Variation principles for obtaining lower bounds by means of density matrices are applied to a number of many-body problems. This is done by means of restrictions derived from the study of exacly solvable models. Two classes of problems are considered: (a) obtaining a lower bound to the ground-state energy of a system of particles, and (b) obtaining a lower bound to the Helmholtz free energy. For the first case an exact lower bound is obtained for the ground-state energy of the particle-conserving Bogoliubov Hamiltonian. Of particular significance in the nonzero temperature case is a rigorous lower bound to the free energy of the Ising model with small external field at low temperatures.
The programme of reducing N -particle problems to one-particle problems is extended to include the Pauli principle. To calculate the ground-state energy of a system of N fermions interacting by pair forces, a rigorous lower-bound shell model is derived. This shell model follows a building-up principle, and tends to improve with increasing number of particles as well as with increasing strength of interaction. For twenty particles antisymmetric in ordinary space interacting by square-well interaction of strength V 0 a 2 = 200 hslash 2 /2 m the shell model differs by less than 8% from the calculated upper bound, and hence a fortiori from the exact energy. In order to test the quality of the approximation for various interactions, it was necessary to calculate upper bounds.
Abstract The problem of upper and lower ground state energy bounds for many‐fermion systems is considered from the viewpoint of reduced density matrices. Model density matrices are used for upper bounds to, first uncoupled, then coupled fermions. Model Hamiltonians are developed for lower bounds in corresponding fashion. Both mathematical and physical models are constructed for setting up universally valid inequalities on density matrices. These are joined by both inequalities and equalities in which the explicit form of the system at hand is used. A few illustrative examples are presented.
Here, we revisit the problem of finding the ground-state energy of an N-fermion fluid under an external field, with molecular structure as the ultimate target. Density-functional methods only have to deal with electron density, but require an empirical functional; reduced density-matrix methods involve a matrix on pair space and do give exact bounds, but require very complex linear programming to achieve their results. The polydensity representation that we introduce has the advantage of dealing only with densities, requires no empirical information, and also gives exact bounds; the major problem is that of accumulating and utilizing conditions on the densities that iteratively improve their realizability in the class of N-fermion systems. We indicate several directions along these lines and make some primitive applications.
If $\ensuremath{\psi}$ is a determinantal eigenfunction for the
N-fermion Hamiltonian,
H, with one- and two-body terms, then ${e}_{0}<~〈\ensuremath{\psi},H\ensuremath{\psi}〉=E(K)$, where
e0 is the ground-state energy,
K is the one-body reduced density matrix of $\ensuremath{\psi}$, and
E(K) is the well-known expression in terms of direct and exchange energies. If an arbitrary one-body
K is given, which does not come from a determinantal $\ensuremath{\psi}$, then
E> e0 does not necessarily hold. This Letter proves, however, that if the two-body part of
H is positive, then in fact
e0< eHF< E(K), where
eHF is the Hartree-Fock ground-state energy.