These are discussed in section 5 of the Tech Ref. The notes below are all covered in Tech Ref but in a slightly different order. You may find these useful.

Imagine a simple case where you want to use some form of soft-restraint to keep a coordinate x1 close to a particular fractional coordinate, e.g. close to 0.137. Here 0.137 represents an “expected value”.

• In topas you could use a penalty equation: penalty = (x1 - 0.137)^2;

• Or you could use a restraint equation: restraint = (x1 - 0.137);

• Internally topas will square the restraint equation. Your overall chi^2 therefore increases regardless of whether x1 is less than or greater than 0.137. It therefore ends up being very similar to using the penalty equation.

• Don’t get too confused by the terms “penalty” and “restraint”. They both try to steer the refinement towards a solution that is consistent with a non-diffraction based observation (e.g. here for some reason you expect x1 to be close to 0.137; more commonly you expect bond distances or angles to have certain values), or some condition such as non-overlap of atoms or an energetically sensible configuration (e.g. Topas anti_bump or grs_interaction).

• Soft-restraints differ from constraints. Constraints are built into the refinement mathematically so that they have to be obeyed exactly.

• Restraint equations are handled by the least squares in exactly the same way as experimental observations (they are “extra observations”). The restraint is an integral part of refinement and would typically be included in esd calculations.

• Penalties are typically used to influence the direction of a refinement. For example, anti bump penalties can reduce the number of minima in Chi^2. Penalties typically aren’t included in esd calculations (though you can override this with do_errors_include_penalties).

• Topas mimimises the overall Chi^2, which is a weighted sum of Chi^2(data) + k1 Chi^2(penalties) + k2*Chi^2(restraints) where k1 and k2 are weights for penalties and restraints (see the manual for exact definitions).

• Topas uses a separate A matrix for the data, penalties and restraints. It chooses how to scale penalties or restraints against the data (i.e. how closely each is obeyed) by considering the relative magnitudes of the inverse error terms in each matrix.

• If you have several penalty equations you can change their relative weighting by using an equation like: penalty = w*(x1 - 0.137)^2. The value w might be different for a soft-restraint on a bond angle compared to a bond distance. Typically you’d weight by 1/sigma^2 where sigma^2 is the standard uncertainty on the quantity. The relative weighting for soft-restraints on distances (sigma = 0.01) and angles (sigma = 1) might therefore be 10000 to 1.

• The relative importance of penalties to data can be increased using the “penalties_weighting_K1 1” keyword. Normally the default value of 1 is appropriate.

• Tech Ref tells you how to change the penalty/restraint weightings to, for example, exactly mimic the formulation in shelx.

• There are differences in how penalties and restraints feed into the minimisation of Chi^2. These are covered in detail in Tech Ref. One difference is that off-diagonal terms in the A matrix are not calculated for penalties (unless approximate_A is defined). This means that you might get quicker minimisation using restraint equations. If penalties have a significant contribution to chi^2 than using approximate_A may give faster convergence as the off diagonal terms are then approximated by the BFGS method.

• Penalty equations can take any form – i.e. they can be more complex than the sum of differences squared which appear in restraint equations. For example the equations in grs_interaction (which calculates a Lennard-Jones or Born-Mayer potentaion) can’t be written as the sum of differences between observed can calculated atomic interactions.

• Depending on how you write your soft-restraint equations they will have different weightings against the diffraction data and the mimimisation pathway and final minimum may differ.

• The built-in topas macros for soft-restraints such as Distance_Restrain() mainly apply penalty equations.