Quantum chemistry applied mathematics

The material model is just a bit of matter - a molecule all the physical interactions are in principle considered (even if some terms are discarded in actual calculations), the modelization is thus reduced to the mathematical part. In addition, the report has the characteristics of an explanation. Making reference to a celebrated sentence opining the textbook on Quantum Chemistry by Eyrmg, Walter, Kimball [17] "In so far as quantum mechanics is correct, chemical questions are problems in applied mathemathics", it may be said that this program is a realization of that sentence. 

Throughout this work, familiarity will be assumed with basic mathematical notation and terminology of quantum chemistry and matrix algebra at the level of a standard text, such as I. N. Levine, Quantum Chemistry, 5th edn. (Englewood Cliffs, NJ, Prentice Hall, 2000) or J. R. Barrante, Applied Mathematics for Physical Chemistry, 2nd edn. (Upper Saddle River, NJ, Prentice Hall, 1998). 

If quantum theory is to be used as a chemical tool, on the same kind of basis as, say, n.m.r. or mass spectrometry, one must be able to carry out calculations of high accuracy for quite complex molecules without excessive cost in computation time. Until recently such a goal would have seemed quite unattainable and numerous calculations of dubious value have been published on the basis that nothing better was possible. Our work has shown that this view is too pessimistic semiempirical SCF MO treatments, if properly applied, can already give results of sufficient accuracy to be of chemical value and the possibilities of further improvement seem unlimited. There can therefore be little doubt that we are on the threshold of an era where quantum chemistry will serve as a standard tool in studying the reactions and other properties of molecules, thus bringing nearer the fruition of Dirac s classic statement, that with the development of quantum theory chemistry has become an exercise in applied mathematics. 

The quantum mechanical methods described in this book are all molecular orbital (MO) methods, or oriented toward the molecular orbital approach ab initio and semiempirical methods use the MO method, and density functional methods are oriented toward the MO approach. There is another approach to applying the Schrodinger equation to chemistry, namely the valence bond method. Basically the MO method allows atomic orbitals to interact to create the molecular orbitals of a molecule, and does not focus on individual bonds as shown in conventional structural formulas. The VB method, on the other hand, takes the molecule, mathematically, as a sum (linear combination) of structures each of which corresponds to a structural formula with a certain pairing of electrons [16]. The MO method explains in a relatively simple way phenomena that can be understood only with difficulty using the VB method, like the triplet nature of dioxygen or the fact that benzene is aromatic but cyclobutadiene is not [17]. With the application of computers to quantum chemistry the MO method almost eclipsed the VB approach, but the latter has in recent years made a limited comeback [18],... 

A. Simoes and K. Gavroglu, Quantum chemistry qua applied mathematics the contributions of Charles Alfred Coulson (1910-1974) , Hist. Stud. Phys. Biol. Sci., 1999, 29, 363 106. 

Still, skepticism remained as to the general power of quantum mechanics applied to complex chemical systems. The situation around 1930 is described by the well-known dictum of Paul Dirac (the Nobel Prize winning physicist at Cambridge) The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. ... 

Quantum chemistry is the fundamental theory of molecules it allows for an all embracing and correct description ofthe structure and properties of single molecules. In principle all experimentally measurable properties could be calculated mathematically by applying the quantum chemical formalism thereby the only empirical parameters are the Planck constant, the velocity oflight, the masses, the spins and the electromagnetic momentum of the electron and of the nuclei (Muller-Herold and Primas 1984, 309). 

These successes did not go unnoticed by industry. Several pharmaceutical companies (1963-1964) became interested in applications of it-electron theory to biochemistry. While it was admittedly premature, it was felt that quantum chemistry was both the wave of the future and the very matrix for rational drug design. Hiickel energies of cephalosporins could be correlated with their biological activities.While companies were applying some mathematical methods of correlation techniques in quantitative structure-activity relationships (QSAR), it was chiefly the Hiickel theory and various forms of semiempirical quantum mechanics that was using a large share of computer time on the IBM 7094 mainframe in 1966. 

Bom and raised in the New York City area, Mark Gordon received his B.S., Ph.D. and postdoctoral education at Rensselaer Polytechnic Institute, Carnegie-Mellon University (with J.A. Pople) and Iowa State University (with K. Ruedenberg). At North Dakota State University, he rose to Distinguished Professor and Department Chair. At Iowa State University, he is Distinguished Professor and Director of the Applied Mathematical Sciences Program in the Ames Laboratory USDOE. He has been the Chair of the Theoretical Chemistry Subdivision of the American Chemical Society, and the Secretary-Treasurer of its Physical Chemistry Division. He is a Fellow of the American Physical Society, a Fulbright Senior Scholar and was recently elected to the International Academy of Quantum Molecular Science. 

Ivan Hubac obtained his Ph.D. in 1971 (applied mathematics) from University of Waterloo, Canada. His supervisors were Professors J. Koutecky, J. Cizek and J. Paldus. After his return to Czechoslovakia from Canada in 1971 he worked at the Department of Mathematics, Chemical faculty, Slovak Technical University, Bratislava. At present he works as Professor of Physics at Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia. His hrst quantum chemistry paper appeared in 1967. 

Josef (Joe) Paldus defended his M.Sc. Thesis in 1958 at the Charles University in Prague, supervised by V. Hanus and J. Koutecky, and the latter also supervised his Ph.D. Thesis, defended in 1961 at the Czechoslovak Academy of Sciences. He did his postdoctoral studies with D.A. Ramsay at the National Research Council in Ottawa. After emigrating to Canada in 1968 he joined the Department of Applied Mathematics of the University of Waterloo and later also its Chemistry Department and Guelph-Waterloo Center for Graduate Work in Chemistry. Since his obligatory retirement in 2001 he continues his research as a Distinguished Professor Emeritus. He became a Fellow of the Royal Society of Canada in 1983, a Member of the International Academy of Quantum Molecular Sciences a year later and, most recently, a Fellow of the Fields Institute for Research in Mathematical Sciences. 

The underlying physical laws necessary for the mathematical theory of...the whole of chemistry are thus completely known, Dirac, P.A.M., Proc. R. Soc., 123, 714, 1929 In so far as quantum mechanics is correct, chemical questions are problems in applied mathematics, Eyring, H., Walter, J., and Kimball, G.E., Quantum Chemistry, Wiley, New York, 1944, p. iii. 

In practice, the Ritz variational method is used most often. One of the technical problems to be solved is the size of the basis set. Enormous progress in computation and software development now facilitates investigations that 20 years ago were absolutely beyond the imagination. The world record in quantum chemistry means a few billion expansion functions. To accomplish this, quantum chemists have had to invent some powerful methods of applied mathematics. 

We recall here the opening statement of a famous book on quantum chemistry In so far as quantum mechanics is correct, chemical questions are problems in applied mathematics. Actually, the mathematics to apply to liquid systems is a hard nut to crack. Fortunately a sequence of many, but reasonable, approximations can be introduced. We shall consider and exploit them in the following section of this chapter. 

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