Successes in early quantum theory

Do not confuse isotope and allotrop. Sulfur exhibits both isotopes and allotropes. Isotopes of sulfur (with percentage naturally occurring abundances) are 15S (95.02%), 15S (0.75%), (4.21%), (0.02%). 

Allotropes of an element are different structural modifications of that element. Allotropes of sulfur include cyclic 

Further examples of isotopes and allotropes appear throughout the book. 

Isotopes of an element have the same atomic number, Z, but different atomic masses. 

Isotopes can be separated by mass spectrometry and Fig. 1.1a shows the isotopic distribution in naturally occurring Ru. Compare this plot (in which the most abundant 

If Aj for Cl is 35.45, what is the ratio of C1 C1 present in a sample of Cl atoms containing naturally occurring Cl  

We saw in Section 1.2 that electrons in an atom occupy a region of space around the nucleus. The importance of electrons in determining the properties of atoms, ions and molecules, including the bonding between or within them, means that we must have an understanding of the 

---

Despite his success with it, quantum theory needed a more solid foundation than Bohr s intuition. Arnold Sommerfeld in Munich was an early contributor to that work after the war the brightest young men, searching out the growing point of physics, signed on to help. Bohr remembered the period as a unique cooperation of a whole generation of theoretical physicists from many countries, an unforgettable experience. He was lonesome no more. 

As shown above, the heat capacity Cy (or at any rate that part of it which is due to the vibrations) may be expected to have a value of 3Jt whenever hvjkT 1. This would be so, even at the lowest temperatures, if Planck s constant h were zero, and this is the case in the classical or pre-quantum mechanics. In fact, classical theory leads to the expectation that, for any crystalline substance, Cy has the constant value of 3R per mole. This is contrary to experiment, and it is known that Cy usually diminishes below 3A, with fall of temperature, and seems to approach zero at the absolute zero. One of the early successes of the quantum theory consisted in finding the reason for this decrease in Cy which is quite inexplicable in classical theory. The explanation is implicit in the previous equations and is due to the fact that the oscillators can only take up finite increments of energy. When a system of oscillators is held at low temperature, most of them are in their lowest energy level, and a small rise oftemperature is insufficient to excite them to the next higher level. Therefore Cy, which measures the intake of energy per unit increase of temperature, is smaller than at higher temperature. 

Id. Bohr s Theory of the Hydrogen Spectrum. Perhaps the greatest of these early successes of the quantum theory was the theory of the hydrogen spectrum as developed by Bohr. Empirically, it has been found that all the lines in the hydrogen spectrum could be represented by the formula... 

In view of this, early quantum mechanical approximations still merit interest, as they can provide quantitative data that can be correlated with observations on chemical reactivity. One of the most successful methods for explaining the course of chemical reactions is frontier molecular orbital (FMO) theory [5]. The course of a chemical reaction is rationali2ed on the basis of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), the frontier orbitals. Both the energy and the orbital coefficients of the HOMO and LUMO of the reactants are taken into account. 

If they were to account for the spectrum of atomic hydrogen and then atoms of the other elements, scientists of the early twentieth century had to revise the nineteenth-century description of matter to take into account wave-particle duality. One of the first people to formulate a successful theory (in 1927) was the Austrian scientist Erwin Schrodinger (Fig. 1.23), who introduced a central concept of quantum theory. 

Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

From mid-1925 through the early months of 1926, events relating to quantum mechanics occurred in rapid succession. After years of intense efforts to develop a coherent quantum theory to explain the results of experimental measurements on atomic sys-... 

Perhaps this is too easy. Perhaps the importance of the duty of universal applicability is as a motivation, rather than as an achievement attempts to unify disparate domains have motivated some of the most ambitious and successful episodes in the history of physics. Newtonian mechanics, we are often told, was the synthesis of terrestrial and astronomical physics. More poignantly for the present discussion, in the early 1920s—the last years of the old quantum theory—attempts to fit atomic models to spectroscopic data required a diverse battery of inexplicable and mutually incompatible quantum conditions. Pauli and Bom, among others, saw in this chaos the need for a radical departure. Hindsight tells us that it was quantum mechanics... 

In early theoretical studies the photodissociation of CH3I was treated as an effective two-mode system with CH3 considered as one particle. As with ICN, a new era began with ab initio calculations performed by Morokuma and coworkers on a reasonably high level of electronic structure theory. These authors determined six- and nine-dimensional adiabatic and diabatic PESs, which were used in classical surface hopping trajectory calculations. Hammerich et al. performed large-scale five-dimensional wave packet calculations using these PESs. As concluded by Eppink and Parker One caveat of the ab initio surfaces is that they do not successfully reproduce the experimental absorption spectrum, which peaks to the red side of and is broader than the spectrum predicted by the theoretical treatments.. .. Furthermore, the predicted I quantum yields of the most... 

A. Pullman In all fairness, it must be said that the quantum chemical computations, as a rule, and quite generally, are much less satisfactory, for excited states properties than for ground state properties. This is true particularly when quantitative evaluation of a property is requested. This inconvenience appears in all its fullness for non-benzenoid hydrocarbons because the dipole moment is a striking example which does not occur in the benzenoid series, but it is true for any quantity related to excitation. As concerns qualitative evolutions of various properties, theory has been extremely successful in the past in the very case of the fulvenes and has even been instrumental in the development of the field since it has often preceded experiment and served as an incentive for the chemist to synthetize the compounds. (See the numerous series of papers by ourselves and Dr Ernst Bergmann in the early 1950 s.)... 

The American quantum chemistry community developed very quickly. As Dresden (1992) pointed out, this could be due to the fact that many of the founders of American quantum chemistry had a very similar development, leading from an early exposure to preliminary quantmn theory via a substantial apprenticeship in formal quantum mechanics to a notable success in atomic and molecular structure (page 28). This common path of development helped enormously to create a well-trained, coherent quantmn chemistry community in the USA. 

---