On the concept of “Partitions” in the perturbation theory of n-electron systems

This paper deals with the perturbation theory of an n-electron Hamiltonian of the general form H = ∑ⁿ ?(i) + λ∑ⁿ g(i, j) = H (f, g; n). In comparison to the Brueckner–Goldstone diagrammatic perturbation theory, we adopt the more general standpoint of admitting, for the construction of an n-particle state, component states of 1, 2, 3, and more particles [O. Sinanoglu, Phys. Rev. 122 , 493 (1961) and C. D. H. Chisholm and A. Dalgarno, Proc. R. Soc. (London) Sec. A 292 , 264 (1966)]. We show that this leads to the concept of a “partition” of a perturbational eigenstate (or energy) of H. A “partition” is a natural decomposition which: (i) is finite; (ii) relates the eigenvalue problem of the system H = H (f, g; n) to those of certain subsystems H (f, g; n₁)(n₁ < n); (iii) uses “nonseparable” components. We domonstrate (under the preliminary assumption of “strict” nondegeneracy) the second-order energy to possess a “partition.” The components therein are second-order energies of two- and three-particle states. The proof uses an extension of Racah's concept of the fractional-parentage expansion.     

Comparison of high-order many-body perturbation theory and configuration interaction for H2O

Diagrammatic many-body perturbation theory, coupled with a recursive computational procedure, is employed to obtain the correlation energy of H₂O within a 39-STO basis set by evaluating all double-excitation diagrams through twelfth order without any approximations. This provides, for the first time, the complete double-excitation diagrams contributions to the correlation energy, which is ?0.28826 hartree, compared with a correlation energy of ?0.27402 hartree obtained from a configuration interaction calculation which includes all double excitations. The difference of 0.0142 hartree includes the “size consistency” correction to the all-double-excitations CI energy, due to the “pathological” unliked-diagram terms remaining in that result, but also involves certain fourth- and higher-order rearrangement diagrams. Contrary to conventional belief, the unshifted, or Møller-Plesset partitioning of the hamiltonian provides a much more rapid convergence of the perturbation series that does the shifted, or Epstein-Nesbet partitioning. In both cases. Padé approximants enhance the convergence of the series considerably. A simple variation-perturbation scheme based on the first-order MBPT wavefunction is sufficient to provide 97.5% of the all-doubles CI correlation energy.     

Site-Directed Surface Derivatization of MCM-41: Use of High-Resolution Transmission Electron Microscopy and Molecular Recognition for Determining the Position of Functionality within Mesoporous Materials

The “staining” cluster–crown compound [Ru₆C(CO)₁₄(η⁶-C₆H₄C₁₀H₂₀O₆)] verifies that the internal walls of mesoporous silica MCM-41 may be selectively functionalized with propylammonium groups (see picture). By the use of high-resolution transmission electron microscopy the presence and position of the cluster and also of the functional groups may be directly determined.     

Electron correlation effects in the 4f14 shell

This paper serves a twofold purpose. First, Löwdin's inner projection in both nonperturbative and perturbative forms is applied to the quartic anharmonic oscillator. Inner projection with perturbation theory yields rational approximations to Brillouin–Wigner-type perturbation expansions. These lower bounds are compared with [N ? 1, N] Padé approximants to the Rayleigh–Schrödinger perturbation series for this problem. These Padés are also expressible as the even convergents, w_2N, of a Stieltjes-type continued fraction. The latter representation has certain advantages with respect to its Padé counterpart. Inner projection without perturbation theory provides significantly better results than the perturbative version. The application of inner projection techniques to a perturbed hydrogen atom is not straightforward. The usual problems associated with the continuum spectrum of hydrogen are present. By means of a nonunitary “tilting” transformation associated with the Lie group SO(4, 2), these problems may be bypassed. In the SO(4, 2)-reformulated eigenvalue problem, a reinterpretation of the basic variables, as developed by Silverstone and Moats, yields a new Hamiltonian that permits direct use of the inner projection method. This method has been applied to the ground state of the hydrogen atom in a magnetic field, using both four- and eight-dimensional basis manifolds. This represents the first application of inner projection to this problem.     

