Open the NH3 H-1s SALC viewer full-screen
``` `--output` selects the HTML file name (default: `SALC_{molecule}_{element}_{orbital}.html`); `--bond EL1 EL2 MAX` draws bonds up to `MAX` Angstroms (repeatable). Since the viewer works on a molecule rather than a crystal, no cell edges are drawn and the compass shows the Cartesian x/y/z axes. The SALC analysis supports the 32 crystallographic point groups; for linear molecules (D\*h/C\*v) analyze a finite subgroup with `crystod-group --decompose` instead. ## 33. Molecular-orbital diagrams (`--diagram`) *Example directory: `example/33_molod` (testsuite section 33)* `crystod-mol --diagram` draws the molecular-orbital diagram of a single-center molecule (one central atom plus its ligands — NH3, CH4, SF6) from **symmetry and overlap alone**: the level ordering follows from the symmetry of the orbitals and from how strongly they overlap in space, with no self-consistent calculation: ```bash crystod-mol --diagram --xyz XYZ_NH3.xyz # -> MolOD_XYZ_NH3.html ``` ``` * Fragments * central atom: N; ligands: 3 H * Valence Atomic Orbital (AO) parameters (single-zeta STO, extended Hueckel) * N 2s: zeta = 1.950 / bohr, H_ii = -26.0 eV N 2p: zeta = 1.950 / bohr, H_ii = -13.4 eV H 1s: zeta = 1.300 / bohr, H_ii = -13.6 eV * Ligand SALCs (standard point-group axes) * -- 3H 1s -- A1: [1s(H1) + 1s(H2) + 1s(H3)] E: [1s(H1) - 1s(H3), 1s(H1) - 2 1s(H2) + 1s(H3)] * Ligand SALC | central AO overlap integrals * A1: < a1 (E = -16.44 eV) | N 2s > S = 0.7205 A1: < a1 (E = -16.44 eV) | N 2p > S = 0.2387 E: < e (E = -11.16 eV) | N 2p > S = 0.5687 * Molecular orbitals (Wolfsberg-Helmholz, K = 1.75) * MO E (eV) occ composition 4a1 21.14 0 70% SALC a1, 25% N 2s, 5% N 2p 2e x2 2.78 0 59% SALC e, 41% N 2p 3a1 -13.75 2 95% N 2p, 4% SALC a1, 1% N 2s 1e x2 -16.49 4 59% N 2p, 41% SALC e 2a1 -28.00 2 73% N 2s, 27% SALC a1 * Electron filling (8 valence electrons) * (2a1)^2 (1e)^4 (3a1)^2 (MO numbering counts the core shells, not shown: N 1s -> a1) HOMO = 3a1 (-13.75 eV), LUMO = 2e (2.78 eV), gap = 16.53 eV ``` The written `MolOD_XYZ_NH3.html` is the diagram below — this is the live output, not a screenshot. Four columns (isolated ligand AOs | ligand-group SALCs | molecular orbitals | central-atom AOs) are joined by dashed correlation lines drawn the more heavily the larger the contribution; electron arrows mark the occupied levels and the HOMO/LUMO are flagged. Hover or click any level to see its composition and a drag-rotatable orbital sketch built from that level's actual eigenvector (buttons flip through the degenerate partners); Ctrl/Cmd + scroll zooms the energy window: ```{raw} html ``` Everything in the report is one small secular problem per irrep: the ligand orbitals are symmetry-adapted into the SALCs of section 32, every valence orbital is a single-zeta Slater-type orbital with a tabulated extended-Hückel exponent `zeta` and diagonal energy `H_ii`, all two-center overlaps `S` are evaluated exactly (ligand-ligand overlap included), and orbitals of different irreps cannot mix. A large overlap therefore means a large bonding/antibonding splitting. ```{seealso} **Theory:** [How the orbital diagrams are computed](theory-orbital-diagrams.md) — the step-by-step extended-Hückel construction, the exact overlap quadrature and the original references. ``` The MO numbering counts the core shells, as in photoelectron spectroscopy. CH4 therefore gives the textbook `(2a1)^2 (1t2)^6` (1a1 = C 1s core), NH3 gives `(2a1)^2 (1e)^4 (3a1)^2` with the 3a1 lone pair as HOMO, and SF6 fills 48 valence electrons up to the nonbonding F 2p block (1t1g/3eg) just below the 6a1g LUMO. `--center EL` selects the central atom explicitly (default: the atom closest to the molecular center); `--output`/`--tolerance` as usual. Elements H-Cl are parameterized, and the energies are semi-quantitative as extended Hückel theory always is — read them for orderings and trends, not as absolute values. ### Two-fragment diagrams (`--ao-left`/`--ao-right`) `--ao-left`/`--ao-right` (without `--pyscf`) replace the central-atom architecture by **any submolecule split by chemical formula**, for molecules without a single center: ```bash crystod-mol --diagram --xyz XYZ_C6H6.xyz --ao-left H6 --ao-right C6 # -> MolOD_XYZ_C6H6.html (H6 MOs | C6H6 MOs | C6 MOs) crystod-mol --diagram --xyz XYZ_CH3OH.xyz --ao-left H4 --ao-right CO crystod-mol --diagram --xyz XYZ_O2.xyz --ao-left O --ao-right O ``` All three columns are solved in the **one molecular AO space**. Because the Wolfsberg-Helmholz off-diagonal depends only on the two orbitals involved, a fragment's (H, S) sub-block *is* the isolated fragment: its eigenstates are the pre-bonding fragment MOs — benzene's H6 column shows the 1a1g < 1e1u < 1e2g < 1b2u ladder of the six H 1s SALCs — and the molecular MOs are projected onto them through the shared overlap matrix. Benzene fills `(2a1g)^2 (2e1u)^4 (2e2g)^4 (2b2u)^2 (3a1g)^2 (3e1u)^4 (1a2u)^2 (1b1u)^2 (3e2g)^4 (1e1g)^4`, i.e. the Hückel π ladder 1a2u < 1e1g (HOMO) < 1e2u (LUMO) < 1b1g. Irrep labels are omitted when a fragment admits no consistent assignment (and for linear molecules, which have no crystallographic character table). ### Quantitative diagrams with PySCF (`--pyscf`) `--diagram --pyscf` replaces the extended-Hückel estimate by **three self-consistent-field calculations sharing one AO space**: the full molecule and its two fragments, the latter carrying **ghost basis functions** on the removed atoms (the Boys-Bernardi counterpoise construction). The fragment levels are therefore true pre-bonding states described in the full molecular basis, the molecular MOs are projected exactly onto them, and the printed interaction energy E(mol) − E(left) − E(right) is BSSE-free. `--pyscf` needs the optional PySCF dependency, `pip install "CrystOD[quantum]"` (see [Installation](install.md)); without it the run stops with a one-line `ERROR:` naming this command: ```bash crystod-mol --diagram --xyz XYZ_H2O.xyz --pyscf # H2 | H2O | O crystod-mol --diagram --xyz XYZ_O2.xyz --pyscf --spin 2 --ao-left O --ao-right O crystod-mol --diagram --xyz XYZ_CH3OH.xyz --pyscf --ao-left H4 --ao-right CO crystod-mol --diagram --xyz XYZ_C6H6.xyz --pyscf --ao-left H6 --ao-right C6 ``` The benzene run produces the diagram below (live output of the last command): the H6 cage on the left, the C6 ring on the right, and the pure-π `1e1g` HOMO / `1e2u` LUMO of benzene in the middle — click the π levels to see the familiar nodal patterns in the orbital sketch: ```{raw} html ``` - The default fragments are the ligand cage | central atom, as in the symmetry-only mode. `--ao-left`/`--ao-right` select **any partition by chemical formula**, which is required for molecules with no single central atom (benzene as `H6`/`C6`) and for homonuclear diatomics (`O` / `O`, displayed as O(L)/O(R)). - Irrep labels come from crystod's own point-group machinery — the characters of every MO evaluated on the PySCF AO basis — so they agree with the symmetry-only diagram and with `crystod-group`: CH4 `1t2` HOMO, H2O `1b2`, benzene `1e1g` HOMO / `1e2u` LUMO. Linear molecules use PySCF's Dooh/Coov labels rendered as σ/π/δ (triplet O2: `(1πu)^4 (1πg)^2`, with two single arrows on the half-filled 1πg). - `--basis` (default def2-svp), `--theory scf|dft`, `--xc` (default b3lyp), `--charge` and `--spin` follow the conventions of `script/calc_pyscf.py`. `--spin` is given as 2S, the number of unpaired electrons, and defaults to the smallest value compatible with the electron count. Open-shell systems are treated with ROHF/ROKS, so the diagram keeps one set of orbital energies. - Core levels are computed and included. When levels lie below −40 eV the default window is clamped to [−40, 15] eV so the valence region stays legible; the **Show all energy levels** button expands it. ```{seealso} **Theory:** [How the orbital diagrams are computed](theory-orbital-diagrams.md) — why a finite basis creates the basis-set superposition error, what the counterpoise correction fixes, why the fragments are spin/spherically averaged, and how ghost-dominated fragment levels are filtered out. ``` The SCF engine is PySCF — a software dependency, not the source of the fragment/irrep construction above. If you use PySCF in your research, please cite: Q. Sun et al., J. Chem. Phys. 153, 024109 (2020); Q. Sun et al., WIREs Comput. Mol. Sci. 8, e1340 (2018); Q. Sun, J. Comput. Chem. 36, 1664 (2015).