A unitary group formulation of many-body theory: Diagram systematics and use of the spin shifts

This paper shows that the spin-shift formalism developed in B. T. Pickup and A. Mukhopadhyay [Int. J. Quantum Chem. 26 , 101 (1984)] supports a one-component diagrammatics which has a systematics akin to that in the spin-orbital many-body theory. The diagrams are neither Goldstone nor Yutsis type, and characterize the chain U(2R) ? U(R)?SU(2) on which the spin-shift formalism is based. Accordingly, while the lines in such diagrams are labeled by the orbital indices, the diagram structure adequately reflects the irreducible representation of the group U(R). In this sense the paper presents a unitary group approach to the natural generalization of the usual many-body theory for the spin-adapted cases. A set of very simple rules is derived; their similarity with the corresponding rules in the ordinary many-body theory and practical utility are discussed in connection with (a) matrix elements over many-electron spin states and (b) closed- and open-shell many-body perturbation theory. A possibility of integral-driven many-body perturbation theory for open-shells is indicated. Connections of this formalism with others are also discussed.     

Excited-state energies from many-body perturbation theory with the excitation-adapted one-particle potential

Calculation of excited-state energies by many-body perturbation theory is discussed. A Hartree-Fock-type potential suitable for a given excited-state configuration is introduced. Advantages which follow from the excitation-adapted one-particle potential are examined. The theory resembles that for external perturbation effects and amounts to removing or minimizing the contribution of non-diagonal one-particle terms.     

A many-body diagrammatic theory of rotation–vibration spectra

A many-body diagrammatic perturbation theory of rotation–vibration spectra is elaborated. The present approach is based on two many-body techniques, namely on the second quantization formalism (a rotating–vibrating molecule is formally treated here as a system of interacting vibrons, obeying the Bose–Einstein statistics) and the many-body diagrammatic theory of a model Hamiltonian, initially suggested in the microscopic theory of nuclei and in the last decade very frequently exploited in the accounting for the correlation effects in many electron systems. In the framework of this theory, the rotation–vibration energies are determined as the eigenvalues of a finite-dimensional model eigenproblem.     

Third-order many-body perturbation theory for the ground state of the carbon monoxide molecule

Many-body perturbation calculations have been performed for the ground state of the carbon monoxide molecule at its equilibrium internuclear separation. The calculations are complete through third order within the algebraic approximation; i.e., the state functions are parameterized by expansion in a finite basis set. All two-, three-, and four-body terms are rigorously determined, and many-body effects are found to be very important. A detailed comparison is made with a previously reported configuration interaction study. Padé approximants to the energy expansion are constructed. The many-body perturbative wave function is used in the Rayleigh quotient to produce upper bounds to the electronic energy.     

Power moments of hydrogenic Green's functions and Green's functions of the second kind

Power moments for all of the reduced hydrogenic Green's functions have been computed. These moments have the same form as the corresponding moments of the free-particle Green's functions. Green's functions of the second kind are defined, and uses for these objects in model potential theory and the theory of many-body Green's functions are pointed out. In the case of the ground state of the hydrogenic atom, the Green's function of the second kind has been given.     

Embedded,graph-theoretically defined many-body approximations for wavefunction-in-DFT and DFT-in-DFT: Applications to gas- and condensed-phase ab initio molecular dynamics,and potential surfaces for quantum nuclear effects

We present a graph-theoretic approach to adaptively compute many-body approximations in an efficient manner to perform (a) accurate post-Hartree–Fock (HF) ab initio molecular dynamics (AIMD) at density functional theory (DFT) cost for medium- to large-sized molecular clusters, (b) hybrid DFT electronic structure calculations for condensed-phase simulations at the cost of pure density functionals, (c) reduced-cost on-the-fly basis extrapolation for gas-phase AIMD and condensed phase studies, and (d) accurate post-HF-level potential energy surfaces at DFT cost for quantum nuclear effects. The salient features of our approach are ONIOM-like in that (a) the full system (cluster or condensed phase) calculation is performed at a lower level of theory (pure DFT for condensed phase or hybrid DFT for molecular systems), and (b) this approximation is improved through a correction term that captures all many-body interactions up to any given order within a higher level of theory (hybrid DFT for condensed phase; CCSD or MP2 for cluster), combined through graph-theoretic methods. Specifically, a region of chemical interest is coarse-grained into a set of nodes and these nodes are then connected to form edges based on a given definition of local envelope (or threshold) of interactions. The nodes and edges together define a graph, which forms the basis for developing the many-body expansion. The methods are demonstrated through (a) ab initio dynamics studies on protonated water clusters and polypeptide fragments, (b) potential energy surface calculations on one-dimensional water chains such as those found in ion channels, and (c) conformational stabilization and lattice energy studies on homogeneous and heterogeneous surfaces of water with organic adsorbates using two-dimensional periodic boundary conditions.     

