Response function theory of electron correlation

The full perturbation expansion for the response (or density—density correlation) function is examined in order to provide a useful general theory of excitation energies, oscillator strengths, dynamic polarizabilities, etc., that is more accurate than the random phase approximation. It is first shown how the formal partition of the diagrammatic version of the perturbation expansion into reducible and irreducible diagrams is generally useless as the latter category contains all the difficult terms which have heretofore resisted analysis in all but a haphazard form. It is then shown how the diagram for the response function can be partitioned into “correlated” and “uncorrelated” subsets. Restricting attention to the particle—hole blocks of the full response function, the “uncorrelated” diagrams desecribe the propagation of a particle—hole pair in an N-electron system where the particle and hole are each interacting with the remaining electrons but they are not interacting with each other. The “correlated” diagrams are those containing the hole—particle interactions, and, by defining a new class of reducible and irreducible diagrams, these are all summed to provide a perturbation expansion of the effective two-body hole—particle interaction that appears in the inverse of the response function. The “uncorrelated” diagrams are further partitioned into two sets, one of which is summed to all orders, while the other set is inverted in an order by order fashion. The final result presents a perturbation expansion for the inverse of the response function that is analogous to the Dyson equation for one-electron Green functions. Maintaining the perturbation expansion through first order for the inverse of the response function yields the eigenvalue equation of the familiar random phase approximation, while truncation at second order provides the most advanced theories that have been generated by the equations-of-motion method.     

Approximate fourth-order perturbation theory of the electron correlation energy

An approximate fourth-order expression for the electron correlation energy in the Møller–Plesset perturbation scheme is proposed. It takes into account all the contributions to the fourthorder energy neglecting only those of the triple-substituted determinants. It is size consistent and correct to fourth order for an assembly of isolated two-electron systems. Illustrative calculations are reported for a series of small molecules.     

Density functional theory treatment of electron correlation in the nuclear-electronic orbital approach

This paper presents the nuclear-electronic orbital density functional theory [NEO-DFT(ee)] method for including electron-electron correlation and nuclear quantum effects self-consistently in quantum chemical calculations. The NEO approach is designed to treat a relatively small number of nuclei quantum mechanically, while the remaining nuclei are treated classically. In the NEO-DFT(ee) approach, the correlated electron density is used to obtain the nuclear molecular orbitals, and the resulting nuclear density is used to obtain the correlated electron density during an iterative procedure that continues until convergence of both the nuclear and electronic densities. This approach includes feedback between the correlated electron density and the nuclear wavefunction. The application of this approach to bihalides and acetylene indicates that the nuclear quantum effects do not significantly impact the electron correlation energy, but the quantum nuclear energy is enhanced in the NEO-DFT(ee) B3LYP method. The excellent agreement of the NEO-DFT(ee)-optimized bihalide structures with the vibrationally averaged geometries from grid-based quantum dynamical methods provides validation for the NEO-DFT(ee) approach. Electron-proton correlation could be included by the development of an electron-nucleus correlation functional. Alternatively, explicit electron-proton correlation could be included directly into the NEO self-consistent-field framework with Gaussian-type geminal functions.     

A two‐scale approach to electron correlation in multiconfigurational perturbation theory

We present a new approach for the calculation of dynamic electron correlation effects in large molecular systems using multiconfigurational second‐order perturbation theory (CASPT2). The method is restricted to cases where partitioning of the molecular system into an active site and an environment is meaningful. Only dynamic correlation effects derived from orbitals extending over the active site are included at the CASPT2 level of theory, whereas the correlation effects of the environment are retrieved at lower computational costs. For sufficiently large systems, the small errors introduced by this approximation are contrasted by the substantial savings in both storage and computational demands compared to the full CASPT2 calculation. Provided that static correlation effects are correctly taken into account for the whole system, the proposed scheme represent a hierarchical approach to the electron correlation problem, where two molecular scales are treated each by means of the most suitable level of theory. © 2014 Wiley Periodicals, Inc.     

The self-interaction error and the description of non-dynamic electron correlation in density functional theory

The self-interaction error (SIE) plays a central role in density functional theory (DFT) when carried out with approximate exchange-correlation functionals. Its origin, properties, and consequences for the development of standard DFT to a method that can correctly describe multi-reference electron systems by treating dynamic and non-dynamic electron correlation on an equal footing, is discussed. In this connection, the seminal work of Colle and Salvetti on wave function-based correlation functionals that do no longer suffer from a SIE is essential. It is described how the Colle–Salvetti correlation functional is an anchor point for the derivation of a functional multi-reference DFT method.     

