\(\renewcommand{\AA}{\text{Å}}\)

# bond_style lepton command¶

Accelerator Variants: *lepton/omp*

## Syntax¶

```
bond_style lepton
```

## Examples¶

```
bond_style lepton
bond_coeff 1 1.5 "k*r^2; k=250.0"
bond_coeff 2 1.1 "k2*r^2 + k3*r^3 + k4*r^4; k2=300.0; k3=-100.0; k4=50.0"
bond_coeff 3 1.3 "k*r^2; k=350.0"
```

## Description¶

New in version 8Feb2023.

Bond style *lepton* computes bonded interactions between two atoms with
a custom function. The potential function must be provided as an
expression string using “r” as the distance variable relative to the
reference distance \(r_0\) which is provided as a bond coefficient.
For example “200.0*r^2” represents a harmonic potential with a force
constant *K* of 200.0 energy units:

The Lepton library, that the
*lepton* bond style interfaces with, evaluates this expression string at
run time to compute the pairwise energy. It also creates an analytical
representation of the first derivative of this expression with respect to
“r” and then uses that to compute the force between the atom pairs forming
bonds as defined by the topology data.

The following coefficients must be defined for each bond type via the bond_coeff command as in the examples above, or in the data file or restart files read by the read_data or read_restart commands:

Lepton expression (energy units)

\(r_0\) (distance)

The Lepton expression must be either enclosed in quotes or must not contain any whitespace so that LAMMPS recognizes it as a single keyword. More on valid Lepton expressions below. The \(r_0\) is the “equilibrium distance”. The potential energy function in the Lepton expression is shifted in such a way, that the potential energy is 0 for a bond length \(r_i == r_0\).

## Lepton expression syntax and features¶

Lepton supports the following operators in expressions:

+ |
Add |
- |
Subtract |
* |
Multiply |
/ |
Divide |
^ |
Power |

The following mathematical functions are available:

sqrt(x) |
Square root |
exp(x) |
Exponential |

log(x) |
Natural logarithm |
sin(x) |
Sine (angle in radians) |

cos(x) |
Cosine (angle in radians) |
sec(x) |
Secant (angle in radians) |

csc(x) |
Cosecant (angle in radians) |
tan(x) |
Tangent (angle in radians) |

cot(x) |
Cotangent (angle in radians) |
asin(x) |
Inverse sine (in radians) |

acos(x) |
Inverse cosine (in radians) |
atan(x) |
Inverse tangent (in radians) |

sinh(x) |
Hyperbolic sine |
cosh(x) |
Hyperbolic cosine |

tanh(x) |
Hyperbolic tangent |
erf(x) |
Error function |

erfc(x) |
Complementary Error function |
abs(x) |
Absolute value |

min(x,y) |
Minimum of two values |
max(x,y) |
Maximum of two values |

delta(x) |
delta(x) is 1 for x = 0, otherwise 0 |
step(x) |
step(x) is 0 for x < 0, otherwise 1 |

Numbers may be given in either decimal or exponential form. All of the following are valid numbers: 5, -3.1, 1e6, and 3.12e-2.

As an extension to the standard Lepton syntax, it is also possible to use LAMMPS variables in the format “v_name”. Before evaluating the expression, “v_name” will be replaced with the value of the variable “name”. This is compatible with all kinds of scalar variables, but not with vectors, arrays, local, or per-atom variables. If necessary, a custom scalar variable needs to be defined that can access the desired (single) item from a non-scalar variable. As an example, the following lines will instruct LAMMPS to ramp the force constant for a harmonic bond from 100.0 to 200.0 during the next run:

```
variable fconst equal ramp(100.0, 200)
bond_style lepton
bond_coeff 1 1.5 "v_fconst * (r^2)"
```

An expression may be followed by definitions for intermediate values that appear in the expression. A semicolon “;” is used as a delimiter between value definitions. For example, the expression:

```
a^2+a*b+b^2; a=a1+a2; b=b1+b2
```

is exactly equivalent to

```
(a1+a2)^2+(a1+a2)*(b1+b2)+(b1+b2)^2
```

The definition of an intermediate value may itself involve other
intermediate values. Whitespace and quotation characters (’'’ and ‘”’)
are ignored. All uses of a value must appear *before* that value’s
definition. For efficiency reasons, the expression string is parsed,
optimized, and then stored in an internal, pre-parsed representation for
evaluation.

Evaluating a Lepton expression is typically between 2.5 and 5 times slower than the corresponding compiled and optimized C++ code. If additional speed or GPU acceleration (via GPU or KOKKOS) is required, the interaction can be represented as a table. Suitable table files can be created either internally using the pair_write or bond_write command or through the Python scripts in the tools/tabulate folder.

Styles with a *gpu*, *intel*, *kk*, *omp*, or *opt* suffix are
functionally the same as the corresponding style without the suffix.
They have been optimized to run faster, depending on your available
hardware, as discussed on the Accelerator packages
page. The accelerated styles take the same arguments and should
produce the same results, except for round-off and precision issues.

These accelerated styles are part of the GPU, INTEL, KOKKOS, OPENMP, and OPT packages, respectively. They are only enabled if LAMMPS was built with those packages. See the Build package page for more info.

You can specify the accelerated styles explicitly in your input script by including their suffix, or you can use the -suffix command-line switch when you invoke LAMMPS, or you can use the suffix command in your input script.

See the Accelerator packages page for more instructions on how to use the accelerated styles effectively.

## Restrictions¶

This bond style is part of the LEPTON package and only enabled if LAMMPS was built with this package. See the Build package page for more info.

## Default¶

none