Douglas Hartree

A Cambridge boy with a gunner’s stopwatch

Douglas Rayner Hartree was born on 27 March 1897 in Cambridge. His father William lectured in engineering; his mother Eva Rayner was the first woman to serve as mayor of Cambridge. The household ran on mathematics and mechanical apparatus, and by his teens Douglas was already building clocks and outpacing his father on the slide rule.

He went up to St John’s College in 1915 to read mathematics, but the First World War interrupted almost everything. By 1916 Hartree had joined a small group of mathematicians working under A. V. Hill on the problem of anti-aircraft gunnery. An aeroplane was fast enough that by the time a shell reached the altitude it was aimed at, the plane was no longer there. The shell had to be fired at where the plane was going to be. Solving that meant predicting trajectories in real time, with a stopwatch and a slide rule, in a field tent. Hartree spent two years on the predictor work and learned to compute the way a working tradesman learns his tools, with a stopwatch attached. The lesson stuck for life.

The atomic problem and a beautiful trick

By the mid-1920s Erwin Schrödinger’s wave mechanics had given physics a single equation that, in principle, described every atom in the periodic table. In practice the equation could be solved exactly only for hydrogen. The moment a second electron was added, the electron-electron repulsion term made the whole problem chase its own tail.

Douglas Rayner Hartree (27 March 1897 – 12 February 1958) was an English mathematician and physicist most famous for the development of numerical analysis and its application to the Hartree–Fock equations of atomic physics and the construction of a differential analyser using Meccano.

In 1927 and 1928 Hartree proposed a beautiful trick. He called it the self-consistent field. You start by guessing a charge cloud for the atom. Each electron then sees only that average cloud, and its wavefunction can be solved by a single-particle Schrödinger equation. Once you have the new wavefunctions you add them up, get a new charge cloud, and feed it back as the next guess. Repeat. When input and output clouds agree, the field is self-consistent and you have your answer. It was the first iterative method in computational quantum physics. Hartree applied it by hand to atoms across the periodic table, sitting at his dining table with graph paper and a slide rule.

Wooden gears and the differential analyzer

By 1933 Hartree had grown impatient with hand calculation. He visited MIT, where Vannevar Bush had built a differential analyzer: an analog mechanical machine of shafts and gears that solved differential equations by spinning parts against each other. Hartree returned to Manchester and, with a research student named Arthur Porter, built a working differential analyzer out of Meccano, the British children’s construction toy. It worked. The Meccano machine solved real differential equations, including his own self-consistent-field iterations, in hours instead of weeks. A larger, funded machine followed in 1935.

Through the late 1930s and the war, Hartree’s analyzer was one of the most capable computing machines in Britain. It was used to design radar magnetrons, compute ballistic tables, and model neutron diffusion in the early atomic-bomb work. Hartree was a Quaker and a pacifist by temperament, and never wore the wartime work as a medal.

ENIAC, atomic units, a quiet professorship

In 1946 Hartree saw ENIAC at Pennsylvania and understood immediately that the Meccano era was over. He threw himself into the new electronic machines, running one of the first stored-program scientific calculations on EDSAC: a self-consistent-field computation for the mercury atom. The same year he was elected to the Plummer Chair of Mathematical Physics at Cambridge.

Almost as a working convenience, his 1928 paper introduced a system of units in which the electron mass, the elementary charge, the reduced Planck constant, and the Coulomb constant are all set to one. The natural unit of energy in this system, the hartree, equals about 27.2 electron volts. The natural unit of length, the bohr, is the radius of the ground-state hydrogen orbit. Every serious electronic-structure code in the world uses Hartree atomic units internally today.

The self-consistent-field method itself, refined in 1930 by Vladimir Fock to honour the Pauli exclusion principle, became the Hartree-Fock method. Every modern protein-folding simulation, battery-electrolyte study, and photovoltaic-material screen begins, somewhere down in its plumbing, with an iteration Hartree first wrote on graph paper in 1928.

Hartree died on 12 February 1958 in Cambridge, aged sixty. He was the figure who first realised that the way to make quantum mechanics useful was not to wait for analytical genius but to iterate, patiently, on a machine, until the numbers stopped moving. Every modern quantum calculation is a child of that insight.