Certain problems connected with the choice of trial variational functions for the quantum many-body problem are discussed. It is convenient to make use of the reduced-density-matrix formulation developed by L\"owdin and Mayer, since a system whose Hamiltonian contains only 1- and 2-body interactions is characterized by the two simplest reduced density matrices ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. It is known that symmetry, Hermiticity, and normalizability conditions are not sufficient to ensure a physically realizable choice of the functions ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. However an additional nontrivial restriction is imposed by the fact that such functions must be possible variational extrema. The implications of this condition are investigated in some detail and it turns out that all the nonphysical choices of ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$ which have been exhibited by various authors are eliminated. The condition is essentially nonquantum inasmuch as an analogous argument should be possible for a classical system. Moreover one of the consequences of this condition is that the system (classical or quantum) may exhibit macroscopic behavior, i.e., order-disorder transitions. The analysis leads to apparently reasonable choices of trial forms for ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$, although it is still not known whether any further restriction must be imposed to complete the sufficiency argument.
In most nuclear many-body methods, observables are calculated using many-body wave functions explicitly. The variational two-particle reduced density matrix method is one of the few exceptions to the rule. Ground-state energies of both closed-shell and open-shell nuclear systems can indeed be evaluated by minimizing a constrained linear functional of the two-particle reduced density matrix. However, it has virtually never been used in nuclear theory, because nuclear ground states were found to be well overbound, contrary to those of atoms and molecules. Consequently, we introduced new constraints in the nuclear variational two-particle reduced density matrix method, developed recently for atomic and molecular systems. Our calculations then show that this approach can provide a proper description of nuclear systems where only valence neutrons are included. For the nuclear systems where both neutrons and protons are active, however, the energies obtained with the variational two-particle reduced density matrix method are still overbound. The possible reasons for the noticed discrepancies and solutions to this problem will be discussed.
Variationally calculating the ground state of a many-electron quantum system using only the two-electron reduced-density-matrix (2-RDM) requires
N-representability conditions that constrain the 2-RDM to correspond to an
N-electron wave function. A systematic hierarchy of
N-representability conditions, known as
p-positivity conditions, has been developed [D. A. Mazziotti and R. M. Erdahl, Phys. Rev. A 63, 042113 (2001)], and many-electron atoms and molecules in nonminimal basis sets have been solved with useful accuracy by a variational 2-RDM method with 2-positivity conditions [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. This paper considers two forms of partial 3-positivity conditions, the lifting conditions and the
T1∕T2 conditions, to further enhance the accuracy of the 2-RDM methods without the computational cost of full 3-positivity conditions. Variational 2-RDM methods with different
N-representability constraints including 2-positivity conditions, the two types of partial 3-positivity conditions, as well as the complete 3-positivity conditions are applied to compute the ground state of the Lipkin spin model. The energies and 2-RDMs are compared to the results from full and truncated configuration interaction, many-body perturbation theory, and couple cluster theory with single and double excitations. Implications of using partial 3-positivity for variational 2-RDM calculations of many-electron atoms and molecules will be discussed.
Because the molecular Hamiltonian contains only one-body and two-body operators, the two-electron reduced density matrix contains all the information needed to evaluate the energy, as well as most of the other properties of interest to chemists and molecular physicists. A straightforward minimization of the energy is confounded by the N-representability problem, which can only be addressed approximately. The resulting theory has both advantages and disadvantages compared to more-traditional wavefunction-based approaches. The biggest advantage is that it performs well even when the molecule of interest has strong multireference character and the “Hartree-Fock plus correction” wavefunction paradigms fail. Also, as a lower-bound method, it provides a complementary tool to variational wavefunction approaches. The biggest disadvantages are the computational cost (which may yet be surmounted) and problems with dissociation and degeneracy that seem to afflict all approaches based on a “reduced” description of the system.
The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix
〈m1m2∣A∣p1p2〉. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix
A. Hole-state energies are given self-consistently in terms of the matrix
A, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\ensuremath{\psi}|H|\ensuremath{\psi}〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.
With the density-matrix variational method, the ground-state energy of a many-electron system is calculated by minimizing the energy expectation value subject to the selected representability conditions for the second-order reduced-density matrix (2-RDM). There is a lack of size extensivity under the
P,
Q, and
G conditions of this method by applying to
M noninteracting atoms and molecules up to
M=32;
BeM,
(CH4)M, and
(N2)M. The energy per molecule
(E/M) monotonically decreases as
M increases, showing the lack of size extensivity in this method. The inextensive contributions to energies are $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ and $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\text{ }\text{a}\text{.u}\text{.}$ for
(CH4)M and
(N2)M using the STO-6G basis set, respectively. In the examples studied,
E/M approaches a finite value as $M\ensuremath{\rightarrow}\ensuremath{\infty}$. The intermolecular elements of 2-RDMs do not satisfy the cumulant expansion.