Applications of the MBPT in the localized representation

Diagrammatic formulation of the MBPT is applied when the occupied and the virtual canonical orbitals are separately localized by unitary transformations. In this localized representation, due to the off-diagonal Fock matrix elements, the perturbation operator contains extra terms generating the so-called localization corrections. These corrections enter the perturbation energy in third and higher orders. Their magnitude depends on the type of localization, but they represent only a small fraction of the canonical corrections. The calculation of the localization corrections, however, does not need a significant amount of extra computer time. It is shown that by introducing an “order of neighborhood” local and nonlocal effects of the electron correlation can be separated and the contribution of the nonlocal effects can be neglected to a good approximation. Ab initio calculations have been carried out for the normal saturated hydrocarbons: C_2n+1H_4n+4 and for the all-trans conjugated polyenes C_2n+2H_2n+4. As to the ratio of the local and nonlocal corrections, it is shown that there is only a quantitative difference for these two kinds of systems (strongly or weakly localizable). Neglecting nonlocal effects, considerable amount of computer time can be saved.     

Wellenmechanische Absolutrechnungen an Molekülen und Atomsystemen mit der SCF?MO?LC(LCGO) Methode. V. Bindungsbeteiligung der Inneren Elektronen des C-Atoms

Using the results of ab initio calculations, by comparison of the “1s orbital energies” of the C atom in the compounds C₆H₆, C₅H, C₃H₆ (cyclopropane), C₂H₄ as well as at the C atom itself the bond electrons were found to have a significant influence on the inner electrons. The reason for this is pointed out and an explanation is given. The connection between the bonding and this “1s orbital energy” change as well as the importance of this result for quantum-chemical “models” is discussed.     

Experimentally determined temperature–concentration phase diagrams of monodisperse alkanes with chains containing between 100 and 200 carbons

The temperature–concentration phase (T–c) diagrams of the uniform n-alkanes C₁₀₂H₂₀₆, C₁₂₂H₂₄₆, C₁₆₂H₃₂₆, and C₁₉₈H₃₉₈ in toluene have been determined for solution concentrations in the range 0.1 to 6% (w/w). The shorter alkanes display a “classical” behavior with the expected, strong dependence of dissolution temperature on solution concentration. The longest alkane displays a very different, “polymeric” type behavior with a concentration independent dissolution temperature (for both extended and folded chain crystals). It is argued that no current theory of polymer dissolution is able to explain this behavior. It is suggested that a locally higher concentration occurs when molecules are partially attached to a crystal either during crystallization or dissolution, and that this increased local concentration accounts for the independence of dissolution temperature on the global concentration. There are some small variations in the dissolution temperature of crystals of the same thickness grown at the same concentration, but at different temperatures. These are ascribed to differences in the stacking of the separate layers. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 3188–3200, 1999     

Supracluster Assembly of Archimedean Cages with 72 Hydrogen Bonds for the Aldol Addition Reaction

Clusters that can be experimentally precisely characterized and theoretically accurately calculated are essential to understanding the relationship between material structure and function. Here, we propose the concept of “supraclusters”, which aim to connect “supramolecules” and “suprananoparticles” as well as reveal the unique assembly behavior of “supraclusters” with nanoparticle size at the molecular level. The implementation of supraclusters is full of challenges due to the difficulty in satisfying the ordered connectivity of clusters due to their abundant and dispersed hydrogen bonding sites. By solvothermal synthesis under a high catechol (H₂CATs) content, we successfully isolated a series of triangular {Al₆M₃} cluster compounds possessing brucite-like structural features. Interestingly, eight {Al₆M₃} clusters form 72-fold strong hydrogen bonding truncatedhexahedron Archimedean {Al₆M₃}₈ supracluster cage (abbreviated as H-tcu ). Surprisingly, the solution stability of the H-tcu was further proved by electrospray ionization mass spectrometry (ESI-MS) characterization. Therefore, it is not difficult to explain the reason for assembly of H-tcu into edge-directed and vertex-directed isomers. These porous supraclusters can be obtained by scale-up synthesis and exhibit a noticeable catalysis effect towards the condensation of acetone and p-nitrobenzaldehyde. As an intermediate state of supramolecule and suprananoparticle, the supracluster assembly can enrich the cluster chemistry and bring new structural types.     

