Theory

QUARTIC2D starts from the physical four-center matrix element and reduces it to reusable angular-harmonic form factors and one-dimensional radial momentum integrals. The reduction separates three ingredients that should remain conceptually distinct:

• the orbital or transition-field structure, represented by \(F_m(q)\);

• the scalar radial interaction, represented by \(U(q)\);

• the relative geometry, represented by translation-Bessel factors and the displacement angle.

This separation is the reason a transition-field transform can be reused across many displacement vectors and, when its represented momentum interval is sufficient, across different radial kernels.

• Four-center reduction
  • From four orbitals to two transition fields
  • Momentum-space form
  • Angular harmonics
  • Displacement integral
  • Computational factorization and reuse
  • Equivalent nonnegative-order form
  • Zero displacement
• Fourier and coordinate conventions
  • Real-space angular expansion
  • Transition-field Fourier transform
  • Kernel Fourier transform
  • Displacement sign
  • Negative Hankel order
  • Units

The derivation is written for two general complex transition fields. Density-density terms are a special case in which the relevant orbital products happen to be densities. The assumptions that delimit the reduction are collected separately in Limitations and scope.