Electron Theory of Chemistry to Solids. 



733 



electrons varying inversely as the cubes of the distance, and 

 also forces which we have represented by f{r) between the 

 positive charges. It will be convenient to divide the potential 

 energy up into two parts — one due to the forces varying 

 inversely as the square of the distance, the other to 

 the supplementary forces — and to calculate these parts 

 separately. 



Let us take first the potential energy possessed by a single 

 electron ; this is e 2 X(l/r), where the summation extends over 

 all the other charges. If we try to calculate %(l/r) by 

 taking first the electrons and then the positive particles we 

 get divergent series which are unmanageable ; if, however, 

 we regard the solid as built up of cubes with a unit positive 

 charge at the centre and 1/8 of a unit negative charge at 

 each of the eight corners, and, taking this as our unit, 

 calculate the potential energy of the electron with reference 

 to each of the cells, we get a series which rapidly converges 

 and which can be calculated without great difficulty. 



Suppose is the electron for which we wish to calculate 

 2(1/ r). will be the meeting-place of eight cells, remove 



Fi<r. 3. 



^ 



s* 



S 









/ 









K 















13 







^ 







y 







~2L 







s 



L 





H 











S 







/ 





/ 



/ 





/ 



/ 





these, leaving a cubical cavity around O. Then 2(l/r) will 

 be the sum of terms due to the cells outside the cavity, plus 

 the part due to the charges we have removed by taking away 

 the eight cells. These charges were —1/8 at each of the 

 eight corners, ABCDEFGrH, -1/4 at each of the twelve 

 points like K where two cells meet, and charges —1/2 at 

 each of the six points like L where four of the cells which 

 we have removed meet ; in addition, there are the eight posi- 

 tive charges at the centres of the cubes which have been 



removed. The value of 

 side oi a cube, 



e 2 %- for these charges is, if d = the 



z&+&+*-%)-~™i 



