MODELING MATTER AT NANOSCALES

Transcription

1 MODELING MATTER AT NANOSCALES 6. The theory of molecular orbitals for the description of nanosystems (part II) 6.0. Ab initio methods. Basis functions. Luis A. Monte ro Firmado digitalmente por Luis A. Montero Nombre de reconocimiento (DN): cn=luis A. Montero, o=universidad de La Habana, ou=dep. de Química Física, quimica.uh.cu, c=cu Fecha: :6:0-05'00'

2 The SCF cycle

3 The SCF cycle The Fock matrix must be constructed: F F F =. F ν F F.. and the problem is to evaluate the F µν matrix elements on the grounds of basis orbitals participating in a nanoscopic system, to diagonalize it, and then obtaining eigenvalues and eigenvectors. This is a symmetric matrix and we are imposing the condition that such basis set can not be orthogonal F F F µ µ. µµ 3

4 The SCF cycle As: 4 = i i i i c n c p ν µ µν ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( σν λµ τ χ χ χ χ νσ λµ τ χ χ χ χ λµσν ν σ µ λ λµνσ σ ν µ λ = = = = K d r r r r r J d r r r r r n m mn n m n m mn n m

5 The SCF cycle As: χ ( r λ χ ( r λ m m ) χ ( r µ ) χ ( r µ n n ) ) r mn r mn p µν χ ( ν χ ( r σ = nic i r ) χ ( r m m σ ) χ ( r ν n n µ i c νi ) dτ = ) dτ = J K λµνσ λµσν = ( λµ νσ ) = ( λµ σν ) Then the formula to calculate every matrix element remains as: [ ] Fµν = hµν + pµν ( λµ νσ ) µ, ν ( λµ σν ) 5

6 The SCF cycle Therefore, the Hartree Fock s solution for molecules means the following integral evaluation: h µν monoelectronic ( λµ νσ ) bielectronic of four centers p µν electron density It can only be achieved after the appropriate selection of a basis set. 6

7 The SCF cycle Therefore, the Hartree Fock s solution for molecules means the following integral evaluation: h µν monoelectronic ( λµ νσ ) bielectronic of four centers p µν electron density It can only be achieved after the appropriate selection of a basis set. The most convenient way is to work with atomic basis functions, i.e. centered on nuclei belonging to the nanoscopic system. 7

8 Ab initio proceeding It is said to be following an ab initio routine or proceeding when all h µν and (λµ νσ) integrals belonging to the Hartree Fock s matrix elements are evaluated consequently with the selected basis set, with no adaptions to the object under study. Therefore, the further iterative solution to find convergent eigenvectors and eigenfunctions give essentially a priori results from the first principles of this theory. 8

9 Ab initio proceeding It is said to be following an ab initio routine or proceeding when all h µν and (λµ νσ) integrals belonging to the Hartree Fock s matrix elements are evaluated consequently with the selected basis set, with no adaptions to the object under study. Therefore, the further iterative solution to find convergent eigenvectors and eigenfunctions give essentially a priori results from the first principles of this theory. The advantage of this working path is that no relevant adaptions of quantum theory is made to fit results of any kind. 9

10 Ab initio proceeding It is said to be following an ab initio routine or proceeding when all h µν and (λµ νσ) integrals belonging to the Hartree Fock s matrix elements are evaluated consequently with the selected basis set, with no adaptions to the object under study. Therefore, the further iterative solution to find convergent eigenvectors and eigenfunctions give essentially a priori results from the first principles of this theory. The advantage of this working path is that no relevant adaptions of quantum theory is made to fit results of any kind. These approaches bring high reliability to the procedure a if calculated physical properties can be appropriately related with direct experimental values. 0

11 Basis sets The appropriately non orthogonal basis set serves for calculating all h µν and (λµ νσ) integrals building the F matrix.

12 Basis sets The appropriately non orthogonal basis set serves for calculating all h µν and (λµ νσ) integrals building the F matrix. It is conveniently orthogonalized: F'= S FS

13 Basis sets The appropriately non orthogonal basis set serves for calculating all h µν and (λµ νσ) integrals building the F matrix. It is conveniently orthogonalized: Diagonalization proceeds: F'= S FS ( F' E) c' = 0

14 Basis sets The appropriately non orthogonal basis set serves for calculating all h µν and (λµ νσ) integrals building the F matrix. It is conveniently orthogonalized: Diagonalization proceeds: F'= And the consequent orthogonal based c matrix is then deorthogonalized for calculated properties: S FS ( F' E) c' = c = S c' 0

15 Basis sets The Slater s hydrogen like basis functions for all multielectronic atoms are called STO (Slater type orbitals) and are considered as references: ζ parameters or exponents determine the orbital size. s e r 3 ζ π ζ χ = s re r 5 3 ζ π ζ χ = r p xe x 5 ζ π ζ χ =

