Diese Präsentation wurde erfolgreich gemeldet.
Die SlideShare-Präsentation wird heruntergeladen. ×

Presentation

Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Wird geladen in …3
×

Hier ansehen

1 von 24 Anzeige
Anzeige

Weitere Verwandte Inhalte

Diashows für Sie (20)

Anzeige

Ähnlich wie Presentation (20)

Presentation

  1. 1. TESTING THE GOODNESS OF APPROXIMATE STATIONARY STATES TANOY DUTTA UNIVERSITY OF CALCUTTA
  2. 2. THE USUAL APPROACH The possible ways are:  Take the known function 𝛹 and calculate the energy E and < 𝐴 >  Take the trial function 𝜙 and calculate average energy <E> and < A>  Compare these results  Idea is to compare all such average properties.  We take for simplicity 𝒙 2=A  Also, compute < 𝜙 Ψ > |2
  3. 3. SAMPLE RESULTS The particle in a box  The exact wave-functions that were, 1. Ψ1 = √ 2 𝐿 sin 𝜋𝑥 𝐿 2. Ψ2 = √ 2 𝐿 sin 2𝜋𝑥 𝐿 3. Ψ3 = √ 2 𝐿 sin 3𝜋𝑥 𝐿 For the known functions the results are Wave-function Exact Energy(E) < 𝒙 2 > Ψ1 9.869/𝐿2 0.2826 𝐿2 Ψ2 39.478/𝐿2 0.3206 𝐿2 Ψ3 88.826/𝐿2 0.3277 𝐿2
  4. 4. SAMPLE RESULTS The particle in a box The trial functions that were taken 1. 𝜙 1 = N𝑥 𝐿 − 𝑥 2. 𝜙 2 = N 𝑥 𝐿 2 − 𝑥 𝐿 − 𝑥 3. 𝜙 3 = N 𝑥 𝐿 3 − 𝑥 2𝐿 3 − 𝑥 𝐿 − 𝑥  For the trial functions the results are Wave- function Average Energy(<E>) < 𝒙 2> 𝜙1 10/𝐿2 0.2857 𝐿2 𝜙2 42/𝐿2 0.3333 𝐿2 𝜙3 100/𝐿2 0.3636 𝐿2
  5. 5. PIB RESULTS Comparison % Error in Energy % Error in ‹ x2› Ψ1 vs. 𝜙1 1.31 1.085 Ψ2 vs. 𝜙2 6.003 3.81 Ψ3 vs. 𝜙3 11.17 10.95
  6. 6. SAMPLE RESULTS The harmonic oscillator The exact wave-functions that were, Ψ0 = ( 1 𝜋 ) ¼ 𝑒− 𝑥2 2 Ψ1 = ( 4 𝜋 ) ¼ 𝑥𝑒− 𝑥2 2 For the known functions the results are Wave-function Exact Energy(E) Ψ0 1 Ψ1 3
  7. 7. SAMPLE RESULTS The harmonic oscillator The trial functions that were taken 𝜙 0 = N (A2 – x2) 𝜙 1 = N x (A2 – x2) For the trial functions the results are Wave-function Average Energy(<E>) 𝜙0 1.1952 𝜙1 3.7416
  8. 8. HO RESULTS Competing functions % Error in Energy Ψ0 vs. 𝜙0 16.33 Ψ1 vs. 𝜙1 19.82
  9. 9. USE OF OVERLAP: THE HYDROGEN ATOM AND STOs Slater Type Orbitals(STO) 1. 𝛹1𝑠 = 1 𝜋 𝑒−𝑟 2. 𝛹2𝑠 = 1 4 2𝜋 (2 − 𝑟)𝑒− 𝑟 2 Advantages  Probability of finding the electron near the nucleus is faithfully represented. Disadvantages  Three and four center integrals cannot be performed analytically.  No radial nodes. These can be introduced by making linear combinations of STOs  Does not ensure rapid convergence with increasing number of functions.
  10. 10. THE HYDROGEN ATOM: GTOs Gaussian Type Orbitals(GTO)  Introduced by Boys (1950)  α is a constant (called exponent) that determines the size (radial extent) of the function  The normalized 1s Gaussian-type function is, 𝜙1𝑠 𝐺𝐹 (𝛼, r) = ( 2𝛼 𝜋 ) 3 4 𝑒−𝛼𝑟2 GTOs are inferior to STOs in these ways:  GTO’s behavior near the nucleus is poorly represented. GTOs diminish too rapidly with distance.  The ‘tail’ behavior is poorly represented. Advantage  GTOs are computationally advantageous. Therefore, we use a linear combination of GTOs to overcome these deficiencies. Overlap is here the key.
