Ground state momentum

Numerical Hartree-Fock calculations, free from basis set artifacts, have been used to establish that the ground state momentum densities of all the atoms and their ions can be classihed into three types [84,85]. Type I and III momentum densities are found almost exclusively in metal atoms He, N, all atoms from groups 1-14 except Ge and Pd, and all the lanthanides and actinides. These momentum densities all have a global maximum at p = 0 and resemble the momentum density shown in Fig. 19.3 for the beryllium atom. The maximum atp = 0 comes mainly from the outermost s-subshell, 2s in this case. Type I and III densities dilfer in that the latter have a secondary maximum that is so small as to be invisible on a diagram such as Fig. 19.3. Type II densities are the norm for non-metallic atoms and are found in Ge, Pd and all atoms from groups 15-18 except He and... 

Figure Al.6.27. Equipotential contour plots of (a) the excited- and (b), (c) ground-state potential energy surfaces. (Here a hamionic excited state is used because that is the way the first calculations were perfomied.) (a) The classical trajectory that originates from rest on the ground-state surface makes a vertical transition to the excited state, and subsequently undergoes Lissajous motion, which is shown superimposed, (b) Assuming a vertical transition down at time (position and momentum conserved) the trajectory continues to evolve on the ground-state surface and exits from chaimel 1. (c) If the transition down is at time 2 the classical trajectory exits from chaimel 2 (reprinted from [52]).

There are significant differences between tliese two types of reactions as far as how they are treated experimentally and theoretically. Photodissociation typically involves excitation to an excited electronic state, whereas bimolecular reactions often occur on the ground-state potential energy surface for a reaction. In addition, the initial conditions are very different. In bimolecular collisions one has no control over the reactant orbital angular momentum (impact parameter), whereas m photodissociation one can start with cold molecules with total angular momentum 0. Nonetheless, many theoretical constructs and experimental methods can be applied to both types of reactions, and from the point of view of this chapter their similarities are more important than their differences. 

Of the variety of quantum effects which are present at low temperatures we focus here mainly on delocalization effects due to the position-momentum uncertainty principle. Compared to purely classical systems, the quantum delocalization introduces fluctuations in addition to the thermal fluctuations. This may result in a decrease of phase transition temperatures as compared to a purely classical system under otherwise unchanged conditions. The ground state order may decrease as well. From the experimental point of view it is rather difficult to extract the amount of quantumness of the system. The delocahzation can become so pronounced that certain phases are stable in contrast to the case in classical systems. We analyze these effects in Sec. V, in particular the phase transitions in adsorbed N2, H2 and D2 layers. 

Split valence basis sets allow orbitals to change size, but not to change shape. Polarized basis sets remove this limitation by adding orbitals with angular momentum beyond what is required for the ground state to the description of each atom. For example, polarized basis sets add d functions to carbon atoms and f functions to transition metals, and some of them add p functions to hydrogen atoms. 

It can now be seen that there is a direct and simple correspondence between this description of electronic structure and the form of the periodic table. Hydrogen, with 1 proton and 1 electron, is the first element, and, in the ground state (i.e. the state of lowest energy) it has the electronic configuration ls with zero orbital angular momentum. Helium, 2 = 2, has the configuration Is, and this completes the first period since no... 

Here Uj and Uj are Cartesian unit vectors, a) and j3) are localized orbitals that are doubly occupied in the HF ground state, jm) and n) are virtual orbitals. Rq is the position vector of the local gauge origin assigned to orbital a) and = (r — R ) x p is the angular momentum relative to Re- Superscript 1 denotes terms to first order in the fluctuation potential, and = [A — is the principal propagator at the zero energy... 

What is the probability density as a function of the momentum p of an oscillating particle in its ground state in a parabolic potential well (First find the momentum-space wave function.)... 

The angular momentum quantum numbers are often given letter designations, so that when they are stated along with principal quantum numbers, less confusion results. The letter designations of importance in the ground states of atoms are the following ... 

This mode of hyperfine interaction will become important only when the impaired electron is able to partially occupy a low-lying excited state (AE small), and the ground state has orbital angular momentum (L 0). The adsorbed nitric oxide molecule and the superoxide ion with 170 are typical examples where hyperfine coupling via spin-orbit interaction may be observed. 

---