Linked Diagram Expansion Research Articles

The linked-diagram expansion of the ground state of a many-electron system: A time-independent derivation

A time-independent derivation of the linked-diagram expansion of the ground-state energy of a many-electron system is given for the case of a nondegenerate unperturbed system. The starting point is the Rayleigh-Schroedinger perturbation theory, and the derivation follows in many respects suggestions by Cizek and Paldus, including the use of the Frantz-Mills factorization theorem for the cancellation of unlinked and renormalization diagrams.

Functional integrals for the Anderson lattice

The infinite U limit of the Anderson lattice model can be formulated as a non-Hermitean Hamiltonian, where the usual hybridization is contained in the unperturbed part. The grand partition function is written as a functional integral, from which one obtains a perturbation expansion which automatically contains a linked diagram expansion. Further typical approximations to the functional integral are discussed.

AN INTRODUCTORY GUIDE TO GREEN’S FUNCTION METHODS IN NUCLEAR MANY-BODY PROBLEMS

We present an elementary and fairly detailed review of several Green’s function methods for treating nuclear and other many-body systems. We first treat the single-particle Green’s function, by way of which some details concerning linked diagram expansion, rules for evaluating Green’s function diagrams and solution of the Dyson’s integral equation for Green’s function are exhibited. The particle-particle hole-hole (pphh) Green’s function is then considered, and a specific time-blocking technique is discussed. This technique enables us to have a one-frequency Dyson’s equation for the pphh and similarly for other Green’s functions, thus considerably facilitating their calculation. A third type of Green’s function considered is the particle-hole Green’s function. RPA and high order RPA are treated, along with examples for setting up particle-hole RPA equations. A general method for deriving a model-space Dyson’s equation for Green’s functions is discussed. We also discuss a method for determining the normalization of Green’s function transition amplitudes based on its vertex function. Some applications of Green’s function methods to nuclear structure and recent deep inelastic lepton-nucleus scattering are addressed.

Diagrammatic many-body perturbation expansion for atoms and molecules: IX. On the use of dynamic load balancing on a multi-processor computer

A formulation of the diagrammatic many-body perturbation theory suitable for effective implementation on multi-processor computers is described. This concurrent computation many-body perturbation theory ( ccMBPT) exploits the additive separability of the correlation energy components resulting from the linked diagram expansion to devise algorithms suitable for a parallel processing environment. A dynamic load balancing technique is employed to exploit the parallel processing capabilities of computers such as the CRAY Y-MP. The performance of the technique is demonstrated in calculations of the most computationally demanding of the fourth order energy components, those involving triply excited intermediate states. Execution rates in excess of 2.28×10 9 floating point operations per second are observed on an eight-processor machine.

Linked diagram expansions for the nuclear transition matrix and the normalization of model-space wave functions

A unified and relatively simple linked-diagram expansion for performing microscopic calculations of the nuclear transition matrix T fi is derived. We first review a folded-diagram effective interaction theory, based on which, the model-space effective hamiltonian can be calculated from realistic nucleon-nucleon interactions. An important ingredient of this theory is the wave-function decomposition theorem which provides a convenient connection between the true and model-space wave functions. By directly making use of this theorem, a folded-diagram expansion of T fi is readily obtained. Using a partial summation method, the folded diagrams of T fi are then eliminated, leading to a considerably simpler expression for T fi. Our formalism requires that T fi must be calculated in strict consistence with the derivation of the model-space effective hamiltonian H eff. The model-space wave function normalization factor contained in our T fi, may play an important role in “quenching” the calculated nuclear transition matrix. A simple method for evaluating this normalization factor is derived, namely it can be obtained readily from the energy derivative of the respective self-consistent eigenvalue of H eff.

Collective Hamiltonian in linked diagram expansions.

A method for obtaining collective Hamiltonians in linked diagram expansions is given. Effects of the Pauli principle have been properly taken into account. An illustration with a two-level pairing force model is given. The derivation of the interacting boson model and truncated quadrupole phonon model Hamiltonians as well as a comparison with the random-phase approximation are discussed.

Summation of ring diagrams of nuclear thermodynamic potential

Using an imaginary-time linked diagram expansion of the thermodynamic potential Ω, we derive methods by which the particle-particle (hole-hole) and the particle-hole ring diagrams of Ω can each be summed up to all orders in a rather convenient way. A model space P and a corresponding finite temperature reaction matrix K are introduced in order to take care of the strong short-range correlations between nucleons. Methods of the finite temperature Green functions are employed. The contribution from the particle-particle (hole-hole) ring diagrams to Ω is given in terms of simple integrals of the form ∝ 0 1d λ tr p { M( λ) K} where λ is a strength parameter, and M a thermal transition matrix which is calculated from a finite-temperature RPA-type secular equation. Solutions of this equation are derived, and their possible connections with phase transitions are discussed. The particle-hole ring diagrams are treated in a similar way. Applications of the present approach and prospects for formulating a Brueckner-type theory of nuclear matter at finite temperature are discussed.

