MODELING
MATTER AT
NANOSCALES
6.
The theory of
molecular orbitals for the
description of nanosystems
(part II)
6.02. Ab initio methods.
Basis functions.
1
The
SCF
cycle
2
The SCF cycle
The Fock matrix must be constructed:
 F11

 F21
F
.

F
 1
F12
F22
.
.
... F1 

... F2  
... . 

... F 
and the problem is to evaluate the F 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.
3
The SCF cycle
As:
p   ni ci ci
i
1
   (rm )   (rn ) r  (rm )  (rn )d  J   ( | )
mn
1
   (rm )   (rn ) r  (rm )  (rn )d  K   ( |  )
mn
4
The SCF cycle
As:
p   ni ci ci
i
1
   (rm )   (rn ) r  (rm )  (rn )d  J   ( | )
mn
1
   (rm )   (rn ) r  (rm )  (rn )d  K   ( |  )
mn
Then the formula to calculate every matrix element remains as:
F  h   p 2( | )  ( |  )
 ,
5
The SCF cycle
Therefore, the Hartree – Fock’s solution for molecules means the
following integral evaluation:
h
monoelectronic
( |  ) bielectronic of four centers
p 
electron density
It can only be achieved after the appropriate selection of a basis
set.
6
The SCF cycle
Therefore, the Hartree – Fock’s solution for molecules means the
following integral evaluation:
h
monoelectronic
( |  ) bielectronic of four centers
p 
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
Ab initio proceeding
It is said to be following an ab initio routine or proceeding when
all h and (|) 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
Ab initio proceeding
It is said to be following an ab initio routine or proceeding when
all h and (|) 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
Ab initio proceeding
It is said to be following an ab initio routine or proceeding when
all h and (|) 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.
10
Basis sets
The appropriately non – orthogonal basis set serves for
calculating all h and (|) integrals building the F matrix.
Basis sets
The appropriately non – orthogonal basis set serves for
calculating all h and (|) integrals building the F matrix.
It is conveniently orthogonalized:
 12
F'  S FS
 12
Basis sets
The appropriately non – orthogonal basis set serves for
calculating all h and (|) integrals building the F matrix.
It is conveniently orthogonalized:
 12
F'  S FS
Diagonalization proceeds:
 12
(F 'E)c'  0
Basis sets
The appropriately non – orthogonal basis set serves for
calculating all h and (|) integrals building the F matrix.
It is conveniently orthogonalized:
 12
F'  S FS
 12
Diagonalization proceeds:
(F 'E)c'  0
And the consequent orthogonal based c´ matrix is then
deorthogonalized for calculated properties:
 12
c  S c'
Basis sets
The Slater’s hydrogen like basis functions for all multielectronic atoms
are called STO (Slater type orbitals) and are considered as references:
1
 z  2 z r
1s    e
 
3
1
1
1
 z  2 z r
 2 s    re
 3 
5
2
2 p
1
 z  2 z r
   xe
 
5
2
x
2
2
z parameters or “exponents” determine the orbital size.
Basis sets
There are routines for calculations of all electronic integrals on
STO basis of the general form:
n 1 zr
2






 nlm  2z
2n ! r e Ylm q , 
n  12
1
where n, l, m are hydrogen like atomic quantum numbers and
Ylm(q,) are the angular components being always the
normalized spherical harmonics.
Basis sets
There are routines for calculations of all electronic integrals on
STO basis of the general form:
n 1 zr
2






 nlm  2z
2n ! r e Ylm q , 
n  12
1
where n, l, m are hydrogen like atomic quantum numbers and
Ylm(q,) 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.
Basis sets
There are routines for calculations of all electronic integrals on
STO basis of the general form:
n 1 zr
2






 nlm  2z
2n ! r e Ylm q , 
n  12
1
where n, l, m are hydrogen like atomic quantum numbers and
Ylm(q,) 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 z parameter is
defined by:
Z
z 
n
Basis sets
Routine calculations with STO’s are avoided because the existing
procedures are very computing time consuming.
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:
1
 z  2 z r
1s    e
 
