MODELING MATTER AT
NANOSCALES
6.
6.05.
The theory of molecular orbitals for the description of
nanosystems (part II)
Variational methods for dealing with correlation energy
1
Optimizing the Hartree - Fock
wave function
Being Fs the multielectronic Hartree-Fock wave function of any s
state corresponding to a molecule or other nanoscopic system it
can always be expressed as a linear combination of associated
wave functions, following the algebraic formalism:
F s   asI I
I
where the sum is over a sufficiently large I series of I
multielectronic reference states and asI are their corresponding
participation coefficients in such s state.
Optimizing the Hartree - Fock
wave function
Being Fs the multielectronic Hartree-Fock wave function of any s
state corresponding to a molecule or other nanoscopic system it
can always be expressed as a linear combination of associated
wave functions, following the algebraic formalism:
F s   asI I
I
where the sum is over a sufficiently large I series of I
multielectronic reference states and asI are their corresponding
participation coefficients in such s state.
 must be an orthonormal basis.
Optimizing the Hartree - Fock
wave function
Obviously, asI participation coefficients could be optimized on
the grounds of a variational approach to obtain the I series
giving a s state of minimal total energy.
Optimizing the Hartree - Fock
wave function
Obviously, asI participation coefficients could be optimized on
the grounds of a variational approach to obtain the I series
giving a s state of minimal total energy.
Therefore, the resulting Fs for each state of the system will
better approach the exact wave function as the total energy
becomes smaller.
Optimizing the Hartree - Fock
wave function
If multielectronic reference functions are antisymmetrized
products or Slater determinants resulting from a Hartree – Fock
calculation, a typical electronic configuration is that where each
spin orbital represents a given existing electron.
y 1 (r1a ) y 1 (r2  ) ... y 1 (rN  )
y 2 (r1a ) y 2 (r2  ) ... y 2 (rN  )
1
I  ( N !)
.
.
...
.
y N (r1a ) y N (r2  ) ... y N (rN  )
 ( N !) 1 dety i (rna )y i 1 (rn1 )
where yi reference functions are spin orbitals describing the
state of an electron of rnan ≡tn spatial and spin coordinates.
Optimizing the Hartree - Fock
wave function
0  ( N !)
y 1 (r1a ) y 1 (r2  ) ... y 1 (r6  )
y 2 (r1a ) y 2 (r2  ) ... y 2 (r6  )
1
.
.
...
.
y 6 (r1a ) y 6 (r2  ) ... y 6 (r6  )
Distribution of different
spin orbital populations
can vary from that of the
ground state 0. Each
one of the I electron
populations could be a
valid basis
representation of the
system serving for a
variational optimization.
y 1 (r1a )
y 2 (r1a )
y (r a )
I  ( N !) 1 3 1
y 5 (r1a )
y 6 (r1a )
y 10 (r1a )
y 1 (r2  )
y 2 (r2  )
y 3 (r2  )
y 5 (r2  )
y 6 (r2  )
y 10 (r2  )
y 1 (r3a )
y 2 (r3a )
y 3 (r3a )
y 5 (r3a )
y 6 (r3a )
y 10 (r3a )
y 1 (r4  )
y 2 (r4  )
y 3 (r4  )
y 5 (r4  )
y 6 (r4  )
y 10 (r4  )
y 1 (r5a )
y 2 (r5a )
y 3 (r5a )
y 5 (r5a )
y 6 (r5a )
y 10 (r5a )
y 1 (r6  )
y 2 (r6  )
y 3 (r6  )
y 5 (r6  )
y 6 (r6  )
y 10 (r6  )
Optimizing the Hartree - Fock
wave function
Each determinant showing a different electron distribution to
that of the ground state is known as an excited configuration.
y 1 (r1a )
y 2 (r1a )
y (r a )
I  ( N !) 1 3 1
y 5 (r1a )
y 6 (r1a )
y 10 (r1a )
y 1 (r2  )
y 2 (r2  )
y 3 (r2  )
y 5 (r2  )
y 6 (r2  )
y 10 (r2  )
y 1 (r3a )
y 2 (r3a )
y 3 (r3a )
y 5 (r3a )
y 6 (r3a )
y 10 (r3a )
y 1 (r4  )
y 2 (r4  )
y 3 (r4  )
y 5 (r4  )
y 6 (r4  )
y 10 (r4  )
y 1 (r5a )
y 2 (r5a )
y 3 (r5a )
y 5 (r5a )
y 6 (r5a )
y 10 (r5a )
y 1 (r6  )
y 2 (r6  )
y 3 (r6  )
y 5 (r6  )
y 6 (r6  )
y 10 (r6  )
Optimizing the Hartree - Fock
wave function
Configurations could correspond to one of the following types:
yLUMO+3
yLUMO+2
yLUMO+1
yLUMO
yHOMO
yHOMO-1
yHOMO-2
HF
S
S
D
D
T
Q
HF: ground state; S: monoexcited configuration; D: double excited configuration;
T: triple excited configuration; Q: quadruple excited configuration
Optimizing the Hartree - Fock
wave function
Departing from a 0( 0 ) solution of the Hartree – Fock
multielectronic wave function a very large number of
configurations can be made depending on the number of
electrons entering different occupations of spin orbitals:
F s   asI I
I
 a0( 0 ) 0( 0 )   asI(1) I(1)   asI( 2 ) I( 2 )   asI( 3) I( 3)   asI( 4 ) I( 4 )  ...
I
I
I
I
where a0( p ) 0( p ) means a p order excited configuration,
redistributing p charges among all initially unoccupied spin
orbitals.
Optimizing the Hartree - Fock
wave function
Therefore, the solution of a certain n electron system under the
Hartree – Fock procedure represented by N spin orbital basis
functions will amount m = N – n non occupied spin orbitals.
It can be shown that the number of possible reference functions
for this development is huge and can be calculated according to:
n  m !
n  m !
 n  m   n  m 



