1. 1 MODELING MATTER AT NANOSCALES 6. The theory of molecular orbitals for the description of nanosystems (part II) 6.05. Variational methods for dealing with correlation energy

2. Optimizing the Hartree - Fock wave function Being  s the multielectronic Hartree-Fock wave function of any s state corresponding to a molecule or other nanoscopic system it can always be expressed as a linear combination of associated wave functions, following the algebraic formalism: where the sum is over a sufficiently large I series of  I multielectronic reference states and a sI are their corresponding participation coefficients in such s state.

3. Optimizing the Hartree - Fock wave function Being  s the multielectronic Hartree-Fock wave function of any s state corresponding to a molecule or other nanoscopic system it can always be expressed as a linear combination of associated wave functions, following the algebraic formalism: where the sum is over a sufficiently large I series of  I multielectronic reference states and a sI are their corresponding participation coefficients in such s state.  must be an orthonormal basis.

4. Optimizing the Hartree - Fock wave function Obviously, a sI participation coefficients could be optimized on the grounds of a variational approach to obtain the I series giving a s state of minimal total energy.

5. Optimizing the Hartree - Fock wave function Obviously, a sI participation coefficients could be optimized on the grounds of a variational approach to obtain the I series giving a s state of minimal total energy. Therefore, the resulting  s for each state of the system will better approach the exact wave function as the total energy becomes smaller.

6. Optimizing the Hartree - Fock wave function If multielectronic reference functions are antisymmetrized products or Slater determinants resulting from a Hartree – Fock calculation, a typical electronic configuration is that where each spin orbital represents a given existing electron. where  i reference functions are spin orbitals describing the state of an electron of r n  n ≡  n spatial and spin coordinates.

7. Optimizing the Hartree - Fock wave function Distribution of different spin orbital populations can vary from that of the ground state  0. Each one of the I electron populations could be a valid basis representation of the system serving for a variational optimization.

8. Optimizing the Hartree - Fock wave function Each determinant showing a different electron distribution to that of the ground state is known as an excited configuration.

9. Optimizing the Hartree - Fock wave function Configurations could correspond to one of the following types: HF: ground state; S: monoexcited configuration; D: double excited configuration; T: triple excited configuration; Q: quadruple excited configuration

10. Optimizing the Hartree - Fock wave function Departing from a solution of the Hartree – Fock multielectronic wave function a very large number of configurations can be made depending on the number of electrons entering different occupations of spin orbitals: where means a p order excited configuration, redistributing p charges among all initially unoccupied spin orbitals.

11. Optimizing the Hartree - Fock wave function Therefore, the solution of a certain n electron system under the Hartree – Fock procedure represented by N spin orbital basis functions will amount m = N – n non occupied spin orbitals. It can be shown that the number of possible reference functions for this development is huge and can be calculated according to:

12. Water as an example If we develop a minimal Slater basis function for water: We will have n = 10 electrons and N = 14 possible basis spin orbitals when the Hartree – Fock molecular problem was solved. Therefore m = 4 orbitals will remain as non occupied. H: 1s    1s  O: 1s  , 1s  , 2s  , 2s  , 2p x  , 2p x  , 2p y  , 2p y    2p z  2p z 

13. Water as an example If we develop a minimal Slater basis function for water: We will have n = 10 electrons and N = 14 possible basis spin orbitals when the Hartree – Fock molecular problem was solved. Therefore m = 4 orbitals will remain as non occupied. H: 1s    1s  O: 1s  , 1s  , 2s  , 2s  , 2p x  , 2p x  , 2p y  , 2p y    2p z  2p z  Therefore, the number of possible electronic configurations will be:

14. Water as an example If the basis function is as simple as 3-21G, basis atomic orbitals would be: We will again have n = 10 electrons but N = 26 possible basis spin orbitals when the Hartree – Fock molecular problem was solved. Therefore m = 16 orbitals will remain as non occupied. H: 1s    1s  1s  ’  1s  ’ O: 1s  , 1s  , 2s  , 2s  , 2p x  , 2p x  , 2p y  , 2p y    2p z  2p z  2s  ’, 2s  ’, 2p x  ’, 2p x  ’, 2p y  ’, 2p y  ’  2p z  ’  2p z  ’

