Stationary state, functionals

Stationary State Functionals of Reactive Systems that... 

It was emphasized in Chapter 6 that the definition of an atomic stationary state property is determined by the form of the atomic stationary state functional fl]. In precisely the same manner, the definition of an atomic property in the general time-dependent case is determined by the form of the atomic Lagrangian integral 2,t]. In both the stationary-state and... 

The system s possible stationary-state functions are given by (7.98) as T = where the functions are the eigenfunctions of H [Eq. 

The function xj/ can be chosen as any smooth function that integrates out with respect to V, I, q over [0, oo] x R x M. We already know from our discussion of Nose-Hoover that p = p, = (p M p - NdkBT)//iT will have as stationary states functions of the form... 

We can imagine a situation where a system in a stationary state is suddenly perturbed to produce a new time-independent hamiltonian. P will change as the system adjusts to this new situation, giving us a case where the hamiltonian is time-independent (after the perturbation, at least) but is not a stationary state function. The way in which P evolves in time is governed by Eq. (6-1). 

Suppose, for instance, v e choose to describe the starting wavefunction (x, 0) as a normalized half sine wave in the left side of the box and zero at the right (Fig. 6-3). Then we can calculate the amount (c ) of each of the stationary-state functions ij/n present in this function as follows ... 

For the quantum mechanical results that we require we shall be concerned only with stationary states, known sometimes as eigenstates. The wave functions for these states may be referred to as eigenfunctions and the associated energies E as the eigenvalues. 

Metastability implies that F E, so that the wave function inside the well is close to that of a stationary state with energy E°. Strictly speaking, the energy spectrum in this case is quasidiscrete with the density of states... 

We note that the virial theorem is automatically fulfilled in the Hartree-Fock approximation. This result follows from the fact that the single Slater determinant (Eq. 11.38) built up from the Hartree-Fock functions pk x) satisfying Eq. 11.46 is the optimum wave function of this particular form, and, since this wave function cannot be further improved by scaling, the virial theorem must be fulfilled from the very beginning. If we consider a stationary state with the nuclei in their equilibrium positions, we have particularly Thf = — Fhf, and for the correlation terms follows consequently that... 

Boys, S. F., Proc. Roy. Soc. London) A200, 542, Electronic wave function. I. A general method of calculation for the stationary state of any molecular systems." a. 

If the hamiltonian is truly stationary, then the wx are the space-parts of the state function but if H is a function of t, the wx are not strictly state functions at all. Still, Eq. (7-65) defines a complete orthonormal set, each wx being time-dependent, and the quasi-eigenvalues Et will also be functions of t. It is clear on physical grounds, however, that to, will be an approximation to the true states if H varies sufficiently slowly. Hence the name, adiabatic representation. 

Thus, the probability density depends only on the position variable x and does not change with time. For this reason the wave function W(x, t) in equation (2.31) is called a stationary state. If (jc, i) is normalized, then xp x) is also normalized... 

For stationary states, the eigenfunctions xp if, Q) may be chosen to be real functions, so that this integral can also be written as... 

Heterogeneous chemical reactions in which adsorbed species participate are not pure chemical reactions, as the surface concentrations of these substances depend on the electrode potential (see Section 4.3.3), and thus the reaction rates are also functions of the potential. Formulation of the relationship between the current density in the stationary state and the concentrations of the adsorbing species in solution is very simple for a linear adsorption isotherm. Assume that the adsorbed substance B undergoes an... 

Two other attempts, without the use of a distribution function, are worth mentioning, as these are operationally related to experiments and serve to give a rough estimate of the thermalization time. Christophorou et al. (1975) note that in the presence of a relatively weak external field E, the rate of energy input to an electron by that field is (0 = eEvd, where vd is the drift velocity in the stationary state. Under equilibrium, it must be equal to the difference between the energy loss and gain rates by an electron s interaction with the medium. The mean electron energy is now approximated as (E) = (3eD )/(2p), where fl = vd /E is the drift mobility and D is the perpendicular diffusion coefficient (this approximation is actually valid for a Maxwellian distribution). Thus, from measurements of fl and D the thermalization time is estimated to be... 

Silinsh et al. (1989) applied their thermalization procedure to naphthalene and anthracene at low temperatures, 35 K or less. A stationary state was envisaged in the presence of an external field. Calculations have been performed for the saturation drift velocity, friction coefficients, and effective mass as functions of the external field. The conclusions are almost the same as for pentacene. 

Solution of (12) gives the complete non-relativistic quantum-mechanical description of the hydrogen atom in its stationary states. The wave function is interpreted in terms of... 

There is of course attenuation of the signal, as shown in Fig. 5, taken from Joyner and Roberts (28) The gas phase spectrum will also be obtained, but this usually can be separated easily from the signal of the solid. This sample cell arrangement thus permits the study of the stationary-state surface during catalysis and also its evolution in response to pulses and step functions in the gas composition. The temperature of the sample should be controlled so that the surface can be studied during temperature-programmed desorption and reaction. 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

For a constant-volume batch reactor operated at constant T and pH, an exact solution can be obtained numerically (but not analytically) from the two-step mechanism in Section 10.2.1 for the concentrations of the four species S, E, ES, and P as functions of time t, without the assumptions of fast and slow steps. An approximate analytical solution, in the form of a rate law, can be obtained, applicable to this and other reactor types, by use of the stationary-state hypothesis (SSH). We consider these in turn. 

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