BMP Potential (Bertani–Menziani–Pedone)¶

Interactive example

See the Melt-Quench Tutorial notebook for worked examples with both bmp-harmonic and bmp-screened-harmonic.

The BMP potential is a Morse-based force field extended with cation–cation Buckingham repulsion and explicit three-body angular terms. It is available in two variants that differ only in how the three-body interactions are handled:

bmp-harmonic — uses nb3b/harmonic (harmonic angle potential). Simpler form, covers Si–O–P and Si–O–Si triplets.
bmp-screened-harmonic — uses nb3b/screened (screened harmonic). Adds distance-dependent screening and covers B and V in addition to Si and P.

Both variants use identical Morse, Buckingham, and electrostatic parameters. The choice of variant controls which LAMMPS three-body style is activated and which element triplets are parameterised.

References¶

M. Bertani, M. C. Menziani, A. Pedone. "Improved empirical force field for multicomponent oxide glasses and crystals", Phys. Rev. Materials 5, 045602 (2021)
G. Malavasi, A. Pedone. "The effect of the incorporation of catalase mimetic activity cations on the structural, thermal and chemical durability properties of the 45S5 Bioglass®", Acta Materialia 229, 117801 (2022)
M. Bertani, A. Pallini, M. Cocchi, M. C. Menziani, A. Pedone. "A new self-consistent empirical potential model for multicomponent borate and borosilicate glasses", J. Am. Ceram. Soc. 105, 7254–7271 (2022)
M. Bertani et al. "Erratum for a new self-consistent empirical potential model for multicomponent borate and borosilicate glasses", J. Am. Ceram. Soc. 106, 5104–5105 (2023)

Functional Form¶

Morse (pair) interaction¶

\[ V_{\text{Morse}}(r_{ij}) = D_{ij}\left[\left(1 - e^{-\alpha_{ij}(r_{ij}-r_{0,ij})}\right)^2 - 1\right] + \frac{A_{ij}}{r_{ij}^{12}} + \frac{q_i q_j}{r_{ij}} \]

where \(D_{ij}\) (eV) is the well depth, \(\alpha_{ij}\) (Å⁻¹) the stiffness, \(r_{0,ij}\) (Å) the equilibrium distance, and \(A_{ij}\) (eV·Å¹²) the short-range repulsive wall.

For Boron, \(D_{\text{B-O}}\) is not a fixed constant — it is computed from the glass composition via the Dell–Bray model (see Boron restriction below).

Cation–cation Buckingham repulsion¶

\[ V_{\text{Buck}}(r_{ij}) = A_{ij}^{\text{Buck}} \, e^{-r_{ij}/\rho_{ij}} \]

Only specific cation–cation pairs (Si–Si, Si–Al, B–Si, etc.) carry Buckingham parameters; most cross-terms are zero.

Three-body angular terms¶

bmp-harmonic (nb3b/harmonic):

\[ V_{\text{3b}}(\theta_{ijk}) = \frac{K}{2}\left(\theta_{ijk} - \theta_0\right)^2 \]

bmp-screened-harmonic (nb3b/screened):

\[ V_{\text{3b}}(r_{ij}, r_{ik}, \theta_{ijk}) = K\left(\theta_{ijk} - \theta_0\right)^2 e^{-r_{ij}/\rho} e^{-r_{ik}/\rho} \]

The screened form damps the angular penalty as bond lengths increase, which is more physically appropriate for the disordered glass network and allows parameterisation of B and V triplets.

LAMMPS pair style:

pair_style hybrid/overlay coul/dsf <alpha> <cutoff> pedone 7.0 buck 7.0 nb3b/harmonic   # bmp-harmonic
pair_style hybrid/overlay coul/dsf <alpha> <cutoff> pedone 7.0 buck 7.0 nb3b/screened   # bmp-screened-harmonic

Three-body Parameters¶

`bmp-harmonic` (`nb3b/harmonic`)¶

Triplet (center–O–center)	\(K/2\) (eV/rad²)	\(\theta_0\) (°)	Cutoff (Å)
Si–O–Si	0.73	109.47	2.0
Si–O–P	2.00	109.47	2.0
P–O–P	2.00	109.47	2.0

`bmp-screened-harmonic` (`nb3b/screened`)¶

Triplet (center–O–center)	\(K\) (eV/rad²)	\(\theta_0\) (°)	Cutoff (Å)
Si–O–Si	25.0	109.47	3.30
Si–O–P	120.0	109.47	2.00
P–O–P	65.0	109.47	2.00
B–O–B	60.0	109.47	3.30
B–O–Si	60.0	109.47	3.30
V–O–V	30.0	109.00	2.50
V–O–P	120.0	109.00	2.00
V–O–Si	120.0	109.00	2.00

All triplets not in the table receive zero parameters (effectively inactive).

