  The formalism of real-time (RT) methods has been well-established during
recent years, while no inclusion beyond the double excitation has been
discussed. In this article, we introduce an implementation of real-time coupled
cluster singles, doubles and approximate triples (CC3) method to explore the
potential of a high excitation level. The CC3 method is well-known for its
advantages in calculating dynamic properties and combining with the response
theory. It is a well-qualified candidate for handling the interaction between
the system and the applied field, and therefore suitable for a RT
implementation. The derivation and implementation are first demonstrated
following applications on calculating frequency-dependent properties. Terms
with triples are calculated and added upon the existing CCSD equations, giving
the method a formally $N^{7}$ scaling. The Graphics Processing Unit (GPU)
accelerated implementation is utilized to reduce the computational cost. It is
been verified that the GPU implementation can speed up the calculation by up to
a factor of 17 for water cluster test cases. Additionally, the single-precision
arithmetic is used and compared to the conventional double-precision
arithmetic. No significant difference is found in the polarizabilities and $G'$
tensor results, but a higher percentage error for the first
hyperpolarizabilities is observed. Compared to the linear response (LR) CC3
results, the percentage errors of RT-CC3 polarizabilities and RT-CC3 first
hyperpolarizabilities are under 0.1% and 1%, respectively for the
$H_2O$/cc-pVDZ test case. Furthermore, a discussion on the calculation of
polarizabilities is included, which compares RT-CC3 with RT-CCSD and
time-dependent nonorthogonal orbital-optimized coupled cluster doubles
(TDNOCCD), in order to examine the performance of RT-CC3 and the
orbital-optimization effect using a group of ten-electron systems.