3
1
e   d1,k e
1
2 p
x
2
N
re   d 2 s ,k e
r
 2 ,k r 2
k 1
1
 z  2 z r
   xe
 
5
2
1 ,k r 2
k 1
 z  2 z r
 2 s    re
 3 
5
2
N
r
1
2
N
xe   d 2 p ,k xe
r
k 1
 2 ,k r 2
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:
1
 z  2 z r
1s    e
 
3
1
e   d1,k e
1
2 p
x
2
N
re   d 2 s ,k e
r
 2 ,k r 2
k 1
1
 z  2 z r
   xe
 
5
2
1 ,k r 2
k 1
 z  2 z r
 2 s    re
 3 
5
2
N
r
1
2
N
xe   d 2 p ,k xe
r
 2 ,k r 2
k 1
d (fixed coefficients) values and  (exponent scaling) parameters
are optimized to fit the original corresponding Slater curve
according the length (N) of the selected series.
Basis sets
This functions are called minimal basis STO-NG, where N = 2 to
6.
As an example, the case of STO-3G basis for carbon, values are:
l
S
S
1
2.23
0.41
0.11
2
0.99
0.23
0.08
P
2
0.99
0.23
0.99
d1s
0.15
0.53
0.44
d2s
-0.10
0.40
0.70
d2p
0.16
0.61
0.39
d parameter is called as “coefficient”, and  parameter as
“exponent”.
Basis sets
Jargon
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).
Basis sets
Jargon
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).
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).
Basis sets
Jargon
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).
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.
Basis sets
Shell Number
Primitive shell
of
primitive
shells
1
S
s
3
P
px, py, pz
6
D
d x 2 , d y 2 , d z 2 , d xy , d xz , d yz
SP
5
d x 2  y 2 , d z 2 , d xy , d xz , d yz
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:
1
1
 z  z r
z  2 N
1s    e  1s     d1,k e  f
 
   k 1
3
1
3
1
2
1
1
2
2
1 s1 , k r
1
2 N
f
z
z 


z r
 2 s    re   2 s     d 2 s ,k e
 3 
 3  k 1
5
2
2 p
2
1
2
2
sp  2 , k r
1
2 N
z
z 


f
z r
   xe   2 p     d 2 p ,k xe
 
   k 1
5
2
x
5
2
2
5
2
2
2
x
2
2
sp 2 , k r
Basis sets
It is interesting that empirical scale factors have been included
by the authors of STO’s to the squared exponents:
1
1
 z  z r
z  2 N
1s    e  1s     d1,k e  f
 
   k 1
3
1
3
1
2
1
1
2
2
1 s1 , k r
1
2 N
f
z
z 


z r
 2 s    re   2 s     d 2 s ,k e
 3 
 3  k 1
5
2
2 p
2
1
2
2
sp  2 , k r
1
2 N
z
z 


f
z r
   xe   2 p     d 2 p ,k xe
 
   k 1
5
2
x
5
2
2
5
2
2
2
2
2
sp 2 , k r
x
As z parameters or “exponents” determine the orbital size, they
became larger with f’s to be adapted for molecules.
Basis sets
STO-3G basis functions for carbon (f1s = 5.67; fsp = 1.72) remain
as:
Capa
S
SP
f1s21
71.6
13.0
3.53
fsp22
d1s
0.15
0.53
0.44
d2s
d 2 px
d 2 py
d 2 pz
2.94
0.68
0.22
-0.10
0.40
0.70
0.16
0.61
0.39
0.16
0.61
0.39
0.16
0.61
0.39
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:
E
total
E
elect
Z AZ B

A B RAB
Z AZ B
 2 Ei   2 J ij  K ij   
i
i, j
A B RAB
Basis sets
Remembering that the total Hartree – Fock’s energy as written in
terms of molecular orbitals is:
E
total
E
elect
Z AZ B

A B RAB
Z AZ B
 2 Ei   2 J ij  K ij   
i
i, j
A B RAB
and it becomes on the grounds of the atomic basis set as:
E
total
 2 Ei    2 p p  |   p p  |  
i
i , j  ,  , ,
Z AZ B