    

n!m!
p 1  p  p 
 n  n!n  m  n !
n
Water as an example
If we develop a minimal Slater basis function for water:
H:
H:
O:
1sa1, 1s
1sa1, 1s
1sa1, 1s1, 2sa1, 2s1, 2pxa1, 2px1, 2pya1, 2py1, 2pza, 2pz
We will have n = 10 electrons and N = 14 possible basis spin
orbitals when the Hartree – Fock molecular problem was solved.
Therefore m = 4 orbitals will remain as non occupied.
Water as an example
If we develop a minimal Slater basis function for water:
H:
H:
O:
1sa1, 1s
1sa1, 1s
1sa1, 1s1, 2sa1, 2s1, 2pxa1, 2px1, 2pya1, 2py1, 2pza, 2pz
We will have n = 10 electrons and N = 14 possible basis spin
orbitals when the Hartree – Fock molecular problem was solved.
Therefore m = 4 orbitals will remain as non occupied.
Therefore, the number of possible electronic configurations will
be:
n  m! 14!

 1001
n!m! 10!4!
Water as an example
If the basis function is as simple as 3-21G, basis atomic orbitals
would be:
H:
H:
O:
1sa1, 1s, 1sa’, 1s’
1sa1, 1s, 1sa’, 1s’
1sa1, 1s1, 2sa1, 2s1, 2pxa1, 2px1, 2pya1, 2py1, 2pza, 2pz, 2sa’, 2s’,
2pxa’, 2px’, 2pya’, 2py’, 2pza’, 2pz’
We will again have n = 10 electrons but N = 26 possible basis
spin orbitals when the Hartree – Fock molecular problem was
solved. Therefore m = 16 orbitals will remain as non occupied.
Water as an example
If the basis function is as simple as 3-21G, basis atomic orbitals
would be:
H:
H:
O:
1sa1, 1s, 1sa’, 1s’
1sa1, 1s, 1sa’, 1s’
1sa1, 1s1, 2sa1, 2s1, 2pxa1, 2px1, 2pya1, 2py1, 2pza, 2pz, 2sa’, 2s’,
2pxa’, 2px’, 2pya’, 2py’, 2pza’, 2pz’
We will again have n = 10 electrons but N = 26 possible basis
spin orbitals when the Hartree – Fock molecular problem was
solved. Therefore m = 16 orbitals will remain as non occupied.
Therefore, the number of possible electronic configurations will
be much higher:
n  m! 26!

