1. 1 MODELING MATTER AT NANOSCALES 6. The theory of molecular orbitals for the description of nanosystems (part II) 6.01. The Hartree-Fock method applied to polyatomic systems

2. Restrictions of one electron Hamiltonians for nanosystems: need to consider electron interactions 2

3. One electron Hamiltonian conventions 1 Electron interactions are excluded on the grounds of the approximate equivalence of binding and repulsive potentials in a Hamiltonian: 3

4. One electron Hamiltonian conventions 2 Atomic cores are “frozen” and contain both nuclei and the most internal electrons to avoid their explicit consideration in calculations. 4

5. One electron Hamiltonian conventions 2 Atomic cores are “frozen” and contain both nuclei and the most internal electrons to avoid their explicit consideration in calculations. 5 The system’s wave function is partitioned and both factors are considered as mutually orthogonal:

6. One electron Hamiltonian conventions 3 Atomic basis sets in the Hilbert space of each nanoscopic system are not explicit, but implicit. Wave functions are taken as the coefficient ( C ) or transformation matrix. 6

7. One electron Hamiltonian conventions Such restrictions can be softened or avoided although it could mean a significant increase of complexity in formulas and a subsequent requirement of computational power. 7

8. One electron Hamiltonian conventions Such restrictions can be softened or avoided although it could mean a significant increase of complexity in formulas and a subsequent requirement of computational power. 8 The goal is to apply more accurate approaches to quantum modeling of nature to gain consistency and reliability.

9. The multielectronic wave function 9

10. Molecular orbital of interacting electrons The energy of an electronic state  k of a given nanoscopic system can be obtained departing from the expression: 10 The  k (R,r) wave function of the k state depends on the coordinate matrices of all nuclei ( R ) and electrons ( r ).

11. Molecular orbital of interacting electrons “The basic concept of the MO method is to find approximate electronic wave functions for a molecule by assigning to each electron a one-electron wave function which in general extends over the whole molecule.” (1) 11 1. Roothaan, C. C. J., New Developments in Molecular Orbital Theory. Rev. Mod. Phys. 1951, 23 (2), 69-89.

12. Molecular orbital of interacting electrons “The basic concept of the MO method is to find approximate electronic wave functions for a molecule by assigning to each electron a one-electron wave function which in general extends over the whole molecule.” (1) 12 It means that the  k (R,r) wave function of the whole system is then conceived as depending on  i ( r n ) molecular orbital wave functions in the state i representing the composition in the Euclidean tridimensional state of an electron with r n coordinates. 1. Roothaan, C. C. J., New Developments in Molecular Orbital Theory. Rev. Mod. Phys. 1951, 23 (2), 69-89.

13. Molecular orbital of interacting electrons Resulting evident that electrons experimentally show a residual field identified as the “spin”, they must be treated as Fermions from the statistical point of view.

14. Molecular orbital of interacting electrons Resulting evident that electrons experimentally show a residual field identified as the “spin”, they must be treated as Fermions from the statistical point of view. In order to implicitly express the spin  ( r n ) wave function factor, it can be evaluated by other implicit  (r n ) and  (r n ) function values as:  (r n ) = c   (r n ) + c   (r n ) where: c  2 + c   

15. Molecular orbital of interacting electrons Resulting evident that electrons experimentally show a residual field identified as the “spin”, they must be treated as Fermions from the statistical point of view. In order to implicitly express the spin  ( r n ) wave function factor, it can be evaluated by other implicit  (r n ) and  (r n ) function values as: Therefore, a spin molecular orbital can be expressed as:  (r n ) = c   (r n ) + c   (r n ) where: c  2 + c   

16. Molecular orbital of interacting electrons The total multielectronic wave function is expressed as the determinant of all N spin orbitals corresponding to the N electrons, being them an orthonormal set of functions describing the nanosystem.

