Perturbation theory for excited states. IV

Two of the perturbation methods developed in Paper I (CIPT and SCPT) are extended for use with uncoupled Hartree-Fock perturbation theory. Corrections to the uncoupled results are discussed and it is found that it is possible to correct using functions which when applied to ground state calculations are equivalent to the geometric approximation. It is also shown that CIPT and SCPT can be applied to ground state calculations and that when coupled perturbed orbitals are used CIPT and SCPT are equivalent to the usual coupled perturbation theory.     

Perturbation theory for excited states of molecules. I

The problem of perturbations of excited states is discussed and three methods are developed. The first of these uses a zero order wave-function made up of a linear sum of singly excited SCF configurations, whereas the second uses just one of these configurations. The third method is restricted to small -systems, the zero order wave-function being a linear sum of all possible determinants formed from the basis set used. The perturbations considered here are one-electron operators. Example calculations are performed on the butadiene molecule within the context of the -electron approximation.

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Zusammenfassung Für das Problem einer Störung von Einelektronen-Operatoren für angeregte Zustände werden drei Verfahren vorgeschlagen: Erstens die Verwendung einer Zustandsfunktion nullter Ordnung, die eine Linearkombination einfach angeregter SCF-Konfigurationen ist, zweitens die entsprechende Verwendung nur einer ausgewählten Konfiguration. Drittens läßt sich, wenn auch nur bei kleinen -Elektronensystemen, als nullte Näherung eine Linearkombination von allen möglichen angeregten Konfigurationen verwenden. Als Beispiel wird das -Elektronensystem des Butadiens gewählt.

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Résumé Discussion du problème de perturbation pour les états excités et développement de trois méthodes. La première utilise une fonction d'ordre zéro combinaison linéaire de configurations SCF monoexcitées, alors que la seconde utilise seulement une de ces configurations. La troisième méthode est restreinte à de petits systèmes , la fonction d'onde d'ordre zéro étant une combinaison linéaire de tous les déterminants construits dans la base utilisée. Les perturbations envisagées ici sont constituées par des opérateurs monoélectroniques. La molécule de butadiène sert d'exemple dans le cadre de l'approximation à électrons .

     

Perturbation analysis of the variation method

It is pointed out that an assumption common to all general discussions of the perturbation analysis of the variation method is sometimes unjustified, and some consequences are discussed.     

Perturbation theory for excited states of molecules. III

The configuration interaction perturbation theory and the single configuration perturbation theory developed in Paper I is applied to the problem of chemical reactivity for conjugated hydrocarbon molecules. Various related hetromolecules are also considered. It is found that the pattern of reactivity for excited states is much the same as for the ground state.I wish to express my appreciation to Dr. A. T. Amos for constant encouragement during the preparation of this paper and to the S.R.C. for the provision of a maintenance grant.     

Electron spin resonance investigation of biradical metal chelates in the slow tumbling region

The ESR spectra of a series of N,N′-disubstituted 1,4-diazabutadiene and of the tetrachloro-o-benzoquinone metal chelates in methyltetrahydrofuran have been recorded at different temperatures. The spectra which displayed the typical pattern of slow tumbling triplet species were reproduced by the discrete jump model of Norris and Weissmann. The model was extended to include the case of molecule having a non-vanishing E parameter. The logarithm of the product of the absolute temperature and of the rotational diffusion rates ln(K_rT) shows a linear dependence on 1/T like that observed for the viscosity of the solvent. However, the slope was found to vary with the molecular species and to be quite different from the corresponding value of the solvent. These results are discussed in terms of recent modifications of the Debye—Stokes—Einstein relation.     

