Energy representation

When about the energy representation, one should restrict to the stationary states (eigen-energies). 

Within this context, the dynamical state t) fulfills the conservation of its norm (scalar product in fact)  

The second application of the temporal vectorial state regards the temporal variation of the average of an operator A, which in Schrodinger equation is constant in time d,A-0, leading with  

The third consequence appears through the temporal-stationary factorization of the dynamical states  

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

We deal with a simple case of a homogeneous system of one component. The energy should be a function of entropy, volume, and mol number, U = U S, V, n). Recall that it is necessary to include the mol number as parameter for sake of homogeneity. In a homogeneous system, we can represent the energy as the sum of the terms 

We deal with the situation of constant energy, thus U(S, V, n) = U. Next, we substitute TS y, —pV - x, and fin - z. With these substitutions, Eq. (2.19) turns into 

we inspect the isentropic path for an ideal gas in terms of TS, —pV, and 

Therefore, for an ideal gas, at constant entropy and mol number, the variables x, y, z are linearly dependent, as the motion is in a plane passing the origin. 

There is still another way to substitute for the natural variables. By substitution 

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In the above discussion of relaxation to equilibrium, the density matrix was implicitly cast in the energy representation. However, the density operator can be cast in a variety of representations other than the energy representation. Two of the most connnonly used are the coordinate representation and the Wigner phase space representation. In addition, there is the diagonal representation of the density operator in this representation, the most general fomi of p takes the fomi... 

The Boltzman probability distribution function P may be written either in a discrete energy representation or in a continuous phase space formulation. 

In both the entropy and energy representations the extensive parameters are the independent variables, whereas the intensive parameters are derived concepts. This is in contrast to reality in the laboratory where the intensive parameters, like the temperature, are the more easily measurable while a variable such as entropy cannot be measured or controlled directly. It becomes an attractive possibility to replace extensive by intensive variables as... 

As in energy representation the fundamental thermodynamic equation in entropy representation (3) may also be subjected to Legendre transformation to generate a series of characteristic functions designated as Massieu-Planck (MP) functions, m. The index m denotes the number of intensive parameters introduced as independent variables, i.e. 

Figure 4.12 (a) Gibbs energy representation of the phases in the system Zr02-Ca0 at 1900 K. McaO - MzrO = TSZ n°t deluded for clarity, (b) Calculated phase diagram of the system Zr02 Ca0. Thermodynamic data are taken from reference [9]. 

In this section, we describe our model, and give a brief, self-contained account on the equations of the non-equilibrium Green function formalism. This is closely related to the electron and particle-hole propagators, which have been at the heart of Jens electronic structure research [7,8]. For more detailed and more general analysis, see some of the many excellent references [9-15]. We restrict ourselves to the study of stationary transport, and work in energy representation. We assume the existence of a well-defined self-energy. The aim is to solve the Dyson and the Keldysh equations for the electronic Green functions ... 

The starting point for thermodynamic description, whether in the calculus-based or the geometry-based formalism, is the Gibbs fundamental equation for a given equilibrium state S. In the energy representation, this is expressed as... 

Find the formulas for the matrix elements of the matrix representatives of x and px in the particle-in-a-box energy representation. (This is the representation with particle-in-a-box wave functions as basis functions.) Use the general formulas to write down the first several elements in the northwest corner of each matrix. 

If the proof in section 2 is studied carefully, it will be seen that there is nothing in the derivation of this law which could not be applied mutatis mutandis to any kind of energy representable in the form mec2. For, apart from general considerations of probability, the only condition assumed in the proof was that the total kinetic energy could be expressed as a sum of three quadratic terms— equation 1 of section 2 multiplied by I/2m expresses this condition. 

The magnitude values of the frequency domain representation are converted to a 1/3-critical band energy representation. This is done by adding the magnitude values within a threshold calculation partition. 

Thus, the fact that there is a well-defined phase relationship between the eigenstates of the Hamiltonian, contained in the wave function, is manifest in the existence of off-diagonal elements in the energy representation. The absence of off-diagonal matrix elements for the thermally eqiulibrated case makes clear that collisions have destroyed matter coherence manifest as quantum correlations between energy eigenstates. 

Given the significance of collisional effects in solution, we introduce the simplest of models for relaxation and concomitant decoherence. That is, the equation motion of px t) in the energy representation is given the form i... 

If the system is in the presence of a radiation field, then Hs in Eq. (5.12) is augmented by the dipole-electric field interaction HUR [Eq. (2.10)]. The result is the so-called optical Bloch equations. Note that this approach focuses explicitly on decoherence in the energy representation. 

Feynman rules for the Green function. In the Furry picture, in addition to the standard Feynman rules in the energy representation (see [24,13]), the following vertices and lines appear (we assume that the Coulomb gauge is used)... 

As one can easily note, a well-defined hierarchy of successive interaction energy approximations, varying from the most expensive MP2 method to the various electrostatic energy representation (the more simplified the theory, the less computationally demanding calculation), demonstrates the utility of this decomposition scheme (Figure 8-3) ... 

The evaluation of the free energy is essential to quantitatively treat a chemical process in condensed phase. In this section, we review methods of free-energy calculation within the context of classical statistical mechanics. We start with the standard free-energy perturbation and thermodynamic integration methods. We then introduce the method of distribution functions in solution. The method of energy representation is described in its classical form in this section, and is combined with the QM/MM methodology in the next section. 

The above is the brief introduction to the density-functional theory of solutions. The mathematical development is quite straightforward. The numerical implementation is difficult, however, in the full coordinate representation. As noted in Section 17.3.2, the full coordinate is multidimensional the solute-solvent distribution is a function over high-dimensional configuration space and cannot be implemented in practice. To overcome the problem of dimensionality, it is necessary to introduce a projected coordinate. In Section 17.3.5, we introduce the energy representation and formulate the density-functional theory in the energy representation. 

The above drawbacks of RISM and its variants are well documented since their first formulations [11,12], They are all related to the point that a molecule is treated as a collection of sites. In the method of energy representation introduced next, each of the solute and solvent molecules is taken to be a single unit as a whole, and those drawbacks vanish. 

In the energy representation, the density-functional theory can be formulated by restricting the set of solute-solvent interaction potentials ux( jr,x) to those which are... 

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