17. Molecular orbital of interacting electrons The total multielectronic wave function is expressed as the determinant of all N spin orbitals corresponding to the N electrons, being them an orthonormal set of functions describing the nanosystem. antisymmetrized product Slater determinant It is known as the antisymmetrized product or the Slater determinant of the system:

18. Molecular orbital of interacting electrons Observe that indexes of spin orbitals and are repeated for the spatial component depending on the spin function, although the electron coordinates not. It leaves clear double occupancy of the spatial component.

19. Molecular orbital of interacting electrons Expressing the wave function as a determinant allows the correspondence with the so called “Pauli exclusion principle”. It means that the wave function representing the state for each electron remains unique because any exchange of rows or columns change the determinant result.

20. Molecular orbital of interacting electrons Expressing the wave function as a determinant allows the correspondence with the so called “Pauli exclusion principle”. It means that the wave function representing the state for each electron remains unique because any exchange of rows or columns change the determinant result. It ascribes such representation of a cloud of electrons to the experimental reality.

21. Molecular orbital of interacting electrons A set of N  i spatial orbitals gives 2N  i orthonormal spin orbitals: It means that a condition is imposed that the spatial coordinates must be shared by two electrons with different spins.

22. Molecular orbital of interacting electrons A set of N  i spatial orbitals gives 2N  i orthonormal spin orbitals: It means that a condition is imposed that the spatial coordinates must be shared by two electrons with different spins. restricted spin orbitals They are known as restricted spin orbitals.

23. Molecular orbital of interacting electrons A set of N  i spatial orbitals gives 2N  i orthonormal spin orbitals: It means that a condition is imposed that the spatial coordinates must be shared by two electrons with different spins. restricted spin orbitals They are known as restricted spin orbitals. restricted determinants The determinants or antisymmetrized products of restricted spin orbitals are known as restricted determinants.

24. Molecular orbital of interacting electrons If a set of N  i spatial orbitals gives also N  i orthonormal spin orbitals, corresponding each one to a unique combination of spatial and spin components: It means that no conditions are imposed to occupancy of spatial coordinates.

25. Molecular orbital of interacting electrons If a set of N  i spatial orbitals gives also N  i orthonormal spin orbitals, corresponding each one to a unique combination of spatial and spin components: It means that no conditions are imposed to occupancy of spatial coordinates. unrestricted spin orbitals They are then known as unrestricted spin orbitals.

26. Molecular orbital of interacting electrons If a set of N  i spatial orbitals gives also N  i orthonormal spin orbitals, corresponding each one to a unique combination of spatial and spin components: It means that no conditions are imposed to occupancy of spatial coordinates. unrestricted spin orbitals They are then known as unrestricted spin orbitals. unrestricted determinants The determinants or antisymmetrized products of restricted spin orbitals are also known as unrestricted determinants.

27. Molecular orbital of interacting electrons Restricted determinants are appropriate to save computational power when a closed shell system is modeled (all electrons are paired by spin), because calculating only one half of the monoelectronic wave functions allows that the state of the total system can be represented.

28. Molecular orbital of interacting electrons Restricted determinants are appropriate to save computational power when a closed shell system is modeled (all electrons are paired by spin), because calculating only one half of the monoelectronic wave functions allows that the state of the total system can be represented. It is under the assumption that every spatial orbital represents the states of (or is occupied by) two electrons with different spin, given their degeneracy in absence of an external field.

29. Molecular orbital of interacting electrons Restricted determinants are appropriate to save computational power when a closed shell system is modeled (all electrons are paired by spin), because calculating only one half of the monoelectronic wave functions allows that the state of the total system can be represented. It is under the assumption that every spatial orbital represents the states of (or is occupied by) two electrons with different spin, given their degeneracy in absence of an external field. Unrestricted determinants must be used when the pairing of electrons is not complete because any reason, as is the case of free radicals or triplets.

