VOLUME 48, NUMBER 24 PHYSICAL REVIEW LETTERS 14 JUNE 1982 Variational Monte Carlo Calculations for Spin-Aligned Deuterium R. M. Panoff and J. W. Clark McDonnell Center for the Space Sciences and Department of Physics, Washington University, St. Louis, Missouri 63130 and Michael A. Lee, (a) K. E. Schmidt, and M. H. Kalos C ourant Institute of Mathematical Sciences, New York University, New York, New York 10012 and G. V. Chester Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853 (Received 1 April 1982) The first variational Monte Carlo calculations of the ground-state energy of spin- aligned deuterium Dt are presented. Using Slater-Jastrow trial wave functions, the cal- culations are based on the accurate theoretical pair potential of Kolos and Wolniewicz for the & 3 2 u + state. For D* with equal population of the three nuclear-spin states, the min- imum theoretical ground-state energy is found to be negative, implying that this system is a liquid at zero temperature under zero external pressure. PACS numbers: 67.50.Dg, 67.90, + z The study of the properties of bulk samples of spin-aligned atomic deuterium, D*, has proven to be a challenge to experimentalists 1 ' 2 and theo- rists 3 " 7 alike. Current experimental efforts are aimed at producing D* at densities such that di- verse quantum effects can be observed. From the viewpoint of computational many-body theory, the D* systems are extremely interesting since they provide new examples of strongly interacting fermion systems displaying a range of level de- generacies. 3 Further, D* has the special advan- tage (with respect to helium or nuclear systems) that the pair potential is known to great accura- cy. 8 One of the first questions to answer about D* is the following: What is the nature of the ground- state phase? We would like to be able to say with certainty which of two eventualities is realized at zero temperature: (i) the system, in the vicinity of the finite-density minimum of the energy ver- sus density curve, is a self-bound liquid; (ii) the system is never self-bound, but will undergo a gas-liquid phase transition under modest pres- sure. 4 Attempts to answer this question have been hampered by an exceptionally close cancellation of relatively large kinetic and potential-energy terms which reduces the magnitude of the calcu- lated (H)/N to a size less than the error bar of the theoretical energy estimate (some 0.1-0.2 K in the best calculations to data). Previous inves- tigations of the ground state of D* have employed either cluster-expansion 6 or integral equation methods 3 for evaluation of the energy expectation value, or both. 4 The errors in such calculations are difficult to assess. Variational Monte Carlo calculations, on the other hand, yield rigorous upper bounds for the energy within a small statistical uncer- tainty which can be reduced as necessary by suf- ficiently long runs. We present here the results of a Monte Carlo study of the system D* 3 . The downward arrow means that all electronic spins are aligned antiparallel to the stabilizing field, and the subscript 3 indicates the level degeneracy. (Recall that the spin of the deuteron is 1 and con- sequently has three possible orientations with re- spect to the reference direction. For D* 3 we as- sume equal population of all three nuclear-spin states.) We adopt the simple trial wave function ip T giv- en by where ^j is a Jastrow correlation factor, ipj(R)=exp (D (2) and D 0 is a product of Slater determinants of plane waves, one for each nuclear-spin state. Re- garding the choice of pair correlations, one op- tion is to assume a suitable analytic form for u(r) containing certain variational parameters. A popular form in helium studies, and one which we shall also use here, is the single-parameter © 1982 The American Physical Society 1675