  In recent years, the skyrmion lattice phase with a short lattice constant has
attracted attention due to its high skyrmion density, making it a promising
option for achieving high-density storage memory and for observing novel
phenomena like the quantized topological Hall effect. Unlike conventional
non-centrosymmetric systems where the Dzyaloshinsky-Moriya interaction plays a
crucial role, the short pitch skyrmion phase requires a quadratic magnetic
interaction $J(q)$ with a peak at finite-$Q$, and weak easy-axis magnetic
anisotropy is also critical. Thus, conducting first-principles evaluations is
essential for understanding the formation mechanism as well as for promoting
the discovery of new skyrmion materials. In this {\it Perspective}, we focus on
recent developments of the first-principles evaluations of these properties and
apply them to the prototype systems Gd$T_2X_2$ and Eu$T_2X_2$, where $T$
denotes a transition metal and $X$ represents Si or Ge. In particular, based on
the spin density functional theory with the Hubbard correction combined with
the Liechtenstein method in the Wannier tight-binding model formalism, we first
show that the Hubbard $U$ and Hund's coupling is essential to stabilize a
skyrmion lattice state by enhancing the easy-axis anisotropy. We then discuss
mechanisms of finite-$Q$ instability and show that competition among Gd-5$d$
orbitals determines whether ferromagnetism or a finite-$Q$ structure is favored
in Gd$T_2$Si$_2$ with $T=$ Fe and Ru. Our systematic calculations reveal that
GdRu$_2$$X_2$, GdOs$_2$$X_2$, and GdRe$_2X_2$ are promising, while GdAg$_2X_2$,
GdAu$_2X_2$, and EuAg$_2X_2$ are possible candidates as the skyrmion host
materials. Analysis based on a spin spiral calculation for the candidate
materials is also presented.
