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Tutorial 3: Intro to Quantum Chemistry

This tutorial shows how to select a quantum chemistry method, visualize orbitals, and analyze the electronic wavefunction.

I. Build and minimize minimal basis set hydrogen

A. Build the molecule

This cell builds H₂ by creating the two atoms, and explicitly setting their positions.

Try editing this cell to:

• Create HeH⁺
• Create H₃⁺
• Change the atoms' initial positions

B. Run a hartree-fock calculation

The next cell adds the RHF energy model to our molecule, then triggers a calculation.

Try editing this cell to:

• Change the atomic basis
• Get a list of other available energy models (type mdt.models. and then hit the [tab] key)

C. Visualize the orbitals

After running the calculation, we have enough information to visualize the molecular orbitals.

D. Minimize the energy

Here, we'll run a quick energy minimization then visualize how the hydrogen nuclei AND the atomic wavefunctions changed.

II. Analyzing the wavefunction

The wavefunction created during QM calculations will be stored as an easy-to-analyze python object:

A. The atomic basis set

First, let's examine the atomic orbitals. The overlaps, fock matrix, coefficents, and density matrix are all available as 2D numpy arrays (with units where applicable):

The 1- and 2-electron parts of the hamiltonian are also available:

B. The canonical molecular orbitals

These are the orbitals that result from Hartree-Fock calculations - they diagonalize the fock matrix. All quantities that we looked at for the atomic orbitals are also available for the canonical orbitals.

MOs are, of course, a linear combination of AOs:

\begin{equation} \left| \text{MO}_i \right \rangle = \sum_j c_{ij} \left| \text{AO}_j \right\rangle \end{equation}

The coefficient $c_{ij}$ is stored at mos.coeffs[i,j]

Most MO sets are orthogonal; their overlaps will often be the identity matrix (plus some small numerical noise)

By definition, the fock matrix should be orthogonal as well; the orbital energies are on its diagonal.

The MolecularOrbitals class also offers several methods to transform operators into different bases. For instance, the overlap method creates an overlap matrix between the AOs and MOs, where olap[i,j] is the overlap between MO i and AO j: \begin{equation} \text{olap[i,j]} = \left\langle MO_i \middle| AO_j \right \rangle \end{equation}

Note: this matrix is NOT the same as the MO coefficient matrix, because AOs are not orthogonal.

C. Individual orbitals

You can work with inidividual orbitals as well. For instance, to get a list (in order) of the four AOs:

Many matrix elements can also be accessed directly from these orbitals.