Representation mixed

Polarization fluctuations of a certain type were considered in the configuration model presented above. In principle, fluctuations of a more complicated form may be considered in the same way. A more general approach was suggested in Refs. 23 and 24, where Eq. (16) for the transition probability has been written in a mixed representation using the Feynman path integrals for the nuclear subsystem and the functional integrals over the electron wave functions of the initial and final states t) and t) for the electron ... 

Then, instead of performing the six-dimensional integral in Eq. (5.19) all at once, we perform successive three-dimensional integrals over s and R. The first step takes us to W R,P), the Wigner representation [130,131] of the density matrix, and the second step to the p-space density matrix, n(P — p/2 P + p/2). The reverse transformation of Eq. (5.20) can also be performed stepwise over P and p to obtain A( , p), the Moyal mixed representation [132], and then the r-space representation V R— s/2 R + s/2). These steps are shown schematically in Figure 5.2. 

In recent work, we have further pursued forms of g ( ) which manifest nonstationary effects directly in t, and other mixed-representations. The first of these representations is the so-called iGLE dynamics that may be characterized by the stochastic differential equation. 

It should be noted that in the RPA, the dipole oscillator strengths calculated in dipole velocity, dipole length, or mixed representation and all sum rules would be identical, and the TRK sum rule, Eq. (13), would be fulfilled exactly, that is, be equal to the number of electrons if the computational basis were complete [30,34,35]. Comparison of the oscillator strengths calculated in the different formulations thus gives a measure of the completeness of the computational basis in addition to the fulfillment of the Thomas-Reiche-Kuhn sum rule (vide infra). 

Clearly, in the subsystem resolution one could also consider all intermediate specifications of the molecular (constrained) equilibria, when only a part of the subsystems remains externally open (characterized by the fixed chemical potentials of a common reservoir) with the remaining, complementary set of subsystems being closed (characterized by the fixed subsystem numbers of electrons) [4,5]. Such mixed representations can be also naturally defined in the CSA approach. We would like to observe, that in the theory of chemical reactivity these partially opened situations do indeed arise, e.g., in the surface reactions, when one adsorbate is opened (chemisorbed) while the other reactant remains externally closed (physisorbed) on the catalyst surface, which acts as the electron reservoir for the reaction. 

This said, the number of the local expansions, their location, the number of expansion terms for each centre, and the method to get the numerical coefficients for each expansion term must be defined. There are no formal constraints, and the strategy should be selected on the basis of its efficacy computer time and precision. A detailed and clear exposition of the problems involved in, and of the options open to the definitions of local expansions has been recently done by F. Vigne-Maeder and the late P. Claverie [72]. We shall follow in part this exposition, giving more emphasis, at the end, to the use of atomic monopole expansions (i.e. atomic charges) and to mixed representations, which represent, in our opinion, the most versatile method for chemical reactivity problems. 

Let us suppose it is possible to treat the nuclear subsystem in a molecule classically and electronic subsystem quantum mechanically. This type of theoretical framework is called a mixed quantum-classical representation. Such a mixed representation can find many applications in science. For instance, a fast mode such as the proton dynamics in a protein should be considered as a quantum subsystem, while the rest of the skeletal structure can be treated as a classical subsystem [3, 484, 485]. It is quite important in this context to establish the correct equations of motion for each of the subsystems and to ask what are their rigorous solutions and how the quantum effects penetrate into the classical subsystems. By studying the quantum-electron and classical-nucleus nonadiabatic dynamics as deeply as possible, we will see how such rigorous solutions, if any, look like qualitatively and quantitatively. This is one of the main aims of this book. 

Mixed quantum-classical mixed representation of the Hamiltonian and wavefunctions... 

Dynamics in the electron-nuclear quantum-classical mixed representation... 

The DVR is related to, but distinct from pseudo-spectral and collocation methods of solving differential equations. For the DVR there is an orthogonal transformation which defines die relation of die DVR to the finite basis representation (FBR). > Thus, for example, the Hermidan character of operators remains obvious in the DVR. Both pseudo-spect and collocation methods, however, use a "mixed" representation operators and, as such, do not display the Hermitian character of operators such as H. Thus the advantages of the DVR are that the accuracy is that of a Gaussian quadrature and it is a true representation, while the collocation methods permit more freedom in the choice of points, a distinct advantage in some multidimensional problems. 

For each partial wave J and parity e, the Hamiltonian and wave packet are discretized in the BF frame in mixed representation [21, 64, 80, 89,160] discrete variable representation (DVR) is employed for the two radial degrees of freedom and finite basis representation (FBR) of normalized associated Legendre function i jK(O) for the angular degree of freedom. Thus the wave packet in the BF frame is written as... 

In general, results of fully atomistic or mixed representations have confirmed the formation of a microphase-separated morphology in PEMs, albeit results on sizes, shapes, and distributions of phase domains have remained inconclusive. In comparison to experiment, molecular models were found to underestimate the sizes of ionic... 

This result also demonstrates clearly that the Wannier functions are not eigenfunctions of the Hamiltonian of a periodic crystal. Therefore in many applications, when one applies the Wannier functions to describe local perturbations, it is more advantageous to start from a mixed representation in which some matrix elements are expressed with the help of Wannier functions and others with the aid of Bloch functions. 

Using the off-diagonal hypervirial relation, Eq. (3.66), one can define two alternative formulations of the oscillator strength (Hansen, 1967), a mixed representation... 

The mixed representation is particular interesting because it does not involve the excitation energies explicitly. It can alternatively also be written in the following two... 

Hint Start with the mixed representation of the oscillator strengths in Eq. (7.73). Use the fact that the set of excited states is complete, i.e. 

Figure 34 Mixed representation of the p53 DNA complex. The p53 protein and the DNA is shown as capped stick model, the protein backbone is represented as ribbon model. Parts of the molecular surface indicate the p53 protein DNA interface region. The surfaces are color coded with respect to the electrostatic potential calculated by a finite difference algorithm solving the Poisson-Boltzmann equation - (blue, negative gray, neutral red. positive). The electropositive parts of the p53 protein fit perfectly in the major and minor groove of the almost electronegative DNA...

T, and (4) Tf. Alternative (1) is said to be fully covariant, (2) is fully cowtravariant, and the other two are mixed representations. In principle, one is free to formulate physical laws and quantum chemical equations in any of these alternative representations, because the results are independent of the choice of representation. Furthermore, by applying the metric tensors, one may convert between all of these alternatives. It turns out, however, that it is convenient to use representations (3) or (4), which are sometimes called the natural representation. In this notation, every ket is considered to be a covariant tensor, and every bra is contravariant, which is advantageous as a result of the condition of biorthogonality in the natural representation, one obtains equations that are formally identical to those in an orthogonal basis, and operator equations may be translated directly into tensor equations in this natural representation. On the contrary, in fully co- or contravariant equations, one has to take the metric into account in many places, leading to formally more difficult equations. 

Note that the integrals on the right-hand side of this equation are in a mixed representation, with one index in the AO basis and the remaining indices in the MO basis. Substituting this expression into the variational conditions (10.6.6), we arrive at the following conditions for the optimized Hartree—Fock state ... 

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