# Approximation of electron density of a small molecule

Starting point for Quantum Theory of Atoms in Molecules (AIM) is the electron density $\rho(r)$. It could be determined experimentally (X-ray diffractions). I want to know how can we estimate them ab initio. In AIM it is often emphasized that the focus is on density rather than a wave function for the later is just a mathematical function while the former is experimentally observable. Are there ways to estimate electron density without first calculating wave function (which will obviously lead to density function).

My interest is to use AIM in quantitative structure-activity relationship. I will typically deal with molecules of around 50 atoms. Usually I will already have the graph (connectivity) of the molecule (eg .SDF file, smile notation etc), how can we incorporate this to speed up electron density calculation?

It would be helpful if brief comment on the computational cost is also included in the answer.

• What are you trying to get out of AIM to to use in QSAR? (I ask because the level of accuracy/realism you require will affect what methods are usable.) Apr 29 '13 at 17:54
• I should be able to discriminate molecules with low MIC (minimum inhibitory concentration) from that of molecules having high MIC values. May 1 '13 at 3:48
• That's not a thing you can get out of AIM. (As far as I know.) May 1 '13 at 19:14

You are looking for density functional theory. The theory is somewhat involved and you may be interested in A Chemist's Guide to Density Functional Theory or similar text book. A full description of the variety of ways to compute the (ground state) electron density is beyond the scope of an answer and better suited to a textbook.

The general gist is to optimize the energy with respect to the electron density numerically.

• can you please give a brief outline (step wise) how to calculate the density. I referred to the book but I still find that they assume density is already given and do further analysis on that. Apr 25 '13 at 5:31
• @DurgaDatta: While DFT does produce a density, it relies on something very much like a wavefunction calculation, and thus is probably not what you are looking for. Apr 29 '13 at 17:53
• @DurgaDatta Additionally, the step by step procedure is going to depend on the software package you choose to use. Ultimately, they will look like "include yadda in the instruction file on line 2 if you want to incorporate diffuse wavefunctions," or "click on Setup Calculation in the Calculation menu". May 1 '13 at 1:37
• @Aesin This is not necessarily true. The most common implementations of DFT that are generally used by chemists are indeed based on orbital descriptions, SCFs etc, but this is to reuse existing codebase and easier interpretations for chemists. Using e.g plane waves, pseudopotentials or first order approximations for the KS equation etc one can create different implementations with less chemistry taste.
– Greg
Jun 20 '14 at 16:29
• @Greg: In most DFT techniques I've seen, even when you're using plane waves and pseudopotentials, you're still creating an SCF. It is my understanding that while Hohenberg-Kohn theory doesn't require it, almost all practically used implementations use Kohn-Sham techniques, which do. Your basis set, whether hydrogenic orbital-like, plane-wave, or wavelet-based, does not affect the meaning of the thing you calculate (though will obv. affect the accuracy), though you may have to do more or fewer transformations to produce something you recognise as a chemistry MO. Jun 22 '14 at 12:34

As far as obtaining the density without first computing the wave function--there's almost no way to do it with any accuracy for anything "realistic." The easiest way to obtain the electron density is to perform an ab initio electronic structure theory calculation.

For example, you can use the GAMESS quantum chemistry program to perform a Hartree-Fock or DFT calculation, and then have it write the resulting wave function to a AIMPAC input deck (WFN file).

The AIMPAC program is the original version of QTAIM, written by the late Richard Bader and his research group. There are more modern versions, such as Todd Keith's AIMALL which is certainly more powerful and user-friendly than the original AIMPAC.

You can probably perform some calculation on fifty or so atoms, assuming a typical organic molecule. It all depends on how fast your computer(s) is and how many basis functions you use. Constructing molecular graphs with AIM is quite fast. Where things get expensive is when you start calculating properties of atomic basins, as these are high-accuracy integrations over "irregular" (but well-defined) regions.

Sorry to ruin your day, but the connectivities in your SDF file are probably of little value. There are no such things as "single"/"double"/"triple" bonds, at least in the context of QTAIM. Either they are bonded or they are not. Bonding appears when the interaction between two atoms lowers the total energy of the system. QTAIM will give you that information unambiguously.

Sometimes the answers are surprising, but it is guaranteed that they are more "correct" than what we unfortunately call "chemical intuition" and/or simple proximity of two atoms.

• Your claims about bonds not being useful is not quite correct. Valence-bond methods can use that information and you can also put that information into your initial guess as in Hueckel approaches. You could always use NBO to enhance QTAIM. Jun 3 '13 at 13:02
• @Deathbreath 1) No modern electronic structure theory codes will need that bond information to do its job. 2) NBO on top of QTAIM is like painting a mustache on the Mona Lisa. QTAIM doesn't need "enhancement" by arbitrary unitary transformations of wave functions (which themselves have arbitrarily been constrained to be real rather than complex). Also, I'm not sure if NBO even maintains constant density. Jun 3 '13 at 13:22