16 Basis sets There are routines for calculations of all electronic integrals on STO basis of the general form: χ nlm = ( ) n+ [( )] ζ ( ) n r ζ n θ, ϑ where n, l, m are hydrogen like atomic quantum numbers and Y lm (θ,ϕ) are the angular components being always the normalized spherical harmonics.! r e Y lm

17 Basis sets There are routines for calculations of all electronic integrals on STO basis of the general form: χ nlm = ( ) n+ [( )] ζ ( ) n r ζ n θ, ϑ where n, l, m are hydrogen like atomic quantum numbers and Y lm (θ,ϕ) are the angular components being always the normalized spherical harmonics. Such spherical harmonics coming from the exact solution of hydrogen atom are used for angular components in the great majority if not all kinds of basis functions for ab initio calculations.! r e Y lm

18 Basis sets There are routines for calculations of all electronic integrals on STO basis of the general form: χ nlm = ( ) n+ [( )] ζ ( ) n r ζ n θ, ϑ where n, l, m are hydrogen like atomic quantum numbers and Y lm (θ,ϕ) are the angular components being always the normalized spherical harmonics. Such spherical harmonics coming from the exact solution of hydrogen atom are used for angular components in the great majority if not all kinds of basis functions for ab initio calculations. Is the Bohr s radius is used as length unit, the ζ parameter is defined by:! ζ = Z n r e Y lm

19 Basis sets Routine calculations with STO s are avoided because the existing procedures are very computing time consuming.

20 Basis sets Routine calculations with STO s are avoided because the existing procedures are very computing time consuming. Exponential component of Slater basis are thus developed in terms of Gaussian series to fit a convoluted Slater function: = N k r k r k e d e, α, = N k r k s r k e d re, α, = N k r k p r k xe d xe, α, s e r 3 ζ π ζ χ = s re r 5 3 ζ π ζ χ = r p xe x 5 ζ π ζ χ =

21 Basis sets Routine calculations with STO s are avoided because the existing procedures are very computing time consuming. d (fixed coefficients) values and α (exponent scaling) parameters are optimized to fit the original corresponding Slater curve according the length (N) of the selected series. Exponential component of Slater basis are thus developed in terms of Gaussian series to fit a convoluted Slater function: = N k r k r k e d e, α, = N k r k s r k e d re, α, = N k r k p r k xe d xe, α, s e r 3 ζ π ζ χ = s re r 5 3 ζ π ζ χ = r p xe x 5 ζ π ζ χ =

22 Basis sets This functions are called minimal basis STO-NG, where N = to 6. As an example, the case of STO-3G basis for carbon, values are: l α d s S S α d s P α d p d parameter is called as coefficient, and α parameter as exponent.

23 Basis sets Jargon Shell: Is the set of atomic basis with common Gaussian exponents, independently of their nature (S and P basis sets in carbon form a shell in Pople s STO s).

24 Basis sets Jargon Shell: Is the set of atomic basis with common Gaussian exponents, independently of their nature (S and P basis sets in carbon form a shell in Pople s STO s). Primitive shell: Is the type of atomic basis functions in a shell and sharing Gaussian exponents (S and P are both primitive shells of their shell).

25 Basis sets Jargon Shell: Is the set of atomic basis with common Gaussian exponents, independently of their nature (S and P basis sets in carbon form a shell in Pople s STO s). Primitive shell: Is the type of atomic basis functions in a shell and sharing Gaussian exponents (S and P are both primitive shells of their shell). Minimal basis: Minimal basis means that the number of STO s are the same as the number of such orbitals in a hydrogen like atom.

26 Basis sets Shell Number Primitive shell of primitive shells S s P 3 p x, p y, p z D 6 d d, d, d, d, 5 x, x y y z z xy xy xz xz d d, d, d, d, SP 4 s, p x, p y, p z d yz yz

27 Basis sets It is interesting that empirical scale factors have been included by the authors of STO s to the squared exponents: = = = N k r f k s r s k s e d e, 3 3, ζ α π ζ χ π ζ χ = = = N k r f k s s r s k sp e d re, 5 5, 3 3 α ζ π ζ χ π ζ χ = = = N k r f k p p r p k sp x x xe d xe, 5 5, ζ α π ζ χ π ζ χ

28 Basis sets It is interesting that empirical scale factors have been included by the authors of STO s to the squared exponents: As ζ parameters or exponents determine the orbital size, they became larger with f s to be adapted for molecules. = = = N k r f k s r s k s e d e, 3 3, ζ α π ζ χ π ζ χ = = = N k r f k s s r s k sp e d re, 5 5, 3 3 α ζ π ζ χ π ζ χ = = = N k r f k p p r p k sp x x xe d xe, 5 5, ζ α π ζ χ π ζ χ

29 Basis sets STO-3G basis functions for carbon (f s = 5.67; f sp =.7) remain as: Capa f s α d s S SP f sp α d s d p x d p d y pz

30 Basis sets Coefficients on how each Gaussian term contributes to the convoluted STO (d s) are fixed parameters that only were optimized when the basis set was set up and were published.

