Improving Φ0 via 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
{Hψi = Εiψi, E0 < E1 <…}, Φ0 ≈ ψ0
• ψ1 Can be Computed Independently of ψ0
(Löwdin)
• Suppose (for normalized functions):
• ψ1 is known
ψ0 is unknown
• Φ0 (with E[Φ0] < E1) is known Φ0 ≈ ψ0
• <ψ1|Φ0> ≠ 0 Φ0 is not orthogonal to ψ1
• Φ1┴Φ0 (various Φ1,s can be computed ┴Φ0)
• One of them: Φ1+(┴ Φ0 ) is closest to ψ1
One: Φ1+ (┴ Φ0) is the closest
(has the largest projection) to ψ1
1
1
0
0
1
0
1
1
2
It has energy:
1
1
1
1
1
1
0
0
0
1
1
2
Among all Φ1,s orthogonal to Φ0,
the closest to ψ1: Φ1+ lies Lower than ψ1
2
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 EΦ1 ORTHOGONALLY to Φ0
leads to Lower Bound of ψ1: Φ1min:
EΦ1min < EΦ1+ < E1
(departing from ψ1 instead of approaching ψ1)
Equalities
• If Φ0 є{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
Bracketing
1+
EΦ
<
1
E
<
1
U
By expanding any Φ in {ψ0, ψ1, ...} basis:
1
U1
0
0
0
0
1
2
2
It is easily verified that
1
U1
0
1
1
1
1
0
0
0
1
1
2
2
Knowing
1+
Φ
improves
0
Φ
• On the plane of {Φ0┴ Φ1+} containing ψ1:
• Τhe highest eigenroot is Ψ+ = ψ1.
• The lowest is Ψ– ≡ Φ0+ (better than Φ0), where:
0
0
1
1
0
1
0
0
0
1
1
1
2
0
0
0
1
1
2
2
0
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…
Analytic atomic orbitals of
NMCSCF accuracy
• STO: α rn e-ζr
• H (Laguerre): (Σν αν rν) 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])
1
EΦ ,
1
Φ(
┴
 0  1  2 0   0
0
{Φ
┴
1  1   1 1   2 , where :
   
1
To leading order :

     
1
1
2
1+
Φ })
2


0
   
1
2
1
1
0
0

0

2
1 2

0.2
0
-0.2
-2.12
-2.14
0.2
0
-0.2
-0.4
1 0
1 2
1  
-2.16
0.4
0