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6.6: The Slater-Condon Rules

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    162851
  • To form Hamiltonian matrix elements \(H_{K,L}\) between any pair of Slater determinants constructed from spin-orbitals that are orthonormal, one uses the so-called Slater-Condon rules. These rules express all non-vanishing matrix elements involving either one- or two- electron operators. One-electron operators are additive and appear as

    \[F = \sum_i \phi(i);\]

    two-electron operators are pairwise additive and appear as

    \[G = \sum_{i< j}g(i,j) = \frac{1}{2} \sum_{i \ne j} g(i,j).\]

    The Slater-Condon rules give the matrix elements between two determinants

    \[| \rangle = |\phi_1\phi_2\phi_3... \phi_N|\]

    and

    \[| '\rangle = |\phi'_1\phi'_2\phi'_3...\phi'_N|\]

    for any quantum mechanical operator that is a sum of one- and two- electron operators (\(F + G\)). It expresses these matrix elements in terms of one-and two-electron integrals involving the spin-orbitals that appear in \(| \rangle\) and \(| '\rangle\) and the operators \(f\) and \(g\).

    As a first step in applying these rules, one must examine \(| \rangle\) and \(| '\rangle\) and determine by how many (if any) spin-orbitals \(| \rangle\) and \(| '\rangle\) differ. In so doing, one may have to reorder the spin-orbitals in one of the determinants to achieve maximal coincidence with those in the other determinant; it is essential to keep track of the number of permutations ( \(N_p\)) that one makes in achieving maximal coincidence. The results of the Slater-Condon rules given below are then multiplied by \((-1)^{N_p}\) to obtain the matrix elements between the original \(| \rangle\) and \(| '\rangle\). The final result does not depend on whether one chooses to permute \(| \rangle\) or \(| '\rangle\) to determine \(N_p\).

    The Hamiltonian is, of course, a specific example of such an operator that contains both one- and two-electron components; the electric dipole operator \(\sum_i e\textbf{r}_i\) and the electronic kinetic energy \(- \frac{\hbar^2}{2m_e}\sum_i\nabla_i^2\) are examples of one-electron operators (for which one takes \(g = 0\)); the electron-electron coulomb interaction \(\sum_{i<j} e^2/r_{ij}\) is a two-electron operator (for which one takes \(f = 0\)).

    The two Slater determinants whose matrix elements are to be determined can be written as

    \[| \rangle = \frac{1}{\sqrt{N!}} \sum_{P=1}^{N!} (-1)^p P \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots \phi_N(N)\]

    \[| '\rangle = \frac{1}{\sqrt{N!}} \sum_{P=1}^{N!} (-1)^q Q \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_N(N)\]

    where the spin-orbitals {\(\phi_j\)} and {\(\phi’_j\)} appear in the first and second determinants, respectively, and the operators \(P\) and \(Q\) describe the permutations of the spin-orbitals appearing in these two determinants. The factors \((-1)^p\) and \((-1)^q\) are the signs associated with these permutations as discussed earlier in Section 6.1.1. Any matrix element involving one- and two-electron operators

    \[\langle |F+G|'\rangle =\frac{1}{\sqrt{N!}} \sum_{P,Q} (-1)^{p+q} \\\langle P \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots \phi_N(N)|F+G|Q \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_N(N)\rangle \]

    needs to be expressed in terms of integrals involving the spin-orbitals in the two determinants and the one- and two-electron operators.

    To simplify the above expression, which contains \((N!)^2\) terms in its two summations, one proceeds as follows:

    a. Use is made of the identity \(\langle P\psi |\psi’\rangle = \langle y|P\psi’\rangle\) to move the permutation operator \(P\) to just before the (\(F+G\))

    \[\langle P \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots \phi_N(N)| F+G |Q \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_N(N)\rangle \\
    =\langle \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots \phi_N(N)| P(F+G) |Q \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_N(N)\rangle
    \]

    b. Because \(F\) and \(G\) contain sums over all \(N\) electrons in a symmetric fashion, any permutation \(P\) acting on \(F+G\) leaves these sums unchanged. So, \(P\) commutes with \(F\) and with \(G\). This allows the above quantity to be rewritten as

    \[\langle \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots \phi_N(N)| F+G |PQ \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_N(N)\rangle \]

    c. For any permutation operator \(Q\), the operator \(PQ\) is just another permutation operator. Moreover, for any \(Q\), the set of all operators \(PQ\) runs over all \(N!\) permutations, and the sign associated with the operator \(PQ\) is the sign belonging to \(P\) times the sign associated with \(Q\), \((-1)^{p+q}\). So, the double sum (i.e., over \(P\) and over \(Q\)) appearing in the above expression for the general matrix element of \(F+G\) contains \(N!\) identical sums over the single operator \(PQ\) of the sign of this operator \((-1)^{p+q}\) multiplied by the effect of this operator on the spin-orbital product on the right-hand side

    \[ \langle |F+G|'\rangle =\frac{1}{\sqrt{N!}}N!\\
    \sum_{P,Q} (-1)^{p+q} \langle \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots \phi_N(N)| F+G |PQ \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_N(N)\rangle \]

