Introduction
The change to a center of variable system is both easier, than expected and harder. I point out an obvious flaw in the results of the first algebra in The equation.htm. Of some interest is ../thoughts/derivs.doc in which numerical derivatives for r – RCMare isscussed. The details of the Non Adiabatic equation for H2+ are detailed below.
- Two Particle – discussion of Messiah’s two particle derivation.
- Extension to many particles Messiah’s extension of the two particle result
- H2+ Bob’s application of Messiah’s result to H2+
- The Wave Function Bob’s discussion of the form for the H2+ wave function
Two Particle
Messiah’s derivation of the correct result is in MessiahCM.doc. The two particle result is MessiahCM.doc - eigensolutions
Equation 1
Equation 2
Equation 3
Equation 4
The functions and respectively satisfying the separate equations
Equation 5
Equation 6
The energy eigenvalue of the overall system is the sum of the energy eigenvalues of the partial systems:
Equation 7
In a footnote, Messiah implies that there are many other ways to arrive at this same result.
Extension to many particles
Messiah’s extension to more than two particles proceeds by first making a two particle transformation between r1and r2yielding equation 2 and a Then a two particle transformation between R and r3yielding -- this is similar to the notes for Henk.
H2+
In this H2+ case r1is the location of the first nucleus, r2 is the location of the second nucleus and r3is the location of the electron. , the first R is . The second rho is , .
Figure 1 The difference variables for the separated wave function.
To find the distances from the nuclei to the electron, impose a cooridinate system with the first nucleus at (0,0,0) and the second at . Then with the electron at , the vector . The distance to the first nucleus is thus . The distance to the second nucleus is . Thus:. Equation 8
Equation 9
Note that for H2+ that and that to a good approximation , though it is actually the reduced electron mass with respect to half the proton mass. An interesting feature of the rigorous transformations here is that there is a different for each , if there were M electrons in the system, the first electron has which is the reduced between half the proton mass and 1 electron and ends with which is the reduced mass between the M’th electron and the reduced nuclear mass + all other electrons.
The Wave Function
Equation 10
Equation 10 is exact. In the Born Oppenheimer approximation
Equation 11
The equation for is the equation that is normally solved for H2+. The distance enters as only as a parameter in the potential, while the vector becomes the electron coordinate measured from the point between the two nuclei. The slight difference between the electron mass and that reduced with ½ the nuclear mass is ignored. In this case
Figure 2 Energy versus internuclear separation. The vibrational levels of the eigenvalue for the nuclear equation are marked in red.
The equation for is
The l,m refers to the angular momentum degrees of rotational freedom while the n refers to the vibrational modes in Figure 2. Equation 11 is a good starting point for the solution of the full wave function