 5311735
n!m! 10!16!
Correlation energy
Being the Hartree – Fock equation of eigenvalues and
eigenfunctions of a given I electronic configuration I :
FˆI  EI I
A certain matrix element on the ground of atomic orbitals can be
written in a simplified way as:
F I     Hˆ    c Ii ciI       

i
where
           1 2 r121   1 2   1  2dt 1dt 2
Correlation energy
The total energy of such I configuration then remains as:
I 
I  I 
EIHF    
  Hˆ   12   
         Vcore


being a matrix element of the density matrix in the I state:
I 
 
  c Ii ciI 
i
on the grounds of ci molecular orbital coefficients for atomic
orbitals  and .
Correlation energy
On the other hand, the “exact” expression for the s state
energy Es, as an output of optimizing linear combinations
of all reference I states:
F s   asI I
I
would be expected to become from:
̂F s  Es F s
Correlation energy
It is evident that finding the Fs wave functions and their
corresponding eigenvalues Es after optimizing Hartree –
Fock electron configurations means that we are trying to
attain a form of correlation energy, given the corresponding
definition, as in the case of a ground state:
HF
Ecorr  E0  Etot
Correlation energy
It is evident that finding the Fs wave functions and their
corresponding eigenvalues Es after optimizing Hartree –
Fock electron configurations means that we are trying to
attain a form of correlation energy, given the corresponding
definition, as in the case of a ground state:
HF
Ecorr  E0  Etot
It is congruent with the definition of correlation energy if
considering the variationally optimized energy as “exact”.
The method of configuration
interaction (CI)
The method of configuration interaction (CI) means the
variational optimization of multielectronic states on the
basis of linear combinations of several different
configurations of the given system. Such configurations are
expressed as their respective determinants being built from
a previous one electron wave function procedure.
The method of configuration
interaction (CI)
The method of configuration interaction (CI) means the
variational optimization of multielectronic states on the
basis of linear combinations of several different
configurations of the given system. Such configurations are
expressed as their respective determinants being built from
a previous one electron wave function procedure.
If the expansion includes all possible configurations, then
this is called as a full configuration interaction procedure
which is said to exactly solve the electronic Schrödinger
equation within the space spanned by the one-particle
basis set.
The method of configuration
interaction (CI)
The matrix representation to solve the problem is:
HA = EA
It means the evaluation and diagonalization of the HIJ
matrix elements in:
 H 00 H 01 ... H 0 N 


 H 10 H 11 ... H 1N 
H
.
. ...
. 


. ... H NN 
 HN0
The method of configuration
interaction (CI)
The matrix representation to solve the problem is:
HA = EA
It means the evaluation and diagonalization of the HIJ
matrix elements in:
 H 00 H 01 ... H 0 N 


 H 10 H 11 ... H 1N 
H
.
. ...
. 


