ISSN:
0020-7608
Keywords:
Computational Chemistry and Molecular Modeling
;
Atomic, Molecular and Optical Physics
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Chemistry and Pharmacology
Notes:
Instead of the Schródinger equation HΨ = EΨ subject to the boundary condition 〈ϕ|Ψ〉 = 1, where ϕ is a normalized reference function in the Hilbert space, one studies the inhomogeneous equation (H - E)ΨE = aϕ, where E is a complex variable, with the same boundary condition, which gives a = 〈ϕ|H|ΨE〉 - E = E1 - E. Introducing the projector P = 1 - |ϕ〉〈ϕ| for the complement to O = |ϕ〉〈ϕ|, one finds easily the explicit solution ΨE = (1 - PH/E)-1ϕ = (1 + TEH)ϕ, where TE = (E - PH)-1P = P(E - PHP)-1P is the reduced resolvent associated with the auxiliary Hamiltonian H̄ = PHP. The existence of these operators is discussed. It is shown that, if the parameter E is real in the “discrete part” of the spectrum to H, then E and E1 = 〈ϕ|H|ΨE〉 = 〈ϕ|H + HTEH|Φ〉 ≡f(E) bracket a true eigenvalue E satisfying the relation E = f(E). The Newton-Raphson solution to the equation F(E) = E - f(E) = 0 is related to the variation principle. Putting H = H0 + V and expanding the inverse (E - PH0 - PV)-1 in terms of powers of V or (V - α), one gets various expansions relating to finite-order perturbation theory. Exact expressions for the ordinary wave and reaction operators are obtained. If A is an arbitrary nonsingular operator and h = {h1,h2,…,hn} is a linearly independent set, the inner projection Án = |h〉 〈h|A-1|h〉-1〈h| is a “rational approximation” to the operator A which converges toward A when n→∞ and the set h becomes complete. If A is positive (or has a finite negative part) the convergence is from below. Applying this principle to the partitioning technique, one gets rational perturbation approximations instead of the standard power series, similar to the Padé approximants but derived in a different way with the remainder term under control. The method has been used to calculate lower bounds to eigenvalues.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/qua.560210105
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