31 Basis sets Coefficients on how each Gaussian term contributes to the convoluted STO (d s) are fixed parameters that only were optimized when the basis set was set up and were published. They are not further optimized in the SCF variational procedure, but considered as they are, only parameters.

32 Basis sets STO s toward their Gaussian simulations:

33 Basis sets Remembering that the total Hartree Fock s energy as written in terms of molecular orbitals is: E total = E = elect + A> B E ( ) i Jij Kij + i i, j Z AZ R AB B A> B Z AZ R AB B

34 Basis sets E total Remembering that the total Hartree Fock s energy as written in terms of molecular orbitals is: E total = E = elect + A> B E ( ) i Jij Kij + i A> B and it becomes on the grounds of the atomic basis set as: = + i A> B E i Z AZ R AB i, j Z AZ R AB B Z AZ R [ p p ( λµ νσ ) p p ( λµ σν )] B i, j λ, µ, ν, σ λν µσ λσ AB µν B

35 Ab initio theoretical consistency These ab initio calculations are theoretically consistent from one basis set to another and even considering different approaches for building Hamiltonians or calculating integrals.

36 Ab initio theoretical consistency These ab initio calculations are theoretically consistent from one basis set to another and even considering different approaches for building Hamiltonians or calculating integrals. Therefore, the total energy values obtained after the SCF iterative loop previously described determine the quality of formulas and procedure: If the total energy decreases, the procedure, formula or basis set is better:

37 Split basis sets Being fulfilled that the number of electrons and nuclei in the system remains as a physical reality to be modeled, the construction of the reference system is not depending on any pre-established kind of atomic basis (as hydrogenoid bases are) nor the number of them.

38 Split basis sets Being fulfilled that the number of electrons and nuclei in the system remains as a physical reality to be modeled, the construction of the reference system is not depending on any pre-established kind of atomic basis (as hydrogenoid bases are) nor the number of them. On the other hand, it is well known that the molecular wave functions are better optimized if the variational procedure accounts a reference basis being as complete as possible.

39 Split basis sets Being fulfilled that the number of electrons and nuclei in the system remains as a physical reality to be modeled, the construction of the reference system is not depending on any pre-established kind of atomic basis (as hydrogenoid bases are) nor the number of them. On the other hand, it is well known that the molecular wave functions are better optimized if the variational procedure accounts a reference basis being as complete as possible. In this way, it is fair to optimize the calculated total energy with respect to the corresponding LCAO coefficients by taking into account all necessary characteristics and details. It could only be provided by extending the basis set as much as to cover all would be expected details.

40 Split basis sets The so called split basis sets use the concepts of multiple ζ basis functions by considering several basis of the same l azimuthal quantum number or hydogenoid reference kind, on the same nucleus or center.

41 Split basis sets The so called split basis sets use the concepts of multiple ζ basis functions by considering several basis of the same l azimuthal quantum number or hydogenoid reference kind, on the same nucleus or center. Split basis functions with the same l azimuthal quantum number on the same center may participate with different coefficients and exponents and they can be as many as necessary.

42 Split basis sets The so called split basis sets use the concepts of multiple ζ basis functions by considering several basis of the same l azimuthal quantum number or hydogenoid reference kind, on the same nucleus or center. Split basis functions with the same l azimuthal quantum number on the same center may participate with different coefficients and exponents and they can be as many as necessary. This is totally independent on the rules and even sometimes on the quantum numbers derived from the solution of hydrogen atom wave functions.

43 Split basis sets The most used kind of split basis functions are Gaussians because the availability of efficient calculation algorithms and they are called as Gaussian type orbitals (GTO): ( ) 4 3, r s e r g α π α α = ( ) , r q qe r g α π α α = ( ) , r qq e q r g α π α α = where q = x, y, z.