. ... H NN 
 HN0
It will output:
– asI eigenvectors for the A transformation matrix
– Es eigenvalues of relative energies corresponding to all s
states, resulting in the E diagonal matrix.
The method of configuration
interaction (CI)
The general formulation for HIJ matrix elements is*:
H IJ   I* t 1t 2 ...t N Hˆ J t 1t 2 ...t N dt
 I* t 1t 2 ...t N  Hˆ J t 1t 2 ...t N 
Depending on the energy of interaction among all and each
one of the I and J configurations, taken by pairs.
* Nesbet, R. K., Configuration interaction in orbital theories. Proc. Roy. Soc. (London) A 1955, 230 (1182 ), 312-321.
The method of configuration
interaction (CI)
The general formulation for HIJ matrix elements is*:
H IJ   I* t 1t 2 ...t N Hˆ J t 1t 2 ...t N dt
 I* t 1t 2 ...t N  Hˆ J t 1t 2 ...t N 
Depending on the energy of interaction among all and each
one of the I and J configurations, taken by pairs.
It builds a symmetrical matrix.
IJ basis functions are Slater determinants or any
orthonormal set of multielectronic wave functions.
* Nesbet, R. K., Configuration interaction in orbital theories. Proc. Roy. Soc. (London) A 1955, 230 (1182 ), 312-321.
The method of configuration
interaction (CI)
H IJ   I* t 1t 2 ...t N Hˆ J t 1t 2 ...t N dt
 I* t 1t 2 ...t N  Hˆ J t 1t 2 ...t N 
From this general formulation, it must be noticed that:
 Each HIJ can exist for interactions between I and J
excited configurations independently of the number of
involved electrons.
The method of configuration
interaction (CI)
H IJ   I* t 1t 2 ...t N Hˆ J t 1t 2 ...t N dt
 I* t 1t 2 ...t N  Hˆ J t 1t 2 ...t N 
From this general formulation, it must be noticed that:
 Each HIJ can exist for interactions between I and J
excited configurations independently of the number of
involved electrons.
 When the basis to build a configuration is orthonormal
interactions between terms differing in more than two
monoelectronic orbitals vanish.
The method of configuration
interaction (CI)
H IJ   I* t 1t 2 ...t N Hˆ J t 1t 2 ...t N dt
 I* t 1t 2 ...t N  Hˆ J t 1t 2 ...t N 
From this general formulation, it must be noticed that:
 Each HIJ can exist for interactions between I and J
excited configurations independently of the number of
involved electrons.
 When the basis to build a configuration is orthonormal
interactions between terms differing in more than two
monoelectronic orbitals vanish.
 When I = J then II means the energy EI of such basis
configuration.
Evaluation of CI matrix elements
Evaluation of HIJ elements of the CI’s H matrix remains as
the key problem, as occurs in all variational quantum
calculations.
Evaluation of CI matrix elements
Evaluation of HIJ elements of the CI’s H matrix remains as
the key problem, as occurs in all variational quantum
calculations.
It must be taken into account that basis functions are no
longer one electron wave functions (molecular or atomic
orbitals) but Hartree – Fock’s multielectronic
configurations after the previous optimization of the initial
occupation, that usually is the ground state:
F s   asI I
I
Evaluation of CI matrix elements
HIJ terms depend on projections of a given I configuration
or multielectronic state on another one denoted as J. It
means an scalar product in the configuration space.
*
*


d
t


 I J
I J
Evaluation of CI matrix elements
HIJ terms depend on projections of a given I configuration
or multielectronic state on another one denoted as J. It
means an scalar product in the configuration space.
*
*


