lecture19_umn

# lecture19_umn - Lecture 19 Variational Calculations on the...

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Lecture 19 October 26, 2009 Variational Calculations on the H Atom Gaussian Functions and their linear combinations

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Variational Calculations on the H Atom Consider a system where the answer is known exactly: H atom. Try different variational protocols . Replace the exact 1s wave function (19-1) with a different functional dependence on r , namely (19-2) N normalization constant and α variational constant . " 100 r , # , \$ ( ) = Z 3 / 2 % e & Zr " 1 s r , # , \$ ; % ( ) = Ne & % r 2
Normalization Constant We can find N (19-3) (19-4) " 1 s r , # , \$ ; % ( ) " 1 s r , # , \$ ; % ( ) = 1 = N 2 e & 2 % r 2 r 2 dr sin # d # d \$ 0 2 ( 0 ( 0 ) ( = 4 N 2 r 2 e & 2 % r 2 dr 0 ) ( = 4 N 2 1 8 % 2 % * + , - . / 1/ 2 0 1 2 2 3 4 5 5 N = 2 ! " # \$ % & ( 3/ 4

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Variational Condition Variational condition for a normalized wave function is (19-5) Evaluate the expectation values of the kinetic and potential energy operators . 0 = d d ! " 1 s r , # , \$ ; ! ( ) H " 1 s r , # , \$ ; ! ( ) [ ] = d d ! " 1 s r , # , \$ ; ! ( ) % 1 2 & 2 % 1 r " 1 s r , # , \$ ; ! ( ) ( ) ) * + , ,
Kinetic Energy Operator In spherical polar coordinates (and a.u.) (19-6) Angular momentum of an s wave function is 0 , we need only consider the r component: (19-7) Take the constants out front, and integrate over θ , φ to get 4 π . " 2 r , # , \$ ( ) = % 1 r 2 & & r r 2 & & r % L 2 ( ) * + , ! 1 s r , " , # ; \$ ( ) % 1 2 & 2 ! 1 s r , " , # ; \$ ( ) = % 1 2 2 \$ ( ) * + , - 3 /2 4 ( ) e %\$ r 2 1 r 2 d dr r 2 d dr ( ) * + , - 0 . / e %\$ r 2 r 2 dr

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Evaluating the Integral The d/dr operators only operate on the wave function. First differentiate the exponential function multiply by r 2 ; differentiate the product once again ; divide by r 2 ; then finally include the volume element (19-8) = 2 " # \$ % & ( ) 3/ 2 2 # ( ) 6 " r 2 e * 2 " r 2 dr 0 + , * 4 " 2 r 4 e * 2 " r 2 dr 0 + , ( ) " 1 2 2 # \$ % & ( ) * 3/ 2 4 \$ ( ) e " # r 2 1 r 2 d dr r 2 d dr % & ( ) * 0 + , e " # r 2 r 2 dr =
Kinetic Energy Integral In an integral table (19-9) This gives (19-10) r 2 n e " ar

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