# Is ground-state structure of a molecule entirely determined by the positions of the nuclei?

I'm a computer science with only a weak undergraduate chemistry background, but have recently become interested in simulation methods for computational chemistry. Many simulation methods, such as DFT, make the Born-Oppenheimer approximation (nuceli are static) and solve for some properties of the electrons (in the case of DFT, the density). In fact, when using Gaussian-16, I only need to specify the locations of the nuclei in space.

What then, is the role of actual bonds? Are bonds, bond order, etc. just a convenient abstraction for what the electrons are doing? If I tell you the location of some nuclei in 3D, can you then (theoretically) uniquely determine the bond structure?

• Yes, bonds are just a convenient abstraction. Nature does not know about them. – Ivan Neretin May 2 '18 at 15:00
• Born-Oppenheimer approximation is only this - approximation. For example ground state of $\ce{CH5+}$ cation is fluxional and position of it's hydrogen atoms is changing constantly no matter how low is the temperature. – Mithoron May 2 '18 at 22:45

Yes, these methods (both wavefunction-based and electron density-based) typically solve a purely electronic system defined by an electrostatic potential which is produced by the nuclei; the Born-Oppenheimer approximation can be understood as treating the electronic system as a function of nuclear geometry, which implies that (1) nuclear dynamics are not strongly affected by electrons and (2) electrons will almost immediately adapt to any new position of the nuclei. Born-Oppenheimer can also be understood in terms of electronic and nuclear motion (and the corresponding energies) being separable; that is, electronic structure takes shape after (and as a consequence of) nuclear dynamics, but does not directly affect nuclear dynamics themselves. Since chemistry is usually interested in stationary solutions as snapshots, these approximations are fitting for pretty much all chemical questions.

In practice, nuclei are not even particles in electronic structure calculations - in both Schrödinger equation methods (HF and post-HF) and Hohenberg-Kohn methods (DFT), the only role of nuclei is to generate an external electrostatic potential, $V_{ext}$, that will be part of the Hamiltonian/energy functional of the system. The specification of nuclear positions that you mention is just a more convenient, and chemically intuitive, way of describing that external electrostatic potential, but there is no mathematical or conceptual difference if we consider that this external potential is arbitrarily imposed from outside; it's just that it's easier to define a number of point charges (with their charge defined by their atomic number) than to provide a complex 3D potential function.

Once we assume that Born-Oppenheimer is a valid approximation and that nuclei merely exert an external electrostatic potential on electrons, the fact that electronic structure for the ground state is entirely determined by the positions of the nuclei is axiomatic: it is a consequence of the postulates of quantum mechanics (and the basis of all variational methods) for wavefunction-based calculations, and one of the Hohenberg-Kohn theorems for density functional calculations.

Geometry optimisation is also performed under the same assumptions - electronic structure is calculated under a given geometry, and then resulting forces over nuclei are estimated; a small motion is performed along these tensions, and the process is repeated iteratively until forces experienced by nuclei are small enough to fall below a certain tolerance level. Note that there is no explicit treatment of the interaction between electrons and nuclei; only two process that interact iteratively through electrostatics: electronic structure is minimised for each geometry, and then geometry is relaxed, but not minimised, according to that electronic structure. We obtain a quantum mechanics depiction of the electronic system for each geometry, but we do not perform a quantum mechanics treatment of the nuclei. Ab initio molecular dynamics is a similar approach, in which nuclei move through classical dynamics but electronic structure is calculated through QM at each step.

In all these cases, chemical bonds are only emergent interpretations of the energetics of electronic structure, but their identification is largely arbitrary, as energetic effects are not localised. While we can calculate the energy of two molecular fragments as they approach from infinity, and we'll observe interactions that we identify with bonding, the limits of what is a bond are not clear, and at a quantum chemical level, it makes more sense to talk about energies associated with different geometries than to think in terms of bond "objects".

It is in principle possible to perform QM computations which treat both nuclei and electrons as quantum particles, thus obtaining a single wavefunction describing both, but the complexity of this treatment (especially given the large difference in scale for electronic and nuclear motion) restricts calculations to a very small number of particles - the highest number I remember reading about is 8, although I might have missed recent studies. In addition, the problem of describing interactions accurately (which is behind the variety of quantum chemistry methods used just to describe electrons) grows considerably worse, to the point that quantum Monte Carlo may be a better option than quantum chemistry methods, as happens with bosons.

As far as Quantum mechanics is concerned, there is no such thing as bonds or bond order. Any QM method is just dependent on finding the wavefunction that satisfies the Schrodinger (or Dirac if concerned with relativity) equation (or some approximation of it). Within the Born-Oppenheimer approximation for a particular method, the only user inputs needed are the location and type of all the nuclei, and the basis set(s) you are using to construct your MOs. Its important to note that basis sets are empirically optimized to describe particular atoms, so if you were making your own electronic structure program from scratch, it wouldn't be enough to just have descriptions of nuclei; you would need to incorporate preexisting basis sets or determine your own. So I would say no, the ground state structure depends not only on the positions of the nuclei, but also on the basis functions and method you use to evaluate the energy/wavefunction.