  The Goodenough-Kanamori-Anderson (GKA) rules have been widely applied for
explaining the magnetic properties induced by super-exchange interaction. As
conclusions of the super-exchange theory, they reveal the antiferromagnetic
(ferromagnetic) ordering along with bond angle of 180 degrees (90 degrees) in
the cation-anion-cation interaction path, in which the theory sets a
pre-condition that the electronic states of cations in all paths are identical.
We observed that the GKA rules are in fact not universal and even invalid to
materials containing anions with different valence states, for example, the
layered CrOCl crystal (with two valence states of anions: O2- and Cl-). In this
study, we propose an extended super-exchange theory (ESET) related to
superposed electronic states of cation in a specific path. ESET is capable of
predicting not only the sign and relative magnitude of magnetic exchange
constants in different cation-anion-cation paths, but also the magnetic ground
state. Through our proposed theory, we conclude that the magnetic ordering
along with bond angle of 90 degrees in Cr-Cl-Cr path is moderately
antiferromagnetic and of 180 degrees in Cr-O-Cr path is strongly ferromagnetic,
which are opposite to the contents of GKA rules. Moreover, we clarify that
monolayer CrOCl has antiferromagnetic ordering rather than ferromagnetic as
reported recently. The reliability of ESET is verified via first-principles
calculation and previous experimental report as well, and its universality is
also demonstrated. Thus, our theory is powerful to predict the magnetic
properties, which makes it possible to design new high Curie temperature
two-dimensional semiconducting ferromagnets with polyvalent anion materials.
