Foundations

Consider a scalar Gaussian function: per dimension, the Gaussian can either be a function of a position or two variables. Higher numbers of variables are possible but not considered. Let us call the localized Gaussian an orbital and function of two variables a geminal. You could imagine mixing and matching across components, which make up a collected bundle of dimensions where each dimension on a component is called a particle.

     At this point, we may have a Gaussian, but the sum and products of Gaussians are extremely nontrivial based on Gaussians. Therefore, one ought to form a consistent basis to place the Gaussian Information. The most general sinusoidal functions are exactly one wave-vector component. Let us collect a plane-waves into bundles and call it a Cardinal Sine function (Sinc). The natural Sinc function is orthonormal at a unit distance along a direction. Thus a Grid basis is naturally manifest.

     Next, we need an overlap of 1 Sinc with an orbital. This overlap is simply the integral of their product. We also need the overlap of a pair of Sincs with a geminal. In this way, we sum arbitrary numbers of Gaussian onto a grid. Let us call these kinds of grids data.

     The single Gaussian function can be represented across components by multiplication; in the exponent of the Gaussian, R^2 = X^2+Y^2. However, as we start adding Gaussians, we find that the summation is no longer a simple Gaussian with this single product structure. To maintain the component product separation notion, we project the Gaussians into the grid per component and decompose. Decomposition will reduce the number of summations keeping the information content in sums of product form. This method is called Alternating Least Squares. We can, therefore, compress data.

     The manipulation of data begins by multiplying data by Gaussians. 2 Sincs will form an orbital action, and 4 Sincs will form the basis of the action of Geminals on the data. These operators are also decomposable as sums of products.

     The mathematical understanding of summing Gaussians into arbitrary functions is called Laplace-transforms. Notice, the traditional Laplace-transform is related to the Gaussian expression by changing variables in an integral.

     The inverse geometric distance between two points is the Coulomb interaction, formed by a uniform summation of Gaussians. We can create decomposable data-operators of the inverse metric.

     The inverse metric is fundamental in physics, and thus all of Chemistry is approachable by writing down Sums of Products Coulombic operators. We need only write down the quantum action of kinetic energy, which associates to each Sine-wave a square of the Sine-wave momentum. We are thus forming a Kinetic energy operator that also lives in the space of data-operators. To understand this quantum action, we eigensolve the data-operators by the Lanczos method: start with data, multiply by the data-operator, and eigensolve the collection of data—repeat until satisfied. The eigenvectors will remain identical under data-operation action.