Is a formula with no bound variables an atomic formula even if it contains a Quantifier?

Question

For example my understanding is:

∃(P(x)) contains no bound variables even though we have a quantifier, it also always evaluates to either true or false.

Does this mean that it is an atomic formula?

Thanks

Answer 1

No. An atomic formula has the form $R(t_1,\dots,t_n)$ or $t_1 = t_2$ (where $R$ is an $n$-ary relation symbol and $t_1,\dots,t_n$ are terms). Nothing else is an atomic formula.

Now a complex formula $\varphi$ may be logically equivalent to an atomic formula. But that does not make $\varphi$ itself atomic.