15. Water as an example If the basis function is as simple as 3-21G, basis atomic orbitals would be: We will again have n = 10 electrons but N = 26 possible basis spin orbitals when the Hartree – Fock molecular problem was solved. Therefore m = 16 orbitals will remain as non occupied. Therefore, the number of possible electronic configurations will be much higher: H: 1s    1s  1s  ’  1s  ’ O: 1s  , 1s  , 2s  , 2s  , 2p x  , 2p x  , 2p y  , 2p y    2p z  2p z  2s  ’, 2s  ’, 2p x  ’, 2p x  ’, 2p y  ’, 2p y  ’  2p z  ’  2p z  ’

16. Correlation energy Being the Hartree – Fock equation of eigenvalues and eigenfunctions of a given I electronic configuration  I : A certain matrix element on the ground of atomic orbitals can be written in a simplified way as: where

17. Correlation energy The total energy of such I configuration then remains as: being a matrix element of the density matrix in the I state: on the grounds of c  i molecular orbital coefficients for atomic orbitals   and .

18. Correlation energy On the other hand, the “exact” expression for the s state energy E s, as an output of optimizing linear combinations of all reference I states: would be expected to become from:

19. Correlation energy It is evident that finding the  s wave functions and their corresponding eigenvalues E s after optimizing Hartree – Fock electron configurations means that we are trying to attain a form of correlation energy, given the corresponding definition, as in the case of a ground state:

20. Correlation energy It is evident that finding the  s wave functions and their corresponding eigenvalues E s after optimizing Hartree – Fock electron configurations means that we are trying to attain a form of correlation energy, given the corresponding definition, as in the case of a ground state: It is congruent with the definition of correlation energy if considering the variationally optimized energy as “exact”.

21. The method of configuration interaction (CI) The method of configuration interaction (CI) means the variational optimization of multielectronic states on the basis of linear combinations of several different configurations of the given system. Such configurations are expressed as their respective determinants being built from a previous one electron wave function procedure.

22. The method of configuration interaction (CI) The method of configuration interaction (CI) means the variational optimization of multielectronic states on the basis of linear combinations of several different configurations of the given system. Such configurations are expressed as their respective determinants being built from a previous one electron wave function procedure. If the expansion includes all possible configurations, then this is called as a full configuration interaction procedure which is said to exactly solve the electronic Schrödinger equation within the space spanned by the one-particle basis set.

23. The method of configuration interaction (CI) The matrix representation to solve the problem is: HA = EA It means the evaluation and diagonalization of the H IJ matrix elements in:

24. The method of configuration interaction (CI) The matrix representation to solve the problem is: HA = EA It means the evaluation and diagonalization of the H IJ matrix elements in: It will output: – a sI eigenvectors for the A transformation matrix – E s eigenvalues of relative energies corresponding to all s states, resulting in the E diagonal matrix.

25. The method of configuration interaction (CI) The general formulation for H IJ matrix elements is*: Depending on the energy of interaction among all and each one of the I and J configurations, taken by pairs. * Nesbet, R. K., Configuration interaction in orbital theories. Proc. Roy. Soc. (London) A 1955, 230 (1182 ), 312-321.

26. The method of configuration interaction (CI) The general formulation for H IJ matrix elements is*: Depending on the energy of interaction among all and each one of the I and J configurations, taken by pairs. It builds a symmetrical matrix.  IJ basis functions are Slater determinants or any orthonormal set of multielectronic wave functions. * Nesbet, R. K., Configuration interaction in orbital theories. Proc. Roy. Soc. (London) A 1955, 230 (1182 ), 312-321.

27. The method of configuration interaction (CI) From this general formulation, it must be noticed that:  Each H IJ can exist for interactions between I and J excited configurations independently of the number of involved electrons.

28. The method of configuration interaction (CI) From this general formulation, it must be noticed that:  Each H IJ can exist for interactions between I and J excited configurations independently of the number of involved electrons.  When the basis to build a configuration is orthonormal interactions between terms differing in more than two monoelectronic orbitals vanish.

29. The method of configuration interaction (CI) From this general formulation, it must be noticed that:  Each H IJ can exist for interactions between I and J excited configurations independently of the number of involved electrons.  When I = J then  II means the energy E I of such basis configuration.  When the basis to build a configuration is orthonormal interactions between terms differing in more than two monoelectronic orbitals vanish.

30. Evaluation of CI matrix elements Evaluation of H IJ elements of the CI’s H matrix remains as the key problem, as occurs in all variational quantum calculations.

