RightNablaD[i] denotes \nabla _{i}, acting to the right.
Overview, ExpandPartialD, FCPartialD, LeftNabalD.
RightNablaD[i]\vec{\nabla }^i
RightNablaD[i] . QuantumField[A, LorentzIndex[\[Mu]]]
ex = ExpandPartialD[%]\vec{\nabla }^i.A_{\mu }
-\left(\partial _iA_{\mu }\right)
ex // StandardForm
(*-QuantumField[FCPartialD[CartesianIndex[i]], A, LorentzIndex[\[Mu]]]*)RightNablaD[i] // StandardForm
(*RightNablaD[CartesianIndex[i]]*)