Interval finite element

Within computational chemistry, 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, or in other words, the original 3N dimensional integral is expressed in terms of many three- and six-dimensional integrals.

The rules are used in deriving the working equations for all methods of approximately solving the Schrödinger equation that employ wavefunctions constructed from Slater determinants. These include Hartree-Fock theory, where the wavefunction is a single determinant, and all those methods which use Hartree-Fock theory as a reference such as Møller-Plesset perturbation theory, and Coupled cluster and Configuration interaction theories.

In 1929 John C. Slater derived expressions for diagonal matrix elements of an approximate Hamiltonian while investigating atomic spectra within a perturbative approach. The following year Edward Condon extended the rules to non-diagonal matrix elements. In 1955 Per-Olov Löwdin further generalized these results for wavefunctions constructed from non-orthonormal orbitals, leading to what are known as the Löwdin rules.

Mathematical background

In terms of an antisymmetrization operator ($\mathcal{𝒜}$) acting upon a product of N orthonormal spin-orbitals (with r and σ denoting spatial and spin variables), a determinantal wavefunction is denoted as

$|\mathrm{\Psi }⟩=\mathcal{𝒜}({\varphi }_1({\mathbf{r}}_1{\sigma }_1){\varphi }_2({\mathbf{r}}_2{\sigma }_2)\cdots {\varphi }_m({\mathbf{r}}_m{\sigma }_m){\varphi }_n({\mathbf{r}}_n{\sigma }_n)\cdots {\varphi }_N({\mathbf{r}}_N{\sigma }_N)).$

A wavefunction differing from this by only a single orbital (the m'th orbital) will be denoted as

$|{\mathrm{\Psi }}_m^p⟩=\mathcal{𝒜}({\varphi }_1({\mathbf{r}}_1{\sigma }_1){\varphi }_2({\mathbf{r}}_2{\sigma }_2)\cdots {\varphi }_p({\mathbf{r}}_m{\sigma }_m){\varphi }_n({\mathbf{r}}_n{\sigma }_n)\cdots {\varphi }_N({\mathbf{r}}_N{\sigma }_N)),$

and a wavefunction differing by two orbitals will be denoted as

$|{\mathrm{\Psi }}_{mn}^{pq}⟩=\mathcal{𝒜}({\varphi }_1({\mathbf{r}}_1{\sigma }_1){\varphi }_2({\mathbf{r}}_2{\sigma }_2)\cdots {\varphi }_p({\mathbf{r}}_m{\sigma }_m){\varphi }_q({\mathbf{r}}_n{\sigma }_n)\cdots {\varphi }_N({\mathbf{r}}_N{\sigma }_N)).$

For any particular one- or two-body operator, Ô, the Slater–Condon rules show how to simplify the following types of integrals:

$⟨\mathrm{\Psi }\left|\hat{O}\right|\mathrm{\Psi }⟩,⟨\mathrm{\Psi }\left|\hat{O}\right|{\mathrm{\Psi }}_m^p⟩,\mathrm{a}\mathrm{n}\mathrm{d}⟨\mathrm{\Psi }\left|\hat{O}\right|{\mathrm{\Psi }}_{mn}^{pq}⟩.$

Matrix elements for two wavefunctions differing by more than two orbitals vanish unless higher order interactions are introduced.

Integrals of one-body operators

One body operators depend only upon the position or momentum of a single electron at any given instant. Examples are the kinetic energy, dipole moment, and total angular momentum operators.

A one-body operator in an N-particle system is decomposed as

$\hat{F}=\sum _{i=1}^N\hat{f}(i).$

The Slater–Condon rules for such an operator are:

$\begin{array}{rl}⟨\mathrm{\Psi }\left|\hat{F}\right|\mathrm{\Psi }⟩& =\sum _{i=1}^N⟨{\varphi }_i\left|\hat{f}\right|{\varphi }_i⟩,\\ ⟨\mathrm{\Psi }\left|\hat{F}\right|{\mathrm{\Psi }}_m^p⟩& =⟨{\varphi }_m\left|\hat{f}\right|{\varphi }_p⟩,\\ ⟨\mathrm{\Psi }\left|\hat{F}\right|{\mathrm{\Psi }}_{mn}^{pq}⟩& =0.\end{array}$

Integrals of two-body operators

Two-body operators couple two particles at any given instant. Examples being the electron-electron repulsion, magnetic dipolar coupling, and total angular momentum-squared operators.

A two-body operator in an N-particle system is decomposed as

$\hat{G}=\frac{1}{2}\sum _{i=1}^N\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^N\hat{g}(i,j).$

The Slater–Condon rules for such an operator are:

$\begin{array}{rl}⟨\mathrm{\Psi }\left|\hat{G}\right|\mathrm{\Psi }⟩& =\frac{1}{2}\sum _{i=1}^N\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^N(⟨{\varphi }_i{\varphi }_j|\hat{g}|{\varphi }_i{\varphi }_j⟩-⟨{\varphi }_i{\varphi }_j|\hat{g}|{\varphi }_j{\varphi }_i⟩),\\ ⟨\mathrm{\Psi }\left|\hat{G}\right|{\mathrm{\Psi }}_m^p⟩& =\sum _{i=1}^N(⟨{\varphi }_m{\varphi }_i|\hat{g}|{\varphi }_p{\varphi }_i⟩-⟨{\varphi }_m{\varphi }_i|\hat{g}|{\varphi }_i{\varphi }_p⟩),\\ ⟨\mathrm{\Psi }\left|\hat{G}\right|{\mathrm{\Psi }}_{mn}^{pq}⟩& =⟨{\varphi }_m{\varphi }_n\left|\hat{g}\right|{\varphi }_p{\varphi }_q⟩-⟨{\varphi }_m{\varphi }_n\left|\hat{g}\right|{\varphi }_q{\varphi }_p⟩,\end{array}$

where

$⟨{\varphi }_i{\varphi }_j\left|\hat{g}\right|{\varphi }_k{\varphi }_l⟩=\int \mathrm{d}\mathbf{r}\int \mathrm{d}{\mathbf{r}}^{\prime }{\varphi }_i^{*}(\mathbf{r}){\varphi }_j^{*}({\mathbf{r}}^{\prime })g(\mathbf{r},{\mathbf{r}}^{\prime }){\varphi }_k(\mathbf{r}){\varphi }_l({\mathbf{r}}^{\prime }).$

References

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