Geometry optimizations with explicit inclusion of electron correlation

Automatic techniques for geometry optimization are applied in conjunction with configuration interaction and perturbation treatments of electron correlation. The computational effort and numerical accuracy of the optimizations are discussed, as well as problems with approximate correlation methods concerning the continuity of the potential surface. The optimized geometries of fourteen molecules obtained with different correlation treatments (MNDO SCF MOs) are compared. The configuration interaction results are reproduced satisfactorily by simple perturbation approaches. The largest change of the optimized SCF geometry is found for hydrogen peroxide.     

Systematic sequences of even-tempered Gaussian primitives in electron correlation studies using many-body perturbation theory

The use of systematic sequences of even-tempered Gaussian primitive functions in electron correlation studies using diagrammatic many-body perturbation theory is examined. The s limit electronic energy of the Be atom and the sp limit energy of the Ne atom have been computed as examples. The use of the Hartree extrapolation procedure to obtain empirical upper bounds for the basis set limit is investigated. The empirical lower bound for the basis set limit suggested by Schmidt and Ruedenberg is examined for calculations which include electron correlation.     

Harmonic electron correlation operator

An appealing way to model electron correlation within the single determinant wave function formalism is through the expectation value of a linear two-electron operator. For practical reasons, it is desirable for such an operator to be universal, i.e., not depend on the positions and types of nuclei in a molecule. We show how a perturbation theory applied to a hookium atom provides for a particular form of a correlation operator, hence called the harmonic correlation operator. The correlation operator approach is compared and contrasted to the traditional ways to describe electron correlation. To investigate the two-electron approximation of this operator, we apply it to many-electron hookium systems. To investigate the harmonic approximation, we apply it to the small atomic systems. Directions of future research are also discussed.     

The electron correlation cusp

The Kais function is an exact solution of the Schrödinger equation for a pair of electrons trapped in a parabolic potential well with r ₁₂ ^?1 electron-electron interaction. Partial wave analysis (PWA) of the Kais function yields E _L = E + C₁(L + \-C ^?1 ₂)^?3 + O(L ^?5) where E is the exact energy and E _L the energy of a renormalized finite sum of partial waves omitting all waves with angular momentum ? > L. Slight rearrangement of an earlier result by Hill shows that the corresponding full CI energy differs from E _L only by terms of order O(L ^?5) with FCI values of C ₁ and \-C ^?1 ₂ identical to PWA values. The dimensionless \-C ₂ parameter is weakly dependent upon the size of the physical system. Its value is 0.788 for the Kais function, and 0.893 for the less diffuse helium atom, and approaches \-C ₂→ 1 in the limit of an infinitely compact charge distribution. The ?th energy increment satisfies an approximate virial theorem which becomes exact in the high ? limit. This analysis, formulated to facilitate use of the Maple system for symbolic computing, lays the mathematical ground work for subsequent studies of the electron correlation cusp problem. The direction of future papers in this series is outlined.     

A generalized any particle propagator theory: Assessment of nuclear quantum effects on electron propagator calculations

In this work we propose an extended propagator theory for electrons and other types of quantum particles. This new approach has been implemented in the LOWDIN package and applied to sample calculations of atomic and small molecular systems to determine its accuracy and performance. As a first application of the method we have studied the nuclear quantum effects on electron ionization energies. We have observed that ionization energies of atoms are similar to those obtained with the electron propagator approach. However, for molecular systems containing hydrogen atoms there are improvements in the quality of the results with the inclusion of nuclear quantum effects. An energy term analysis has allowed us to conclude that nuclear quantum effects are important for zero order energies whereas propagator results correct the electron and electron-nuclear correlation terms. Results presented for a series of n-alkanes have revealed the potential of this method for the accurate calculation of ionization energies of a wide variety of molecular systems containing hydrogen nuclei. The proposed methodology will also be applicable to exotic molecular systems containing positrons or muons.     

Localizability of dynamic electron correlation

The convergence of the intrapair correlation energy for a localized internal orbital is investigated as the virtual subspace is enlarged. At variance with previous investigations of this kind, the virtual subspace is represented in atomic orbitals. This allows to define spatial relations between the orbitals involved. Typically, over 98% of the pair correlation energy is recovered by a small local basis set, consisting of the valence orbitals of the atoms with which the electron pair is associated. This opens the possibility of an efficient Cl procedure based on localized pairs.     