A B RAB
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 sets
The 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 sets
The 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.
Split basis sets
The 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 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):
2  r

g s  , r     e
 
 128 3 
r
g q  , r   
 qe
3
  
3
4
2
1
4
1
2
 2048  2 r
g qq  , r   
 qe
3
 9 
7
where q = x, y, z.
4
2
Split basis sets
gs has the same symmetry as s Slater orbitals
Split basis sets
gs has the same symmetry as s Slater orbitals
gx, gy, gz have the same symmetry as px, py and pz Slater orbitals
Split basis sets
gs has the same symmetry as s 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
dxz
dyz
dz
2
dx y
2
2
 gxy
 gxz
 gyz
 g 3 zz rr  12 2 g zz  g xx  g yy 
 g xx yy   34  g xx  g yy 
1
2
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 sets
Pople´s group obtained a set of GTO’s as expansions of primitive
Gaussians. For example:
Carbon
K


1s r    d1s ,i g s 1i , r 
Hydrogen
K


1s ' r    d i g s  i ' , r 
L


 2 s ' r    d 2 s ,i g s  2i ' , r 


1s ' ' r   g s  ' ' , r 
i 1
i 1


 2 s ' ' r   g s  2 ' ' , r 
L


 2 p x ' r    d 2 p ,i g p x  2i ' , r 
i 1


 2 p x ' ' r   g p  2 ' ' , r 
i 1
Split basis sets
Pople´s group obtained a set of GTO’s as expansions of primitive
Gaussians. For example:
Carbon
K


1s r    d1s ,i g s 1i , r 
Hydrogen
K


1s ' r    d i g s  i ' , r 
L


 2 s ' r    d 2 s ,i g s  2i ' , r 


1s ' ' r   g s  ' ' , r 
i 1
i 1
i 1


 2 s ' ' r   g s  2 ' ' , r 
L


 2 p x ' r    d 2 p ,i g p x  2i ' , r 
i 1


 2 p x ' ' r   g p  2 ' ' , r 
Naming of these functions follows the rule of the acronym KL1G (ej. 3-21G, 4-31G, 6-31G, etc.).
Split basis sets
If we consider that there are scaling factor, too, in such split
functions used as n = fnl2nl , then the 3-21G basis set for
carbon remains as:
Shell
S
1s
172.3
25.9
5.53
2'
d1s
0.062
0.359
0.701
d 2s '
d 2 px '
d2 py '
d 2 pz '
SP
3.66
0.771
2''
-0.396
1.22
d 2s ' '
0.236
0.861
d 2 px ' '
0.236
0.861
d2 py ' '
0.236
0.861
d 2 pz ' '
SP
0.196
1.00
1.00
1.00
1.00
Split basis sets
The so – called 6-31G basis set for carbon is:
6-31G
Shell
S
1s
d1s
3047.5 0.002
457.4 0.014
103.9 0.069
29.21 0.232
9.287 0.468
3.164 0.362
d 2s '
2'
d 2 px '
d2 py '
d 2 pz '
SP
7.868
1.881
0.544
2''
-0.119
-0.161
1.143
d 2s ' '
0.069
0.316
0.744
d 2 px ' '
0.069
0.316
0.744
d2 py ' '
0.069
0.316
0.744
d 2 pz ' '
SP
0.169
1.00
1.00
1.00
1.00
Split basis sets
And the so – called 6-311G basis set for carbon is:
6-311G
Shell
S
SP
SP
SP
1s
4563.2
682.0
155.0
44.46
13.03
1.828
 2'
20.96
4.803
1.459
2''
0.483
2'''
0.146
d1s
0.002
0.015
0.076
0.261
0.616
0.221
d2s´
0.115
0.920
-0.303
d2s´´
1.00
d2s´´´
1.00
d2px´
0.040
0.238
0.816
d2px´´
1.00
d2px´´´
1.00
d2py´
0.040
0.238
0.816
d2py´´
1.00
d2py´´´
1.00
d2pz´
0.040
0.238
0.816
d2pz´´
1.00
d2pz´´´
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.
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.
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.
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 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.
Polarization basis sets
A “polarized” basis set of carbon at the 3-21G level remains as:
Shell
S
1s
172.3
25.9
5.53
2'
d1s
0.062
0.359
0.701
d 2s '
d 2 px '
d2 py '
d 2 pz '
SP
3.66
0.771
2''
-0.396
1.22
d 2s ' '
0.236
0.861
d 2 px ' '
0.236
0.861
d2 py ' '
0.236
0.861
d 2 pz ' '
SP
0.196
3
1.00
d d xx
1.00
d d yy
1.00
d d zz
1.00
d d xy
d d xz
d d yz
0.800
1.00
1.00
1.00
1.00
1.00
1.00
D
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:
3-21G
3-21G*
3-21G**
Non polarized split basis
Split basis with polarization for
heavy atoms
Split basis with polarization for all
atoms
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:
3-21G
3-21G*
3-21G**
Non polarized split basis
Split basis with polarization for
heavy atoms
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
d1s
1s
172.3
25.9
5.53
2'
0.062
0.359
0.701
d 2s '
d 2 px '
d2 py '
d 2 pz '
SP
3.66
0.771
2''
-0.396
1.22
d 2s ' '
0.236
0.861
d 2 px ' '
0.236
0.861
d2 py ' '
0.236
0.861
d 2 pz ' '
SP
0.196
2+
1.00

