the free atom, and as a result you get the molecular orbital coefficients that you use to pick the contraction.

In addition to the contracted functions, basis sets have a large number of additional functions that add flexibility to represent changes in the charge density in a molecular environment, i.e. polarization. These do *not* have atomic orbital form. In addition to polarization functions, many basis sets also add functions to describe correlation effects for post-HF methods such as Møller-Plesset perturbation, configuration interaction or coupled-cluster theory. Again, these are *not* atomic orbitals.

While the above discussion was for GTOs, similar remarks also hold for STO basis sets [2] that are used in e.g. the ADF program.

[1] J. Chem. Phys. 22, 148 (1954); doi: 10.1063/1.1739831

[2] Int. J. Quantum Chem. 62, 1 (1997), doi: 10.1002/(SICI)1097-461X(1997)62:13.0.CO;2-#

Dear Susi,

exp(-kx) is not only asymptotic for Schreodinger equation, but for any bounded (finite) system. It is very common statement, as well as exp(-kt) for decay. You can never describe long range interecations (Van der Vaals and so on) by gaussians. It is well known problem.

Can You enumerate complete gaussian system?