^{1,2,3,a)}, D. Alfè

^{1,2,3,4}and M. J. Gillan

^{1,2,3}

### Abstract

We present calculations of the free energy, and hence the melting properties, of a simple tight-binding model for transition metals in the region of -band filling near the middle of a -series, the parameters of the model being designed to mimic molybdenum. The melting properties are calculated for pressures ranging from ambient to several megabars. The model is intended to be the simplest possible tight-binding representation of the two basic parts of the energy: first, the pairwise repulsion due to Fermi exclusion; and second, the -band bonding energy described in terms of an electronic density of states that depends on structure. In addition to the number of -electrons, the model contains four parameters, which are adjusted to fit the pressure dependent -band-width and the zero-temperature pressure-volume relation of Mo. We show that the resulting model reproduces well the phonon dispersion relations of Mo in the body-centered-cubic structure, as well as the radial distribution function of the high-temperature solid and liquid given by earlier first-principles simulations. Our free energy calculations start from the free energy of the liquid and solid phases of the purely repulsive pair potential model, without -band bonding. The free energy of the full tight-binding model is obtained from this by thermodynamic integration. The resulting melting properties of the model are quite close to those given by earlier first-principles work on Mo. An interpretation of these melting properties is provided by showing how they are related to those of the purely repulsive model.

The work was supported by the EPSRC-GB Grant No. EP/C534360, which is 50% funded by DSTL(MOD). The work was conducted as part of a EURYI scheme award to DA as provided by the EPSRC-GB (see www.esf.org/euryi).

I. INTRODUCTION

II. THE TIGHT-BINDING TOTAL-ENERGY MODEL

A. The canonical -band tight-binding Hamiltonian

B. The repulsive pair potential

III. SIMULATION TECHNIQUES AND TESTS OF THE MODEL

A. Molecular dynamics simulation

B. Tests of the model

IV. FREE ENERGY AND MELTING PROPERTIES OF THE MODEL

A. Free energy and phase diagram of the pure exponential model

B. Free energy and melting properties of the TB model

V. ANALYSIS OF MELTING RELATIONSHIPS

VI. DISCUSSION AND CONCLUSIONS

### Key Topics

- Density functional theory
- 33.0
- Melting
- 33.0
- Molybdenum
- 30.0
- Free energy
- 28.0
- Transition metals
- 26.0

## Figures

-component of the electronic density of states of bcc Mo calculated at using DFT and TB at pressures of 0 GPa (top panel) and 350 GPa (bottom panel). The Fermi energy is set to zero (vertical lines).

-component of the electronic density of states of bcc Mo calculated at using DFT and TB at pressures of 0 GPa (top panel) and 350 GPa (bottom panel). The Fermi energy is set to zero (vertical lines).

Equation of state of bcc Mo obtained from the present TB model (solid line) and DFT (dashed line); experimental data (dots) from Ref. 6 are shown for comparison.

Equation of state of bcc Mo obtained from the present TB model (solid line) and DFT (dashed line); experimental data (dots) from Ref. 6 are shown for comparison.

Phonon dispersion relations of bcc Mo calculated with the present TB model (solid lines) and DFT (dashed lines) at the experimental equilibrium volume . Experimental data (dots) from Ref. 55 are shown for comparison.

Phonon dispersion relations of bcc Mo calculated with the present TB model (solid lines) and DFT (dashed lines) at the experimental equilibrium volume . Experimental data (dots) from Ref. 55 are shown for comparison.

Radial distribution function of solid bcc Mo at and from long DFT and TB m.d. runs. Bottom: Radial distribution function of liquid Mo at and obtained from long DFT and TB m.d. runs.

Radial distribution function of solid bcc Mo at and from long DFT and TB m.d. runs. Bottom: Radial distribution function of liquid Mo at and obtained from long DFT and TB m.d. runs.

-band electronic density of states calculated by DFT and TB m.d. simulation for bcc Mo at , (top panel) and for liquid Mo at , (bottom panel).

-band electronic density of states calculated by DFT and TB m.d. simulation for bcc Mo at , (top panel) and for liquid Mo at , (bottom panel).

Phase diagram of the pure exponential model obtained from coexisting solid and liquid phase (, , ) simulations. The solid line in the figure corresponds to the bcc-liquid phase boundary while the dashed and dotted lines are the fcc-liquid and hcp-liquid ones, respectively. Dots symbolize points obtained directly from the phase coexistence simulations.

Phase diagram of the pure exponential model obtained from coexisting solid and liquid phase (, , ) simulations. The solid line in the figure corresponds to the bcc-liquid phase boundary while the dashed and dotted lines are the fcc-liquid and hcp-liquid ones, respectively. Dots symbolize points obtained directly from the phase coexistence simulations.

Thermal average of the TB energy as a function of in an adiabatic thermodynamic integration calculation of the free energy difference between the and REP systems. The plot shows from a simulation in which executes a double cycle , the rate of variation being .

Thermal average of the TB energy as a function of in an adiabatic thermodynamic integration calculation of the free energy difference between the and REP systems. The plot shows from a simulation in which executes a double cycle , the rate of variation being .

Melting curve of TB model at -band fillings (dashed line) and 5.0 (dotted line). The melting curve of the pure exponential model and that of Mo from DFT simulations (Ref. 25) are show for comparison.

Melting curve of TB model at -band fillings (dashed line) and 5.0 (dotted line). The melting curve of the pure exponential model and that of Mo from DFT simulations (Ref. 25) are show for comparison.

Melting curve of the repulsive pure exponential potential (solid line), pure exponential potential plus a bonding energy term depending just on volume (short-dashed line), and full TB model at (long-dashed line) and 5.0 (dotted line).

Melting curve of the repulsive pure exponential potential (solid line), pure exponential potential plus a bonding energy term depending just on volume (short-dashed line), and full TB model at (long-dashed line) and 5.0 (dotted line).

Top: Free energy difference of the total TB model and repulsive pure exponential potential in the liquid and solid phases at temperature and for . Bottom: Quantity in the liquid and solid phases at temperature and for .

Top: Free energy difference of the total TB model and repulsive pure exponential potential in the liquid and solid phases at temperature and for . Bottom: Quantity in the liquid and solid phases at temperature and for .

## Tables

Calculated -band-width , second moment and energy difference from DFT and TB at ( in parentheses). Energies are in eV, and the number of electrons is .

Calculated -band-width , second moment and energy difference from DFT and TB at ( in parentheses). Energies are in eV, and the number of electrons is .

Melting temperature as a function of pressure , volumes per atom and in coexisting liquid and solid, relative volume change , and entropy of fusion of the pure exponential system for coexisting bcc solid and liquid. Estimated errors are given in parentheses.

Melting temperature as a function of pressure , volumes per atom and in coexisting liquid and solid, relative volume change , and entropy of fusion of the pure exponential system for coexisting bcc solid and liquid. Estimated errors are given in parentheses.

Melting pressure as a function of temperature , volumes per atom and of coexisting liquid and solid, relative volume of fusion , and entropy of fusion , for TB model at -band fillings .

Melting pressure as a function of temperature , volumes per atom and of coexisting liquid and solid, relative volume of fusion , and entropy of fusion , for TB model at -band fillings .

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