Transition Dipoles for Excitations to Singlet and Triplet States

9 Transition Dipoles for Excitations to Singlet and Triplet States 

As we discussed in Chap. 2 (Sect. 2.4), electrons have an intrinsic angular momentum or spin that is characterized by spin quantum numbers s = 1/2 and = 1/2. The different values of Wj can be described by two spin wavefunctions, a ( spin up ) for nig =+1/2 and p ( spin down ) for m = —l/2. For systems with more than one electron, wavefunctions that include the electronic spins must be written in a way so that the complete wavefunction changes sign if we interchange any two electrons. We skipped over this point quickly when we derived expressions for the transition dipole for forming an excited singlet state (Eqs. 4.22a-4.22e), so let s check whether we get the same expressions if we write the spin wavefunctions explicitly. We ll also examine the transition dipoles for excitation to triplet states. 

Using the notation of Eqs. (2.47), the transition dipole for excitation from a singlet ground state to an excited singlet state of a system with two electrons takes the form 

Because the electric dipole operator does not act on the spin wavefunctions, integrals such as (a(l)la(l)) and (a(l)ip(l)) can be factored out of the overall integral. By doing this and making the same approximations we made above in Eqs. (4.22a-4.22c), we obtain 

The transition dipoles for transitions from the singlet ground state to the excited triplet states are very different from the transition dipole for forming the excited singlet state. The transition dipoles for forming the triplet states aU evaluate to zero because terms with opposite signs cancel or because each term includes an integral of the form (a(l)ip(l)). For we have  

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