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