Following the the wikipedia article about the adiabatic theorem, and Sakurai's Modern QM, we start with the definition of the geometric phase that we get when doing a loop with a parameter R which depends on time :
$$ \gamma_n(T)=i \int_0^T\left\langle n(R(t)) \mid \frac{\partial}{\partial t}(n(R(t)))\right\rangle d t $$
Let's calculate this time derivative. I use the chain rule with a differential :
$$ \frac{\partial}{\partial t} n(R(t))=dn_{R(t)}\left(R^{\prime}(t)\right) $$
In order to get the notations with a gradient (which are in direction of $R$), we can put the components of the differential, and then those differentials are those of real-valued functions so we can write them using gradients :
$$ =\left(\nabla_R n_1(R(t)) \cdot R^{\prime}(t), \ldots, \nabla_{R}n_m(R(t)) \cdot R^{\prime}(t)\right) $$
And now, if I understand correctly, the authors use the following ambiguous notation :
$$ =\nabla_R n(R(t)) \cdot R^{\prime}(t) $$
Finally, our geometric phase is :
$$ \gamma_n(T)=i \int_0^T\left\langle m(R(t)) \mid \nabla_R n(R(t)) \cdot R^{\prime}(t)\right\rangle d t $$
Here, we're making a change of variables, but honestly I'm not sure how to formalize it. Using $R^{\prime}(t)=d R / d t$, and cancelling $dt$s, we get the usual definition of Berry's phase :
$$ \gamma_m(T)=i \int_{R(0)}^{R(T)}\left\langle n(R) \mid \nabla_R n(R)\right\rangle \cdot dR $$
I think something doesn't add up because what we're integrating now is a sort of "inner product which takes as the second argument a vector of vectors" and that definitely doesn't sound right. The author also mentions that the gradient is an operator, a bit like in QM, so I might miss something obvious. I'm not at ease with the notion of a gradient of a vector field. Is there something wrong with my formulas ? Thanks in advance !