Coupled perturbed Kohn-Sham theory

F. Neese. Metal and ligand hyperfine couplings in transition metal complexes The effect of spin-orbit coupling as studied by coupled perturbed Kohn-Sham theory. /. Chem. Phys., 118(9) (2003) 3939-3948. 

In principle, density functional theory calculations should be able to give answers that are more reliable than Hartree-Fock but at similar cost. Static a and can be calculated by finite field methods or by coupled perturbed Kohn-Sham theory (CPKS) and give answers that are broadly comparable with MP2. In 1986 Sennatore and Subbaswamy did some calculations of the dynamic polarizability and second hyperpolarizability of rare gas atoms, but there have been no calculations of frequency dependent polarizabilities or hyperpolarizabilities of molecules until very recently. 

Hesselmann A, Jansen G (2002) Intermolecular induction and exchangeinduction energies from coupled-perturbed Kohn—Sham density functional theory. Chem Phys Lett 362 319—325... 

Kohn-Sham formalism, which suffers from the spin contamination problem, and on the sum-over-states or coupled perturbed Kohn-Sham approaches. In recent articles devoted to computations of elechonic g-tensors we advocated the use of an alternative approach, namely linear response theory based on the spin-restricted open-shell Kohn-Sham formalism, which is free from spin contamination problem (see Theory section). In the following we briefly review the applicability of this approach for some paramagnetic compounds. 

Complete Active Space (CAS) 141 configuration interaction singles (CIS) 89, 93-95 coupled cluster (CC) theory 140, 142 coupled perturbed Hartree-Fock (CPHF) 19 coupled perturbed Kohn-Sham (CPKS) 19 CPCM 9... 

Hesselmann, A., 8c Jansen, G. (2002b). Intermolecular induction and exchange-induction energies from coupled-perturbed Kohn-Sham density functional theory. Chemical Physics Letters, 362, 319-325. 

Shedge, S. V., Carmona-Espindola, J., Pal, S., Koster, A. M. (2010). Comparison of the auxiliary density perturbation theory and the noniterative approximation to the coupled perturbed Kohn-Sham method Case study of the polarizabilities of disubstituted azoarene molecules. Journal of Physical Chemistry A, 114, 2357. 

Neese F. 2001. Prediction of electron paramagnetic resonance g values using coupled perturbed Hartree-Fock and Kohn-Sham theory. J Chem Phys 115 11080-11096. 

Most molecular quantum-mechanical methods, whether SCF, Cl, perturbation theory (Section 16.3), coupled cluster (Section 16.4), or density functional (Section 16.5), begin the calculation with the choice of a set of basis functions Xn which are used to express the MOs (pi as = IiiCriXr [Eq. (14.33)]. (Density-functional theory uses orbitals called Kohn-Sham orbitals P that are expressed as (pf = 1,iCriXn see Section 16.5.) The use of an adequate basis set is an essential requirement for success of the calculation. 

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. 

Table 1 Electric dipole polarizabilities of benzene and naphthalene in the coupled and uncoupled Kohn-Sham perturbation theory...

Unlike the true propagator, the UCHF approximation is given by a simple closed formula and reqnires only minimum computational effort to evalnate on the fly if the orbitals are available. The nnconpled Hartree-Fock/Kohn-Sham approximation has almost completely vanished from the chemistry literature about 40 years ago when modem derivative techniques became available because of the poor results it produced for second-order properties. Some systematic expositions of analytical derivative methods still use it as a starting point, but it is in our opinion pedagogi-cally inappropriate, as it requires considerable effort to recover the coupled-perturbed Hartree-Fock results which can be derived in a simpler way. UCHF/UCKS is still used in some approximate theories, but we suspect that its only merit is easy computability. According to Geerlings et al. [29], the polarizabilities derived from the uncoupled density response function correlate well with accurate results but can be off by up to a factor of 2, and thus they are only qualitatively useful. Our results in Table 1 confirm this. 

Figure 1 A family tree of quantum chemistry DFT, density functional theory QMC, quantum Monte Carlo RRV, Rayleigh-Ritz variational theory X-a, X-alpha method KS, Kohn-Sham approach LDA, BP, B3LYP, density functional approximations VQMC, variational QMC DQMC, diffusion QMC FNQMC, fixed-node QMC PIQMC, path integral QMC EQMC, exact QMC HF, Hartree-Fock EC, explicitly correlated functions P, perturbational MP2, MP4, Maller-Plesset perturbational Cl, configuration interaction MRCI, multireference Cl FCI, full Cl CC, CCSD(T), coupled-cluster approaches. Other acronyms are defined in the text.

The ESR hyperfine coupling is determined by triplet perturbations. Thus, in principle one should use an unrestricted wave function to describe the reference state. However, it is also possible to use a spin-restricted wave function (Fernandez et al. 1992) and take into account the triplet nature of the perturbation in the definition of the response. Within such a (e.g., SCF or MCSCF) restricted-unrestricted approach, first-order properties are given as the sum of the usual expectation value term and a response correction that takes into account the change of the wave function induced by the perturbation (of the type (0 H° 0)). This restricted-unrestricted approach has also been extended to restricted Kohn-Sham density functional theory (Rinkevicius et al. 2004). 

Ferrero, M., Civalleri, B., Rerat, M., Orlando, R., and Dovesi, R. (2009) The calculation of the static first and second susceptibilities of crystalline urea A comparison of Hartree-Fock and density functional theory results obtained with the periodic coupled perturbed Hartree-Fock/Kohn-Sham scheme. 

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