d 2s
1.00

d 2 px
1.00

d2 py
1.00

d 2 pz
SP
0.044
1.00
1.00
1.00
1.00
S
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
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-L1G). They provide large energy to
the system, but the importance of them for the most common
phenomena is controversial.
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-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.
65
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 :
Huzinaga / Dunning’s full double z :
Dunning’s correlation consistent:
D95V
D95
cc-pVDZ, cc-pVTZ,
cc-pVQZ, cc-pV5Z,
cc-pV6Z
66
The “ab initio” energy landscape
The 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.
67
The “ab initio” energy landscape
The 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:
E total  2 Ei
i
   2 p p  |   p p  |  
i , j  ,  , ,
Z AZ B

A B R AB
68
The “ab initio” energy landscape
The 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:
E total  2 Ei
i
   2 p p  |   p p  |  
i , j  ,  , ,
Z AZ B

A B R AB
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
69
electron term Ei also contains nuclear coordinates.
The “ab initio” energy landscape
As expected, the goal is finding a certain Req giving the lowest
Etotal.
70
The “ab initio” energy landscape
As expected, the goal is finding a certain Req giving the lowest
Etotal.
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.
71
The “ab initio” energy landscape
As expected, the goal is finding a certain Req giving the lowest
Etotal.
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 total 

0
 dR 
72
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
73
to the previous step.
Some examples
In the case of formaldehyde with minimal basis:
STO-2G
STO-3G
Total energy
-109.0244 -112.3525
(Hartrees)
rCO (Å)
1.220
1.217
rCH (Å)
1.110
1.101
<HCH
113.1
114.5
1.118
1.520
 (debye)
Relative
computer
1
2
time
STO-4G
STO-5G
STO-6G
-113.1611
-113.3752
-113.4408
1.216
1.099
114.8
1.592
1.216
1.098
114.8
1.596
1.216
1.098
114.8
1.596
3
6
10
Exp.
1.210
1.102
121.1
2.33
Some examples
In the case of formaldehyde with split basis:
STO-6G
Total energy
(Hartrees)
rCO (Å)
rCH (Å)
<HCH
m (debye)
Relative
computer
time
-113.4408
3-21G
6-21G
6-31G
-113.2218 -113.6985 -113.8084
6-311++G**
Exp.
-113.9029
1.216
1.098
114.8
1.596
1.207
1.083
114.9
2.657
1.209
1.083
115.1
2.673
1.210
1.082
116.6
3.304
1.180
1.094
116.0
2.806
1
1
1
0.8
1.4
1.210
1.102
121.1
2.33
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 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.
The valence electrons are then self-consistently solved for in
the Hilbert space orthogonalized to the frozen core orbitals.
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.
The valence electrons are then self-consistently solved for in
the Hilbert space orthogonalized to the frozen core orbitals.
Resulting pseudo – valence atomic orbitals v are then
expressed in terms of its corresponding eigenfunctions f in
their original configuration together with a linear combination of
all C core fc atomic orbitals:
C
 v  fv   acfc
c 1
Pseudopotentials
A pseudo atomic Hamiltonian Ĥ acting on a certain v pseudo
– valence orbital can be defined as:
PS
Z

