Electron Correlation In Molecules Research Articles

Periodic Bootstrap Embedding.

Bootstrap embedding (BE) is a recently developed electronic structure method that has shown great success at treating electron correlation in molecules. Here, we extend BE to treat surfaces and solids where the wave function is represented in periodic boundary conditions using reciprocal space sums (i.e., k-point sampling). The major benefit of this approach is that the resulting fragment Hamiltonians carry no explicit dependence on the reciprocal space sums, allowing one to apply traditional nonperiodic electronic structure codes to the fragments even though the entire system requires careful consideration of periodic boundary conditions. Using coupled cluster singles and doubles (CCSD) as an example method to solve the fragment Hamiltonians, we present minimal basis set CCSD-in-HF results on 1D conducting polymers. We show that periodic BE-CCSD can typically recover ∼99.9% of the electron correlation energy. We further demonstrate that periodic BE-CCSD is feasible even for complex donor-acceptor polymers of interest to organic solar cells─despite the fact that the monomers are sufficiently large that even a Γ-point periodic CCSD calculation is prohibitive. We conclude that BE is a promising new tool for applying molecular electronic structure tools to solids and interfaces.

Dynamical electron correlation and the chemical bond. II. Recoupled pair bonds in the a4Σ- states of CH and CF.

We extended our studies of the effect of dynamical electron correlation on the covalent bonds in the AH and AF series (A = B-F) to the recoupled pair bonds in the excited a4Σ- states of the CH and CF molecules. Dynamical correlation is energetically less important in the a4Σ- states than in the corresponding X2Π states for both molecules, which is reflected in smaller changes in bond energies (De). Changes in the equilibrium bond distance (Re) and vibrational frequency (ωe), on the other hand, are influenced by the changes in the slope and curvature of the dynamical electron correlation energy as a function of the internuclear distance R, EDEC(R). In the CH(a4Σ-) state, these changes are much smaller than in the CH(X2Π) state, but in the CF(a4Σ-) state, they are larger, reflecting a significant difference in the shapes of EDEC(R) curves. At large R, the shape of EDEC(R) curves for covalent and recoupled pair bonds is similar although different in magnitude. For the CH(a4Σ-) state, EDEC(R) has a minimum at R = Re + 0.72 Å as the orbitals associated with the formation of the recoupled pair bond switch places. EDEC(R) for the CF(a4Σ-) state decreases continuously throughout the bound region of the potential energy curve because the dynamical correlation energy associated with the electrons in the lone pair orbitals is increasing. These results support our earlier conclusion that we still have much to learn about the nature of dynamical electron correlation in molecules.

Atom-Based Bootstrap Embedding For Molecules.

Recent developments in quantum embedding have offered an attractive approach to describing electron correlation in molecules. However, previous methods such as density matrix embedding theory (DMET) require rigid partitioning of the system into fragments, which creates significant ambiguity for molecules. Bootstrap embedding (BE) is more flexible because it allows overlapping fragments, but when done on an orbital-by-orbital basis, BE introduces ambiguity in defining the connectivity of the orbitals. In this Letter, we present an atom-based fragment definition that significantly augments BE's performance in molecules. The resulting method, which we term atom-based BE, is very effective at recovering valence electron correlation in moderate-sized bases and delivers near-chemical-accuracy results using extrapolation. We anticipate atom-based BE may lead to a low-scaling and highly accurate approach to electron correlation in large molecules.

Density matrix embedding in an antisymmetrized geminal power bath.

Density matrix embedding theory (DMET) has emerged as a powerful tool for performing wave function-in-wave function embedding for strongly correlated systems. In traditional DMET, an accurate calculation is performed on a small impurity embedded in a mean field bath. Here, we extend the original DMET equations to account for correlation in the bath via an antisymmetrized geminal power (AGP) wave function. The resulting formalism has a number of advantages. First, it allows one to properly treat the weak correlation limit of independent pairs, which DMET is unable to do with a mean-field bath. Second, it associates a size extensive correlation energy with a given density matrix (for the models tested), which AGP by itself is incapable of providing. Third, it provides a reasonable description of charge redistribution in strongly correlated but non-periodic systems. Thus, AGP-DMET appears to be a good starting point for describing electron correlation in molecules, which are aperiodic and possess both strong and weak electron correlation.

