Dimensional Analysis (working with parmed.unit)

The unit package was originally developed by Christopher M. Bruns at SimTK (Stanford University) for the simtk Python package in OpenMM.

Different programs use different systems of units internally while performing calculations on quantities with specific dimensions. For instance, the Amber and CHARMM programs deal with energies in kcal/mol and lengths in Angstroms, while GROMACS and OpenMM deal with energies in kJ/mol and distances in nanometers. Electronic structure packages, on the other hand, tend to use so-called atomic units, with energies in Hartrees and lengths in Bohrs.

Comparing the results between these programs requires that the results be converted to a common set of units beforehand. More importantly—interconverting these packages, or preparing common input for both, requires that the input for each program conform to its expected system of units. Not doing so can result in a very costly mistake

In all of the examples below, I will assume that the unit package has been imported as follows:

from parmed import unit as u

What does parmed.unit do?

The main classes in the unit module are shown below:

Quantity([value, unit]) Physical quantity, such as 1.3 meters per second.
is_quantity(x) Returns True if x is a Quantity, False otherwise.
Unit(base_or_scaled_units) Physical unit such as meter or ampere.
BaseUnit(base_dim, name, symbol) Physical unit expressed in exactly one BaseDimension.
UnitSystem(units) A complete system of units defining the base unit in each dimension

The unit package allows you to turn numbers into a Quantity, that has both a value and a unit. It allows you to do basic math with units and it automatically performs the requisite dimensional analysis, raising an exception if you try to illegally combine incompatible units. For example:

>>> x = 1.0 * u.nanometers
>>> x + 1.0 * u.angstroms
Quantity(value=1.1, unit=nanometer)
>>> # Find the volume of a cube with side 1 cm
>>> x = 1 * u.centimeters
>>> v = x ** 3
>>> v
Quantity(value=1, unit=centimeter**3)
>>> v + 1.0 * u.milliliters
Quantity(value=1.9999999999999998, unit=centimeter**3)
>>> # Now try to combine incompatible units
>>> 1*u.millimeters + 1*u.kelvin
TypeError: Cannot add two quantities with incompatible units "millimeter" and "kelvin".

Two units can be multiplied together to form a composite unit with the corresponding dimensionality—in the example above, cubing 1 centimeter resulted in a volume, which is a cubic length dimension. This works for much more complex units as well:

>>> x = 1 * u.kilograms * u.meters**2 / u.seconds**2
>>> x
Quantity(value=1, unit=kilogram*meter**2/(second**2))
>>> x + 1 * u.joule
Quantity(value=2, unit=kilogram*meter**2/(second**2))

How do I use units?

There are a variety of tasks you may want to do with units – the more popular of which is shown in the sections below.

Checking to see if an object is a Quantity

There may be times when you want to see if a Python object is a Quantity instance, or if it is just a raw number. Many functions in ParmEd do this when taking user input, converting to its internal AKMA unit system (see below). Checking that an object is a Quantity is done using the is_quantity() function in the parmed.unit namespace:

>>> x = 1
>>> y = 1 * u.nanometers
>>> u.is_quantity(x)
False
>>> u.is_quantity(y)
True

Creating units

You can create a unit in one of two ways:

  1. Use the Quantity constructor, as shown below:

    >>> u.Quantity(1, u.nanometers)
    Quantity(1, nanometer)
    >>> u.Quantity(1, u.meters**-3) # You can raise units to exponents
    Quantity(value=1, unit=/(meter**3))
    
  2. Multiply (or divide!) by a specific unit, as shown below:

    >>> 1 * u.nanometers
    Quantity(1, nanometer)
    >>> 1 / u.meters**3 # You can raise units to exponents
    Quantity(value=1, unit=/(meter**3))
    

Seems easy enough, and I usually use method 2 when turning a float, int, list, or tuple into a Quantity.

Converting to another unit

Aside from using units as a type-checker to make sure you are not making an arithmetic mistake with your transformations, the most common reason to use the unit package is to handle unit conversions for you. The Quantity class defines two functions for this:

  1. in_units_of(unit): This method returns a new Quantity instance with the specified unit (a TypeError is raised).
  2. value_in_unit(unit): This method returns a copy of the underlying data without an attached unit (i.e., it is not a Quantity instance). This should be used when a scalar is needed from a quantity for use in a particular API, or if the performance hit from the dimensional analysis is unacceptable.

