Cardinal number

When you request an orbital, yon can use the cardinal number of the orbital (ordered by energy and starting with number=l) or an offset from either the highest occupied molecular orbital (HOMO) or the lowest unoccupied molecular orbital (LL MO). Offset from the HOMO are negative and from the LUMO are positive. Often these frontier orbitals are the ones of most chemical interest. 

Grund-versueb, m. fundamental experiment, -wasser, n. (under)ground water, -wasser-splegel, m. ground-water level, water table, -zabl, /. unit base number, base cardinal number, -zug, m. leading feature, characteristic pi.) outline, -zustand, m. ground (level) state. 

As can be seen from the table, the number of AOs increases rapidly with the cardinal number X. Thus, with each increment in the cardinal number, a new shell of valence AOs is added to the cc-pVXZ set since the number of AOs added in each step is proportional to X2, the total number (Nbas) of AOs in a correlation-consistent basis set is proportional to X3. The core-valence sets cc-pCVXZ contain additional AOs for the correlation of the core electrons. As we shall see later, the hierarchy of correlation-consistent basis sets provides a very systematic description of molecular electronic systems, enabling us to develop a useful extrapolation technique for molecular energies. 

This equation contains two unknowns and we can thus extrapolate to the basis-set limit from two separate calculations with different cardinal numbers X and Y. This gives us the following simple expression for the energy at the basis-set limit [32, 33] ... 

Dunning s correlation consistent basis sets cc-pVAZ [27] augmented with diffuse functions [28] were used in the calculations. We considered cardinal numbers X—D, T, Q, 5, 6 and single (s), double (d), triple (t), and quadruple (q) augmentations. The orbitals were not allowed to relax in the coupled cluster response calculations. 

The Cauchy moments of Ne at the CCSD/q-aug-cc-pV5Z level were found in Ref. [4] to be converged within 1 % compared to the basis-set limit result. We have calculated the Cauchy moments also for the X—6 cardinal number. From the results in Table 1 it appears that the Cauchy moments at this level are significantly less than 1 % from the basis-set limit result. 

For any three elements p, q, and r of a scheme S, the cardinal number apqr will be called the structure constant of p, q, and r in S. Structure constants, complex products, and the relationship between these two notions form the subject of the first chapter. Most of these results, in particular those of Section 1.4, are fundamental for the further development of the theory. 

With items other than units of time or measure, use words for cardinal numbers less than 10 use numerals for 10 and above. Spell out ordinals first through ninth use numerals for 10th or greater. 

CASSI gives the number of a meeting in ordinal form. Convert this number to an italic cardinal number, and use it as the volume number in the citation, unless CASSI has already indicated another volume number. 

How big are sets There are infinitely many natural numbers and there are infinitely many irrational numbers. The natural numbers can be counted, the irrational numbers cannot. Are there two different kinds of infinity It was Cantor s genius to answer with a resounding yes . He defined a sequence of transfinite cardinals to characterize the different types of infinity. Cantor first looked for the lowest type of infinity and found it in the set of natural numbers, N. This set, as everybody intuitively knows, is infinite . This is so, because there is no end to the natural numbers. Given a number n 6 iV, one can always come up with n+. which is also in N. Cantor assigned the transfinite cardinal number 1 0 to the set of integers. The cardinality or size , of a set S is denoted by I 5. Therefore, N = Uq. 

Numeration systems also define a person or unit s order in a series, for example, to determine who crosses a finish line in a race in first, second, or third place. Numbers that define order are known as the ordinal numbers (first, second, etc.) and contrast with the cardinal numbers (one, two, three, etc.) which express a tally or total of units. 

The ordinal name of a number differs somewhat from the cardinal name. In most instances the cardinal name can be converted into the ordinal name by adding th. Thus the cardinal number one thousand becomes the ordinal number one thousandth four becomes fourth and so on. hi the case of 1,2, and 3, however completely different names are used. 

Table 3. The f em symmetric double dissociation of water (into 2H(ls 2S) + 0(2p4 3P) cut (ii)). The H-O-H angle is kept fixed at its equilibrium value taken from Ref. [139] (ae = 104.501 degree). R is an O-H distance and Re = 0.95785 A is the equilibrium value of R [139]. All energies E (in cm-1) are reported as E - E(Re, ae), where E(Re, ae) are the corresponding values of E at the equilibrium geometry. X is a cardinal number defining the aug-cc-pCVXZ basis sets used in the calculations. In all CC calculations, all electrons were correlated.

Table 5. The differences between CC/CR-CC energies, calculated relative to their equilibrium values (the CC/CR-CC E - E(Re, ore)] values in Table 3) and the corresponding MRCI(Q) relative energies (the MRCI(Q) [E — E(Re, ae)] values in Table 1) for the dissociation of a single O-H bond in water (into H(l.s 2S) 4- OH(X 2 n) cut (i)). X is a cardinal number defining the aug-cc-pCVXZ basis sets used in the calculations.

At the Hartree-Fock level the hyperpolarizabilities usually increase if the augmentation level and also if the cardinal number X are increased. For the correlated contribution to "yn (0) the convergence pattern is dominated by different effects At the CCSD level an increase of "yn (0) with the cardinal number beyond T is only found for the lower augmentation levels. In particular for molecules we observe, as illustrated in Table S for N2 and CH4, a monotonic decrease of Are second hyperpolarizability when the correlation treatment is improved in the series X = T, Q, 5, etc. The results for X = T typically overestimate the correlation contribution to yn (0) by a few percent. Many correlated hyperpolarizability calculations in the literature were performed with basis sets of triple- or similar quality and basis set convergence was often only explored with respect to augmentation with diffuse functions. From the above observations one may conclude that many of these studies obtained too large results for 7 (0). 

Quantum mechanics has to a large extent resolved an antinomy inherited from the discussions of Heraclit and Democrit. The wave-functions are continuous and extended in the former sense, but at the same time, the indivisible parts of Democrit have been replaced by normalization conditions, the numbers K, Z, N,.. . of electrons, protons, neutrons,.. . being cardinal numbers without any possible way of assigning ordinal numbers to the individual, indiscernible entities. This trend has been further accentuated by most particles having anti-particles, with exception of some bosons (such as the photon and the neutral pion, but not the a-particle). 

Clearest, amplest introduction in English, well suited for independent study. Subdivision of main theory, such as theory of sets of points, are discussed, but emphasis is on general theory. Partial contents rudiments of set theory, arbitrary sets and their cardinal numbers. ordered sets and their order types, well-ordered sets and their cardinal numliers. Bibliography. Key to symbols. Index, vii 4. 144pp. 5Vs x 8- Paperltound 1.50... 

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