The extended Koopmans' theorem Fock operator

By expanding the wave function of a system of N particles in terms of products of functions of one and (N-1) particles, the one-particle, nonlocal operator F?_EKT (extended Koopmans' theorem) is determined. It is shown that although this operator is nonhermitian, its eigenvalues and eigenfunctions represent the ionization energies and occupied orbitals, respectively. The eigenfunctions of F?_EKT are the one-particle functions that enter into the expansion of the wave function of the system as partners of the (N-1)-particle wave functions. The eingenvalues are also one-particle energies that, multipled by the orbital occupancy probalities, enter the expression for the total N-particle energy of the system.     

Vertical ionization potential in hydrogen molecule with many-body Green's function method

The many-body Green's function method is applied to the vertical ionization potential of the hydrogen molecule. The ionization potential is calculated iteratively by expanding the self-energy part up to third order. The effects of higher-order correlation corrections and nondiagonal self-energy elements on the solutions of the Dyson equation are examined with some techniques and approximations, by means of which a Koopmans' defect of 97.7% of the accurate value is obtained.     

Dyson-corrected orbital energies for the perturbative treatment of electron correlation

The effect of replacing the Hartree–Fock one-particle energies with ionization potentials obtained from inverse Dyson equation when calculating electron correlation energies perturbatively is investigated. Though the energy shifts vary from system to system, the slight decrease of the resulting excitation energies at around equilibrium geometries leads to a slight increase of the correlation energies in most cases. In the dissociation limit the inverse Dyson equation opens the gap, thus nondiverging potential curves emerge even at the restricted Hartree–Fock (RHF)+RS2 level. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 713–719, 1998     

Application of many-body green's functions to the calculation of molecular ionization potentials

We present a method for calculating the one-particle Green's function for molecules. The scheme is essentially that proposed by Schneider and Taylor [1]. The Green's function is obtained through the Dyson equation. Closed expressions result by using the functional derivation technique to truncate an infinite set of coupled equations. A physical interpretation of the approximation is given and a connection with the equations-of-motion method is pointed out. In a numerical application the ionization potentials are obtained for the molecules N₂, H₂O, NH₃ and CH₄.     

The extended Koopmans' theorem Fock operator and the generalized overlap amplitude one-electron operator

The wave function of a system may be expanded in terms of eigenfunctions of the N −1 electron Hamiltonian times one-particle functions known as generalized overlap amplitudes (GOAS). The one-electron operator whose eigenfunctions are the GOAS is presented, without using an energy-dependent term as in the one-particle Green function or propagator approach. It is shown that this operator and the extended Koopmans' theorem (EKT) one-electron operator are of similar form, but perform complementary roles. The GOA operator begins with one-electron densities and total energies of N −1 electron states to generate the two-matrix and total energy of an N-electron state. The EKT operator begins with the two-matrix of an N-electron state to generate one-electron densities and ionization potentials (or approximations thereto) for N −1 electron states. However, whereas the EKT orbitals must be linearly independent, no such restriction applies to the GOAS. © 1996 John Wiley & Sons, Inc.     

Binding energies at different momenta for the valence orbitals of HCl by the binary (e, 2e) method

The binding energy spectra for the valence orbitals of hydrogen chloride have been obtained using the binary (e,2e) method at 1200 eV. The strength of the innermost valence orbital (4σ) is severely split among several ion states in the energy range 25 to 41 eV. The measured cross sections are compared with results of calculations using contracted Gaussian basis sets of double-zeta quality, and with a one-particle Green's function calculation.     

Ordinary field-theoretic methods for self-consistent wave functions which describe bond formation and dissociation. II. Commutative coupling case

Hartree–Bogoliubov–Valatin (HBV ) theory may be implemented with Lipkin Hamiltonians to obtain self-consistent BCS wave functions which describe bond formation and dissociation. These wave functions are in turn vacuua for Nambu's representation of Feynman–Dyson–Goldstone diagrammatic perturbation theory, and hence provide suitable references for the many-body treatment of correlation. Exact SCF solutions of the HBV equations are equivalent to special even-replacement MC –SCF solutions. The latter are similar to generalized valence bond theory, and require one Fock operator for each one-particle shell. The commutative coupling case of HBV theory is realized when the number-conserving renormalized one-body and number-nonconserving pairing operators commute. In this case, a set of orbital equations which involves a single Fock operator may be solved. Since this could prove to be a significant simplification for large systems, the commutative coupling and exact solutions are compared here for the fragmentation of H₂ and F₂. Results suggest that commutative coupling orbitals will be useful for the aforementioned many-body theory.     

