Special Quantum Mechanical Approaches

As the context should be clear, the operator designation is neglected. 

Because the p operator commutes with each component operator, it must commute with the raising operator and with the lowering operator. 

we can work out the effect of the raising and lowering operators by applying them to an angular momentum function. Using T to designate any particular eigenfunction of 

H and V (times Planck s constant) are the eigenvalues for P. To understand the effect of a raising (or lowering) operator, we investigate what happens when it is applied to T  

A new function, Q, is whatever is produced. Applying ptoQ. and using the commutation property in Equation C.2 shows that Q s eigenvalue is the same as P s. 

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Powerfui streamlined development of group theory and advanced topics in quantum mechanics, via appendices covering molecular symmetry and special quantum mechanical approaches... 

Powerful streamlined development of group theory and advanced topics in quantum mechanics Appendix B (Molecular Symmetry) and Appendix C (Special Quantum Mechanical Approaches) cover topics that many physical chemistry courses include and that could each in fact be their own chapter. However, they are not essential to the flow of fhe remaining maferial, and so fhese appear af fhe end for inclusion in a course af the instructor s discretion. 

The standard quantum mechanical approach to any problem is to begin with a set of basis functions that could describe the spin system under certain specialized... 

In 1991, the Administration Board of the University of Namur asked for the opening of a second laboratory specialized in theoretical chemistry with the principal aim to foster on the increasingly important aspects of molecular modeling that complemented the already well-established quantum mechanical approaches. A new laboratory, called Laboratoire de Physico-Chimie Informatique (PCI for... 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

Most of the AIMD simulations described in the literature have assumed that Newtonian dynamics was sufficient for the nuclei. While this is often justified, there are important cases where the quantum mechanical nature of the nuclei is crucial for even a qualitative understanding. For example, tunneling is intrinsically quantum mechanical and can be important in chemistry involving proton transfer. A second area where nuclei must be described quantum mechanically is when the BOA breaks down, as is always the case when multiple coupled electronic states participate in chemistry. In particular, photochemical processes are often dominated by conical intersections [14,15], where two electronic states are exactly degenerate and the BOA fails. In this chapter, we discuss our recent development of the ab initio multiple spawning (AIMS) method which solves the elecronic and nuclear Schrodinger equations simultaneously this makes AIMD approaches applicable for problems where quantum mechanical effects of both electrons and nuclei are important. We present an overview of what has been achieved, and make a special effort to point out areas where further improvements can be made. Theoretical aspects of the AIMS method are... 

Of course, in reality new chemical substances are not synthesized at random with no purpose in mind—the numbers that have still not been created are too staggering for a random approach. By one estimate,1 as many as 10200 molecules could exist that have the general size and chemical character of typical medicines. Instead, chemists create new substances with the aim that their properties will be scientifically important or useful for practical purposes. As part of basic science, chemists have created new substances to test theories. For example, the molecule benzene has the special property of aromaticity, which in this context refers to special stability related to the electronic structure of a molecule. Significant effort has gone into creating new nonbenzenoid aromatic compounds to test the generality of theories about aromaticity. These experiments helped stimulate the application of quantum mechanical theory to the prediction of molecular energies. 

A well defined theory of chemical reactions is required before analyzing solvent effects on this special type of solute. The transition state theory has had an enormous influence in the development of modern chemistry [32-37]. Quantum mechanical theories that go beyond the classical statistical mechanics theory of absolute rate have been developed by several authors [36,38,39], However, there are still compelling motivations to formulate an alternate approach to the quantum theory that goes beyond a theory of reaction rates. In this paper, a particular theory of chemical reactions is elaborated. In this theoretical scheme, solvent effects at the thermodynamic and quantum mechanical level can be treated with a fair degree of generality. The theory can be related to modern versions of the Marcus theory of electron transfer [19,40,41] but there is no... 

The decision of which quantum mechanical model to use boils down to what size molecule you want to calculate, how reliable an answer you want, and how much time are you willing to wait for the results. Fortunately, as software and hardware improve, the tipping point of the balance weighing the pros and cons of semiempirical vs. DFT vs. ab initio is shifting such that larger molecules can be handled by the better methods. In special situations, a molecule with a couple of hundred atoms can be treated by an ab initio method (46,47), but the typical molecule of interest to theorists, spectrosco-pists, and physicists is smaller than what a pharmaceutical chemist usually wants to treat. Large molecular systems are often best left to one of the FF approaches (see next section). 

Free atoms are spherically symmetrical, which implies conservation of their angular momenta. Quantum-mechanically this means that both Lz and L2 are constants of the motion when V = V(r). The special direction, denoted Z, only becomes meaningful in an orienting field. During a chemical reaction such as the formation of a homonuclear diatomic molecule, which occurs on collisional activation, a local held is induced along the axis of approach. Polarization also happens in reactions between radicals, in which case it is directed along the principal symmetry axes of the activated reactants. When two radicals interact they do so by anti-parallel line-up of their symmetry axes, which ensures that any residual angular momentum is optimally quenched. The proposed sequence of events is conveniently demonstrated by consideration of the interactions between simple hydrocarbon molecules. 

Modem quantum-mechanical Valence-Bond (VB) theory has firm roots back to classical ideas even of a century and a half ago. These connections are of special interest, especially if greater general insight and extension of the classical concepts can be made. The interconnecting simpler semiempirical approaches, such as are of the prime focus here, are historically inextricably mixed with that of the ab initio theory, and the development has been via a peculiarly torturous road toward quantitative relevance. Thence here some brief historical commentary which also sets some nomenclature and ideas is first made. 

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