30. Spin operators The operator for total spin angular momentum of a system of N electrons as a function of spin operators for each electron of r n coordinates is: 30

31. Spin operators The operator for total spin angular momentum of a system of N electrons as a function of spin operators for each electron of r n coordinates is: 31 and the corresponding operator for total quadratic spin angular momentum is a scalar product:

32. Spin operators The operator for total spin angular momentum of a system of N electrons as a function of spin operators for each electron of r n coordinates is: 32 and the corresponding operator for total quadratic spin angular momentum is a scalar product: When the values of sub indexes are the same ( m = n ) it means a monoelectronic quadratic spin operator and when they are different ( m ≠ n ) means that it is bielectronic quadratic spin operator.

33. Spin operators Scalar (quadratic) spin operators commute with the system’s Hamiltonian in non – relativistic quantum theory: 33 and therefore they have the same eigenfunctions.

34. Spin operators Scalar (quadratic) spin operators commute with the system’s Hamiltonian in non – relativistic quantum theory: 34 and therefore they have the same eigenfunctions. It means that the system’s wave function for calculating the energy is also an eigenfunction of scalar spin operators.

35. Spin operators It can be demonstrated that for restricted multielectronic wavefunctions: 35 where S is the spin quantum number describing the total spin and M s that describing the component in the z axis for a state associated to the  k wave function.

36. Spin operators Some significant values are: 36 Quantum number S Multiplicity 2S+1 Eigenvalue of S 2 S(S+1) 010 ½20.75 132 3/243.75

37. Spin operators Unrestricted determinants are not eigenfunctions of because in this case spin orbitals are each essentially dependant from the spin component. 37

38. Spin operators Unrestricted determinants are not eigenfunctions of because in this case spin orbitals are each essentially dependant from the spin component. 38 When closed shells are represented by unrestricted determinants reach their minimal energy when the spatial components of each pair of electrons only differing on spin are identical.

39. Self Consistent Field (SCF) theory for nanosystems 39

40. Self Consistent Field theory of Roothaan and Hall The most generalized theory for treating nanoscopic systems including all formal components providing energy to the system (electrons and nuclei) is based on the Hartree 1 – Fock 2 scheme (1928 – 1930) for atoms (systems with a central – symmetrical field) as was applied to molecules (nanoscopic systems) by Roothaan 3 and Hall 4 in 1951. 40 1.- D. R. Hartree, Proc. Cambridge Phil. Soc. 24, 89 (1928). 2.- J. C. Slater, Phys. Rev. 35, 210 (1930); V. Fock, Z. Physik 61, 126 (1930) 3.- C. C. J. Roothaan, Rev. Mod. Phys. 23 (2), 69 (1951) 4.- G. Hall, Proc. R. Soc. London, A 205, 541 (1951).

41. Self Consistent Field theory of Roothaan and Hall The total Hamiltonian operator is defined (in atomic units) as: 41 where is the Hamiltonian operator for a certain n electron moving alone in the field of nuclei and r nm is the distance between n and any other m electron.

42. Atomic units Mass unit: electron mass = m e = 9.1094 · 10 −31 kg Charge unit: proton charge = e = 1.6022 · 10 −19 C Length unit: 1 bohr = a 0 = = 0.52918 Å = 5.2918 · 10 −11 m Energy unit: 1 hartree= = 627.51 kcal/mol = 4.3597 · 10 −18 J = 27.22 eV 42

43. Working units Length: 1 nm = 1000 pm = 10 Å 1 Å= 100 pm Energy: 1 eV= 23.061 kcal/mol = 8066 cm -1 = 96.49 kJ/mol 1 kcal/mol= 4.184 kJ/mol = 349.8 cm -1 43

44. Operators The one electron term of the Hartree – Fock operator can be developed according to: 44 where: Kinetic energy of nuclei of mass M A and non depending on electrons- Kinetic energy of electrons Potential energy of attraction between all electrons and all nuclei of Z A charge, separated by an R nA distance. Pairwise repulsive potential of nuclei with Z A and Z B charge and separated at an R AB distance. It does not depend on electrons.

45. Operators It must be observed that operators: 45 for monoelectronic kinetic and potential energies are formulated in absence of external fields.

