# Variational principle, LCAO & basis sets

The [previous chapter](many-electron-and-bo.md) left us with the electronic Schrödinger equation — exact,
but unsolvable. This chapter introduces the strategy that makes quantum chemistry *computable*: stop
searching for the exact wavefunction, restrict it to a form we can handle, and **minimize the energy**
within that form. Two ingredients make this concrete — the **variational principle** (why minimizing
works) and **basis sets** (the form we restrict to). Together they turn a differential equation into a
matrix problem a computer can solve.

## The variational principle

Suppose we cannot find the true ground-state wavefunction $\Psi_0$, so we guess a **trial wavefunction**
$\Psi$ instead. How good is our guess? The variational principle gives a remarkable answer: the energy of
*any* trial wavefunction is an upper bound to the true ground-state energy.

:::{prf:theorem} Variational principle
:label: thm-variational

For any well-behaved trial wavefunction $\Psi$, the **Rayleigh quotient**

$$
E[\Psi] = \frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}
$$

satisfies $E[\Psi] \ge E_0$, where $E_0$ is the exact ground-state energy, with equality **iff**
$\Psi = \Psi_0$.
:::

This single fact reshapes the whole problem. Because a lower energy always means a *better* wavefunction,
we do not need to solve the Schrödinger equation directly — we can **parameterize** $\Psi$ with some
adjustable numbers and **minimize $E[\Psi]$** over them. Calculus (solving a differential equation) becomes
optimization (finding the lowest energy). Every method in this manual — Hartree–Fock, DFT, and the
correlated methods — is a particular choice of trial form plus this minimization.

## Molecular orbitals and the LCAO idea

The parameters have to come from somewhere. The organizing idea is the **molecular orbital (MO)**: a
one-electron function $\psi_i(\mathbf r)$ that describes how a single electron is spread over the whole
molecule. We will build the many-electron wavefunction out of MOs (the next chapter makes this precise with
a Slater determinant); for now the question is just how to *represent* one MO.

We cannot store a general 3-D function exactly, so we **expand each MO in a fixed set of simple functions**
— the **basis set**. The functions $\{\phi_\mu\}$ are atom-centered "atomic orbitals," and an MO is a
**Linear Combination of Atomic Orbitals** (LCAO):

$$
\psi_i(\mathbf r) = \sum_{\mu=1}^{K} C_{\mu i}\,\phi_\mu(\mathbf r) .
$$

Now the unknowns are just the numbers $C_{\mu i}$ — the **MO coefficients**. Minimizing the energy over
them is a finite problem: an MO is fixed once you know its $K$ coefficients.

## What the basis functions are

In practice the atomic orbitals $\phi_\mu$ are built from **Gaussian-type orbitals (GTOs)**. A primitive
Gaussian centered on atom $A$ has the form

$$
g(\mathbf r) = N\,x_A^{\,a} y_A^{\,b} z_A^{\,c}\; e^{-\alpha\,|\mathbf r - \mathbf A|^2},
$$

where $(x_A,y_A,z_A)=\mathbf r-\mathbf A$, the exponents $a,b,c$ set the angular shape (their sum is the
angular momentum: $0=s$, $1=p$, $2=d$, …), $\alpha$ controls the width, and $N$ normalizes it. Real basis
functions are **contracted** — fixed linear combinations of several primitives that mimic an atomic
orbital,

$$
\phi_\mu(\mathbf r) = \sum_{k} d_{k\mu}\,g_k(\mathbf r),
$$

with the contraction coefficients $d_{k\mu}$ baked into the basis-set definition.

:::{note} Why Gaussians?
A hydrogen-like orbital actually decays like $e^{-\zeta r}$ (a **Slater** function), which Gaussians
reproduce only imperfectly (they are too flat at the nucleus and fall off too fast). Gaussians win anyway
for one decisive reason: the **product of two Gaussians on different centers is a single Gaussian on a
point between them**. That identity makes the millions of two-electron integrals a molecule needs
*analytically* fast — which is exactly why qc-rs (via the libcint library) uses them.
:::

## Basis-set families

Bigger, more flexible basis sets describe the electrons better — at higher cost. The vocabulary you will
meet in `ao=`:

- **Minimal** (e.g. `sto-3g`) — one function per occupied atomic orbital. Cheap, qualitative.
- **Split-valence / multiple-zeta** (e.g. `cc-pvdz` = double-zeta, `cc-pvtz` = triple-zeta) — two, three, …
  functions per valence orbital, letting orbitals shrink or expand in different bonding environments.
- **Polarization functions** — higher angular momentum than the atom needs in isolation (a $d$ on oxygen, a
  $p$ on hydrogen), letting orbitals distort along bonds. The "P" in cc-pV*D*Z is these.
- **Diffuse functions** — extra broad functions (a `aug-` prefix) for anions, lone pairs, and long-range
  interactions.

The **correlation-consistent** family `cc-pVXZ` ($X=$ D, T, Q, …) is designed to approach the exact answer
smoothly as $X$ grows.

## From coefficients to a matrix problem

Insert the LCAO expansion into the Rayleigh quotient and minimize over the coefficients $C_{\mu i}$. Because
the atomic orbitals are **not orthogonal** (neighbouring atoms' functions overlap), the result is a
*generalized* matrix eigenvalue problem,

$$
\mathbf{F}\,\mathbf{C} = \mathbf{S}\,\mathbf{C}\,\boldsymbol{\varepsilon},
\qquad S_{\mu\nu} = \langle \phi_\mu | \phi_\nu \rangle ,
$$

where $\mathbf S$ is the **overlap matrix**, $\mathbf F$ encodes the energy, $\mathbf C$ holds the MO
coefficients, and $\boldsymbol\varepsilon$ the orbital energies. The exact form of $\mathbf F$ is the
subject of the [next chapter](hartree-fock.md); the point here is that **the impossible differential
equation has become a $K\times K$ matrix problem**, and $K$ — the number of basis functions — is what you
choose when you pick `ao=`.

## Basis-set convergence, in numbers

Here is the whole idea in one experiment. The RHF energy of water, in three basis sets of growing size:

| `ao=` | kind | basis functions ($K$) | RHF energy / $E_h$ |
|---|---|---|---|
| `sto-3g` | minimal | 7 | −74.963023 |
| `cc-pvdz` | double-zeta | 24 | −76.026772 |
| `cc-pvtz` | triple-zeta | 58 | −76.057127 |

The energy drops steadily toward the true Hartree–Fock limit as the basis grows — exactly as the
variational principle promises (a bigger, more flexible trial form can only lower the energy). But notice
$K$ grows fast (7 → 24 → 58), and the cost of the two-electron integrals grows *much* faster than $K$, so
in practice you choose the smallest basis that is accurate enough. This basis-set incompleteness is one
half of the "basis and method" caveat behind the slightly-too-large water dipole in the
[tutorial](../00-intro/tutorial-dft-to-properties.md); the other half — the *method* — is next.

:::{tip} What qc-rs does with `ao=`
`ao="cc-pvdz"` tells qc-rs which contracted GTOs to place on each atom. `ao_rep="spherical"` (the default)
uses the $2\ell+1$ spherical harmonics per shell (5 $d$-functions, not 6); `ao_rep="cartesian"` keeps the
Cartesian set. The count $K$ above is for the spherical default.
:::

With the wavefunction reduced to a finite matrix problem, we can finally specify $\mathbf F$ — the
[Hartree–Fock](hartree-fock.md) method.