d
t


 I J
I J
As each configuration is expressed as a determinant of one
electron wave functions:
*
*
 I J dt  I J
 y  I 1 t 1  *y  I 2  t 2  * ... det y  J 1y  J 2  ...dt
 det y k t 1  *y l t 1 dt 1  dety k t 1  * y l t 1    det k l 
 DIJ
where orbitals of I determinant are numbered by k sub
indexes and those of J by l’s.
Evaluation of CI matrix elements
Such determinant can be developed in terms of minors of p
order, according the “p” one electron functions differencing
I and J configurations.
DIJ k | l 
DIJ k1k2 | l1l2 
DIJ k1k2 ...k p | l1l2 ...l p 
For single excitations
For double excitations
For any “p” order excitation
Evaluation of CI matrix elements
As one electron basis functions (spin orbitals) building
Slater determinants are orthonormal, the sole none
vanishing elements of DIJ determinant are those where k
and l are the same, although appearing in two separate
configurations.
Evaluation of CI matrix elements
As one electron basis functions (spin orbitals) building
Slater determinants are orthonormal, the sole none
vanishing elements of DIJ determinant are those where k
and l are the same, although appearing in two separate
configurations.
If spin orbital ordering of both I and J determinants are
the same, non vanishing terms will appear in the diagonal
of DIJ. Any occupation difference between them will also
brought vanishing diagonal terms when l ≠ k.
Evaluation of CI matrix elements
As one electron basis functions (spin orbitals) building
Slater determinants are orthonormal, the sole none
vanishing elements of DIJ determinant are those where k
and l are the same, although appearing in two separate
configurations.
If spin orbital ordering of both I and J determinants are
the same, non vanishing terms will appear in the diagonal
of DIJ. Any occupation difference between them will also
brought vanishing diagonal terms when l ≠ k.
Therefore, for all determinant minors in general:
DIJ k1k2 ...k p | l1l2 ...l p    IJ
Evaluation of CI matrix elements
Thus, the CI matrix element is generally expressed as:
H IJ   I* t 1t 2 ...t N Hˆ J t 1t 2 ...t N dt
 I* t 1t 2 ...t N  Hˆ J t 1t 2 ...t N 
 H DIJ    y k t 1  * Hˆ 1 y l t 1  DIJ k | l 
(0)
N
N
k
l
1 N N
   y k t 1  *y k t 2  * Hˆ 1, 2 y l t 1 y l t 2  DIJ k1k2 | l1l2 
2! k k l l
1 N N
   y k t 1  *y k t 2  *y k t 3  * Hˆ 1, 2 y l t 1 y l t 2 y k t 3  DIJ k1k2 ...k p | l1l2 ...l p 
3! k k k l l l
 ...
1
2
1
2
1 2 12
1
1 2 3 123
2
3
1
2
3
Evaluation of CI matrix elements
And in the case of orthonormal spin orbitals DIJ’s become
Kroneker’s deltas:
H IJ   I* t 1t 2 ...t N Hˆ J t 1t 2 ...t N dt
 I* t 1t 2 ...t N  Hˆ J t 1t 2 ...t N 
 H DIJ    y k t 1  * Hˆ 1 y l t 1   kl
(0)
N
N
k
l
1 N N
   y k t 1  *y k t 2  * Hˆ 1, 2 y l t 1 y l t 2   k l  k l
2! k k l l
1 N N
   y k t 1  *y k t 2  *y k t 3  * Hˆ 1, 2 y l t 1 y l t 2 y k t 3   k l  k l  k l
3! k k k l l l
 ...
1
2
1
2
11
2 2
1 2 1 2
1
2
3
1 2 3 1 2 3
cancelling most terms of all sums.
1
2
3
11
2 2
3 3
Evaluation of CI matrix elements
If in the case of interactions between configurations where
i and j are initially filled spin orbitals transferring electrons
to certain a and b initially empty spin orbitals, the
calculation of matrix elements only involve the populated
and unpopulated spin orbitals by:
H IJia  y i t 1  Hˆ 1 y a t 1 

  y i t 1 y j t 2  y a t 1 y j t 2   y i t 1 y j t 2  y j t 1 y a t 2 
j
in the case of single excitations and
H IJia , jb  y i t 1 y j t 2  y a t 1 y b t 2   y i t 1 y j t 2  y b t 1 y a t 2 
for double excitations. Similar terms can be developed for
triples and quadruples.

Evaluation of CI matrix elements
Single electron excitations from the ground state can be
expressed as:
H 0iaI  y i t 1  hˆ y a t 1 

  y i t 1 y j t 2  y a t 1 y j t 2   y i t 1 y j t 2  y j t 1 y a t 2 
j
 y i t 1  Fˆ y a t 1 
leading to the matrix element between two spin orbitals
of the same monoelectronic solution.

Evaluation of CI matrix elements
Being both spin orbitals eigenfunctions of the Fock
operator:
Fˆy a t 1    ay a t 1 
y i t 1  Fˆ y a t 1    a y i t 1  y a t 1 
  a ia
because all yi and ya orbitals are orthonormal by definition.
It means that:
H 0iaI  0
Evaluation of CI matrix elements
Being both spin orbitals eigenfunctions of the Fock
operator:
Fˆy a t 1    ay a t 1 
y i t 1  Fˆ y a t 1    a y i t 1  y a t 1 
  a ia
because all yi and ya orbitals are orthonormal by definition.
It means that:
H 0iaI  0
It is known as the Brillouin’s theorem that establish that the
ground state do not intervene in optimization of singly excited
states.
Evaluation of CI matrix elements
As single excited configurations do not participate in
ground state energy optimizations by CI procedures, such
singly excited determinants only serve for energy
optimizations of excited states.
Evaluation of CI matrix elements
As single excited configurations do not participate in
ground state energy optimizations by CI procedures, such
singly excited determinants only serve for energy
optimizations of excited states.
Double and quadruple excited configurations are important to
optimize energy of the ground state by accounting electron
correlation.
Evaluation of CI matrix elements
In the case of singly excited configuration interaction we
arrived to:
H IJia  y i t 1  Hˆ 1 y a t 1 

  y i t 1 y j t 2  y a t 1 y j t 2   y i t 1 y j t 2  y j t 1 y a t 2 
j
Here, the term:
Eia  y i t 1  Hˆ 1 y a t 1 
is the associated transition energy of an electron with space
and spin coordinates t1 in spin orbital yi changing to occupy
the formerly empty ya being defined by certain I singly
excited configuration (as expressed by the corresponding
Slater determinant).