Time-dependent Hartree–Fock calculations in the Pariser–Parr–Pople model. Applications to aniline,azulene and pyridine

Propagator or Green's function methods are used to analyze the time-dependent Hartree–Fock model. The non-hermitian matrix problem for the time-dependent Hartree–Fock solution is reduced to a problem related only to hermitian matrices. Particular attention is given to the calculation of oscillator strength in different approximations. The connection between the stability of the Hartree–Fock solution and the solution of the time-dependent Hartree–Fock problem is demonstrated. The results of numerical calculations are given for aniline, azulene and pyridine.     

Koopmans' defects in metal group VIII carbonyls—a many-body approach

The ionization potentials of iron pentacarbonyl ( 1 ), ethylene–iron tetracarbonyl ( 2 ), cobalt tetracarbonylhydride ( 3 ), and nickel tetracarbonyl ( 4 ) have been calculated using a Green's function perturbation method based on the INDO approximation. It is shown that the deviations from Koopmans' theorem are largest in the Co complex, while the smallest reorganization energies of strongly localized MO s with predominant metal 3d character are found in the Fe carbonyls 1 and 2 . The calculated Koopmans' defects are analyzed by an investigation of the relaxation terms of the self-energy part and are compared with previous INDO results for Cr, Mn, and Fe tricarbonyl derivatives. Additionally, orbital energies, bond indices, and net charges for the ground states of 1 – 4 are calculated and compared with experimental data.     

EHF Description of superexchange: Relations to Anderson's model

It is shown that the treatment of the superexchange in insulators, based on the spin projected extended Hartree–Fock (EHF ) method may be considered as a direct generalization of Anderson's ligand field (or kinetic exchange) theory: the EHF methods permits to construct explicitly the effective orbitals for the magnetic electrons and to determine all the parameters occurring in Anderson's theory from only one variational problem.     

A correlation study of Li ground state by the MCHF procedure

Results are reported for multiconfiguration Hartree–Fock studies of correlation in the lithium ground state, which maintain orthogonality of orbitals within a configuration. It is shown that when the 1s- and 2s-orbitals are fixed at their Hartree–Fock value, configurations for which Brillouin's theorem holds may be important, particularly for atomic properties other than energy. The Fermi contact term is considered as an example.     

Extended Koopmans' theorem ionization potentials for beryllium atom shake-up transitions

The ionization potentials were calculated for Be using the extended Koopmans' theorem (EKT ) using several full configuration interaction (CI ) and multiconfigurational-self-consistent-field (MCSCF ) wave functions as reference wave functions. The wave functions used account for 89.7–96.7% of the correlation energy. Comparisons are made with experimental values and with δCI values calculated as the difference in energy obtained from CI wave functions for Be and Be⁺. The best EKT IP differed from the δCI value by 0.0003 eV for the lowest IP and by 0.0006 eV for ionization into the lowest ²P state of Be⁺. A calculation of ionization into the second ²P state of Be⁺ requires diffuse orbitals that are unimportant in the wave function for the ground state of Be. This results in small natural orbital occupation numbers for natural orbitals needed in the EKT calculation. © 1994 John Wiley & Sons, Inc.     

The perturbation theory of the extended Hartree–Fock approximation for two-electron atoms

The first-order 1/Z perturbation theory of the extended Hartree–Fock approximation for two-electron atoms is described. A number of unexpected features emerge: (a) it is proved that the orbitals must be expanded in powers of Z^?1/2, rather than in Z^?1 as expected; (b) it is shown that the restricted Hartree–Fock and correlation parts of the orbitals can be uncoupled to first order, so that second-order energies are additive; (c) the equation describing the first-order correlation orbital has an infinite number of solutions of all angular symmetries in general, rather than only one of a single symmetry as expected; (d) the first-order correlation equation is a homogeneous linear eigenvalue-type equation with a non-local potential. It involves a parameter μ and an eigenvalue ω(μ) which may be interpreted as the probability amplitude and energy of a virtual correlation state. The second-order correlation energy is 2μ²ω. Numerical solutions for the first-order correlation orbitals, obtained variationally, are presented. The approximate second-order correlation energy is nearly 90% of the exact value. The first-order 1/Z perturbation theory of the natural-orbital expansion is described, and the coupled first-order integro-differential perturbation equations are obtained. The close relationship between the first-order extended Hartree–Fock correlation orbitals and the first-order natural correlation orbitals is discussed. A comparison of the numerical results with those of Kutzelnigg confirms the similarity.     

