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 Variation Functionals for Excited States (a)  Inherent restrictions of the Standard Methods Lowercase letters:                                     Exact eigenfunctions (unknown):                                                                         g: Exact ground state (unknown)                                                                         a: Exact 1st excited state (unknown)                                                                         e: another Exact excited state (unknown) CAPITAL LETTERS:                            Approximate wave functions (known)                                                                         G: Approximate Ground state (known)= g Orthogonal to  G:          A, F, H, M, R:    Approximate 1st excited states (known)                                                                         E: higher Approximate excited state (known) A: The Closest  to a.        EVIDENTLY:                 ENERGY[A] < ENERGY[a]   (A below a level) E:                                                                                 ENERGY[E] > ENERGY[a]    (E above a level) F:                                                                                 ENERGY[F] = ENERGY[a]     (F at a level) H: Standard Hylleraas-Undheim-MacDonald:  ENERGY[H] ³ ENERGY[F]    (H above F, above a level) M: Standard Minimization of the Energy:       ENERGY[M] < ENERGY[A]   (M below A, below a level) R: OR-THOG-O-NAL to...a !        Alas !! :            ENERGY[R] = ENERGY[a] !!!  (R at a level)!!! If  G -> g,  then   all   of   {A, F, H, M}   APPROACH   the   Exact   a Except in exceptionally simple cases, in large systems always G .ne. g, never G -> g accuratey enough. But if  G .ne. g,  then   NONE   of   {A, F, H, M, R}   APPROACHES   the   Exact   a QUESTION:   If  G .ne. g,   What   APPROACHES  the Exact   a? TPCI ANSWER:   Ωn: Variation Functionals for Excited States (VFES)(1) (1) "Inherent restrictions of the Hylleraas-Undheim-MacDonald higher roots, and minimization functionals at the excited states", N.C. Bacalis, Z. Xiong and D. Karaoulanis, J. Comput. Meth. Sci. Eng. 8, 277 (2008) (Category: Invited Paper) Collaborators: Dr. Z. Xiong, AMS Research Center, Southeast University, Nanjing, China. (b) Ωn VFES Variation Functionals for Excited States Φi: (known) Approximations of the(unknown) Exact eigenfunctions ψi of increasing energy E[ψ0] < ...< E[ψi] < ...< E[ψn] Variation of Φn to minimize Ωn (the rest of Φi unvaried) approaches the exact ψn Why? The energy E[Φn] has Downward paraboloids PL , and Upward paraboloids PH.                            Standard treatments (H, M):                                     Diminish (approximately) the Downward paraboloids PL : E[ψn] + PH (then  Φ0 =  G  always has better quality than fn = H or M)                            TPCI treatment:                                     Invert (exactly) the Downward paraboloids PL: E[ψn] + PH  + PL: This leads to the VFES  Ωn with minimum at ψn. If Φ1 ->ψ1 = a  has better quality than Φ0 =  G ->ψ0 = g then G can be improved orthogonally to Φ1(1) (1) ''Utilizing the fact that among all trial functions Orthogonal to an approximate ground state, Φ0, the closest, Φ1+, to the exact first excited state,ψ1, has lower energy than the exact: E[Φ1+]1]'', N.C. Bacalis, Computation In Modern Science And Engineering, AIP Conf. Proc. 963. Proc. on Computational Methods in Sci. and Eng. 2007 2 Part A, 2007, pp. 6-9     © National Hellenic Research Foundation (NHRF), 48 Vassileos Constantinou Ave., 11635 Athens, Greece, Tel. +302107273700, Fax. +302107246618 