Fundamental high pressure calibration from all-electron quantum Monte Carlo calculations K. P. Esler, 1,2 R. E. Cohen, 1 B. Militzer, 3 Jeongnim Kim, 2 R.J. Needs, 4 and M.D. Towler 4 $^{1}$Carnegie Institution of Washington, Geophysical Laboratory $^{3}$University of California, Berkeley, Dept. of Earth and Planetary Science and of Astronomy $^{2}$University of Illinois at Urbana-Champaign, NCSA ∗ $^{4}$Theory of Condensed Matter Group, Cavendish Laboratory, Cambridge CB3 0HE, United Kingdom (Dated: November 1, 2018) We develop an all-electron quantum Monte Carlo (QMC) method for solids that does not rely on pseudopo- tentials, and use it to construct a primary ultra-high pressure calibration based the equation of state of cubic boron nitride(c-BN). We compute the static contribution to the free energy with QMC, and obtain the phonon contribution from density functional theory, yielding a high-accuracy calibration up to 900 GPa usable directly in experiment. Furthermore, we compute the anharmonic Raman frequency shift with QMC as a function of pressure and temperature, allowing optical pressure calibration in table-top experiments. In contrast to present experimental approaches, small systematic errors in the theoretical EOS do not increase with pressure, and no extrapolation is needed. This all-electron methodology is generally applicable to first-row solids, and can be used to provide a new reference for ab initio calculations of solids and to benchmark pseudopotential accuracy. PACS numbers: 64.30.Jk,81.05.Je,02.70.Ss,63.20.dk Although the number of studies of materials using density functional theory (DFT) continues to grow explosively, the accuracy of their predictions is variable, sometimes excellent and sometimes poor, limiting the confidence one may place in DFT results as a quantitative calibration for experimen- tal studies. Quantum Monte Carlo (QMC), particularly dif- fusion Monte Carlo (DMC), is the highest-accuracy method for finding the ground state of a many-electron Hamiltonian. For solids with atoms heavier than He, however, the Hamilto- nian itself is approximated using pseudopotentials (PPs) based on a lower-accuracy theory, limiting its reliability, and as we show, commonly used PPs give disparate results. In this Let- ter, we push the state of the art in accuracy by introducing a method for an all-electron DMC simulation of solids, elimi- nating the bias from pseudopotentials, and apply it to create a high-accuracy pressure calibration scale. ulus, in conjunction with X-ray diffraction measurements of volume, can be integrated to provide an EOS, but a correc- tion must be made to transfer from an adiabatic to an isother- mal path[3]. New approaches such as quasi-adiabatic Z-pinch based experiments[4, 5] also hold future promise for a pri- mary scale. There have been attempts to refine the ruby scale[6], and new calibrations have also been suggested[3]. Cubic boron nitride(c-BN) has been identified as a promising material for a new scale[7]. In this Letter, we provide a new pressure scale based on DMC calculations of the EOS and Ra- man frequency of c-BN. This theoretical approach has the ad- vantage that, in contrast to present experimental approaches, the method works equally well under high compression, and uncertainty does not grow with pressure. The combination of ultra-high pressure mineralogy with seismology has yielded a wealth of insight into the internal structure of our planet. Pressure is the key that links these disciplines, mapping phase transitions and mineral properties to planetary depth. Establishing an absolute pressure cali- bration at multi-megabar pressures, however, poses a funda- mental and continuing problem for high-pressure experiment. Current primary calibrations are based on data from shock- wave experiments, which infer pressure from conservation of momentum and energy as the shock traverses the sample. Ex- perimental uncertainty, combined with errors in model extrap- olation, yield scales which differ from each other by as much to 7% at room temperature, with even greater discrepancies at high temperature[1]. Such disparity remains a serious obsta- cle to a quantitative understanding of Earth’s interior. Pressure is the negative volume derivative of the Helmholtz free energy. In a wide-gap insulator such as c-BN, the free energy can be written as a sum of the frozen lattice enthalpy, dependent only on volume, and a phonon thermal free energy, which depends on both volume and temperature. Since the static enthalpy is the dominant contribution at ordinary tem- peratures, errors in a theoretical EOS can most often be at- tributed to the static part. Previous calculations of the EOS of c-BN have been based on DFT[7], which use approximate functionals to treat electron exchange and correlation. Several functionals are in common use, each giving rise to a different EOS, and there is no a priori way to predict which will give the most reliable result. Quantum Monte Carlo simulation methods explicitly treat electron exchange and correlation instead of resorting to ap- proximate functionals. Variational Monte Carlo (VMC) com- putes properties by a Metropolis sampling of a trial wave func- tion. Diffusion Monte Carlo samples the many-body ground state of the Hamiltonian through a stochastic projection of the trial function. In practice, a fixed node approximation i s used for fermions, such as electrons, which tests have shown give relatively small error when the nodes are obtained from A pressure calibrant is a material with a known equation of state (EOS), a small sample of which may be included in hy- drostatic equilibrium with the test subject (e.g. ruby[2]). As an alternative to shock experiments, high-pressure Brillouin scattering, which provides a measurement of the bulk mod- arXiv:1001.2079v1[cond-mat.mtrl-sci]13 Jan 2010