Partial melting of the mantle has been studied using many tools, including those of analytical geochemistry, observational geophysics, experimentation, and application of physical and chemical theory. Through combination of these tools, substantial progress has been made in illuminating partial melting processes and basalt generation, particularly beneath mid-ocean ridges. One of the newer tools used to study these processes is MELTS, which applies thermodynamic models of minerals and silicate liquid to calculate phase equilibria as a function of intensive variables and bulk composition [Ghiorso and Sack, 1995]. For a full description of the MELTS computational algorithm, constituent thermochemical models and considerations of application to partial melting problems, see Ghiorso and Sack  and Hirschmann et al. [1998b].
 Major topics addressed with earlier versions of MELTS include (1) the composition of near-solidus partial melts of peridotite [Baker et al., 1995; Hirschmann et al., 1998b, 1999a], (2) the production of melt during adiabatic upwelling [Asimow et al., 1995, 1997, 2001; Hirschmann et al., 1999b], (3) the effect of melt-rock reaction on melt and residual peridotite composition and the origin of dunites [Asimow and Stolper, 1999; Kelemen and Dick, 1995], (4) the effect of addition of water to hot peridotite on melting [Eiler et al., 2000; Hirschmann et al., 1999b] and (5) the relationship between mantle heterogeneity and melt composition [Hirschmann et al., 1998b, 1999a; Schiano et al., 2000]. These studies have demonstrated the unique value of the thermochemical modeling approach in exploring processes that cannot be reproduced with experiments or described with simple parametarizations. Insights gained from these efforts have changed the way we look at mantle melting and have inspired new avenues for experimental investigation.
 Despite these successes and prospects, application of MELTS to mantle melting has been limited by a number of critical problems. Predicted compositions of partial melts of peridotite are systematically displaced from experimentally determined compositions and the relationship between temperature and melt fraction is offset by ∼100°C relative to experimental results [Baker et al., 1995; Hirschmann et al., 1998b]. Also, although the effect of water on solid-liquid phase relations is incorporated into MELTS, the adopted equation of state for water fails above 1 GPa. Several less prominent imperfections are also present, as discussed in detail by Hirschmann et al. [1998b]. These deficiencies have limited the value of MELTS as a quantitative, as opposed to qualitative tool for exploring the consequences of partial melting during adiabatic upwelling.
 In this contribution we outline a number of refinements to the thermodynamic model for the silicate liquids incorporated in MELTS that render an improved ability to calculate phase equilibria related to partial melting of the shallow mantle. This revised model is packaged into a computational algorithm with equivalent functionally to MELTS that we will term pMELTS (the “p” stands for pressure).
1.1. Overview of the Problem
 From afar, recalibration of MELTS to better predict partial melting of peridotite and related lithologies may seem like a simple matter. One may seek to reproduce a selected set of relevant existing experiments on partial melting of peridotite by using those experiments as calibrates. However, a number of considerations prevent such a simple solution from being effective. The objections are both practical and philosophical.
 From a practical perspective, there are simply not enough experimental constraints of sufficient quality on mantle bulk compositions over the temperature (T), pressure (P) range of interest to calibrate a set of thermodynamic models that adequately describe phase equilibria in this system. Because of the complex nature of igneous phases, the underlying thermodynamic models used to describe them necessarily contain too many adjustable parameters to be unambiguously fitted to the limited existing experimental data set. This problem is compounded by the important factor that it is not possible to acquire an experimental data set where T, P, and phase composition are uncorrelated. Systematic shifts in mineral and liquid compositions with T and P substantially inhibit isolation of compositional variables from intensive variables. This situation exacerbates the tendency for model parameters calibrated from such a restricted data set to be correlated and for the resulting model (thermodynamic or otherwise) to be unsuitable for extrapolation beyond the domain-space of the set. In addition, the complexity of performing experiments at high pressure, the difficulty of characterizing the intensive variables of the experiments, and systematic interlaboratory discrepancies in experimental results render our knowledge of the phase-equilibrium relations involved in mantle melting at best incomplete and at times contradictory. Systematic interlaboratory errors in unrelated experimental studies can generate fictitious geochemical trends that are difficult to recognize and eliminate during the fitting process.