31. Evaluation of CI matrix elements Evaluation of H IJ elements of the CI’s H matrix remains as the key problem, as occurs in all variational quantum calculations. It must be taken into account that basis functions are no longer one electron wave functions (molecular or atomic orbitals) but Hartree – Fock’s multielectronic configurations after the previous optimization of the initial occupation, that usually is the ground state:

32. Evaluation of CI matrix elements H IJ terms depend on projections of a given I configuration or multielectronic state on another one denoted as J. It means an scalar product in the configuration space.

33. Evaluation of CI matrix elements H IJ terms depend on projections of a given I configuration or multielectronic state on another one denoted as J. It means an scalar product in the configuration space. As each configuration is expressed as a determinant of one electron wave functions: where orbitals of I determinant are numbered by k sub indexes and those of J by l ’s.

34. Evaluation of CI matrix elements Such determinant can be developed in terms of minors of p order, according the “ p ” one electron functions differencing I and J configurations.

35. Evaluation of CI matrix elements As one electron basis functions (spin orbitals) building Slater determinants are orthonormal, the sole none vanishing elements of D IJ determinant are those where k and l are the same, although appearing in two separate configurations.

36. Evaluation of CI matrix elements As one electron basis functions (spin orbitals) building Slater determinants are orthonormal, the sole none vanishing elements of D IJ determinant are those where k and l are the same, although appearing in two separate configurations. If spin orbital ordering of both I and J determinants are the same, non vanishing terms will appear in the diagonal of D IJ. Any occupation difference between them will also brought vanishing diagonal terms when l ≠ k.

37. Evaluation of CI matrix elements As one electron basis functions (spin orbitals) building Slater determinants are orthonormal, the sole none vanishing elements of D IJ determinant are those where k and l are the same, although appearing in two separate configurations. If spin orbital ordering of both I and J determinants are the same, non vanishing terms will appear in the diagonal of D IJ. Any occupation difference between them will also brought vanishing diagonal terms when l ≠ k. Therefore, for all determinant minors in general:

38. Evaluation of CI matrix elements Thus, the CI matrix element is generally expressed as:

39. Evaluation of CI matrix elements And in the case of orthonormal spin orbitals D IJ ’s become Kroneker’s deltas: cancelling most terms of all sums.

40. Evaluation of CI matrix elements If in the case of interactions between configurations where i and j are initially filled spin orbitals transferring electrons to certain a and b initially empty spin orbitals, the calculation of matrix elements only involve the populated and unpopulated spin orbitals by: for double excitations. Similar terms can be developed for triples and quadruples. in the case of single excitations and

41. Evaluation of CI matrix elements Single electron excitations from the ground state can be expressed as: leading to the matrix element between two spin orbitals of the same monoelectronic solution.

42. Evaluation of CI matrix elements Being both spin orbitals eigenfunctions of the Fock operator: because all  i and  a orbitals are orthonormal by definition. It means that:

43. Evaluation of CI matrix elements Being both spin orbitals eigenfunctions of the Fock operator: because all  i and  a orbitals are orthonormal by definition. It means that: It is known as the Brillouin’s theorem that establish that the ground state do not intervene in optimization of singly excited states.

44. Evaluation of CI matrix elements As single excited configurations do not participate in ground state energy optimizations by CI procedures, such singly excited determinants only serve for energy optimizations of excited states.

45. Evaluation of CI matrix elements As single excited configurations do not participate in ground state energy optimizations by CI procedures, such singly excited determinants only serve for energy optimizations of excited states. Double and quadruple excited configurations are important to optimize energy of the ground state by accounting electron correlation.

46. Evaluation of CI matrix elements In the case of singly excited configuration interaction we arrived to: Here, the term: is the associated transition energy of an electron with space and spin coordinates  1 in spin orbital  i changing to occupy the formerly empty  a being defined by certain I singly excited configuration (as expressed by the corresponding Slater determinant).

47. Evaluation of CI matrix elements In the case of singly excited configuration interaction we arrived to: Here, the term: is the associated transition energy of an electron with space and spin coordinates  1 in spin orbital  i changing to occupy the formerly empty  a being defined by certain I singly excited configuration (as expressed by the corresponding Slater determinant). If we re dealing with Hartree – Fock spin orbitals:

48. Evaluation of CI matrix elements On the other hand, for the two electron terms in: we have the general expression: as the corresponding general matrix element expressing the interaction of electrons 1 and 2 that occupied  i and  j as changing to their new occupation in  a and  b spin orbitals.

