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Coulombic operator

  1. Jun 14, 2007 #1

    I am trying to understand this equation where the coulombic operator is given by

    J = \int d\tau \phi(2) \frac{1}{r_{12}}\phi(2)

    so I integrate over \tau but what is tau and the number I get from the equation is the energy I pressume?
    Any hints or help appreciated.

    Thanks in advance
  2. jcsd
  3. Jun 14, 2007 #2

    Meir Achuz

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    [tex]d\tau[/tex] just represents the differential volume element.
    In spherical coordinates, it is [tex]r^2 dr d\Omega[/tex].
  4. Jun 14, 2007 #3
    what is that more exactly? I know from spherical geometry that the volume is calculate
    dV = \rho^2 sin \phi d\phi d\rho d\theta

    If I integrate over two wave functions representing two electrons how to interpreted it?

    Any help or advice appreciated

    Thanks in advance
  5. Jun 15, 2007 #4
    Are you trying to integrate the expectation of the coulombic interaction operator over two wavefunctions? <a(r1)|J|b(r2)> ? (with r1, r2 position vectors so in general dependent on r, theta and phi).

    If so, which wavefunctions a and b are you using? You'd also need to rexpress the 1/r term in J as some function r1-r2, which will be fairly complicated in 3D. I'd guess this problem is best solved computationally...
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