44 Split basis sets g s has the same symmetry as s Slater orbitals

45 Split basis sets g s has the same symmetry as s Slater orbitals g x, g y, g z have the same symmetry as p x, p y and p z Slater orbitals

46 Split basis sets g s has the same symmetry as s Slater orbitals g x, g y, g z have the same symmetry as p x, p y and p z Slater orbitals g xx, g yy, g zz, g xy, g xz, g yz do not have same symmetry as d z, d x -y and d xy, d xz, d yz Slater type orbitals. However they can be transformed to provide a set of 5 orbitals with the corresponding STO symmetry: d xy g xy d xz g xz d yz g yz ( ) d g z 3zz rr = g zz g xx g yy 3 d g = g g x y xx yy 4 ( ) ( ) xx yy

47 Split basis sets GTO s do not well behave at close distances from the nucleus. To solve this problem they are again developed in series of Gaussian functions (as it was previously made for STO s) with participating coefficients that are originally optimized for Hartree Fock s calculations of isolated atoms. Such basis function expansion coefficients of primitive Gaussians are not further optimized when a LCAO procedure is followed, as well as was in the case of those for STO s.

48 Split basis sets Pople s group obtained a set of GTO s as expansions of primitive Gaussians. For example: χ Carbon K s s, i s i, i= Hydrogen K r = d g α ' χ ( r ) = d g ( α r ) χ '( ) ( r ) L s i s i, i= χ '( r ) = d g ( α ' r ) χ ''( r ) = g ( '', r ) s s, i s i, i= χ s ''( r ) = g s ( α '', r ) L p ' x p, i px i, i= χ ''( r ) = g p ( α '', r ) ( r ) = d g ( α ' r ) p x s s α

49 Split basis sets Pople s group obtained a set of GTO s as expansions of primitive Gaussians. For example: χ Carbon K s s, i s i, i= Hydrogen K r = d g α ' χ ( r ) = d g ( α r ) χ '( ) ( r ) L Naming of these functions follows the rule of the acronym K- LG (ej. 3-G, 4-3G, 6-3G, etc.). s i s i, i= χ '( r ) = d g ( α ' r ) χ ''( r ) = g ( '', r ) s s, i s i, i= χ s ''( r ) = g s ( α '', r ) L p ' x p, i px i, i= χ ''( r ) = g p ( α '', r ) ( r ) = d g ( α ' r ) p x s s α

50 Split basis sets If we consider that there are scaling factor, too, in such split functions used as α n = f nl α nl, then the 3-G basis set for carbon remains as: Shell α s d s S α ' d ' d ' s p x d p y ' d p z SP α '' d '' d '' d '' d '' s p x SP p y p z '

51 Split basis sets The so called 6-3G basis set for carbon is: 6-3G Shell α s d s S α ' d ' d ' d ' s p x p y d p z SP α '' d '' d '' d '' s p x p y SP d p z ' ''

52 Split basis sets And the so called 6-3G basis set for carbon is: 6-3G Shell α s d s S α ' d s d px d py d pz SP α '' d s d px d py d pz SP α ''' d s d px d py d pz SP

53 Polarization basis sets In the way to further improvements of the quality of results and on the grounds that there are no restrictions regarding the number of atomic orbitals on each nucleus in the molecule, other basis corresponding even to non occupied atomic shells can be added.

54 Polarization basis sets In the way to further improvements of the quality of results and on the grounds that there are no restrictions regarding the number of atomic orbitals on each nucleus in the molecule, other basis corresponding even to non occupied atomic shells can be added. As an example, there are several bonding and structural phenomena among atoms of the first rows that require more spatial options than those of pure s and p Slater type orbitals to be properly represented.

55 Polarization basis sets Atomic orbitals belonging to upper non filled shells in atoms that are included in atomic basis are called as polarization orbitals or basis, and the resulting set of augmented atomic basis functions are known as polarized basis functions.

56 Polarization basis sets Atomic orbitals belonging to upper non filled shells in atoms that are included in atomic basis are called as polarization orbitals or basis, and the resulting set of augmented atomic basis functions are known as polarized basis functions. In the case of hydrogen, usual polarization basis are p x, p y and p z orbitals with only one Gaussian primitive function each. In the case of the nd and 3 rd rows, usual polarization basis are upper level d functions with only one Gaussian primitive function each.

57 Polarization basis sets A polarized basis set of carbon at the 3-G level remains as: Shell α s d s S α ' d ' d ' d ' s p x p y d p z SP α '' d '' d '' d '' d '' s p x p y SP d d α 3 d d xx d yy p z d d zz d xy ' d d d xz d yz D

58 Polarization basis sets Jargon The normal naming for Pople s polarized basis are: One star at the end of the basis set acronym means that second row heavy atoms (C, O, S, etc.) are polarized An additional star means that hydrogen atoms are also polarized: 3-G Non polarized split basis 3-G* Split basis with polarization for heavy atoms 3-G** Split basis with polarization for all atoms

59 Polarization basis sets Jargon The normal naming for Pople s polarized basis are: One star at the end of the basis set acronym means that second row heavy atoms (C, O, S, etc.) are polarized An additional star means that hydrogen atoms are also polarized: 3-G Non polarized split basis 3-G* Split basis with polarization for heavy atoms 3-G** Split basis with polarization for all atoms A more informative and also accepted naming could be: 3-G* = 3-G(d) 3-G** = 3-G(d,p)

60 Diffuse basis sets In those cases where a system could be overcharged with electrons, as anions are, it can be convenient to add some SP shell basis with small exponents. They are called as diffuse basis.

