Method and system of computing and rendering the nature of the chemical bond of hydrogen-type molecules and molecular ions
Provided is a system of computing and rendering a nature of a chemical bond based on physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions. The system includes a processor for processing Maxwellian equations representing charge, mass, and current density functions of hydrogen-type molecules and molecular ions and an output device in communication with the processor for displaying the nature of the chemical bond.
1. A system of computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, said system comprising:
processing means for processing Maxwellian equations representing charge, mass, and current density functions of hydrogen-type molecules and molecular ions, and;
an output device in communication with the processing means for displaying the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, wherein the physical, Maxwellian solutions of the charge, mass, and current density functions of hydrogen-type molecules and molecular ions comprises a solution of the classical wave equation
[
∇
2
-
1
v
2
∂
2
∂
t
2
]
ρ
(
r
,
θ
,
ϕ
,
t
)
=
0.
2. A system of computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, said system comprising:
processing means for processing Maxwellan equations representing charge, mass, and current density functions of hydrogen-type molecules and molecular ions, and;
an output device in communication with the processing means for displaying the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, wherein the hydrogen molecular ion charge and current density functions, bond distance, and energies are solved from the Laplacian in ellipsoidal coordinates:
(
η
-
ζ
)
R
ξ
∂
∂
ξ
(
R
ξ
∂
ϕ
∂
ξ
)
+
(
ζ
-
ξ
)
R
η
∂
∂
η
(
R
η
∂
ϕ
∂
η
)
+
(
ξ
-
η
)
R
ζ
∂
∂
ζ
(
R
ζ
∂
ϕ
∂
ζ
)
=
0.
with the constraint of nonradiation.
3. A system of computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydroqen-type molecules and molecular ions, said system comprising:
processing means for processing Maxwellian equations representing charge, mass, and current density functions of hydrogen-type molecules and molecular ions, and;
an output device in communication with the processing means for displaying the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, wherein the hydrogen molecular charge and current density functions, bond distance, and energies are solved from the Laplacian in ellipsoidal coordinates:
(
η
-
ζ
)
R
ξ
∂
∂
ξ
(
R
ξ
∂
ϕ
∂
ξ
)
+
(
ζ
-
ξ
)
R
η
∂
∂
η
(
R
η
∂
ϕ
∂
η
)
+
(
ξ
-
η
)
R
ζ
∂
∂
ζ
(
R
ζ
∂
ϕ
∂
ζ
)
=
0
with the constraint of nonradiation.
4. A system of computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, said system comprising:
processing means for processing Maxwellian equations representing charge, mass, and current density functions of hydrogen-type molecules and molecular ions, and;
an output device in communication with the processing means for displaying the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, wherein the hydrogen molecular ion charge and current density functions, bond distance, and energies are solved from the Laplacian in ellipsoidal coordinates:
(
η
-
ζ
)
R
ξ
∂
∂
ξ
(
R
ξ
∂
ϕ
∂
ξ
)
+
(
ζ
-
ξ
)
R
η
∂
∂
η
(
R
η
∂
ϕ
∂
η
)
+
(
ξ
-
η
)
R
ζ
∂
∂
ζ
(
R
ζ
∂
ϕ
∂
ζ
)
=
0
with the constraint of nonradiation.
5. A system of computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, said system comprising:
processing means for processing Maxwellian equations representing charge, mass, and current density functions of hydrogen-type molecules and molecular ions, and;
an output device in communication with the processing means for displaying the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, wherein the hydrogen molecular charge and current density functions, bond distance, and energies are solved from the Laplacian in ellipsoidal coordinates:
(
η
-
ζ
)
R
ξ
∂
∂
ξ
(
R
ξ
∂
ϕ
∂
ξ
)
+
(
ζ
-
ξ
)
R
η
∂
∂
η
(
R
η
∂
ϕ
∂
η
)
+
(
ξ
-
η
)
R
ζ
∂
∂
ζ
(
R
ζ
∂
ϕ
∂
ζ
)
=
0
with the constraint of nonradiation.
