I want to understand these things in both intuitive and non-intuitive ways. In some cases, we constrain hydrogens so that we can run molecular dynamics at a higher time step than we normally would, also tricks like using virtual sites. I can understand that these tricks allow us to propagate our system so that it becomes computationally stable. Non-bonded interactions shouldn't be affected by these, but what about the bonded interactions and eventual dynamics. Won't it affect the eventual dynamics of the system, also some properties of the system. I hope I made my question clear. If not, I can elaborate more. Thanks.
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This question is quite broad and I can only give you some examples from the literature. If you want an example of how different treatment of force constants can affect system properties, you can have a look at the Ferguson water model from 1995, where a anharmonic component in the bond stretching term allows the model to capture some of the (IR) vibrational modes of water. These models have some applications, such this work by Caleman and van der Spoel.
Constraints, as you correctly pointed out, are mainly used to remove the fastest vibrational modes which allows for the use of a larger timestep. Additionally it removes the potential energy component of that vibration, which affects properties like heat capacity.* Highly constrained models can lead to artifacts as well; in 2022 Thallmair et al. published a nice overview of how nonconverged constraints lead to temperature gradients in cholesterol containing bilayer systems.
