Restating the question, what probability mass function with finite support $ P(X=k) $ most excellently approximates the normal distribution $ X \sim \mathcal{N}(\mu, {\sigma}^2) $;
mimicking its symmetrical shape, statistical measures, and general behavior?
The Normal Mass Function
$$ P(X=k) = \frac{g(k)}{C} $$
$$ g(k) = e^{\frac{-(k-\mu)^2}{2{\sigma}^2}} $$
$$ C = \sum_{i=1}^{n} g(k_i) \approx \sqrt{2 \pi {\sigma}^2 } $$
The Normal Mass Function (NMF) is a PMF that is exactly proportional to the Normal PDF, and if given an adequate support, is virtually the same function.
It performs so well, and shares such a tight connection with its continuous brother, that I had to name it authoritatively.
But some limitations must be respected in order to achieve admirable performance, as follows.
The support of the Normal Mass Function must consist of equally spaced points.
$$ \delta = k_{i+1} - k_i $$
I have three free parameters in creating the support $ \{k_1, \ldots, k_n\} $.
The parameters are the spacing, the start, and the number of points, namely $ \delta $, $ k_1 $ and $ n $.
These parameters may be chosen for one to one comparison with other discrete distributions.
However, for best agreement to the normal distribution, the support should not only have equally spaced points, but also be defined symmetrically, with $ \text{median}(k_i) = \mu $, or equivalently:
$$ k_1 = \mu - (n-1) \delta / 2 $$
$$ k_n = \mu + (n-1) \delta / 2 $$
This refined support in context of the same function defines the Symmetric Normal Mass Function.
I have two free parameters in creating the symmetric support: the spacing, and the number of points, namely $ \delta $ and $ n $.
If $ (n-1) \delta / \sigma \ge 12 $ and $ \sigma / \delta \ge 1 $ then I have an acceptable support: the Symmetric Normal Mass Function agrees closely with the Normal Distribution.
An acceptable support spans at least 6 standard deviations away from the mean (in both directions) and has at least one point per standard deviation.
By these rules, the minimum acceptable support has 13 points spaced at 1 point per standard deviation.
Given the minimum acceptable support, the Symmetric Normal Mass Function compared to the Normal Distribution, has
- identical mean,
- approximate variance (low, within 2.2 parts per 10 million),
- identical skewness, and
- approximate kurtosis (high, within 2.4 parts per million).
$$ \mu, \sigma^2, \frac{\mu_3}{\sigma^3}, \frac{\mu_4}{\sigma^4} \approx \mu, \sigma^2, 0, 3 $$
Except for this minor decimal disagreement, the Symmetric Normal Mass Function meets all 4 requirements of the original question marvelously.
The approximation improves as the support's span and density increases relative to the standard deviation.
I expound further about why the Normal Mass Function is the best choice as the discrete variant of the Normal Distribution in a linked article.