Only through the observables could we explore the unknown and determine the value of the model; $p(t|M)$ as the output of the model is the only thing that matters, or within our capability, that affects the value of the model. General chemical experiments have a frame (eg. Van der Waals Equation, Vant Hoff equation, Beer-Lambert) which links the observable and the parameter that represents its nature beneath. This could be represented as a detailed balance where given the link between the known and the unknown, the unknown can be sketched. Specifically, we observe the varying pattern of the unknown as we vary the known.

Two existential shortcoming exist in learning through experiment: 1) what if there are two unobservable unknowns 2) model accuracy is the function of the parameter and model validity cannot be tested in every possible combination of the observable. To elaborate, since it is difficult to count the number of molecules within a system, we came up with the linking model in PV = nRT and instead measure the pressure, volume, and temperature to indirectly count the molecule. However, a setting where the pressure of the environment cannot be measured and therefore becomes unknown. Also, this linking model is afterall a model and therefore its accuracy depends on the region of its components. In high pressure, interaction between molecules become an obstacle.

Computational chemistry is an aid to this shortcoming of real experiments where iteration-based self-correction becomes possible. Multiple unobservable s are alternately updated using self-consistent field approximation and since the updates are finished based on the accuracy of the model, inhomogeneous model quality across the parameter space is no longer an issue. The principle behind computational chemistry is: start from a simple model that evolves with increased complexity but decreased closeness to the real world. Molecular dynamics (MD) and quantum chemistry (QC) are two main branch of computational chemistry. The latter is the topic of this writing.

The linking model in QC’s is Schrodinger’s equation $i \hbar \frac{\partial}{\partial t} \Psi=\hat{H} \Psi$

For a complex system other than hydrogen molecule, a perfect solution for the above equation cannot be obtained. This leads to the necessity of multiple approximate solutions including self-consistency field approximation. In the theory, Atom’s nucleus is fixed (Born-Oppenheimer) and the energy level is searched. The solution is first expressed as the combination of basis functions whose distance to the real solution becomes closer for every iteration. Electron distribution of molecules like He, N2, O2 could be traced upon which the bond length and energy could be determined.

A self-consistent field (SCF) methodology is used to solve the problem by converging the solution so that there is no contradiction between the solution and the equation. There are a variety of methodologies for constructing a self-consistent field describing chemistry within a molecule as follows:

Methodologies for constructing a self-consistent field describing chemistry within a molecule includes:

  • Hartree-Fock methodology, which divides the Schrodinger equation into equations for individual orbitals
  • post-HF, which adds perturbation terms to HF
  • density functional theory (DFT), which converts the equation for the density of electrons into an equation for the density of electrons instead of the Schrodinger equation

General atomic and molecular electronic structure system (GAMESS ) offers computational results based on ab initio- the first principle in quantum chemistry. Two modes are offered:

  • OPTIMIZE is a geometric optimization
  • ENERGY is energy calculation at current location

Reference

(1) GAMESS manual

(2) Schmidt, Michael W., et al. “General atomic and molecular electronic structure sys tem.” Journal of computational chemistry 14.11 (1993): 1347-1363.

(3) Gordon, Mark S., and Michael W. Schmidt. “Advances in electronic structure theory: GAMESS a decade later.” Theory and applications of computational chemistry . Elsevier, 2005. 1167-1189.