1. MODELING MATTER AT NANOSCALES 4. Introduction to quantum treatments 4.03. Eigenvectors and eigenvalues of a matrix

2. Simultaneous equations in the variational method The problem of simultaneous equations in the variational method can be written as: HC – SEC = 0 The solution can be achieved when the reference functions to the arbitrary solution are orthonormal. In that case the overlap matrix becomes a unitary |  mn | = 1, and the expression becomes: HC – 1EC = 0

3. Simultaneous equations in the variational method The problem of simultaneous equations in the variational method can be written as: HC – SEC = 0 The solution can be achieved when the reference functions to the arbitrary solution are orthonormal. In that case the overlap matrix becomes a unitary |  mn | = 1, and the expression becomes: HC – 1EC = 0

4. Simultaneous equations in the variational method The problem of simultaneous equations in the variational method can be written as: HC – SEC = 0 The solution can be achieved when the reference functions to the arbitrary solution are orthonormal. In that case the overlap matrix becomes a unitary |  mn | = 1, and the expression becomes: HC – 1EC = 0

5. Simultaneous equations in the variational method Identity matrix can be omitted and then: HC – EC = 0 HC = EC H = CEC -1 secular equation The determinantal equation corresponding to the orthonormal reference is known as the system’s secular equation: |H – E| = 0

6. Simultaneous equations in the variational method Identity matrix can be omitted and then: HC – EC = 0 HC = EC H = CEC -1 secular equation The determinantal equation corresponding to the orthonormal reference is known as the system’s secular equation: |H – E| = 0

7. Diagonalization diagonalized A given matrix H becomes diagonalized when both a diagonal matrix E and a reversible matrix C exists and are found, such that: H = CEC -1 It relates the problem of obtaining expected or exact values for the Schrödinger equation if H is the Hamiltonian matrix and C corresponds to coefficients of participation in the linear combination of system’s reference wave functions.

8. Diagonalization diagonalized A given matrix H becomes diagonalized when both a diagonal matrix E and a reversible matrix C exists and are found, such that: H = CEC -1 It relates the problem of obtaining expected or exact values for the Schrödinger equation if H is the Hamiltonian matrix and C corresponds to coefficients of participation in the linear combination of system’s reference wave functions.

9. Diagonalization Diagonalization of H matrix can mostly be performed when it is symmetric ( H  = H  ). It is a routine treatment of numeric mathematics. Several algorithms and programs in software libraries are nowadays highly optimized to fast solving the problem of even huge matrices.

10. Diagonalization Diagonalization of H matrix can mostly be performed when it is symmetric ( H  = H  ). It is a routine treatment of numeric mathematics. Several algorithms and programs in software libraries are nowadays highly optimized to fast solving the problem of even huge matrices.

11. Diagonalization Diagonalization of H matrix can mostly be performed when it is symmetric ( H  = H  ). It is a routine treatment of numeric mathematics. Several algorithms and programs in software libraries are nowadays highly optimized to fast solving the problem of even huge matrices.

12. Eigenvalues and eigenvectors eigenvalues and eigenvectors Diagonalization is also known as the case of finding eigenvalues and eigenvectors of a symmetric matrix. Eigenvalues are the E k terms of the diagonal E matrix. Eigenvectors are given by the c i  coefficients in C matrix.

13. Eigenvalues and eigenvectors eigenvalues and eigenvectors Diagonalization is also known as the case of finding eigenvalues and eigenvectors of a symmetric matrix. Eigenvalues are the E k terms of the diagonal E matrix. Eigenvectors are given by the c i  coefficients in C matrix.

14. Eigenvalues and eigenvectors eigenvalues and eigenvectors Diagonalization is also known as the case of finding eigenvalues and eigenvectors of a symmetric matrix. Eigenvalues are the E k terms of the diagonal E matrix. Eigenvectors are given by the c i  coefficients in C matrix.

15. Eigenvalues and eigenvectors Eigenvectors resulting from routine diagonalization procedures are orthonormal: C matrices are non-symmetric.

16. Eigenvalues and eigenvectors Eigenvectors resulting from routine diagonalization procedures are orthonormal: C matrices are non-symmetric.

17. Orbitals as a case of reference

18. MO as LCAO One of the most relevant applications of linear algebra for understanding the nanoscopic universe is the consideration of wave functions as references that can enter variational optimizations for building solutions to the state wave function of complex systems. orbitals Such reference wave functions usually correspond to the state of a single electron, either in an atom, a bond, a molecule, a unit crystal, etc. and are defined as orbitals.

19. MO as LCAO One of the most relevant applications of linear algebra for understanding the nanoscopic universe is the consideration of wave functions as references that can enter variational optimizations for building solutions to the state wave function of complex systems. orbitals Such reference wave functions usually correspond to the state of a single electron, either in an atom, a bond, a molecule, a unit crystal, etc. and are defined as orbitals.

20. MO as LCAO Molecular Orbitals Linear Combination of Atomic Orbitals In order to optimize one-electron wave functions of molecules, or Molecular Orbitals (MO), the Linear Combination of Atomic Orbitals (LCAO) has often been used: where coefficients c i  provide the participation of the basis or reference atomic orbital   in the state or molecular orbital  i.

21. MO as LCAO Molecular Orbitals Linear Combination of Atomic Orbitals In order to optimize one-electron wave functions of molecules, or Molecular Orbitals (MO), the Linear Combination of Atomic Orbitals (LCAO) has often been used: where coefficients c i  provide the participation of the basis or reference atomic orbital   in the state or molecular orbital  i.

22. MO as LCAO If the coefficient or eigenvector matrix is considered as: and the system’s matrix is given by energies in terms of the atomic orbital structure is given as: Then, a product can be obtained as: HC = EC

23. MO as LCAO If the coefficient or eigenvector matrix is considered as: and the system’s matrix is given by energies in terms of the atomic orbital structure is given as: Then, a product can be obtained as: HC = EC

24. MO as LCAO If the coefficient or eigenvector matrix is considered as: and the system’s matrix is given by energies in terms of the atomic orbital structure is given as: Then, a product can be obtained as: HC = EC

25. MO as LCAO E is then a diagonal matrix terming E k eigenvalues of all considered one-electron wave functions of the system:

26. MO as LCAO E can be obtained by the coefficient C matrix from H after a linear transformation, or diagonalization: E = CHC -1

27. MO as LCAO H is constructed by energies of each basis set component (basis orbitals) in the system’s context. E is a result of a linear transformation of H energy matrix by C eigenvector matrix of MO’s coefficients giving the minimal total energy. finding H on a given basis to transform it in an optimized energy expression of the complete system This step is considered as the leitmotif of almost all quantum mechanical calculations of nanosystems: finding H on a given basis to transform it in an optimized energy expression of the complete system.

28. MO as LCAO H is constructed by energies of each basis set component (basis orbitals) in the system’s context. E is a result of a linear transformation of H energy matrix by C eigenvector matrix of MO’s coefficients giving the minimal total energy. finding H on a given basis to transform it in an optimized energy expression of the complete system This step is considered as the leitmotif of almost all quantum mechanical calculations of nanosystems: finding H on a given basis to transform it in an optimized energy expression of the complete system.

29. MO as LCAO H is constructed by energies of each basis set component (basis orbitals) in the system’s context. E is a result of a linear transformation of H energy matrix by C eigenvector matrix of MO’s coefficients giving the minimal total energy. finding H on a given basis to transform it in an optimized energy expression of the complete system This step is considered as the leitmotif of almost all quantum mechanical calculations of nanosystems: finding H on a given basis to transform it in an optimized energy expression of the complete system.