Quantum Approach

Plllet P, Crubelller A, Bleton A, Dulleu O, Nosbaum P, Mourachko I and Masnou-Seeuws F 1997 Photoassociation In a gas of cold alkali atoms I. Perturbative quantum approach J.Phys.B At.MoLOpt.Phys. 30 2801-20... 

There are several examples in the literature of recent years of convincing numerical demonstrations that a compound not yet observed has a stable structure. It must be remarked that these studies usually regard compounds of marginal chemical interest, and that for innovative problems the quantum approach has always been late with respect to the experiments. This delay decreases, but it is unlikely to expect that the leadership in the search of new compounds will be assumed by in-depth calculations. 

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

In the full quantum mechanical approach [8], one uses Eq. (22) and considers both the slow and fast mode obeying quantum mechanics. Then, one obtains within the adiabatic approximation the starting equations involving effective Hamiltonians characterizing the slow mode that are at the basis of all principal quantum approaches of the spectral density of weak H bonds [7,24,25,32,33,58, 61,87,91]. It has been shown recently [57] that, for weak H bonds and within direct damping, the theoretical lineshape avoiding the adiabatic approximation, obtained directly from Hamiltonian (22), is the same as that obtained from the RR spectral density (involving adiabatic approximation). 

Note that this last expression is nothing but the closed form [90] of the autocorrelation function obtained (as an infinite sum) in quantum representation III by Boulil et al.[87] in their initial quantum approach of indirect damping. Although the small approximation involved in the quantum representation III and avoided in the quantum representation II, both autocorrelation functions are of the same form and lead to the same spectral densities (as discussed later). 

The pure quantum approach of the strong anharmonic coupling theory performed by Marechal and Witkowski [7] gives the most satisfactorily zeroth-order physical description of weak H-bond IR lineshapes. 

The quantum approaches of direct and indirect dampings, respectively by Rosch and Ratner [58] and by Boulil et al. [90], may be considered as the well-behaved extension of the zeroth-order approach of Marechal and Witkowski [7]. 

Taylor, P.L. (1970), A Quantum Approach to the Solid State, Prentice-Hall, Englewood Cliffs, New Jersey. 

Such work could be used to test the validity of a quantum approach to preferred conformations of Lewis adducts. We therefore decided to perform such a test by means of semi-empirical methods in the hope of providing chemists with an inexpensive, rapid, reliable tool for the study of large series of compounds. 

As stated above, CNDO formalism was able to predict for many methyl derivatives (containing numerous hydrogen atoms) preferred conformations fully identical to those obtained by the most appropriate experimental techniques, electron diffraction and microwave spectroscopy. This was the case, for example, for each term of the (CH3)2M (14) and (CH3)3M (15) series. This quantum approach appeared likely to help experimentalists to locate accurately, and in a simpler way than usual, the light atoms - mainly hydrogen — in a molecule. 

Electron diffraction provides experimental diffraction spectra for comparison with computed spectra obtained from various intuitive geometrical models, but this technique alone is generally insufficient to locate the hydrogen atoms. A quantum approach, on the other hand, indicates the positions of the H atoms, which can then be introduced into the calculation of the theoretical spectra in order to complete the determination of the geometry. 

The so-called quantum approach to the Periodic Table is based on the following principles (Ostrovsky 2004) ... 

Thus, we could prove that the semi-empirical CNDO/2 quantum approach of Pople and Segal was a convenient tool for solving conformational problems in the... 

Even with the progress that has been made in rigorous quantum approaches, it is nevertheless possible to carry out such calculations only for relatively simple chemical systems. For example, the largest molecular system for which such calculations have been carried out is for the reaction H2 + OH — H2O + H. There is clearly interest, therefore, in the development of approximate versions of the approach that can be applied to more complex systems. Section III describes a semiclassical approximation for doing this, and Section IV concludes. 

This remark is associated with the amount of calculation performed and is not intended as a criticism. This work provides a valuable quantum mechanical analysis of a three-dimensional system. The artificial channel method (19,60) was employed to solve the coupled equations that arise in the fully quantum approach. A progression of resonances in the absorption cross-section was obtained. The appearance of these resonances provides an explanation of the origin of the diffuse bands found... 

