Ground State of the He Atom 1s State

Transcription

1 Ground State of the He Atom s State First order perturbation theory Neglecting nuclear motion H m m 4 r 4 r r Ze Ze e o o o o 4 o kinetic energies attraction of electrons to nucleus electron electron repulsion - electron - electron r - distance of to nucleus r - distance of to nucleus r - distance between two electrons Copyright Michael D. Fayer, 9

2 Substituting a h o me o Bohr radius r a r a Distances in terms of Bohr radius r a x a X, etc. spatial derivatives in units of Bohr radius Gives H Ze Ze e m a 4 a 4 a 4 a o o o o Copyright Michael D. Fayer, 9

3 In units of e 4 oa e 4 a o ground state, s, energy of H atom H Z Z (Nothing changed. Substitutions simplify writing equations.) Take H Z Z Zeroth order Hamiltonian. No electron electron repulsion. H ' Perturbation piece of Hamiltonian. Electron electron repulsion. Copyright Michael D. Fayer, 9

4 Need solutions to zeroth order equation H E Take () () and d E E () E () H has terms that depend only on and. No cross terms. Can separate zeroth order equation into () () Z () E Z () E () () These are equations for hydrogen like atoms with nuclear charge Z. Copyright Michael D. Fayer, 9

5 For ground state (s) 3/ () Z e () Z 3/ Z e Z Hydrogen s wavefunctions for electrons and but with nuclear charge Z. The zeroth order solutions are 3 (,) () () Z e Z e Z product of s functions () () ( ) E E E Z Es H sum of s energies with nuclear charge Z Copyright Michael D. Fayer, 9

6 Correction to energy due to electron electron repulsion E H H expectation value of perturbation piece of H nn s,s H ' d d * d d e Z e e 4 a 6 Z Z d d o d sin d d d d sin d d d Electron electron repulsion depends on the distance between the two electrons. spherical polar coordinates Copyright Michael D. Fayer, 9

7 This is a tricky integral. The following procedure can be used in this and analogous situations. cos is the angle between the two vectors and. e - + Let > be the greater of and < be the lesser of and e - Then x x cos x Copyright Michael D. Fayer, 9

8 x xcos( ) x Expand in terms of Legendre polynomials (complete set of functions in cos()). n ap n n cos( ) The a n can be found. an x n Therefore n xpn cos( ) n Copyright Michael D. Fayer, 9

9 Now express the Pn cos( ) in terms of the & ; & the absolute angles of the vectors rather than the relative angle. The position of the two electrons can be written in terms of the Spherical Harmonics, the solutions to the Φ(φ) and Θ(θ) equations in the H atom. P m n P m n im cos e im cos e Complete set of angular functions. The result is m m! P cos P cos e m! m m im Copyright Michael D. Fayer, 9

10 m m! P cos P cos e m! m m im Here is the trick. The ground state hydrogen wavefunctions involve P P (cos ) e im ( ) (cos ) e im s wavefunctions have spherical harmonics with m These are constants. Each is ust the normalization constant. A constant times any spherical harmonic except the one with, m which is a constant, integrated over the angles, gives zero. Therefore, only the m term in the sum survives when doing integral of each term. The entire sum reduces to for s state or any s state. For other states, t t limited it d number b of f terms. t Group G theory. Full F rotation t ti group. Integral of product of three functions. Direct product of representations of function must contain totally symmetric rep. Formulas exist. Copyright Michael D. Fayer, 9

11 Then e Z e e E d d 6 Z Z 4oa The integral over angles yields 6. Z Z 6 e e e 4 a o E 6Z d d This can be written as 6 6Ze e d d 4oa Z Z Z E e e d e d Copyright Michael D. Fayer, 9

12 Doing the integrals yields E 5 e Z 8 4 a o Putting back into normal units 5 E Z E ( ) s H 4 E s e ( H) 3.6eV 4 a o Therefore, E E E Z Z E ( ) s H 4 5 negative number Electron repulsion raises the energy. For Helium, Z =, E = ev Copyright Michael D. Fayer, 9

13 atom exp. value (ev) calc. value (ev) %Error He Li Be B C Experimental values are the sum of the first and second ionization energies. Ionization energy positive. Binding energy negative. Copyright Michael D. Fayer, 9

14 The Variational Method The Variational Theorem: If is any function such that * dd (normalized) and if the lowest eigenvalue of the operator H is E, then * H H d E The expectation value of H or any operator for any function is always great than or equal to the lowest eigenvalue. Copyright Michael D. Fayer, 9

15 Proof Consider H E H E H E The true eigenkets of H are i H i E i i Expand in terms of the i orthonormal basis set Copyright Michael D. Fayer, 9

16 c Expansion in terms of the eigenkets of H. i i i Substitute the expansion H E c H E c i i i The are eigenkets of (H E ). Therefore, the double sum collapses into a single sum. cc H E Operating H on returns E. H E c c E E Copyright Michael D. Fayer, 9

17 H E c c E E cc A number times its complex conugate is positive or zero. E E An eigenvalue is greater than or equal to the lowest eigenvalue. Then, and E i E cc E E Therefore, H E Finally H E H E * H H d E The lower the energy you calculate, the closer it is to the true energy. The equality holds only if, the function is the lowest eigenfunction. Copyright Michael D. Fayer, 9

18 Using the Variational Theorem Pick a trial function Calculate,, normalized * J H d J is a function of the s. Minimize J (energy) with respect to the s. The minimized i i J - Approximation to E. The obtained from minimizing with respect to the s - approximation to. Method can be applied to states above ground state with minor modifications. Pick second function normalized and orthogonal to first function. Minimize. i i If above first calculated l energy, approximation to next highest h energy. If lower, it is the approximation to lowest state and initial energy is approx. to the higher state energy. Copyright Michael D. Fayer, 9

19 Example - He Atom Trial function 3 Z e Z The zeroth order perturbation function but with Z Z a variable. Writing H as in perturbation treatment after substitutions H Z Z ewrite by adding and subtracting Z Z H Z Z ( Z Z) Copyright Michael D. Fayer, 9

20 H Z Z ( Z Z) Want to calculate * J H d using H from above. The terms in red give Z Es( H) Zeroth order perturbation energy with Z Z Therefore, Z Z d d d d d d s J Z E H These two integrals have the same value; only difference is subscript. Copyright Michael D. Fayer, 9

21 For the two integrals in brackets, performing integration over angles gives 6 Z Z Z 6 e d e d Z In conventional units e Z 4 a o The last term in the expression for J / term was evaluated in the perturbation problem except Z Z. The result is (in conventional units) 5 e Z 8 4 a o Copyright Michael D. Fayer, 9

22 Putting the pieces together yields e 5 e s 4oa 8 4oa J Z E ( H ) ( Z Z ) Z Z 5 Z Z Z Z Z E H 4 4 ( ) s( ) E s ( H) e 4 a o 5 Z ZZ Z E H 4 4 s( ) Copyright Michael D. Fayer, 9

23 To get the best value of E for the trial function, minimize J with respect to. Z J 5 4Z 4 Z Es( H) Z 4 Solving for Z yields Z 5 Z 6 Using this value to eliminate Z in the expression for J yields E Z E H s( ) J Z 4 ZZ Z Es ( H ) 4 5 This is the approximate energy of the ground state of the He atom (Z = ) or two electron ions (Z > ). Copyright Michael D. Fayer, 9

24 atom exp. value (ev) calc. value (ev) % Error He Li Be B C He perturbation theory value 74.8 ev Copyright Michael D. Fayer, 9