Mode-coupling theory for multiple-time correlation functions of tagged particle densities and dynamical filters designed for glassy systems

The theoretical framework for higher-order correlation functions involving multiple times and multiple points in a classical, many-body system developed by Van Zon and Schofield [Phys. Rev. E 2002, 65, 011106] is extended here to include tagged particle densities. Such densities have found an intriguing application as proposed measures of dynamical heterogeneities in structural glasses. The theoretical formalism is based upon projection operator techniques which are used to isolate the slow time evolution of dynamical variables by expanding the slowly evolving component of arbitrary variables in an infinite basis composed of the products of slow variables of the system. The resulting formally exact mode-coupling expressions for multiple-point and multiple-time correlation functions are made tractable by applying the so-called N-ordering method. This theory is used to derive for moderate densities the leading mode coupling expressions for indicators of relaxation type and domain relaxation, which use dynamical filters that lead to multiple-time correlations of a tagged particle density. The mode coupling expressions for higher order correlation functions are also successfully tested against simulations of a hard sphere fluid at relatively low density.     

Soliton states in polyacetylene chains with consideration of electron correlation

The stationary states of a longtrans-(CH)_N chain have been considered in the ppp-Peierls model; electron correlation was taken into account with the aid of a wave function constructed from variable geminals. The states of a chain with a pair of neutral solitons, which have local minima of the total energy lying 1.6 eV above the ground state, have been determined numerically by finding a self-consistent approximation of the electronic system and the lattice deformations. The soliton pair is stable only when the kinks in the lattice are at a sufficient distance, and it is stabilized significantly by electron correlation. The influence of the correlation of the electrons and the kinks in the lattice on the position of excited levels of different symmetry and alternant character has been considered.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 22, No. 4, pp. 385–393, July–August, 1986.We thank A. A. Ovchinnikov for some useful discussion.     

On the theory of electron correlation in atoms and molecules: Relation between cluster expansion theory and the correlated wave functions method

A relation between the cluster expansion theory of many electron wave functions and the correlated wave functions method is established. In this way, the theoretical basis of the method is elucidated and the approximations involved in its application become apparent. General forms of the correlated wave function, differing in certain important respects from that form usually assumed, are derived.     

Polarization potential in three particle theory

The exact analytical expression for the electric dipole polarizability of the two-particle charged bound system with a short-range interaction of an arbitrary form is obtained within the three-particle theory.     

On electron correlation in NaCl2

This article reports the intrapair and interpair electron correlation energies of the radical NaCl₂. The total interpair correlation energy dominates. Hence, the interpair electron correlation energy must be considered in building models for correcting computed correlation energies. The 6-311+G* basis set recovers only 32% of the total estimated correlation energy and 44% of this amount came from the core electrons. © 1995 John Wiley & Sons, Inc.     

Structural and electronic properties of infinite cis and trans polyenes: Perturbation theory of electron correlation effects

The geometrical structure of five infinite polyene models (equidistant and alternating trans, equidistant cis, cis-transoid, and trans-cisoid) have been optimized using three different atomic basis sets at the Hartree–Fock level and by including electron correlation effects within the second order of Møller–Plesset perturbation theory. The single-particle energy bands have also been corrected for correlation effects applying the electron polaron method. Bond alternation has always reduced the energy (both for trans and cis) polyenes but this stabilization energy has decreased with an increasing basis set and has saturated about 3 mH for trans and trans-cisoid and about 7.5 mH for the cis-transoid model. On the absolute energy scale, however, the alternating trans structure has always turned out to be more stable than either the cis-transoid or the trans-cisoid one. The energetic order is in all cases trans < cis-transoid < trans-cisoid. Using the largest basis set with correlation, the corresponding energy differences are ΔE(trans → cis-transoid) = 1.69 mH and ΔE(trans → trans-cisoid) = 6.12 mH, respectively. The HF values of the valence-band widths vary in the region of 4–6 eV for the cis models and about 7 eV for the trans one. Correlation reduced them by about 20%, and the centers of the bands were shifted upward by about 2 eV due to self-energy renormalization. The best values for the electron polaron valence bandwidths are 6.4, 4.2, 3.3, and 5.2 eV for the alternating trans, trans-cisoid, cis-transoid, and equidistant cis models, respectively. The vertical ionization potentials of the above four structures are 4.80, 5.47, 5.71, and 4.18 eV, respectively. The conduction bands have a width of 6–8 eV at the HF level (except equidistant cis with 4.6 eV for the larger basis sets) that will be reduced by about 15–20% due to polaron formation. The bands are shifted in this case uniformly downward by about 2–3 eV due to correlation. The best values obtained for the conduction bandwidths are 4.7, 6.4, 5.6, and 3.7 eV, respectively. These band shifts reduce the HF value of the fundamental gap by several eV's for all models. For the largest basis set, the ΔE_gap values change from 6.6, 6.9, 7.7, and 5.3 eV (HF results for the alternating trans, trans-cisoid, cis-transoid, and equidistant cis models, respectively) to 2.7, 3.2, 4.5, and 2 eV, respectively, at the correlated level.