Equation of Motion (EOM) derivation of RPA

The Equation of Motion derivations of excited state and response properties are elegant, and (in my opinion) very direct. Here I’ll give an example of how it can be used to derive TDHF/RPA. We’ve already done it two other ways, and the agreement between them is important if we wish to extend these methods further.

Given an exact ground state, we can say that

Define operator and :

These operators generate excited states from the ground state (not excited determinants, as in the case of post-HF correlation methods). So it is clear that, when acting on an exact ground state:

Multiply on left by arbitrary state of form , giving

Where we have made use of the fact that . Note that is we express by particle-hole operators , with coefficients and , then is given by for arbitrary variations . These are in principle exact, since exhausts the whole Hilbert space, such that the above equation corresponds to the full Schrödinger equation. Tamm-Dancoff (or Configuration Interaction Singles) can be obtained by approximating and the operator , restricting ourselves to 1p-1h excitations. Thus , ( cancels), and

These are the CIS equations. Put another way:

Similarly, for RPA/TDHF, if we consider a ground state containing 2p-2h correlations, we can not only create a p-h pair, but also destroy one. Thus (choosing the minus sign for convenience):

So instead of the basis of only single excitations, and therefore one matrix , we work in a basis of single excitations and single de-excitations, and have two matrices and . We also have two kinds of variations , namely and . This gives us two sets of equations:

These contain only expectation values of our four Fermion operators, which cannot be calculated since we still do not know . Thus we assume . This gives

The probability of finding states and in excited state , that is, the p-h and h-p matrix elements of transition density matrix are:

Thus altogether

with

and

which are the TDHF/RPA equations.