2
PS 
1
ˆ
ˆ
ˆ
ˆ
H  v    2    Vcore  Vval  V   v   v  v
r


PS
81
Pseudopotentials
A pseudo atomic Hamiltonian Ĥ acting on a certain v pseudo
– valence orbital can be defined as:
PS
Z

2
PS 
1
ˆ
ˆ
ˆ
ˆ
H  v    2    Vcore  Vval  V   v   v  v
r


PS
where:
Vˆcore express the non – local potentials involving both
Coulomb and exchange interactions of v with core
orbitals
82
Pseudopotentials
A pseudo atomic Hamiltonian Ĥ acting on a certain v pseudo
– valence orbital can be defined as:
PS
Z

2
PS 
1
ˆ
ˆ
ˆ
ˆ
H  v    2    Vcore  Vval  V   v   v  v
r


PS
where:
Vˆcore express the non – local potentials involving both
Coulomb and exchange interactions of v with core
orbitals
Vˆval express the non – local potentials involving both
Coulomb and exchange interactions of v with other
valence pseudo – orbitals.
83
Pseudopotentials
A pseudo atomic Hamiltonian Ĥ acting on a certain v pseudo
– valence orbital can be defined as:
PS
Z

2
PS 
1
ˆ
ˆ
ˆ
ˆ
H  v    2    Vcore  Vval  V   v   v  v
r


PS
where:
Vˆcore express the non – local potentials involving both
Coulomb and exchange interactions of v with core
orbitals
Vˆval express the non – local potentials involving both
Coulomb and exchange interactions of 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.
84
Pseudopotentials
The pseudopotential operator Vˆ PSis usually expressed as:
Vˆ  v   fc  v   c  fc  v
PS
C
c
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
Pseudopotentials
The pseudopotential operator Vˆ PSis usually expressed as:
Vˆ  v   fc  v   c  fc  v
PS
C
c
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
Pseudopotentials
In 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.
87
Pseudopotentials
As an example, the case of AgF (rexp = 1.983 Å, IPexp = 7.574 ev):
Basis set or PS
STO-3G
LanL2MB
rcalc
1.633
2.031
-eHOMO (ev)
5.731
8.027
Etot (Hartrees) rel. calc. time
-5244.5645993
1
-242.793708
0.7
88
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.
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.
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.
Relativistic effects
The functional form of the Dirac’s equation for the state of an
electron is:
 (R, t )
c  pˆ  mc  (R, t )  i
t
2
where the  and  spin elements are 4 x 4 matrices and the
wave function is four component.
Relativistic effects
The functional form of the Dirac’s equation for the state of an
electron is:
 (R, t )
c  pˆ  mc  (R, t )  i
t
2
where the  and  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.
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.
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.
Relativistic effects
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.
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).
97
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”):
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”):
• An independent calculation of both  and  electron spin
manifolds, the so – called non – restricted Hartree – Fock
treatment (UHF).
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 2, a “doublet”) or because two of them are
unpaired ( or multiplicity 3, a “triplet”):
• An independent calculation of both  and  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).
100
Open shell systems: UHF
It is based in evaluating and diagonalizing two ground state
Slater determinants, independently, with different electron
occupations.



2( | )  ( |  )
F  h
   p
 p
 ,



2( | )  ( |  )
F  h
   p
 p
 ,
101
Open shell systems: UHF
It is based in evaluating and diagonalizing two ground state
Slater determinants, independently, with different electron
occupations.



