# 6.6: The Slater-Condon Rules

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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.

6.6: The Slater-Condon Rules is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.