61 Diffuse basis sets In those cases where a system could be overcharged with electrons, as anions are, it can be convenient to add some SP shell basis with small exponents. They are called as diffuse basis. Every diffuse basis is indicated by a + sign in their acronym: 3-G with a diffuse basis = 3-+G

62 Diffuse basis sets A diffuse basis set of carbon at the 3-+G level then remains as: Shell α s d s S α ' d ' d ' d ' s p x p y d p z SP α '' d '' d '' d '' d '' s p x p y SP α d s d SP p x d p y + d p z p z ' +

63 An interesting point Valence electrons are the most important for physics of chemical and biological phenomena at nanoscopic levels because the involved range of energies (between 5 and 40 ev). 63

64 An interesting point Valence electrons are the most important for physics of chemical and biological phenomena at nanoscopic levels because the involved range of energies (between 5 and 40 ev). Core electrons are represented in split basis by the first K number in the acronym (K-LG). They provide large energy to the system, but the importance of them for the most common phenomena is controversial. 64

65 An interesting point Valence electrons are the most important for physics of chemical and biological phenomena at nanoscopic levels because the involved range of energies (between 5 and 40 ev). Core electrons are represented in split basis by the first K number in the acronym (K-LG). They provide large energy to the system, but the importance of them for the most common phenomena is controversial. Therefore, split basis set with many Gaussians to represent core electrons use to drop the total energy of the system with a high computational cost, although could be non important for describing the truly interesting problems. 65

66 Most popular wave functions Pople s basis: Huzinaga / Dunning s valence double ζ : Huzinaga / Dunning s full double ζ : Dunning s correlation consistent: STO-NG, 3-G, 6-G, 4-3G, 6-3G, 6-3G D95V D95 cc-pvdz, cc-pvtz, cc-pvqz, cc-pv5z, cc-pv6z 66

67 The ab initio energy landscape The hypersurface of an ab initio calculated nanoscopic system is given in a general expression by E total = E total (R) where R = [R AB ] is the distance matrix in the Euclidean space of all nuclei or reference centers of the system. 67

68 The ab initio energy landscape The hypersurface of an ab initio calculated nanoscopic system is given in a general expression by E total = E total (R) where R = [R AB ] is the distance matrix in the Euclidean space of all nuclei or reference centers of the system. The developed formula is: total E = E + i i, j λ, µ, ν, σ A> B i Z AZ R ( λµ νσ ) p p ( λµ σν ) [ p p ] AB B λν µσ λσ µν 68

69 The ab initio energy landscape The hypersurface of an ab initio calculated nanoscopic system is given in a general expression by E total = E total (R) where R = [R AB ] is the distance matrix in the Euclidean space of all nuclei or reference centers of the system. The developed formula is: total E = E i i i, j λ, µ, ν, σ ( λµ νσ ) p p ( λµ σν ) [ p p ] λν Z AZ B + A> B RAB It must be realized that R explicitly appears in the core repulsion term, although is implicit in all other because the convention is that electrons have the coordinates of nuclei and the one electron term E i also contains nuclear coordinates. µσ λσ µν 69

70 The ab initio energy landscape As expected, the goal is finding a certain R eq giving the lowest E total. 70

71 The ab initio energy landscape As expected, the goal is finding a certain R eq giving the lowest E total. It is achieved after finding or guessing the G gradient matrix and Hessians (the H matrix), which terms are located in each and all centers of reference (usually nuclei) by the same or similar procedures as those with classical potentials. 7

72 The ab initio energy landscape As expected, the goal is finding a certain R eq giving the lowest E total. It is achieved after finding or guessing the G gradient matrix and Hessians (the H matrix), which terms are located in each and all centers of reference (usually nuclei) by the same or similar procedures as those with classical potentials. The condition to find the optimized geometry is then: E total = de dr total = 0 7

73 Calculations are performed by iterating the whole SCF routine on the energies and electron densities for each geometry. After each SCF step gradients and Hessians are calculated, the geometry is modified in the direction to reduce energy and cancelling forces acting on each atom. The calculation is finished when the force and energies are at tollerable levels of differences with respect to the previous step. 73

74 Some examples In the case of formaldehyde with minimal basis: STO-G STO-3G STO-4G STO-5G STO-6G Exp. Total energy (Hartrees) r CO (Å) r CH (Å) <HCH µ (debye) Relative computer time 3 6 0

75 Some examples In the case of formaldehyde with split basis: STO-6G 3-G 6-G 6-3G 6-3++G** Exp. Total energy (Hartrees) r CO (Å) r CH (Å) <HCH m (debye) Relative computer time 0.8.4

76 Pseudopotentials Internal electrons of atoms and molecules show scarce importance for modeling nanoscopic processes, although they must be considered for the sake of performing a complete modeling.

