Two-body wave functions of the harmonic oscillator

Polazeći sa stajališta funkcionalne analize, valne funkcije obično shvaćamo kao vektore u Hilbertovom prostoru kvantnih stanja. Kad je broj identičnih čestica veći od jedan, pojavi se specifična algebarska struktura, tako da je isti Hilbertov prostor graduirana algebra nad prstenom simetričnih polinoma (bozonskih pobuđenja). Generatori algebre (vakuumi) su od prije poznati u Kartezijevoj bazi, a u ovom radu ih prevodimo u bazu dobrog zakretnog impulsa za poseban slučaj dvije čestice. Tada tri generatora (Ψ1; Ψ2; Ψ3) čine vektor osnovnog stanja, a četvrti Ψ4 = Ψ1Ψ2Ψ3 je pseudoskalar i nalazi se medu dvostruko pobuđenim stanjima, kojih ima 28. Kad se sva ta stanja napišu u obliku bozonskih pobuđenja vakuuma i u bazi dobrog zakretnog impulsa, otkrije se da je Ψ4 komponenta određenog pobuđenog stanja zakretnog impulsa l=3 i projekcije m = ±2. Time se to stanje identificira kao glava vrpce pobuđenih stanja drugačije simetrije od vrpce koja nastaje pobuđivanjem osnovnog stanja.From the point of view of functional analysis, wave functions are understood as vector in Hilbert’s space of quantum states. When the number of identical particles is greater than one, there appears a specific algebraic structure, where the same Hilbert space is a graded algebra over the ring of symmetric polynomials (bosonic excitations). Generators of the algebra (vacuums) are known in the Cartesian basis. In this work we translate them to the basis of good angular momentum for the specific case of two particles. Then three generators (Ψ1; Ψ2; Ψ3) make up the ground-state vector and the fourth Ψ4 = Ψ1Ψ2Ψ3 is a pseudoscalar found among doubly excited states, of which there are 28. When all those states are written in the form of bosonic excitations of the vacuums and in the basis of good angular momentum, it is revealed that Ψ4 is a component of a specific excited state of angular momentum l=3 and projection m = ±2. Therefore this state is identified as a band head of excited states with different symmetry then the band which is generated by excitations of the ground state

Određivanje strukture natrijem kationiranih aminokiselina u plinskoj fazi pomoću H/D izmjene

Taking into account the difference in structure of a sodiated gas phase amino acid in its zwitterionic and charge solvated forms, gas phase structures of sodiated histidine, lysine, phenylalanine, proline, tryptophan and tyrosine were probed by H/D exchange reactions. Experiment results obtained from the site-specific reaction rate constants indicated the zwitterionic structure for all the sodiated amino acids studied. In contrast, B3LYP calculations for sodiated lysine, along with the already presented theoretical results for other amino acids studied, show the charge solvated form to be stable, except for proline which is in the zwitterionic form. Absence of H/D exchange in sodiated methyl esters of phenylalanine and histidine, along with the discrepancy between experimental and theoretical data for the amino acids studied, indicates that H/D exchange occurs only when sodiated amino acids are in the zwitterionic form.Obzirom na različitu strukturu natrijem kationirane aminokiseline u zwitterionskoj formi od one u formi solvatiranoga naboja, moguće je promatrajući H/D izmjenu razlikovati te dvije strukture. Informacije dobivene određivanjem konstanti brzine kemijske reakcije na specifična mjesta u molekuli ukazuju da se promatrane natrijem kationirane aminokiseline: histidin, lizin, fenilalanin, prolin, triptofan i tirozin nalaze u zwitterionskoj formi. Nasuprot tome, B3LYP rezultati za natrijem kationirani lizin zajedno s već publiciranim rezultatima za ostale promatrane aminokiseline ukazuju da se iste u plinskoj fazi nalaze u formi solvatiranoga naboja, uz iznimku prolina koji se nalazi u zwitterionskoj formi. Izostanak H/D izmjene na natrijem kationiranim metilnim esterima histidina i fenilalanina zajedno s neslaganjem teorijskih i eksperimentalnih rezultata upućuje da u eksperimentima H/D izmjene do iste dolazi samo kada je natrijem kationirana aminokiselina u zwiterionskoj formi. .