Compact two-electron wave function for bond dissociation and Van der Waals interactions: a natural amplitude assessment.

Electron correlations in molecules can be divided in short range dynamical correlations, long range Van der Waals type interactions, and near degeneracy static correlations. In this work, we analyze for a one-dimensional model of a two-electron system how these three types of correlations can be incorporated in a simple wave function of restricted functional form consisting of an orbital product multiplied by a single correlation function f (r12) depending on the interelectronic distance r12. Since the three types of correlations mentioned lead to different signatures in terms of the natural orbital (NO) amplitudes in two-electron systems, we make an analysis of the wave function in terms of the NO amplitudes for a model system of a diatomic molecule. In our numerical implementation, we fully optimize the orbitals and the correlation function on a spatial grid without restrictions on their functional form. Due to this particular form of the wave function, we can prove that none of the amplitudes vanishes and moreover that it displays a distinct sign pattern and a series of avoided crossings as a function of the bond distance in agreement with the exact solution. This shows that the wave function ansatz correctly incorporates the long range Van der Waals interactions. We further show that the approximate wave function gives an excellent binding curve and is able to describe static correlations. We show that in order to do this the correlation function f (r12) needs to diverge for large r12 at large internuclear distances while for shorter bond distances it increases as a function of r12 to a maximum value after which it decays exponentially. We further give a physical interpretation of this behavior.

Probing electron correlation in molecules via quantum fluxes

We present quantum simulations of a vibrating hydrogen molecule H-2 and address the issue of electron correlation. After appropriately setting the frame and the observer plane, we were able to determine precisely the number of electrons and nuclei which actually flow by evaluating electronic and nuclear fluxes. This calculation is repeated for three levels of quantum chemistry, for which we account for no correlation, Hartree-Fock, static correlation, and dynamic correlation. Exciting each of these systems with the same amount of energy, we show that the electron correlation can be revealed with the knowledge of quantum fluxes. This is evidenced by a clear sensitivity of these fluxes to electron correlation. In particular, we find that this correlation remarkably enhances more electronic yield than the nuclear one. It turns out that less electrons accompany the nuclei in Hartree-Fock than in the correlation cases.

Electron correlation methods based on the random phase approximation

In the past decade, the random phase approximation (RPA) has emerged as a promising post-Kohn–Sham method to treat electron correlation in molecules, surfaces, and solids. In this review, we explain how RPA arises naturally as a zero-order approximation from the adiabatic connection and the fluctuation-dissipation theorem in a density functional context. This is contrasted to RPA with exchange (RPAX) in a post-Hartree–Fock context. In both methods, RPA and RPAX, the correlation energy may be expressed as a sum over zero-point energies of harmonic oscillators representing collective electronic excitations, consistent with the physical picture originally proposed by Bohm and Pines. The extra factor 1/2 in the RPAX case is rigorously derived. Approaches beyond RPA are briefly summarized. We also review computational strategies implementing RPA. The combination of auxiliary expansions and imaginary frequency integration methods has lead to recent progress in this field, making RPA calculations affordable for systems with over 100 atoms. Finally, we summarize benchmark applications of RPA to various molecular and solid-state properties, including relative energies of conformers, reaction energies involving weak and covalent interactions, diatomic potential energy curves, ionization potentials and electron affinities, surface adsorption energies, bulk cohesive energies and lattice constants. RPA barrier heights for an extended benchmark set are presented. RPA is an order of magnitude more accurate than semi-local functionals such as B3LYP for non-covalent interactions rivaling the best empirically parametrized methods. Larger but systematic errors are observed for processes that do not conserve the number of electron pairs, such as atomization and ionization.

The nature of electron correlation in molecules

The nature of electron correlation in molecules

Probing electron correlations in molecules by two-dimensional coherent optical spectroscopy.

The nonlinear optical signal generated in phenol by three femtosecond pulses with wavevectors k1, k2, and k3 in the phase-matching direction k1 + k2 - k3 is simulated. This two-dimensional coherent spectroscopy (2DCS) signal has a rich pattern containing information on double-excitation states. The signal vanishes for uncorrelated electrons due to interference among quantum pathways and, thus, provides direct signatures of correlated many-electron wavefunctions. This is illustrated by the very different 2DCS signals predicted by two levels of electronic structure calculations: state-averaged complete active space self-consistent field (SA-CASSCF) and multistate multiconfigurational second-order perturbation theory (MS-CASPT2).

