Density functional derivatives

Rauhut, G., Pulay, P., 1995, Transferable Scaling Factors for Density Functional Derived Vibrational Force Fields , J. Phys. Chem., 99, 3093. 

In order to determine atomic charges, Su performed a least-squares fit to the EP values calculated at sites of atomic nuclei [75], Thus, the approximating character of fitting on grid points outside the van der Waals surface of the molecule was eliminated. For calculating atomic charges, electron density functions derived from XRD data [76] were applied. 

This formula was used by Slater [385] to define an effective local exchange potential. The generally unsatisfactory results obtained in calculations with this potential indicate that the locality hypothesis fails for the density functional derivative of the exchange energy Ex [294],... 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

Thesis Are density functional derivatives really local functions 2... 

THESIS ARE DENSITY FUNCTIONAL DERIVATIVES REALLY LOCAL FUNCTIONS ... 

A deeper argument is that local density functional derivatives appear to be implied by functional analysis [2,21,22]. The KS density function has an orbital structure, p = Y.i niPi = X fa- For a density functional Fs, strictly defined only for normalized ground states, functional analysis implies the existence of functional derivatives of the form SFj/ Sp, = e, — v(r), where the constants e, are undetermined. On extending the strict ground-state theory to an OFT in which OEL equations can be derived, these constants are determined and are just the eigenvalues of the one-electron effective Hamiltonian. Since they differ for each different orbital energy level, the implied functional derivative depends on a direction in the function-space of densities. Such a Gateaux derivative [1,2] is equivalent in the DFT context to a linear operator that acts on orbital functions [23]. 

The mathematical issues relevant to the definition of density functional derivatives can be considered in the simple model of noninteracting electrons. As in the KSC [4], this singles out the kinetic energy. The /V-electron Hamiltonian operator is H = T + V. Orbital functional derivatives determine the noninteracting OEL equations... 

For fixed normalization the Lagrange multiplier terms in 8Ts vanish. If these constants are undetermined, it might appear that they could be replaced by a single global constant pt. If so, this would result in the formula [22] 8Ts = J d3r p, — v(r) 8p(r). Then the density functional derivative would be a local function vr(v) such that STj/Sp = Vj-(r) = ix — v(r). This is the Thomas-Fermi equation, so that the locality hypothesis for vT implies an exact Thomas-Fermi theory for noninteracting electrons. 

To apply this theory to calculate surface tensions one simply uses the free energy density function derived from equation (2.5.8) in conjunction with the square gradient theory of section 2.4. As usual this leads to a smoothly var3dng density profile between the liquid and vapour phase, which may be visualised in terms of the FOV model as a variation in cell size through the stnface, as shown schematically in figure 2.24. 

The relaxation dynamics of junctions in polymer networks have not been well-known until the advent of solid-state NMR spin-lattice relaxation measurements in a series of poly(tetrahydrofuran) networks with tris(4-isocyanatophenyl)-thiophosphate junctions [100]. The junction relaxation properties were studied in networks with molecular weights between crosslinks. Me, ranging from 250 to 2900. The dominant mechanism for nuclear spin lattice relaxation times measured over a wide range of temperatures were fit satisfactorily by spectral density functions, /( ), derived from the Fourier transforms of the Kohlrausch stretched exponential correlation functions... 

Rappoport D, Furche F (2005) Analytical time-dependent density functional derivative methods within the RI-J approximation, an approach to excited states of large molecules. J Chem Phys 122(064105) 1-8... 

Density Functional Derivative Methods within the Rl-J Approximation, an Approach to Excited States of Large Molecules. 

Y. Qin and R. A. Wheeler,/. Phys. Chem., 100, 10554 (1996). Density-Functional-Derived Structures, Spin Properties, and Vibrations for Phenol Radical Cation. 

Most density functionals derive from arguments that are complicated, dubious or both. The elegant derivation of TF27 is an exception, however, and is worthy of close examination. Consider a cube with sidelength L and volume V and place the origin at one comer of the cube. The Schrodinger equation for the state m of a single electron confined within the cube is... 

---