46. Operators It must be observed that operators: 46 for monoelectronic kinetic and potential energies are formulated in absence of external fields. If any other physical influence on electron energy could occur, as the presence of an external field, which could even be an interaction with another external particles, is possible to add other operator terms related to it in this Hamiltonian.

47. Operators In the original Hamiltonian operator: 47 The second term is essentially bielectronic and it is the pairwise consideration of their repulsive potential energy:

48. Operators In the original Hamiltonian operator: 48 The second term is essentially bielectronic and it is the pairwise consideration of their repulsive potential energy: electron correlation It must be observed that it is a very comprehensive consideration of the charge between all electrons, although the collective interactions (those originated on the mutual effect of more than two electrons) or electron correlation is missing.

49. Operators Taking into account that the mass of nuclei is largely bigger than that of electrons, M. Born and R. Oppenheimer 1 proposer an approximation that is widely used in the quantum mechanical treatment of multinuclear systems. It implies that nuclei and electrons can be separately considered without significant damage for the calculated results, at least for chemical effects. 49 1. Born, M.; Oppenheimer, R., Zur Quantentheorie der Molekeln. Ann. Physik 1927, 389 (20), 457-484.

50. Operators Therefore according the so – called Born – Oppenheimer approximation the Hamiltonian is constrained to electronic energy contributions in the following way: 50 1.- Kinetic energy of nuclei is considered as the reference energy state:

51. Operators Therefore according the so – called Born – Oppenheimer approximation the Hamiltonian is constrained to electronic energy contributions in the following way: 51 1.- Kinetic energy of nuclei is considered as the reference energy state: 2.- The electron repulsion term is considered invariant with respect to the electron energy:

52. Operators The Born – Oppenheimer approximation means that energy calculations must be performed as the nuclei and the mass center of the system are fixed in the space. 52

53. Operators The Born – Oppenheimer approximation means that energy calculations must be performed as the nuclei and the mass center of the system are fixed in the space. 53 Simulations of molecular vibrations, reactions and translations where nuclei are evidently moving, are then performed step by step of fixed geometries.

54. Operators Therefore, after the Born – Oppenheimer approximation the Hamiltonian can be separated as: 54 1.- A Hamiltonian for describing the potential of nuclear interactions in the field of electrons with an E NN eigenvalue equivalent to a constant or fixed value for each geometry.

55. Operators Therefore, after the Born – Oppenheimer approximation the Hamiltonian can be separated as: 55 1.- A Hamiltonian for describing the potential of nuclear interactions in the field of electrons with an E NN eigenvalue equivalent to a constant or fixed value for each geometry. 2.- An electronic Hamiltonian only depending explicitly on the coordinates of each particle ( r nm ) and parametrically on nuclear coordinates ( R nA ) and with the form:

56. Total energy The value of total energy E tot when nuclei are fixed can be expressed as the sum of electronic energy E elect and the considered invariant nuclear repulsion term E NN : 56

57. The Self Consistent Field A reduced expression of the multielectronic wave function could be written as: 57 and this function is normalized:

58. The Self Consistent Field A reduced expression of the multielectronic wave function could be written as: 58 and this function is normalized: Developing the Fock matrix element in terms of spatial and orthonormal molecular orbitals (doubly occupied) it can yield the expression for expectation values of the system’s energy :

59. The Self Consistent Field Both summation terms must be understood separately: 59 is the sum of the energy for each electron motion in the field of nuclei, being doubly occupied.

60. The Self Consistent Field Both summation terms must be understood separately: 60 is the sum of the energy for each electron motion in the field of nuclei, being doubly occupied. is originated in electron interactions among different pairs of molecular orbitals i-j, that could even be between two electrons in the  i orbital and none in the  j. As means the probability of the electron with coordinates r m at the state described by function  i, then:

61. The Self Consistent Field It must be observed that: 61 is an integro-differential equation, being the kinetic energy operator a differential term and the electron interactions expressed by integral operators.

