# Basis Functions for Hartree Fock, and Configuration Interaction

I'm about halfway through the book Szabo and Ostlund, and while I think I understand the rough idea of Hartree-Fock and configuration interaction, there is something I would like to clarify.

For one, I'd like to note the fact that while the book often refers to the s, p, and such orbitals, the actual basis functions used to find the SCF are completely arbitrary, and picking a basis of any old functions (don't even have to be orthonormal) yield a valid Slater-Determinant where I can minimize coefficients to obtain a Hartree-Energy. This is true even if I picked really stupid functions, such as Gaussian orbitals centered 100's of A.U. away from any of the nuclei. Then the Hartree-Energy is just a really bad approximation to the ground state.

Furthermore, unless I only have basis functions equal to the number of electrons (minimal basis), my actual orbital solutions ($$|\psi_i\rangle$$) that I obtain in the end can be linear combinations of the basis sets ($$|\phi_j\rangle$$) through coefficients $$c_{ij}$$.

$$| \psi_i \rangle = \sum_{j}c_{ij}|\phi_j \rangle$$

By orbitals here I suppose I mean the eigenfunctions to the Fock-operator. These are ultimately found by being put into a slater determinant $$|\Psi\rangle = A|\psi_1\psi_2...\psi_N\rangle$$ (where $$A$$ is the anti-symmetrizer) and after acted on by the Fock-operator solved iteratively until convergence for the coefficients.

Now, of course we'd expect something closer to the actual energy if we use a larger basis set, but I'm a bit confused about the distinction between this and the configuration interaction. I read that the configuration ground state is give by:

$$|\Phi\rangle =c_0|\Psi\rangle+\sum_ac_a|\Psi_a\rangle + \sum_{ab} c_{ab}|\Psi_{ab}\rangle +\cdots$$

Where $$a$$ denotes a replacement of one of the orbitals by some single other (presumably orthogonal to rest) orbital. Never mind the fact that the book starts calling these "excitations" which to me invokes some sort of physical picture.

My question is that didn't this already happen when we expanded $$| \psi_i \rangle$$ in terms of the basis functions? Why do we need further determinants to correct the ground state? If we used a complete basis before it should be exact? It's not immediately clear to me that summing over Slater determinants creates a bigger basis than taking a Slater determinant over arbitrary sums. I read that the energy is lowered due to 'correlation' effects, but I'm afraid I don't understand the nature of these or encountered a good simple example of what's really happening.

Edit: Disregard spin for this question

• Hartree-Fock method with its variational slater determinental ansatz gives an approximate many electron wave function, which is not a true eigen function of many electron Hamiltonian. True eigen function can be expanded (CI) in terms of Hartree Fock determinants. This expansion can be organized in terms of number of excitations (in terms of Roothans orbitals) over hartree fock ground state. – Sunyam Feb 11 '19 at 20:59
• You construct a different Hamiltonian in CI and the groundstate of this Hamiltonian is closer to the true energy of the groundstate, in principle exact if you are doing full CI plus infinite basis set. The CI matrix does restore the correlation/coupling/interaction of the electrons which is lost in Hartree-Fock by using the effective average potentials. So what matters is not the expansion alone but the fact that this expansion leads to a CI-Matrix(= representation of the Hamilton operator with determinants as basis) which contains new interactions that are completely missing in normal HF. – Hans Wurst Oct 1 '20 at 14:53

## 2 Answers

1) Quantum chemists are not interested in bad solutions. That is why they take atomic like basis functions. 2) the minimal basis is half the number of electrons, plus one if the latter is odd. 3) The hfs equations are an approximation to the Schrödinger equation valid for the case of a single determinant. In general a single determinant does not solve the many electron Schrödinger equation. The approximation can therefore be improved by including more determinants in a linear combination. CI is a procedure to select and combine more determinants.

• 1) I'm more interested in a theoretical understanding here. 2) I'm aware. 3) I understand, but I'm curious why sum over determinants =/= determinant of sums. – Connor Dolan Feb 11 '19 at 22:14
• 1) atomic orbitals are a good starting point. 2) the why do you state something else. 3) a sum of determinants cannot in general be written as a single determinant. – my2cts Feb 11 '19 at 23:37
• I just edited the OP, I'm not regarding spin. And ok, if there situations where a sum of determinants can't be written as a single determinant that answers my question as to the distinction and necessity of the configuration interaction. It wasn't obvious to me that this is the case. – Connor Dolan Feb 12 '19 at 0:27

Consider a single Helium atom. Clearly for each electron, the probability of finding it on the "left" or "right" side of the nucleus is the same; 50%. However, the probability that one electron is on the left side given that the other is on the left side is less than 50%. This is correlation, and accounts for the reduction in energy associated with a multi-particle wavefunction ansatz - i.e. one that can take account of particle distributions that are conditional on the location of other particles. The HF solution optimizes the spatial distribution of each electron in the presence of the mean field of the other, and so excludes this effect.