Evaluation of CI matrix elements
In the case of singly excited configuration interaction we
arrived to:
H IJia  y i t 1  Hˆ 1 y a t 1 

  y i t 1 y j t 2  y a t 1 y j t 2   y i t 1 y j t 2  y j t 1 y a t 2 
j
Here, the term:
Eia  y i t 1  Hˆ 1 y a t 1 
is the associated transition energy of an electron with space
and spin coordinates t1 in spin orbital yi changing to occupy
the formerly empty ya being defined by certain I singly
excited configuration (as expressed by the corresponding
Slater determinant).
If we re dealing with Hartree – Fock spin orbitals:
Eia  ea  ei

Evaluation of CI matrix elements
On the other hand, for the two electron terms in:
H IJia  y i t 1  Hˆ 1 y a t 1 

  y i t 1 y j t 2  y a t 1 y j t 2   y i t 1 y j t 2  y j t 1 y a t 2 

j
we have the general expression:
y i t 1 y j t 2  y a t 1 y b t 2    y i t 1 y j t 2  r1 y a t 1 y b t 2  y b t 1 y a t 2 dt 1dt 2
12
as the corresponding general matrix element expressing the
interaction of electrons 1 and 2 that occupied yi and yj as
changing to their new occupation in ya and yb spin orbitals.
Evaluation of CI matrix elements
By taking into account previous considerations, after some
algebraic deductions*, the practical evaluation of CI singly
excited matrix becomes from the following for diagonal
elements:
1
H II 1H IIia  ea  ei  y i t 1 y a t 2  r1 y i t 1 y a t 2   2 y i t 1 y a t 2  r1 y a t 1 y i t 2 
3
H II 3H IIia  ea  ei  y i t 1 y a t 2  r1 y i t 1 y a t 2 
12
12
12
HI0  0
and for off diagonal elements (being i  j and a  b) become:
1
H IJ 1H IJia , jb  2 y j t 1 y a t 2  r1 y b t 1 y i t 2   y j t 1 y a t 2  r1 y i t 1 y b t 2 
12
3
12
H IJ 3H IJia , jb   y j t 1 y a t 2  r1 y i t 1 y b t 2 
12
* Pople, J. A., The electronic spectra of aromatic molecules. II. A theoretical treatment of excited states of alternant
hydrocarbon molecules based on self-consistent molecular orbitals. Proc. Phys. Soc. London 1955, 68A, 81-9.
Some CI results
H2O*
A standard contracted Gaussian double z basis set of Dunning
– Huzinaga was used with a scale factor of 1.2 for H.
Geometry was previously optimized at the same level.
* Saxe, P.; Schaeffer III, H. F.; Handy, N. C., Exact solution (within a double-zeta basis set) of the Schrödinger
electronic equation for water. Chem. Phys. Lett. 1981, 79 (2), 202-204.
Some CI results
H2 equilibrium bond distance (Bohr’s)*
Atomic basis set
STO-3G
4-31G
6-31G**
“Exact” value**
SCF
1.346
1.380
1.385
“Full CI”
1.389
1.410
1.396
1.401
* Szabo, A.; Ostlund, N. S., Modern quantum chemistry: introduction to advanced electronic structure theory. First
edition, revised ed.; McGraw-Hill: New York, 1989; p 466.
** Kolos, W.; Wolniewicz, L., Improved Theoretical Ground-State Energy of the Hydrogen Molecule. J. Chem. Phys.
1968, 49 (1), 404-410 after an extensive quantum calulation over all possible variables of the molecule where
vibrational energy remain only 0.9 cm-1 above experimental value.
Some CI results
H2 equilibrium bond distance (Bohr’s)*
* Szabo, A.; Ostlund, N. S., Modern quantum chemistry: introduction to advanced electronic structure theory. First
edition, revised ; McGraw-Hill: New York, 1989; p 466.
Some CI results
CH2O (formaldehyde)
Total energy
(Hartrees)
rCO (Å)
rCH (Å)
<HCH
 (debye)
Relative calc.
time
STO-6G
6-31G
6-311++G**
CID|
6-311++G**
CISD|
6-311++G**
-113.4408
-113.8084
-113.9029
-114.2245
-114.2279
1.216
1.098
114.8
1.596
1
1.210
1.082
116.6
3.304
0.8
1.180
1.094
116.0
2.806
1.4
1.197
1.101
116.1
2.893
10.9
1.199
1.102
116.0
2.905
12.6
Exp.
1.210
1.102
121.1
2.33
Single excitations and light
Single excitation configuration interactions (CIS) are mostly
used to optimize excited state energies for simulating UV
spectra of molecules.
Single excitations and light
Single excitation configuration interactions (CIS) are mostly