The following example shows both of the above in action, demonstrating their differences:

>>> x = 1 * u.kilocalorie
>>> x.in_units_of(u.kilojoule)
Quantity(value=4.184, unit=kilojoule)
>>> x.value_in_unit(u.kilojoule)
4.184

Converting units in batch – the UnitSystem

The previous section described how to convert a Quantity into another Quantity in a different, but compatible unit. The argument to in_units_of and value_in_unit methods must be a compatible unit, which can be limiting if you want to write a function that changes the units of an argument with unknown dimensionality into a particular system of units.

This is where UnitSystem comes in. Each Quantity instance also has a corresponding methods for conversion to units compatible within a particular UnitSystem:

  1. in_unit_system(UnitSystem): This method returns a new Quantity instance composed of the base units defined in that unit system.
  2. value_in_unit_system(UnitSystem): This method returns a scalar of the value in that particular unit system.

For example:

>>> x = 1 * u.kilocalorie
>>> x.in_unit_system(u.md_unit_system)
Quantity(value=4.184, unit=nanometer**2*mole*dalton/(picosecond**2))
>>> x.value_in_unit_system(u.md_unit_system)
4.184

This can be useful when designing a function that needs to handle arbitrary units, but it comes with some drawbacks (see below), so you should avoid unit system conversions when you can.

A warning about UnitSystem

Note that this method does not provide the added security of checking that the dimensions are correct – it simply converts all base dimensions to the ones defining the unit system. For instance, consider the two functions below:

def area(x, y):
    """ Computes the area of a rectangle

    Parameters
    ----------
    x : distance Quantity
        The length of the first side
    y : distance Quantity
        The length of the second side

    Returns
    -------
    area : float
        Area in nanometers^2
    """
    return (x.value_in_unit_system(u.md_unit_system) *
                y.value_in_unit_system(u.md_unit_system))

def area2(x, y):
    return (x.value_in_unit(u.nanometers) *
                y.value_in_unit(u.nanometers))

As long as you provide the correct dimensions in the arguments, both functions yield the same result:

>>> area(1*u.angstroms, 1*u.millimeters)
100000.0
>>> area2(1*u.angstroms, 1*u.millimeters)
100000.0

Consider what happens if you make a small mistake, though:

>>> area(1/u.angstroms, 1*u.millimeters)
9999999.999999998
>>> area2(1/u.angstroms, 1*u.millimeters)
TypeError: Unit "/angstrom" is not compatible with Unit "nanometer".

Converting 1/angstrom to the md_unit_system converts the value to 10/nanometer (rather than 1 angstrom to 0.1 nanometer), leading to a result that is 2 orders of magnitude larger than expected! The area2 function catches the error and raises an exception, though, thereby making the mistake easier to find.

Available UnitSystem options

The unit module contains a number of UnitSystem instances available in the parmed.unit namespace. They are summarized in the table below, along with the units defining them (the energy unit is a composite of the mass unit times the square of the ratio of the length-to-time units):

UnitSystem Length unit Mass unit Time unit Charge unit [1] Temperature unit Amount unit Energy unit
si_unit_system meters kilograms second Ampere kelvin mole joule
cgs_unit_system centimeters gram second Ampere kelvin mole 1e-7 joule
md_unit_system nanometers daltons picosecond q electron kelvin mole kilojoule
planck_unit_system pl. length pl. mass pl. time pl. charge pl. temperature item pl. energy
akma_unit_system angstroms daltons akma time [2] q electron kelvin mole kilocalorie

[1] Some unit systems use current instead of charge as a base unit, while others use the charge.

[2] The time in the AKMA unit system (Angstroms, Kilocalories per Mole, Atomic mass units) does not have an official name, and is roughly equal to 20.455 picoseconds.

Different programs tend to use different unit systems. As a general reference, the table below summarizes a subset of programs and the unit systems they use:

Program Unit System
AMBER akma_unit_system
CHARMM akma_unit_system
Tinker akma_unit_system
Desmond akma_unit_system
LAMMPS akma_unit_system
NAMD akma_unit_system
AceMD akma_unit_system
OpenMM md_unit_system
Gromacs md_unit_system
Gromos md_unit_system