Linker IT Software
Google
Web www.oraxcel.com
menubar-top-links menubar-top-rechts
Home Help Search Login
Welcome, Guest. Please Login.
SQL*XL: Database to Excel bridge litLIB: Excel power functions pack ExcelLock: Locking and securing your valuable Excel spreadsheets encOffice: Protect your Excel file easy and safe encOffice: Protect your Excel file easy and safe
Pages: 1
The Energy of a Slater Deterimant (Read 6202 times)
Gerrit-Jan Linker
YaBB Administrator
*****




Posts: 75
The Energy of a Slater Deterimant
22.08.12 at 13:07:23
 
The Energy of a Slater Deterimant
 
The energy of a Slater determinant can be calculated from the expectation value of the Hamiltonian:
E = <Slater det.|H|Slater det.> provided that the Slater determinant is properly normalised: <Slater det.|Slater det.>=1
 
E = < |Φ1(1)Φ2(2)...ΦN(N)| |Ĥ| |Φ1(1)Φ2(2)...ΦN(N)| >
  = < |Φ1(1)Φ2(2)...ΦN(N)| | ∑iĥ(i) + ∑i<j1/rij| |Φ1(1)Φ2(2)...ΦN(N)| >
 
We identify one and two electron integrals:
 
One electron integrals:
 ∑i< |Φ1(1)Φ2(2)...ΦN(N)| | ĥ(i) |Φ1(1)Φ2(2)...ΦN(N)| >  
= ∑i< Φi(i) | ĥ(i) |Φi(i) >  
= ∑i hii
 
Two electron integrals:
i<j < |Φ1(1)Φ2(2)...ΦN(N)| | 1/rij| |Φ1(1)Φ2(2)...ΦN(N)| >
= ∑i<j [ < Φi(i)Φj(j) | 1/rij| Φi(i)Φj(j) > - < Φi(i)Φj(j) | 1/rij| Φj(j)Φi(i) > ]
= ∑i<j [ < ij||ij > - < ij | ji > ]
= ∑i<j [ Jij - Kij ]
 
Using the one and two electron integrals, we can write:
 
E = ∑i hii + ∑i<j [ Jij - Kij ]
Back to top
 
« Last Edit: 10.10.13 at 12:02:26 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
Email WWW Gerrit-Jan Linker   IP Logged
Gerrit-Jan Linker
YaBB Administrator
*****




Posts: 75
Calculating the energies of Slater Determinants
Reply #1 - 27.08.12 at 07:22:52
 
Calculating the energies of Slater Determinants
 
Let's consider 3 two electron Slater determinants:
 
Closed shell singlet: |aa|: E=2ha + Jaa
Triplet:                   |ab| : E=ha + hb + Jab - Kab
and             : |ab|: E=ha + hb + Jab
 
The last determinant is not an eigenfunction of Ŝ2 and but the following combinations are:
Triplet: |ab| + |ab| : E = ha + hb + Jab - Kab
Singlet: |ab| - |a|b: E = ha + hb + Jab + Kab
 
Note that h1 < 0, J12 > 0, K12 > 0  
 
Derivation:
 
Closed shell singlet: |aa|: E=2ha + Jaa
 
E = < |aa| | Ĥ | |aa| > (note: the factor is due to the normalisation constants)
  = < aa - aa | Ĥ | aa - aa >
  = [ < aa | Ĥ | aa > - < aa | Ĥ | aa > - < aa | Ĥ | aa > + < aa | Ĥ | aa > ]
           (spin integration makes the 2nd and 3rd term zero)
  = [ < aa | Ĥ | aa > + < aa | Ĥ | aa > ]
  = [ < aa | ĥ(1) + ĥ(2) + 1/r12 | aa > + < aa | ĥ(1) + ĥ(2) + 1/r12 | aa > ]
  = [ ha + ha + <aa || aa > + ha + ha + <aa || aa> ]
                              (note: <aa | aa> is short for <a(1)a(2) |1/r12| a(1)a(2)> )
  = 2ha + <aa | aa >  
  = 2ha + Jaa  
 
Triplet:                   |ab| : E=ha + hb + Jab - Kab
 
E = < |ab| | Ĥ | |ab| >  
  = < ab - ba | Ĥ | ab - ba >
  = [ < ab | Ĥ | ab > - < ab | Ĥ | ba > - < ba | Ĥ | ab > + < ba | Ĥ | ba > ]
  = [ < ab | ĥ(1) + ĥ(2) + 1/r12 | ab > - < ab | ĥ(1) + ĥ(2) + 1/r12 | ba >  
           - < ba | ĥ(1) + ĥ(2) + 1/r12 | ab > < ba | ĥ(1) + ĥ(2) + 1/r12 | ba > ]
            (note: <ba|ĥ(1)|ab> = <b|ĥ(1)|a><a|b>=0)
  = [ ha + hb + <ab | ab > - <ab|ba> - <ba|ab> + hb + ha + <ba | ba> ]
  = ha + hb + <ab|ab> - <ab|ba> - <ba|ab> + <ba|ba>  
  = ha + hb +<ab|ab> - <ab|ba>  
  = ha + hb + Jab - Kab  
 
 
 
|ab|: E=ha + hb + Jab (determinant is not a proper eigenfunction of S^2!)
 
E = < |ab| | Ĥ | |ba| >  
  = < ab - ba | Ĥ | ab - ba >
  = [ < ab | Ĥ | ab > - < ab | Ĥ | ba > - < ba | Ĥ | ab > + < ba | Ĥ | ba > ]
  = [ < ab | Ĥ | ab > + < ba | Ĥ | ba > ]
  = [ < ab | ĥ(1) + ĥ(2) + 1/r12 | ab > + < ba | ĥ(1) + ĥ(2) + 1/r12 | ba > ]
  = [ ha + hb + <ab | ab > + ha + hb + <ab | ab> ]
  = ha + hb + <ab | ab >  
  = ha + hb + Jab  
 
Triplet: |ab| + |ab| : E = ha + hb + Jab - Kab
Singlet: |ab| - |a|b: E = ha + hb + Jab + Kab
Back to top
« Last Edit: 14.01.15 at 11:23:50 by Gerrit-Jan Linker »  

SD_SandT_energy.png

Gerrit-Jan Linker
Linker IT Software
Email WWW Gerrit-Jan Linker   IP Logged
Pages: 1