# Simple question about Hartree-Fock algorithm

Classically in Hartree-Fock implementations we define the wave-function in term of a basis set:

$$\phi_i = \sum_{\alpha} C_{\alpha i}\chi_\alpha$$

If I understand correctly the problem is reduced to find a set of these coefficients which minimise the total energy. Again, if my understanding is correct these coefficients are initially given in the basis set. If you look at STO-3G for example, there are always two numbers: the coefficient and the exponent.

What I am confused about is: In the algorithm we have to get a "first guess" of the density matrix. The density matrix is defined as:

$$D_{uv} = \sum_{i} C_{ui}C_{vi}$$

Why not just take the one given in the basis set as an initial guess?

• No, the initial guesses for the basis set coefficients are not part of the basis set. Gaussian basis sets usually include two parameters per Gaussian primitive. One is the Gaussian exponent and the other is the expansion coefficient for the current basis function. This expansion coefficient is fixed, and is used to combine multiple Gaussian primitives into a single basis function. Jun 19, 2022 at 21:57
• Ok that's what I understood after I saw that the matrix C is a square matrix (which doesn't make sense with my original understanding). The expansion coefficients (written "C" in many books...) for each gaussian primitive function are not the same as the one in the first equation of the question, which are called basis set coefficients... Jun 19, 2022 at 22:30

The question is confusing two things: the linear combination of atomic orbitals (LCAO) scheme where the molecular orbitals are expanded as a sum of the atomic orbitals $$\begin{equation} \psi_{i\sigma} ({\bf r}) = \sum_{\alpha} C_{\alpha i} \chi_\alpha ({\bf r}) \label{eq:lcao} \end{equation}$$ and the actual definition for the atomic basis functions, which in the case of contracted Gaussian basis functions is $$\begin{equation} \chi_\alpha ({\bf r}) = r^{l_\alpha} \left[\sum_{k} d_{k \alpha} \exp(-\zeta_kr^2)\right] Y_{l_\alpha}^{m_\alpha}(\hat{\bf r}) \end{equation}$$ where I have assumed a pure spherical basis for ease of notation. The LCAO scheme is not limited to Gaussian basis sets; the linear expansion ansatz is used in every basis set method, regardless of the nature of the basis functions (e.g. Gaussian or Slater-type orbital, numerical atomic orbital, plane waves, finite elements, etc). I have recently written two open access review articles on these topics, see Int. J. Quantum Chem. 119, e25968 (2019) for discussion on various kinds of basis functions and Molecules 25, 1218 (2020) for the self-consistent field algorithm.
I would like to note that unlike stated in the comments, the matrix $${\bf C}$$ is not square in general; this is unfortunately still a very common misconception in computational chemistry. There are often linear dependencies in the basis set (e.g. a basis function on one atom can be also well-described by the sum of basis functions on the neighboring atoms), and removing them removes some columns from $${\bf C}$$. Stereotypical cases of this happening are when you use diffuse functions that are needed to describe long-range interactions and weakly bound electrons, or when two nuclei are close together. However, even such pathological linear dependencies can be cured and Coulomb barriers to nuclear fusion modeled when suitable numerical algorithms are chosen to pick a suitable sub-basis (see the links). The choice of the linearly independent basis is also discussed in the overview on SCF theory linked above.