1 MODELING MATTER AT NANOSCALES 6.The theory of molecular orbitals for the description of...

Luis A. MonteroUniversidad de La Habana, Cuba, 2012

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MODELING MATTER AT

NANOSCALES

6. The theory of molecular orbitals for the description of nanosystems (part II)6.02. Ab initio methods. Basis functions.

2

The SCF

cycle

3

The SCF cycle

The Fock matrix must be constructed:

FF

FFF

FFF

....

......

...

...

1

22221

11211

F

and the problem is to evaluate the Fmn 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.

4

The SCF cycle

As:

iiii ccnp

)|()()(1

)()(

)|()()(1

)()(

Kdrrr

rr

Jdrrr

rr

nm

mn

nm

nm

mn

nm

5

The SCF cycle

As:

Then the formula to calculate every matrix element remains as:

i

iii ccnp

)|()()(1

)()(

)|()()(1

)()(

Kdrrr

rr

Jdrrr

rr

nm

mn

nm

nm

mn

nm

)|()|(2,

phF

6

The SCF cycle

Therefore, the Hartree – Fock’s solution for molecules means the following integral evaluation:

It can only be achieved after the appropriate selection of a basis set.

h monoelectronic

)|( bielectronic of four centers

p electron density

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The SCF cycle

Therefore, the Hartree – Fock’s solution for molecules means the following integral evaluation:

It can only be achieved after the appropriate selection of a basis set.

h monoelectronic

)|( bielectronic of four centers

p electron density

The most convenient way is to work with atomic basis functions, i.e. centered on nuclei belonging to the nanoscopic system.

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Ab initio proceeding

It is said to be following an ab initio routine or proceeding when all hmn and (lm|ns) 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.

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The advantage of this working path is that no relevant adaptions of quantum theory is made to fit results of any kind.

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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.

Basis sets

The appropriately non – orthogonal basis set serves for calculating all hmn and (lm|ns) integrals building the F matrix.

Basis sets


It is conveniently orthogonalized:2

12

1

FSSF '

Basis sets


It is conveniently orthogonalized:2

12

1

FSSF '

Diagonalization proceeds:0')'( cEF

Basis sets


It is conveniently orthogonalized:

And the consequent orthogonal based c´ matrix is then deorthogonalized for calculated properties:

21

21

FSSF '

Diagonalization proceeds:0')'( cEF

'21

cSc

Basis sets

The Slater’s hydrogen like basis functions for all multielectronic atoms are called STO (Slater type orbitals) and are considered as references:

z parameters or “exponents” determine the orbital size.

rs e 1

21

31

1

rs re 2

21

52

2 3

rp xe

x

2

21

52

2

Basis setsThere are routines for calculations of all electronic integrals on STO basis of the general form:

where n, l, m are hydrogen like atomic quantum numbers and Ylm( ,q ) are the angular components being always the normalized spherical harmonics.

,!22 121

21

lmrnn

nlm Yern



,!22 121

21

lmrnn

nlm Yern

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.



,!22 121

21

lmrnn

nlm Yern

Is the Bohr’s radius is used as length unit, the z parameter is defined by:

nZ

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.

Basis setsRoutine 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

kr kede

1,1

2,1

N

k

r

ksr kedre

1,2

2,2

N

k

r

kpr kxedxe

1,2

2,2

rs e 1

21

31

1

rs re 2

21

52

2 3

rp xe

x

2

21

52

2


d (fixed coefficients) values and a (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

kr kede

1,1

2,1

N

k

r

ksr kedre

1,2

2,2

N

k

r

kpr kxedxe

1,2

2,2

rs e 1

21

31

1

rs re 2

21

52

2 3

rp xe

x

2

21

52

2

Basis setsThis functions are called minimal basis STO-NG, where N = 2 to 6.

d parameter is called as “coefficient”, and a parameter as “exponent”.

As an example, the case of STO-3G basis for carbon, values are:l a1 d1s

S 2.23 0.15 0.41 0.53 0.11 0.44S a2 d2s

0.99 -0.10 0.23 0.40 0.08 0.70P a2 d2p

0.99 0.16 0.23 0.61 0.99 0.39

Basis sets

Shell: Is the set of atomic basis with common Gaussian exponents, independently of their nature (2S and 2P basis sets in carbon form a shell in Pople’s STO’s).

Jargon

Basis sets


Primitive shell: Is the type of atomic basis functions in a shell and sharing Gaussian exponents (2S and 2P are both primitive shells of their shell).

Jargon

Basis sets


Primitive shell: Is the type of atomic basis functions in a shell and sharing Gaussian exponents (2S and 2P 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.

