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mented Debye model, 

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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 

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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 

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 

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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]. 

Both Refs. [25] and [26] give a ruby-like calibration for- mula, which can be expressed as 

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, 

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 . 

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 

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