FIG. 1: The c-BN EOS at 300 K, computed with DMC, compared to experiment. We plot the full EOS and the pressure residuals with respect to DMC with the BFD PP. (a) gives the corrected EOS resulting from combining the EOS and PP data, while (b) gives the pressure residuals for the same data. (c) gives the uncorrected pressure residuals from the PP simulations only. Shaded areas represent the one- σ statistical confidence region from QMC data. (a) Corrected EOS (b) Corrected pressure residuals (c) Uncorrected pressure residuals 3 mented Debye model, temperature. The matrix element for the transition from n to n − 1 is proportional to √$_{n}$ and is thermally weighted by the Boltzmann occupation of state n , so the intensity-averaged frequency, 〈 ν 〉 , can be given as in which the Debye temperature, θ , is a function of both V and T (Table I(b)). The Debye free energy per two-atom cell, excluding the zero-point term, is given by c-BN can be used to calibrate pressure optically by mea- suring the frequency shift of the TO Raman mode, allowing bench-top experiments. We compute the pressure and temper- ature dependence of this frequency. Within the quasiharmonic approximation, phonon frequencies have explicit dependence on volume only. At constant pressure, however, these frequen- cies have an implicit temperature dependence resulting from thermal expansion. The dependence due to thermal expan- sion accounts for only about half the total T -dependence of the Raman mode, as observed in [25] and [26]. The remain- ing T -dependence can be attributed to significant anharmonic effects in c-BN, and is included in our calculations. Since the optical branch has small dispersion, we treat the anharmonicity as a one-dimensional on-site anharmonic oscil- lator, in a similar approach to the QMC computation of the TO Raman frequency of diamond[9]. At each volume, we com- pute the effective Born-Oppenheimer potential well for the TO mode with DMC and the BFD PP at nine displacements along the mode eigenvector in the 64-atom supercell. We fit the data to a quartic polynomial, and numerically solve the 1D Schr¨odinger equation in this analytic potential. This results in a set of single-phonon energy levels, { E$_{n}$ } , with nonuniform separation. From { E$_{n}$ } , we compute an intensity-weighted average Raman frequency, ¯ ν , as a function of pressure and The main axis of Figure 2(b) gives the room-temperature Raman frequency versus pressure. There is good agreement in the relatively low-pressure region in which the Raman fre- quency was measured. At very high pressure, the deviation with respect to the extrapolation in [25] increases with a max- imum discrepancy of 38 cm − 1 or, conversely, a deviation in with parameters in Table I(c) and plotted in Figure 2(a). Note that this formula cannot be analytically inverted, but a very simple iterative solution can be used for calibrating pressure from ν and T . where R , ¯ ν , and b have quadratic T -dependence. This de- pendence is sufficient below 2000 K, but cannot represent our data at high temperature. We use a form which captures the Boltzmann occupation of phonon excitations, Both Refs. [25] and [26] give a ruby-like calibration for- mula, which can be expressed as The excess thermal softening, i.e. beyond that from thermal expansion alone, is accounted for by the thermal average of the anharmonic frequencies. Figure 2 shows the computed Raman frequencies compared with the experimental data re- ported in Refs. [25–27]. 〈 ν 〉 = ∑ ∞ n =1 I$_{n}$ E $_{n}$− E $^{n$_{−}$}$1 hc ∑ ∞ n =1 I$_{n}$ , where I$_{n}$ = n e − E n k B T . (7) P = ( R/b ) [ ( ν/ ¯ ν ) b − 1 ] , (8) ν ( P, T ) = ν 0 ( P ) + ν 1 ( P ) exp [ − ν 2 ( P ) T ] (9) ν$_{n}$ ( P ) = c$_{n}$ [ bP R$_{n}$ + 1 ] 1 /b , n = 1 , 2 , 3 (10) F$_{D}$ = 6 k$_{B}$T [ ln ( 1 − e θ T ) − ( θ T ) $^{3}$∫ θ T 0 x 3 e x − 1 dx ] . (6) P$_{th}$ ( V, T ) = − ∂F$_{D}$ ( θ, T ) ∂θ ∂θ ∂V (3) θ ( V, T ) = θ $^{0}$( V ) + β ( V ) exp( − α ( V ) T ) (4) x ( V ) = x 0 + x 1 V + x 2 V $^{2}$+ x 3 V $^{3}$, x ∈ { θ $^{0}$, α, β } (5)