Jargon

Basis sets

Shell Number of

primitive shells

Primitive shell

S 1 s P 3 px, py, pz D 6

yzxzxyzyxdddddd ,,,,, 222

5 yzxzxyzyx

ddddd ,,,, 222 SP 4 s, px, py, pz

Basis sets

It is interesting that empirical scale factors have been included by the authors of STO’s to the squared exponents:

N

k

rf

ksr

sksede

1,1

21

31

1

21

31

1

2,1

211

N

k

rf

kssr

skspedre

1,2

21

52

2

21

52

2

2,2

2

2

33

N

k

rf

kppr

pksp

xxxedxe

1,2

21

52

2

21

52

2

2,2

22

Basis sets

It is interesting that empirical scale factors have been included by the authors of STO’s to the squared exponents:

As z parameters or “exponents” determine the orbital size, they became larger with f’s to be adapted for molecules.

N

k

rf

ksr

sksede

1,1

21

31

1

21

31

1

2,1

211

N

k

rf

kssr

skspedre

1,2

21

52

2

21

52

2

2,2

2

2

33

N

k

rf

kppr

pksp

xxxedxe

1,2

21

52

2

21

52

2

2,2

22

Basis sets

Capa f1s21 sd1

S 71.6 0.15 13.0 0.53 3.53 0.44 SP fsp

22 sd2 xpd2 ypd2 zpd2

2.94 -0.10 0.16 0.16 0.16 0.68 0.40 0.61 0.61 0.61 0.22 0.70 0.39 0.39 0.39

STO-3G basis functions for carbon (f1s = 5.67; fsp = 1.72) remain as:

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.

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.

Basis sets

STO’s toward their Gaussian simulations:

Basis sets

Remembering that the total Hartree – Fock’s energy as written in terms of molecular orbitals is:

BA AB

BA

i jiijiji

BA AB

BAelecttotal

RZZ

KJE

RZZ

EE

,

22

Basis sets

Remembering that the total Hartree – Fock’s energy as written in terms of molecular orbitals is:

and it becomes on the grounds of the atomic basis set as:

BA AB

BA

i jiijiji

BA AB

BAelecttotal

RZZ

KJE

RZZ

EE

,

22

BA AB

BA

i jii

total

RZZ

ppppEE, ,,,

||22

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.

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:

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.

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.

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.

Split basis setsThe so – called “split basis sets” use the concepts of “multiple z” 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.

“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.

Split basis setsThe 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):

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3

2, r

s erg

24

1

3

3128, r

q qerg

24

1

2

3

7

92048

, rqq eqrg

where q = x, y, z.

Split basis setsgs has the same symmetry as s Slater orbitals


gx, gy, gz have the same symmetry as px, py and pz Slater orbitals


gx, gy, gz have the same symmetry as px, py and pz Slater orbitals

gxx, gyy, gzz , gxy, gxz, gyz do not have same symmetry as dz2, dx2-y2

and dxy, dxz , dyz Slater type orbitals. However they can be transformed to provide a set of 5 orbitals with the corresponding STO symmetry: dxy gxy

dxz gxz

dyz gyz yyxxzzrrzzzggggd 22

132

yyxxyyxxyxgggd

21

22 43

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.

Split basis setsPople´s group obtained a set of GTO’s as expansions of primitive Gaussians. For example:

Carbon Hydrogen

K

iisiss rgdr

11,11 ,

K

iisis rgdr

11 ,''

L

iisiss rgdr

12,22 ,'' rgr ss

,''''1

rgr ss

,'''' 22

L

iipipp rgdr

xx1

2,22 ,''

rgr ppx

,'''' 22

Split basis setsPople´s group obtained a set of GTO’s as expansions of primitive Gaussians. For example:

Naming of these functions follows the rule of the acronym K-L1G (ej. 3-21G, 4-31G, 6-31G, etc.).

Carbon Hydrogen

K

iisiss rgdr

11,11 ,

K

iisis rgdr

11 ,''

L

iisiss rgdr

12,22 ,'' rgr ss

,''''1

rgr ss

,'''' 22

L

iipipp rgdr

xx1

2,22 ,''

rgr ppx

,'''' 22

Split basis sets

If we consider that there are scaling factor, too, in such split functions used as an = fnl

2anl , then the 3-21G basis set for carbon remains as:

Shell 1s sd1

S 172.3 0.062 25.9 0.359 5.53 0.701 2' '2sd '2 xpd

'2 ypd

'2 zpd

SP 3.66 -0.396 0.236 0.236 0.236 0.771 1.22 0.861 0.861 0.861 2'' ''2sd ''2 xpd

''2 ypd

''2 zpd

SP 0.196 1.00 1.00 1.00 1.00

Split basis setsThe so – called 6-31G basis set for carbon is:

