In the late 19th and early 20th century, two theosophists tried to discover the secrets of matter. Theosophy was a fairly new religious movement based on the teachings of Helena Blavatsky, who claimed to have met a group of Ascended Masters — spiritually advanced superhumans — in Tibet. Those who studied the teachings of these Masters could hope to gain some of the Masters’ abilities, including ESP. Occult Charles Webster Leadbeater and feminst and socialist Annie Besant devised an experimental protocol in which they obtained samples of the chemical elements, placed them on a table before them, and attempted to psychically discern their inner structure. Their results became Occult Chemistry: A Series of Clairvoyant Observations on the Chemical Elements. The book shows a good grasp of then-current chemistry, and even promised reproducibility — since the process just required “a very slight intensification of ordinary vision,” anyone capable of basic clairvoyance should be capable of testing the atomic hypotheses.
The structure of the atom was at this point still deeply mysterious. Electrons had been discovered, but not protons. The dominant model — put forward just a few years before, in 1904 — was the “plum pudding” theory, in which electrons were imagined to sort of float around in a positively charged field. The periodic table itself was more or less brand new, and it was not yet understood that what crucially distinguished one element from another was its proton count. We know now that there is no element between hydrogen (1 proton) and helium (2 protons) because you can’t have 2.5 protons, but they didn’t know that, so it was not totally implausible that there was a secret element between them.
The atomic structure the theosophists devised (or divined) is… complicated. There are “chemical elemental atoms” which can be divided into four subunits, the most fundamental of which is the “ultimate physical atom,” which cannot be further divided although it may “vanish from the physical plane” (?). But there are also two kinds of ultimate physical atoms — male and female — which “whorl” in opposite directions and which also do something complicated involving higher dimensions and what I think is magic. Through male atoms a life force pours into the physical world from fourth-dimensional space; through female atoms it seeps back out again. Interestingly, they posit that fundamental particles are a manifestation of this energy: where it concentrates in the void, the particles appear. This is actually a not-too-bad summary of quantum field theory, in which fields — not particles — are really fundamental, and particles are excitations or perturbations in a field. QFT has nothing to say about gendered atoms, though.

Chemical elemental atoms are formed from arrangements of ultimate physical atoms, with three intervening stages (proto-elemental, meta-proto-elemental, hyper-meta-proto-elemental), each of which have their own unique shapes that vary by element.

The illustrations of the chemical atoms are the most beautiful and detailed in the book. Some shapes, the authors write, are found again and again in diverse elements: the ‘dumbbell’ in the top left, for example, appears in sodium, copper, silver, and gold, as well as chlorine, bromine, and iodine. The tetrahedron appears in occultum, zinc, and cadmium among others, although the form it takes in selenium is “particularly beautiful...with a star floating across the mouth of each funnel; this star is extremely sensitive to light, and its rays tremble violently and bend if a beam of light falls on it.” It’s these repeating shapes that give the table of the elements its periodicity — the tendency for properties of elements to reappear regularly across the table and sort themselves into regular groups. The noble gases, for example, appear in group 18, on the right side of the table, and ‘nobility’ (the tendency not to form chemical bonds with other elements) recurs at atomic numbers 2, 10, 18, 36, 54, 86, and 118 (and beyond that, if we can ever reach the island of stability and create superheavy elements that don’t immediately disintegrate).

One more:
The internal structure of each chemical element varies, even when they have similar outer shapes. Elements break down into subunits, and also sometimes other elements: gold, for example, contains occultum. Much of the book is given to very careful elucidations of these structures.

The book did not exactly make a splash in scientific circles, but it’s worth reading if for no other reason than its bizarre and beautiful vision of the lower world. Oxygen, write Besant and Leadbetter, contains two spiralling snake-like forms (one positive, one negative), roiling with waves which in turn carry brilliantly-shining beads on their crests until they break up, and “twist and writhe about...reminding one of fire-flies stimulated to wild gyrations.”
You can read the full text of Occult Chemistry for free here.