Atoms expectation values

Recurrence relation for hydrogen-atom expectation values... 

The pressure is usually calculated in a computer simulation via the virial theorem ol Clausius. The virial is defined as the expectation value of the sum of the products of the coordinates of the particles and the forces acting on them. This is usually written iV = X] Pxi where x, is a coordinate (e.g. the x ox y coordinate of an atom) and p. is the first derivative of the momentum along that coordinate pi is the force, by Newton s second law). The virial theorem states that the virial is equal to —3Nk T. 

There is no reason to believe that the off-diagonal Q integrals which yield this increase in U over I should be equal for longitudinal and transverse polarization but neither is there at present any basis for selecting different values. Hence we assume U1 = U2 and by use of the a values given above for the chlorine atom, the value of —El is calculated to be about 30 per cent smaller than was obtained in Table VI. Since the effect of anisotropy would be expected to be about the same for all of the halogens, the qualitative conclusions drawn from the results in Table VI are not affected. It is clear that anisotropy may be important, however, and must be considered in quantitative work. 

Desclaux, J.P. (1973) Relativistic Dirac-Fock expectation values for atoms with Z = 1 to, Z = 120. Atomic Data and Nuclear Data Tables, 12, 311—406. 

For ionic compounds, crystal field theory is generally regarded a sufficiently good model for qualitative estimates. Covalency is neglected in this approach, only metal d-orbitals are considered which can be populated with zero, one or two electrons. To evaluate (Vzz)vai 4t the Mdssbauer nucleus, one may simply take the expectation value of the expression — e(3cos 0 — for every electron in a valence orbital i/, of the Mdssbauer atom and sum up,... 

The expectation values of powers and inverse powers of r for any arbitrary state of the hydrogen-like atom are defined by... 

Show explicitly for a hydrogen atom in the Is state that the total energy is equal to one-half the expectation value of the potential energy of interaction between the electron and the nucleus. This result is an example of the quantum-mechanical virial theorem. 

The expectation values of various powers of the radial variable r for a hydrogenlike atom with quantum numbers n and I are given by equation (6.69)... 

The crystal data compared to expected values assuming no distortions are summarized in Table 18.1. Inspection of the atomic coordinates reveals that the distortions of the packing of spheres are only marginal. As expected, the greatest deviations are observed for the molecular compounds PI3 and NMe3. 

In accord with the resonance structure drawn, there is little B=B bonding in this diborane(4) derivative and the B—B distance is found to be 1.859 A,68 which is considerably longer than even the normal expected value of 1.7 A for a boron-boron single bond.67 This is consistent with the normal repulsion of negative charges on adjacent atoms not stabilized by 7r-bonding. 

---