Coupled perturbed Kohn-Sham equation

Using Eqs. (4.61) and (4.63), matrix U is calculated to give the response properties in terms of the uniform electric field dipole moments, polarizabilities, hyperpolarizabilities, and so forth. Equation (4.61) is called the coupled perturbed Kohn-Sham equation. Other response properties are calculated by solving Eq. (4.61) after setting the first derivative of the Fock operator, F, in terms of each perturbation. Note, however, that this method has problems in actual calculations similarly to the time-dependent response Kohn-Sham method. For example, using most functionals, this method tends to overestimate the electric field response properties of long-chain polyenes. 

The polarizability and first hyperpolarizability of p-nitroaniline and its methyl-substituted derivatives have been calculated using a non-iterative approximation to the coupled-perturbed Kohn-Sham equation where the first-order derivatives of the field-dependent Kohn-Sham matrix are estimated using the finite field method" . This approximation turns out to be reliable with differences with respect to the fully coupled-perturbed Kohn-Sham values smaller than 1% and 5% for a and p, respectively. The agreement with the MP2 results is also good, which enables to employ this simplified method to deduce structure-property relationships. 

The If1 coefficients corresponding to occupied-virtual orbital pairs can be obtained from the magnetic form of the coupled-perturbed Kohn-Sham (CPKS) equations... 

In the coupled perturbed Kohn-Sham method, the first wavefunction derivatives are given by calculating the first derivatives of the orbitals in terms of perturbations. The Kohn-Sham method is based on the Slater determinant. Therefore, since the Kohn-Sham wavefunction is represented with orbitals, the corresponding first wavefunction derivatives are also described by the first derivatives of the orbitals. For simplicity, let us consider the Kohn-Sham-Roothaan equation in Eq. (4.13), which is a matrix equation using basis functions based on the Roothaan method. 

The first derivative of the density matrix with respect to the magnetic induction (dPfiv/dBi) is obtained by solving the coupled-perturbed Hartree-Fock (or Kohn-Sham) equations to which the first derivative of the effective Fock (or Kohn-Sham) operator with respect to the magnetic induction contributes. Due to the use of GIAOs, specific corrections arising from the effective operator Hcnv describing the environment effects will appear. We refer to Ref. [28] for the PCM model and to Ref. [29] for the DPM within either a HF or DFT description of the solute molecule. 

Frequently, noncollinearity is due to spin-orbit coupling. Although spin-orbit terms can be added as a perturbation to the equations of SDFT, a complete description requires a relativistic formulation. A generalization of DFT that does account for spin-orbit coupling and other relativistic effects is rdft. ° Here the fundamental variable is the relativistic four-component current and the Kohn-Sham equation is now of the form of the single-particle Dirac equation, instead of the Schrodinger equation. 

On the contrary, the density derivatives are necessary to get higher order free energy derivatives in particular the second free energy derivatives require the first derivative of the density matrix P , which can be obtained by solving an appropriate coupled perturbed Hartree-Fock (CPHF) or Kohn-Sham (CPKS) equations. 

The systematic derivation of implicit correlation functionals is discussed in Sect. 2.4. In particular, perturbation theory based on the Kohn-Sham (KS) Hamiltonian [16,17,18] is used to derive an exact relation for l xc- This expression is then expanded to second order in the electron-electron coupling constant in order to obtain the simplest first-principles correlation functional [18]. The corresponding OPM integral equation as well as extensions like the random phase approximation (RPA) [19,20] and the interaction strength interpolation (ISI) [21] are also introduced. 

---