    By assumption, as explained earlier, the two Slater determinants have been compared and arranged in an order of maximal coincidence and the factor \((-1)^{N_p}\) needed to bring them into maximal coincidence has been determined. So, let us begin by assuming that the two determinants differ by three spin-orbitals and let us first consider the terms arising from the identity permutation \(PQ = E\) (i.e., the permutation that alters none of the spin-orbitals’ labels). These terms will involve integrals of the form

    \[\langle \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots\phi_j(j)\cdots\phi_N(N)| F+G |PQ \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_j(j)\cdots​\phi'_N(N)\rangle\]

    where the three-spin orbitals that differ in the two determinants appear in positions \(k\), \(n\), and \(j\). In these \(4N\)-dimensional (3 spatial and 1 spin coordinate for each of \(N\) electrons) integrals:

    a. Integrals of the form (for all \(i\ne k\), \(n\), or \(j\))

    \[\langle \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots\phi_j(j)\cdots\phi_N(N)| f(i) | \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_j(j)\cdots​\phi'_N(N)\rangle\]

    and (for all i and \(l \ne k\), \(n\), or \(j\))

    \[\langle \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots\phi_j(j)\cdots\phi_N(N)| g(i,l) | \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_j(j)\cdots​\phi'_N(N)\rangle\]

    vanish because the spin-orbitals appearing in positions \(k\), \(n\), and \(j\) in the two determinants are orthogonal to one another. For the \(F\)-operator, even integrals with \(i = k\), \(n\), or \(j\) vanish because there are still two spin-orbital mismatches at the other two locations among \(k\), \(n\), and \(j\). For the \(G\)-operator, even integrals with \(i\) or \(l = k\), \(n\), or \(j\) vanish because two mismatches remain; and even with both \(i\) and \(l = k\), \(n\), or \(j\), the integrals vanish because one spin-orbital mismatch remains. The main observation to make is that, even for \(PQ = E\), if there are three spin-orbital differences, neither the \(F\) nor \(G\) operator gives rise to any non-vanishing results.

    b. If we now consider any other permutation \(PQ\), the situation does not improve because any permutation cannot alter the fact that three spin-orbital mismatches do not generate any non-vanishing results.

    If there are only two spin-orbital mismatches (say in locations \(k\) and \(n\)), the integrals we need to evaluate are of the form

    \[\langle \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots\phi_N(N)| f(i) |PQ \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_N(N)\rangle\]

    and

    \[\langle \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_n(n)\cdots\phi_N(N)| g(i,l) |PQ \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_n(n)\cdots\phi'_N(N)\rangle\]

    c. Again, beginning with \(PQ = E\), we can conclude that all of the integrals involving the \(F\)-operator (i.e., \(\phi(i)\), \(\phi(k)\), and \(\phi(n)\)) vanish because the two spin-orbital mismatch is too much even for \(\phi(k)\) or \(\phi(n)\) to overcome; at least one spin-orbital orthogonality integral remains. For the \(G\)-operator, the only non-vanishing result arises from the \(i = k\) and \(l = n\) term \(\langle \phi_k(k)\phi_n(n)| g(k,n) | \phi'_k(k)\phi'_n(n)\rangle\).

    d. The only other permutation that generates another non-vanishing result is the permutation that interchanges \(k\) and \(n\), and it produces \(-\langle \phi_k(k)\phi_n(n)| g(k,n) | \phi'_n(k)\phi'_k(n)\rangle\)

    , where the negative sign arises from the \((-1)^{p+q}\) factor. All other permutations would interchange other spin-orbitals and thus generate orthogonality integrals involving other electrons’ coordinates.

    If there is only one spin-orbital mismatch (say in location \(k\)), the integrals we need to evaluate are of the form

    \[\langle \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_N(N)| f(i) |PQ \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_N(N)\rangle\]

    and

    \[\langle \phi_1(1)\phi_2(2)\cdots\phi_k(k)\cdots\phi_N(N)| g(i,l) |PQ \phi'_1(1)\phi'_2(2)\cdots\phi'_k(k)\cdots\phi'_N(N)\rangle.\]

    e. Again beginning with \(PQ = E\), the only non-vanishing contribution from the \(F\)-operator is \(\langle \phi_k(k)|f(k)|\phi'_k(k) \rangle\). For all other permutations, the \(F\)-operator produces no non-vanishing contributions because these permutations generate orthogonality integrals. For the \(G\)-operator and \(PQ = E\), the only non-vanishing contributions are

    \[\langle \phi_k(k)\phi_j(j)| g(k,j) | \phi'_k(k)\phi_j(j)\rangle\]

    where the sum over \(j\) runs over all of the spin-orbitals that are common to both of the two determinants.

    f. Among all other permutations, the only one that produces a non-vanishing result are those that permute the spin-orbital in the kth location with another spin-orbital, and they produce

    \[-\langle \phi_k(k)\phi_j(j)| g(k,j) | \phi'_j(k)\phi_k(j)\rangle.\]

    The minus sign arises from the \((-1)^{p+q}\) factor associated with this pair wise permutation operator.

