A metric $d(\cdot,\cdot)$ of a space $S$ is said to be of negative type, if for $\forall n \geq 2, z_{1}, \ldots, z_{n} \in S$, and $\alpha_{1}, \ldots, \alpha_{n} \in \mathbb{R}$ with $\sum\limits_{i=1}^{n} \alpha_{i}=0$ $$ \sum_{i=1}^{n} \sum_{j=1}^{n} \alpha_{i} \alpha_{j} d(z_{i}, z_{j}) \leq 0. $$
And we know that the spherical geodesic distance defined as $d(x, y)=\arccos(x^{T}y)$ for $x, y\in \mathbb{S}^{p-1}$ where $\mathbb{S}^{p-1}$ is the unit spherical surface of $\mathbb{R}^{p}$, or in general defined as $d(x, y)=\arccos(\frac{x^{T}y}{\|x\|\|y\|})$ for $x, y\in \mathbb{R}^{p}$, is a metric of negative type. (See for example Section 6 of https://arxiv.org/pdf/0710.2063.pdf)
Now I'm curious about whether there exists some $a>0$ s.t. the metric defined as $d(x, y)=\arccos(\frac{a+x^{T}y}{(a+x^{T}x)^{1/2}(a+y^{T}y)^{1/2}})$ is of negative type. Here I'm attemping to construct a link between a (semi)metric space of negative type and a reproducing kernel hilbert space (RKHS). (Similar to https://www.jstor.org/stable/23566550?seq=1#metadata_info_tab_contents)
Let us refer to your base space as $\mathcal{X}$ (either $\mathbb{S}^{p-1}$ or $\mathbb{R}^p$ as in your question), and let $x_0\in \mathcal{X}$ be an arbitrary point. Define the function $k_{d, x_0}$ by $$k_{d, x_0}(x,y) = d(x,x_0)+d(y,x_0)-d(x,y). $$ From this paper, we know that $d$ is a metric of negative type if and only if $k_{d,x_0}$ is positive definite. This is simpler to check, as you can simply evaluate the eigenvalues of the Gram matrix of $k_{d, x_0}$.