Hamiltonian eigenvalues

By Eq. (6) the sum on the right-hand side of the above equation is equal to the energy E, and from Eq. (2) we realize that the sums on the left-hand side are just Hamiltonian operators in the second-quantized notation. Hence, when the 2-RDM corresponds to an A -particle wavefunction i//, Eq. (12) implies Eq. (13), and the proof of Nakatsuji s theorem is accomplished. Because the Hamiltonian is dehned in second quantization, the proof of Nakatsuji s theorem is also valid when the one-particle basis set is incomplete. Recall that the SE with a second-quantized Hamiltonian corresponds to a Hamiltonian eigenvalue equation with the given one-particle basis. Unlike the SE, the CSE only requires the 2- and 4-RDMs in the given one-particle basis rather than the full A -particle wavefunction. While Nakatsuji s theorem holds for the 2,4-CSE, it is not valid for the 1,3-CSE. This foreshadows the advantage of reconstructing from the 2-RDM instead of the 1-RDM, which we will discuss in the context of Rosina s theorem. 

According to (1.22) and (1.26), the eigenvalues v of the Liouvillian L are distributed symmetrically around the point v = 0, and this implies that, even if the Hamiltonian H in physics is bounded from below, H > a 1, the Liouvillian L is as a rule unbounded. Except for this difference, practically all the Hilbert-space methods developed to solve the Hamiltonian eigenvalue problem in exact or approximate form may be applied also to the Liouvillian eigenvalue problem. In the time-dependent case, the L2 methods developed to solve the Schrodinger equation are now also applicable to solve the Liouville equation (1.7). 

In order to discuss also this aspect of the problem, we will return to the original approximate solution D = Bd. The two functions and [ = D are usually neither orthogonal nor noninteracting with respect to H and, in such a case, one can use the adaption procedure developed in the preceding subsection. Using the functions and f as a basis of order 2 for a Hamiltonian eigenvalue problem, one obtains... 

Here we will assume that the reference function has been chosen so that all the functions k are linearly independent and form a truncated basis = < of order m in the carrier space. We will again study the Hamiltonian eigenvalue problem in terms of this basis. The Hamiltonian and metric matrices are given by the expressions (2.24) with the difference that they are now of order m x m. The approximate eigenfunctions are now given by the relations... 

The coefficients a(q) and /% determine the fraction of light (with wavevector q) and the fraction of the excitation on the th molecule in the polaritonic states respectively. From the Hamiltonian eigenvalue problem we find that the coefficients a(q) and /% obey the following system of equations ... 

From the Hamiltonian eigenvalue problem, N , NF, and Np(k) obey the equations... 

Revise your solution to the double-well problem 8.52 so as to treat even and odd functions separately. Do this by introducing a parameter whose value is 1 or 0, depending on whether we are treating the even or the odd wave functions. Calculate the lowest two Hamiltonian eigenvalues using the first 16 even pib functions. Repeat with the lowest 16 odd pib functions. Compare your results with those of Problem 8.52. 

For simultaneous analysis of the Hamiltonian eigenvalues and eigenvectors (related to the valence zone, as the most interesting for chemical purposes), the local density of the electronic states (LDES)... 

We can then associate finding the approximate Hamiltonian eigenvalues with its matrix in the T sd basis ... 

We can now relate Eq to one particular Hamiltonian eigenvalue in the following manner... 

---