Displacement chromatography of chemotactic peptides

Displacement chromatography was successfully used to separate a binary peptide mixture, n-formyl-Met–Phe and n-formyl-Met–Trp, on a reversed-phase column. Displacement parameters such as choice of displacer, displacer concentration, mobile phase organic level, and flow-rate were critically examined in the context of maximizing productivity. Since the feed composition was limited by solubility, optimal productivity was sought as a function of feed volume. The impurities contained in the commercial displacers used in this study did not seem to affect the overall separation quality. In most cases the final pattern of contiguous rectangular bands was not attained; nevertheless, separations of high productivity were achieved using benzethonium chloride and tris[2-(2-methoxyethoxy)ethyl]amine as displacers. In some cases further increase in productivity was not possible only because of solubility constraints. Loading of feed at low initial organic modulator level coupled with displacements at higher modulator level was found to give efficient separations.     

Permeability of the outer membrane of bacteria.

The outer membrane of Gram-negative bacteria is an attractive system for the study of the structure-function relationships in biological membranes. This membrane has, inter alia, the task of regulating the inflow of nutrients and outflow of waste products. Investigations with the aid of mutants showed that there are at least two general pathways for the diffusion of small molecules across the outer membrane: one for hydrophobic and one for hydrophilic compounds. In the case of the “hydrophobic pathway” the hydrophobic compound dissolves in the interior of the membrane and then crosses the membrane in accordance with the partition coefficient. In wild-type forms of enteric bacteria this pathway cannot be used—presumably owing to the absence of regions with phospholipid bilayers. Small hydrophilic molecules, on the other hand, penetrate the membrane through water-filled pores.     

Perturbation theory within the variation method

Some of Moccia's formulas (Int. J. Quant. Chem. 8 , 293 (1974)) for corrections to “not completely optimized variational parameters” are compared with those resulting from a perturbation theory analysis. It is not possible to choose the zero-order Hamiltonian (at least not in any simple way) so as to get exact agreement, but one choice does come close.     

Studies on catalytically active surface compounds, X. Alkali metal induced slow tumbling of vanadyl complexes on the surface of aerosil

The effect of alkali metal admixtures (Li, Na, K, Rb, and Cs) on V/SiO₂ catalysts was studied by ESR spectroscopy. Alkali metals strongly change the VO²⁺ spectra. Reduction leads to spectra showing a slow rotational motion of VO²⁺ species on the support surface. The proposed explanation includes former structural models and the formation of salts of polyvanadium acid.

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(Li, Na, K, Rb Cs) V/SiO₂ . VO²⁺. , VO²⁺ . .

     

Adiabatic perturbation theory for the degenerate case. I. Perturbation of isolated degenerate or quasidegenerate levels

Adiabatic formulae for secular operators and contracted Hamiltonians in an arbirary combination of degenerate or quasidegenerate subspaces are derived. A detailed consideration of the adiabatic limit in the power series is given, and “stability” of proper linear combinations with respect to a transformation produced by the S_α-matrix is proved.     

硬质粒子扰动下铜电沉积研究

使用常规旋转电极电沉积技术,引入陶瓷球等一类硬质粒子,在旋转电极的带动下,使陶瓷球不断磨擦和撞击阴极表面而实现电铸铜.对比酸性溶液电铸铜和碱性溶液电铸铜,发现硬质粒子在电沉积过程中能扰动离子的放电过程,并影响电铸层的组织结构.但由于二者放电机理不同,前者形成的电铸层表面布满尖状毛刺,而后者则表面尖刺消失,但脆性大.SEM和XRD测试表明,由碱性电铸液沉积的电铸铜层,表面光亮平整,晶粒致密,大小约为100~300 nm,其结晶形态接近无序取向.     

