Excited States and the Multiplet Problem

How can we obtain an energy for the excited singlet The normal prescription is not applicable since there is no single determinant on which a Kohn-Sham calculation could be based. However, the determinant that intuitively comes closest to this state is 

This determinant has the desired Ms = 0, but its total spin is not defined. Now comes the trick we recognize that equation (5-34) is actually a mixture of the functions (5-32) and (5-33) of the Ms = 0 states (which can easily be verified by expanding the determinants), 

Now the procedure to get the energy of the singlet is outlined after reordering equation (5-35) and changing to energies rather than determinants we have 

Here tOj is the excitation energy ErE0 and the sum runs over all excited states I of the system. From equation (5-37) we immediately see that the dynamic mean polarizability a(ro) diverges for co,=co, i. e., has poles at the electronic excitation energies o ,. The residues f are the corresponding oscillator strengths. Translated into the Kohn-Sham scheme, the exact linear response can be expressed as the linear density response of a non-interacting 

17 Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in die corresponding time-dependent Kohn-Sham equations and which is rigorously defined as die functional derivative of die exchange-correlation action Axc[p] with respect to die time-dependent electron-density is approximated as die functional derivative of the standard, time-independent Exc with respect to die charge density at time t, i. e., 

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