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11.3: The Slater-Condon Rules

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  • The Slater–Condon rules express integrals of one- and two-body operators over wavefunctions constructed as Slater determinants of orthonormal orbitals in terms of the individual orbitals. In doing so, the original integrals involving N-electron wavefunctions are reduced to sums over integrals involving at most two molecular orbitals

    i. If |> and |'> are identical, then

    \[ \langle|F + G|\rangle = \sum \limits_i \langle\phi_i| f |\phi_i\rangle + \sum\limits_{i>j}\left[ \langle\phi_i \phi_j | g | \phi_i \phi_j - \langle \phi_i \phi_j | g | \phi_j \phi_i \rangle \right], \]

    where the sums over i and j run over all spin-orbitals in | >;

    ii. If | > and | '> differ by a single spin-orbital mismatch ( \( (\phi_p \neq \phi_p' \) ),

    \[ \langle | F + G |' \rangle = \langle \phi_p | f | \phi_p' \rangle + \sum\limits_j \left[ \langle \phi_p\phi_j | g | \phi_p'\phi_j \rangle - \langle \phi_p\phi_j | g | \phi_j\phi_p' \rangle \right], \]

    where the sum over j runs over all spin-orbitals in | > except \(\phi_p\) ;

    iii. If | > and | '> differ by two spin-orbitals ( \( \phi_p \neq \phi_p' \) ),

    \[ \langle | F + G | '\rangle = \langle \phi_p \phi_q | g | \phi_p' \phi_q' \rangle - \langle \phi_p \phi_q | g | \phi_q' \phi_p' \rangle \]

    (note that the F contribution vanishes in this case);

    iv. If | > and | '> differ by three or more spin orbitals, then \[ < | F + G | '> = 0; \]

    v. For the identity operator I, the matrix elements < | I | '> = 0 if | > and | '> differ by one or more spin-orbitals (i.e., the Slater determinants are orthonormal if their spin-orbitals are).

    Recall that each of these results is subject to multiplication by a factor of \((-1)^{N_p}\) to account for possible ordering differences in the spin-orbitals in | > and | '>.

    In these expressions, \[ < \phi_i | f | \phi_j > \] is used to denote the one-electron integral \[ \int \phi_i^{\text{*}}(r)f(r)\phi_j(r)dr \] and \( < \phi_i\phi_j | g | \phi_k\phi_l > \) (or in short hand notation < i j| k l >) represents the two-electron integral \[ \int \phi_i^{\text{*}}(r)g(r,r')\phi_k(r)\phi_l(r')drdr' \]

    The notation < i j | k l> introduced above gives the two-electron integrals for the g(r,r') operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and l indices label the spin-orbitals referring to coordinates r'. The r and r' denote \( r, \theta, \phi, \sigma \text{ and } r', \theta ', \phi ', \sigma ' \) (with \(\sigma \text{ and } \sigma ' \text{ being the } \alpha \text{ or } \beta\) spin functions). The fact that r and r' are integrated and hence represent 'dummy' variables introduces index permutational symmetry into this list of integrals. For example,

    \[ < i j | k l > = < j i | l k > = < k l | i j >^{\text{*}} = < l k | j i >^{\text{*}}; \]

    the final two equivalences are results of the Hermitian nature of g(r,r').

    It is also common to represent these same two-electron integrals in a notation referred to as Mulliken notation in which:

    \[ \int\phi_i^{\text{*}}(r)\phi_j^{\text{*}}(r') g(r,r')\phi_k(r)\phi_l(r')drdr' = (i k | j l). \]

    Here, the indices i and k, which label the spin-orbital having variables r are grouped together, and j and l, which label spin-orbitals referring to the r' variables appear together. The above permutational symmetries, when expressed in terms of the Mulliken integral list read:

    \[ (i k | j l) = (j l | i k) = (k i | l j)^{\text{*}} = (l j | k i)^{\text{*}}. \]

    If the operators f and g do not contain any electron spin operators, then the spin integrations implicit in these integrals (all of the \(\phi_i\) are spin-orbitals, so each \(\phi\) is accompanied by an \(\alpha\) or \(\beta\) spin function and each \(\phi^{\text{*}}\) involves the adjoint of one of the \(\alpha\) or \(\beta\) spin functions) can be carried out as \(<\alpha|\alpha>=1, <\alpha |\beta>=0, <\beta |\alpha >=0, <\beta |\beta>=1\), thereby yielding integrals over spatial orbitals. These spin integration results follow immediately from the general properties of angular momentum eigenfunctions detailed in Appendix G; in particular, because \(\alpha \text{ and } \beta\) are eigenfunctions of \(S_Z\) with different eigenvalues, they must be orthogonal \( <\alpha |\beta > = <\beta | \alpha > = 0\).

    The essential results of the Slater-Condon rules are:

    1. The full N! terms that arise in the N-electron Slater determinants do not have to be treated explicitly, nor do the N!(N! + 1)/2 Hamiltonian matrix elements among the N! terms of one Slater determinant and the N! terms of the same or another Slater determinant
    2. All such matrix elements, for any one- and/or two-electron operator can be expressed in terms of one- or two-electron integrals over the spin-orbitals that appear in the determinants.
    3. The integrals over orbitals are three or six dimensional integrals, regardless of how many electrons N there are.
    4. These integrals over mo's can, through the LCAO-MO expansion, ultimately be expressed in terms of one- and two-electron integrals over the primitive atomic orbitals. It is only these ao-based integrals that can be evaluated explicitly (on high speed computers for all but the smallest systems).

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