# How can I calculate the ground state energy?

Jul 20, 2017

Well, you would have to know the Hamiltonian $\hat{H}$ for your system, which in general is not easy...

Then you would have to go through this mess of solving the Schrodinger equation for your system, which is analytically impossible for atoms with more than one electron due to electron correlation (which is instantaneous and unpredictable without further approximations). It sure would be analytically impossible for molecules!

And as such, these days we resort to iterative, self-consistent methods to computationally minimize ground state energies in accordance with the variational method:

E_(phi) = (int_("allspace") psi^"*"hatH psid tau)/(int_("allspace") psi^"*"psid tau) >= E_(grd),

where ${E}_{g r d}$ is the ground-state energy, ${E}_{\phi}$ is the energy obtained at each iteration, and $\psi$ is the wave function of the system.

The common general procedure is:

$1.$ Do a reference calculation, with some form of a Hartree-Fock level of theory (unrestricted, restricted, restricted open-shell, i.e. UHF, RHF, ROHF, or simply HF).

This usually makes some sort of mean-field approximation, i.e. a smear of electron density that approximates instantaneous electron interaction. $2.$ Start from the orbital guess that is generated from that calculation to do a calculation at the multi-configurational self-consistent field (MCSCF) level of theory. where the rectangled set of orbitals are in your selected active space (the set of orbitals important to chemical bonding).

This is the most accurate and efficient way to account for so-called $\boldsymbol{\textsf{\text{static}}}$ $\boldsymbol{\textsf{\text{correlation}}}$ (which arises from multiple degenerate electron configurations), as it allows interaction between multiple degenerate states to provide a more accurate description of the wave function.

$3.$ From here it's a free-for-all on how to account for $\boldsymbol{\textsf{\text{dynamic}}}$ $\boldsymbol{\textsf{\text{correlation}}}$ (short-range electron repulsions).

The most accurate way to do it (and also the most time-consuming) is to do a full configuration interaction (full CI)... In this, excited-state "character" is mixed into the wave function to allow a more accurate description of the molecule across the entire potential energy curve.

But no one ever does that in full... They always do some sort of approximation to it, like MRCI, MRCIS, MRCISD, CCSD(T), CCSD, etc., to save time and still get as accurate a ground-state energy as possible.