Vectorizable algorithm for green function and many-body perturbation methods

The principles of an efficient, fast algorithm for the calculation of diagrams appearing in Green function and many-body perturbation methods are discussed and timing examples are given. Within the suggested algorithm, the third order-diagrams required in the Green function approach are evaluated by arranging computations in such a way that the most inner loops contain only simple scalar products and multiplication of vector by scalar operations. The molecular symmetry is taken into account for abelian groups. © 1993 John Wiley & Sons, Inc.     

Force constants and many-body electrostatic interactions in two-dimensional colloidal crystal

A model of a two-dimensional colloidal crystal with a hexagonal lattice, the electrostatic interactions in which are described by the nonlinear Poisson-Boltzmann equation, is considered. The calculation procedure for force constants of this crystal is treated in detail. Properties of system symmetry, which make it possible to significantly decrease the volume of calculations and to classify force constants, are analyzed. Numerical data for force constants of a crystal as functions of lattice parameters at different particle sizes are reported. A method that allows us to disclose the presence of many-body interactions in a system by the behavior of force constants at some interval of the values of lattice parameters is proposed. The application of this method to the system under consideration demonstrated that electrostatic interparticle interactions in the system cannot be reduced to simply a pair interaction of any kind; the introduction of many-body potentials is required for the adequate representation of the elastic properties of a crystal.     

The shifted scheme in the general-model-space diagrammatic perturbation theory

The shifted scheme of many-body perturbation theory is applied to open-shell states within the framework of the general-model-space theory. Rules for shifting the denominators of folded diagrams. which appear in open-shell perturbation expansions, are given. The finite-order energies in the shifted scheme obtained in two equivalent representations may differ. This happens, for instance, in the case of triplet states. For ³Σ_u⁺ states of the He₂, differences up to 0.07 mhartree have been found in third order. A similar phenomenon is the size inconsistency of the shifted scheme observed by Silver in the ground state of He₂. A possible advantage of the shifted scheme is its faster convergence for excited states.     

Many-body perturbation theory for open-shell systems. Expansion through fourth-order

All of the diagrams which arise in the many-body perturbation theory of open-shell systems using a restricted Hartree-Fock reference function are given through fourth-order in the energy. New effects which arise in fourthorder are discussed.S.E.R.C. Advanced Fellow     

Many-body force field models based solely on pairwise Coulomb screening do not simultaneously reproduce correct gas-phase and condensed-phase polarizability limits

It is demonstrated that many-body force field models based solely on pairwise Coulomb screening cannot simultaneously reproduce both gas-phase and condensed-phase polarizability limits. Several many-body force field model forms are tested and compared with basis set-corrected ab initio results for a series of bifurcated water chains. Models are parameterized to reproduce the ab initio polarizability of an isolated water molecule, and pairwise damping functions are set to reproduce the polarizability of a water dimer as a function of dimer separation. When these models are applied to extended water chains, the polarization is over-predicted, and this over-polarization increased as a function of the overlap of molecular orbitals as the chains are compressed. This suggests that polarizable models based solely on pairwise Coulomb screening have some limitations, and that coupling with non-classical many-body effects, in particular exchange terms, may be important.     

