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Condensed Matter Theory

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] 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: n: 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) n VFES
Variation Functionals for Excited States


i: (known) Approximations of the(unknown) Exact eigenfunctions i
of increasing energy E[0] < ...< E[i] < ...< E[n]

Variation of n to minimize n (the rest of i unvaried) approaches the exact n

Why?

The energy E[n] has Downward paraboloids PL , and Upward paraboloids PH.

Standard treatments (H, M):
Diminish (approximately) the Downward paraboloids PL : E[n] + PH
(then 0 = G always has better quality than fn = H or M)

TPCI treatment:
Invert (exactly) the Downward paraboloids PL: E[n] + PH + PL:
This leads to the VFES n with minimum at n.

If 1 ->1 = a has better quality than 0 = G ->0 = g then G can be improved orthogonally to 1(1)

(1) ''Utilizing the fact that among all trial functions Orthogonal to an approximate ground state, 0, the closest, 1+, to the exact first excited state,1, has lower energy than the exact: E[1+]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

 

 

 

 

 

 

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