2( | )  ( |  )
F  h
   p
 p
 ,



2( | )  ( |  )
F  h
   p
 p
 ,
However, the corresponding results of density matrices are
summed together for producing the Fock’s matrix elements
of the following SCF cycle.
102
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.
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.
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.
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.
This problem is known as “spin contamination” and some
special numerical treatments are useful for their solution.
105
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:
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:
F  F C , F O 
and it is assumed to be orthonormal, so that the two sets
FC and FO are orthonormal and mutually orthogonal.
106
Open shell systems: ROHF
2. The expectation value of the energy is given by:
E total  2 H k   2 J kl  K kl 
k
kl
 f 2 H m  f  2aJ mn  bK mn    2 J km  K km 
 m

mn
km
where a, b, and f are numerical constants depending on the
specific case.
107
Open shell systems: ROHF
2. The expectation value of the energy is given by:
E total  2 H k   2 J kl  K kl 
k
kl
 f 2 H m  f  2aJ mn  bK mn    2 J km  K km 
 m

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.
108
Open shell systems: ROHF
The “super” Fock matrix FROHF with merging subsets of
molecular orbitals can be expressed, according to one of the
current approximations as:
Filled orbitals
Filled
orbitals
Half filled
orbitals
Empty
orbitals
a
F(p)  K (p Open )
2
1
F (p)  K (p Open )
2
F (p )
Half filled
orbitals
1
F (p)  K (p Open )
2
b
F (p)  K (p Open )
2
1
F (p)  K (p Open )
2
Empty orbitals
F (p )
1
F (p)  K (p Open )
2
c
F(p)  K (p Open )
2
109
Open shell systems: ROHF
The “super” Fock matrix FROHF with merging subsets of
molecular orbitals can be expressed, according to one of the
current approximations as:
Filled orbitals
Filled
orbitals
Half filled
orbitals
Empty
orbitals
a
F(p)  K (p Open )
2
1
F (p)  K (p Open )
2
F (p )
Half filled
orbitals
1
F (p)  K (p Open )
2
b
F (p)  K (p Open )
2
1
F (p)  K (p Open )
2
Empty orbitals
F (p )
1
F (p)  K (p Open )
2
c
F(p)  K (p Open )
2
• 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)
110
Open shell systems: ROHF
The “super” Fock matrix FROHF with merging subsets of
molecular orbitals can be expressed, according to one of the
current approximations as:
Filled orbitals
Filled
orbitals
Half filled
orbitals
Empty
orbitals
a
F(p)  K (p Open )
2
1
F (p)  K (p Open )
2
F (p )
Half filled
orbitals
1
F (p)  K (p Open )
2
b
F (p)  K (p Open )
2
1
F (p)  K (p Open )
2
Empty orbitals
F (p )
1
F (p)  K (p Open )
2
c
F(p)  K (p Open )
2
• 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)
• K(pOpen) is the exchange matrix as calculated with the density
matrix of half filled levels.
111
Open shell systems: ROHF
The “super” Fock matrix FROHF with merging subsets of
molecular orbitals can be expressed, according to one of the
current approximations as:
Filled orbitals
Filled
orbitals
Half filled
orbitals
Empty
orbitals
a
F(p)  K (p Open )
2
1
F (p)  K (p Open )
2
F (p )
Half filled
orbitals
1
F (p)  K (p Open )
2
b
F (p)  K (p Open )
2
1
F (p)  K (p Open )
2
Empty orbitals
F (p )
1
F (p)  K (p Open )
2
c
F(p)  K (p Open )
2
• 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)
• K(pOpen) is the exchange matrix as calculated with the density
matrix of half filled levels.
• (a,b,c) : (0,0,0) in normal calculations; (0,-1,0) for certain cases
112
and (2,0,-2) for others.
Open shell systems
Different approaches to open shell systems can be overviewed
as:
113
Open shell systems
The case of NO (rexp = 1.151 Å) is illustrative:
Method
ROHF [4-31G(d)]
UHF [4-31G(d)]
rcalc
1.123
1.125
Etot (Hartrees)
-129.1185143
-129.1249945
Rel. time
1
1
114