In mathematics and mathematical physics, Slater integrals are certain integrals of products of three

spherical harmonic In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a ...

s. They occur naturally when applying an

orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...

of functions on the

unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...

that transform in a particular way under rotations in three dimensions. Such integrals are particularly useful when computing properties of atoms which have natural spherical symmetry. These integrals are defined below along with some of their mathematical properties.

Formulation

In connection with the

quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ...

atomic structure Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, an ...

John C. Slater John Clarke Slater (December 22, 1900 – July 25, 1976) was a noted American physicist who made major contributions to the theory of the electronic structure of atoms, molecules and solids. He also made major contributions to microwave electroni ...

defined the integral of three spherical harmonics as a coefficient

c

.John C. Slater, Quantum Theory of Atomic Structure, McGraw-Hill (New York, 1960), Volume I These coefficients are essentially the product of two Wigner 3jm symbols. :

c^k(\ell,m,\ell',m')=\int d^2\Omega \ Y_\ell^m(\Omega)^* Y_^(\Omega) Y_k^(\Omega)

These integrals are useful and necessary when doing atomic calculations of the Hartree–Fock variety where matrix elements of the

Coulomb operator The Coulomb operator, named after Charles-Augustin de Coulomb, is a quantum mechanics, quantum mechanical operator (mathematics), operator used in the field of quantum chemistry. Specifically, it is a term found in the Hartree–Fock, Fock operato ...

and

Exchange operator In quantum mechanics, the exchange operator \hat, also known as permutation operator, is a quantum mechanical operator that acts on states in Fock space. The exchange operator acts by switching the labels on any two identical particles describ ...

are needed. For an explicit formula, one can use Gaunt's formula for associated Legendre polynomials. Note that the product of two spherical harmonics can be written in terms of these coefficients. By expanding such a product over a

basis with the same order :

Y_\ell^m Y_^ = \sum_ \hat^(\ell,m,\ell',m',) Y_^,

one may then multiply by

Y^*

and integrate, using the conjugate property and being careful with phases and normalisations: :

\int Y_\ell^m Y_^ Y_^ d^2\Omega = (-1)^\hat^(\ell,m,\ell',m') = (-1)^c^L(\ell,-m,\ell',m').

Hence :

Y_\ell^m Y_^ = \sum_(-1)^ c^(\ell,-m,\ell',m',) Y_^,

These coefficient obey a number of identities. They include ::

\begin
c^k(\ell,m,\ell',m') &= c^k(\ell,-m,\ell',-m')\\
&=(-1)^c^k(\ell',m',\ell,m)\\
&=(-1)^\sqrtc^\ell(\ell',m',k,m'-m)\\
& = (-1)^\sqrtc^(k,m-m',\ell,m).\\
\sum_^ c^k(\ell,m,\ell,m)  &=  (2\ell+1)\delta_.\\
\sum_^\ell \sum_^ c^k(\ell,m,\ell',m')^2  &=  \sqrt\cdot c^k(\ell,0,\ell',0).\\
\sum_^\ell c^k(\ell,m,\ell',m')^2 & =  \sqrt\cdot c^k(\ell,0,\ell',0).\\
\sum_^\ell c^k(\ell,m,\ell',m')c^k(\ell,m,\tilde\ell,m')  &=  \delta_\cdot\sqrt\cdot c^k(\ell,0,\ell',0).\\
\sum_m c^k(\ell,m+r,\ell',m) c^k(\ell,m+r,\tilde\ell,m)  &=  \delta_ \cdot \frac\cdot c^k(\ell,0,\ell',0).\\
\sum_m c^k(\ell,m+r,\ell',m)c^q(\ell,m+r,\ell',m)  &= \delta_\cdot\frac\cdot c^k(\ell,0,\ell',0).
\end

References

Atomic physics Quantum chemistry Rotational symmetry {{Quantum-chemistry-stub