Aspects of Many-Body Effects in Molecules and Extended Systems

Since few problems of interest admit of solutions in closed form in quantum mechanics, we are obliged to use approximate methods based on expansion schemes of one form or another. Such expansions, when truncated at some finite order, often do not reflect some of the inherent properties of the systems under study. A typical example of this type of failure is the well known “symmetry dilemma” encountered in the Hartree — Fock calculation. The broken- symmetric solutions, occasionally faced, stem from the use of a restricted variation of the function in the subspace of single determinants. A full CI calculation will obviously restore the symmetry, but since that is neither practicable nor desirable, additional conditions are imposed to ensure that the resultant functions are symmetry-adapted even at the trial function level.

The many electron-problem of atomic or molecular theory can be formulated either in configuration space (where operators act on the coordinates of the particles) or in Fock space (where operators act on the occupation numbers of spin orbitals). The representation of operators in Fock space (often also referred to as occupation number representation or 2^nd quantization) is essentially equivalent to the representation in configuration space, it is, however, more general insofar as a Fock space Hamiltonian has eigenstates with arbitrary numbers of electrons, while in configuration space the number of electrons is fixed. So Fock space is, in a sense, a direct sum of Hilbert spaces for various particle numbers. Processes like ionization or electron attachment, in which the number of electrons is changed are hence better described in Fock space than in configuration space (i.e. in n- particle Hilbert space).

Truncated configuration interaction (CI) calculations are known¹ to yield total electronic energies which do not scale properly with the number of electrons in the system. This so called “size extensivity” (SE) problem causes truncated CI energies (e.g., those which include all excitations up through some specified level) energies to be of questionable value, for example, in computing thermodynamic energy differences such as chemical reaction energies. Specifically, for the reaction truncated-CI energies of the individual species (C, D, A, and B) can not be used to evaluate ΔE; one must evaluate, within the truncated-CI method, the energy of the C + D and A + B “supermolecules” at large C — D and A — B separations, and then subtract these two energies to evaluate ΔE.

Applications of the formalism of effective Hamiltonians in Fock space are presented using complete and (in most cases) incomplete model spaces. Applicatory examples are given for the H₂ molecule, the He atom, the molecular ion He ₂ ²⁺ , and for the three-electron ions C³⁺ to F⁶⁺ in highly excited states.

A fully linked multi-reference coupled-cluster (MRCC) method is used for the calculation of ionization potentials (IP), electron affinities (EA) and excitation energies (EE) of CH⁺. The model space for the IP/EA calculations consists of determinants where an electron out of an active set of orbitals has been removed/added to an RHF determinant. For the EE calculation the model space is constructed from all particle-hole excitations obtained from the IP/EA active orbitals. Results using the MRCCSD approximation are presented and compared to results obtained from CI, MCSCF, GF, single-reference CC calculations, and experiment.

The application of the open-shell coupled-cluster method to the evaluation of electronic transition energies in atomic and molecular systems is described. Examples are given for ionization potentials, electron affinities, and excitation energies. The problem of model (P) space structure is discussed, and the potential functions for LiH excited states are presented as examples for applications requiring incomplete model spaces.

The open-shell Many-Body Perturbation Theory (MBPT) is now being used quite extensively to generate valence-shell hamiltonians [1] and potential surfaces [2] for a variety of systems of chemical interest. The underlying theoretical strategy in the MBPT is the partition of the space of basis functions into a model space and a virtual space, and the generation of an effective hamiltonian H_eff, which acts on the model space and furnishes N eigenvalues of the hamiltonian H where N is the dimension of the model space. It is highly desirable to have the calculated energies scaling properly with the number of electrons (size-extensivity [3]), and this is ensured if H_eff is completely connected. There is also a related (but distinct) demand that H_eff should describe the molecular dissociation process correctly (size-consistency [4]).This is also satisfied with a connected H_eff provided the zeroth order functions have the product-separability. Starting from the earliest work of Brandow[5], almost all the later developments [6] classified the orbital basis into “core”, “particle” and “valence” orbitals, and the model space was taken to contain determinants with completely filled core orbitals and partially filled valence orbitals.

