Chiral quantum mechanical theory

A particularly useful probe of remote-substituent influences is provided by optical rotatory dispersion (ORD),106 the frequency-dependent optical activity of chiral molecules. The quantum-mechanical theory of optical activity, as developed by Rosenfeld,107 establishes that the rotatory strength R0k ol a o —> k spectroscopic transition is proportional to the scalar product of electric dipole (/lei) and magnetic dipole (m,rag) transition amplitudes,... 

The next section (Sect. 2) is devoted to a lengthy discussion of the molecular hypothesis from the point of view of quantum field theory, and this provides the basis for the subsequent discussion of optical activity. Having used linear response theory to establish the equations for optical activity (Sect. 3), we pause to discuss the properties of the wavefunctions of optically active isomers in relation to the space inversion operator (Sect. 4), before indicating how the general optical activity equations can be related to the usual Rosenfeld equation for the optical rotation in a chiral molecule. Finally (Sect. 5), there are critical remarks about what can currently be said in the microscopic quantum-mechanical theory of optical activity based on some approximate models of the field theory. 

Moffitt (9) introduced the first quantum mechanical theory of optical activity in chiral transition metal complexes. He... 

In Equation 516 the f/ = 1 (/ = 1,2) are sign factors that may be determined experimentally or on the basis of a quantum-mechanical theory. The fi term in Equation 51 b has a special feature. It vanishes for chiral allenes with two identical ligands. For such kinds of allenes the X component of Equation 516 represents a qualitatively complete chirality function and is, from the point of view of the general theory, the strict expression for the calculations of molar rotations of allenes. 

Hund, one of the pioneers in quantum mechanics, had a fundamental question of relation between the molecular chirality and optical activity [78]. He proposed that all chiral molecules in a double well potential are energetically inequivalent due to a mixed parity state between symmetric and antisymmetric forms. If the quantum tunnelling barrier is sufficiently small, such chiral molecules oscillate between one enantiomer and the other enantiomer with time through spatial inversion and exist in a superposed structure, as exemplified in Figs. 19 and 24. Hund s theory may be responsible for dynamic helicity, dynamic racemization, and epimerization. 

No such non-trivial ( chirality ) operator can exist in a quantum mechanical system with a finite number (n) of degrees of freedom because any operator commuting with all observables must be a multiple of the identity operator if n < °°51). Thus this symmetry argument leads us back naturally to a quantum field theory context for understanding optical activity. 

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 discovery of quantum mechanics was seen as a dramatic departure from classical theory because of the unforeseen appearance of complex functions and dynamic variables that do not commute. These effects gave rise to the lore of quantum theory as an outlandish mystery that defies comprehension. In our view, this is a valid assessment only in so far as human beings have become evolutionary conditioned to interpret the world as strictly three dimensional. The discovery of a 4D world in special relativity has not been properly digested as yet, because all macroscopic structures are three dimensional. Or, more likely, minor discrepancies between 4D reality and its 3D projection are simply ignored. In the atomic and molecular domains, where events depend more directly on 4D potential balance, projection into 3D creates a misleading image of reality. We argue this point on the basis of different perceptions of chirality in 3D and 2D, respectively. 

Any theory of molecular structure must be consistent with quantum mechanics. But should every true story about molecules follow from quantum mechanics The non-reduction of molecular chemistry has often been argued for by pointing out that quantum chemistry borrows the notion of molecular structure from classical chemistry (see the article on Atoms and Molecules in Classical Chemistry and Quantum Mechanics in Part 5 of this Volume).In quantum theory an atom or molecule has no extension in space or time neither electrons nor nuclei exist as individual objects electrons are indistinguishable, not identifiable or localizable entities. In particular, chirality has been pointed out as something not found in quantum mechanics, but any form of asymmetry seems to be problematic. 

Computational methods such as molecular mechanics (MM) and quantum theory calculations have become convenient and reliable techniques in the analysis of CD data. A common approach in chiral supramolecular structural study is to examine if a supramolecular conformation simulated by MM method is consistent with the observed CD spectrum of the sample. However, in order to use this approach, one needs to correlate the stereostructures and CD data by using, for example, exciton chirality method or reference CD spectra of analogs with known structures. Thus, it should be noted that the application of this method has some limitations for evaluation in supramolecular systems. 

In the same way, the chirality of a three-dimensional tetrahedron is resolved in four dimensions, which means that the three-dimensional chiral forms are identical when described four dimensionally. Small wonder that all efforts to find a wave-mechanical difference between laevo and dextro enantiomers are inconclusive. The linear superposition principle, widely acclaimed as a distinctive property of quantum systems, is now recognized as no more than a partially successful device to mimic four-dimensional behavior. This includes one of the pillars of chemical-bonding theory, known as the resonance principle. 

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