6. A system of computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, said system comprising:
processing means for processing Maxwellian equations representing charge, mass, and current density functions of hydrogen-type molecules and molecular ions, and;
an output device in communication with the processing means for displaying the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, wherein a vibrational energy of hydrogen molecular ion H 2 + (1/p) having a central field of +pe at each focus of the prolate spheroid molecular orbital is
E vib (1)= p 2 0.270 eV;
and the rotational energy absorbed by a hydrogen-type molecular ion with the transition from the state with the rotational quantum number J to one with the rotational quantum number J+1 is
Δ
E
=
E
J
+
1
-
E
J
=
p
2
ℏ
2
m
p
2
a
H
2
[
J
+
1
]
=
p
2
[
J
+
1
]
1.186
×
10
-
21
J
=
p
2
[
J
+
1
]
0.00740
eV
7. A system of computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, said system comprising:
processing means for processing Maxwellian equations representing charge, mass, and current density functions of hydrogen-type molecules and molecular ions, and;
an output device in communication with the processing means for displaying the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, wherein a vibrational energy of hydrogen-type molecule H 2 (1/p) having a central field of +pe at each focus of the prolate spheroid molecular orbital is
E vib (1)= p 2 0.517 eV;
and the rotational energy absorbed by a hydrogen-type molecule with the transition from the state with the rotational quantum number J to one with the rotational quantum number J+1 is
Δ
E
=
E
J
+
1
-
E
J
=
p
2
h
2
m
p
a
o
2
[
J
+
1
]
=
p
2
[
J
+
1
]
2.37
×
10
-
21
J
=
p
2
[
J
+
1
]
0.0148
eV
8. A system of computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, said system comprising:
processing means for processing Maxwellian equations representing charge, mass, and current density functions of hydrogen-type molecules and molecular ions, and;
an output device in communication with the processing means for displaying the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, wherein a chemical shift
Δ
B
B
of H 2 relative to a bare proton is given by
Δ
B
B
=
Δ
B
B
=
-
μ
0
(
p
3
4
a
0
3
-
p
3
2
a
0
3
ln
2
+
1
2
-
1
)
a
0
3
ⅇ
2
36
p
2
m
e
=
-
μ
0
(
4
-
2
ln
2
+
1
2
-
1
)
pⅇ
2
36
a
0
m
e
=
-
p28
.01
ppm
where p=1 for H 2 .
9. A system of computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hydrogen-type molecules and molecular ions, said system comprising:
processing means for processing Maxwellian equations representing charge, mass, and current density functions of hydrogen-type molecules and molecular ions, and;
an output device in communication with the processing means for displaying the nature, of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of hvdroaen-type molecules and molecular ions, wherein a total NMR shift
Δ
B
T
B
,
of the hydrogen molecule H 2 (1/p) having a central field of +pe at each focus of the prolate spheroid molecular orbital is given by the sum of that of H 2 plus a relativistic component:
Δ
B
T
B
=
-
μ
0
(
4
-
2
ln
2
+
1
2
-
1
)
ⅇ
2
36
a
0
m
e
(
1
+
πα
p
)
Δ
B
T
B
=
-
(
28.01
+
0.64
p
)
ppm
where p=integer>1.
10. The system of any one of claims 1 , 2 , 3 , 4 , 5 , and 6 – 9 , wherein the output device is a display that displays at least one of visual or graphical media.
11. The system of claim 10 wherein the display is at least one of static or dynamic.
12. The system of claim 11 wherein at least one of vibration and rotation is be displayed.
13. The system of any one of claims 1 , 2 , 3 , 4 , 5 , and 6 – 9 , wherein displayed information is used to model reactivity and physical properties.
14. The system of any one of claims 1 , 2 , 3 , 4 , 5 , and 6 – 9 , wherein the output device is a monitor, video projector, printer, or three-dimensional rendering device.
15. The system of any one of claims 1 , 2 , 3 , 4 , 5 , and 6 – 9 , wherein displayed information is used to model other molecules and provides utility to anticipate their reactivity and physical properties.
16. The system of any one of claims 1 , 2 , 3 , 4 , 5 , and 6 – 9 , wherein the processing means is a general purpose computer.
17. The system of claim 16 wherein the general purpose computer comprises a central processing unit (CPU), one or more specialized processors, system memory, a mass storage device such as a magnetic disk, an optical disk, or other storage device, an input means.
18. The system of claim 17 , wherein the input means comprises a serial port, usb port, microphone input, camera input, keyboard or mouse.
19. The system of any one of claims 1 , 2 , 3 , 4 , 5 , and 6 – 9 , wherein the processing means comprises a special purpose computer or other hardware system.
