# The many-electron problem & Born–Oppenheimer

Everything qc-rs computes is, at bottom, an approximate solution of one equation — the **Schrödinger
equation** for a molecule. This chapter writes that equation down, explains why it cannot be solved
exactly, and takes the first essential step toward making it tractable: separating the motion of the
nuclei from that of the electrons.

## A molecule, quantum-mechanically

A molecule is a collection of positively charged **nuclei** and negatively charged **electrons**, held
together by electrostatic (Coulomb) forces. In quantum mechanics its stationary states are described by a
**wavefunction** $\Psi$ and obey the time-independent Schrödinger equation

$$
\hat H\,\Psi = E\,\Psi ,
$$

where $\hat H$ is the **Hamiltonian** operator (the total energy) and $E$ is the energy of the state. Solve
this and you know, in principle, everything: the energy, the structure, and every property.

:::{note} Atomic units
Quantum chemistry works in **atomic units**, chosen so that the electron mass, elementary charge, $\hbar$,
and the Coulomb constant are all $1$. Two you will see constantly: energies are in **hartree**
($1\ E_h \approx 27.211\ \text{eV} \approx 627.5\ \text{kcal/mol}$) and lengths in **bohr**
($1\ a_0 \approx 0.529\ \text{Å}$). This is why the quickstart energy was a plain number like
`-76.026772` — it is in hartree. When you pass `unit="angstrom"`, qc-rs converts your coordinates to bohr
internally.
:::

## The molecular Hamiltonian

For $M$ nuclei (labelled $A,B,\dots$, with charges $Z_A$ and masses $M_A$) and $N$ electrons (labelled
$i,j,\dots$), the Hamiltonian in atomic units is a sum of five physically distinct terms:

$$
\hat H
= \underbrace{-\sum_{A}\frac{1}{2M_A}\nabla_A^2}_{\text{nuclear kinetic}}
  \;\underbrace{-\sum_{i}\frac{1}{2}\nabla_i^2}_{\text{electron kinetic}}
  \;\underbrace{-\sum_{A,i}\frac{Z_A}{r_{Ai}}}_{\text{electron–nucleus attraction}}
  \;\underbrace{+\sum_{i<j}\frac{1}{r_{ij}}}_{\text{electron–electron repulsion}}
  \;\underbrace{+\sum_{A<B}\frac{Z_A Z_B}{R_{AB}}}_{\text{nucleus–nucleus repulsion}} ,
$$

where $r_{Ai}=|\mathbf r_i-\mathbf R_A|$ is an electron–nucleus distance, $r_{ij}=|\mathbf r_i-\mathbf r_j|$
an electron–electron distance, and $R_{AB}$ a nucleus–nucleus distance. Every term is either kinetic energy
or a Coulomb interaction — nothing exotic. The difficulty is entirely in how they are *coupled*.

## Why it cannot be solved exactly

The villain is the **electron–electron repulsion** term $\sum_{i<j} 1/r_{ij}$. It ties the coordinates of
every electron to those of every other, so the $N$-electron wavefunction
$\Psi(\mathbf r_1,\mathbf r_2,\dots,\mathbf r_N)$ does **not** separate into a product of one-electron
functions. This coupling — that each electron's best move depends on where all the others are — is called
**electron correlation**, and it is the reason there is no closed-form solution for anything past the
one-electron hydrogen atom. Worse, the wavefunction lives in $3N$ spatial dimensions: even storing it on a
grid is hopeless for more than a few electrons. The whole enterprise of quantum chemistry is the search for
*good, controlled approximations*.

Electrons carry one more essential rule. They are **fermions**, so the wavefunction must be
**antisymmetric**: swapping the full coordinates (space and spin) of any two electrons flips its sign,

$$
\Psi(\dots,\mathbf x_i,\dots,\mathbf x_j,\dots) = -\,\Psi(\dots,\mathbf x_j,\dots,\mathbf x_i,\dots) .
$$

This is the **Pauli principle**; the next chapters build it in from the start (via Slater determinants).

## The Born–Oppenheimer approximation

The first great simplification exploits a huge disparity: a nucleus is at least ~1800 times heavier than an
electron, so electrons move far faster and effectively "see" the nuclei as stationary. We may therefore
**clamp the nuclei** in place and solve for the electrons alone.

:::{prf:definition} Born–Oppenheimer approximation
:label: def-bo

Treat the nuclear positions $\{\mathbf R_A\}$ as fixed parameters. The **electronic Hamiltonian** drops the
nuclear kinetic term and treats the constant nuclear repulsion separately,

$$
\hat H_{\text{elec}}
= -\sum_{i}\frac{1}{2}\nabla_i^2 - \sum_{A,i}\frac{Z_A}{r_{Ai}} + \sum_{i<j}\frac{1}{r_{ij}} ,
$$

and one solves the **electronic Schrödinger equation** for a fixed geometry,

$$
\hat H_{\text{elec}}\,\Psi_{\text{elec}} = E_{\text{elec}}\,\Psi_{\text{elec}} .
$$

The **total energy** at that geometry is the electronic energy plus the (now constant) nuclear repulsion,

$$
E(\mathbf R) = E_{\text{elec}}(\mathbf R) + \sum_{A<B}\frac{Z_A Z_B}{R_{AB}} .
$$
:::

Two ideas fall out of this that you already met in Part I:

- **The potential energy surface (PES).** Because $E(\mathbf R)$ depends on the fixed geometry, sweeping the
  nuclei traces out an energy landscape. A **minimum** of this surface is an equilibrium structure — exactly
  what the [geometry-optimization tutorial](../00-intro/tutorial-dft-to-properties.md) walked downhill to
  find. Its slope is the force on the nuclei (the *gradient*).
- **Nuclear repulsion is just an additive constant.** For a single geometry the $\sum_{A<B} Z_A Z_B/R_{AB}$
  term is a number added at the end. This is why the total energy the quickstart printed already includes
  it — the electronic problem is where all the difficulty lives.

## Where this leaves us

Born–Oppenheimer reduces "solve a molecule" to "solve the electrons for a fixed set of nuclei," but the
electronic Schrödinger equation is *still* an intractable many-body problem, thanks to correlation and the
$3N$ dimensions. The next chapter shows the standard escape: stop trying to find $\Psi$ exactly, and instead
**expand it in a finite basis and minimize the energy** — turning calculus into linear algebra.