77 Pseudopotentials Internal electrons of atoms and molecules show scarce importance for modeling nanoscopic processes, although they must be considered for the sake of performing a complete modeling. In order to solve the problem of considering core electrons of heavy atoms without spending the huge computational effort that they require, the pseudopotential methods have been developed for taking into account the effects of all core electrons, being avoided the explicit consideration of their molecular wave functions in the SCF process.

78 Pseudopotentials The basic assumption of every ab initio pseudopotential is the frozen-core approximation, i.e., the core orbitals φ c are frozen to be those of the atom in some fixed electronic state.

79 Pseudopotentials The basic assumption of every ab initio pseudopotential is the frozen-core approximation, i.e., the core orbitals φ c are frozen to be those of the atom in some fixed electronic state. The valence electrons are then self-consistently solved for in the Hilbert space orthogonalized to the frozen core orbitals.

80 Pseudopotentials The basic assumption of every ab initio pseudopotential is the frozen-core approximation, i.e., the core orbitals φ c are frozen to be those of the atom in some fixed electronic state. The valence electrons are then self-consistently solved for in the Hilbert space orthogonalized to the frozen core orbitals. Resulting pseudo valence atomic orbitals χ v are then expressed in terms of its corresponding eigenfunctions φ ν in their original configuration together with a linear combination of all C core φ c atomic orbitals: χ = φ + v v C c= a φ c c

81 Pseudopotentials Ĥ A pseudo atomic Hamiltonian acting on a certain χ v pseudo valence orbital can be defined as: Z r PS PS ˆ χ v = + Vˆ core + Vˆ val + Vˆ χv = H PS ε v χ v 8

82 Pseudopotentials Ĥ A pseudo atomic Hamiltonian acting on a certain χ v pseudo valence orbital can be defined as: Z r PS PS ˆ χ v = + Vˆ core + Vˆ val + Vˆ χv = H where: Vˆ express the non local potentials involving both core Coulomb and exchange interactions of χ v with core orbitals PS ε v χ v 8

83 Pseudopotentials Ĥ A pseudo atomic Hamiltonian acting on a certain χ v pseudo valence orbital can be defined as: Z r PS PS ˆ χ v = + Vˆ core + Vˆ val + Vˆ χv = H where: Vˆ express the non local potentials involving both core Coulomb and exchange interactions of χ v with core orbitals Vˆ val express the non local potentials involving both Coulomb and exchange interactions of χ v with other valence pseudo orbitals. PS ε v χ v 83

84 Pseudopotentials Ĥ A pseudo atomic Hamiltonian acting on a certain χ v pseudo valence orbital can be defined as: Z r PS PS ˆ χ v = + Vˆ core + Vˆ val + Vˆ χv = H where: Vˆ express the non local potentials involving both core Coulomb and exchange interactions of χ v with core orbitals Vˆ val express the non local potentials involving both Coulomb and exchange interactions of χ v with other valence pseudo orbitals. V PS is a pseudopotential depending on how important is the interaction between the pseudo valence atomic orbital and the core orbitals projected onto it. PS ε v χ v 84

85 Pseudopotentials The pseudopotential operator Vˆ PS χ v C = c φ c Vˆ PS is usually expressed as: ε ε φ χ ( ) v c c v or the projection of the core orbitals onto the pseudo - atomic valence orbitals that is determined by the energy differences between them and their overlapping. 85

86 Pseudopotentials The pseudopotential operator Vˆ PS χ v C = c φ c Vˆ PS is usually expressed as: ε ε φ χ ( ) v c c v or the projection of the core orbitals onto the pseudo - atomic valence orbitals that is determined by the energy differences between them and their overlapping. V PS is the object of several formulations in literature that depend on elements and the used methods. It must be observed that pseudo atomic valence orbitals are not considered as orthogonal with respect to the original core and valence atomic orbitals, although they must be themselves an orthogonal set. 86

87 In a graphical way, Pseudopotentials Roughly, consideration of the effects of core orbitals on the pseudo valence atomic orbitals modify their energy and shape, reducing the computational effort by considering the relevant effects of core electrons on lowest energy levels that are important for commonly measured properties. 87

88 Pseudopotentials As an example, the case of AgF (r exp =.983 Å, IP exp = ev): Basis set or PS r calc -e HOMO (ev) E tot (Hartrees) rel. calc. time STO-3G LanLMB

89 Relativistic effects The special theory of relativity, relating time and spatial dimensions by the speed of light as a limit, is necessary to understand the residual field of unpaired electrons, known as spin.