6-31G Shell 1s sd1

S 3047.5 0.002 457.4 0.014 103.9 0.069 29.21 0.232 9.287 0.468 3.164 0.362 2' '2sd '2 xpd

'2 ypd

'2 zpd

SP 7.868 -0.119 0.069 0.069 0.069 1.881 -0.161 0.316 0.316 0.316 0.544 1.143 0.744 0.744 0.744 2'' ''2sd ''2 xpd

''2 ypd

''2 zpd

SP 0.169 1.00 1.00 1.00 1.00

Split basis setsAnd the so – called 6-311G basis set for carbon is:

6-311G Shell a1s d1s S 4563.2 0.002 682.0 0.015 155.0 0.076 44.46 0.261 13.03 0.616 1.828 0.221 a2' d2s´ d2px´ d2py´ d2pz´ SP 20.96 0.115 0.040 0.040 0.040 4.803 0.920 0.238 0.238 0.238 1.459 -0.303 0.816 0.816 0.816 a2'' d2s´´ d2px´´ d2py´´ d2pz´´ SP 0.483 1.00 1.00 1.00 1.00 a2''' d2s´´´ d2px´´´ d2py´´´ d2pz´´´ SP 0.146 1.00 1.00 1.00 1.00

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.


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.


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.


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 2px, 2py and 2pz orbitals with only one Gaussian primitive function each.In the case of the 2nd and 3rd rows, usual polarization basis are upper level d functions with only one Gaussian primitive function each.


A “polarized” basis set of carbon at the 3-21G level remains as:Shell 1s sd1

S 172.3 0.062 25.9 0.359 5.53 0.701 2' '2sd '2 xpd

'2 ypd

'2 zpd

SP 3.66 -0.396 0.236 0.236 0.236 0.771 1.22 0.861 0.861 0.861 2'' ''2sd ''2 xpd

''2 ypd

''2 zpd

SP 0.196 1.00 1.00 1.00 1.00 3

xxdd yydd

zzdd xydd

xzdd yzdd D 0.800 1.00 1.00 1.00 1.00 1.00 1.00

Polarization basis setsJargonThe 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 polarizedAn additional star means that hydrogen atoms are also polarized:3-21G Non polarized split basis3-21G* Split basis with polarization for

heavy atoms

3-21G** Split basis with polarization for all atoms

Polarization basis setsJargonThe 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 polarizedAn additional star means that hydrogen atoms are also polarized:3-21G Non polarized split basis3-21G* Split basis with polarization for

heavy atoms

3-21G** Split basis with polarization for all atoms

A more informative and also accepted naming could be:3-21G* = 3-21G(d)

3-21G** = 3-21G(d,p)

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.

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:

3-21G with a diffuse basis = 3-21+G

Diffuse basis sets

A “diffuse” basis set of carbon at the 3-21+G level then remains as:

Shell 1s sd1

S 172.3 0.062 25.9 0.359 5.53 0.701 2' '2sd '2 xpd

'2 ypd

'2 zpd

SP 3.66 -0.396 0.236 0.236 0.236 0.771 1.22 0.861 0.861 0.861 2'' ''2sd ''2 xpd

''2 ypd

''2 zpd

SP 0.196 1.00 1.00 1.00 1.00 2

+ sd2

xpd2

ypd2

zpd2

SP 0.044 1.00 1.00 1.00 1.00

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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).

64



Core electrons are represented in split basis by the first K number in the acronym (K-L1G). They provide large energy to the system, but the importance of them for the most common phenomena is controversial.

65



Core electrons are represented in split basis by the first K number in the acronym (K-L1G). 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.

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Most “popular” wave functions

Pople’s basis: STO-NG, 3-21G, 6-21G, 4-31G, 6-31G, 6-311G

Huzinaga / Dunning’s valence double z : D95VHuzinaga / Dunning’s full double z : D95Dunning’s correlation consistent: cc-pVDZ, cc-pVTZ,

cc-pVQZ, cc-pV5Z, cc-pV6Z

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The “ab initio” energy landscapeThe hypersurface of an ab initio calculated nanoscopic system is given in a general expression by Etotal = Etotal(R) where R = [RAB] is the distance matrix in the Euclidean space of all nuclei or reference centers of the system.