62. The Self Consistent Field The solution to obtain the energy value is reached from explicit molecular orbitals that can only be known after solving the equations. 62

63. The Self Consistent Field The solution to obtain the energy value is reached from explicit molecular orbitals that can only be known after solving the equations. 63 This contradiction can be solved by means of an iterative process, where a test function is used to start and becomes gradually improved after each cycle, to lower total energy values, until achieving a certain pre-established threshold limit of convergence between the new and a previous cycle.

64. The Self Consistent Field The solution to obtain the energy value is reached from explicit molecular orbitals that can only be known after solving the equations. 64 This contradiction can be solved by means of an iterative process, where a test function is used to start and becomes gradually improved after each cycle, to lower total energy values, until achieving a certain pre-established threshold limit of convergence between the new and a previous cycle. Self – Consistent Field (SCF) The process to calculate molecular energies by means of this routine based in the Hartree – Fock considerations is named as a Self – Consistent Field (SCF) calculation because the field created by electrons becomes stationary and stable when the energy converges to a variational minimum.

65. Matrix elements The remaining monoelectronic operator for the kinetic energy and the attraction towards all A nuclei by the n electron is: 65

66. Matrix elements Defining matrix elements for monoelectronic molecular orbitals: 66 where: H i is energy of the state i of the n electron in the field of nuclei

67. Matrix elements Defining matrix elements for monoelectronic molecular orbitals: 67 where: H i is energy of the state i of the n electron in the field of nuclei J ij is the repulsion between two electrons sharing states i and j

68. Matrix elements Defining matrix elements for monoelectronic molecular orbitals: 68 where: H i is energy of the state i of the n electron in the field of nuclei J ij is the repulsion between two electrons sharing states i and j K ij is the electron exchange between two electrons sharing states i and j

69. Matrix elements Defining matrix elements for monoelectronic molecular orbitals: 69 where: H i is energy of the state i of the n electron in the field of nuclei J ij is the repulsion between two electrons sharing states i and j K ij is the electron exchange between two electrons sharing states i and j r mn is the distance between electrons n and m

70. Matrix elements The corresponding bielectronic operators are: 70

71. Matrix elements The corresponding bielectronic operators are: 71 In that way: mean that bielectronic terms appear as were monoelectronic matrix elements.

72. Matrix elements The former expression: 72 can then be simplified to: that is the main equation of Fock’s energy for nanoscopic systems with double occupied monoelectronic state functions.

73. Matrix elements It can now be defined a certain monoelectronic Fock’s operator for a closed shell system where each  i spatial orbital is shared by two electrons as:

74. Matrix elements It can now be defined a certain monoelectronic Fock’s operator for a closed shell system where each  i spatial orbital is shared by two electrons as: In this conditions, the problem of an i molecular orbital energy is stated as:

75. Matrix elements It can now be defined a certain monoelectronic Fock’s operator for a closed shell system where each  i spatial orbital is shared by two electrons as: In this conditions, the problem of an i molecular orbital energy is stated as: and that of all molecular orbitals as: f  =  E

76. The Fock’s matrix The way to find molecular orbitals to solve the previous equations paths through establishing atomic “basis functions” being equivalent to the linear combination of atomic orbitals (LCAO), although the basis can be as extended as desired: 76

77. The Fock’s matrix The way to find molecular orbitals to solve the previous equations paths through establishing atomic “basis functions” being equivalent to the linear combination of atomic orbitals (LCAO), although the basis can be as extended as desired: 77 Coeffients c  i provide the participation of the basis function   in the  i state, or molecular orbital.

78. The Fock’s matrix The way to find molecular orbitals to solve the previous equations paths through establishing atomic “basis functions” being equivalent to the linear combination of atomic orbitals (LCAO), although the basis can be as extended as desired: 78 Coeffients c  i provide the participation of the basis function   in the  i state, or molecular orbital. It must be remembered that molecular orbitals: remain as monoelectronic spatial wave functions occupied by two electrons with opposed spins in the case of closed shells. are considered in Hartree – Fock theory with an integer charge.

