Integral Transformations

In this section the integral transformations will be described.

AO to MO basis 2 electron integrals

The integral transformation of the AO two electron integrals to the MO two electron integrals is given as:

\[\left(ij\left.\right|kl\right)=\sum_{\mu}\sum_{\nu}\sum_{\lambda}\sum_{\sigma}C_{\mu}^{i}C_{\nu}^{j}\left(\mu\nu\left.\right|\lambda\sigma\right)C_{\lambda}^{k}C_{\sigma}^{l}\]

FUNCTION:

  • IntegralTransform.Transform2eMO(C, Vee)
  • return VeeMO

Input:

  • C, MO coefficients from SCF calculation
  • Vee, two electron integrals in AO basis

Output:

  • VeeMO, two electron integrals in MO basis

References:

MO spatial to MO spin basis 2 electron integrals

The integral transformation from MO spatial to MO spin orbitals is given by the following equation:

\[\left\langle \left.ij\right|kl\right\rangle =\left(\left.\sigma_{1}i\sigma_{2}k\right|\sigma_{3}j\sigma_{4}l\right)\delta_{\sigma_{1}\sigma_{2}}\delta_{\sigma_{3}\sigma_{4}}\]

FUNCTION:

  • IntegralTransform.Transform2eSPIN(Vee)
  • return VeeSpin

Input:

  • Vee, two electron integrals

Output:

  • VeeSpin, two electron integrals in spinbasis

References:

  • None