Quantum mechanics problem reduction

The historical development of the electronic configuration model is traced and the status of the model with respect to quantum mechanics is examined. The successes and problems raised by the model are explored, particularly in chemical ah initio calculations. The relevance of these issues to whether chemistry has been reduced to quantum mechanics is discussed, as are some general notions on reduction. 

The straightforward solution for these problems would be the calculation of energies/forces on the fly at every simulation step by means of quantum mechanics. However, even for a very modest simulation box of a few hundred molecules, this approach is computationally not affordable. Two ways have been proposed, therefore, to solve this dilemma the further reduction of the simulation box and the use of approximate QM calculation procedures, and the partition of the system into a QM and a classical part. 

Another reduction is possible when one wishes to treat only one or a few degrees of freedom quantum-mechanically while the rest of the system can be treated still in a classical way. First pioneering studies along such lines treated the problem of electron solvation in molten salts and liquid ammonia. But, it must be noted that, when one studies the dynamics of quantum degrees of freedom coupled to a classical environment, particular care is required This mixed quantum-classical dynamics has subtle features, and is still an active area of research. 

And there is the philosophical problem. If such intertheoretic connections are important because, in company with experimental data, they test and confirm the reducing theory, then the inferential steps in such a reduction are critical to the role. In other words, we had better know why our approximations work it is not adequate to employ any old maneuver that produces agreement with observable data when such agreement serves as the benchmark by which to assess the approximate truth of the reducing theory, in this case, quantum mechanics. Much the same is true if we seek to codify connections to ensure logical consistency between our theories. 

For many chemical problems, it is crucial to consider solvent effects. This was demonstrated in our recent studies on the hydration free energy of U02 and the model reduction of uranyl by water [232,233]. The ParaGauss code [21,22] allows to carry out DKH DF calculations combined with a treatment of solvent effects via the self-consistent polarizable continuum method (PCM) COSMO [227]. If one aims at a realistic description of solvated species, it is not sufficient to represent an aqueous environment simply as a dielectric continuum because of the covalent nature of the bonding between an actinide and aqua ligands [232]. Ideally, one uses a combination model, in which one or more solvation shells (typically the first shell) are treated quantum-mechanically, while long-range electrostatic and other solvent effects are accounted for with a continuum model. Both contributions to the solvation free energy of U02 were... 

One seemingly sensible approach to the relativistic electronic structure theory is to employ perturbation theory. This has the apparent advantage of representing supposedly small relativistic effects as corrections to a familiar non-relativistic problem. In Appendix 4 of Methods of molecular quantum mechanics, we find the terms which arise in the reduction of the Dirac-Coulomb-Breit operator to Breit-Pauli form by use of the Foldy-Wouthuysen transformation, broken into electronic, nuclear, and electron-nuclear effects. FVom a purely aesthetic point of view, this approach immediately looks rather unattractive because of the proliferation of terms at the first order of perturbation theory. To make matters worse, many of the terms listed are singular, and it is presumably the variational divergences introduced by these operators which are referred to in [2]. Worse still, higher-order terms in the Foldy-Wouthuysen transformation used in this way yield a mathematically invalid expansion. 

In a separate contribution [11], we have analysed within the present framework an assessment of the various arrows of time and the possible symmetry violations instigated by gravitation including the fundamental problem of molecular chirality [12]. Other related developments involve Penrose s concept of objective reduction (OR), i.e. gravity s role in quantum state reduction and decoherence as a fundamental concept that relates micro-macro domains including theories of human consciousness [13], see also Ref. [3] for more details. Note also efforts to derive quantum mechanics from general relativity [14]. 

The measurement process is one of the most controversial areas in quantum mechanics. Just how and at what stage in the measurement process reduction occurs is unclear. Some physicists take the reduction of as an additional quantum-mechanical postulate, while others claim it is a theorem derivable from the other postulates. Some physicists reject the idea of reduction [see M. Jammer, The Philosophy of Quantum Mechanics, ley, 1974, Section 11.4 L. E. Ballentine, Am. J. Phys, 55, 785 (1987)]. Ballentine advocates Einstein s statistical-ensemble interpretation of quantum mechanics, in which the wave function does not describe the state of a single system (as in the orthodox interpretation) but gives a statistical description of a collection of a large number of systems each prepared in the same way (an ensemble) in this interpretation, the need for reduction of the wave function does not occur. [See L. E. Ballentine, Am. J. Phys, 40,1763 (1972) Rev. Mod. Phys, 42,358 (1970).] There are many serious problems with the statistical-ensemble interpretation [see Whitaker, pp. 213-217 D. Home and M. A. B. Whitaker, Phys Rep., 210,223 (1992) Problem 10.3], and this interpretation has been largely rejected. 

In a paper devoted to discussing the problem of the existence of the orbitals, we speak of a conceptual breakdown or conceptual discontinuity between molecular chemistry and quantum mechanics Whereas in quantum mechanics orbitaV is a non-referring term, in molecular chemistry orbitals exist as spatial regions on the basis of which the shape of the local and individual molecules can be explained (Labarca and Lombardi 2010a, p. 155). In that paper we stress that, in the last decades, many authors have recognized the conceptual discontinuity between the two theories (Woolley 1978 Primas 1983,1998 Amaim 1992). More recently, Hitme Hettema (2012, p. 368) talks about the ontological discontinuity between the terms of chemistry and those of physics certain terms used both in chemistry and in physics seem to refer to different items in the two disciplines. According to this author, such discontinui is one of the central problems in the philosophy of chemistry, around which many other problems, such as that of reduction, revolve. (Hettema 2012, p. 368). 

Abstract. Four prophetic statements in the introductory paragraph of Dirac s probably most cited paper are analyzed. Not only has his claim been disproved that the quantum mechanical equations needed to solve chemical problems are too complicated to ever be solved, even the reduction of chemistry from quantum mechanics is a tricky epistemological problem. Most surprising is that Dirac believed that relativistic effects are unimportant for chemistry. 

Nevertheless, in his stimulating review, Primas [17] criticizes Dirac as a naive reductionist. According to Primas, Dirac was wrong because his postulate of re-ductionism was based on what Primas calls the pioneer quantum mechanics which he contrasted with modern nonrelativistic quantum mechanics , and that Dirac did not consider the complicated epistemological problems related to the reduction of chemistry from quantum mechanics. It is not the scope of our perspective to comment on this criticism. Note, however, that the two generations of quantum theory differ more in the interpretation than in the operative formalism, that was, in fact, fully formulated in 1929. As to the relevance of problems of interpretation for the application of quantum mechanics, the reader is referred to a refreshing paper by Levy-Leblond [18]. 

The first quantum mechanical approach to the problem of siting of Fe in FAU was performed by Beran et al. [186]. They have modeled a FAU with Fe, Fe and Fe(OH)+ ions localized in Sn and Sf extra-framework positions or with Fe in the framework sites, using the CNDO/2 method on a cluster representing the six-membered ring opening. No difference was found for the framework properties when replacing Al with Fe and the framework Fe was predicted to be quite stable also in the presence of a reduction to Fe +. 

Of course, we must here begin by stating what we mean by the blanket term reduction , and what we take to be some of the problems it faces. First of all, we will not be primarily concerned with the ontological dependence of chemistry upon physics. As stated, we believe the ontological dependence of chemistry on physics to be almost a foregone conclusion. Rather, our concern will be with the epistemological reduction of chemistry to physics - with the question of whether our current description of chemistry can be reduced to our most fundamental current description of physics, namely quantum mechanics - and with its explanatory consequences. ... 

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