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The “ab initio” energy landscapeThe hypersurface of an ab initio calculated nanoscopic system is given in a general expression by Etotal = Etotal(R) where R = [RAB] is the distance matrix in the Euclidean space of all nuclei or reference centers of the system.The developed formula is:

BA AB

BA

ji

ii

total

RZZ

pppp

EE

, ,,,

||2

2

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The “ab initio” energy landscapeThe hypersurface of an ab initio calculated nanoscopic system is given in a general expression by Etotal = Etotal(R) where R = [RAB] is the distance matrix in the Euclidean space of all nuclei or reference centers of the system.The developed formula is:

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 Ei also contains nuclear coordinates.

BA AB

BA

ji

ii

total

RZZ

pppp

EE

, ,,,

||2

2

70

The “ab initio” energy landscape

As expected, the goal is finding a certain Req giving the lowest Etotal.

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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.

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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:

0

RddE

Etotal

total

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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.

Some examples

In the case of formaldehyde with minimal basis:

STO-2G STO-3G STO-4G STO-5G STO-6G Exp.Total energy (Hartrees) -109.0244 -112.3525 -113.1611 -113.3752 -113.4408

rCO (Å) 1.220 1.217 1.216 1.216 1.216 1.210rCH (Å) 1.110 1.101 1.099 1.098 1.098 1.102<HCH 113.1 114.5 114.8 114.8 114.8 121.1m (debye) 1.118 1.520 1.592 1.596 1.596 2.33Relative computer time

1 2 3 6 10

Some examples

In the case of formaldehyde with split basis:

STO-6G 3-21G 6-21G 6-31G 6-311++G** Exp.Total energy (Hartrees) -113.4408 -113.2218 -113.6985 -113.8084 -113.9029

rCO (Å) 1.216 1.207 1.209 1.210 1.180 1.210rCH (Å) 1.098 1.083 1.083 1.082 1.094 1.102<HCH 114.8 114.9 115.1 116.6 116.0 121.1m (debye) 1.596 2.657 2.673 3.304 2.806 2.33Relative computer time

1 1 1 0.8 1.4

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.

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.

Pseudopotentials

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

Pseudopotentials


The valence electrons are then self-consistently solved for in the Hilbert space orthogonalized to the frozen core orbitals.

Pseudopotentials


The valence electrons are then self-consistently solved for in the Hilbert space orthogonalized to the frozen core orbitals.

Resulting pseudo – valence atomic orbitals cv are then expressed in terms of its corresponding eigenfunctions fn in their original configuration together with a linear combination of all C core fc atomic orbitals:

C

cccvv a

1

81

PseudopotentialsA pseudo atomic Hamiltonian acting on a certain cv pseudo – valence orbital can be defined as:

PSH

vvvPS

valcorevPS VVV

rZ

H

ˆˆˆˆ 2

21

82


PSH

vvvPS

valcorevPS VVV

rZ

H

ˆˆˆˆ 2

21

where:coreV express the non – local potentials involving both

Coulomb and exchange interactions of cv with core orbitals

83


PSH

vvvPS

valcorevPS VVV

rZ

H

ˆˆˆˆ 2

21



valV express the non – local potentials involving both Coulomb and exchange interactions of cv with other valence pseudo – orbitals.

84


PSH

vvvPS

valcorevPS VVV

rZ

H

ˆˆˆˆ 2

21



valV express the non – local potentials involving both Coulomb and exchange interactions of cv with other valence pseudo – orbitals.

PSV is a pseudopotential depending on how important is the interaction between the pseudo – valence atomic orbital and the core orbitals projected onto it.

85

PseudopotentialsThe pseudopotential operator is usually expressed as: PSV

vccv

C

ccv

PSV ˆ

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.

86

PseudopotentialsThe pseudopotential operator is usually expressed as: PSV

vccv

C

ccv

PSV ˆ

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.

PSV 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.

87

PseudopotentialsIn a graphical way,

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.

88

Pseudopotentials

As an example, the case of AgF (rexp = 1.983 Å, IPexp = 7.574 ev):

Basis set or PS rcalc -eHOMO (ev) Etot (Hartrees) rel. calc. timeSTO-3G 1.633 5.731 -5244.5645993 1LanL2MB 2.031 8.027 -242.793708 0.7

Relativistic effectsThe 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.


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.

Relativistic effects

The functional form of the Dirac’s equation for the state of an electron is:

t

titmcc

),(

),(ˆ 2 RRp

where the a and b spin elements are 4 x 4 matrices and the wave function is four component.


The functional form of the Dirac’s equation for the state of an electron is:

t

titmcc

),(

),(ˆ 2 RRp

where the a and b 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.


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.


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.


