Derivation of general polarization propagator methods

Most higher-order response methods use some form of the polarization propagator, which is what I intend to derive here. It can then be used to derive other methods, such as TDHF/RPA and SOPPA.

We have derived earlier a response function, or frequency dependent polarizability,

where is the applied perturbation, and is the observable, and both are assumed to be Hermitian. is the excitation energy for the change between states and . It should be clear that the response function has poles when — the applied field frequency – equals to the excitation energy . Finding these poles is precisely the goal of polarization propagator methods. In the polarization propagator approach, the above equation has set to 0, and the response function (the `propagator’), defined as:

Now we want to describe the propagator in terms of commutators between and . Make the observation that , and applying to the first term of the above yields:

Do the same for the second term and combine, recognizing that the term vanishes in the first part (thus we get a sum over all ), and making use of the fact that and and :

Which is to say that

Or, as we will use it:

As you may have started to see, we can define the propagator iteratively in terms of commutator expectation values of ever-increasing complexity. This is what is known as the so-called ‘‘moment expansion’’ of the propagator. Thus by iteration:

We introduce the ‘‘superoperator’’ (analogous to the Liouville operator in Statistical Mechanics), which acts on operators to give their commutator:

With this definition, we have the power series

At this point we make two useful observations. First, recognize that

and so can be applied to instead of insofar as we introduce a factor of . Furthermore, note that the power series is equivalent to

Making use of these two observations (and using and , where is the unit superoperator), we have

Which is merely a cosmetic change at this point, as the superoperator resolvent is defined by the series expansion. We need to find a matrix representation of the resolvent, which implies that we find a complete basis set of operators. To do this, we are going to develop an operator space, where is defined by its effect on operators instead of vectors. Introducing the notation

and it follows that . As defined, we now have

Which is formally exact, albeit useless until we develop approximations. However, the form of the above equation does look similar to ordinary vector spaces in Hartree-Fock, etc. methods. Truncation of a basis in linear vector space to elements produces a subspace , and truncation of a general vector corresponds to finding its projection onto the subspace. It follows, then, that we need to find a projection operator , associated with the truncated basis. If the basis (, say) is orthonormal we write

which in a complete basis gives:

If it is not an orthonormal basis, we must include the metric matrix (or L"{o}wdin basis ):

When using a truncated basis in operator space, two kinds of projections are useful (Löwdin, 1977, 1982),

which are the outer projection and inner projection, respectively, onto space defined by . Note that does not imply does not imply . Plugging the metric into :

and we define

We assume that is Hermitian and positive-definite, so that can be defined. Note that . Because is arbitrary, replace it with , and since with :

As the basis , the inner projection , else it is simply a finite basis approximation to the inverse. This is the operator inverse in terms of a matrix inverse. Since was an arbitrary basis defining , let define n-dimensional subspace . Thus:

Thus the inner projection leads to an approximation for the projector. Let us define the (as of yet undefined) operator basis:

Given that the binary product (compare with for vectors)

then for our resolvent superoperator we have

where and are the analogues of and in operator space. Finally, if

then we have

which is the key to calculating approximations to response properties. The matrix M is determined once we have chosen an operator basis. This approximation depends on two things: 1) the basis (and its truncations), and 2) the reference function, that is not the exact ground state. Any approximations to these two things are where we get out various response methods.