Green's function calculation of through‐bond electronic coupling in donor–bridge–acceptor model systems II: Importance factors applied to atomic sites

Importance factors, associated with the Green's function formalism, are introduced. They are applied for the determination of the relative atomic site contribution to the electronic interaction propagation in a molecular system. The calculation is performed at the Hartree–Fock (self‐consistent) level, using ab initio STO‐3G, 4‐31G, and D95 basis sets. The results are compared with those obtained from the charge densities of the appropriate molecular orbitals at the ab initio STO‐3G level. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Theoretical study of molecular conduction: I. Effective Green's function based on perturbation theory

There have been many experimental and theoretical studies on molecular conduction, as it is a fundamental parameter in the study of molecular‐scale electronics. We have investigated the features of molecular conduction using a Green's function method, which has often been used to solve problems in quantum transport and is also effective in elucidating electron transport in molecules. We have obtained the novel effective Green's functions, including the first‐order energy corrections, by accommodating the self‐energy of the electrodes as perturbation terms. Although these approximate Green's functions only provide information on the first‐order energy corrections, they can involve the elementary properties of molecular conduction. We propose a scheme for the analysis of the relations between molecular orbitals and their roles in molecular conduction and present analytical calculations for normal and cyclic polyenes. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Many-electron effects in the x-ray emission spectra and the one-particle green's function. Correspondence theorem

Within the framework of reasonable approximation the intensity of spectral bands in the X-ray emission spectrum was shown to be proportional to the imaginary part of the one-particle Green's function. From this ensues a statement about a correspondence between the distribution of the intensity in the X-ray emission and photoelectron spectra — correspondence theorem.     

Localization measure and maximum delocalization in molecular systems

A mathematically well-defined measure of localization is presented based on Mulliken's orbital populations. It is shown that this quantity equals 1 for core- and lone-pair orbitals, 2 for two-atomic bonds, 6 for benzene rings, etc., and it is applicable for delocalized canonical HF orbitals as well. The definition of this quantity is general in the sense that ab initio MOS with overlapping AO expansion, and semiempirical wave functions using the ZDO approximation as well, can be treated. The localization quantity is essentially “intrinsic,” i.e., no subdivision of the molecule is required. For N-electron wave functions, mean delocalization can be defined. This measure is not invariant to unitary transformations of the one-electron orbitals, characterizing in this way the localized or extended representation of the N-electron wave function. It can be proven, however, that for unitary transformed wave functions a maximum delocalization exists which depends only on the physical (N-electron) properties of the molecule. It is shown that inhomogeneous charge distribution can cause strong electron localization in molecular systems. The delocalization of the canonical Hartree–Fock orbitals, the Parr–Chen circulant orbitals, and the optimum delocalized orbitals is studied by numerical calculations in extended systems.     

Extended Hartree–Fock calculations for the helium ground state

The extended Hartree–Fock (EHF) wave function of an n-electron system is defined (Löwdin, Phys. Rev. 97 , 1509 (1955)) as the best Slater determinant built on one-electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the ¹S state of a two-electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p₀), 2(d₀) and 1 (f₀) Slater orbitals has produced 90% of the correlation energy.     

The AGP model for fermion systems

A mathematically rigorous treatment of the antisymmetrized geminal power (AGP) model is given. The model is advocated as an extremely flexible tool for studying a wide variety of fermion systems, characterized by a single Slater determinant, on the one hand, or Yang's most highly correlated wave function, on the other. When the first-order reduced density matrix of the actual system has eigenvalues which are evenly degenerate, the model can treat one-particle properties exactly. A formula is obtained for the precise energy of an AGP state in the Born-Oppenheimer approximation. An estimate is given of the error which occurs when the second-order density matrix of an AGP state is approximated by the widely used expression involving the so-called anomalous Green's function. © 1997 John Wiley & Sons, Inc.