Let
L
=
−
Δ
H
n
+
V
\mathcal {L}=-\Delta _{\mathbb {H}^n}+V
be a Schrödinger operator with the nonnegative potential
V
V
belonging to the reverse Hölder class
B
Q
B_{Q}
, where
Q
Q
is the homogeneous dimension of the Heisenberg group
H
n
\mathbb {H}^n
. In this paper, we obtain pointwise bounds for the spatial derivatives of the heat and Poisson kernels related to
L
\mathcal {L}
. As an application, we characterize the space
B
M
O
L
(
H
n
)
BMO_{\mathcal {L}}(\mathbb {H}^n)
, associated to the Schrödinger operator
L
\mathcal {L}
, in terms of two Carleson type measures involving the spatial derivatives of the heat kernel of the semigroup
{
e
−
s
L
}
s
>
0
\{e^{-s\mathcal {L}}\}_{s>0}
and the Poisson kernel of the semigroup
{
e
−
s
L
}
s
>
0
\{e^{-s\sqrt {\mathcal {L}}}\}_{s>0}
, respectively. At last, we pose a conjecture about the converse characterization of
B
M
O
L
(
H
n
)
BMO_{\mathcal {L}}(\mathbb {H}^n)
.