Electron interaction energy in isospin basis

The coefficients of the exchange integrals of electrostatic interaction are equal to the matrix elements of the operator 

in the limiting case, where the expansion of the electrostatic interaction operator in terms of the multipoles (see (19.6)) includes only the central-symmetric part (i.e. only the terms with k = 0), dependent on the term in (18.52) is only the summand with the operator T2.1116 eigenvalues of the operator T2, according to (18.28), are equal to T(T + 1), i.e. in this approximation we obtain the spectrum of energy levels rotational with respect to isospin. 

It follows from the general behaviour of isospin basis that the isospin quantum number will be good if an electron belonging to one shell interacts with another electron belonging to the same shell in exactly the same way as with an electron in another shell. In so doing, we should include only the part of electron-electron interaction that depends on the term. This is described by (k 0) 

By way of example, Table 18.3 provides the squares of the largest weight coefficients a ax, / ax of the wave functions that are obtained using radial hydrogen orbitals after the matrix of the electrostatic interaction operator has been diagonalized in a conventional basis 

When the isospin quantum number is good , the states characterized by a larger value of the isospin T have a higher energy, which is mainly conditioned by the presence of the rotational term T(T + l)Go(nil,n2l) in diagonal (in isospin basis) matrix elements of the electrostatic energy operator. 

In the general case, this equation does not hold, of course. But for 

The calculations of these integrals show that there are notable cases for n ln2lNl configurations at n2 = + 1 where the isospin quantum number 

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Relativistic energy of electron-electron interaction in isospin basis... 

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