79. The Fock’s matrix Hydrogenoid Slater type atomic orbitals are usually taken as reference basis functions although there are quite no formal restrictions for any atomic basis representation. They can be: Non – orthogonal 79

80. The Fock’s matrix Hydrogenoid Slater type atomic orbitals are usually taken as reference basis functions although there are quite no formal restrictions for any atomic basis representation. They can be: Non – orthogonal Centered at any place (not only on nuclei positions) 80

81. The Fock’s matrix Hydrogenoid Slater type atomic orbitals are usually taken as reference basis functions although there are quite no formal restrictions for any atomic basis representation. They can be: Non – orthogonal Centered at any place (not only on nuclei positions) Independent on the number of electrons in the system 81

82. The Fock’s matrix Hydrogenoid Slater type atomic orbitals are usually taken as reference basis functions although there are quite no formal restrictions for any atomic basis representation. They can be: Non – orthogonal Centered at any place (not only on nuclei positions) Independent on the number of electrons in the system Shaped by any other consistent functional form, like plane waves or Gaussians. 82

83. The Fock’s matrix Hydrogenoid Slater type atomic orbitals are usually taken as reference basis functions although there are quite no formal restrictions for any atomic basis representation. They can be: Non – orthogonal Centered at any place (not only on nuclei positions) Independent on the number of electrons in the system Shaped by any other consistent functional form, like plane waves or Gaussians. 83 The essential condition for a basis function is that its expression permits an appropriate representation of properties of electrons in atoms.

84. The Fock’s matrix In terms of a matrix notation: 84 A matrix of M basis functions representing the nanoscopic system (generally centered on nuclei). A vector giving the participation of 1, 2, …, m basis functions in the  i molecular orbital Coefficient matrix for transforming the basis of M atomic functions in N molecular functions. The basis set must be normalized, although it can be worked with non – orthogonal basis sets after an appropriate processing.

85. The Fock’s matrix After the previous definitions: 85 A  i molecular orbital in terms of the basis function is: The expression of all molecular orbitals in terms of the basis functions is:  =  C The matrix of expectation values of any monoelectronic operator: in terms of the basis functions is expressed by:

86. The Fock’s matrix Developing definite integrals on molecular orbitals in terms of the atomic basis functions it can be achieved that all molecular matrix elements can be expressed from the atomic basis functions by means of the coefficient matrix: 86

87. Matrix elements By substituting the expression for molecular orbitals: 87 in: It remains:

88. Matrix elements If such expression is multiplied by to both sides, then: 88

89. Matrix elements If such expression is multiplied by to both sides, then: 89 Being:

90. Matrix elements If such expression is multiplied by to both sides, then: 90 Being: and:

91. Matrix elements If such expression is multiplied by to both sides, then: 91 Being: and: Then:

92. Matrix elements Written in matrix form: 92 being a typical non orthogonal basis statement of the variational problem.

93. Matrix elements As in the case of the simplest methods, the system’s matrix must be built as a Fock’s matrix on the basis functions to be variationally optimized: 93

94. Matrix elements As in the case of the simplest methods, the system’s matrix must be built as a Fock’s matrix on the basis functions to be variationally optimized: 94 Then the problem is finding values for each F  term or matrix element corresponding to the atomic basis, diagonalize it, and then using the resulting eigenvalues and eigenvectors, given that it is a symmetrical matrix and can be non – orthogonal.

95. Matrix elements Consequently developing all previous terms on an atomic basis, remains that the Fock’s matrix is evaluated as: 95

96. Matrix elements If: 96

97. Matrix elements If: 97 then:

98. Matrix elements The Hartree – Fock’s solution for molecules means the following integral evaluation: 98 according the basis function used in each system.