Multi‐state multi‐reference Møller–Plesset second‐order perturbation theory for molecular calculations

AbstractThis work presents multi‐state multi‐reference Møller–Plesset second‐order perturbation theory as a variant of multi‐reference perturbation theory to treat electron correlation in molecules. An effective Hamiltonian is constructed from the first‐order wave operator to treat several strongly interacting electronic states simultaneously. The wave operator is obtained by solving the generalized Bloch equation within the first‐order interaction space using a multi‐partitioning of the Hamiltonian based on multi‐reference Møller–Plesset second‐order perturbation theory. The corresponding zeroth‐order Hamiltonians are nondiagonal. To reduce the computational effort that arises from the nondiagonal generalized Fock operator, a selection procedure is used that divides the configurations of the first‐order interaction space into two sets based on the strength of the interaction with the reference space. In the weaker interacting set, only the projected diagonal part of the zeroth‐order Hamiltonian is taken into account. The justification of the approach is demonstrated in two examples: the mixing of valence Rydberg states in ethylene, and the avoided crossing of neutral and ionic potential curves in LiF. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Charge transfer driven by electron correlation: A non-Dyson propagator approach

A hole charge created in a molecular system, for instance, by ionization, can migrate through the system solely driven by electron correlation. This charge transfer due to electron correlation is referred to as charge migration. We introduce in this work a new ab initio method analyzing charge migration due to electron correlation in molecules. This method, a third-order "non-Dyson" propagator approach, aims in the long run, in particular, at the calculation of charge migration in relatively large molecules such as oligopeptides. First results of the new non-Dyson method are compared with a previously used propagator approach.

Explication and revision of generalized Van Vleck perturbation theory for molecular electronic structure

A revision of second-order Generalized Van Vleck Perturbation Theory (GVVPT2) for the description of dynamic electron correlation in molecules is presented. It is shown that the basic formulas of the suggested method are well-defined approximations to the theoretically carefully constructed self-consistent quasidegenerate perturbation theory. Furthermore, it is shown that nonlinear responses to the perturbations can be obtained by explicit formulas. The revised GVVPT2 makes active use of the recently introduced concept of macroconfigurations, whereby vast numbers of null Hamiltonian matrix elements are prescreened with minimal computational cost and the remainders are organized for facile computation by Table-CI-like methodology. Moreover, use of macroconfigurations allows the efficient use of incomplete model spaces, which extends drastically the applicability of the method. Representative calculations on model systems studied previously with the original formulation show close agreement and on additional model systems show the wide applicability of the revised formulation.

Explicitly correlated SCF study of anharmonic vibrations in (H2O)2

AbstractModeling solvation in high‐pressure liquid chromatography (HPLC) requires calculation of anharmonic vibrational frequencies of solvent clusters for a statistical partition function. An efficient computational method that includes electron correlation is highly desirable for large clusters. A modified version of the “soft Coulomb hole” method of Chakravorty and Clementi has recently been implemented in a Gaussian‐lobe‐orbital (GLO) program (PCLOBE) to include explicit electron–electron correlation in molecules. The soft Coulomb hole is based on a modified form of Coulomb's law: An algorithm has been developed to obtain the parameter “w” from a polynomial in the effective scaling of each primitive Gaussian orbital relative to the best single Gaussian of the H1s orbital. This method yields over 90% of the correlation energy for molecules of low symmetry for which the original formula of Chakravorty and Clementi does not apply. In this work, all the vibrations of the water dimer are treated anharmonically. A quartic perturbation of the harmonic vibrational modes is constrained to be equal to the exact Morse potential eigenvalue based on a three‐point fit. This work evaluates the usefulness of fitting a Morse potential to a hydrogen bond vibrational mode and finds it to be slightly better than using MP2 vibrational analysis for this important dimer. A three‐point estimate of the depth, De, of a Morse potential leads to a correction formula for anharmonicity in terms of the perturbed harmonic frequency: When scaled by 0.9141, the harmonic Morse method leads to essentially the same results as scaling the BPW91 local density method by 0.9827. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Explicitly correlated SCF study of small hydrides

