We test the coherent correlated pair method with the soluble SU(3) Hamiltonian of Li, Klein, and Dreizler. We compare the results for the ground state energy with the exact solution and with the Hartree-Fock results as a function of coupling parameters and number of particles. For all cases the coherent correlated pair method yields the better upper bound to the ground state energy.NUCLEAR STRUCTURE Correlated pairs; coherent states; upper bounds to ground state energies; applications to exactly soluble SU(3)
N-fermion model.
The well known and analytically solvable Lipkin–Meshkov–Glick model was devised to test several many-body approximation approaches used in nuclear physics. Here, we use this model to illustrate a method that sets lower bounds for the ground state energy of a quantum Hamiltonian. The idea of the method consists of dividing the Hamiltonian into identical groups of terms that represent subdivisions in the particle population of the system, and then to make an appropriate use of the ground energy of one of these groups. The bounds clearly improve as the size of these groups increases.
We develop a method for treating the
N-fermion problem as a system of correlated pairs of particles. Coherent pair states are introduced and a variational as well as a secular method are discussed for providing upper bounds to the exact ground state and approximations to the low lying spectra. Specific applications to two soluble models are presented. For these models the coherent correlated pair method is superior to the Bardeen-Cooper-Schrieffer approximation and to the Hartree-Fock approximation and in fact yields the exact solutions for all particle numbers and sufficiently large coupling constants.NUCLEAR STRUCTURE Correlated pairs; coherent states; upper bound to ground state energy; applications to soluble
N-fermion models.
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.
Two methods which give the lower bound for the ground-state energy of a system with N interacting particles are described. The methods exploit the solution of the Schrodinger equation for the comparison system with K<N particles. An illustrative example of fermions interacting with the harmonic force is discussed. More realistic interactions used in nuclear physics are treated numerically with K=2, 3 and 4, showing that the lower bound tends to the exact value of the energy when K increases. The lower bound is strongly dependent on the choice of the interaction.
We establish a new iterative method for computing rigorous lower bounds for the ground-state energies of all atoms and ions. The accuracy of the resulting bounds depends on two ingredients: the quality of the initial energy estimate and the quality of the used upper bound for the maximum number of bound electrons,
Nc(Z), as a function of the nuclear charge
Z>0. In particular, this new method allows us to establish the explicit lower bound
E(N,Z)≥−0.8710N1/3Z2 for the ground-state energy
E(N,Z) of the atom/ion with
N∈N electrons and nuclear charge
Z∈(0,N], thereby considerably improving the best known constant
A=(3/2)1/3≈1.1447 in the estimate
E(N,Z)≥−AN1/3Z2 for
0<Z≤N and coming much closer to the best possible constant given by the Thomas-Fermi asymptotics
limN→∞∣E(N,N)∣N−7/3=−ETF(1)≈0.7687 derived in Adv. Math., 23(1) by Lieb and Simon. To demonstrate the potential and the limits of our method, we also compute explicit lower bounds for the ground-state energies
E(N,N) of all atoms with
2≤N≤86 and we compare them with the bounds obtained in J. Chem. Phys., 62(8) by Hertel, Lieb, Thirring and the bounds obtained by Seco in his dissertation. Finally, we also provide a uniqueness result for the expectation values of the kinetic, potential and interaction energies on the ground-state space.