Spectroscopic studies utilize probes of different kinds, as electro-magnetic fields, charged and neutral particles, to study molecular systems and measure the response of the system or the probe. The probabilities for a particular change to occur are reflected in the intensities of associated spectral lines or bands. There is then no surprise that the natural theoretical tools for the treatment of spectroscopy are response functions and propagators or Green’s functions, which permit direct calculations of characteristic changes in the state of the system or the probe and the probability for such changes to occur due to the system—probe interaction.

We have investigated whether it is possible to approximate RPA transition probabilities by a standard state representation like <λ,app|M | v,app>. Using |HF> and 0 _λ ⁺ |HF> as the approximate eigenstates we find that the approximation seems to work best for transition between excited states. In this case only the first order correction, which involves a sum over all excited RPA states, gives an appreciable correction to the state representation of the transition probability. Parallel conclusions are obtained in both the dipole length and the dipole velocity formulation.

The methodology and applications of the one electron propagator theory have been diacuaaed elsewhere in these proceedings[1,2] and their effectiveness in the direct calculation of ionization energies and eleotron affinities needs no further mention. The energy and width of elctron scattering resonances may be associated with electron affinity and ionization energy where the electron attachment/removal involves a metastable state.

The extended coupled cluster method (ECCM) is applied to the zero-temperature condensed Bose fluid for the general case of a non-uniform and time-dependent condensate wavefunction. The principal aim is to derive the appropriate balance equations of fluid dynamics for such local observables as the number density, current density and energy density. This is achieved by coupling the system to scalar and vector gauge fields, so that the theory may be formulated in a completely gauge-invariant fashion to take fully into account the underlying U(l) symmetry imposed by number conservation. The ECCM formalism is based on the equations of motion for a set of linked-cluster amplitudes which characterise the system completely. The off-diagonal one- and two-body density matrices are studied in terms of these amplitudes, and the various balance equations of fluid dynamics are thereby derived. Furthermore, we show how these are also exactly satisfied by most practical approximation schemes applied to the otherwise exact ECCM description. The fact that the ECCM amplitudes all strictly obey the cluster property, further implies that our gauge-invariant formalism is capable in principle of a complete fluid-dynamical description of the system at zero temperature, including possible states of topological excitation or deformation and of broken symmetry.

Extended electronic systems (crystals, polymers, etc.) can be conveniently described using the language of the quantum field theory^1,2. In this approach one treats the ground state of a system as a “vacuum” with respect to which quasiparticles, representing excited states, are defined. A quasiparticle is characterized by its energy and, usually, by some symmetry labels such as wave vector, spin, charge, etc. When an extended system is studied within the Born-Oppenheimer approximation (i.e., it is assumed that the electrons are decoupled from the phonons) the basic quasiparticles are “electrons” and “holes” (the quotation marks are used here to distinguish between quasiparticles and free particles), treated as elementary particles corresponding to the Fermi-Dirac statistics. Depending on the system studied other quasiparticles: excitons, plasmons, or magnons may also appear.

Huckel model (1) in quantum chemistry has been one of the earliest models proposed for the electronic structure studies of π — systems. Because of the ease of solving these models for comparatively large π- systems, these models have found a great deal of acceptance among chemists in general and organic chemists in particular. The Huckel model is parametrized by two types of parameters, the resonance integral (also called the transfer or hopping integral) t_ij and the orbital energy (also called site energy) ε _i.

Undoubtedly, a majority of living processes take place only for a reason that they occur in the aqueous environment and are reliable only due to the so called “schizophrenia” of water. A schizophrenia of water is manifested in its “anomalous” properties. For example, one of the most striking anomalous property of water is its high electric conductivity arising, as usually emphasized in the text book of physical chemistry, due to the large mobility of the H⁺ ion.

The cluster decomposition property of the time evolution operator is defined and some of the wave packet propagation methods which satisfy this are reviewed.

The electronic excitations of a metal form a continuum and a careful study of their role in the dynamics of electrochemical electron transfer reactions is needed. We suggest a method for accounting for these non~adiabaticity effects. It approximates the electronic excitations as bosons and makes use of quantum transition state theory to calculate the rate of the reaction.