49. Evaluation of CI matrix elements By taking into account previous considerations, after some algebraic deductions*, the practical evaluation of CI singly excited matrix becomes from the following for diagonal elements: and for off diagonal elements (being i  j and a  b ) become: * Pople, J. A., The electronic spectra of aromatic molecules. II. A theoretical treatment of excited states of alternant hydrocarbon molecules based on self-consistent molecular orbitals. Proc. Phys. Soc. London 1955, 68A, 81-9.

50. Some CI results H 2 O* * Saxe, P.; Schaeffer III, H. F.; Handy, N. C., Exact solution (within a double-zeta basis set) of the Schrödinger electronic equation for water. Chem. Phys. Lett. 1981, 79 (2), 202-204. A standard contracted Gaussian double  basis set of Dunning – Huzinaga was used with a scale factor of 1.2 for H. Geometry was previously optimized at the same level.

51. Some CI results H 2 equilibrium bond distance (Bohr’s)* ** Kolos, W.; Wolniewicz, L., Improved Theoretical Ground-State Energy of the Hydrogen Molecule. J. Chem. Phys. 1968, 49 (1), 404-410 after an extensive quantum calulation over all possible variables of the molecule where vibrational energy remain only 0.9 cm-1 above experimental value. * Szabo, A.; Ostlund, N. S., Modern quantum chemistry: introduction to advanced electronic structure theory. First edition, revised ed.; McGraw-Hill: New York, 1989; p 466. Atomic basis setSCF“Full CI” STO-3G1.3461.389 4-31G1.3801.410 6-31G**1.3851.396 “Exact” value ** 1.401

52. Some CI results H 2 equilibrium bond distance (Bohr’s)* * Szabo, A.; Ostlund, N. S., Modern quantum chemistry: introduction to advanced electronic structure theory. First edition, revised ; McGraw-Hill: New York, 1989; p 466.

53. Some CI results CH 2 O (formaldehyde) STO-6G6-31G6-311++G**CID| 6-311++G** CISD| 6-311++G** Exp. Total energy (Hartrees) -113.4408-113.8084-113.9029-114.2245-114.2279 r CO (Å)1.2161.2101.1801.1971.1991.210 r CH (Å)1.0981.0821.0941.1011.102 <HCH114.8116.6116.0116.1116.0121.1  (debye) 1.5963.3042.8062.8932.9052.33 Relative calc. time 10.81.410.912.6

54. Single excitations and light Single excitation configuration interactions (CIS) are mostly used to optimize excited state energies for simulating UV spectra of molecules.

55. Single excitations and light Single excitation configuration interactions (CIS) are mostly used to optimize excited state energies for simulating UV spectra of molecules. Therefore, it allows applications to spectroscopy, photochemistry and photonics, the science for generating, controlling and detecting photons.

56. Single excitations and light An electronic state of a given nanoscopic system can be understood as characterized by a corresponding electron density map. Porphirin ground state charge distribution as calculated at the CNDOL/2CC||MP2|4- 31G(d,p) level. Reds are negative and blues positive charges.

57. Single excitations and light The ground state shows the most stable charge cloud distribution while excited states are different, by using excitation energy to stabilize it and returning to the original ground state density map upon deactivation. Acrolein excitation from the S 0 (ground state) to S 1 (first excited singlet state) as calculated at the CNDOL/2CC||MP2|4-31G(d,p) level. Reds are negative and blues positive charge s. h

58. Multiconfigurational Hartree – Fock procedures (MC-SCF) Are there certain cases where small energy variations decide a nanoscopic procedure, as those with: Biradicals Non saturated transition metals Excited states perturbing ground states Bond breakings Transition states in swinging transitions

59. Multiconfigurational Hartree – Fock procedures (MC-SCF) Are there certain cases where small energy variations decide a nanoscopic procedure, as those with: Biradicals Non saturated transition metals Excited states perturbing ground states Bond breakings Transition states in swinging transitions It could be useful that interacting configurations could also be included for Hartree – Fock wave function optimizations.

60. Multiconfigurational Hartree – Fock procedures (MC-SCF) Then, a more or less limited number of configurations can be chosen to be treated by independent Hartree – Fock ‘s iterative procedures, representing the active space (AS), to optimize both CI and one electron wave function coefficients.

