MacDonald’s theorem

This proof shows that any approximate wave function will have an energy above or equal to the exact ground-state energy. There is a related theorem, known as MacDonald s Theorem, which states that the nth root of a set of secular equations (e.g. a Cl matrix) is an upper limit to the n — l)th excited exact state, within the given symmetry subclass. In other words, the lowest root obtained by diagonalizing a Cl matrix is an upper limit to the lowest exact wave functions, the 2nd root is an upper limit to the exact energy of the first excited state, the 3rd root is an upper limit to the exact second excited state and so on. 

Exercise. Let Y(t) be the fluctuating part of an electrical current. It is often easier to measure the transported charge Z(t) = jo Y(t ) dt. Show that the spectral density of Y is related to the charge fluctuations by MacDonald s theorem ... 

MacDonald s Theorem, which states that the nth root of a set of secular equations (e.g. a positive definite everywhere) which integrates to the number of electrons, N, the energy ... 

MacDonald s theorem, 408 Natural Atomic Orbital (NAO), 230 Pauli spin matrices, 205 (QM-MM) methods, 50 ... 

Proof Let S be either a two dimensional abelian variety or a geometrically ruled surface over an elliptic curve over C. Let S be a good reduction of S modulo q, where gcd(q, n) = 1 such that the assumptions of lemma 2.4.7 hold. Then A 5n iis a good reduction of KSn- modulo q. (3) now follows by lemma 2.4.10 and remark 1.2.2. (1) and (2) follow from this by the formula of Macdonald for p(S n z) (see the proof of theorem 2.3.10). ... 

In the local-scaling procedure embodied in Eq. (60) because the optimization is performed with respect to the particular density pn corresponding to the excited state j ]p( ri, s, ), one is searching for an energy [pn r) Wj which is an upper bound to the exact energy En>exact. This upper-bound character of the calculated energy is guaranteed by the Hylleraas-Undheim-MacDonald theorem. 

Note that in the present case the matrix elements depend on the final density p . Moreover, because this density is obtained from the transformed wavefunction, they also depend on the expansion coefficients. For this reason, Eq. (177) must be solved iteratively. Such a procedure has been applied - in a sample calculation - to the 2 S excited state of the helium atom. The upper-bound character of the energy corresponding to the energy functional for the transformed wavefunction B,p( r,- ) with respect to the exact energy is guaranteed by the Hylleraas-Undheim-MacDonald theorem. 

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