lecture21_umn

lecture21_umn - Lecture 21 Two-electron wave functions...

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Lecture 21 October 30, 2009 Two-electron wave functions Hartree products The Slater determinant Many-electron wave functions Spin orbitals
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Spin-Free Many-Electron Wave Functions and Antisymmetry Two electrons in two orbitals a and b : acceptable antisymmetric wave function is (21-1) A different, but completely equivalent way to write this is (21-2) If a and b are orthonormal, let us normalize Ψ . " 1,2 ( ) = a 1 ( ) b 2 ( ) # a 2 ( ) b 1 ( ) " 1,2 ( ) = a 1 ( ) b 1 ( ) a 2 ( ) b 2 ( )
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Two Electrons in Two Orbitals (21-3) a and b are orthogonal : integral over the coordinates of e1 or e2 (or both) that involves the product a * b is zero. a and b are normalized : if the only products in the integrals are a * a or b * b , they will be equal to one. Thus, the value of eq. 21-3 is 1 0 0+1=2 " * 1,2 ( ) #$ $ % " 1,2 ( ) #$ $ % dr 1 dr 2 = a * 1 ( ) #$ $ % b * 2 ( ) #$ $ % a 1 ( ) b 2 ( ) dr 1 dr 2 # a * 1 ( ) #$ $ % b * 2 ( ) #$ $ % a 2 ( ) b 1 ( ) dr 1 dr 2 # a * 2 ( ) #$ $ % b * 1 ( ) #$ $ % a 1 ( ) b 2 ( ) dr 1 dr 2 + a * 2 ( ) #$ $ % b * 1 ( ) #$ $ % a 2 ( ) b 1 ( ) dr 1 dr 2
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Two Electrons in Two Orbitals Normalized form for eq 21-2 is (21-4) Now, consider a case of more than two electrons (21-5) " 1,2 ( ) = 1 2 a 1 ( ) b 1 ( ) a 2 ( ) b 2 ( ) ! 1,2, , N ( ) = a 1 ( ) b 1 ( ) ! n 1 ( ) a 2 ( ) b 2 ( ) ! n 2 ( ) " " # " a N ( ) b N ( ) ! n N ( )
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Many Electron Case Swapping the coordinates of any two electrons is equivalent to swapping two rows in the determinant In a determinant when two rows are swapped, the value of the determinant changes sign . It satisfies antisymmetry perfectly! ! 1,2, , N ( ) = a 1 ( ) b 1 ( ) ! n 1 ( ) a 2 ( ) b 2 ( ) ! n 2 ( ) " " # " a N ( ) b N ( ) ! n N ( ) (21-5)
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Many Electron Case Normalization, (orthonormality of the orbitals): Integrate Ψ * Ψ , the only products between pairs of terms that will not be zero will be the square moduli of each term with itself , all equal to one. N
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lecture21_umn - Lecture 21 Two-electron wave functions...

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