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Let $h$ be a three times partially differentiable function on ${ℝ}^n$, let $X=(X_1,...,X_n)$ be a collection of real-valued random variables and let $Z=(Z_1,...,Z_n)$ be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference $𝔼h\left(X\right)-𝔼h\left(Z\right)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n\to \infty$. In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic occupancy,...