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.     

Ivar Waller 90 years on June 11, 1988

Linkage properties of the diagrammatic representation of the energies obtained in the multireference many-body perturbation calculations with respect to the incompleteness or completeness of the model space are discussed. The case of not completely degenerate model space is considered for which a comparison with the standard single-reference many-body perturbation expansion is possible. The Hose–Kaldor type of graphical representation of the perturbation expansion for the effective Hamiltonian is used in this comparison. It is shown that for an incomplete model space the perturbation expansion is not size-extensive. In this case, for a truncated expansion of the effective Hamiltonian, the energies obtained by diagonalization of the effective Hamiltonian matrix are represented by both linked and unlinked irreducible contributions. The unlinked ones do not appear when the complete model space is used.     

Calculation of correlation energy by many-body diagrammatic perturbation theory

The many-body diagrammatic perturbation theory is used for calculation of the correlation energy of closed-shell molecular systems. We apply Brueckner's concept of the two-particle renormalized interaction defined by a non-linear diagrammatic expression containing all possible (diagonal and/or non-diagonal) particle-particle, hole-hole and particle-hole intermediate elementary processes. Then, a “second-order” simple diagrammatic expression for the correlation energy can be formed, where the correlation energy is approximated by all the diagrams with biexcited intermediate states. An illustrative numerical application for the LiH molecule is presented. This article is dedicated to the memory of our friends and colleagues Dr. Jarka Surá and Dr. Marta Černayová, who tragically died in July 1976.     

A second-quantization representation of the Epstein-Nesbet partitioning of the full hamiltonian

A second-quantization representation of the Epstein-Nesbet partitioning of the total electronic hamiltonian is suggested. In this approach, the unperturbed hamiltonian contains not only the one-particle orbital energies but also the Coulomb and corresponding exchange two-particle terms. Such a hamiltonian can advantageously be used in all branches of the many-body diagrammatic perturbation theory for simple and correct inclusion of the diagonal ladder and ring diagrams in all orders of perturbation theory.     

A complete diagrammatic construction of PCILO method

In the present short communication, the diagrammatic many-body perturbation theory for the rederivation of PCILO method (perturbation configuration interaction using localized orbitals, see Theoret. Chim. Acta (Berl.)13, 1 (1969) and15, 100 (1969)) is consequently used. Using the Goldstone-Hugenholtz linked-cluster theorem, the diagrammatic expression for the exact ground-state energy is obtained. An application of the present approach to the low-lying excited and/or ionized states is also discussed.     

Application of many-body rayleigh-schrödinger perturbation theory to calculation of ionization potentials and electron affinities

The ionization potentials and electron affinities are calculated by the use of the many-body Rayleigh-Schrödinger perturbation theory. This approach is elaborated up to the third order, where each perturbation contribution can be interpreted by the diagrammatic method. Some simple illustrative calculations of π-electron systems are carried out.     

The one-particle Green's function method in the Dirac-Hartree-Fock framework. I. Second-order valence ionization energies of Ne through Xe

The one-particle Green's function theory in its various implementations is a well-established many-body approach for the calculation of electron ionization and attachment energies in atoms and molecules. In order to describe not only scalar-relativistic effects but also spin-orbit splitting on an equal footing an embedding of this theory in the four-component framework was carried out and fully relativistic ionization energies of the noble gas atoms Ne through Xe were calculated using the second-order algebraic diagrammatic construction [ADC2] approximation scheme. Comparison with nonrelativistic ADC2 results and experimental data was made.     

Universal systematic sequence of even-tempered Gaussian primitive functions in electronic correlation studies

The possibility of developing a universal systematic sequence of eventempered Gaussian primitive functions for atomic and molecular electronic structure studies is examined. The radial beryllium-like ions are used to demonstrate this approach both within the Hartree-Fock model and by including correlation effects. Correlation energies are computed using the diagrammatic many-body perturbation theory. The Hartree extrapolation procedure is used to obtain empirical upper bounds to the basis set limit and the procedure of Schmidt and Ruedenberg is employed to obtain empirical lower bounds for the basis set limit. The convergence properties of the calculations with respect to the size of the basis set are examined.     

Many-body calculation of the electric field gradient in the hydrogen molecule

The electric field gradient in the hydrogen molecule has been calculated by diagrammatic many-body perturbation theory (MBPT ) in Gaussian basis sets. The procedure through third order in electron correlation gives a value for the field gradient of 0.34041 a.u., which is 0.8% greater than the accurate value. The result is discussed in terms of the completeness of the basis sets and the convergence of the perturbation expansion.     

Localized bond model for molecular energy to fourth order in perturbation theory

The localized bond model of Malrieu, Diner, and Claverie is extended to fourth order in perturbation theory. Single, double, triple, and quadruple replacements from the doubly occupied bonding reference function are included utilizing a symmetric form of diagrammatic perturbation theory. The fourth order theory derived executes on a computer as quickly as does the third order theory. Results are examined utilizing the Pariser–Parr–Pople and CNDO/2 model Hamiltonians, and are compared with third order results and with either exact results where they are known, or with a configuration interaction of all singles and doubles. The influence of the initial hybridization, localization, and bond polarization is discussed. In general, the fourth order corrections are of comparable size to third order. Improvement in results appears to be marginal in the Nesbet–Epstein scheme in passing to fourth order because of the oscillating nature of the series; for Moller–Plesset theory errors are approximately halved. The relative energies as a function of modest geometry change about minima is about the same at third order as it is at fourth for most cases examined.     

