  Hohenberg-Kohn (HK) theorem is a cornerstone of modern electronic structure
calculations. For interacting electrons, given that the internal part of the
Hamiltonian ($\hat H_{int}$), containing the kinetic energy and Couloumb
interaction of electrons, has a fixed form, the theorem states that when the
electrons are subject to an external electrostatic field, the ground-state
density can inversely determine the field, and thus the full Hamiltonian
completely. For a general quantum system, a HK-type Hamiltonian in the form of
$\hat H_{hk}\{g_i\}=\hat H_{int}+\sum_i g_i \hat O_i$ can always be defined, by
grouping those terms with fixed or preknown coefficients into $\hat H_{int}$,
and factorizing the remaining as superposition of a set of Hermitian operators
$\{\hat O_i\}$. We ask whether the HK theorem can be extended, so that the
ground-state expectation values of $\{\hat O_i\}$ as the generalized density
can in principle be used as the fundamental variables determining all the
properties of the system. We show that the question can be addressed by
introducing the concept of generalized density correlation matrix (GDCM)
defined with respect to the $\{\hat O_i\}$ operators. The invertibility of the
GDCM represents a mathematically rigorous and practically useful criterion for
the extension of HK theorem to be valid. We apply this criterion to several
representative systems, including the quantum Ising dimer, the frustration-free
systems, N-level quantum systems with fixed inter-level transition amplitude
and tunable level energies, and a fermionic Hubbard chain with inhomogeneous
on-site interactions. We suggest that for a finite-size system, finding an
invertible GDCM under one single $\{g_i\}$ configuration is typically
sufficient to establish the generic extensibility of the HK theorem in the
entire parameter space.
