Should I avoid using personal pronouns in an abstract? I was somewhat surprised when reviewing the Full Paper Template from the Journal of Computational Chemistry, which states that the abstract text should contain "no personal pronouns".  I have seen many articles which go against this and even several that have been published in this journal recently (http://onlinelibrary.wiley.com/doi/10.1002/jcc.24217/full,  http://onlinelibrary.wiley.com/doi/10.1002/jcc.24225/full, http://onlinelibrary.wiley.com/doi/10.1002/jcc.24252/full).

In this specific case, I plan to follow whatever instructions the editor provides.  In general though, is it better to avoid using personal pronouns in the abstract for scientific writing?
The requirement you have found refers to the text of the graphical abstract, not to the textual abstract. A graphical abstract is an image with little text: thus, it is reasonable to have stricter requirements, with respect to standard abstracts, on what is allowed or not.  

See also this page on graphical abstracts.

As for standard abstracts, I quickly checked a few author guides from IEEE, IOP and AIP and none of them seem against the usage of personal pronouns in abstracts. Also The Chicago Manual of Style, 16ed, does not mention anything about the usage of pronouns in abstracts. I've used the pronoun "we" in several abstracts and no one objected.

In general, the main requirement is that the abstract should be self-contained: no citations (if really necessary they should be written inline), or references to figures and tables.