A Nu-Theory
(Hartree units are used throughout.
Experimental values adapted from Moore’s Tables, CRC Handbook or other such sources.)
The job of the theoretician is discovery!
Atomic Systems
Postulate:
The strange shapes generated by the square-magnitude Schrödinger function of one-electron atomic systems are the distortion of space-time by the presence of charges in resonance.
- Corollary: Excess negative charge produces space-time distortion in one form while excess positive produces the opposite.
- Corollary: Neutral systems are flat (Euclidian) outside of a localized area.
- Corollary: There are few inertial frames in atomic and molecular systems.
- General relativity modified to replace mass with charge, and the cosmological constant with a repulsive tensor, is appropriate.
- Spin is a special-relativity concept and must be replaced by an effective general-relativity concept.
- Corollary: The model is stochastic, averaging over ensembles, and the averaged distortions represent strange attractors.
From this postulate we develop a technique for calculating atomic and molecular parameters.
Theorem 1:
The distortion of space-time due to atomic or molecular systems may be found by the spectral sum of the individual distortions:
- These distortions exist in normal space-time, and Hilbert space is not necessary.
- Individual distortions are produced by the interaction of a positive charge with a negative charge and are thus not centered over either.
- Multiple distortions will reside as far apart as possible according to the geometry of the system, but the overall distortion is the sum of all.
- Each distortion is produced by the interaction of a positive with a negative charge in a two-body manner, but the resultant distortions interact.
- A parameter, , serves to locate the disturbances produced by the individual charges relative to a mass concentration. It is time-like, but not time, as it has both positive and negative values. A thought: In this model time may locally bend in on itself in a closed loop thus making the system steady-state. This would explain why, for an eigen-state, there is a maximum value of ε which a stable system must not exceed.
- Only the square magnitude of the total distortion has physical meaning. To this end one might consider the distortion functions to be pure imaginary.
- On a pair-wise basis the square magnitude has the form of Schrödinger functions-squared located a distance apart, and an overlap product centeredbetween them.
- From the standpoint of the Schrödinger equation the overlap product may be treated as a separate entity.
- For atomic and molecular systems the real-valued form of the Schrödinger functions may be used, thus giving a visual model of the distorted space-time, but the resultant must be integrated over all possible values of ε.
Theorem 2:
Coulomb’s law must be modified for small distances.
- The quantity (charge-times-mass) in the exponent of the Schrödinger functions is replace by a generalized quantity ().
- This quantity is determined by the Virial Theorem for each value of .
- This modification cause the model to indicate that mass is generated by charges in resonance just as magnetic fields are generated by charges in motion.
Theorem 3:
The maximum value of , r, is determined by minimizing the calculated eigen-energy, , as a function of various trial values of .
- In some instances there will be a non-zero r.
- In other cases, particularly in the one-electron atomic systems, r = 0, and the minimum of E() is the eigen-energy.
Theorem 4:
Excited states are bound but not stable. For this reason the time-dependent form of the Schrödinger equation may be needed in such cases.
A Statement of Method
Throughout this document you will note I have “tweaked” many of the individual calculations. For example I have required the electron-electron interaction in the two-electron, ground-state series to act as if they were twice as far apart as the interaction diagram would admit, and the Laplacian of the overlap takes on a new definition central from the origin of the system. With these tweaks I calculate values that are probably the best that can be done with this numerical-methods software, i.e. the model is better than the numbers.
I justify this with two ideas. First, the tweak must apply directly to more than one property or calculation. For example, the linear regression calculation for the first excited state of the two-electron series applies directly to the second excited state, and the tweaks necessary to get good values of the energy of diatomic hydrogen also give good results for the mean nuclear separation, i.e., bond length.
Second, I expect to modify Einstein’s general relativity to account for these tweaks. In other words the tweaks define the manifold required for the applicable differential geometry. For example, the tweaks necessary for the first and second excited states define a manifold necessary when the 2s Schrödinger function is used in a calculation, and I expect to develop another, more accurate, definition of “spin”.
I have had great joy from this “toy”. I think as of now I have an excellent proof-of-principle if anyone is ever interested. For now I will continue enjoying my folly.
