1. TESTING THE
GOODNESS OF
APPROXIMATE
STATIONARY STATES
TANOY DUTTA
UNIVERSITY OF CALCUTTA

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. 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. 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. 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. 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. 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. HO RESULTS
Competing
functions
% Error in Energy
Ψ0 vs. 𝜙0 16.33
Ψ1 vs. 𝜙1 19.82

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. 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. 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. 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. 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. 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. 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.

17. 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

18. 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

19. 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

20. 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

22. 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.

23. ACKNOWLEDGEMENTS
Prof. Kamal Bhattacharya
Department of Chemistry
University of Calcutta
Kolkata

24. THANK YOU