Quantum-Mechanical Approaches

We consider a molecule containing a certain number of electrons moving in the field of fixed nuclei (Bom-Oppenheimer approximation). Let H be the electronic Hamiltonian depending on a set of parameters X (1 k 3N — 5 or 3N - 6) specifying nuclear positions. If ift and u represent the exact normalized wave function and energy for a nondegenerate electronic state, the generalized Hellmann-Feynman theorem states that [Pg.10]

As shown by various authors (Hurley, 1964, and references therein), for a given choice of parameters, the consequences of this theorem depend upon the coordinate system used for describing p and //. For illustrating this remark we consider the simplest molecule, H2+. [Pg.10]

It is easy to show that the force acting on nucleus a is given by [Pg.11]

The above formalism may be generalized to any homonuclear diatomic molecule (and to all other molecular species, but those will not be explicitly considered here). Then, the force on nucleus a is given by [Pg.11]

Equation (40) is usually called the electrostatic theorem. This states that the force acting on any nucleus can be calculated by classical electrostatics from the quantum-mechanical charge distribution p of the electrons and the point charges Z of the other nuclei. [Pg.11]

Bethe provided the theoretical basis for understanding the scattering of fast electrons by atoms and molecules [3, 4]. We give below an outline of the quantum-mechanical approach to calculating the scattermg cross section. [Pg.1314]

The modern quantum-mechanical approach to bonding indicates... [Pg.39]

The ideal way to simulate reactions (and indeed many other processes where we might wish to derive properties dependent upon the electronic distribution) would of course be to use a fully quantum mechanical approach. [Pg.632]

The vibration of molecules is best described using a quantum mechanical approach. A harmonic oscillator does not exactly describe molecular vibra-... [Pg.92]

Another progress in our understanding of the ideally polarizable electrode came from theoretical works showing that the metal side of the interface cannot be considered just as an ideal charged plane. A simple quantum-mechanical approach shows that the distribution of the electron gas depends both on the charge of the electrode and on the metal-solution coupling [12,13]. [Pg.805]

A more accurate quantum mechanical approach makes use of the variational method. - The goal is the solution of the basic wave equation... [Pg.195]

So the popular polarization equations of the type (6.5) for electrochemical reactions thus acquire some physical basis. However, according to current concepts the nature of the activated state is different, and quantum-mechanical approaches must be used for a theoretical calculation of the values and These concepts are discussed in more detail in Chapter 34. [Pg.244]

Combining Classical and Quantum-Mechanical Approaches in Describing Polymer Systems... [Pg.51]

Quantum mechanical approaches, such as those applied by Merz [84], use traditional quantum mechanics to calculate free energies of binding. Though relatively successful, the computational time required to calculate the energies make this approach impractical to large data sets. [Pg.71]

It may be shown [8] that both semiclassical [83,84], and full quantum mechanical approaches [7,32,33,58,87] of anharmonic coupling have in common the assumption that the angular frequency of the fast mode depends linearly on the slow mode coordinate and thus may be written... [Pg.287]

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). [Pg.287]

Finally, it must be remembered that DFT and AIMD can be incorporated into the so-called mixed quantum mechanical/molec-ular mechanical (QM/MM) hybrid schemes [12, 13]. In such methods, only the immediate reactive region of the system under investigation is treated by the quantum mechanical approach -the effects of the surroundings are taken into account by means of a classical mechanical force field description. These DFT/MM calculations enable realistic description of atomic processes (e.g. chemical reactions) that occur in complex heterogeneous envir-... [Pg.47]

Quantum mechanical approaches have been successfully used to predict hydrogen abstraction potentials and likely sites of metabolism of drug molecules [78-81]. AMI, Fukui functions, and density functional theory calculations could identify potential sites of metabolism. Activation energies for hydrogen abstraction were calculated by Olsen et al. [81] to be below 80 kj/mol, suggesting most CH groups can be metabolized which particular one depends on steric accessibility and intrinsic reactivities. [Pg.463]

In this chapter a quantum mechanical approach encompassing solvent effects and... [Pg.283]

The reader can see now that experimental conditions are progressing in such a way that would allow for verifications of the quantum theories of solvent effects. The important theoretical fact is the possibility of recasting the standard theory of solvent effects, based upon classical statistical mechanics, into a more complete quantum mechanical approach. [Pg.311]

In the quantum mechanical approach, a transition moment is introduced for characterizing the transition between an initial state and a final state (see Box 2.2). The transition moment represents the transient dipole resulting from the displacement of charges during the transition therefore, it is not strictly a dipole moment. [Pg.27]

It is interesting that there is a relation between the oscillator strength f (given by Eq. 2.6) and the square of the transition moment integral, which bridges the gap between the classical and quantum mechanical approaches ... [Pg.28]

An additional concern arises in regard to any differences which may exist between the classical theory and the quantum-mechanical approach in the calculation of the Franck-Condon factors for symmetrical exchange reactions. In fact, the difference is not very large. For a frequency of 400 cm for metal-ligand totally symmetric vibrational modes, one can expect... [Pg.251]

The quantum mechanical approach to the problem is based on the Hamiltonian of Equation 3.19b with velocities Qi replaced by momenta P conjugate to the coordinate Qi. P is classically defined... [Pg.65]

Turning to the fully quantum mechanical approach, we find that the lowest order rate theory for general non-radiative relaxation processes also provides a factorized rate expression ... [Pg.58]

In a recent upsurge of studies on electron transfer kinetics, importance was placed on the outer shell solvent continuum, and the solvent was replaced by an effective model potential or a continuum medium with an effective dielectric constant. Studies in which the electronic and molecular structure of the solvent molecules are explicitly considered are still very rare. No further modem quantum mechanical studies were made to advance the original molecular and quantum mechanical approach of Gurney on electron and proton (ion) transfer reactions at an electrode. [Pg.72]

A. Lammerts van Bueren, A. Ardevol, J. Fayers-Kerr, B. Luo, Y. Zhang, M. Sollogoub, Y. Bleriot, C. Rovira, and G. J. Davies, Analysis of the reaction coordinate of a-L-fucosidases A combined structural and quantum mechanical approach,./. Am. Chem. Soc., 132 (2010) 1804-1806. [Pg.292]

These various descriptors may be calculated using various molecular mechanics and quantum mechanics approaches, as discussed in chapter 1. [Pg.144]

McMaster (1954) takes a quantum-mechanical approach to the Stokes parameters and polarized light. Two books are devoted entirely to polarization Shurcliff (1962) and Clarke and Grainger (1971). And a splendid collection of papers on many aspects of polarized light has been edited by Gehrels (1974a). Another collection worth consulting is that compiled by Swindell (1975) it contains several of the classical papers on polarization. [Pg.56]

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