Density functional theory Fourier transform

In density functional theories the potential is determined by the density, and consequently its Fourier components are related to those of the density. One can therefore connect the symmetry properties of the momentum funetions, in other words the transformation... 

Fourier-transform microwave spectroscopy, pulsed-nozzie, 46 114-115 Four-iron clusters, see Density, functional theory... 

Naumov S, Beckert D (2002) Reply to the Comment on A Fourier transform EPR study of uracil and thymine radical anions in aqueous solution)/ by DM Close. Phys Chem Chem Phys 4 45 Naumov S, Barthel A, Reinhold J, Dietz F, Geimer J, Beckert D (2000) Calculation of spin densities of radicals of pyrimidine-type bases by density functional theory. Influence of solvent and comparison with EPR results. Phys Chem Chem Phys 2 4207-4211 Naumov S, Hildenbrand K, von Sonntag C (2001) Tautomers of the N-centered radical generated by reaction of SO4 - with N(1)substituted cytosines in aqueous solution. Calculation of isotropic hyperfine coupling constants by a density functional method. J Chem Soc Perkin Trans 2 1648-1653... 

Finally, we consider density functional theory (DFT) computations of p-space properties. A naive way of calculating p-space properties is to use the Kohn-Sham orbitals obtained from a DFT computation to form a one-electron, r-space density matrix Fourier transform / according to Eq. (14), and proceed further. This approach is incorrect because the Kohn-Sham density matrix F is not the true one and, in fact, corresponds to a fictitious non-interacting system with the same p(r) as the true system. On the other hand, Hamel and coworkers [112] have shown that if the exact Kohn-Sham exchange potential is used, then the spherically averaged momentum densities of the Kohn-Sham orbitals should be very close to those of the Hartree-Fock orbitals. Of course, in practical computations the exact Kohn-Sham exchange potential is not used since it is generally not known. 

CW = continuous-wave CW-NQR = continuous wave NQR DFT = density functional theory EFG = electric field gradient IR = infrared NMR = nuclear magnetic resonance NQR = nuclear quadrupole resonance OSSE = octahedral site stabilization energy PAC = perturbed angular correlation pulse-FT = pulse-fourier transform TMED = tetramethylethylenediamine. 

R. M. Osuna, R. P. Ortiz, V. Hernandez, J. T. L. Navarrete, M. Miyasaka, S. Rajca, A. Rajca and R. Glaser, Helically annelated and cross-conjugated beta-ohgothiophenes a Fourier transform Raman spectroscopic and quantum chemical density functional theory study, J. Phys. Chem. C, 111, 4854-4860 (2007). 

Another approach to the calculation of IR spectra of hydrogen-bonded complexes is based on linear response theory, in which the spectral density is the Fourier transform of the autocorrelation function of the dipole moment operator involved in the IR transition [62,63]. Recently Car-Parrinello molecular dynamics (CPMD) [73] has been used to simulate IR spectra of hydrogen-bonded systems [64-72]. 

Akama, T, and Nakai, H. [2010] Short-time Fourier transform analysis of real-time time-dependent Hartree-Fock and time-dependent density functional theory calculations with Gaussian basis functions, / Chem. Phys., 132, 054104/1-11. 

It has been shown by computer simulation [109-111] and density functional theory [106, 108] that the soft, purely repulsive, radially symmetric potential, V(r), will form cluster crystals at sufficiently high density if its Fourier transform, V(fe), becomes negative for a range of wave vectors. Within mean field approximation, the stability limit of the homogeneous liquid is given by the X-line [108]... 

Z. Feng, C. Liang, W. Wu, Z. Wu, A. R. Santen and C. Li. Carbon monoxide adsorption on molybdenum phosphides Fourier transform infrared spectroscopic and density functional theory studies. /. Phys. Chem. B 107, 2003,13698—13702. 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

Using linear response theory and noting (according to the results at the end of Section 5.1.3) that the (complex) electrical conductivity a is the Fourier transform of the current density autocorrelation function, we obtain from Eqn. (5.75) (see the equivalent Eqn. (5.21))... 

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