Slater–Condon rules

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

$|\Psi \rangle ={\mathcal {A}}(\phi _{1}(\mathbf {r} _{1}\sigma _{1})\phi _{2}(\mathbf {r} _{2}\sigma _{2})\cdots \phi _{m}(\mathbf {r} _{m}\sigma _{m})\phi _{n}(\mathbf {r} _{n}\sigma _{n})\cdots \phi _{N}(\mathbf {r} _{N}\sigma _{N})).$

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

$|\Psi _{m}^{p}\rangle ={\mathcal {A}}(\phi _{1}(\mathbf {r} _{1}\sigma _{1})\phi _{2}(\mathbf {r} _{2}\sigma _{2})\cdots \phi _{p}(\mathbf {r} _{m}\sigma _{m})\phi _{n}(\mathbf {r} _{n}\sigma _{n})\cdots \phi _{N}(\mathbf {r} _{N}\sigma _{N})),$

and a wavefunction differing by two orbitals will be denoted as

$|\Psi _{mn}^{pq}\rangle ={\mathcal {A}}(\phi _{1}(\mathbf {r} _{1}\sigma _{1})\phi _{2}(\mathbf {r} _{2}\sigma _{2})\cdots \phi _{p}(\mathbf {r} _{m}\sigma _{m})\phi _{q}(\mathbf {r} _{n}\sigma _{n})\cdots \phi _{N}(\mathbf {r} _{N}\sigma _{N})).$

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

$\langle \Psi |{\hat {O}}|\Psi \rangle ,\langle \Psi |{\hat {O}}|\Psi _{m}^{p}\rangle ,\ \mathrm {and} \ \langle \Psi |{\hat {O}}|\Psi _{mn}^{pq}\rangle .$

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:

${\begin{aligned}\langle \Psi |{\hat {F}}|\Psi \rangle &=\sum _{i=1}^{N}\ \langle \phi _{i}|{\hat {f}}|\phi _{i}\rangle ,\\\langle \Psi |{\hat {F}}|\Psi _{m}^{p}\rangle &=\langle \phi _{m}|{\hat {f}}|\phi _{p}\rangle ,\\\langle \Psi |{\hat {F}}|\Psi _{mn}^{pq}\rangle &=0.\end{aligned}}$

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

${\hat {G}}={\frac {1}{2}}\sum _{i=1}^{N}\sum _{{j=1} \atop {j\neq i}}^{N}\ {\hat {g}}(i,j).$

The Slater–Condon rules for such an operator are:

${\begin{aligned}\langle \Psi |{\hat {G}}|\Psi \rangle &={\frac {1}{2}}\sum _{i=1}^{N}\sum _{j=1 \atop {j\neq i}}^{N}\ {\bigg (}\langle \phi _{i}\phi _{j}|{\hat {g}}|\phi _{i}\phi _{j}\rangle -\langle \phi _{i}\phi _{j}|{\hat {g}}|\phi _{j}\phi _{i}\rangle {\bigg )},\\\langle \Psi |{\hat {G}}|\Psi _{m}^{p}\rangle &=\sum _{i=1}^{N}\ {\bigg (}\langle \phi _{m}\phi _{i}|{\hat {g}}|\phi _{p}\phi _{i}\rangle -\langle \phi _{m}\phi _{i}|{\hat {g}}|\phi _{i}\phi _{p}\rangle {\bigg )},\\\langle \Psi |{\hat {G}}|\Psi _{mn}^{pq}\rangle &=\langle \phi _{m}\phi _{n}|{\hat {g}}|\phi _{p}\phi _{q}\rangle -\langle \phi _{m}\phi _{n}|{\hat {g}}|\phi _{q}\phi _{p}\rangle ,\end{aligned}}$

where

$\langle \phi _{i}\phi _{j}|{\hat {g}}|\phi _{k}\phi _{l}\rangle =\int \mathrm {d} \mathbf {r} \int \mathrm {d} \mathbf {r} '\ \phi _{i}^{*}(\mathbf {r} )\phi _{j}^{*}(\mathbf {r} ')g(\mathbf {r} ,\mathbf {r} ')\phi _{k}(\mathbf {r} )\phi _{l}(\mathbf {r} ').$

Any matrix elements of a two-body operator with wavefunctions that differ in three or more spin orbitals will vanish.