On multireference superdirect configuration interaction in third order

The superdirect configuration interaction (Sup-CI ) method has the usual versatility and stability of the CI methods with computational efficiency typical to that of the many-body methods, such as the many-body perturbation theory (MBPT ). The Hamilton operator is projected into a space of a few trial vectors, such as Krylov, Nesbet, or Møller–Plesset correction vectors. In this space, Hamiltonian matrix elements may be directly computed in the many-body fashion, as weighted sums of integral products over orbital indices. The variation-perturbation method based on the first-order wave function is equivalent to the Sup-CI method with a single correction vector of the Møller–Plesset type. Different points of view on the superdirect CI method are discussed and a version in which third-order contributions are computed for a relatively small (10–100) space of reference and correction vectors is tested. Selection of the best “effective first-order spaces” and size-extensivity corrections in Sup-CI are briefly discussed. Møoller–Plesset, Epstein–Nesbet, and other correction vectors are included in the model calculations on the symmetric stretch of bonds in water, acetylene, and the NH₂ molecule. Errors are almost independent of molecular geometry and the method appears to be superior than the multireference second-order perturbation methods. © 1994 John Wiley & Sons, Inc.     

Restricted Hartree-Fock and unrestricted Hartree-Fock as reference states in many-body perturbation theory: a critical comparison of the two approaches

Summary The paper deals with two topics related to the problem which reference state is better for many-body perturbation theory: restricted Hartree-Fock (RHF) or unrestricted Hartree-Fock (UHF)? The first topic concerns the potential surfaces. Several examples are presented to show shortcomings of the two approaches and a simple way is presented which seems to give a useful potential curve in the whole range of interatomic distances by a composition of RHF and UHF potential curves. The second topic concerns the many-body perturbation theory for open-shell systems in the RHF formalism. The method is critically examined and compared with the ordinary many-body perturbation theory using UHF as the reference. This examination of many-body techniques provides also some insight into the problems inherent of the SCF theory: spin contamination from higher multiplets, localization of orbitals, and self-consistency effects.     

Unique continuation for the magnetic Schrödinger equation

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schrödinger equation. This property guarantees that if a solution of the Schrödinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one-body or two-body functions, typical for Hamiltonians in many-body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique-continuation property plays an important role in density-functional theories, which underpins its relevance in quantum chemistry.     

A proposed definition of resonant states

The widely cited definition of quantization in terms of square-integrable wave functions does not apply to continuum wave functions, to such phenomena as metastable states, or many-body resonances. A better philosophical foundation for quantum mechanics separates the probabilistic aspects based on square integrable Hilbert space functions from the dynamical aspects based upon the solutions of Shroedinger's (or Dirac's) equation. A Hilbert space may have a non-Hilbert space basis, which may be described by Stieltjes integrals and a spectrum measure. This viewpoint is expounded by reference to a very detailed analysis of a simple model, through which a precise definition of a Bohr–Feshbach resonance can be given. We propose a definition of a “metastable state,” showing that it is consistent with accepted usage, and that it overcomes a series of objections which have been catalogued by Simon. Its rate of decay is given by the Fourier–Stieltjes transform of the spectral density function; it is moreover the longest-lived initially localized state which can be formed from a small span of energy eigenfunctions near its mean energy.     

Application of multiconfigurational many-body perturbation theory to the calculation of ionization potentials,electron affinities and excitation energ

Multiconfigurational many-body perturbation theory is applied to the problem of calculating ionization potentials, electron affinities, and excitation energies. H₂O, C₂H₄, and H₂ are studied, with correlation corrections through third order and inclusive of certain higher-order terms. Results are compared with those by other many-body theoretical methods.     

A new method of solving the many-body Schrödinger equation

A new method of solving the many-body Schrödinger equation is proposed. It is based on the use of constant particle-particle interaction potential surfaces (IPSs) and the representation of the many-body wave function in a configuration interaction form with coefficients depending on the total interaction potential. For these coefficients the corresponding set of linear ordinary differential equations is obtained. A hierarchy of approximations is developed for IPSs. The solution of a simple exactly solvable model and He-like ions proves that this method is more accurate than the conventional configuration interaction method and demonstrates a better convergence with increasing basis set.     

Many-body effects in X-ray photoemission spectroscopy and electronic properties of solids

Photoemission from a solid is evidently a many-body process since the motion of each electron cannot be independent of the motions of other electrons. In this article we review the reported many-body effects in X-ray photoemission such as extra-atomic relaxation energy, charge transfer satellite and energy loss structure which are informative in relation to the characteristics of solids.