20. The system of any one of claims 1 , 2 , 3 , 4 , 5 , and 6 – 9 , further comprising computer program products.
21. The system of claim 20 comprising computer readable medium having embodied therein program code means.
22. The system of claim 21 wherein the computer readable media is any available media which can be accessed by a general purpose or special purpose computer.
23. The system of claim 22 wherein the computer readable media comprises at least one of RAM, ROM, EPROM, CD ROM, DVD or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which can embody the desired program code means and which can be accessed by a general purpose or special purpose computer.
24. The system of claim 23 wherein the program code means comprises executable instructions and data which cause a general purpose computer or special purpose computer to perform a certain function of a group of functions.
25. The system of claim 24 wherein the program code is Mathematica programmed with an algorithm based on the physical solutions, and the computer is a PC.
26. The system of claim 25 wherein the algorithm is ParametricPlot3D[{2*Sqrt[1−z*z]*Cos[u],Sqrt[(1−z*z)]*Sin [u],z}{u,0,2*Pi},{z,−1,9999}], and the rendering is viewed from different perspectives.
27. The system of claim 26 wherein the algorithms for viewing from different perspectives comprises Show[Out[1], ViewPoint- 22 {x,y,z}] where x, y, and z are Cartesian coordinates.
28. The system of claim 1 wherein the boundary constraint of the wave equation solution is nonradiation according to Maxwell's equations.
29. The system of claim 28 wherein a boundary condition is met for an ellipsoidal-time harmonic function when
ω
n
=
π
ℏ
m
e
A
=
ℏ
m
e
ab
where the area of an ellipse is
A=πab
where 2b is the length of the semiminor axis and 2a is the length of the semimajor axis.
30. The system of claim 2 wherein a force balance equation for the hydrogen molecular ion is
ℏ
2
m
e
a
2
b
2
2
ab
2
X
=
e
2
4
πɛ
o
X
where
X
=
1
ξ
+
a
2
1
ξ
+
b
2
1
c
ξ
2
-
1
ξ
2
-
η
2
.
31. The system of claim 30 wherein the force balance equation has the parametric solution
r ( t )= a cos ω t+jb sin ω t
when the semimajor axis, a, is
a=2a o .
32. The system of claim 31 wherein an internuclear distance, 2c′, which is the distance between the foci is
2c′=2a o ;
the semiminor axis is
b √{square root over (3)} a o , and
the eccentricity, e, is
e
=
1
2
.
33. The system of claim 32 wherein a potential energy of the electron in the central field of the protons at the foci is
V
e
=
-
4
e
2
8
π
ɛ
0
a
2
-
b
2
ln
a
+
a
2
-
b
2
a
-
a
2
-
b
2
=
-
59.7575
eV
;
The potential energy of the two protons is
V
p
=
e
2
8
π
ɛ
0
a
H
=
13.5984
eV
,
and
The kinetic energy of the electron is
T
=
2
ℏ
2
m
e
a
a
2
-
b
2
ln
a
+
a
2
-
b
2
a
-
a
2
-
b
2
=
29.8787
eV
.
34. The system of claim 33 wherein during bond formation, the electron undergoes a reentrant oscillatory orbit with vibration of the protons, and the corresponding energy Ē osc is the difference between the Doppler and average vibrational kinetic energies:
E
_
osc
=
E
_
D
+
E
_
Kvib
=
(
V
o
+
T
+
V
p
)
2
E
_
K
Mc
2
+
1
2
h
k
μ
,
and
The total energy is
E
T
=
V
e
+
T
+
V
p
+
E
_
osc
E
T
=
-
{
e
2
8
π
e
o
a
H
(
4
ln
3
-
1
-
2
ln
3
)
[
1
+
2
ℏ
2
e
2
4
pe
o
(
2
a
H
)
3
m
e
m
e
c
2
]
-
1
2
ℏ
k
m
}
=
-
16.2803
eV
-
0.118811
eV
+
1
2
(
0.29282
eV
)
=
-
16.2527
eV
35. The system of claim 34 wherein a bond dissociation energy, E D , is the difference between the total energy of the hydrogen atom and E T :
E D =E ( H )− E T =2.654 eV
wherein the total energy of a hydrogen atom is
E ( H )=−13.59844 eV.
36. The system of claim 3 wherein a force balance equation for the hydrogen molecule is
ℏ
2
m
e
a
2
b
2
2
ab
2
X
=
e
2
4
π
ɛ
o
X
+
ℏ
2
2
m
e
a
2
b
2
2
ab
2
X
where
X
=
1
ξ
+
a
2
1
ξ
+
b
2
1
c
ξ
2
-
1
ξ
2
-
η
2
.
