Non-interacting

One can utilize some very simple cases to illustrate this approach. Suppose one considers a solution for non-interacting electrons i.e. in equation A1.3.1 the last temi in the Hamiltonian is ignored. In diis limit, it is... 

Perhaps the simplest description of a condensed matter system is to imagine non-interacting electrons contained within a box of volume, Q. The Scln-ddinger equation for this system is similar to equation Al.3.9 with the potential set to zero ... 

Thus many aspects of statistical mechanics involve techniques appropriate to systems with large N. In this respect, even the non-interacting systems are instructive and lead to non-trivial calculations. The degeneracy fiinction that is considered in this subsection is an essential ingredient of the fonnal and general methods of statistical mechanics. The degeneracy fiinction is often referred to as the density of states. 

We first consider tlnee examples as a prelude to the general discussion of basic statistical mechanics. These are (i) non-mteracting spin-i particles in a magnetic field, (ii) non-interacting point particles in a box,... 

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

As a prelude to discussing mean-field theory, we review the solution for non-interacting magnets by setting J = 0 in the Ising Flamiltonian. The PF... 

This can be seen by considering the second-order MPPT energy of two non-interacting Be atoms. The reference CSF is = lsi 2s- ls 2s as discussed earlier, only doubly-excited CSFs contribute to the... 

Kolm and Sham [25] decompose G[p] into the kinetic energy of an analogous set of non-interacting electrons with the same density p(r) as the interacting system. 

Here the transition state is approximated by the lowest crossing pomt on the seam intersecting the diabatic (non-interacting) potential energy surfaces of the reactant and product. The method was originally developed... 

The kinetic theory of gases has been used so far, the assumption being that gas molecules are non-interacting particles in a state of random motion. This... 

This equation has a different temperature coefficient from that calculated from kinetic tlreory, and shows a shortcoming in the theory which is due to the assumption of non-interaction between the molecules. 

The equation of state for a gas consisting of non-interacting point particles has the form... 

The classical kinetic theoty of gases treats a system of non-interacting particles, but in real gases there is a short-range interaction which has an effect on the physical properties of gases. The most simple description of this interaction uses the Lennard-Jones potential which postulates a central force between molecules, giving an energy of interaction as a function of the inter-nuclear distance, r. 

Solving now the Heisenberg equations of motion for the a operators perturbatively in the same way as in the weak-coupling case, one arrives (at = 0) at the celebrated non-interacting blip approximation [Dekker 1987b Aslangul et al. 1985]... 

The dispersion of a solute band in a packed column was originally treated comprehensively by Van Deemter et al. [4] who postulated that there were four first-order effect, spreading processes that were responsible for peak dispersion. These the authors designated as multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase and resistance to mass transfer in the stationary phase. Van Deemter derived an expression for the variance contribution of each dispersion process to the overall variance per unit length of the column. Consequently, as the individual dispersion processes can be assumed to be random and non-interacting, the total variance per unit length of the column was obtained from a sum of the individual variance contributions. 

This review is structured as follows. In the next section we present the theory for adsorbates that remain in quasi-equilibrium throughout the desorption process, in which case a few macroscopic variables, namely the partial coverages 0, and their rate equations are needed. We introduce the lattice gas model and discuss results ranging from non-interacting adsorbates to systems with multiple interactions, treated essentially exactly with the transfer matrix method, in Sec. II. Examples of the accuracy possible in the modehng of experimental data using this theory, from our own work, are presented for such diverse systems as multilayers of alkali metals on metals, competitive desorption of tellurium from tungsten, and dissociative... 

We start with a non-interacting molecular adsorbate for which the chemical potential is given by... 

To gain some qualitative insight into the elfect of lateral interactions it is useful to employ simple analytical approximations in the calculation of the chemical potential, of which the quasichemical approximation is the best suited. We split the chemical potential into a non-interacting part, Eq. (10), and a term due to lateral interactions, and get for the... 

FIG. 11 Adsorbed amount as a function of bulk concentration for a non-interacting (empty symbols) and adsorbing (full symbols) wall. Diamonds and triangles correspond to a system with semi-rigid chains, circles and squares for flexible chains [28]. 

Suppose now that we have an ensemble of N non-interacting particles in a thermally insulated enclosure of constant volume. This statement means that the number of particles, the internal energy and the volume are constant and so we are dealing with a microcanonical ensemble. Suppose that each of the particles has quantum states with energies given by i, 2,... and that, at equilibrium there are Ni particles in quantum state Su particles in quantum state 2, and so on. 

If we deal with N isolated non-interacting entities such as the molecules in a gas at low density, we can further divide up molecular energies with reasonable accuracy into their electronic, vibrational and rotational contributions... 

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