Spherical to Cartesian coordinates transformation for solid harmonics revisited: Construction of the Hartree...

Area of Science:

• Quantum chemistry
• Computational physics
• Mathematical physics

Background:

• Spherical harmonic Gaussian-type orbitals and Slater functions are fundamental in quantum chemistry.
• Representing these functions in both spherical and Cartesian coordinates is crucial for various calculations.
• Existing methods for transformation can be computationally intensive or limited in scope.

Purpose of the Study:

• To derive general expressions for transformation coefficients between spherical and Cartesian representations of spherical harmonic Gaussian-type orbitals and Slater functions.
• To tabulate these coefficients for efficient use in computational methods.
• To demonstrate the application of these coefficients in constructing the Hartree potential.

Main Methods:

• Derivation of general analytical expressions for transformation coefficients.
• Tabulation of coefficients up to quantum number ℓ = 10.
• Application of derived formulas to multipole expansion of the Hartree potential.

Main Results:

• General formulas for transformation coefficients are presented.
• Tabulated values for coefficients up to ℓ = 10 are provided.
• The method is successfully applied to construct the Hartree potential using multipole expansion.

Conclusions:

• The derived transformation coefficients offer a computationally efficient way to switch between coordinate systems for atomic orbitals.
• The tabulated values simplify the implementation of these transformations in quantum chemical software.
• This work facilitates more accurate and efficient calculations of electronic structure, particularly in methods involving multipole expansions.