Eigenvalue equations coupling

The Fermi coupling operator l is given by (90). Owing to the eigenvalue equation... 

From the A -electron Hilbert-space eigenvalue equation, Eq. (2), follows a hierarchy of p-electron reduced eigenvalue equations [13, 17, 18, 47] for 1 < p < N — 2. The pth equation of this hierarchy couples Dp,Dp+, and and can be expressed as... 

The simultaneous optimization of the latter two points can be done by solving coupled eigenvalue equations. These equations can be derived from imposing variational conditions on the energy [2] ... 

We note that the basis orbitals k(r) are common for both a and / spin electrons, therefore, the evaluation of one- and two-electron integrals has to be performed only once (or they can be taken over from the corresponding closed-shell calculations). The procedure analogous to the one applied for the closed-shell case leads to a set of coupled complex pseudo-eigenvalue equations of the form... 

Substitution of (1-3) in the eigenvalue equation leads to the following coupled differential equations for the expansion coefficients / (Q) [48]... 

Substituting the state vector f) in (20), the above equations provide the relations among Cj f) and Cj (/). There are two ways to couple the two components Cj f) and Cj (/). If we couple the X component to the nearest-neighbor Y component and vice versa, we obtain a set of eigenvalue equations given by... 

For such a coupled spin system the density matrix is conveniently expressed in the product space of the eigenfunctions of the 1 operators of the uncoupled spins. Denoting the eigenfunctions for the spin-up and the spin-down states as a) and 1/6), respectively, the following eigenvalue equations apply for each of the coupled spins. 

The form of Hmuit having been obtained, it now remains to discuss how to solve the subsequent eigenvalue equation. For radiation-matter coupling that is small relative to intramolecular Coulomb potential energies, the interaction Hamiltonian may be considered as a perturbation on the particle-field sysfem. A perfurbafion fheory solufion is then the most obvious choice. The first two terms of Eq. (6) are faken fo constitute the unperturbed Hamiltonian Ho, so that... 

The c , necessary to compute P are obtained by self-consistent solution (SCF) of the (unrestricted PPP) coupled eigenvalue equations... 

The Contracted Schrodinger Equation is studied here in a spin-orbital representation coupled with the S2 eigenvalue equation as an auxiliary condition. A set of new algorithms for approximating RDM s in terms of the lower order ones are reported here. These new features improve significantly the method. 

In the low-density limit and with 0 = 0 these equations correspond to the eigenvalue equation (13.2) with the coupling matrix element given by (13.19) since for ls-exciton... 

The modes of a dielectric-metal-dielectric waveguide can be found by solving the eigenvalue (Eq. 35). Numerical solutions of this eigenvalue equation for a symmetric waveguide structure ( di = d2) are shown in Fig. 8. For any thickness of the metal film, there are two coupled surface plasmons, which are referred as to the symmetric and antisymmetric surface plasmons. 

One way to derive this is to eliminate the coefficients for the upper set between Eqs. (19-19). This leads to an eigenvalue equation for eigenvalues of the matrix <5o /) -hZj, W y Wyi,. The sum of eigenvalues (sum of E ) over this lower set is exactly equal to the trace of this matrix. Similarly, the sum of E over the set is exactly equal to the trace of the squared matrix. One can also write these sums as sums over , = — ITj H- A,-, noticing that A,- has second-order and fourth-order terms, and solve for the sum of Eq. (19-20). It is also readily confirmed that this is correct to fourth order for the special case of only two coupled states by expanding the exact solution, >/lTf2 + VVj, in Wi2-... 

Here, the e, value is the orbital s energy eigenvalue. Equation (6.9) is remarkably similar to the original Schrodinger equation. Equation (6.2), but the wave functions have been replaced with the KS orbitals and the exchange and correlation terms have been isolated. Thus, we have replaced the iV-body coupled electronic wave function with a collection of uncorrelated wave fimctions while at the same time defining precisely what the uncertain many-body terms in need of approximation are. 

From the definitions (3.316) and (3.318) of the two Fock operators / and /, we can see that the two integro-differential eigenvalue equations (3.312) and (3.313) are coupled and cannot be solved independently. That is,/ depends on the occupied orbitals, through Jj, and depends on the occupied a orbitals, through JJ. The two equations must thus be solved by a simultaneous iterative process. 

It contains two p E, i) producing terms (either from different energy levels of A, term 1, or from p E, t), term 4) and two consuming terms, in which p E, t) is lost to other energy levels of A (term 2) or to B (term 3). For the population density of B an analogous ME exists. Both populations are coupled by the mass conservation requirement and therefore the set of coupled differential equations contains both species. We can define a new vector p E, t) which contains the populations of both isomers. This leads to the same eigenvalue equation as discussed earlier. 

Edmonds classic text on the theory of angular momentum is recommended for further studies on this subject. Our presentation was very brief in many respects. For instance, we have neither introduced ladder operators to explicitly constmct the eigenfunctions of angular momentum eigenvalue equations nor have we deduced the expression for total angular momentum eigenstates in the coupled product basis. Edmonds book fills all these gaps. 

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