used to optimize excited state energies for simulating UV
spectra of molecules.
Therefore, it allows applications to spectroscopy,
photochemistry and photonics, the science for generating,
controlling and detecting photons.
Single excitations and light
An electronic state of a given nanoscopic system can be
understood as characterized by a corresponding electron
density map.
Porphirin ground state charge distribution as calculated at the CNDOL/2CC||MP2|431G(d,p) level. Reds are negative and blues positive charges.
Single excitations and light
The ground state shows the most stable charge cloud
distribution while excited states are different, by using
excitation energy to stabilize it and returning to the original
ground state density map upon deactivation.
h
Acrolein excitation from the S0 (ground state) to S1 (first excited singlet state) as calculated at the
CNDOL/2CC||MP2|4-31G(d,p) level. Reds are negative and blues positive charges.
Multiconfigurational Hartree –
Fock procedures (MC-SCF)
Are there certain cases where small energy variations
decide a nanoscopic procedure, as those with:
• Biradicals
• Non saturated transition metals
• Excited states perturbing ground states
• Bond breakings
• Transition states in swinging transitions
Multiconfigurational Hartree –
Fock procedures (MC-SCF)
Are there certain cases where small energy variations
decide a nanoscopic procedure, as those with:
• Biradicals
• Non saturated transition metals
• Excited states perturbing ground states
• Bond breakings
• Transition states in swinging transitions
It could be useful that interacting configurations could also
be included for Hartree – Fock wave function optimizations.
Multiconfigurational Hartree –
Fock procedures (MC-SCF)
Then, a more or less limited number of configurations can
be chosen to be treated by independent Hartree – Fock ‘s
iterative procedures, representing the active space (AS), to
optimize both CI and one electron wave function
coefficients.
Multiconfigurational Hartree –
Fock procedures (MC-SCF)
Then, a more or less limited number of configurations can
be chosen to be treated by independent Hartree – Fock ‘s
iterative procedures, representing the active space (AS), to
optimize both CI and one electron wave function
coefficients.
This procedure is called as a multiconfigurational self
consistent field (MC-SCF) routine, meaning that:
F MCSCF   AK K
K
is the multiconfigurational wave function and AK are the
optimizing coefficients on the basis of previously optimized
K Hartree – Fock’s configurations.
Multiconfigurational Hartree –
Fock procedures (MC-SCF)
Each optimized K , or configurational state function (CSF),
can be:
• A unique Slater determinant with a given and convenient
electron occupation of spin orbitals.
• A combination of Slater determinants to correct spin
influences.
Multiconfigurational Hartree –
Fock procedures (MC-SCF)
CSF’s can be expressed by orbital occupations of the initial
ground state Hartree – Fock solution and their
corresponding Slater determinants:
K  ( N !)
L  ( N !)
y 1 (t 1 )... y d (t 1 ) ... y N (t 1 )
y 1 (t 2 )... y d (t 2 ) ... y N (t 2 )
1
.
.
...
.
y 1 (t N )... y d (t N ) ... y N (t N )
y 1 (t 1 )... y f (t 1 ) ... y N (t 1 )
y 1 (t 2 )... y f (t 2 ) ... y N (t 2 )
1
.
.
...
.
y 1 (t N )... y f (t N ) ... y N (t N )
;
;...
Multiconfigurational Hartree –
Fock procedures (MC-SCF)
Each one electron spin orbital of a given K configuration is
defined as the LCAO of Hartree – Fock’s method:
y i K  (t 1 )   c Ki    (t 1 )
i
Multiconfigurational Hartree –
Fock procedures (MC-SCF)
Each one electron spin orbital of a given K configuration is
defined as the LCAO of Hartree – Fock’s method:
y i K  (t 1 )   c Ki    (t 1 )
i
At the end, both coefficient sets of A CI’s and CK Hartree –