On the HALA effect in the NMR carbon shielding constants of the compounds containing heavy p‐elements

The heavy atom (HA) effect on the NMR isotropic carbon shielding constants is computationally investigated in the series of model ethanes, ethylenes, and acetylenes, C_βH₃? C_αH₂? XH_n, C_βH₂? C_αH? XH_n, C_βH?C_α? XH_n (n = 0, 1, 2, or 3 depending on X), where X covers p‐elements in the 13–17 groups of the 3–6 periods in as many as 60 compounds. Compounds under study provide diverse bonding situations for the α‐ and β‐carbons, which are characterized by the consecutive increase of the s‐character of the C_β? C_α and C_α? X bonds, being one of the factors influencing spin‐orbit part of the HA on light atom effect (SO‐HALA). The “chalcogen dependence,” “pnictogen dependence,” “tetrel dependence,” and “triel dependence” are established for the 16th, 15th, 14th, and 13th groups, respectively. A well‐known “normal halogen dependence” for the ¹³C NMR chemical shifts, established much earlier for the compounds containing 17th group elements, also revealed itself in all three series under investigation. The dependence of the spin‐orbit effects size depending on the number of the lone electron pairs (LEPs) on HA X has also been investigated. The comparison of theoretical ¹³C NMR chemical shifts with experiment is performed for three representative tellurides. The HALA effect in this series has been shown to be strongly dependent on the number of tellurium LEPs.     

Bond types of molecular orbitals and the photoelectron spectrum

The vibrational structures of the photoelectron spectra for diatomic molecules can be accounted for in terms of the slope of the orbital energy curve in the conventional correlation diagram with respect to internuclear distance. The vibrational structures of the photoelectron spectra for simple polyatomic molecules HCN, C₂H₂, and AH₂ type of hydrides can also be accounted for in terms of the slopes of the orbital energy curves in the correlation diagrams with respect to angles, as well as distances. Among all correlation diagrams, the slopes in the distance correlation diagram are related to the criterion for bond type—the positive for “bonding,” the negative for “antibonding,” while slopes with small magnitudes for “nonbonding.” The Fock matrix elements within the bond orbital basis provide heuristic and systematic rationalization of the slopes for the orbital energy curves. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 53–65, 2001     

Valence bond theoretical study for chemical reactivity

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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.     

Perturbation theory for open-shell systems: Simulation of the UHF states by means of the perturbed RHF wave functions

Second-, third-, and selected fourth-order contributions to the correlation energy were calculated for a series of simple open-shell systems by means of the previously developed double-perturbation theory in the restricted MO formalism. It was found possible to assign some of the diagrams to self-consistency effects and to approximate in this way the E – E energy difference. A comparison is made with a more rigorous approach, in which the UHF ground-state wave function is expressed as a first-order perturbation expansion based on the RHF reference wave function. Distinguishing between “self-consistency” and “correlation” diagrams for open-shell systems in the RHF formulation represents a special case of a more general problem met in any double-perturbation treatment, such as, e.g., treatments of systems in the external field or perturbation expansions with noncanonical orbitals.     

Comments on the convergence of the ordinary Rayleigh-Schrödinger perturbation expansion

The convergence properties of the ordinary Rayleigh-Schrödinger perturbation expansion are discussed for the polarization expansion of H⁺₂ like systems. It is shown that this expansion may still be useful if the perturbed hamiltonian has an additional symmetry.     

Orbital correspondence analysis in maximum symmetry: Formulation and conceptual framework

A recently proposed method for the analysis of the course of chemical reactions, based on the maximal use of available symmetry, is formulated as a set of procedural rules. The application of these rules is illustrated with a simple prototype reaction: CH₂+C₂H₄ fcyclo-C₃H₆. They are then derived, using the formalism of time-dependent perturbation theory within the Born-Oppenheimer approximation, thus bringing out the method's underlying assumptions and its relation to the widely used Woodward-Hoffmann procedure.     

Studies in perturbation theory XIII. Treatment of constants of motion in resolvent method,partitioning technique,and perturbation theory

