﻿$62 Mr. J. H. Van Vleck on the normal Helium Atom 



the power series development in the parameter w. For w = 0,. 

 the perturbations p and s vanish, and C = *875 ; this is an 

 exact solution, which is nothing else than the Bohr helium 

 atom, in which the two electrons move in the same plane* 

 As the constant o£ integration w is given increasing values,, 

 the orbits intersect each other at greater angles, and the 

 perturbations become larger. 



Evaluation of Energy. 



After the orbits have been determined, the next step is to 

 compute the energy, so that the ionization potential may be 



calculated. If we eliminate cj> by means of the relation (3) 

 and use the reduced units of (7), the expressions for the 

 kinetic and potential energies given in equations (1) and (2) 

 become 



T _8*r (>+£) ., oi 



A L B 2 r 2 / 



v = e2 ri i=l 



A|_2r v V+r-J' 



Next, making use of the relations r — rj -t-p, f=S + s, expand 

 the various powers of r and f as power series in the perturba- 

 tions p and 5, neglecting terms of third and higher orders* 

 This gives 



! f ° \ +W + ji (2v'p + 2Ss +? + p 2 ) 



-Z(p 2 v 2 -±2 v Sps+ SV)]. 



If w = 0, so that the two electrons move in the same plane, 

 p = s = and C = '875 : the above expressions then reduce to 



T- li* V- _ If* 

 4a ' ~ 2a' 



so that we have the familiar expressions for the energy of 

 the Bohr helium atom. We thus see that practically all the 

 terms in the above expressions for the energy are pertur- 

 bative terms of small magnitude. 