An Isolated Charge
Let’s assume we have an isolated charge, either positive or negative. If its normalized distortion-function is
, where is the acknowledgement that the normalization takes place after any operation on the function; then Coulomb’s law in its potential form may be reproduced by the operator
For a test charge of the same sign α = -1/m, but for opposite sign α = 1/m. The -factor will be the charge-times-mass as before, but this shows the electrostatic interaction may be thought of as effected by the spectral interaction of the Schrödinger disturbance functions.
Things are different for the excited states. For example
Check the integral (I’m having some trouble with my n-m software):
Okay!
The -form of the Schrödinger Equation
Invoking the invariance of the Laplacian, and requiring the equation to exist in 4-space we have
, where we require
for each and all values of ε.
We require the geometry of “as far apart as possible” and β(ε) determined by the Virial theorem which translates
The way the models are constructed, the interaction of the electron with the resultant distortion field turns out to be central.
The time-like nature of ε results in the final equation for the eigen-energy:
(Integrate both sides of the equation. The normalization condition yields the above.)
In practice the form
may sometimes be used. This may not be true in cases of non-zero orbital angular momentum.
The Time Dependent Schrödinger Equation
For excited states I expect there may be two parameters ε: one for the stable, ionized configuration, and one for the excited electron, but this analysis may not be necessary.
The quantity ε is a cyclic measure of time, even for ground state configurations so I will approximate this with the idea
, so
Thus
I suspect this will require putting the “normalization” factor outside of the derivative. I would like this to reduce to the steady state situation when, but that will only occur iff
I’m not sure how that would go.
Spectral Interference of the Schrödinger Functions
(The following is for atoms. Molecules will have to be revised appropriately.)
Notation:
Two alternating (superscript-subscript) identical, sequential, English indices means the sum over N components, N is the number of electrons present. Corresponding Greek indices indicate no sum is to be made.
Similarly
For some cases there is a cleardistinction in meaning between covariance and contravariance.
Indistinguishability:
There must be no way to determine locus or identity of any electron in the system. This is handled by the fact that may have both positive and negative values, and the integral is over all possible . If necessary we also can further average over an ensemble.
SpinConsiderations:
There appears to be a necessary modification to my original notion of the spin-involvement. Spin is fundamentally a special-relativity concept, and if my surmise about the twin paradox is correct, wrong. There must be a more accurate version from general relativity. In the following I will put together a coupling scheme, but not limit it to possible spin relationships. I will say only that the system normalization must reduce to the number of electrons, N, at ε = 0.
Let be constants related to the spin of the relevant space-time distortion. A total disturbance function may be found from
The normalization constant,, is found from the condition
We define (Note: with this definition ). The cofactor may be adjusted to suit the system, so
The Kinetic Energy Operator
From Schrödinger’s postulates, for each electron (using the infinite nuclear mass approximation)
whereΔ is the Laplacian. Thus the total kinetic energy averaged over the ensemble is
The Schrödinger functions obey the equation
Defining we have and
Unfortunately this does not quite work. Realizing that the overlap product takes on something of an independent identity, we can modify it. The integral must be done with respect to the center of mass of the system, i.e. at the nucleus. Define the integral
, then
Note: the definition of seems to indicate a space-time distortion of the manifold such that the Laplacian behaves differently for the overlap portion.
The Potential Energy Operator
The electrostatic potential is a two-body interaction with a primitive of the form.
Electron-Nucleus Interaction
Note: the nucleus serves only to create the net disturbance for each electron-function. It is strictly two-body, and thus only the part of the disturbance function not associated with the overlap can be used.
Electron-Electron Interaction
For atomic systems, the way the disturbance function is constructed the overlap, and the total function, is centered on the origin (nucleus). Just as the change in the integral for T had to be made a similar change must be made to the electron-electron interaction, but the e-e interaction is viewed by each “electron disturbance” as taking place a distance of away. First we must eliminate all self-energy. If the square-magnitude-total function is
The factor of ½ is to prevent double counting. The total potential energy is then
Degrees of Freedom
One must multiply the total energy, found from summing the above, by the number of ways the system can be configured. For example, while two 1s functions can only be configured one way, a 1s, 2s set may be configured two.
The Virial Ratio
In spherical polar coordinates the Virial translates into
, where is the potential (operator) experienced by the electron. Thus, by the nature of the model,
Two Electron Atomic Systems
Ground State
Spectral Interference
Then
If
This gives
Note: I have required that the o’s may be either positive or negative. This will apply only at the overlap terms of the system and in this case, where the Schrödinger functions are equivalent, produces the Pauli Exclusion Principle.
