Atoms central force system

The hydrogen atom is an example of a central-force system... 

The Yim functions and the possible values of and are the same for any central-force system, no matter what the potential energy function fir) is. In a later chapter we will study a model for the rotation of a diatomic molecule that is a central-force system. The angular momentum of this model takes on the same values as that of the hydrogen atom. 

The Finnis-Sinclair type potentials (Finnis and Sinclair 1984) are central-force potentials but have a many-body character in that the energy of a system of particles is not merely a sum of pair interactions between individual atoms. In this scheme, modified for binary alloys by Ackland and Vitek (1990), the total energy of a system of N atoms is written as... 

Now, Lz is a constant because of central forces, and electron/positron spin is also a constant given by Eq. (87). Hence the magnitude of photon spin is also a constant, which has been equated to nh in Eq. (90). Parameter n allows for different levels of energy of the system. Quantization is introduced as in Bohr s atom assuming that orbits are stable for values of energy corresponding to integer values of n. 

The nuclear model of the atom, as envisioned by Rutherford and Bohr, had much in common with the solar system. In each there is a massive core that exerts a controlling influence over less massive satellites orbiting around the central core. In both the solar system and the atom, the force between the central core and the orbiting satellites decreases as the square of their separation. In the case of the solar system, it was Johannes Kepler, early in the seventeenth century, who first allowed hard data—data he knew to be accurate—to sit in judgment on his speculations about the orbits of the Sun s planets. 

In an exposition which aims to encompass general systems and ensembles, it is appropriate to make use of the Hamiltonian version of dynamics. In this view forces do not appear explicitly and the dynamics of the system evolve so as to keep the Hamiltonian function constant. In Newtonian dynamics forces appear explicitly and molecules move as a response to the forces they experience. For our purposes, the Newtonian view is sufficient since we will illustrate the large scale computational aspects with simplest possible particles, atoms with spherical, central force fields. The same principles hold for molecules with internal degrees of freedom as well. 

We prefer to use the gaussian system in applications to atomic phenomena. (In any event, it will not matter once we change to atomic units.) Since the Coulomb attraction is a central force (dependent only on / ), the potential energy is related by... 

Consider a system of N identical atoms interacting through central forces (Skryshevsky, 1980). Choose a set of n atoms (n = 1, 2,..., 1V — 1) inside it. The probability of these atoms being placed in space units dVy, dVt,..., dV near the ends of the vectors f), f, ... jf), from a fixed atom when the others N — n) are placed arbitrarily is represented by... 

One of the simplest approximations employed in reducing the number of independent force constants is the assumption of central forces. It is assumed that the forces holding the atoms in their ciiuilibrium positions act only along the lines joining pairs of atoms and that every pair of atoms is connected by such a force. This type of force function would result if the molecule were held together by purely ionic interactions. Also, this type of force yields only diagonal terms in the force constant matrix when the internal coordinate system is the complete set of interatomic distances (the central force coordinates). 

Two central problems remain. One is that one needs the potential which governs the motion. In many-atom systems, even if the motion is confined to the ground electronic state, this potential is a function of the spatial configuration of all the atoms. It is therefore a function of many variables, so its analytical form is far from obvious, nor do we necessarily want to know it everywhere. Indeed, we really only want it at each point along the actual trajectory of the system (so that the forces can be computed and thereby the next point to which the system will move to can be determined). Such an approach has been implemented [25] and applied to many-atom systems, and an extension to a multi-electronic state dynamics will be important... 

The central point within our consciousness, our "spirit" in the hermetic sense, can now be seen as an entity that can work to control quantum probabilities. To our "spirits" our brain is a quantum sea providing a rich realm in which it can incarnate and manifest patterns down into the electrical/chemical impulses of the nervous system. (It has been calculated that the number of interconnections existing in our brains far exceeds the number of atoms in the whole universe - so in this sense the microcosm truly mirrors the macrocosm ). Our "spirit" can through quantum borrowing momentarily press a certain order into this sea and this manifests as a thought, emotion, etc. Such an ordered state can only exist momentarily, before our spirit or point of consciousness is forced to jump and move to other regions of the brain, where at that moment the pattern of probability waves for the particles in these nerve cells, can reflect the form that our spirit is trying to work with. 

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