The contribution of relativistic effects to atoms with a fully relativistic approach can be illustrated by a Clementi´s paper:

Element Erel (Hartrees) EHF (total) %

He -0.000071 -2.8616785 0.0025

Ne -0.131292 -128.54698 0.0010

Ar -1.766130 -526.81705 0.0034

Hartmann, H.; Clementi, E., Relativistic Correction for Analytic Hartree-Fock Wave Functions. Phys. Rev. 1964, 133 (5A), A1295.

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 1 (2S + 1 = 1).

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 2, a “doublet”) or because two of them are unpaired ( or multiplicity 3, a “triplet”):

99

Open shell systems


• An independent calculation of both a and b electron spin manifolds, the so – called non – restricted Hartree – Fock treatment (UHF).

100

Open shell systems


• An independent calculation of both a and b 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).

101

Open shell systems: UHF

It is based in evaluating and diagonalizing two ground state Slater determinants, independently, with different electron occupations.

)|()|(2,

pphF

)|()|(2,

pphF

102


It is based in evaluating and diagonalizing two ground state Slater determinants, independently, with different electron occupations.

However, the corresponding results of density matrices are summed together for producing the Fock’s matrix elements of the following SCF cycle.

)|()|(2,

pphF

)|()|(2,

pphF

103


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.

104



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.

105



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.

106

Open shell systems: ROHFThis procedure divide molecular orbitals in those filled and unpaired and treat both separately for further merging results on the grounds of: 1. The total wave function is, in general, a sum of several Slater determinants, each of which contains a (doubly occupied) closed-shell core FC, and a partially occupied open shell chosen from a set FO. The total wave function could be then expressed as:

and it is assumed to be orthonormal, so that the two sets FC

and FO are orthonormal and mutually orthogonal.

OC ,

107

Open shell systems: ROHF

2. The expectation value of the energy is given by:

where a, b, and f are numerical constants depending on the specific case.

mn kmkmkmmnmn

mm

klklkl

kk

total

KJbKaJfHf

KJHE

222

22

108

Open shell systems: ROHF

2. The expectation value of the energy is given by:

where a, b, and f are numerical constants depending on the specific case.

mn kmkmkmmnmn

mm

klklkl

kk

total

KJbKaJfHf

KJHE

222

22

The first two sums in the energy expression represent the closed-shell energy, the next two sums the open-shell energy, and the last sum the interaction energy of the closed and open shell.

109

Open shell systems: ROHFThe “super” Fock matrix FROHF with merging subsets of molecular orbitals can be expressed, according to one of the current approximations as:

Filled orbitals Half filled orbitals

Empty orbitals

Filled orbitals )(

2)( OpenpKpF

a )(2

1)( OpenpKpF )(pF

Half filled orbitals )(

2

1)( OpenpKpF )(

2)( OpenpKpF

b )(

2

1)( OpenpKpF

Empty orbitals )(pF )(

2

1)( OpenpKpF )(

2)( OpenpKpF

c

110


• F(p) = h + J - K/2 is the Fock’s matrix as calculated with the total density matrix p (including all filled and half filled shells)


Empty orbitals

Filled orbitals )(

2)( OpenpKpF

a )(2

1)( OpenpKpF )(pF


2

1)( OpenpKpF )(

2)( OpenpKpF

b )(

2

1)( OpenpKpF


2

1)( OpenpKpF )(

2)( OpenpKpF

c

111



• K(pOpen) is the exchange matrix as calculated with the density matrix of half filled levels.


Empty orbitals

Filled orbitals )(

2)( OpenpKpF

a )(2

1)( OpenpKpF )(pF


2

1)( OpenpKpF )(

2)( OpenpKpF

b )(

2

1)( OpenpKpF


2

1)( OpenpKpF )(

2)( OpenpKpF

c

112



• K(pOpen) is the exchange matrix as calculated with the density matrix of half filled levels.


Empty orbitals

Filled orbitals )(

2)( OpenpKpF

a )(2

1)( OpenpKpF )(pF


2

1)( OpenpKpF )(

2)( OpenpKpF

b )(

2

1)( OpenpKpF


2

1)( OpenpKpF )(

2)( OpenpKpF

c

• (a,b,c) : (0,0,0) in normal calculations; (0,-1,0) for certain cases and (2,0,-2) for others.

113

Open shell systems

Different approaches to open shell systems can be overviewed as:

114

Open shell systems

The case of NO (rexp = 1.151 Å) is illustrative:

Method rcalc Etot (Hartrees) Rel. timeROHF [4-31G(d)] 1.123 -129.1185143 1UHF [4-31G(d)] 1.125 -129.1249945 1

1 MODELING MATTER AT NANOSCALES 6.The theory of molecular orbitals for the description of...

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