37. The system of claim 36 wherein the force balance equation has the parametric solution
r ( t )= ia cos ω t+jb sin ω t
when the semimajor axis, a, is
a=a o .
38. The system of claim 37 wherein an internuclear distance, 2c′, which is the distance between the foci is
2 c′= √{square root over (2)} a o ;
the semiminor axis is
b
=
1
2
a
o
,
and
the eccentricity, e, is
e
=
1
2
.
39. The system of claim 38 wherein a potential energy of the two electrons in the central field of the protons at the foci is
V
e
=
-
2
e
2
8
π
ɛ
o
a
2
-
b
2
ln
a
+
a
2
-
b
2
a
-
a
2
-
b
2
=
-
67.836
e
V
;
The potential energy of the two protons is
V
p
=
e
2
8
π
ɛ
o
a
2
-
b
2
=
19.242
e
V
;
The kinetic energy of the electrons is
T
=
ℏ
2
2
m
e
a
a
2
-
b
2
ln
a
+
a
2
-
b
2
a
-
a
2
-
b
2
=
33.918
e
V
,
and
The energy, V m , of the magnetic force between the electrons is
V
m
=
ℏ
2
4
m
e
a
a
2
-
b
2
ln
a
+
a
2
-
b
2
a
-
a
2
-
b
2
=
-
16.959
e
V
.
40. The system of claim 39 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the protons, and the corresponding energy Ē osc is the difference between the Doppler and average vibrational kinetic energies:
E
_
osc
=
E
_
D
+
E
_
Kvib
=
(
V
e
+
T
+
V
m
+
V
p
)
2
E
_
K
Mc
2
+
1
2
ℏ
k
μ
,
and
The total energy is
E
T
=
V
e
+
T
+
V
m
+
V
p
+
E
_
osc
E
T
=
-
e
2
8
πɛ
o
a
0
[
(
2
2
-
2
+
2
2
)
ln
2
+
1
2
-
1
-
2
]
[
1
+
2
ℏ
e
2
4
πɛ
o
a
0
3
m
e
m
e
c
2
]
.
-
1
2
ℏ
k
μ
=
-
31.689
eV
41. The system of claim 40 wherein a bond dissociation energy, E D , is the difference between the total energy of the hydrogen atoms and E T :
E D =E (2 H[a H ])− E T =4.478 eV
wherein the total energy of two hydrogen atoms is
E (2 H[a H ])=−1.27.21 eV.
42. The system of claim 4 wherein a force balance equation the hydrogen molecular ion H 2 + (1/p) having a central field of +pe at each focus of the prolate spheroid molecular orbital is
ℏ
2
m
e
a
2
b
2
2
ab
2
X
=
p
e
2
4
πɛ
o
X
where
X
=
1
ξ
+
a
2
1
ξ
+
b
2
1
c
ξ
2
-
1
ξ
2
-
η
2
.
43. The system of claim 42 wherein a force balance equation has the parametric solution
r ( t )= ia cos ω t+jb sin ω t
when the semimajor axis, a, is
a
=
2
a
0
p
.
44. The system of claim 43 wherein an internuclear distance, 2c′, which is the distance between the foci is
2
c
′
=
2
a
o
p
;
the semiminor axis is
b
=
3
p
a
o
,
and
the eccentricity, e, is
e
=
1
2
.
45. The system of claim 44 wherein a potential energy of the electron in the central field of +pe at the foci is
V
e
=
-
4
p
2
e
2
8
πɛ
o
a
o
ln
3
;
The potential energy of the two protons is
V
p
=
p
2
e
2
8
π
ɛ
o
a
o
,
and
The kinetic energy of the electron is
T
=
2
p
2
e
2
8
π
ɛ
o
a
o
ln
3.