99. Matrix elements It must be observed that the representation  =  C requires that the  basis was as wide as be required for representing the electron behavior in  i molecular orbitals according to the Hartree – Fock’s approximation. 99

100. Matrix elements It must be observed that the representation  =  C requires that the  basis was as wide as be required for representing the electron behavior in  i molecular orbitals according to the Hartree – Fock’s approximation. 100 Hartree – Fock’s limit It is said that to achieve an “exact” C matrix, the basis must be infinite. It is known as the “Hartree – Fock’s limit”.

101. Matrix elements It must be observed that the representation  =  C requires that the  basis was as wide as be required for representing the electron behavior in  i molecular orbitals according to the Hartree – Fock’s approximation. 101 Hartree – Fock’s limit It is said that to achieve an “exact” C matrix, the basis must be infinite. It is known as the “Hartree – Fock’s limit”. It is clear, however, that even in this case the method’s approximations are present. And it is not intrinsically exact to the quantum limit of exact wave functions.

102. Finding the HF solution Diagonalizing the F matrix to find C and E is not possible on the grounds of a non – orthogonal basis set. 102

103. Finding the HF solution Diagonalizing the F matrix to find C and E is not possible on the grounds of a non – orthogonal basis set. 103 It must be applied an internal transformation to orthogonalize the basis, diagonalize and the deorthogonalize…

104. Finding the HF solution In this case each term of the F matrix requires the evaluation of the density matrix p. 104

105. Finding the HF solution In this case each term of the F matrix requires the evaluation of the density matrix p. 105 It can be observed that p can only be evaluated after having available a C matrix.

106. Finding the HF solution In this case each term of the F matrix requires the evaluation of the density matrix p. 106 The only way to do it is after an iterative process starting from a guess C 0, matrix and the consequent density matrix p 0, to look for convergence of the self – consistent (SCF) result after the appropriate number of cycles or loops. It can be observed that p can only be evaluated after having available a C matrix.

107. Finding the HF solution as: 107 In these conditions it holds the equations: where S  1

108. Finding the HF solution as: 108 In these conditions it holds the equations: where S  1 Because this, it is not a secular determinant, nor can be diagonalized.

109. Finding the HF solution The Roothaan – Hall’s HF method requires that the F matrix was built with a non orthogonal basis set of atomic orbitals. 109

110. Finding the HF solution The Roothaan – Hall’s HF method requires that the F matrix was built with a non orthogonal basis set of atomic orbitals. 110 Then, an orthogonalization procedure is applied to the system:

111. Finding the HF solution The Roothaan – Hall’s HF method requires that the F matrix was built with a non orthogonal basis set of atomic orbitals. 111 Then, an orthogonalization procedure is applied to the system: The secular matrix system is resolved:

112. Finding the HF solution The Roothaan – Hall’s HF method requires that the F matrix was built with a non orthogonal basis set of atomic orbitals. 112 Then, an orthogonalization procedure is applied to the system: The secular matrix system is resolved: Then, it is diagonalized and the orthogonal coefficient matrix elements based c i  ’ is again transformed to bring the non – orthogonal molecular wave function:

113. The total energy The diagonal E matrix, that remains invariant to deorthogonalization, provides a way to find the system’s total energy.

114. The total energy The diagonal E matrix, that remains invariant to deorthogonalization, provides a way to find the system’s total energy. By definition: which integral of the corresponding expectation values in the whole Hilbert’s space provide the energy of one electron with r n coordinates being described by in the  i (r n ) monoelectronic state: given that the molecular orbitals constitute an orthonormal basis.

115. The total energy As the definition of elements intervening in the Fock’s operator:

116. The total energy Then, substituting it by its equivalent, we can express the orbital energy eigenvalue as a function of all intervening matrix elements: As the definition of elements intervening in the Fock’s operator:

117. The total energy Then, substituting it by its equivalent, we can express the orbital energy eigenvalue as a function of all intervening matrix elements: As the definition of elements intervening in the Fock’s operator: This expression can be understood as if one electron in the  i state is considered as being stabilized by the own kinetic energy and that of the attraction by all nuclei given by H i and balanced with all Coulomb and exchange interactions with other electrons, except itself ( J ii – K ii = 0 when i = j ).