AbstractThe “soft Coulomb hole” method of Chakravorty and Clementi has been implemented in a Gaussian lobe‐orbital (GLO) program to include explicit electron–electron correlation in molecules according to a modified form of Coulomb's law in a program for workstations and personal computers (PCLOBE): f(R12)=(1−exp (−w(R12)2))/R12, w=scaling parameter. Twice as many two‐electron integrals must be calculated compared to the usual Hartree–Fock–Roothaan algorithm, but this “correlated self‐consistent field (SCF)” method may be embedded within well‐known SCF computer codes and yields more than 90% of the correlation energy in run times less than twice that of the restricted Hartree–Fock (RHF) method. A two‐parameter model of correlation is calibrated on HF and LiH using the 6‐311G** basis as Gaussian lobe orbitals. The quantum Monte Carlo energies of LiH and HF were fitted to within 150 cal/mol, but the scaling model for first‐row atoms is less good for H2. This work shows promise as a fast way to modify the Hartree–Fock–Roothaan method for accuracy approaching quantum Monte Carlo methods. This proof‐of‐concept work offers extension to modeling solvation properties of water solutions in high‐pressure liquid chromatography (HPLC). © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

On the possible role of the nitrogen atom in living matter

Molecules of NH 3 are capable of emitting stimulated radiation (MASER). Their organic derivates could have a similar effect, as is suggested by the new results published in Electron Correlations in Molecules and Solids [P. Fulde, Electron Correlations in Molecules and Solids, Springer-Verlag, Berlin, Heidelberg, 1993].

The Biological Universe , reviewed by J. N. Tatarewicz * Electron Correlation in Molecules and Condensed Phases , S. Das Sarma * Vignettes * Browsings

<i>The Biological Universe</i> , reviewed by J. N. Tatarewicz * <i>Electron Correlation in Molecules and Condensed Phases</i> , S. Das Sarma * Vignettes * <i>Browsings</i>

Electron Correlations in Molecules and Solids

Electron Correlations in Molecules and Solids

Natural bond orbitals in multiconfigurational expansions: Local treatment of electron correlation in molecules

We describe a localized treatment of electron correlation in terms of complete active-space wave functions derived from natural bond orbitals, closely related to the localized transferable units of molecular structure. We find that in many cases the procedure leads to solutions which retain the essentially localized character of their ‘‘parent’’ orbitals. Such localized solutions lead to an extremely selective treatment of electron correlation in molecules, allowing one to readily extend the concepts of electron correlation in diatomic molecules (‘‘left-right,’’ ‘‘in-out,’’ ‘‘angular,’’ etc.) to the general polyatomic case using a familiar chemical language. We illustrate the procedure with applications to various diatomic (Li2, CO), triatomic (H2O, symmetric and unsymmetric dissociation), and polyatomic (formaldehyde ground and excited states; malonaldehyde) species, in order to demonstrate the important advantages of this procedure with respect to numerical behavior (improved convergence, reduced dimensionality, easy implementation with standard program packages) and conceptual utility (selectivity, transferability, description in a language common to all chemists).

Electron Correlations in Molecules and Solids

Electron Correlations in Molecules and Solids

P. Fulde. Electron Correlations in Molecules and Solids. Springer Series in Solid‐State Sciences 100, Springer‐Verlag, Berlin u. a., 1991; 422 Seiten, 127 Abbildungen, DM 98,00, ISBN 3–540–53623–X

Crystal Research and TechnologyVolume 27, Issue 6 p. 850-850 Book Review P. Fulde. Electron Correlations in Molecules and Solids. Springer Series in Solid-State Sciences 100, Springer-Verlag, Berlin u. a., 1991; 422 Seiten, 127 Abbildungen, DM 98,00, ISBN 3–540–53623–X W. John, W. JohnSearch for more papers by this author W. John, W. JohnSearch for more papers by this author First published: 1992 https://doi.org/10.1002/crat.2170270620AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume27, Issue61992Pages 850-850 RelatedInformation