Reduced resolvent

We shall call S(l) the reduced resolvent of H with respect to is°i[Pg.8]

Since (3.18) has the same form as (3. 1), we can apply the results obtained above to each eigenvalue of B0) in place of the eigenvalue do of H0 hitherto considered. By hypothesis B y) has to distinct eigenvalues k, . .., Xm when considered in the subspace Eofy. Let normalized eigenvectors. We introduce the reduced resolvent of Bm belonging to Xj (considered in the subspace Eq X) ... 

Then the reduced resolvent of B Vj considered in the whole space is easily seen... 

The induction energy, E, is obtained when the reduced resolvent R0, Eq. (1-24), is restricted to terms where one of the monomers is in the ground state and the other in the excited state. The corresponding expression is given by... 

Equation 31 -2 tells us that we can achieve faster separations by using short columns, higher-than-usual carrier gas velocities, and small retention factors. The price to be paid is reduced resolving power, caused by increased band broadening and reduced peak capacity (that is, the number of peaks that will fit in the chromatogram). 

Let the eigenstates and eigenvalues of the zero order Hamiltonian, , be and E, (ji = 0,1,2,oo), respectively, which we assume to be discrete. Note that = . The spectral representation of the reduced resolvent then takes the form... 

The effective Hamiltonian ff (z) is the sum of the projected Hamiltonian and of the level-shift operator [7]. For any initial state belonging to the inner space, the relevant information is contained in the resolvent projected, on the right, in the inner space. The dependence of W fz) on z can be reduced by time-filtering (see Section 2.2.1) or by extending the dimension of fhe inner space. P/(z-ff z) and Q/(z-H) are short-hand notations for fhe inversion of P z-W fz))P in fhe inner space and Q(z-H)Q in the outer space, respectively. If the initial state belongs to the inner space, the dynamics is obtained from the reduced resolvent... 

Dissipation appears through the second term in the right-hand side of (B.9). Since we are inferesfed in the projected dynamics (subdynamics [71,72]), one has to consider the reduced resolvent... 

Reduced Resolvent or the Almosf Inverse of MBPT Machinery Part 1 Energy Equation MBPT Machinery Part 2 Wave Function Equation BriUouin-Wigner Perturbation Theory Rayleigh-Schrodinger Perturbation Theory... 

The definition says that the reduced resolvent represents an operator that from an arbitrary vector d> of the Hilbert space, takes the following actions ... 

In the equation (12.83) for Hamiltonian, the shielding constants occur in the term I a H. The perturbation operator contains a lot of terms, but most of them, when inserted into the above formula, are unable to produce terms that behave like I a H. Only some very particular terms could produce such a dot product dependence. A minute of reflection leads directly to 83,84, 85 and Bio as the only terms of the Hamiltonian that have any chance of producing the dot product form. Therefore, using the definition of the reduced resolvent of Eq. (10.76), we have ... 

Note that whenever the reduced resolvent appears in a formula, infinite summation over unperturbed states is involved. 

We recognize on the right side (the first two terms) the complete first-order contribution i.e., the electrostatic energy (Feist) and the valence repulsion energy (F pj,). From the definition of the induction and dispersion energies [Eqs. (13.13) on p. 808], as well as fi-om the reduced resolvent Rq of Eq. (10.76) on p. 644. (applied here to the individual molecules), one may write... 

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