Sum-over-states density-functional perturbation theory

Sum-Over-States Density Functioned Perturbation Theory 

To describe the system in the presence of a small perturbation, we will use the SOS-DFPT approach of ref. [28] from which this section is, for a large part, taken. Here only a brief outline will follow. The interested reader can find more details in the original paper [28]. We will consider a system in the presence of an infinitesimal purely imaginary perturbation with operator i u Hi (i = y/-i and A is the parameter of the perturbation). Therefore, the first-order correction to the many-electron wave function 

As an approximation to the exact many-electron wave function of the real system, we will use the Slater determinant built from the occupied KS MO s [64]. Although it is only an approximation to the exact many-electron wave function of the real system, it seems to be a reasonable one especially if one is interested in calculations of one-electron matrix elements only (in the case of a local multiplicative operator such an approximation yields the exact values of the matrix elements). To describe an excited state corresponding to the transition of an electron from the occupied MO k into the virtual MO a , we will use the many-electron wave function of the excited state in the form of a Slater determinant that differs from the ground state determinant by replacing the occupied MO k by the virtual MO a . 

To evaluate the denominator in Eqn. (21) one has to calculate the energy difference between Eg, the energy of the ground state, and E%, the energy of the excited state k — a corresponding to the density pk a, where 

In the HF method the evaluation of the energy difference between the ground and the singlet excited state k - a using SOS perturbation theory leads to the following expression 

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Malkin, V. G., Malkina, 0. L., Casida, M. E., Salahub, D. R., 1994, Nuclear Magnetic Resonance Shielding Tensors Calculated With a Sum-Over-States Density Functional Perturbation Theory , J. Am. Chem. Soc., 116, 5898. 

S = 103 ppm, which is in excellent agreement with the chemical shifts found earlier in other dioxiranes such as lb, Ic, and Id. The experimental H, and NMR spectra are in good agreement with calculated (sum-over-states density-functional perturbation theory, SOS-DFPT) chemical shifts. The oxygen nuclei in 4 are somewhat more shielded in the NMR spectrum (5 = 321 ppm) than those in DMDO lb. 

The PSO contribution is associated with the interactions between the operators H010 for nuclei TV and M (see Eqn. (7)), respectively. To calculate the PSO contribution, we used Sum-Over-States Density Functional Perturbation Theory (SOS-DFPT). The SOS-DFPT approach leads to the well-known equations... 

Malkin VG, Malkina OL, Casida ME, Salahub DR (1994) Nuclar magnetic resonance shielding tensors calculated with a sum-over-states density functional perturbation theory. J Am Chem Soc 116 5898-5908... 

Density functional theory (DFT) calculations employing sum-over-states DF perturbation theory were applied to calculate both the H and chemical shifts of 1,3-dioxane, 1,3-oxathiane, 1,3-dithiane, and the parent cyclohexane <1997JMT(418)231>. Both normal and anomalous trends in the H chemical shifts could be reproduced well and. 

We saw earlier that a very simple form of the dispersion energy is obtained from frequency-dependent polarizabilities at the so-called uncoupled Hartree-Fock level. The sum over states appearing in second order RS perturbation theory is simply a sum over (occupied and virtual) orbitals. A first improvement of this simple model is obtained by including apparent correlation [140], i.e. by using frequency-dependent polarizabilities obtained from the TDCHF method [36,141]. This method was initially proposed in the context of the multipole expansion, but could be generalized [142-146] to charge density susceptibility functions (or polarization propagators), which avoids the use... 

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