 Aside from the practical issues that prohibit calibrating a thermodynamic model of mantle melting solely from experiments on partial melting of peridotite and related lithologies, there is the underlying philosophical issue that motivates work on MELTS and its descendents. Thermodynamic models are useful and worth constructing because they provide a framework for extrapolating experimental results beyond the direct objectives of the experiments. If one is only interested in interpolating, smoothing, or systematizing experimental observations, then the best method to use is cubic splines [Press et al., 1992]; the advantages gained in fitting observations to a thermodynamic formalism do not outweigh the difficulties and inconvenience if interpolation is all that is to be done with the model. In practice, however, pragmatic limitations dictate that experiments almost always involve some simplification and abstraction of the natural process they are designed to explore. Therefore most comparisons of experiments to natural occurrences involve some degree of extrapolation. Furthermore, if we are interested in performing calculations that are of a completely different nature than the experimental results (e.g., predicting the proportion of liquid during adiabatic, polybaric melting when we have available experiments detailing coexisting phase compositions at various T and P), then a thermodynamic formalism is worth the trouble.
 Such an ability to extrapolate comes at a cost. The underlying thermodynamic models for the solid and liquid phases that constitute the system must be complex enough to adequately characterize the energetics of these phases over a compositional, T and P range that generally far exceeds that of the experiments. The models must be internally consistent, follow the rules of thermodynamics derived from the first and second laws, and consequently satisfy experimental constraints in addition to those foremost of interest (e.g., calorimetric constraints on reference state properties). The consequence for a computational thermodynamics package like MELTS (or pMELTS) is that the experimental database of phase equilibria used to calibrate the underlying thermodynamic models must have as broad a range of bulk compositions and intensive variables as is feasible. Our philosophy [Ghiorso and Carmichael, 1980; Ghiorso, 1983; Hirschmann and Ghiorso, 1994; Ghiorso and Sack, 1995] has been to use experimental data of liquid-solid phase equilibria on bulk compositions that span the entire range of silicate magma types found in nature and to reference model calibration to an internally consistent compilation of reference state properties of minerals [Berman, 1988]. Our specific goal in this paper is to improve the ability of MELTS to predict mantle phase-equilibria. We will proceed by refining the underlying thermodynamic models to achieve this objective, but we will not do so by excluding experimental data on other systems. Our goal is to make pMELTS a superset not a subset of MELTS.
1.2. Brief Thermodynamic Background
 MELTS uses methods of computational thermodynamics and thermodynamic models of minerals and melts to compute an equilibrium assemblage as a function of composition and T, P, or other intensive thermodynamic variable [Ghiorso, 1997]. The thermodynamic underpinnings of MELTS and their relevance to modeling partial melting of the mantle have been extensively discussed and reviewed in a number of previous papers [Ghiorso et al., 1983; Ghiorso and Sack, 1995; Hirschmann et al., 1998b]. Here we briefly review the salient characteristics of MELTS that are relevant to the revisions of the model presented in this work. The thermodynamic models for minerals used in pMELTS are the same as those used in MELTS [Ghiorso and Sack, 1995] and therefore will not be reviewed here. Differences between pMELTS and MELTS are the result of (1) revisions of the thermodynamic model for the silicate liquid, (2) adoption of a new reference model for H2O, and (3) adoption of the Birch-Murnaghan volume equation of state.
 In MELTS and pMELTS, the molar Gibbs free energy of formation of silicate liquids is given by the function
where μio is the standard state chemical potentials of a set of linearly independent compositional end-members and Xi are their mole fractions ( refers specifically to water), is the molar configurational entropy,
arising from a set of chemical species taken to be equivalent to thermodynamic components, and Wij are a set of adjustable parameters (taken to be independent of T and P) that account for the nonideal contributions to mixing as inferred from experimentally determined compositions of silicate liquids and coexisting solids. Contributions to as a function of T, P, and composition in pMELTS differs from that of MELTS in the following ways: (1) some standard state properties of liquid components (in particular the reference state enthalpy and entropy) are optimized during model calibration, whereas in MELTS these quantities are not adjusted. (2) The P dependence of μio is modeled using a third-order Birch-Murnaghan equation of state (EOS). (3) Definitions of thermodynamic components are modified, resulting in a different and. (4) The Wij have been refit using an expanded database of experimental statements of mineral-liquid equilibria and a modified methodology. The details of these differences are described in the section 2.