90 Relativistic effects The special theory of relativity, relating time and spatial dimensions by the speed of light as a limit, is necessary to understand the residual field of unpaired electrons, known as spin. The Dirac s equation, is a relativistic wave equation describing particles as quantum fields which fluctuate like waves.

91 Relativistic effects The special theory of relativity, relating time and spatial dimensions by the speed of light as a limit, is necessary to understand the residual field of unpaired electrons, known as spin. The Dirac s equation, is a relativistic wave equation describing particles as quantum fields which fluctuate like waves. It introduced special relativity into the Schrödinger equation explaining not only the spin (the intrinsic angular momentum of the electrons), a property which can only be stated, but not explained by non-relativistic quantum mechanics, and led to the prediction of the antiparticle of the electron, the positron.

92 Relativistic effects The functional form of the Dirac s equation for the state of an electron is: ( ) ψ ( cα pˆ + βmc ψ ( R, t) = i t R, t) where the α and β spin elements are 4 x 4 matrices and the wave function is four component.

93 Relativistic effects The functional form of the Dirac s equation for the state of an electron is: ( ) ψ ( cα pˆ + βmc ψ ( R, t) = i t R, t) where the α and β spin elements are 4 x 4 matrices and the wave function is four component. As well as atoms have more electrons, the behaviors congruent with the Schrödinger non relativistic equation become altered because interactions originated in their nature of fluctuating quantum fields become relevant with respect to the single particle model.

94 Relativistic effects The most relevant relativistic effect is the break down of azimuthal spin quantum number relationships found for the solution of hydrogen atom, known as spin orbit interactions.

95 Relativistic effects The most relevant relativistic effect is the break down of azimuthal spin quantum number relationships found for the solution of hydrogen atom, known as spin orbit interactions. Among the ways to treat this problem are: the full relativistic treatment of multielectronic atoms. parameterizing pseudopotentials to take into account such spin orbit interactions.

96 Relativistic effects The contribution of relativistic effects to atoms with a fully relativistic approach can be illustrated by a Clementi s paper: Element E rel (Hartrees) E HF (total) % He Ne Ar Hartmann, H.; Clementi, E., Relativistic Correction for Analytic Hartree-Fock Wave Functions. Phys. Rev. 964, 33 (5A), A95.

97 Open shell systems The restricted Hartree Fock theory for molecules is that applied to systems where all electrons are paired with antiparalell spins, and therefore the resulting total spin quantum number is zero (S = 0) and the corresponding multiplicity is (S + = ). 97

98 Open shell systems There are two ways for treating cases of unpaired electrons, even being originated from one excess or missing particle (S = ½ or multiplicity, a doublet ) or because two of them are unpaired ( or multiplicity 3, a triplet ): 98

99 Open shell systems There are two ways for treating cases of unpaired electrons, even being originated from one excess or missing particle (S = ½ or multiplicity, a doublet ) or because two of them are unpaired ( or multiplicity 3, a triplet ): An independent calculation of both α and β electron spin manifolds, the so called non restricted Hartree Fock treatment (UHF). 99

100 Open shell systems There are two ways for treating cases of unpaired electrons, even being originated from one excess or missing particle (S = ½ or multiplicity, a doublet ) or because two of them are unpaired ( or multiplicity 3, a triplet ): An independent calculation of both α and β electron spin manifolds, the so called non restricted Hartree Fock treatment (UHF). An independent treatment of filled orbitals with respect to the unfilled or unpaired, the so called extended or restricted open shell Hartree Fock procedure (ROHF). 00

101 Open shell systems: UHF It is based in evaluating and diagonalizing two ground state Slater determinants, independently, with different electron occupations. F F ( α β p + )[ ( λµ νσ ) ( λµ σν )] α α µν = hµν + µν pµν µ, ν ( α β p + )[ ( λµ νσ ) ( λµ σν )] β β µν = hµν + µν pµν µ, ν 0

102 Open shell systems: UHF It is based in evaluating and diagonalizing two ground state Slater determinants, independently, with different electron occupations. F F ( α β p + )[ ( λµ νσ ) ( λµ σν )] α α µν = hµν + µν pµν µ, ν ( α β p + )[ ( λµ νσ ) ( λµ σν )] β β µν = hµν + µν pµν µ, ν However, the corresponding results of density matrices are summed together for producing the Fock s matrix elements of the following SCF cycle. 0

103 Open shell systems: UHF The main problem is that each determinant with a different occupation gives an individual minimum of energy upon diagonalization and both density matrices are not equivalent. 03