Summation of particle-particle and particle-hole ring diagrams in binding energy calculations and Lipkin models

In this paper we study methods for calculating the particle-particle (pp) and particle-hole (ph) ring diagrams in the linked diagram expansion of the ground-state energy shift Δ E 0 for many-body systems. Analytic expressions for summing up the pp ring diagrams to all orders are derived, given as an integral involving G pp( ω, λ) ν, where G pp is the pp Green function, υ the interaction hamiltonian (taken to be two-body) and λ a strength parameter to be integrated over from 0 to 1. Similar expressions for summing up the ph ring diagrams are also derived, their integral form involving G ph (ω, λ) ν, where G ph is the ph Green function and ν the particle-hole interaction. T energy variable ω is to be integrated over from −∞ to +∞. Two methods for carrying out the ω-integration are suggested. One is the multipole expansion method, and the other is the transition amplitude method where the Green functions G ppand G ph are calculated using RPA equations from which the physical transition amplitudes can be obtained. Our methods for summing up the pp and ph ring diagrams to all orders are applied to several Lipkin-model many-body problems. The effect of using Hartree-Fock self-consistent single-particle wave functions on the contribution of high-order ring diagrams to Δ E 0 is studied and found to be very important. Possible applications of our methods to actual nuclear many-body problems are discussed.

Molecular hyperpolarizabilities. I. Theoretical calculations including correlation

Static polarizabilities and hyperpolarizabilities for molecules are investigated at the correlated level. The finite-field, coupled Hartree-Fock theory is used as a zeroth-order approximation, with correlation included by using the linked-diagram expansion and many-body perturbation theory, that includes single, double, and quadruple excitation diagrams. The theory is illustrated by studying the hydrogen fluoride molecule. It is demonstrated that the correlation effect for the hyperpolarizabilities $\stackrel{\ensuremath{\leftrightarrow}}{\ensuremath{\beta}}$ and $\stackrel{\ensuremath{\leftrightarrow}}{\ensuremath{\gamma}}$ can be quite large. The average polarizability and dipole moment of HF are in excellent agreement with experiment. The relative importance of the various types of diagrams contributing to electric field properties are discussed. The dependence of the computed hyperpolarizability on basis sets is also investigated.

Convergence properties of the effective Hamiltonian for the excitation energies in closed-shell systems

Methods for calculations of the effective Hamiltonian for particle–hole excitations of closed shell molecular systems, based on the linked-diagram expansions of the Rayleigh–Schrödinger (the folded diagram expansion) and Brillouin–Wigner (the Bloch–Horowitz diagram expansion) perturbation schemes for a quasidegenerate multiconfigurational model space, are studied. The convergence properties of these perturbative methods up to third order, in particular paying attention to the presence of a crossing of levels, are numerically studied for excitation energies of a π-electron system of trans-butandiene. The Padé approximant consistent with the continued fraction approach is applied to the folded diagram expansion treatment. The results of the low-order Padé approximants are similar to those of the Bloch–Horowitz diagram expansions. The singularity of the [2/1] Padé approximant is not necessarily related to the crossing of levels. A strong energy dependence of the effective interaction is shown to be attributed to the crossing of levels.

Many-body calculations of the hyperfine interaction of some excited states of alkali atoms, using approximate Brueckner or natural orbitals

The theory of maximum-overlap or Brueckner orbitals and of natural Orbitals is reviewed for closed-shell systems. A technique is described to calculate approximate Brueckner or natural orbitals for systems with a single valence electron, and this is applied together with the linked-diagram expansion of effective operators to evaluate the hyperfine interaction of some excited states of alkali atoms. It is found that it is in this way possible to reproduce reasonably well also the highly perturbed interactions of the excitedd states, which is not possible in a low-order expansion based on HF orbitals. Numerical results are given for the 5d state in K and the 4d state in Rb, where good experimental information is available.

A folded-diagram perturbation expansion for effective operators

Linked-diagram expansions for the model-space effective operators are derived for both open-shell nuclei and closed-shell nuclei. These expansions are derived using time-dependent perturbation theory and are given as a diagrammatic series which contains folded diagrams. Our results are in general agreement with the results obtained earlier by Brandow, but the general structure of the present expansions is perhaps simpler and more convenient for calculations. The projections of the true nuclear eigenfunctions onto the model space are required for the derivation of effective operators. These projections are obtained from the solution of the model-space eigenvalue problem defined by the energy-independent effective Hamiltonian. Methods for calculating the above diagrammatic series for effective operators are discussed.

Collective effects in atomic photoabsorption spectra. III. Collective resonance in the 4d10shell in Xe

For pt. II see abstr. A595 of 1972. The collective properties of the 4d10 shell in Xe are studied, making use of a previously developed method for analysing the linked diagram expansion for the polarizability. The infinite series of bubble and ladder diagrams (RPA with exchange) are considered for the (4d;nf)1P channel and sum the on the energy shell part exactly. The off the energy shell part is given by a reaction matrix integral equation. An approximate, closed solution of Fredholm type is found to have two poles for energies between the continuum threshold and 2 Ryd above, in which region the exact reaction matrix equation cannot be solved by iteration. This result is interpreted in terms of a broad collective resonance in the (4d;nf)1P channel, very clearly seen in the photoabsorption cross section. The theory is extended to include some effects of intershell interaction, representing core relaxation and improved agreement is found with experiments for low energies above the 4d9 threshold.

