The 126 "dimensions" are not dimensions of space but coordinates of particles which are modeled as dimensions of state space. The paper is
open access, and this is explained before it gets too technical:
"However, we may inspect the 3N-dimensional electronic wavefunction (3N=126 in the case of benzene) that results from any theoretical framework
(including MO theory) to regain chemical insight."
There are C6H6 = 6*6 + 1*6 = 42 electrons in a benzene molecule, and each of these has an x, y, and z-coordinate, so in the usual quasistatic
approximation where the nuclei are presumed stationary, the Schroedinger equation has 42*3 = 126 variables. In the study of differential
equations, we often use an analogy called state space with one dimension per variable so that the solution can be viewed as a
manifold, which is a mathematical term that roughly means "smooth object". This representation gives rise to the field of mathematics called
differential geometry.
The paper finds an approximate, or variational, solution to this 126-dimensional equation by using a stochastic algorithm that works within
the quotient space obtained by "equating" any two points in R^126 which differ only by swapping the coordinates around [1]. For example, if I
take the quotient space of R^2, the plane, up to a change of (the two) coordinates, I get half of a plane, because any point in the other half can
undergo a change of coordinates that flips it to the other side. In R^3, the permutation-quotient space is bounded by the planes x = y and y = z, and
is shaped like a "wedge" with interior angle of 60 degrees and edge along the line x = y = z (where the planes intersect). In order to understand this
quotient space in the case of a 126-variable equation, we end up with a problem in high-dimensional geometry.
[1]What the authors do is a little different -- they only allow interchanging the coordinates of electrons with the same spin.
Ultimately this is really just a math/CS paper with good marketing. The only important conclusion is that benzene is, in fact, symmetric. But we
already knew that by XRD :p
[Edited on 7-4-2020 by clearly_not_atara] |
