## Ground State of the Electron Gas by a Stochastic Method

(D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45 (1980) 566), 3548 citations

*This is the eighth in a series of articles by James Riordon. The first article appeared in the November 2002 issue.*

Although the paper "Ground State of the Electron Gas by a Stochastic Method" is third in our list of the ten most highly cited *Physical Review Letters*, David Ceperley has one regret regarding the work he co- authored with computational physics legend Berni Alder, "Well, the letter was so successful that I never wrote the long paper," says Ceperley. "That was a mistake actually. I thought I should have been able to do a better job, but it was harder than I thought to improve on it." For Alder, on the other hand, choosing not to follow up on the paper with a longer *Physical Review* submission was in keeping with his lifelong approach to physics. "I like to find the new problems and skim the cream off the top," laughs Alder, "and I think we really creamed that one. After I finish a problem, I like to move on."

The cream, in this particular case, was the first important application of a type of quantum many-body algorithm now known as the Quantum Diffusion Monte Carlo method, or quantum DMC. Ceperley and Alder applied DMC to determine the properties of electron gases at intermediate densities.

Previous work had led to solutions to the problem at high and low electron densities. But other than crude estimates by the likes of Eugene Wigner and others dating back to the 1930s, the problem of intermediate densities had lingered for more than half a century.

"The basic bosonic algorithm was developed by Malvin Kalos, my thesis advisor, and we partially extended it to fermions as part of my thesis in 1976" says Ceperley. "What I'd done as a postdoc was make it into a much more convenient and accurate form. Then the ground work paid off when I came to Berkeley and collaborated with Berni Alder. We had access to orders-of-magnitude more computing time than I had ever had before…except it was all behind the fence at Livermore."

Although Ceperley did not have a clearance he managed to run simulations, which required thousands of hours of computer time, through an intermediary. "I had to communicate my instructions to Alder's assistant, Mary Ann Mansigh, over the telephone. We'd spend half an hour talking every day and she would set up five or ten different runs for the evening. The next morning she would tell me what the results were, or mail me back the output." Surprisingly, Ceperley notes, "It was actually rather efficient after the algorithm was working."

By the time Ceperley arrived in California, Alder was already a renowned pioneer in computational physics and the author of at least two other highly cited papers. His previous work primarily centered on the classical dynamics of hard spheres, but in the late 1970s he was eager to tackle quantum many-body problems. "I have a knack for being at the right place at the right time," says Alder, "You have to sort of smell what are the right problems in physics. And I think I may have that smell. And you also must have the tools to follow through that smell, that's the key. In the early days, one of the tools was big computers, and in this case certainly big computers helped. It was also important to think about physics in a numerical way, differently than people who did not have big computers."

To solve the electron gas problem, the researchers began with a restricted, fixed-node problem.

"The node," explains Alder, "is where the wave function goes from positive to negative. And if you knew the nodes exactly you could solve the quantum Monte Carlo problem exactly." Generally, however, the exact nodes are unknown and the researchers must guess where they might lie from some approximate theory.

To determine their true position, the researchers release the nodes so that they can shift about and lower the energy of the system. In an electron gas, the energy converges nicely and the equilibrium solution can be precisely determined. For more complex systems, DMC is plagued by an instability known as the fermion sign problem. "The fermion wave function has, of course, a positive and a negative part," says Alder. "When you release the nodes you allow both the negative and positive part to exponentially grow."

Despite the instability, DMC is still useful provided that a system's boson energy and fermion energy are comparable. In such cases says Alder, the answer appears as the difference between the positive and negative populations in the calculation, if the two portions of the solution don't grow too quickly. "But if you go to other systems," says Alder, "like chemical systems where the difference between the fermi energy and the boson energy is very large, you can no longer accurately project out the difference." The fermion sign instability is one of the outstanding problems in computational physics.

Nonetheless, DMC is one of a handful of quantum many-body methods that Alder and Ceperley say can apply in principle to any equilibrium quantum problem. "Whenever you want to calculate things ab initio," says Ceperley, "starting with the positions and charges of the nuclei, and with many electrons, then the state of the art is density functional theory. Density functional theorists use the electron gas result because they are perturbing their system about the uniform electron system."

Ceperley is now a faculty member in the physics department and a researcher with the National Center for Supercomputing Applications at the University of Illinois, where he spends some of his time searching for solutions to DMC instabilities.

Alder is retired from Livermore, but is keeping his hand in as well, working on the problem part-time with Ceperley's former thesis advisor, Malvin Kalos. Alder, however, is primarily interested these days in extending molecular dynamics to help explain the origin of hydrodynamic turbulence on molecular scales.

Strangely enough, unlike other powerful methods that made our top ten PRL list, quantum DMC is not named after the researchers who were so instrumental in its development. "There have been a fairly large number of contributions to the problem," says Alder, "which distributes the credit."

Nevertheless, Ceperley chuckles when he admits that he was able to leave one personal and indelible mark on the method, "I did manage to embed my initials in the acronym." DMC, it happens, also stands for David M. Ceperley.

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