118. The total energy where summations are only on occupied orbitals. It can be easily deduced that the total energy in terms of the doubly occupied molecular orbitals is:

119. The total energy where summations are only on occupied orbitals. It can be easily deduced that the total energy in terms of the doubly occupied molecular orbitals is: In terms of atomic basis functions:

120. SCF cycle 120

121. The total energy SCF procedure The total Hartree – Fock energy, after the Roothaan – Hall adaption for polyatomic systems, more well known as the SCF procedure, can be expressed in terms of molecular orbitals and within the Born – Oppenheimer approximation remains:

122. The total energy SCF procedure The total Hartree – Fock energy, after the Roothaan – Hall adaption for polyatomic systems, more well known as the SCF procedure, can be expressed in terms of molecular orbitals and within the Born – Oppenheimer approximation remains: And in terms of atomic basis functions:

123. Ionization and excitation energies 123 In the case of calculations of molecular ionization and excitation energies, we have to set up approximate wave functions for either the ionized or excited states.

124. Ionization and excitation energies The basic idea of this procedure is that for an ionized or excited state we do not set up and solve the appropriate variational problem by which all the MO's have to be determined for that particular state, but we make use of the MO's which were found from the variational problem for the ground state. In the case of calculations of molecular ionization and excitation energies, we have to set up approximate wave functions for either the ionized or excited states.

125. Ionization energies 125 The wave function of the ionized state is obtained by omitting the selected MO that we want to investigate from the ground state Slater determinant.

126. Ionization energies The wave function of the ionized state is obtained by omitting the selected MO that we want to investigate from the ground state Slater determinant. Being the ground state:

127. Ionization energies The ionized state of the N spin orbital, that would be the HOMO, is expressed as.

128. Ionization energies 128 The corresponding total electronic energies for each state are:

129. Ionization energies The differences between both terms represents the ionization potential of the spin orbital and is: 129 The corresponding total electronic energies for each state are:

130. Ionization energies The differences between both terms represents the ionization potential of the spin orbital and is: 130 The corresponding total electronic energies for each state are: It means a theoretical justification of the Koopman’s theorem.

131. Ionization energies Comparisons with experimental XPS ionization spectra and other methods favor the modeling of ionization of water valence bands by Hartree – Fock procedures Stowasser, R.; Hoffmann, R., What Do the Kohn−Sham Orbitals and Eigenvalues Mean? J. Am. Chem. Soc. 1999, 121 (14), 3414-3420.

132. Excitation energies 132 The wave function of any state in the Hartree – Fock theory can be represented by distribution in the Slater determinant.

133. Excitation energies 133 The wave function of any state in the Hartree – Fock theory can be represented by distribution in the Slater determinant. Being the ground state:

134. Excitation energies 134 An excited state can be a singlet, if the spin remains as in the ground state.

135. Excitation energies 135 An excited state can be a singlet, if the spin remains as in the ground state. A certain k excited state corresponding to missing a previous occupied molecular orbital  and replacing it by a previously empty level with the same spin could be:

136. Excitation energies 136 An excited state can also be a triplet, if the total spin multiplicity becomes originated in an occupation with a different spin.

137. Excitation energies 137 An excited state can also be a triplet, if the total spin multiplicity becomes originated in an occupation with a different spin. A certain l triplet excited state corresponding to missing a previous occupied molecular orbital  and adding a previously empty level could be:

138. Excitation energies 138 The corresponding energies of both singlet and triplet k state only differentiates in signs of the exchange integral: where the plus sign holds for the singlet and the minus for the triplet.

139. Excitation energies This expression grants that electronic transitions depend on energy differences between the monoelectonic MO eigenvalues as corrected by two electron terms J and K, depending on multiplicity. 139 It can be further developed to the energy of excitation :

140. Excitation energies 140 It must be noticed that electron repulsion J here contributes to drop the energy of excited states, while K electron exchange favors their energy increase. The split expressions for each multiplicity are