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lecture18_umn - Lecture 18 Some Simple Variational...

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Lecture 18 October 23, 2009 Some Simple Variational Calculations The Helium Atom
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The Variational Approach The variational approach: MOs as linear combinations of AOs. The coefficients of the linear combinations are the variational parameters subject to optimization. System with only a single variational parameter: ground state for a particle having mass 1 a.u. in a box of length L = 1 a.u. Eigenfunction and eigenvalue (in a.u.) for this particle-in-a-box (18-1) ! 1 x ( ) = sin " x ( ) 0 # x # 1 E 1 = " 2 2
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How to derive the correct eigenfunction The function should have no nodes From the boundary conditions it must be zero at both ends of the box. One very simple choice for a trial function (18-2) This function has the correct behavior at x = 0 and x = 1 and it is everywhere positive in between. ! x ( ) = x 1 " x ( )
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The Variational Condition No variational parameter in ξ , nothing there to optimize. Choose instead (18-3) where a is the variational parameter to be optimized . Minimization of the energy of the trial wave function: upper bound on the ‘true’ energy. (18-4) ! x ; a ( ) = x a 1 " x ( ) 0 = d da ! x ; a ( ) H ! x ; a ( ) ! x ; a ( ) ! x ; a ( ) " # $ $ % &
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Optimization Steps Square modulus of ξ . (18-5) ! x ; a ( ) ! x ; a ( ) = x a 1 " x ( ) 0 1 # x a 1 " x ( ) dx = x 2 a 0 1 # " 2 x 2 a + 1 0 1 # + x 2 a + 2 0 1 # = x 2 a + 1 2 a + 1 0 1 " 2 x 2 a + 2 2 a + 2 0 1 + x 2 a + 3 2 a + 3 0 1 = 1 2 a + 1 " 2 2 a + 2 + 1 2 a + 3 = 2 2 a + 1 ( ) 2 a + 2 ( ) 2 a + 3 ( )
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Expectation value of H (18-6) The second derivative of ξ with respect to x is (18-7) ! x ; a ( ) H ! x ; a ( ) = x a 1 " x ( ) " 1 2 d 2 dx 2 # $ % % & ( ( 0 1 ) x a 1 " x ( ) [ ] d 2 2 x a 1 ! x ( ) [ ] = d d x a 1 ! x ( ) [ ] " # $ % & = d ax a ! 1 1 ! x ( ) ! x a [ ] = a a ! 1 ( ) x
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lecture18_umn - Lecture 18 Some Simple Variational...

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