    Finally, if there is no mismatch (i.e., the two determinants are identical), then

    g. The identity permutation generates

    \[-\langle \phi_k(k)| f(k) | \phi_k(k)\rangle.\]

    from the \(F\)-operator and

    \[\frac{1}{2}\sum_{j \ne k=1}^N \langle \phi_j(j)\phi_k(k)| g(k,j) | \phi_j(j)\phi_k(k)\rangle\]

    from the \(G\)-operator.

    h. The permutation that interchanges spin-orbitals in the kth and jth location produces

    \[-\frac{1}{2}\sum_{j \ne k=1}^N \langle \phi_j(j)\phi_k(k)| g(k,j) | \phi_k(j)\phi_j(k)\rangle .\]

    The summations over \(j\) and \(k\) appearing above can, alternatively, be written as

    \[\sum_{j < k=1}^N \langle \phi_j(j)\phi_k(k)| g(k,j) | \phi_j(j)\phi_k(k)\rangle\]

    and

    \[-\sum_{j < k=1}^N \langle \phi_j(j)\phi_k(k)| g(k,j) | \phi_k(j)\phi_j(k)\rangle .\]

    So, in summary, once maximal coincidence has been achieved, the Slater-Condon (SC) rules provide the following prescriptions for evaluating the matrix elements of any operator \(F+G\) containing a one-electron part \(F = \sum_i \phi(i)\) and a two-electron part \(G = \sum_{i< j}g(i,j)\).:

    1. If \(| \rangle\) and \(| '\rangle\) are identical, then \[\langle | F+G | \rangle = \sum_i \langle \phi_i| f | \phi_i\rangle +\sum_{i\rangle j} [\langle \phi_i \phi_j | g | \phi_i \phi_j \rangle - \langle \phi_i \phi_j | g | \phi_j \phi_i​ \rangle ],\] where the sums over \(i\) and \(j\) run over all spin-orbitals in \(| \rangle\) ;
    2. If \(| \rangle\) and \(| '\rangle\) differ by a single spin-orbital mismatch ( \(\phi_p \ne \phi'_p\) ), \[\langle | F+G | '\rangle = (-1)^{N_p} {\langle \phi_p | f | \phi'_p \rangle +\sum_j [\langle \phi_p\phi_j | g | \phi'_p\phi_j \rangle - \langle \phi_p\phi_j | g | \phi_j\phi'_p \rangle ]},\] where the sum over \(j\) runs over all spin-orbitals in \(| \rangle\) except \(\phi_p\);
    3. If \(| \rangle\) and \(| '\rangle\) differ by two spin-orbitals ( \(\phi_p \ne \phi'_p\) and \(\phi_q \ne \phi'_q\)), \[\langle | F+G | '\rangle = (-1)^{N_p} {\langle \phi_p \phi_q | g | \phi'_p \phi'_q \rangle - \langle \phi_p \phi_q | g | \phi'_q \phi'_p \rangle }\](note that the \(F\) contribution vanishes in this case);
    4. If \(| \rangle\) and \(| '\rangle\) differ by three or more spin orbitals, then \[\langle | F+G | '\rangle = 0;\]
    5. \(\Phi\) or the identity operator \(I\), the matrix elements \(\langle | I | '\rangle = 0\) if \(| \rangle\) and \(| '\rangle\) differ by one or more spin-orbitals (i.e., the Slater determinants are orthonormal if their spin-orbitals are).

    In these expressions,

    \[\langle \phi_i| f | \phi_j \rangle \]

    is used to denote the one-electron integral

    \[\int \phi^*_i(r) f(r) \phi_j(r) dr\]

    and

    \[\langle \phi_i \phi_j | g | \phi_k\phi_l \rangle \]

    (or, in short hand notation, \(\langle i j| k l \rangle\) ) represents the two-electron integral

    \[\int \phi^*_i(r) \phi^*_j(r') g(r,r') \phi_k(r)\phi_l(r') drdr'.\]

    The notation \(\langle i j | k l \rangle\) 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 \(r,\theta,\phi,\sigma\) and \(r',\theta',\phi',\sigma'\) (with \(\sigma\) and \(\sigma'\) being the \(\alpha\) or \(\beta\) spin functions).

    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 \(\phi^*\) involves the adjoint of one of the \(\alpha\) or \(\beta\) spin functions) can be carried out using \(\langle a|a\rangle =1\), \(\langle a|b\rangle =0\), \(\langle b|a\rangle =0\), \(\langle b|b\rangle =1\), thereby yielding integrals over spatial orbitals.