Chemistry Reference
In-Depth Information
is available for Windows and Linux platforms. Some programs also calculate non-
linear functions of the descriptors (e.g., ratios, products, parabolic, exponential, loga-
rithmic functions). Comparative molecular field analysis (CoMFA) provides 3D grid
descriptors (Cramer et al. 1988). Descriptors are calculated according to certain char-
acteristics of the atoms (e.g., Cartesian coordinates, atomic number, mass, volume,
or net charge) and chemical bonds (e.g., length, bond order, or type of bound atoms).
4.3.1 B
ond
o
rders
The sum of the kinetic energy and the potential energy for a specific system of
atomic nuclei and electrons (such as an organometallic complex) can be estimated.
H = p
2
/2m + V(r) = - (ħ
2
/2m)∇
2
+ V(r)
(4.1)
The equation of vectors and eigenvalues for energy is:
[- (ħ
2
/2m)∇
2
+ V(r)]
|n
〉 = E
n
|n
〉
(4.2)
where
|n
〉 are eigenvectors corresponding to the E
n
eigenvalues.
Equation (4.2), called the
static Schrödinger equation
(Schrödinger 1926), is a
special case of the equation of vectors and eigenvalues, and is written in a particu-
lar representation (the representation of position). The E
n
energies are solutions of
Equation (4.2) and reflect
possible
energies of the system. On a case-by-case basis,
each system is characterized by a different form of potential energy V(r).
Equation (4.2) provides analytical solutions for mono-electronic systems only.
Such solutions have been calculated for the hydrogen atom and for H
2
+
, He
+
, and Li
+2
ions. For common organic species comprising many atomic nuclei and electrons, the
expression of the potential energy is complicated because the electrostatic repulsion
between nuclei, the electrostatic repulsion between electrons, and the electrostatic
attraction between nuclei and electrons should be considered. Solving Equation (4.2)
is very difficult for many common organic species. For systems including many
electrons, the solutions of Equation (4.2) are obtained by the
variational method.
According to this method, every molecular orbital |m〉 is considered a linear combi-
nation of atomic orbitals |φ
i
〉.
|m〉 = Σ c
i
|φ
i
〉
(4.3)
The c
i
coefficients must have values that give a minimum value to the energy E
m
of that specific molecular orbital. In order to have other solutions besides the com-
mon (ordinary) solution, c
1
= c
2
= c
3
= ... = 0, these must be solutions of the system of
Equation (4.4).
Σ c
mi
· (F
ij
− E
m
· S
ij
) = 0
(4.4)
and E
m
is a solution of Equation (4.5)
|F
ij
− E
ij
· S
ij
| = 0
(4.5)