  11. 11. LINEAR COMBINATION OF GTOs Contracted Gaussian functions:  Need for using better basis functions  Fixed linear combinations of primitive Gaussian functions-- Contracted Gaussian functions  We use the following basis functions for further calculation 𝜙1𝑠 𝐶𝐺𝐹 (ζ =1.0, STO-1G) = 𝜙1𝑠 𝐺𝐹 𝛼11 𝜙1𝑠 𝐶𝐺𝐹 (ζ =1.0, STO-2G) =𝑐12 𝜙1𝑠 𝐺𝐹 (𝛼12) + 𝑐22 𝜙1𝑠 𝐺𝐹 (𝛼22) 𝜙1𝑠 𝐶𝐺𝐹 (ζ =1.0, STO-3G) =𝑐13 𝜙1𝑠 𝐺𝐹 (𝛼13) + 𝑐23 𝜙1𝑠 𝐺𝐹 (𝛼23) + 𝑐33 𝜙1𝑠 𝐺𝐹 (𝛼33)
  12. 12. THE OVERLAP BETWEEN 1s-SLATER & CGF’s 1s Basis Functions Overlap STO-1G 0.978400 STO-2G 0.998420 STO-3G 0.999907 Standard textbook stuff
  13. 13. THE OVERLAP BETWEEN 2s-SLATER & CGFs 2s Basis Functions Overlap STO-1G 0.89664340 STO-2G 0.94512792 STO-3G 0.95788308 Our observation
  14. 14. THE MOTIVATION: NEED FOR A DIFFERENT APPROACH  Standard schemes require exact 𝛹 and E  Errors are measured ‘on average’ (integration process)  Knowledge of excited 𝛹 is difficult to gather  Necessity of employing some kind of self-check process  No need of any knowledge of exact function or its property
  15. 15. A SELF-ASSESSMENT POLICY What we do actually  TISE: Ψ″ = (𝑉 − 𝐸)Ψ  Assume: 𝜙′′ ≈ (𝑉 −< 𝐸 >)𝜙  Differentiate once more: 𝜙′′′ ≈ 𝑉 −< 𝐸 > 𝜙′ + 𝑉′𝜙  Check the nodes, boundaries, origin and minima/maxima.
  16. 16. SAP Contd.  Where V(x)=0, −𝜙′′ /(𝜙 < 𝐸 >) ≈1. Not applicable to box and any odd states for symmetric potentials.  At nodes, 𝜙′′′ 𝜙′(𝑉−<𝐸>) ≈ 1  At boundaries, − 𝜙′′′ 𝜙′<𝐸> ≈1 (only for the box system)  At minima/maxima, 𝜙′′′ 𝜙 𝑉′ ≈1  Judge the goodness using these relations
  17. 17. PERFORMANCE The PIB and harmonic oscillator Taking the same trial functions mentioned before, for PIB, For harmonic oscillator, Wave- function − 𝝓′′′ 𝝓′<𝑬> at node % Error − 𝝓′′′ 𝝓′<𝑬> at boundaries % Error 𝜙1 --- --- --- --- 𝜙2 0.57143 0.42857 - 0.28571 128.57 𝜙3 0.54 0.46 -0.54 154.00 Wave-function − 𝝓′′′ 𝝓′<𝑬> at node % Error 𝝓 𝟐 0.2857 71.43
  18. 18. PERFORMANCE The anharmonic oscillators  V = x2 + x4 state = even 4th lowest Potential Basis 𝐿 𝑜𝑝𝑡 <E> % error in <E> <x2> % Error in <x2>1 2 1 1 6 8 10 12 2.8639 3.0964 3.6172 3.6772 28.845714 28.835718 28.8353399 28.8353385 0.03598 0.00132 0.000021 0.00 2.267767 2.261797 2.261703 2.26170396 0.26807 0.00411 0.000042 0.00 Usual strategy
  19. 19. PERFORMANCE The anharmonic oscillators  V = x2 + x4 state = even 4th lowest Potential Basis Node at % error Node at % Error Node at % Error 1 2 1 1 6 8 10 12 ±0.2925 ±0.2923 ±0.2926 ±0.29255 1.97 × 10−3 5.10 × 10−3 3.8 × 10−4 6.6 × 10−6 ±0.8835 ±0.8826 ±0.8827 ±0.88276 2.17 × 10−2 7.8 × 10−3 1.02 × 10−3 2.6 × 10−5 ±1.5098 ±1.5082 ±1.5079 ±1.50788 5.4 × 10−3 4.9 × 10−4 1.6 × 10−3 9.1 × 10−5 Our strategy
  20. 20. CONCLUSION  We have introduced certain criteria to judge the quality of an optimized approximate stationary state.  The criteria do not require any knowledge of exact function or energy.  This self-check is more useful for higher excited states.  Our equations reduce to exactness as 𝜙 → 𝛹.
  21. 21. ACKNOWLEDGEMENTS Prof. Kamal Bhattacharya Department of Chemistry University of Calcutta Kolkata
  22. 22. THANK YOU

×