Perturbation theory of relativistic corrections

Methods for perturbation theory of relativistic corrections for an electron in a Coulomb field are divided into three categories: (1) in terms of 4-component spinors; (2) in terms of the ‘large components’ of the Dirac spinor; (3) involving a Foldy-Wouthuysen type transformation, where one attempts to obtain a two-component spinor different from the ‘large component’. In methods of category 1 (the ‘direct perturbation theory’ of paper I of this series, the related approaches by Rutkowski as well as by Gesteszy, Grosse, and Thaller and a somewhat different one by Moore) the wave function, the energy and the Hamiltonian are analytic inc ^?2. No divergent terms arise. In methods of category 2 (that of the elemination of the small component as well as a similarity transformation in intermediate normalization) wave function and energy are still analytic inc ^?2, but the effective Hamiltonian no longer is. Regularized results can be obtained by controlled cancellation of divergent terms. In category 3 both the effective Hamiltonian and the wave function are highly singular and non-analytic inc ^?1. A controlled cancellation of divergent terms is at least very difficult. These pathologic feature survive in the non-relativistic limit and have hence little to do with relativistic effects. They are related to the fact that forr → 0 the sign of the quantum number κ rather than that of the energy determines which component of the Dirac spinor is large and which is small. In the limitr → 0 andc → ∞ the Foldy-Wouthuysen wave function of a 2p _1/2 state is a 1p wave function. Hierarchies of transformations of the Dirac equation and its non-relativistic limit are presented and discussed. Finally the problem of the regularization of effective Hamiltonians on 2-component level ‘for electrons only’ is addressed.     

Adiabatic perturbation theory for the degenerate case. II. Perturbation of degenerate bound states embedded in the continuum

An adiabatic formula for the contracted Hamiltonian in a reference space containing bound-state eigenfunctions of degenerate energy levels embedded in the continuum is derived. A general factorization theorem for the dynamic operatorS_α(0, – ∞/λ) is proved, and the cancellation of the pole singularities in the perturbation series of the contracted Hamiltonian in adiabatic form is discussed.     

Perturbation improvement of SCF orbitals

A simple method for the perturbation improvement of SCF orbitals, based on the appropriate variation principle, is described. The method allows for the optimal extension of the original basis set without re-optimization of all the orbital exponents and guarantees the best lowering of the total energy. The numerical illustration of the proposed variation scheme is also given.     

Physical genome analysis of bacteria.

Pulsed-field gel electrophoresis (PFGE) is a general analytical tool to separate large DNA molecules and may therefore be applied to problems from all areas of bacteriology. The genome size of bacteria covers the range of 0.6 to 10 megabase pairs. For genome fingerprinting, the bacterial chromosome is cleaved with a restriction endonuclease that gives a resolvable and informative number of five to one hundred fragments on the PFGE gel. Restriction enzymes are chosen according to GC content, degree of methylation, and codon usage of the respective bacterial genus. Macrorestriction fingerprinting allows the identification of bacterial strains and the distinction between related and unrelated strains. If fragment patterns of several restriction digestions are quantitatively evaluated, strains can be classified according to genetic relatedness at the level of genus, species, and biovar. In particular, members of a clonal lineage can be uncovered. Hence, any problem from applied, environmental, and clinical microbiology may be addressed by PFGE restriction analysis where the spatiotemporal spread of a bacterial clone is of interest. In bacterial genomics, PFGE is employed for the top-down construction of macrorestriction maps of the chromosome which yields data about genome organization, mobile genetic elements, and the arrangement of gene loci and gene families. The genomic diversity of a bacterial species is elucidated by comparative chromosome mapping. Map positions of restriction sites and gene loci of interest serve as landmarks to assess the extent of gross chromosomal modification, namely insertions, deletions and inversions. Intra- and interspecies comparisons of genome organization provide insights into the structure and diversity of bacterial populations and the phylogeny of bacterial taxa.     

Perturbation theory of relativistic corrections

After a discussion of the problems associated with the non-relativistic limit of the Dirac equation and of the expansion of the exact eigenvalues and eigenfunctions of the H atom in powers ofc ^?2 the traditional approaches for a perturbation theory of relativistic effects are critically reviewed. Then a direct perturbation theory is presented, that is characterized by a change of the metric in 4-component spinor space such that the Lévy-Leblond equation appears as the straightforward non-relativistic limit of the Dirac equation. The various orders in perturbation theory of the energy and the wave function are derived first in a direct way, then in a resolvent formalism. The formulas are very compact and easily generalizeable to arbitrary order. All integrals that arise to any order exist, and no controlled cancellation of divergent terms (as in other approaches) is necessary. In the same philosophy an iterative approach towards the solution of the Dirac equation is derived, in which the solution of the Schrödinger equation is the first iteration step.