This is scary! I did a test calculation using a new worksheet and typed in then I copied the expression from an existing worksheet and got totally different answers. The green set was what I would have expected. Here is a screen capture to show what actually happened.
I tried to make this happen again on 12/06/2012 but it wouldn’t happen. The only thing I changed was to defrag the disc.
Kinetic Energy
Potential Energy
Electron-Nucleus
Electron-Electron
To get a near-perfect fit with the ground state, two-electron isoelectronic series we have to compute the electron-electron interaction as if the two disturbances were twice as far apart. Let
Total Energy
Virial
Results:Two Electrons, Ground State
Z / Eexp / εr / / pct1 / -0.5274 / 2.6896 / -0.5222 / 99.01
2 / -2.9034 / 0.8946 / -2.8971 / 99.78
3 / -7.2798 / 0.5040 / -7.2604 / 99.73
4 / -13.6564 / 0.3425 / -13.6277 / 99.79
5 / -22.0345 / 0.2563 / -22.0033 / 99.86
6 / -32.4156 / 0.2034 / -32.3920 / 99.93
7 / -44.8009 / 0.1678 / -44.7947 / 99.99
8 / -59.1912 / 0.1424 / -59.2120 / 100.04
First Excited State
Spectral Interference
We would like , but . I shall use o = 1 so
Kinetic Energy
; I would expect a cofactor of 5/8 if this were a screened-charge model.
Potential Energy
Electron-Nucleus
; I would expect a cofactor of 5/4 if this were a screened-charge model.
Electron-Electron
In this case α = 1. Neutral He gives spot-on results, but to do the ions with extra positive charge I have to introduce a factor λ(Z)
; this puts much of the weight onto the overlap.
Total Energy
Remember, there are two ways the system may be set up: with the 2s on the +z axis, and with the 2s on the –z axis, so
Virial
Results:Two Electrons, First Excited State
Z / Eexp / / εr / / pct2 / -2.1750 / 0 / 1.205 / -2.1739 / 99.95
*3 / -5.1107 / 1.265 / 0.7705 / -5.0984 / 99.76
*4 / -9.2983 / 2.676 / 0.5390 / -9.2722 / 99.72
*5 / -14.7375 / 4.087 / 0.4081 / -14.7085 / 99.80
*6 / -21.4291 / 5.498 / 0.3253 / -21.4104 / 99.91
*7 / -29.3736 / 6.909 / 0.2688 / -29.3801 / 100.02
*8 / -38.5722 / 8.320 / 0.228 / -38.6197 / 100.12
*Unlike the ground state extra positive charge does something strange to the above equations. I have to find out how to compensate before I can effectively evaluate these ions. For now I have a formula that seems to describe the process.
Second Excited State
Spectral Interference
We would like , but . I shall use o = 1 so
Kinetic Energy
; I would expect a cofactor of 5/8 if this were a screened-charge model.
Potential Energy
Electron-Nucleus
; I would expect a cofactor of 5/4 if this were a screened-charge model.
Electron-Electron
In this case α = 1. Neutral He gives spot-on results, but to do the ions with extra positive charge I have to introduce a factor λ(Z). It is of note that this is the same as for the first excited state. To achieve the second excited state I have to add one more . This appears strange, but the requirement of the Virial Theorem sometimes makes something expected to increase the bound-state energy will actually decrease it, and the reverse.
; this puts much of the weight onto the overlap.
Total Energy
Remember, there are two ways the system may be set up: with the 2s on the +z axis, and with the 2s on the –z axis, so
Virial
Results:Two Electrons, Second Excited State
Z / Eexp / / εr / / pct2 / -2.14577 / 0 / 1.3684 / -2.1352 / 99.51
3 / -5.046802 / 1.265 / 0.8433 / -5.0339 / 99.74
4 / -9.298264* / 2.676 / 0.5829 / -9.1736 / ---
5 / -14.73755* / 4.087 / 0.4381 / -14.5729 / ---
6 / -21.429055* / 5.498 / 0.3472 / -21.2369 / ---
7 / -29.373646* / 6.909 / 0.2857 / -29.1675 / ---
8 / -38.572249* / 8.320 / 0.2415 / -38.3668 / ---
- Values for the second excited state are not found in the tables. These are for the first excited state.