After a brief survey of some basic concepts in the theory of linear spaces, the eigenvalue problem is formulated in the resolvent technique based on the introduction of a reference function φ and a complex variable ?. This leads to a series of fundamental concepts including the trial wave function, the inhomogeneous equation, and finally the transition and expectation values of the Hamiltonian, of which the former renders a “bracketing function” for the energy. In order to avoid the explicit limiting procedures in this approach, the eigenvalue problem is then reformulated in terms of the partitioning technique which, in turn, leads to a closed form of infinite-order perturbation theory. The eigenvalue problem is greatly simplified if the Hamiltonian H has a constant of motion Λ or has symmetry properties characterized by the group G = {g}, and the question is now how these simplifications can be incorporated into the partitioning technique and into perturbation theory. In both cases, there exists a set of projection operators {Q_k} which lead to a splitting of the Hilbert space into subspaces which have virtually nothing to do with each other. It is shown that, in the partitioning technique, it is sufficient to consider one of these subspaces at a time, and the results are then generalized to perturbation theory. It turns out that the finite-order expansions are no longer unique, and the commutation rules connecting the various forms are derived. The infinite-order results are finally presented in such a form that they are later suitable for the evaluation of upper and lower bounds to the energy eigenvalues.     

Darstellung der Koordinatenräume der Gleichgewichtslösungen der Ein-Bodensalz-Gebiete des Systems H2O/Na2/K2/Mg/SO4/Cl2 bei 25°C durch Schalen,die sich durch Höhenlinien der „Basisfläche”︁ der Räume ergeben

Coordinate Spaces of the Equilibrium Solutions of one-sediment Regions in the System H₂O/Na₂/K₂/Mg/SO₄/Cl₂ at 25°C by Shells, which are represented by Niveau lines of “basal Areas” The coordinate spaces of 15 regions of the above mentioned system are constructed by one-dimensional arraying of their limiting areas. The two groups of lower and upper limiting areas merge within the so constructed “basal area”. For spacial representation, the two groups are assigned by niveau lines, thus leading to shells, which circumscribe the coordinate spaces. For evaluating these shell diagrams, the niveaus are substituted by nomographic scales allowing the incorporation of further “supplemental coordinates”. Practical applications of the shell diagrams are communicated.     

Rates of Reactive Removal of Anthropogenic Emissions in the Atmosphere

Anthropogenic emissions such as SO₂, CO, NO_x, and hydrocarbons constitute a perturbation of the natural equilibrium of the atmosphere. On a global scale, the extent of this perturbation does not significantly exceed the natural background levels as a result of direct or indirect photochemical “clean-up” processes in the atmosphere. Regionally, however, considerably increased concentrations of these trace components may occur. They are degraded by free radical reactions having widely differing yet substance-specific rates.     

Hydration of GABA and of the GABA·ZN2+ complex: A Monte Carlo simulation study

With empirical and theoretical atom–atom potentials the GABA·nH₂O, n = 25, 192 and GABA·Zn²⁺ · nH₂O, n = 25, 50, 100 complexes are simulated at 298.15 K by the Monte Carlo technique. The results show that the carboxyl group of GABA coordinates six water molecules. Two geometries of the GABA·Zn²⁺ complex, corresponding to the “direct” and “through-water” interaction of Zn²⁺ with the carboxyl group of GABA were found. For the latter interaction a GABA·Zn²⁺ · 6H₂O complex was found whereas the hydration of the former interaction leads to a GABA·Zn²⁺ · 5H₂O complex. Here the carboxyl group of GABA displaces only one water molecule in the first hydration shell of Zn²⁺. Energetically the “direct” and “through-water” geometries seem to be competitive, the former being slightly favored.     

Synthesis and Crystal Structure of the Palladium(IV) Polyoxomolybdate,K0.75Na3.75[PdMo6O24H3.5]·17H2O

The first example of a heteropolyoxomolybdate containing palladium(IV) was isolated and characterized by X‐ray crystallography. The palladium(IV) hexamolybdate, K_0.75Na_3.75[PdMo₆O₂₄H_3.5]·17H₂O, was isolated from an aqueous solution at pH 4.5 in the space group P\bar{1} , a 10.790(2), b 12.244(3), c 14.086(3) Å, α 113.77(1), β 90.41(1),γ 107.86(1)°, and the structure was determined using X‐ray diffraction methods, refining to a residual of 0.0301 for 5334 reflections. A formal “[PdMo₆O₂₄H₃]^5–” subunit exhibits the basic Anderson structure, with two [PdMo₆O₂₄H₃]^5– cluster anions in the structure bridged by a hydrogen atom (formally an H⁺) situated on a center of symmetry to give a “[Pd₂Mo₁₂O₄₈H₇]^9–” dimeric anion. The palladium(IV) atom occupies a slightly distorted octahedral environment, with Pd–O distances ranging from 1.968 to 2.009 Å.