The energies and wave functions within the most commonly used theories of atomic structure are assumed to arise as solutions of a Schrodinger equation whose hamiltonian h(r) possess three kinds of energies Kinetic energy, whose average value is computed by taking the expectation value of the kinetic energy operator-A/2m V with respect to any particular solution o, (r) to the Schrodinger equation: KE=<oJ-h/2m V-dJ> 2. Coulombic attraction energy with the nucleus of charge Z: <pl-Ze/r lo 3 And Coulomb repulsion energies with all of the n-l other electrons, which are assumed to occupy other atomic orbitals(AOs)denoted ox, with this energy computed as ∑<φr)φs(r”)e2/r-r)|()d(r) The so-called Dirac notation <o(r)x(r)l(e/r-rD,(ox(r)> is used to represent the six-dimensional Coulomb integral JIx= J%(r) l%k(r)(e2/r-r,)drdr'that describes the Coulomb repulsion between the charge density lo, (rl for the electron in pj and the charge density p(r)l for the electron in x. Of course, the sum over K must be limited to exclude k=J to avoid counting a self-interaction"of the electron in orbital o with itself The total energy a of the orbital o ,, is the sum of the above three contributions E=-h22mV2|>+<-ze/r伸 +Σx<φr)(r)e/rrD)|ψr)(r)5 The energies and wave functions within the most commonly used theories of atomic structure are assumed to arise as solutions of a Schrödinger equation whose hamiltonian he (r) possess three kinds of energies: 1. Kinetic energy, whose average value is computed by taking the expectation value of the kinetic energy operator – h2 /2m Ñ 2 with respect to any particular solution fJ (r) to the Schrödinger equation: KE = <fJ | – h2 /2m Ñ 2 |fJ>; 2. Coulombic attraction energy with the nucleus of charge Z: <fJ | -Ze2 /r |fJ>; 3. And Coulomb repulsion energies with all of the n-1 other electrons, which are assumed to occupy other atomic orbitals (AOs) denoted fK, with this energy computed as SK <fJ (r) fK(r’) |(e2 /|r-r’|) | fJ (r) fK(r’)>. The so-called Dirac notation <fJ (r) fK(r’) |(e2 /|r-r’|) | fJ (r) fK(r’)> is used to represent the six-dimensional Coulomb integral JJ,K = ò|fJ (r)|2 |fK(r’)|2 (e2 /|r-r’) dr dr’ that describes the Coulomb repulsion between the charge density |fJ (r)|2 for the electron in fJ and the charge density |fK(r’)|2 for the electron in fK. Of course, the sum over K must be limited to exclude K=J to avoid counting a “self-interaction” of the electron in orbital fJ with itself. The total energy eJ of the orbital fJ , is the sum of the above three contributions: eJ = <fJ | – h2 /2m Ñ 2 |fJ> + <fJ | -Ze2 /|r |fJ> + SK <fJ (r) fK(r’) |(e2 /|r-r’|) | fJ (r) fK(r’)>