Third Excited State
The system as envisaged here is asymmetrical. I use the symmetrical form by invoking a negative of the disturbance function for all ε < 0. The other “orbitals” would probably eliminate the need for this stratagem, but require 3D computing, which takes much longer.
Spectral Interference
We would like , but . I shall use o = 1, so
Kinetic Energy
Potential Energy
Electron-Nucleus
Electron-Electron
In this case α = 1. Neutral He gives spot-on results, but to do the ions with extra positive charge I have to introduce a factor λ(Z)
Total Energy
Remember, there are two ways the system may be set up: with the 2s on the +z axis, and with the 2s on the –z axis, so
Virial
Results:Two Electrons, Third Excited State
Z / Eexp / α / λ / εr / / pct2 / -2.1330 / 1.25 / 0.5* / 0.4114 / -2.1362 / 100.15
3 / -5.0277 / 1.25 / 1.265 / 0.2560 / -5.0112 / 99.67
4 / -9.1758 / 1.25 / 2.000 / 0.1807 / -9.1765 / 100.01
5 / -14.5763 / 1.25 / 2.750 / 0.1390 / -14.5859 / 100.07
6 / -21.2290 / 1.25 / 3.500 / 0.1128 / -21.2415 / 100.06
7 / -29.1350 / 1.25 / 4.250 / 0.0948 / -29.1430 / 100.03
8 / -38.2950 / 1.25 / 5.000 / 0.0817 / -38.2901 / 99.99
*Original value of 0; this gives a nearly perfect fit for the line representing . Unfortunately it gives only 98.35% of the true value of E. With this λ I must use an α of 1.5 to get 99.67% of the true value. This may relate to my method of determining both α and λ, but study will be deferred to later.
Three Electrons: Neutral Lithium
Ground State
Spectral Interference
Then
Kinetic Energy
Potential Energy
Electron-Nucleus
Electron-Electron
, but with the caveat’s associated with the other systems on the separation, and weighting of the Λ’s.
There are three ways these can interact so
Total Energy
There are four ways the system can be configured, so
Results:ThreeElectrons, Ground State
Because of the three-dimensional integrals these calculations take “for ever”. was chosen, but a more thorough calculation would probably give a slightly different value as the calculated values seem to get worse with increasing Z.
Z / Eexp / λ / εr / / pct3 / -7.4779 / 0 / 3.7722 / -7.5191 / 100.55
4 / -14.3256 / 0.7500 / 2.6941 / -14.3855 / 100.42
5 / -23.4285 / 1.500 / 2.0932 / -23.4673 / 100.17
6 / -34.7857 / 2.250 / 1.7126 / -34.7654 / 99.94
7 / -48.3983 / 3.000 / 1.448 / -48.2798 / 99.76
8 / -64.2670 / 3.750 / 1.2548 / -64.0109 / 99.60
Molecular Systems
Postulate:
Once the resonance-disturbance is established within the atom, the covalent portion of chemical bonding is effectedthrough the overlap of disturbances between atomic systems. Little change occurs in the component atoms or ions.
Spectral Interference
This instance is closer to molecular orbital theory and might later be revised accordingly.
Diatomic Molecules
Hydrogen—Vibrating Disturbances Model
Spectral Interference
Then
If
This gives
Kinetic Energy
Potential Energy
Electron-Nucleus
Electron-Electron
To get a near-perfect fit with the ground state, two-electron isoelectronic series we have to compute the electron-electron interaction as if the two disturbances were twice as far apart. Let
Nucleus-Nucleus
Note: the nucleus-nucleus interaction is purely Coulomb, except for the need for β.
Total Energy
Note: the normalization constant, γ, is not used.
Virial
Average Energy
, where are the values which produce as the bound-state range for .
Average Half-Separation of the Nuclei
Results:Diatomic Hydrogen, Ground State
Model / α / / εr / δ / / /Simple / 2 / 1.245 δ / 0.723 / 0.724 / 103.33 / -0.6585 / 99.92
Vibrating / 1.5 / 0.582 / -0.968 / 0.654 / 93.36 / -0.637 / 96.65
1.75 / 0.651 / -1.013 / 0.657 / 93.72 / -0.6550 / 99.37
2 / 0.649 / -0.996 / 0.649 / 92.58 / -0.6722 / 101.99
More than likely these results could be improved by allowing vibration perpendicular to the line-of-centers as well.
Such a calculation is beyond what I want to do right now.