6.2 Populations, orbitals, and density cubes
These are unexecuted teaching inputs and starting models. Original diagrams are schematics, not calculated results. Validate version-specific syntax, licensed or authorized data, numerical convergence and the scientific model before using this workflow.
6.2.1 Model, units and provenance
Geometry in Å; electronic energy in hartree; vibrational wavenumbers in cm⁻¹. Check each printed field and keep thermal and standard-state terms distinct.
Shared inputs, conventions and evidence
6.2.2 Unexecuted inputs and explicit deltas
Use the accompanying instructions to identify the parent calculation and placement of every delta; a snippet is not automatically a standalone input. Preserve all blank-line and file-provenance requirements.
6.2.2.1 Input block 1
%oldchk=formaldehyde20_gs.chk
%chk=formaldehyde22_analysis.chk
%mem=2GB
%nprocshared=2
#p CAM-B3LYP/aug-cc-pVDZ Geom=AllCheck Guess=Read
Pop=(Full,Hirshfeld) Density=SCF SCF=(Tight,XQC) Int=UltraFine
6.2.2.2 Input block 2
formchk formaldehyde22_analysis.chk formaldehyde22_analysis.fchk
cubegen 0 Density=SCF formaldehyde22_analysis.fchk formaldehyde22_density.cube 80 h
cubegen 0 MO=HOMO formaldehyde22_analysis.fchk formaldehyde22_homo.cube 80 h
cubegen 0 MO=LUMO formaldehyde22_analysis.fchk formaldehyde22_lumo.cube 80 h
cubegen 0 Potential=SCF formaldehyde22_analysis.fchk formaldehyde22_esp.cube -1 h formaldehyde22_density.cube
6.2.3 Worked investigation
6.2.3.1 Question, intuition, and prerequisites
What does a molecular “charge picture” actually show? An electron density is a spatial field; an atomic charge is a chosen partition of that field; an orbital is a signed amplitude. These are different objects. Formaldehyde provides a compact way to compare oxygen polarization, frontier orbitals, and two population schemes without pretending that a single atomic charge is observable. Prerequisites: the ground-state formaldehyde20_gs.chk, the installed Gaussian formchk and cubegen utilities, and a viewer that reads Gaussian cube files.
6.2.3.2 Exact analysis delta
See input block 1 above.
The source checkpoint is explicitly the ground-state minimum, not the excited geometry or TD density. Pop=Full increases orbital/population output; Hirshfeld requests that partitioning analysis. The current Gaussian documentation also associates CM5 output with this option. Record the actual headings in your own output, rather than treating every printed charge column as interchangeable.
6.2.3.3 Reproducible cube commands
Run these in the licensed Gaussian environment after the analysis job ends
See input block 2 above.
These are shell commands, not Gaussian route lines. The last command reuses the density cube's grid for electrostatic-potential mapping. h retains the cube header. 80 requests an automatically defined rectangular grid with 80 points along each direction; it is not a physical box length. The leading zero is a supported processor/default argument, not an instruction to allocate zero memory. Do not assume all cube types run faster just because the Gaussian SCF used several cores.
6.2.3.4 Numbered workflow
- Verify the final SCF solution and that formaldehyde has 16 electrons: eight doubly occupied spatial orbitals in this closed-shell calculation. Locate occupied/virtual eigenvalues.
- Record Mulliken and Hirshfeld charges by atom, including their sums. Discuss agreement in qualitative polarization separately from agreement in numeric magnitude.
- Generate the cubes. Open HOMO and LUMO with the same absolute amplitude isovalue, for example ±0.03 atomic units, and label both phase colors.
- Show electron density at a separately stated density isovalue, for example 0.001 e bohr⁻³, then map the electrostatic potential onto that surface with a symmetric, explicitly stated color range.
- Repeat a cube at a finer grid and inspect whether nodal features and the plotted surface are stable. For quantitative integration, refine both resolution and box extent; visual smoothness alone is insufficient.
6.2.3.5 Read and check
Atomic charges should sum to the molecular charge, here zero, within printed rounding/integration error. A density integrated over an adequate, sufficiently fine box should approach 16 electrons; a coarse 80³ box is not guaranteed to yield high-precision core-density integrals. The occupied-orbital count is the more immediate bookkeeping check. The total electron density is nonnegative; opposite signs on an MO surface indicate wavefunction phase, not positively and negatively charged electron clouds. Reversing all signs of one orbital leaves its density unchanged.
The HOMO is often interpreted as oxygen lone-pair-like and the LUMO as carbonyl π*-like in this molecule, but verify the actual nodal pattern and method-dependent ordering. A plotted LUMO is an unoccupied mathematical orbital, not occupied electron density. Mulliken charges can vary strongly with basis; different partitioning schemes may disagree without either job being broken. Electrostatic potential includes nuclear and electronic contributions; its sign legend must be stated. Never subtract two cubes until their grids, origins, orientation, and density definitions match. A Kohn–Sham orbital gap remains distinct from an optical excitation or fundamental gap.
6.2.3.6 Exercise
EN exercise: Draw the same HOMO at ±0.02 and ±0.05 atomic units. Explain why the visible lobe size changes without the molecule gaining or losing electrons. Compare the oxygen charge across the two schemes and state which conclusion is robust enough to describe as a qualitative trend.