How to calculate molecular orbitals of octaoxygen under high pressure?

Question

This question about octaoxygen:

Why is octaoxygen diamagnetic?,

was asked in chemistry, but I really don't have an answer in terms of molecular orbitals. I think it could be a mixing of the O$_2$'s $\pi^*$ orbitals that have been bent inwards the monoclinic structure due to the high pressure of the $\varepsilon$-phase (about 10 GPa), thus greatly reducing sample volume, but I can't find a ressonant combination of 8 $\pi^*$ mixed orbitals that works.

I know that Gaussian can calculate molecular orbitals, but I don't have access to it.

My question is: is there any package out there, preferably free open-source software, which can calculate molecular orbitals of octaoxygen, which works for the gigapascal range (perhaps simulating cluster pressure with external electrostatic field)?

Answer 1

TL;DR

Two recently published and easy-to-use methods come to mind:

• X-HCFF (1)
• GOSTSHYP (2)

Both are available in Q-Chem 5 (6). The user can simply input a desired pressure, and the algorithms take care of the rest. Note that X-HCFF does not let the electrons interact with the pressure, but GOSTSHYP does. X-HCFF only works for molecules (i.e. multi-nuclei systems), but GOSTSHYP can be used for both atoms and molecules.

Introduction

There are several methods available that can simulate the electronic structure of molecules under high pressure. How to include pressure in an electronic structure calculation is not necessarily obvious or straightforward; no terms in the Hamiltonian operator depend explicitly on the pressure. So we need to smuggle it in, somehow.

As far as my understanding goes, there are essentially two main categories of ways to include pressure: 1) By confining the nuclei to a smaller volume, and 2) By confining the electron density to a smaller volume.

Confining the nuclei

This is the simplest method, and also the more physically incorrect way. The general idea is to compute external mechanical forces on each nuclei, that point inward either toward the molecular centroid or inward perpendicular to the molecular surface. The methods differ in how these forces are defined and how they are computed. The external forces are then added to the nuclear gradient, and the effect of pressure is simulated due to the compression that takes place during a geometry optimization. Such methods belong to the class "quantum mechanochemistry", and I refer you to the literature for more information about this topic (3-5). Note that the electron density is not directly affected; one only modifies the potential energy surface by adding external forces to the nuclear gradient, and then the wave function is relaxed to the new geometry during the SCF.

A modern quantum mechanochemical method is the eXtended Hydrostatic Compression Force Field (X-HCFF) (1). The idea is to build a van der Waal's surface of the molecule, which is tesselated by a Lebedev grid. The user inputs a pressure (e.g. 100 GPa), and the forces perpendicular to the surface are computed in such a way that the requested pressure is exerted on the molecule. This method is available in Q-Chem 5 (6) (free trial version). Note that this method cannot be used for single atoms, since one relies on being able to compute the nuclear gradient (one cannot geometry optimize a single atom).

Confining the electron density

A more physically correct strategy is to have the electrons directly interact with whatever is causing the pressure. This can be done by confining the electron density. Electron confinement is quite fundamental to quantum mechanics; recall that the quantized energy levels in a particle-in-a-box is a consequence of the electron being confined. Electron confinement has a long history in quantum chemistry, and typically various confinement potentials have been used. However, a flexible and easy-to-use method was recently published: Gaussians On Surface Tesserae Simulate Hydrostatic Pressure (GOSTSHYP) (2).

The idea in GOSTSHYP is to build a van der Waal's surface of the molecule and tesselate it by a Levedev grid. The Gaussian potentials are added at each grid point. The width of each Gaussian is determined in a way to ensure that the Gaussian field is as uniform as possible. The user inputs a certain pressure, and the amplitudes of each Gaussian is determined in a way to ensure that the requested pressure is indeed applied. The pressure arises from the repulsive interactions due to the electrons and the Gaussian potentials. The GOSTSHYP method is also available in Q-Chem 5. Also note that the GOSTSHYP method can be used for both atoms and molecules, since one does not rely on the nuclear gradient.

Another strategy for confining the electrons is the eXtreme Pressure Polarizable Continuum Model (XP-PCM) (7). This is based on the PCM approach for simulating solvent effects. Here, pressure is simulated by the repulsive Pauli interactions between the molecule's electron density and the solvent's electron (uniform) electron density. However, the XP-PCM is not available in any release version of quantum chemistry programs, as far as I am aware.

References

1. T. Stauch. Int J Quantum Chem. 2020, 153, 134503. doi: 10.1063/5.0024671

2. M. Scheurer et al. Int J Quantum Chem. 2021, 17, 583–597. doi: 10.1021/acs.jctc.0c01212

3. L. Takacs. Chem Soc Rev. 2013, 42, 7649-7659. doi: 10.1039/C2CS35442J

4. T. Stauch and A. Dreuw. Chem Rev. 2016, 116, 14137–14180. doi: 10.1021/acs.chemrev.6b00458

5. T. Stauch. Int J Quantum Chem. 2021, 121, e26208. doi: 10.1002/qua.26208

6. https://www.q-chem.com/

7. R. Cammi et al. Chem Phys. 2008, 344, 135-141. doi: 10.1016/j.chemphys.2007.12.010