Subspace of Hilbert space as manifold for variational state

Fundamentally, quantum states are rays in Hilbert space $mathcal{H}$, i.e. of the form $r = mathbb{C} Psi$ for $Psi in mathcal{H}$. Therefore, a time-dependent state is a path of rays $r(t)$. However, assuming $r(0)$ to not be the zero ray (i.e. there is a non-zero vector in $r(0)$), we can pick a representative $Psi_0 in r(0)$ of unit length. As time evolution is by unitaries, we get a function $tmapsto Psi_t in r(t)$, where all the $Psi_t$ have unit length. Therefore, we can consider $tmapsto Psi_t$ a map into the unit sphere $Smathcal{H}$ inside $mathcal{H}$. Usually one assumes time-evolution to be strongly continuous, and therefore $tmapsto Psi_t$ is continuous. It might happen that this curve is in fact smooth, for example if $mathcal{H}$ is finite dimensional.

Suppose we have a map $Psi : mathbb{R}^n mapsto Smathcal{H}$, i.e. we have a map that assigns to a given value of the parameters $x in mathbb{R}^n$ a wavefunction $Psi(x) in Smathcal{H}$. We can now simply define a subset $Smathcal{H}supsetmathcal{M} := Psi(mathbb{R}^n)$. Note that $mathcal{M}$ is not, in general, a manifold. This has to be checked for the specific $xmapsto Psi(x)$ given. However, if this has been checked, we automatically get a basis of the tangent space $T_pmathcal{M}$. There are two equivalent definitions, one in terms of germs of curves, the other in terms of derivations. For the definition in terms of curves, consider curves $(-epsilon,epsilon) ni t mapsto Phi_t in mathcal{M}$ such that $Phi_0 = p$. Such maps are called smooth if $varphi circ Phi_t$ is smooth for every chart $varphi$. Two curves $Phi^1_t, Phi^2_t$ are equivalent if the derivatives of $varphi circ Phi^1_t$ and $varphi circ Phi^2_t$ coincide at $t=0$ for every chart. The equivalence class is denoted as $frac{d Phi_t}{dt}|_{t=0}$. To see that this is a vector space, i.e. that the sum of two vectors and the multiple of a vector are again vectors, one has to introduce a chart. In $mathbb{R}^n$, one can to any given inital vector, $v$, find a curve through the origin having $v$ as its tangent vector at the origin. Then this curve can be lifted to the manifold. Furthermore, if we have a tangent vector in this sense (an equivalence class of curves), we can also consider them as derivations on the smooth functions; indeed, let $fin C^infty(mathcal{M})$. Then for a given curve $Phi_t$ define a derivation $D(f):= frac{d}{dt}|_{t=0} (fcircPhi_t)$. The definition in terms of curves is useful for our purposes, because by defintion of mathcal{M}, all curves $(Phi_t)_t subset mathcal{M}$ are of the form $Phi_t = Psi(x(t))$, where $t mapsto x(t)$ is a curve in $mathbb{R}^n$. Consequently, $$ frac{dPhi_t}{dt}|_{t=0} = frac{partial Psi(x)}{partial x_i}|_{x = x(0)} frac{d x^i(t)}{dt}|_{t=0} . $$

Crucially, for a general $Phi_t in Smathcal{H}$, we will not be able to find a parametrization $Phi_t = Psi(x(t))$, simply as the latter exists only in the subset $mathcal{M}$.