Comutational Chemistry - From Basis Set to MOs

[IPnano]

I am wondering if someone can explain exactly how we (or the computer more specifically) move from the basis set to the molecular orbitals.

For example,

If we use a 3-21G basis set this means the following:

1. We are approximating three slater type orbitals (STOs) using contracted Gaussian functions.

2. The 3 in the basis set means we are using three contracted Gaussians to approximate the STO of the inner electrons, the 2 in the basis set means two contracted Gaussians for the inner-valence electrons and the 1 in the basis set is for using 1 contract Gaussian for the outer-valence electrons.

3. These three STOs, which we have approximated using contracted Gaussians, are brought together using a Linear Combination of Atomic Orbitals (LCAO) in order to approximate the molecular orbitals. In other words, the STOs are our basis functions, which we can assume are our atomic orbitals.

Now, if we use LCAO to create the molecular obritals (MOs), we should have 3 MOs since we are combining 3 atomic orbitals (the three STOs). Number of MOs equals the number of AOs.

Questions:

1. How does the computer generate three MOs? I can see how the LCAO would give us a SINGLE new MO for a SINGLE linear combination of basis functions. Where do the other two come from? If it has to do with manipulating the exponents, how are those exponents decided? Even if they are optimized, where do they come from and how are they limited to 3 MOs?

2. If we have a molecule with, say, 7 carbon atoms and we want to make an MO for this molecule, the inner electrons (1s2) of the carbon atom will be approximated using the 3 from the basis set. Is this saying that we will take an LCAO approach with SEVEN STOs all approximated using the 3 from our basis set? And the inner-valence electrons would be the 2s2 electrons approximated taking a linear combination of SEVEN STOs using the 2 from our basis set? And finally our outer-valence electrons using a linear combination of SEVEN STOs using the 1 from our basis set?

In other words, if we have 7 carbons atoms, are we using the basis set SEVEN times for EACH linear combination of STOs?

3. If the above is true, what does it mean to use the 3-21G on hydrogen since hydrogen only has 1 electron? By default is it only using the 3 from the basis set?

Thanks,

 

[jbsteele]

I appreciate any help anyone can provide. 1. The three MOs are generated by taking linear combinations of the three STOs (which have been approximated by contracted Gaussians) using the LCAO approach. The coefficients of the linear combinations are determined by solving the secular equation which is a matrix equation in the basis of the STOs. The exponents of the contractions are determined by optimization methods such as the Hartree-Fock method. The number of MOs is limited by the number of AOs (STOs in this case).2. Yes, if you have 7 carbon atoms, you will use the basis set seven times for each linear combination of STOs. 3. Yes, if you are using the 3-21G basis set on hydrogen, it will only use the 3 from the basis set.