Both the classical and quantum approaches ultimately lead to a model in which the polarizability is related to the ease with which the electrons can be displaced within a potential well. The quantum mechanical picture presents a more quantitative description of the potential well surface, but because of the number of electrons involved in nonlinear optical materials, theoreticians often use semi-empirical calculations with approximations so that quantitative agreement with experiment is not easily achieved. 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

The fits of Janev et al. [12] stem from a compilation of the results obtained with different theoretical approaches (i) semi-classical close-coupling methods with a development of the wave function on atomic orbitals (Fritsch and Lin [16]), molecular orbitals (Green et al [17]), or both (Kimura and Lin [18], (ii) pure classical model - i.e. the Classical Trajectory Monte Carlo method (Olson and Schultz [19]) - and (iii) perturbative quantum approach (Belkic et al. [20]). In order to get precise fits, theoretical results accuracy was estimated according to many criteria, most important being the domain of validity of each technique. 

A similar translation scheme from the full quantum approach to a mixed quantum classical description has been used recently in Ref. [26-29] to calculate infrared absorption spectra of polypeptides within the amide I band (note that the translation scheme has been also used in the mentioned references to compute nonlinear response functions). 

All the aforesaid also applies to the classical limit of the quantum density operator p, namely to the probability density p(0, tensor operators to which the spherical functions Ykq 0, different ways of expansion are used by different authors, both in the quantum approach and in the classical limit. This complicates considerably comparison of the results obtained by these authors, including experimental data. 

Auzinsh, M.P. and Ferber, R.S. (1991). Optical pumping of diatomic molecules in the electronic ground state Classical and quantum approaches, Phys. Rev. A, 43, 2374-2386. 

Following the wave-mechanical reformulation of the quantum atomic model it became evident that the observed angular momentum of an s-state was not the result of orbital rotation of charge. As a result, the Bohr model was finally rejected within twenty years of publication and replaced by a whole succession of more refined atomic models. Closer examination will show however, that even the most refined contemporary model is still beset by conceptual problems. It could therefore be argued that some other hidden assumption, rather than Bohr s quantization rule, is responsible for the failure of the entire family of quantum-mechanical atomic models. Not only should the Bohr model be re-examined for some fatal flaw, but also for the valid assumptions that led on to the successful features of the quantum approach. 

In quantum approaches the ACF Ggu(t) of the dipole moment operator may be given by the following trace, tr, to be performed within the base spanning the operators describing the system. 

Equation (153) is the semiclassical limit of the quantum approach of indirect damping. Now, the question may arise as to how Eq. (153) may be viewed from the classical theory of relaxation in order to make a connection with the semiclassical approach of Robertson and Yarwood, which used the classical theory of Brownian motion. 

First, one may observe that the asymmetry of the line shape appears only in the quantum approach corresponding to the top of the figure. One may also remark about the similarity in the line shapes that are intermediate between that of the quantum model and that of the semiclassical Robertson and Yarwood model appearing at the bottom. 

In Section IE, a theoretical approach of the quantum indirect damping of the H-bond bridge was exposed within the strong anharmonic coupling theory, with the aid of the adiabatic approximation. In Section III, this theory was shown to reduce to the Marechal and Witkowski and Rosch and Ratner quantum approaches. In Section IV, this quantum theory of indirect damping was shown to admit as an approximate semiclassical limit the approach of Robertson and Yarwood. 

In addition to the quantum approaches mentioned above, classical optimal control theories based on classical mechanics have also been developed [3-6], These methods control certain classical parameters of the system like the average nuclear coordinates and the momentum. The optimal laser held is given as an average of particular classical values with respect to the set of trajectories. The system of equations is solved iteratively using the gradient method. The classical OCT deals only with classical trajectories and thus incurs much lower computational costs compared to the quantum OCT. However, the effects of phase are not treated properly and the quantum mechanical states cannot be controlled appropriately. For instance, the selective excitation of coupled states cannot be controlled via the classical OCT and the spectrum of the controlling held does not contain the peaks that arise from one- and multiphoton transitions between quantum discrete states. 

---