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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