INDEX Eigenvalues

Here E, are the Dirac eigenvalues, index s runs over the complete spectrum of the Dirac Hamiltonian. This spectrum includes the solutions with positive and negative energy values E,. For the bound-state solutions the... 

Those derived from the adjacency matrix the total adjacency index, the Zagreb group indices, the Randic connectivity index, the Platt index, the compatibility code, and the largest eigenvalue index ... 

This is because no four-indexed two-electron integral like expressions enter into the integrals needed to compute the energy. All such integrals involve p(r) or the product p(/)p(r) because p is itself expanded in a basis (say of M functions), even the term p(r)p(r) scales no worse than tvF. The solution of the KS equations for the KS orbitals ([). involves solving a matrix eigenvalue problem this... 

The electronic Hamiltonian and the comesponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on m. The index i in Eq. (9) can span both discrete and continuous values. The q ) form... 

In the following, we will consider a certain eigenvalue X and drop the lower index k. Expanding each one of the functions 0m (= 0km) in the complete set we obtain... 

Some or all of the eigenvalues may be degenerate, but each eigenfunction has a unique index i. Suppose further that the system is in state aj), one of the eigenstates of A. If we measure the physical observable A, we obtain the result aj. What happens if we simultaneously measure the physical observable B To answer this question we need to calculate the expectation value (B) for this system... 

Thus, the operators H and have the same eigenfunctions, namely, the spherical harmonics Yj iO, q>) as given in equation (5.50). It is customary in discussions of the rigid rotor to replace the quantum number I by the index J m the eigenfunctions and eigenvalues. 

In addition, we should beware that the indexing of state variables in MATLAB is in reverse order of textbook examples. Despite these differences, the inherent properties of the model remain identical. The most important of all is to check the eigenvalues ... 

It is convenient to take the U(2) z> 0(2) symmetry of the preceding section as the starting point for approximations. Since it is unnecessary to carry the index z or x, the wave functions can be written simply as IN, m >. Denoting by C2 the Casimir operator of 0(2) with eigenvalues... 

Note that the sum over i it/means over all states except that indexed i =/, which we choose to denote as the state having the largest eigenvalue. 

We make a small digression and note that the spin-degeneracy problem we have alluded to before is evident in Eq. (5.102). It will be observed that / = 1,..., /x in the index of e s pnp these functions are linearly independent since the efj are. There are, thus, fi linearly independent spin eigenfunctions of eigenvalue S(S + 1). Each of these has a full complement of Ms values, of course. In view of Eq. (5.40) the number of spin functions increases rapidly with the number of electrons. Ultimately, however, the dynamics of a system governs if many or few of these are important. 

The above equations have to be complemented with rules for dealing with the singularities emerging when PHP has eigenvalues at energy E, which is the case primarily when the Wp) states are continuum states. In such a case, we denote /p) = c,E 1) and ) = Ic.ii"), with c standing for all quantum numbers related to a continuum channel (the channel index). The notation E = E - iO, serves to remind of the incoming boundary conditions used for example, for the c,E ) states one has... 

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