Improving Φ 0 via ψ 1 (independently known)

1. Improving Φ0via ψ1(independently known) Exactly orthogonal wave functions N.C. Bacalis Theoretical and Physical Chemistry Institute National Hellenic Research Foundation Athens, Greece ICCMSE-2007, Corfu, Greece, 29 Sep 2007 Partial support: ENTEP2004/04EP111, GSRD, Greece

2. {Hψi= Εiψi, E0 < E1 <…}, Φ0≈ψ0 • ψ1 Can be Computed Independently of ψ0 (Löwdin) • Suppose (for normalized functions): • ψ1is known ψ0is unknown • Φ0(with E[Φ0] < E1) is known Φ0≈ψ0 • <ψ1|Φ0> ≠ 0 Φ0is not orthogonal to ψ1 • Φ1┴Φ0 (various Φ1,s can be computed ┴Φ0) • One of them: Φ1+(┴Φ0) is closest to ψ1

3. One: Φ1+ (┴Φ0) is the closest (has the largest projection) to ψ1 It has energy: Among all Φ1,s orthogonal to Φ0, the closest to ψ1: Φ1+lies Lower than ψ1

4. In a Hilbert-subspace, the lowest Hamiltonian root has the lowest energy in the subspace (Eckart).On any plane {Φ1+┴Φ1} ┴ Φ0, if <Φ1+|H|Φ1> ≠ 0, the Hamiltonian opens the gap |EΦ1+- EΦ1|. ~> EΦ1min < EΦ1+ Therefore: (unless Φ0┴ψ1): Minimizing EΦ1ORTHOGONALLY to Φ0leads to Lower Bound of ψ1: Φ1min: EΦ1min < EΦ1+ < E1 (departingfrom ψ1 instead of approaching ψ1)

5. Equalities • If Φ0є{ψ0,ψ1} then Φ1+ є{ψ0,ψ1}, Φ1(┴{Φ0┴Φ1+}) є{ψ2, ψ3,... } ~> • < Φ1+|H| Φ1> = 0 ~> EΦ1min = EΦ1+ < E1 • If Φ0┴ψ1 then Φ1+ = ψ1, Φ1(┴{Φ0┴ Φ1+})є{ψ0, , ψ2, ψ3,... } ~> • < Φ1+|H| Φ1> = 0, ~> EΦ1min = EΦ1+ = E1

6. Bracketing EΦ1+ < E1 < U1 By expanding any Φ in {ψ0, ψ1, ...} basis: It is easily verified that

7. Knowing Φ1+improves Φ0 • On the plane of {Φ0┴Φ1+} containing ψ1: • Τhe highest eigenroot is Ψ+ = ψ1. • The lowest is Ψ– ≡ Φ0+ (better than Φ0), where:

8. Rotating Φ0+ around ψ1 improves Φ0+ • Introduce Φ0++┴ {Φ0+┴ ψ1} • On the plane of {Φ0+┴ Φ0++} ┴ ψ1: • The lowest eigenroot Ψ– ≡ Φ0 –has energy EΦ0 –< EΦ0 + (closer to E0). • Introduce another Φ0+++┴ {Φ0 –┴ ψ1} • On the plane of {Φ0 –┴ Φ0+++} ┴ ψ1: • The lowest eigenroot Ψ– ≡ Φ0 – –has energy EΦ0 – –< EΦ0 – (even closer to E0), etc…

9. Analytic atomic orbitals of NMCSCF accuracy • STO: α rn e-ζr • H (Laguerre): (Σν ανrν) e-ζr, αν= constants (same ζ, not contraction of different ζs) • Variationally optimized αν lead to concise analytic atomic orbitals of NMCSCF accuracy (reducing O[106] of configurations to O[102])

10. EΦ1, Φ1(┴{Φ0┴Φ1+})