In this exercise we will confirm that both the variational principle is working and that the LCAO approach works. We can do this by numerically calculating 1-electron systems for which analytically exact solutions exist and for which the LCAO-MO Hartree-Fock approach should give a numerically exact solution as well. 1-electron systems are the only system where the Hartree-Fock method is in principle exact and thus if the basis set expansion is close to complete, the energy should converge at the exact result.
The LCAO approximation makes use of basis functions from a basis set. A basis set is simply a collection of suitable (always Gaussians in ORCA) functions that will be used in the calculation. The LCAO-MO procedure will take linear combinations of all the functions in the basis set and variationally solve the Hartree-Fock equations. In a later lecture we will learn more details about what these basis sets consist of.
Example Gaussian basis sets, available in ORCA are:
MINIX, def2-SVP, def2-TZVP, def2-QZVPP, cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z, cc-pV6Z
We will look at 2 systems:
- The hydrogen atom. Exact energy is -0.5 hartree
- The H2+ diatomic molecule. The only molecule that can be solved exactly. Exact energy is -0.602634214495 hartree. Taken from: DOI: 10.1016/j.chemphys.2011.09.013
1. Calculate the H atom using the Hartree-Fock method (HF keyword; not HF-3c keyword) and all or most of the listed basis sets (above). Feel free to try others (see manual).
2. Calculate the H2+ diatomic molecule using HF method and the same basis sets. Use a bond length of 2 Bohrs.
Note: You can either convert Bohrs to Angstrom or tell ORCA that you are using Bohrs in the coordinate block. See ORCA Input Library or manual regarding this.
3. Calculate the bond dissociation energy of H2+. Is it homolytic or heterolytic ?
How close can you get to the exact result for both systems?
At what basis set is the accuracy good enough you think ?
Hint: You need to convert the energy difference to kcal/mol or eV and compare to something to answer this.
Does the energy always get lower when you use a larger basis set?
Does the LCAO-MO approach work?
How much is the nuclear repulsion energy for H and H2+ ? What is the electronic energy without the nuclear repulsion?
The report should give a table of basis set name, the number of functions in the basis set and the energy in hartrees.
Also a plot of energy as a function of the number of basis functions.
Notes:
- The number of basis functions can be found in the ORCA outputfile for each calculation. Look for "contracted basis functions".
- For both H and H2+ you should not include the "Opt" keyword. There is nothing to optimize for H (no degree of freedom) and for H2+ we want the geometry to have exactly a bond length of 2 Bohrs.