Determination the potential function of the diatomic moleules by the applying the Hartree-Fock-Roothan methods

2.  The total electron energy has calculated on
various values of the internuclear distances
by the quantum mechanical calculations.
Then table of the total electron energy is
made for distance. It was found potential
function of diatomic molecules by applying
the least squares method to table.
 In references, it was found the potential
function of diatomic molecule as following:

3.  Least squares problems appear very naturally when one
would like to estimate values of parameters of a mathematical
model from measured data, which are subject to errors. The
method of the Least Squares has for it mechanics adjustment
and comparison of observations. . It is hold to condition on
deviation that, experimental curve at result enough to
describe givens more exactly. At first sight, the condition
smaller sum of the all the deviations is suitable.
Although, concrete deviation is different than zero:

4.  We can’t get differention for the minimum
points though the sum of the absolute
values are differ than zero. So, therefore,
it is best to request squared sum of the
deviations to be minimal in this case.
Formula can be differentiated and hereby one can
determine the functional dependence. In this case
,finding the functional dependence by this way is called
the least squares method.

6.  The total electronic energy of
the molecule was calculated at
different values on the
internuclear distance.
Calculations were made by
Hartree-Fock-Roothan method .
Slater functions are used
Instead of atomic orbitals during
calculations . These
calculations are based on the
computer programs which is
constituted by the employees of
the department “Chemical
Physics of Nanomaterials”.
Hartree-Fock-Roothan methods
were applied to these
calculations . Value of the
internuclear distance are
chosen by the 0.05 atomic unit
step around the equilibrium
distance.

8. Our matrix is as below : We will find Δ for all
matrices according
to Cramer’s rule.
Afterthem, by
replacing the
methods we will
find the respectively
for the given
matrices. We will
calculate the ratio of
these to Δ. Let’s
apply our method to
thegiven
determinant

9. WAS WRITTEN DOWN THE FIRST MAIN MATRIX AND
WAS FOUND ITS DETERMINANT.

10.  Was replaced the 1st column of the main matrix with the
solution vector and was found its first determinant:

11.  Was replaced the 2nd column of the main matrix
with the solution vector and was found its
determinant:

12.  Was replaced 3rd column of the main matrix with the
solution vector and was found its determinan:

13.  REPLACE 4TH COLUMN OF THE MAIN MATRIX WITH
THE SOLUTION VECTOR AND FIND ITS DETERMINANT:

14. The Cramer’s method
was applied for the
solution of the Linear
equaions. By the applying
specific steps for the
calculations we get the
unknown constant values.
Then replace the a,b,c,d
unknown constants into
the given formula we get
main expression of the
total electron energy
A=-15.30131526286984408
B= 0.13254561246770884
C=-0.13254561246770884
D=-0.000011272447659941127

16. THE CRAMER’S METHOD WAS APPLIED TO THE
SOLUTION OF THE LINEAR EQUATIONS. BY THE
APPLYING SPECIFIC STEPS TO THE CALCULATIONS
WE GET THE UNKNOWN CONSTANT VALUES. THEN
REPLACE THE A,B,C,D UNKNOWN CONSTANTS INTO
THE GIVEN FORMULA WE GET MAIN EXPRESSION OF
THE ELECTRON ENERGY. AS A RESULT OF
CALCULATIONS, THE FOLLOWING DEPENDENCY
HAVE BEEN TAKEN:

17. -15.131
-15.1305
-15.13
-15.1295
-15.129
-15.1285
-15.128
-15.1275
-15.127
-15.1265
-15.126
2.4 2.6 2.8 3
E(a.u)

18. FREQUENCY OF THE
HARMONIC OSCILLATIONS.
ROTATION STATE.
DISSOCIATION ENERGY.

19.  1. The Schrodinger’s equation for molecules was explained and
the difficulties in its solution were investigated.
 2. The dependence of the potential function on the internuclear
distance of diatomic molecules has been studied.
 3. The least squares method has been applied.
 4. The total electron energy of molecule has been calculated on
dfferent values of internuclear distances by quantum mechanical
calculation and table of the dependence of electron energy on the
internuclear distance is made. Unknown constans of the
expression of the potential function was calculated by the
applying the least squares method. The dependence graph of the
potential function on the internuclear distance of diatomic
molecules has been made.