The behaviour of relativistic many-electron wave functions for arbitrary atoms or molecules at the Coulomb singularities of a point nucleus and of the electron interaction is derived. The wave functions have weak singularities of the type r ^v with v < 0. No singularity arises for extended nuclei. Solutions of the Klein-Gordon equation, the ‘square-root equation’ or of Sucher’s nopair equation with free-particle projectors’ have a different behaviour than those of the Dirac equation. While v is quadratic in Z/c for the Dirac equation, it is linear in Z/c for both the square-root operator and the free-particle no-pair-projected Dirac operator. For the latter two cases there is no smooth non-relativistic limit. For a Dirac-Coulomb Hamiltonian there is also a singularity of the type r _kl ^λ with λ < 0 at the point of coalescence of two electrons, while for the Dirac-Gaunt (or Dirac-Brown) operator one finds λ > 0 (in view of the higher singularity of the magnetic interaction one might have expected a more singular behaviour of the wave function in its presence), and for the Dirac Breit operator λ = 0. Like in the nuclear cusp, A is in these cases quadratic in 1/c, while it is linear in 1/c for the no-pair projected operators. In spite of the complexity of these results, they only affect an extremely small neighborhood of the point of coalescence of two electrons. A large region close to r ₁₂ = 0 is governed by the behaviour in the non-relativistic limit, i.e. by Kato’s correlation cusp.

A review is given of the non-relativistic many-body perturbation theory (MBPT), emphasizing the diagrammatic expansion and the all-order procedures, such as the coupled-cluster approach. Illustrative numerical examples are given. More recent formal developments, concerning incomplete model space and hermitian effective hamiltonian, are also mentioned. Various approximate schemes currently in use for relativistic many-body calculations are discussed and analysed from the point of QED. The “no-virtual-pair approximation”, which contains the all-order non-relativistic MBPT as well as the leading relativistic corrections, is applied to He-like systems. Possibilities are discussed of going beyond that scheme for general many-electron systems in a rigorous and systematic way.

A systematic approach to a relativistic theory of electronic excitations in atoms and molecules is discussed. The methodology is based on the relativistic equations of motion for excitation processes. For low-Z systems, relativistic and radiative corrections can be calculated in Breit-Pauli approximation. For medium and high-Z systems the “no-pair” Hamiltonian is chosen as the unperturbed Hamiltonian, the residual interaction is regarded as a perturbation and quantum electrodynamical effects are calculated within the framework of perturbation theory. The second-order correction to the excitation energy, due to creation and destruction of virtual electron-positron pairs, is discussed. Radiative corrections to the excitation energy are calculated as first-order corrections arising from the use of the Breit operator. Detailed expressions for the Breit interaction contributions are derived when the excitation operator is approximated by the first-order perturbation theory solution of the RPA equations.

For a non-degenerate quantum mechanical system to possess a permanent electric dipole moment (EDM), there must be a simultaneous violation of parity (P) and time-reversal (T) symmetries [1]. Since P-violation in weak interactions is now well-established, the significance of observing a permanent EDM is that it would be a direct confirmation of T violation. An atomic EDM can arise due to: While all of the above-mentioned sources of atomic EDM are interesting in their own right, I shall concentrate only on the first case. Several particle physics models which call for CP violation in the lepton sector predict an intrinsic electric dipole moment of the electron [2].

Spinor representations of generators of the Lie algebra of SO(N)(N=2n, 2n+1; n integer, have played a key role in a number of areas of Physics [1–5]. A general approach to these representations in a form suitable for practical applications has been of recent origin [6–8]. Starting with the unitary algebra of U(2ⁿ), the generators of SO(N) and U(n) were realised in the chain U(2ⁿ) ⊃ SO(2n+1) ⊃ SO(2n) ⊃ U(n). It was found that the symmetric bispinor basis spanning the representation [2 Ȯ] of U(2ⁿ) could be used to subduce the spin-free configurations spanning the representations [2^N/2-s, 1^2s, Ȯ] of U(n). Some preliminary studies of generating the configuration space in this manner have recently been carried out for basis adapted for the chains U(n)⊃...⊃ U(1) [8] and U(n) ⊃ SO(n) ...⊃ SO(2) [9]. From a practical point of view this approach has a basic drawback. This is the fact that the simple one electron orbital description of spin-free configurations is masked in using the spinor basis. In the present study we examine the possibility of inducing a basis spanning the representations of U(2ⁿ) and SO(N) starting with the antisymmetric representations, [1^N, Ȯ] (0≤N≤n; N integer) of U(n). The aim is to provide a simple interpretation of the spinor basis in terms of the tensor (integer) representations of U(n).

