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The Journal of Chemical Physics

On Moment Conserving Decoupling Techniques for Electron Propagators

S. V. Babu, Mark A. Ratner

Published Oct 15, 1972• 2 citations
Publisher

Abstract

Moment-conserving decoupling procedures for the one-electron Green's function (electron propagator) are investigated. By analysis of one particular system (Hubbard model) it becomes clear that the moment-conserving techniques so far proposed for interacting systems are all special cases of [1, 0], [2, 1], or [3, 2] Padé approximants differing only in which terms they omit, how they define self-consistency, and which operator algebra they utilize. The number of formally conserved moments appears not to be a particularly useful criterion of the accuracy of a decoupling procedure, since a two-moment scheme (Hartree-Fock) may be numerically closer to the exact answer than some three-moment procedures. Use of the complete Padé approximant, as suggested by Lukman and Goscinski for the particle-hole propagator, appears to be the most consistent method.

DOI
10.1063/1.1678732

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Physical Review•Jul 15, 1967•20 citations
Moment-conserving decoupling procedures for the one-electron Green's function (electron propagator) are investigated. By analysis of one particular system (Hubbard model) it becomes clear that the moment-conserving techniques so far proposed for interacting systems are all special cases of [1, 0], [2, 1], or [3, 2] Padé approximants differing only in which terms they omit, how they define self-consistency, and which operator algebra they utilize. The number of formally conserved moments appears not to be a particularly useful criterion of the accuracy of a decoupling procedure, since a two-moment scheme (Hartree-Fock) may be numerically closer to the exact answer than some three-moment procedures. Use of the complete Padé approximant, as suggested by Lukman and Goscinski for the particle-hole propagator, appears to be the most consistent method.
The Hubbard I decoupling approximation [J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963)] is extended by introducing a k-dependent self-energy which describes a mobile correlation hole of opposite spin propagating along with the electron in its way around the lattice. The theory, despite its simplicity, (1) becomes exact in the strong-coupling limit; and (2) reproduces with great accuracy the ground-state energy, the double occupancy, the on-site correlation functions, and the effective hopping for all the values of the Hubbard Coulomb repulsion U. It also gives reasonable spectral functions leading to (3) a single-particle Green function showing the well-known spectral-weight transfer between the Hubbard bands as soon as the electron density deviates from half-filling, and (4) a momentum distribution as well as magnetic and charge structure factors in qualitative agreement with quantum Monte Carlo simulations, slave-boson calculations, and conserving approximations like fluctuation exchange and parquet. On the other hand, as in most decoupling schemes, liftime effects and the near-Fermi-level additional structure characteristic of strongly correlated electron systems are missing.
Ab initio electron propagator methods are developed to study electronic properties of molecular systems with strong electron-electron and electron-phonon interactions. For the calculation of electron Green's functions we apply a canonical small polaron transformation that intrinsically contains strong electron-phonon effects. In the transformed Hamiltonian, the energy levels for the noninteracting particles are shifted down by the relaxation (solvation) energies. The Coulomb integrals are also renormalized by the electron-phonon interaction. For certain values of the electron-phonon coupling constants, the renormalized Coulomb integrals can be negative which implies the attraction between two electrons. Within the small polaron transformation we develop a diagrammatic technique for the calculation of electron Green's function in which the electron-phonon interaction is already included into the multiple phonon correlation functions. Since the decoupling of the phonon correlation functions is impossible, and therefore, a Wick's theorem for such correlation functions is invalid, there is no Dyson equation for the electron Green's function. To find the electron Green's function, we use different approximations. One of them is a link-cluster approximation that includes diagonal transitions for the renormalized zeroth Green's function. In the linked-cluster approach the Dyson equation is derived in the most general case, where the self-energy operator is an arbitrary functional (not only in the Hartree-Fock approximation). It is shown that even a Hartree-Fock electron (hole) is not a particle any longer. It is a quasiparticle with a finite lifetime that depends on energy of particle and hole states in different ways. As a consequence of this, a standard description of a Hartree-Fock approximation in terms of wave functions becomes inappropriate in this problem. To challenge the linked-cluster approximation we develop a different approach: a sequential propagation approximation where scattering events occur only for sequential transitions. A self-consistent Hartree-Fock equation for a four-index Green's function matrix is derived. In conclusion, the proposed schemes can be considered for future method developments for quantum chemical calculations for large molecules with strong nonadiabatic effects, e-e correlated electron transfer reactions, and electron transport in molecular transport junctions.
A nonperturbative, controllable, and moment-conserving decoupling approach is proposed. Our assumption is that a higher-order Green function (GF) essentially contains two-part information. One part relates to the single-particle GF, which is involved in the corresponding moment; the other part is a higher-order quantity, which is independent of the single-particle GF. Thus the higher-order GF may be separated into two corresponding terms. We derive the chain of equations of motion (EOM) of the latter term from the usual EOM for GF's. To solve the equations a truncation approximation is made at the level desired. The single-band Hubbard model is studied using this approach. Moment-conserving solutions up to the order m=2m=2m=2 are obtained. The properties of a solution corresponding to the Hubbard-I approximation level are discussed in detail. The quasiparticle bands and the momentum distribution are in excellent agreement with the results from the quantum Monte Carlo simulation.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix
Introduction In this and the next two chapters we lay down the basis to go beyond the Hartree–Fock approximation. We develop a powerful approximation method with two main distinctive features. The first feature is that the approximations are conserving . A major difficulty in the theory of nonequilibrium processes consists in generating approximate Green's functions yielding a particle density n (x, t ) and a current density J (x, t ) which satisfy the continuity equation, or a total momentum P ( t ) and a Lorentz force F ( t ) which is the time-derivative of P ( t ), or a total energy E ( t ) and a total power which is the time-derivative of E ( t ), etc. All these fundamental relations express the conservation of some physical quantity. In equilibrium the current is zero and hence the density is conserved, i.e., it is the same at all times; the force is zero and hence the momentum is conserved; the power fed into the system is zero and hence the energy is conserved, etc. The second important feature is that every approximation has a clear physical interpretation. In Chapter 7 we showed that the two-particle Green's function in the Hartree and Hartree–Fock approximation can be represented diagrammatically as in Fig. 7.1 and Fig. 7.2 respectively. In these diagrams two particles propagate from (1′, 2′) to (1, 2) as if there was no interaction.
A simple cubic hydrogenic lattice with the Hubbard Hamiltonian is used as a model for Mott-type insulators. A new self-consistent approximation is developed by the decoupling of four-particle Green's functions, leading to a higher-order Hartree-like eigenvalue equation for electron-hole Green's functions. This equation is an improvement over the previously used $\ensuremath{\Omega}$ approximation. The solution of the eigenvalue equation leads to the conjecture that an exact solution (i.e., one without decoupling) would result in each atom having exactly one electron.