Multibody Energy States With exact electron-electron interactions

4481 | P a g e D e c e m b e r 2 0 1 6 w w w . c i r w o r l d . c o m Multibody Energy States With exact electron-electron interactions James W Goodman Louisiana State University Baton Rouge, Louisiana Abstract In a class on quantum mechanics in 1964 at LSU a question was asked whether the potential energies of a multibody could be added up to give the energy state. It was stated that Vanagas was somewhat successful in using 2 body states. In 1971 two hypotheses were made using current Hilbert Space applied to the multibody Schrödinger’s Equation and the wave functions derived. The ground states of atoms with atomic number Z=2-20 were calculated and agreed with HartreeFox iterations. The success was due to the exact electron-electron interactions.


Introduction
The material here presented uses the Texas Method of solving mathematical theorems.A few lemmas are required before the derivation of the solution of the multibody Schrödinger's Equation The derivative of a function of x offset by a constant c with respect to x is equal to the derivative with respect to x offset by c.From the definition of the derivative: lim |F(x-c)-F(x'-c)|/||x-x'|=lim|.|F(x-c)-F(x'-c)|/|(x-c)-(x'-c)|.
From Goldstein Classical Mechanics near the end of chapter 1 can be found the Virial Theorem which says statistically 2T=-V or 2E=V.This holds for the S orbitals and the peak of the P orbitals.
From Hilbert space with T,V and E as Hermitian operators with positive signed eigenvalues, the Schrödinger's Equation for the hydrogen atom is given by T-V=-E and with the same eigenfunctions V-T=E by multiplying both sides of the equation by -1.
The possibility of independent distances between N points where n=N*(N-1)/2 is shown by a rotation tensor acting on an n dimensional vector yielding an n+1 dimensional vector conserving the distance d except for the last distance which varies independently.

Derivation
Hypothesis: The wave function of a system of N charged particles is the multiple of functions of the vector distance between each pair of particles.
Hypothesis: The distances between pairs of particles are independent of each other.

Schrödinger's Equation with constant E:
Terms from T: Therefore with independent and : +…+V Each of the above terms divided by is independent and can be set equal to a constant or Assuming the dot product is a perturbation we have the Schrödinger equations for hydrogen atoms: We now know that the dot product of the momentums using the hydrogen wave functions is exactly zero so that we have an exact ab initio solution for the quantum mechanical many body problem.

Dialogue
Start with n(n-1)/2 unknown functions of the vector distance between each pair of particles.Use Schroedinger's equation in Cartesian coordinates to solve for the functions.The first derivatives Dij operating on functions without an i nor a j are zero.A lemma is d/dr=d/(r-c) so that d/drij=d/d(rij-rjk) since rij is independent of rjk so that rjk can be treated as a constant with respect to rij.Only when solving the functions for a separated hydrogen atom are spherical coordinates used.This D e c e m b e r 2016 w w w .c i r w o r l d .c o m solution is well known and is not done explicitly in the derivation.The second derivative returns the equation to a dot product scalar sum of the two varieties of terms d^2 and d*d as shown in the derivation.Setting the equation plus the potential energy V to an unknown E times psi gives the separated Schrödinger's Equation by dividing by the function psi resulting in a series of independent terms each of which may be set equal to a constant Eij whose sum is a constant E. Each term is a hydrogen equation and is given.E is the energy state of the atom.Note the e-e solution is below.The psi function is solved to be the product of hydrogen wave functions.

Results
Ground states of atoms found in the Handbook of Chemistry and Physics compared to ground states calculated from the above derivation.The main quantum number n is 1 for the first row of the periodic table and n=2 for the second row and is thought to change to 3 for the third row for new states.However the nuclear spin flips providing 8 new states with n=2 in the third row.Each higher row is off also.

Electron -Electron Solution
Solve the electron -positron Schrödinger's Equation noting the energy is 1/2 because of the reduced mass.From group theory the energy equation is T -V = -E and multiplying both sides by -1 the energy is found to be positive: V -T = E.
The wave functions are the same and the energy is 13.6x/(n^2)/2.x is unknown but may be solved for in a element of atomic number Z. From the Virial Theorem (Classic Mechanics by Goldstein (index)) statistically: The table gives the sawf ground state E0 for the first 20 atoms of atomic number Z.The next column is the energy state with the electrons in the 6th orbital E6.The next column is E sawf = E0-E6 which compares to about 1% with the next column ground state from the Handbook of Chemistry and Physics.It appears the E6 correction is due to accepted calculations stopping at the 5th or 6th orbital.The difference for Z=11-20 is explained by the nuclear spin: Main quantum number n correction from nuclear spin..Orbital capacities are 2,16,36...Not 2,8,8,18,18...