61. Multiconfigurational Hartree – Fock procedures (MC-SCF) Then, a more or less limited number of configurations can be chosen to be treated by independent Hartree – Fock ‘s iterative procedures, representing the active space (AS), to optimize both CI and one electron wave function coefficients. This procedure is called as a multiconfigurational self consistent field (MC-SCF) routine, meaning that: is the multiconfigurational wave function and A K are the optimizing coefficients on the basis of previously optimized  K Hartree – Fock’s configurations.

62. Multiconfigurational Hartree – Fock procedures (MC-SCF) Each optimized  K, or configurational state function (CSF), can be: A unique Slater determinant with a given and convenient electron occupation of spin orbitals. A combination of Slater determinants to correct spin influences.

63. Multiconfigurational Hartree – Fock procedures (MC-SCF) CSF’s can be expressed by orbital occupations of the initial ground state Hartree – Fock solution and their corresponding Slater determinants:

64. Multiconfigurational Hartree – Fock procedures (MC-SCF) Each one electron spin orbital of a given K configuration is defined as the LCAO of Hartree – Fock’s method:

65. Multiconfigurational Hartree – Fock procedures (MC-SCF) At the end, both coefficient sets of A CI’s and C K Hartree – Fock’s coefficient matrix for the chosen K configurations are involved in a MC-SCF optimization. They are optimized iteratively in cycles until a desired convergence limit. Each one electron spin orbital of a given K configuration is defined as the LCAO of Hartree – Fock’s method:

66. Multiconfigurational Hartree – Fock procedures (MC-SCF) At the end, both coefficient sets of A CI’s and C K Hartree – Fock’s coefficient matrix for the chosen K configurations are involved in a MC-SCF optimization. They are optimized iteratively in cycles until a desired convergence limit. Each one electron spin orbital of a given K configuration is defined as the LCAO of Hartree – Fock’s method: MC-SCF procedures are particularly demanding of computational resources.

67. The particular case of CAS-SCF The complete active space self-consistent field (CAS-SCF) procedure was developed* to select an active space given by a certain N A number of electrons to participate in the variational optimization and a given M number of spin orbitals of convenience, by following certain rules. *Roos, B. O.; Taylor, P. R.; Siegbahn, P. E. M., A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys. 1980, 48 (2), 157-173..

68. The particular case of CAS-SCF The total of participating orbitals are divided in three sets: Inactive orbitals always taken as double occupied with a total of N – N A electrons, where N is the total number of system’s electrons. They are all spin orbitals below the last reaching up to complete N – N A electrons.

69. The particular case of CAS-SCF The total of participating orbitals are divided in three sets: Inactive orbitals always taken as double occupied with a total of N – N A electrons, where N is the total number of system’s electrons. They are all spin orbitals below the last reaching up to complete N – N A electrons. Active orbitals, giving the active space, consisting in all configurations where the chosen N A electrons can participate in M spin orbitals above the inactive space. It is recommended that the selected M orbitals were those more involved in the modeled phenomenon (i.e.,  orbitals near to HOMO). Their average occupation will be a real number, between 0 and 2.

70. The particular case of CAS-SCF The total of participating orbitals are divided in three sets: Inactive orbitals always taken as double occupied with a total of N – N A electrons, where N is the total number of system’s electrons. They are all spin orbitals below the last reaching up to complete N – N A electrons. Active orbitals, giving the active space, consisting in all configurations where the chosen N A electrons can participate in M spin orbitals above the inactive space. It is recommended that the selected M orbitals were those more involved in the modeled phenomenon (i.e.,  orbitals near to HOMO). Their average occupation will be a real number, between 0 and 2. Virtual orbitals, always empty of electrons. They will be all those remaining above the active orbitals to reach the total number of spin orbitals of the system.

71. The particular case of CAS-SCF Graphical representation: After selecting the orbitals to consider, CI is only performed on the active space, and the rest of the system is treated at a normal Hartree – Fock level.

72. The particular case of CAS-SCF The example of formaldehyde: STO-6G6-311++G**CISD| 6-311++G** CASSCF(6,8)| 6-311++G Exp. Total energy (Hartrees) -113.4408-113.9029-114.2279-114.0260 r CO (Å)1.2161.1801.1991.2121.210 r CH (Å)1.0981.0941.1021.0941.102 <HCH114.8116.0 116.1121.1  (debye) 1.5962.8062.9051.0052.33 Relative calc. time 11.412.6313.9 CASSCF(m,n): n = number of electrons in active space; m = number of spin orbitals in active space.