The electrostatic interaction H

The electrostatic or Coulombic interaction between two electrons is given by (equation 3.3)

H, = 1 e*/r,j 1>4

« ________________ ef________________ 3.41

<<ri)a + (rj) 2 - 2r,rjcos(w)),/a

where w is the angle between the radii vectors r, and r, of the two electrons.

Alternatively the denominator in equation 3.41 can bo expanded as a series of Legendre polynomials P* C173

<<r.)* ♦ <rj)* - 2r,rjcos(w)>-*'* - I i r ' l r * * 1)Pkcos(w) 3.42

where r< it the lesser and r> is the greater of r, and r t . Furthermore the

spherical harmonics addition theorem C51 states that

PkCOS(w) 3 <4TT/Ckl) l Yl* (0. ,0t) Y2 (0j, p j )

q

* l (-ll'MClf’), <Cy»lj

giving Pkcos(w) * < C y ,.C,Jk»> 3.43

where the tensor operator C ‘k> is defined by

Ci,k* = (4n/Ckl) 1 / 3 Y! (0,0) 3.44

The subscript to C ‘k> in equation 3.43 indicates that it is a function of the coordinates of a particular electron. The electrostatic interaction is therefore expressed in the tensor operator form as

H, = ea I (r<k/r,k*•1) (C.,k’. C ‘,k ’ ) 3.45

k *

The matrix elements of Hi are eost easily calculated for an LS-coupling scheme where the matrix is diagonal. Considering the angular component initially, the general matrix element is given by

< (e.Sa)S, (1 11 s ) L , J , M I C V ' . C V ’ l(SiSa)8‘,(lila)L',J‘,M'> 3.46

A simplification is afforded by the fact that for a two electron system it is only possible to have S * 0 or 1. This gives rise to two sets of terms called singlet and triplet terms respectively. Each term is given by

(equation 3.38)

< 11,1 a , LI t V ' - i V ’ 111,1,,L' > 47

3.47 - ( _ n i, * i , . . . j j * j * J j ] a ( L , L - K l , | | C*"‘ H l . X l . l l CV» 111,)

- f„ say

The reduced matrix elements are given in general by C161

<1 II C ‘k* 111 ' > » (-1)» (Cl! II • 3> •'* ( l k 1 \ 3.48 VO 0 0 /

so that fa - ¡- ¡J n. Hl.lft* 0 o K o2 0 O2) 3 ’ 49

The 3-j symbols of the type have the property that they are

2ero if the sum j. + j, + j3 is odd. If the s u b is even then the 3-j symbol

can be evaluated according to the following algebraic formula C161

(j * j * j a ( - n j'a ( J-2 j , )! ( J-2j,)! ( J-2 j_3)!]'

<J + 1>! ) ( J / 2 ) ! 3.30

where J ■ j. ♦ j, ♦ j3 . The total number of k values necessary for the description of a particular configuration is limited by the specific fora of the 3-j symbols. This is also known as the triangular condition and states that the matrix element is zero unless the sua 1 + k ♦ 1 is even and k < 2 1. Therefore all k values are even and k*a* > 21am.

The radial component is normally left in the fore of a radial integral designated by Fk

Fk <r<k/r£*‘) (R^<r. IR^.(rj)l2 dr. dr., 3.31

o o

where R„i(r) is the radial function for the state with quantum numbers n,l. Fk is known as the direct Slater integral.

While the above calculation would suffice for two equivalent electrons, that is n, > n, and 1. » 1,, a core general case would involve electrons in

different atomic shells. In this case it is necessary to consider the

quantum mechanical exchange term in addition to the direct term evaluated above. The exchange interaction involves the general matrix element

3.52

and has a sign which is dependent on the way in which the electron spins are coupled. 1f both electrons have their spin vectors parallel giving S ■ 1

(triplet state), then the exchange term is negative. If the spins are antiparallel giving a singlet state with S > 0 then it is positive. To distinguish between these two possibilities we Introduce the factor -(1/2 + 2(si.sr )>. The matrix element is therefore given by

The radial integral 6“ , also known as the indirect Slater integral, is similarly

Once again only certain values of k satisfying the relations 1. ♦ k + I2 « even and k < 1, + 12 need be considered. However in this case k may be either even or odd.

Nhen the term energies for a given configuration have been determined

it is observed that they involve a commonly repeated denominator D„. In

order to simplify the presentation of these terms the Slater-Condon parameters

F„ and 6k are introduced. These parameters are related to the radial

integrals Fk and 6h by - U / 2 + 2(Si.sa) ) (-1 ) »• *»»*■- 6 (L,L ' ) <1 , Il C ,‘" Il la>*fT. 1. Li . X. k J 3.53 6- » e* j (r^/rï**) (Rni,,(r. >R„.,,(r j >IW,(r* »R^lr j ) ) dr.dr., 3.54

:i f

m

F * = F k / Dl r

G* = Gk / D k

3.55

It is these corrected parameters which are quoted in most experimental studies.

A determination of the matrix elements of Hi in a jj-coupling scheme is more complicated. Probably the easiest method is to take the terms as calculated in the LS-couplinq scheme and then transform them into the jj-couplinq scheme usinq the recouplinq equation (equation 3.25). The complete transformation is qiven by

< ( • » ! i ) j i , < S z l z > j * , J | H I ( S t 1 i ) j , ' , ( s z l z ) j * ' , J •> * 6 < J , J ‘ > U j i H j z H j t ' 3 C j 2 ' 3 ) Z [ S 3 C L ] < S i l i S p s 1 2 L S i 1 1 S z S l z L 8 , L. j . j 2 j i ’ j z ' J_ x < < S i S z ) S , ( 1 X12 > L , JI H I ( S i S z ) S ' , ( 1 , U ) L • , J •> 3 . 5 6