Coupled perturbed Kohn-Sham method

let us consider how to calculate chemical properties other than excitation energies. The key is that most spectroscopic properties are response properties, which are proportional to the derivatives of energy in terms of various perturbations (Jensen 2006), 

Energy derivatives are represented as perturbation terms. Assume the perturbed Hamiltonian operator of the Kohn-Sham method as 

Based on the Rayleigh-Schrodinger perturbation theory (Schrodinger 1926), the first and second energy derivatives are written using the notation in Sect. 3.4 as 

However, it seems impossible to obtain the derivative exactly, because Eq. (4.48) contains the sum for all excitations created from a Kohn-Sham electron configuration. What makes it possible is the coupled perturbed Kohn-Sham method. 

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. 

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The vector potential is also incorporated in the coupled-perturbed Kohn-Sham method. In this method, the following term is supplemented to matrix (F ) in Eq.(4.61) ... 

Kamiya M, Sekino H, Tsuneda T, Hirao K (2005) Nonlinear optical property calculations by the long-range-corrected coupled-perturbed Kohn-Sham method. J Chem Phys 122(23) 234111... 

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. 

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. 

N(C2H2)bNH2 with respect to the number of units n. The DFT (LC-BOP, BOP, and B3LYP) results were obtained by the coupled-pertuibed Kohn-Sham method (see Sect. 4.7), the HF result was given by the coupled-perturbed Hartree-Fock method, and the ab initio results were provided by the finite-field method (see Sect. 4.7). The aug-cc-pVDZ basis functions tire used. See Kamiya et al. (2005)... 

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. 

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. 

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. 

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.

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