1. 1 MODELING MATTER AT NANOSCALES 4. Introduction to quantum treatments 4.02. The variational method

2. Schrödinger Equation The solution of the Schrödinger equation gives the exact wave function of a system by means of the Hamiltonian operator that comes from the total energy of a system in classical mechanics: where the “spectrum” of the operator is the set of eigenvalues E k of all the K states of the system.

3. Schrödinger Equation However, exact solutions for a Hamiltonian are scarce, although very relevant, and were mostly obtained for ideal conditions.

4. The variational theorem

5. The minimal energy as a criterion of accuracy A wave function is more accurate or near to the exact eigenfunction when the expectation values of energy are minimal: where  arb is an arbitrary wave function and  0 would be the eigenfunction (exact wave function) of the system.

6. Variational Theorem Any expectation value of energy corresponding to a test wave function is equal or the upper limit to the lowest exact value of the system

7. Variational Theorem Optionally, the variational theorem expresses that the best approximate wave function is that minimizing the expression: where the denominator is a normalizing factor. This expression must depend on parameters to minimize energy with respect to them.

8. Variational Theorem Optionally, the variational theorem expresses that the best approximate wave function is that minimizing the expression: where the denominator is a normalizing factor. This expression must depend on parameters to minimize energy with respect to them.

9. Variational Theorem To this purpose, the arbitrary function can be developed in terms of   possible states of the system that must be linearly independent, although not necesarily eigenfunctions of the operator: where c  are coefficients of participation in the linear combination and can be considered as parameters serving to optimize the function.

10. A simple case A typical simple case where the arbitrary function depends on two states  1 and  2 :

11. A simple case This example could be referred to the electronic component of hydrogen molecule. The wave function is to be represented as a linear combination of functions describing the behavior of each electron as  1 and  2.

12. A simple case The variational function of the system can be written as: and the conditions to yield the minimal energy with respect to coefficients would be:

13. A simple case The energy expression in terms of the previous linear combination is then:

14. A simple case Defining the term or matrix element: expressing the energy of interaction between   and  local functions, and: meaning the overlap between both functions in the space defined by  coordinates, we can write:

15. A simple case Defining the term or matrix element: expressing the energy of interaction between   and  local functions, and: meaning the overlap between both functions in the space defined by  coordinates, we can write:

16. A simple case Defining the term or matrix element: expressing the energy of interaction between   and  local functions, and: meaning the overlap between both functions in the space defined by  coordinates, we can write:

17. A simple case Reordering, and deriving with respect to c 1 and c 2, we are ready to obtain their values for a minimal energy: where the unknown are c  ´s that will approach the exact energy according the variational principle. It must be observed that (H  – S  E) terms are the coefficients of the c  variables.

18. A simple case Reordering, and deriving with respect to c 1 and c 2, we are ready to obtain their values for a minimal energy: where the unknown are c  ´s that will approach the exact energy according the variational principle. It must be observed that (H  – S  E) terms are the coefficients of the c  variables.

19. A simple case In matrix terms, it means: or (H – SE)C = 0 HC – SEC = 0 HC = SEC

20. A simple case In matrix terms, it means: or (H – SE)C = 0 HC – SEC = 0 HC = SEC

21. A simple case The only non – trivial solution of this system, that is, the necessary and sufficient condition for the solution, is that the determinant of coefficients H  – S  E must vanish: or |H – SE| = 0 It is an equation of n th degree (in this case second degree) with n solutions for E.

22. A simple case The only non – trivial solution of this system, that is, the necessary and sufficient condition for the solution, is that the determinant of coefficients H  – S  E must vanish: or |H – SE| = 0 It is an equation of n th degree (in this case second degree) with n solutions for E.

23. A simple case Developing the determinant: If we substitute some terms to H 11 = H 22 = , H 21 = H 12 =  and the wave functions are normalized (S 11 =S 22 =1), the equation simplifies to a simple second degree equation in E :

24. A simple case Developing the determinant: If we substitute some terms to H 11 = H 22 = , H 21 = H 12 =  and the wave functions are normalized (S 11 =S 22 =1), the equation simplifies to a simple second degree equation in E :

25. A simple case Solutions must be: and therefore the energy only depends on parameters ,  and S 12. It must be remembered that these solutions were deduced under the condition that the energy was minimal with respect to coefficients c , not being them explicitly there.

26. A simple case Solutions must be: and therefore the energy only depends on parameters ,  and S 12. It must be remembered that these solutions were deduced under the condition that the energy was minimal with respect to coefficients c , not being them explicitly there.

27. The variational wave function Solving these equations, the spectrum of eigenvalues (energies) of the system’s Hamiltonian is obtained. They must correspond to the corresponding E k different states. The variational optimal wave function must also be deduced after finding c  ’s.

28. The variational wave function Solving these equations, the spectrum of eigenvalues (energies) of the system’s Hamiltonian is obtained. They must correspond to the corresponding E k different states. The variational optimal wave function must also be deduced after finding c  ’s.

29. The variational wave function If the values of S 12,  and  were considered known since the initial steps and we found values for E 1 and E 2, then c  values can be also calculated: However, it is easy to realize that this system is unable to provide solutions different from 0 for c 1 and c 2 if other condition is not applied.

30. The variational wave function If the values of S 12,  and  were considered known since the initial steps and we found values for E 1 and E 2, then c  values can be also calculated: However, it is easy to realize that this system is unable to provide solutions different from 0 for c 1 and c 2 if other condition is not applied.

31. The variational wave function The solution of this problem can only be achieved if the state wave functions are also mutually orthogonal, giving: where   = 1 when  =   = 0 when  ≠

32. The variational wave function Then: And consequently, solving the system:

33. The variational wave function Then: And consequently, solving the system:

34. The variational solution In our case of interest the variational wave function is then:

35. Notes and Conclusions To obtain a reliable variational wave function it is also necessary to select appropriate  1 and  2 basis functions. For the variational method to obtain a wave function of a system approaching the exact it is necessary to find an appropriate parameter dependence of the basis reference functions As much appropriate parameters can be considered, providing accountable details of the object system, the resulting variational wave functions will be more reliable and near to the exact solution.

36. Notes and Conclusions To obtain a reliable variational wave function it is also necessary to select appropriate  1 and  2 basis functions. For the variational method to obtain a wave function of a system approaching the exact it is necessary to find an appropriate parameter dependence of the basis reference functions As much appropriate parameters can be considered, providing accountable details of the object system, the resulting variational wave functions will be more reliable and near to the exact solution.

37. Notes and Conclusions To obtain a reliable variational wave function it is also necessary to select appropriate functional forms for the  1 and  2 basis. For approaching the exact wave function of a system by the variational method holds that as much appropriate parameters can be considered, the resulting variational wave functions will be more reliable and near to the exact solution. It works if such parameters can account energy relevant details of the object system.

38. Notes and Conclusions To obtain a reliable variational wave function it is also necessary to select appropriate functional forms for the  1 and  2 basis. For approaching the exact wave function of a system by the variational method holds that as much appropriate parameters can be considered, the resulting variational wave functions will be more reliable and near to the exact solution. It works if such parameters can account energy relevant details of the object system.

39. Notes and Conclusions To obtain a reliable variational wave function it is also necessary to select appropriate functional forms for the  1 and  2 basis. For approaching the exact wave function of a system by the variational method holds that as much appropriate parameters can be considered, the resulting variational wave functions will be more reliable and near to the exact solution. It works if such parameters can account energy relevant details of the object system.