Boron Restriction¶

When Boron is present in the composition, the B–O well depth \(D_{\text{B-O}}\) is computed from the glass composition using the Dell–Bray model, which expresses \(D\) as a function of two ratios:

\[ R = \frac{[\text{alkali oxide}] + [\text{alkaline-earth oxide}]}{[\text{B}_2\text{O}_3]}, \qquad K = \frac{[\text{SiO}_2]}{[\text{B}_2\text{O}_3]} \]

This model is only valid for systems containing B, Si, O and alkali/alkaline-earth modifiers. If any other element is present alongside Boron, generate_potential raises a ValueError.

Elements allowed with Boron:

Category	Elements
Network formers	B, Si, O
Alkali modifiers	Li, Na, K
Alkaline-earth modifiers	Mg, Ca, Sr, Ba

Compositions such as B + Al, B + Ti, or B + Fe will raise an error. Remove Boron or use bmp-harmonic with a B-free composition in those cases.

bmp-harmonic vs. boron: The harmonic three-body style has no B triplets in its parameter set, so bmp-harmonic works correctly with B-free glasses. To simulate borosilicate or borate systems, use bmp-screened-harmonic.

Supported Elements¶

The BMP potential covers 36 elements:

Element	Charge (\(e\))	Role
Li, Na, K	+0.6	Alkali modifiers
Be, Mg, Ca, Sr, Ba	+1.2	Alkaline-earth modifiers
B	+1.8	Network former (composition-dependent \(D\))
Al	+1.8	Intermediate / former
Si	+2.4	Network former
P	+3.0	Network former
Ge, Sn	+2.4	Network formers
Ti, Zr, Sc	+2.4 / +1.8	Intermediate
Fe²⁺, Fe³⁺	+1.2 / +1.8	Transition metal
Mn, Mn³⁺, Mn⁴⁺	+1.2 / +1.8	Transition metal
Cr	+1.8	Transition metal
Cu, Cu²⁺	+0.6 / +1.2	Transition metal
Zn, Ni, Co	+1.2	Transition metal
Ag	+0.6	Noble metal
Ce³⁺, Ce⁴⁺	+1.8 / +2.4	Rare earth
Nd, Gd, Er	+1.8	Rare earth
V⁴⁺, V⁵⁺	+2.4 / +3.0	Vanadium (shrm only for 3b)
O	−1.2	Anion (fixed)

Usage¶

from amorphouspy import get_structure_dict
from amorphouspy.potentials import generate_potential

structure_dict = get_structure_dict(
    {"SiO2": 70, "Na2O": 20, "CaO": 10},
    target_atoms=3000,
)

# BMP with harmonic three-body (Si/P glasses, no boron)
potential = generate_potential(structure_dict, potential_type="bmp-harmonic")

# BMP with screened three-body (borosilicate/borate glasses)
structure_dict_b = get_structure_dict(
    {"SiO2": 60, "B2O3": 20, "Na2O": 20},
    target_atoms=3000,
)
potential_b = generate_potential(structure_dict_b, potential_type="bmp-screened-harmonic")

Melt pre-equilibration¶

Pass melt=True when the structure was generated by get_structure_dict. The BMP potential can produce large forces on random configurations, particularly from the \(r^{-12}\) repulsive wall. The pre-equilibration stage runs 10,000 steps of Langevin dynamics with a maximum displacement of 0.5 Å per step at 4000 K to gently relax atomic overlaps.

potential = generate_potential(structure_dict, potential_type="bmp-harmonic", melt=True)

When melt=False (default), the pre-equilibration block is omitted — useful when starting from an already-equilibrated structure.

Electrostatics options¶

BMP supports all four Coulomb solvers. The default is DSF with \(\alpha = 0.25\) Å⁻¹ and a cutoff of 8.0 Å.

from amorphouspy import DsfConfig, WolfConfig, PppmConfig, EwaldConfig

# DSF (default) — real-space, no k-space
potential = generate_potential(structure_dict, potential_type="bmp-harmonic",
                               electrostatics=DsfConfig(alpha=0.25, long_range_cutoff=8.0))

# Wolf summation
potential = generate_potential(structure_dict, potential_type="bmp-harmonic",
                               electrostatics=WolfConfig(alpha=0.25, long_range_cutoff=8.0))

# PPPM — reciprocal-space, more accurate for large systems
potential = generate_potential(structure_dict, potential_type="bmp-harmonic",
                               electrostatics=PppmConfig(long_range_cutoff=12.0))

# Ewald — reciprocal-space
potential = generate_potential(structure_dict, potential_type="bmp-harmonic",
                               electrostatics=EwaldConfig(long_range_cutoff=12.0))

The short-range (Morse + Buckingham) cutoff is fixed at 7.0 Å.

When to Use Each Variant¶

	`bmp-harmonic`	`bmp-screened-harmonic`
Three-body style	`nb3b/harmonic`	`nb3b/screened`
Boron support	No	Yes (with alkali/AE only)
Vanadium 3b	No	Yes
Angular screening	None	Distance-dependent
Best for	Multi-modifier silicate/phosphate oxides	Borate and borosilicate glasses

Both variants support the full 36-element library for non-three-body interactions. The choice only affects which angular triplets are active.