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I am trying to understand how to quantify the electronic interaction/conduction/overlap between two metal cores at varying distances. As someone who works primarily with classical MD, and has less experience in first-principles simulations, I wanted to know if those more experienced could give some recommendations on the most appropriate simulation/software to do this, and the type of output that would be used to quantify this.

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I recommend the use of Multiwfn package. This software is free with Windows/Linux versions. It uses the wave function calculated from other software.

From the site:

Briefly speaking, Multiwfn can perform wavefunction analyses based on outputted file of almost all well-known quantum chemistry programs, such as Gaussian, ORCA, GAMESS-US, Molpro, NWChem, Dalton, xtb, PSI4, Molcas, Q-Chem, MRCC, deMon2k, Firefly, CFour, Turbomole...

This imply you have to set-up your calculations using one of those software above and then, use Multiwfn to run the analysis you want.

Some of the analysis it is able to do:

  • Topology analysis for any real space function, such as electron density (AIM analysis), Laplacian, ELF, LOL, electrostatic potential and so on. Critical points (CPs) can be located, topology paths and interbasin surfaces can be generated, and then they can be directly visualized in a 3D GUI window or be plotted in plane map. Value of various real space functions can be calculated at critical points or along topology paths. CP properties can be decomposed as orbital contributions.
  • Population analysis. Hirshfeld, Hirshfeld-I, VDD, Mulliken, Löwdin, Modified Mulliken (including three methods: SCPA, Stout & Politzer, Bickelhaupt), Becke, ADCH (Atomic dipole moment corrected Hirshfeld), CM5, CHELPG, Merz-Kollmann, RESP (Restrained ElectroStatic Potential), RESP2, AIM (Atoms-In-Molecules), EEM (Electronegativity Equalization Method) and PEOE (Gasteiger) are supported. Electrostatic interaction energy of two given fragments can be calculated based on atomic charges.
  • Orbital composition analysis. Mulliken, Stout & Politzer, SCPA, Hirshfeld, Hirshfeld-I, Becke, natural atomic orbital (NAO) and AIM methods are supported to obtain orbital composition. Orbital delocalization index (ODI) can be outputted to quantify extent of spatial delocalization of orbitals.
  • Plotting total, partial, overlap population density-of-states (TDOS, PDOS, OPDOS) and MO-PDOS. Up to 10 fragments can be very flexibly and conveniently defined. Local DOS (LDOS) can also be plotted for a point as curve map or for a line as color-filled map. Furthermore, plotting photoelectron spectrum (PES) based on (generalized) Koopmans' theorem is fully supported.
  • Analyzing real space functions in fuzzy atomic spaces (defined by Becke, Hirshfeld or Hirshfeld-I partitions). These quantities can be computed: Integral of real space functions in atomic spaces or in overlap region between atomic spaces, dipole and multipole moments of atom/fragment/molecule, atomic overlap matrix (AOM), localization and delocalization indices (LI, DI), condensed linear response kernel, multi-center DI, as well as four aromaticity indices, namely FLU, FLU-pi, PDI and information-theoretic index.

... and many more.

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By the "metal cores", If you're referring to the overlapping of wave functions of electrons of metal ions separated by a distance, then you can in principle use quantum ESPRESSO to get some insights. You can define a CIF file with the ions separated at varying distances from each other and visualize the electronic density.

Apart from quantum ESPRESSO, you can also use CP2K to create STM images in conduction mode. I'm not quite sure about the latter but it's worth a try.

Hope this helps :)

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