Collective effects in atomic photoabsorption spectra. II. 5p6shell in Xe

For pt. I see abst. A63958 of 1971. The author analyses the linked diagram expansion for the polarization propagator, describing the atomic response to an external perturbation. In particular an application is made to the 5p6 shell in Xe and excitations into the 5p-nd 1P channel discussed. These are described by an effective RPA expansion in which ladder diagrams, representing particle-hole interaction, are included. The expansion can be separated into two parts, one which describes absorption and one which describes virtual fluctuations or screening. The absorption part stems from the expansion 'on the energy shell' and forms a divergent series for the continuum threshold. The screening part forms a convergent series and can be expressed as an integral equation for a reaction matrix, describing 'off the energy shell' fluctuations. The author also discusses how to apply the theory to the case of many channels and presents a method for treating ground state correlations. The results for the dynamical response of the shell in terms of intuitively defined dielectric functions are discussed.

A folded-diagram expansion of the model-space effective hamiltonian

A linked-diagram expansion for the model-space effective Hamiltonian is derived. This expansion which contains the folded diagrams is independent of the energy eigenvalue. Our result is generally in agreement with the expansion obtained earlier by Brandow, Morita and Oberlechner et al., but our method of derivation is different from theirs and is probably simpler. The properties of the model-space wave functions and their connection with the true eigen-functions are investigated. In particular, a linked-diagram expansion is obtained for the evaluation of the expectation value of a general operator with respect to the true eigenfunction. This expansion also contains the folded diagrams and is independent of the energy eigenvalues.

Green's function perturbation technique for the Kondo system

With the aid of a linked diagram expansion for the one electron scattering matrix of the Kondo system the singular log ¦ω−contributions are calculated up to forth order perturbation theory. Infinite order perturbation theory is carried out for a special class of generalized scattering matrix diagrams. The basic equation of Suhl's dispersion theory is derived under the restriction on one-particle intermediate states. From perturbation theory it is shown that the validity of Suhl's equation is restricted in the sense that it reproduces only the highest order logarithmic singularities exactly.

Wick theorem and linked-diagram expansion for the Heisenberg model ferromagnet

A finite-temperature perturbation theory for the Heisenberg model of ferromagnetism is presented. A modified Wick theorem proved for S = ½ and finite temperature allows the terms in the expansion for the temperature-dependent Green function to be expressed by linked diagrams of the Feynman type. These differ from the normal boson and fermion diagrams in the appearance of time(temperature)-dependent broken lines between propagators, a modification caused by the spin statistics.

Wick's Theorem for Spin-½ Operators, with an Application to Spin Waves in Antiferromagnets

An analog of Wick's theorem is developed for spin-\textonehalf{} operators, and a linked diagram expansion for spin Green's functions is derived. As an application we derive the familiar Anderson approximation for spin waves in antiferromagnets, and we then obtain the leading dynamical and kinematical corrections to that approximation. The Oguchi form of correction, usually obtained by a formal expansion in $\frac{1}{S}$ extrapolated to $S=\frac{1}{2}$, is found here as the leading term of an expansion in powers of $\frac{1}{z}$, where $z$ is the number of nearest neighbors. However, the Oguchi result is here found to be valid only for spin waves with wavelengths greater than two or three interspin distances.

Wick's Theorem for Spin Operators and Its Relation to the Coupled-Fermion Representation

It is shown that Wick's theorem and the linked diagram expansion theorem for S = ½ operators (applied elsewhere by the authors to discuss antiferromagnetic spin waves) can be derived by a direct application of conventional theorems to the coupled-fermion representation of spin operators.

Statistical Theory of the Dielectric Constant of an Imperfect Gas

By means of the linked-diagram expansion of the grand partition function of a molecular gas in an electrostatic field, an expression for the polarization P(R) of the gas is obtained. Spatial variation of the external electric field E0(R) requires an explicit treatment of long-range cooperative interactions between ``clusters'' of molecules. For fields that vary appreciably over microscopic dimensions, an integral relation is found relating the polarization P(R) to the electric field E(R). For fields varying negligibly over microscopic regions, an expression for the dielectric constant K of the gas is obtained: (K − 1)/(K + 2) = (4/3) π Σ∞m=1 nmαm(θ). This generalization of the Clausius-Mossotti formula involves the density n and the temperature-dependent polarizabilities of m-molecule linked clusters: α1(θ) is the effective polarizability of a single (possibly polar) molecule; α2(θ) is the diagonal element of the scalar tensor α2(θ) = ∫ d3R exp [−βφ(θ, R)][½α2(θ, R) − α1(θ)]. In this expression α2(θ, R) is the effective polarizability tensor of two molecules with fixed positions of their centers of mass; φ is the free energy of the two-molecule system, relative to infinite separation.