46. The system of claim 45 wherein during bond formation, the electron undergoes a reentrant oscillatory orbit with vibration of the protons, and the corresponding energy Ē osc is the difference between the Doppler and average vibrational kinetic energies:
E
_
osc
=
E
_
D
+
E
_
Kvib
=
(
V
c
+
T
+
V
P
)
2
E
_
K
Mc
2
+
1
2
h
k
μ
,
and
The total energy of the hydrogen molecular ion H 2 + (1/p) having a central field of +pe at each focus of the prolate spheroid molecular orbital is
E
T
=
V
e
+
T
+
V
P
+
E
_
osc
E
T
=
-
p
2
{
e
2
8
π
ɛ
o
a
H
(
4
ln
3
-
1
-
2
ln
3
)
[
1
+
p
2
h
2
e
2
4
π
ɛ
o
(
2
a
H
)
3
m
e
m
c
c
2
]
-
1
2
h
k
μ
}
.
=
-
p
2
16.13392
eV
-
p
3
0.118755
eV
47. The system of claim 46 wherein a bond dissociation energy, E D , is the difference between the total energy of the hydrogen atom H(1/p) and E T :
E
D
=
-
p
2
13.59844
-
E
T
=
-
p
2
13.59844
-
(
-
p
2
16.13392
eV
-
p
3
0.118755
eV
)
=
p
2
2.535
eV
+
p
3
0.118755
eV
wherein the total energy of a hydrogen atom H(1/p) is
E ( H (1 /p ))=− p 2 13.59844 eV.
48. The system of claim 5 wherein a force balance equation for the hydrogen molecule H 2 (1/p) having a central field of +pe at each focus of the prolate spheroid molecular orbital is
h
2
m
e
a
2
b
2
2
ab
2
X
=
p
e
2
4
π
ɛ
o
X
+
h
2
2
m
e
a
2
b
2
2
ab
2
X
where
X
=
1
ξ
+
a
2
1
ξ
+
b
2
1
c
ξ
2
-
1
ξ
2
-
η
2
.
49. The system of claim 48 wherein the force balance equation has the parametric solution
r ( t )= ia cos ω t+jb sin ω t
when the semimajor axis, a, is
a
=
a
o
p
.
50. The system of claim 49 wherein an internuclear distance, 2c′, which is the distance between the foci is
2
c
′
=
2
p
a
o
;
the semiminor axis is
b
=
c
′
=
1
p
2
a
o
,
and
the eccentricity, e, is
e
=
1
2
.
51. The system of claim 50 wherein a potential energy of the two electrons in the central field of +pe at the foci is
V
e
=
-
2
p
e
2
8
π
ɛ
o
a
2
-
b
2
ln
a
+
a
2
-
b
2
a
-
a
2
-
b
2
;
The potential energy of the two protons is
V
p
=
p
8
π
ɛ
o
e
2
a
2
-
b
2
;
The kinetic energy of the electrons is
T
=
h
2
2
m
e
a
a
2
-
b
2
ln
a
+
a
2
-
b
2
a
-
a
2
-
b
2
,
and
The energy, V m , of the magnetic force between the electrons is
V
m
=
-
ℏ
2
4
m
e
a
a
2
-
b
2
ln
a
+
a
2
-
b
2
a
-
a
2
-
b
2
.
52. The system of claim 51 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the protons, and the corresponding energy Ē osc is the difference between the Doppler and average vibrational kinetic energies:
E
_
osc
=
E
_
D
+
E
_
Kvib
=
(
V
e
+
T
+
V
m
+
V
p
)
2
E
_
K
Mc
2
+
1
2
ℏ
p
2
k
μ
,
and
=
-
p
3
0.326469
eV
+
1
2
p
2
(
0.56764
eV
)
The total energy is
E
T
=
V
e
+
T
+
V
m
+
V
P
+
E
_
osc
E
T
=
-
p
2
{
e
2
8
π
ɛ
o
a
O
[
2
2
-
2
+
2
2
ln
2
+
1
2
-
1
-
2
]
[
1
+
p
2
h
e
2
4
π
ɛ
o
a
O
3
m
e
m
e
c
2
]
-
1
2
h
k
μ
}
.
=
-
p
2
31.351
eV
-
p
3
0.326469
eV
53. The system of claim 52 wherein a bond dissociation energy, E D , is the difference between the total energy of the hydrogen atoms H(1/p) and E T :
E
D
=
E
(
2
H
(
1
/
p
)
)
-
E
T
=
-
p
2
27.20
eV
-
E
T
=
-
p
2
27.20
eV
-
(
-
p
2
31.351
eV
-
p
3
0.326469
eV
)
=
p
2
4.151
eV
+
p
3
0.326469
eV
wherein the total energy of two hydrogen atoms H(1/p) is
E (2 H (1 /p ))=− p 2 27.20 eV.