104 Open shell systems: UHF The main problem is that each determinant with a different occupation gives an individual minimum of energy upon diagonalization and both density matrices are not equivalent. It means that monoelectronic states of filled orbitals become splitting from one spin manifold to the other during the SCF iterative process, and it can conduct to non congruent results. 04

105 Open shell systems: UHF The main problem is that each determinant with a different occupation gives an individual minimum of energy upon diagonalization and both density matrices are not equivalent. It means that monoelectronic states of filled orbitals become splitting from one spin manifold to the other during the SCF iterative process, and it can conduct to non congruent results. This problem is known as spin contamination and some special numerical treatments are useful for their solution. 05

106 Open shell systems: ROHF This procedure divide molecular orbitals in those filled and unpaired and treat both separately for further merging results on the grounds of:. The total wave function is, in general, a sum of several Slater determinants, each of which contains a (doubly occupied) closed-shell core Φ C, and a partially occupied open shell chosen from a set Φ O. The total wave function could be then expressed as: Φ = Φ, Φ C O ( ) and it is assumed to be orthonormal, so that the two sets Φ C and Φ O are orthonormal and mutually orthogonal. 06

107 Open shell systems: ROHF E. The expectation value of the energy is given by: total = + f k H k m + H ( J ) kl Kkl kl m + f ( aj ) + ( ) mn bkmn J km Kkm mn km where a, b, and f are numerical constants depending on the specific case. 07

108 Open shell systems: ROHF E. The expectation value of the energy is given by: total = + f k H k m + H ( J ) kl Kkl kl m + f ( aj ) + ( ) mn bkmn J km Kkm mn km where a, b, and f are numerical constants depending on the specific case. The first two sums in the energy expression represent the closedshell energy, the next two sums the open-shell energy, and the last sum the interaction energy of the closed and open shell. 08

109 Open shell systems: ROHF The super Fock matrix F ROHF with merging subsets of molecular orbitals can be expressed, according to one of the current approximations as: Filled orbitals Filled Open orbitals F( p) K( p ) Half filled orbitals Empty orbitals a Open + F ( p) + K( p ) F (p) Open b Open Open F ( p) + K( p F( p) + K( p ) F( p) K( p ) Open c Open F (p F( p) K( p ) F( p) + K( p ) Half filled orbitals ) Empty orbitals ) 09

110 Open shell systems: ROHF The super Fock matrix F ROHF with merging subsets of molecular orbitals can be expressed, according to one of the current approximations as: Filled orbitals Filled Open orbitals F( p) K( p ) Half filled orbitals Empty orbitals a Open + F ( p) + K( p ) F (p) Open b Open Open F ( p) + K( p F( p) + K( p ) F( p) K( p ) Open c Open F (p F( p) K( p ) F( p) + K( p ) Half filled orbitals ) Empty orbitals ) F(p) = h + J - K/ is the Fock s matrix as calculated with the total density matrix p (including all filled and half filled shells) 0

111 Open shell systems: ROHF The super Fock matrix F ROHF with merging subsets of molecular orbitals can be expressed, according to one of the current approximations as: Filled orbitals Filled Open orbitals F( p) K( p ) Half filled orbitals Empty orbitals a Open + F ( p) + K( p ) F (p) Open b Open Open F ( p) + K( p F( p) + K( p ) F( p) K( p ) Open c Open F (p F( p) K( p ) F( p) + K( p ) Half filled orbitals ) Empty orbitals ) F(p) = h + J - K/ is the Fock s matrix as calculated with the total density matrix p (including all filled and half filled shells) K(p Open ) is the exchange matrix as calculated with the density matrix of half filled levels.

112 Open shell systems: ROHF The super Fock matrix F ROHF with merging subsets of molecular orbitals can be expressed, according to one of the current approximations as: Filled orbitals Filled Open orbitals F( p) K( p ) Half filled orbitals Empty orbitals a Open + F ( p) + K( p ) F (p) Open b Open Open F ( p) + K( p F( p) + K( p ) F( p) K( p ) Open c Open F (p F( p) K( p ) F( p) + K( p ) Half filled orbitals ) Empty orbitals ) F(p) = h + J - K/ is the Fock s matrix as calculated with the total density matrix p (including all filled and half filled shells) K(p Open ) is the exchange matrix as calculated with the density matrix of half filled levels. (a,b,c) : (0,0,0) in normal calculations; (0,-,0) for certain cases and (,0,-) for others.

113 Open shell systems Different approaches to open shell systems can be overviewed as: 3

114 Open shell systems The case of NO (r exp =.5 Å) is illustrative: Method r calc E tot (Hartrees) Rel. time ROHF [4-3G(d)] UHF [4-3G(d)]