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.     

Correlation effects in externally perturbed many-electron systems

Different methods for the calculation of the electron correlation contribution to atomic and molecular properties are analyzed and evaluated. The methods based on the self-consistent solution of the external perturbation problem are shown to offer several formal and computational advantages. The analysis of the correlation perturbation series for properties of many-electron systems indicates the importance of the appropriate treatment of unlinked diagrammatic contributions. In particular, the standard limited configuration interaction scheme based on single and double substitutions in the reference function may significantly suffer from the erratic treatment of unlinked clusters and needs to be corrected appropriately. The basis set choice for the calculation of highly accurate values of properties is also discussed. In order to circumvent the dimensionality problem the use of basis sets with explicit dependence on the external perturbation strength is recommended and methods for their choice and optimization are presented. A particular attention is paid to the many-body perturbation theory involving singly and doubly substituted intermediate states and based on the coupled Hartree–Fock solution for the one-electron perturbation problem. Different computational aspects of this method are discussed and compared with other techniques currently in use.     

On the internal rotations in p-cresol in its ground and first electronically excited states

The overall rotation and internal rotation of p-cresol (4-methyl-phenol) has been studied by comparison of the microwave spectrum with accurate ab initio calculations using the principal axis method in the electronic ground state. Both internal rotations, the torsions of the methyl and the hydroxyl groups relative to the aromatic ring, have been investigated. The internal rotation of the hydroxyl group can be approximately described as the motion of a symmetrical rotor on an asymmetric frame. For the methyl group it has been found that the potential barrier hindering its internal rotation is very small with the first two nonvanishing Fourier coefficients of the potential V(3) and V(6) in the same order of magnitude. Different splittings of b-type transitions for the A and E species of the methyl torsion indicate a top-top interaction between both internal rotors through the benzene ring. An effective coupling potential for the top-top interaction could be estimated. The hindering barriers of the hydroxyl and methyl rotation have been calculated using second-order Moller-Plesset perturbation theory and the approximate coupled-cluster singles-and-doubles model (CC2) in the ground state and using CC2 and the algebraic diagrammatic construction through second order in the first electronically excited state. The results are in excellent agreement with the experimental values.     

Diagram approach to group algebraic methods

Class sum theory, the duality with IRREP methods and tensor operators in the group algebra are discussed by generalizing the diagrammatic approach of conventional IRREP theory to include group label manipulation. Concepts such as invariant nodes and Jucys–Levinson–Vanagas reduction theorems generalize straightforwardly. The results are capable of unique simplification for certain nodes, when the group rearrangement theorem is useable or when a class sum is performed. A duality transformation (between IRREP –partner and class–element labels) emerges as an important concept.     

Global analysis of composite particles

The theory of vibrations of a composite particle when vibrational amplitudes are not constrained to be small according to the Eckart conditions is developed using the methods of differential topology. A global classical Hamiltonian appropriate for this system is given, and for the case of the molecular vibration–rotation problem, it is transformed into a global quantum Hamiltonian operator. It is shown that the zeroth-order term in the global Hamiltonian operator is identical to the Wilson–Howard Hamiltonian; higher-order terms are shown to give successively better approximations to the large amplitude problem. Generalized Eckart conditions are derived for the global classical Hamiltonian; the quantum equivalent of these conditions along with the quantum equivalent of the Eckart conditions are given. The spectrum of the global Hamiltonian operator is discussed and it is shown that the calculation of the vibration–rotation energy states of the system reduces to the same straight-forward procedure, the solution of a secular determinant, as was carried out for the Wilson–Howard Hamiltonian at a later time by Nielsen.     

Quasidegenerate coupled-cluster approach with hermitian model hamiltonian

The quasidegenerate coupled-cluster approach with a hermitian model Hamiltonian is constructed. This theory provides an efficient method for simultaneous construction and evaluation of the hermitian diagrammatic model Hamiltonian up to an arbitrary high order of perturbation theory and prescribed form of topology.     

Construction of an effective hamiltonian for open-shell molecular systems

An effective Hamiltonian for open-shell molecular systems is constructed. The unrestricted Hartree–Fock orbitals are applied as a basis set of one-particle functions. This effective Hamiltonian is determined as a simple product of the original total Hamiltonian and the spin annihilator. The second-quantization formalism and the Feynman–Goldstone diagrammatic technique are used. The resulting effective hamiltonian is composed of zero- to four-particle terms. A possibility of applying the nondegenerate diagrammatic perturbation theory constructed over this effective Hamiltonian is discussed.     

Calculation of single-beam two-photon absorption rate of lanthanides: effective operator method and perturbative expansion

Perturbative contributions to single-beam two-photon transition rates may be divided into two types. The first, involving low-energy intermediate states, require a high-order perturbation treatment, or an exact diagonalization. The other, involving high-energy intermediate states, only require a low-order perturbation treatment. We show how to partition the effective transition operator into two terms, corresponding to these two types, in such a way that a many-body perturbation expansion may be generated that obeys the linked cluster theorem and has a simple diagrammatic representation.