Aspartic Acid Side Chain Effect—Experimental and Theoretical Insight

Gas-phase H/D exchange and density functional theory study of the Asp and Glu side-chain carboxylic group intrinsic reactivity is reported. H/D exchange site specific treatment and some additional theoretical calculations showed that a side-chain carboxylic group may initiate proton transfer along with bond formation to one of its oxygens, i.e., possibility to initiate selective of cleavage peptide bond (“aspartic acid effect”). That finding is used to select aspartic acid cleavage mechanisms (side-chain proton transfer either to backbone carbonyl or to amide nitrogen) for further computational study. B3LYP/6-31G(d) and G3(MP2)//B3LYP potential energy profiles of both mechanisms on a model system CH3CO-Asp-NHCH3 were constructed. Although energy employed in low-energy collision induced dissociation suffices for both mechanisms thresholds, energy transferred to specific modes suggests a complex one-step mechanism of proton transfer (from the side-chain carboxylic group to the backbone amide group), bond formation (between deprotonated carboxylic group and carbon atom of the backbone carbonyl), and peptide bond cleavage as favorable

Matrix-assisted Laser Desorption Ionization

Veliki zamah u razvoju spektrometrije masa omogućilo je uvoðenje matricom potpomognute ionizacije laserskom desorpcijom (MALDI) te ionizacije elektroraspršenjem. Usprkos velikoj popularnosti i analitičkoj primjeni MALDI metode, temeljni procesi formiranja iona i desorpcije još uvijek nisu razjašnjeni. Kemijski procesi u MALDI procesu događaju se kako prilikom pripreme uzorka, tako i tijekom desorpcije/ionizacije pa se oba procesa odražavaju na spektar masa. U radu su opisani priprema uzorka i model procesa desorpcije/ionizacije kojeg je predložila istraživačka skupina M. Karasa početkom 2000. godine.Since its invention, matrix-assisted laser desorption ionization (MALDI) has found wide application in mass spectrometry of high molecular weight compounds such as synthetic polymers and biopolymers. Despite widespread application of MALDI, the fundamental processes of ion formation and desorption are still poorly understood. The chemistry of the MALDI process, occurring both during sample preparation and during ionization is reflected in the mass spectrum.As the MALDI technique now stands, a low concentration of analyte molecules, which usually exhibit only moderate absorption per molecule, is embedded in matrix crystals consisting of a small, highly absorbing species. In this manner the efficient and controllable energy transfer is retained while the analyte molecules are separated from excessive energy that would lead to their decomposition.The matrix is believed to serve two major functions: adsorption of energy from the laser light and the isolation of analyte molecules from each other. There are 3 major metods for the preparation of samples for analysis which are quite quick and simple: dried droplet, surface and sandwich preparation. Experiments with pH indicator dyes serve as proof that analyte´s charge state in the matrix crystals is the same as in solution.Upon laser desorption a sudden and explosive phase transition occurs and a dense plume of desorbed material is formed. The initial velocity of analyte ions in the plume depends only on the matrix used. Initial species formed as a result of laser desorption are tiny clusters. They consist of a matrix, analyte and other ionic species embedded in the matrix crystals all held together by hydrogen bonds and coulombic interactions. The first essential charging and thus ionization process is the statistical occurrence of clusters with a deficit/excess of anions or cations. Very small initial clusters are a likely to be highly charged. Highly charged initial clusters cannot survive in the matrix plume and their charge drops. Clusters shrink by evaporation of neutral molecules. This paper presents only those cases leading to analyte ions and compares MALDI and the electrospray ionization technique

The gas-phase H/D exchange mechanism of protonated amino acids

A mass spectrometry and Density Functional Theory study of gas-phase H/D exchange in protonated Ala, Cys, Ile, Leu, Met and Val is reported. Site-specific rate constants were determined and results identify the α − amino group as the protonation site. Lack of exchange on the Cys thiol group is explained by the absence of strong intramolecular hydrogen bonding within the reaction complex. In aliphatic amino acids the presence of a methyl group at the β − C atom was found to lower the site-specific H/D exchange rate for amino hydrogens. Study of the exchange mechanism showed that isotopic exchange occurs in two independent reactions: in one only the carboxylic hydrogen is exchanged and in the other both carboxylic and amino group hydrogens exchange. The proposed reaction mechanisms, calculated structures of various species and a number of structural findings are consistent with experimental data