A field theoretic formulation of many-electron problems relying on the principles of the orbital unitary group adaptation of N-electron states is presented. Fock-space one- and two-operators are provided to create and annihilate such adapted electron states. The associated algebra is utilised to effect orbital level-wise factorisation of the matrix-elements of any number-conserving spinless operator such as the Hamiltonian. Other potential uses of the operators are indicated.

An ab-initio solution of the many body problem using the hyperspherical harmonics expansion method has been reviewed. Results of typical calculations has been compared with those by other methods. Merits and demerits of this method, especially when the number of particle increases, have been discussed. Finally a modification of this method, in which an integro-differential equation in two variables is set up, has been discussed. The integro-differential equation approach appears to be very convenient for the many body problem. Some results of calculation by this method have also been presented.

Several quantum mechanical theories and statistical and field theoretic models of physical interest possess continuous group symmetry often in some internal space. Many of these theories admit straightforward extensions in which the number of internal degrees of freedom N may be treated as a free variable parameter. Inflating the given number N to infinity then entails, quite surprisingly, a drastic simplification in the analyses of a diverse class of such theories. The method of large-N expansion hinges on the fact that if the large-N limit of such a problem can be obtained explicitly then the finite-N corrections can be incorporated by introducing a systematic expansion in powers of 1/N and contact with the original theory may be made by substituting for N its given fixed value at the end of the calculation. This inverse-N expansion technique has emerged in recent years as a powerful approximation scheme in fields as disparate as quantum mechanics, nuclear physics, critical phenomena, laser physics and quantum chromodynamics [1,2]. Here we shall however restrict our discussion mainly to quantum mechanics where the large-N expansion was first applied by Ferrei and Scalapino [3] in 1974 and since then interest in this subject has continued unabated.

There are many applications of catastrophe theory in different fields of science. We will apply this theory to molecules. The theory deals with the mathematical methods of investigation of analytic behavior of a smooth function and it was developed by Arnold and Thorn [1–4]. The theory of singularity of the function( the other name of it is the catastrophe theory) is the generalization of the investigation of the minimum and maximum of the function. The application of this theory to molecules is based on finding of the extremal points of the function p(r,X), which is the total density of the electron charge of a molecule. Basing on such an investigation, which was done by Srebnik and Bader [5], Collard and Hall have suggested to use the catastrophe theory for the investigation of structural instabilities in a molecule [6]. This suggestion was developed in brilliant works of Bader with collaborators[7–11]. They have shown, that in a three-atom molecule and in some cases, in more complicated molecules the structural instabilities can be described by a catastrophe, which was called by Thorn as the elliptic ombilic.

It is stressed that the conceptual crisis due to duality in the origin of quantum chemistry can be naturally resolved in terms of the energy density functional theory. The requirements imposed for formulating the density-functional ground-state variational principle within the Schrödinger picture of quantum mechanics are discussed. The rigorous construction of the energy density functional theory of many-electron systems, based on the concept of local-scaling transformations, is proposed.

A review of procedures for the estimation of electron momentum density for atoms, molecules and solids exclusively with the position space density as the starting point, is presented, These procedures are based on semiclassical and density functional schemes and their extensions. Inequality type relations between information entropies in the two spaces are also discussed, followed by projections regarding future work.

Some recent developments in the theoretical foundations of density functional formalism for many-electron systems are discussed. The time-dependent systems characterised by scalar as well as vector potentials are shown to be amenable to a density-description using charge and current densities as basic variables. The fluid dynamical transcription corresponding to this time-dependent density functional theory is also obtained. The electron fluid picture leads to the definition of several thermodynamic-like quantities of chemical significance. The electron density in position space also describes the properties in momentum space for which a practical scheme is presented.