Fock’s coefficient matrix for the chosen K configurations
are involved in a MC-SCF optimization. They are optimized
iteratively in cycles until a desired convergence limit.
Multiconfigurational Hartree –
Fock procedures (MC-SCF)
Each one electron spin orbital of a given K configuration is
defined as the LCAO of Hartree – Fock’s method:
y i K  (t 1 )   c Ki    (t 1 )
i
At the end, both coefficient sets of A CI’s and CK Hartree –
Fock’s coefficient matrix for the chosen K configurations
are involved in a MC-SCF optimization. They are optimized
iteratively in cycles until a desired convergence limit.
MC-SCF procedures are particularly demanding of
computational resources.
The particular case of CAS-SCF
The complete active space self-consistent field (CAS-SCF)
procedure was developed* to select an active space given
by a certain NA number of electrons to participate in the
variational optimization and a given M number of spin
orbitals of convenience, by following certain rules.
*Roos, B. O.; Taylor, P. R.; Siegbahn, P. E. M., A complete active space SCF method (CASSCF)
using a density matrix formulated super-CI approach. Chem. Phys. 1980, 48 (2), 157-173.
.
The particular case of CAS-SCF
The total of participating orbitals are divided in three sets:
• Inactive orbitals always taken as double occupied with a
total of N – NA electrons, where N is the total number of
system’s electrons. They are all spin orbitals below the last
reaching up to complete N – NA electrons.
The particular case of CAS-SCF
The total of participating orbitals are divided in three sets:
• Inactive orbitals always taken as double occupied with a
total of N – NA electrons, where N is the total number of
system’s electrons. They are all spin orbitals below the last
reaching up to complete N – NA electrons.
• Active orbitals, giving the active space, consisting in all
configurations where the chosen NA electrons can
participate in M spin orbitals above the inactive space. It
is recommended that the selected M orbitals were those more
involved in the modeled phenomenon (i.e., p orbitals near to
HOMO). Their average occupation will be a real number, between 0
and 2.
The particular case of CAS-SCF
The total of participating orbitals are divided in three sets:
• Inactive orbitals always taken as double occupied with a
total of N – NA electrons, where N is the total number of
system’s electrons. They are all spin orbitals below the last
reaching up to complete N – NA electrons.
• Active orbitals, giving the active space, consisting in all
configurations where the chosen NA electrons can
participate in M spin orbitals above the inactive space. It
is recommended that the selected M orbitals were those more
involved in the modeled phenomenon (i.e., p orbitals near to
HOMO). Their average occupation will be a real number, between 0
and 2.
• Virtual orbitals, always empty of electrons. They will be all
those remaining above the active orbitals to reach the total number
of spin orbitals of the system.
The particular case of CAS-SCF
Graphical representation:
← Virtual orbitals
← Active orbitals: NA
electrons and a total of M
orbitals.
← Inactive orbitals
inactivos
After selecting the orbitals to consider, CI is only performed on the
active space, and the rest of the system is treated at a normal
Hartree – Fock level.
The particular case of CAS-SCF
The example of formaldehyde:
Total energy
(Hartrees)
rCO (Å)
rCH (Å)
<HCH
 (debye)
Relative calc.
time
CASSCF(6,8)|
6-311++G
-114.0260
Exp.
-113.9029
CISD|
6-311++G**
-114.2279
1.180
1.094
116.0
2.806
1.4
1.199
1.102
116.0
2.905
12.6
1.212
1.094
116.1
1.005
313.9
1.210
1.102
121.1
2.33
STO-6G
6-311++G**
-113.4408
1.216
1.098
114.8
1.596
1
CASSCF(m,n): n = number of electrons in active space; m =
number of spin orbitals in active space.