Density functional eigenvalues

C.-O. Almbladh, U. von, and Barth, Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues, Phys. Rev. B 31, 3231-3244 1985. 

Wi.th.in the same app3roxi]nation we can derive also an explicit expression for the gap correction A to the density-functional eigenvalues (see sec. II). A is found to be... 

In recent years density-functional methods32 have made it possible to obtain orbitals that mimic correlated natural orbitals directly from one-electron eigenvalue equations such as Eq. (1.13a), bypassing the calculation of multi-configurational MP or Cl wavefunctions. These methods are based on a modified Kohn-Sham33 form (Tks) of the one-electron effective Hamiltonian in Eq. (1.13a), differing from the HF operator (1.13b) by inclusion of a correlation potential (as well as other possible modifications of (Fee(av))-... 

Thus, the response kernel for the interacting system can be obtained from that of the noninteracting system if one has a suitable functional form for the XC energy density functional for TD systems. The standard form for the kernel yo(r, r" Kohn Sham orbitals (/ (r), their energy eigenvalues sk, and the occupation numbers nk, is given [17,19] by... 

Consequently, from the density the Hamiltonian can be readily obtained, and then every property of the system can be determined by solving the Schrodinger equation to obtain the wave function. One has to emphasize, however, that this argument holds only for Coulomb systems. By contrast, the density functional theory formulated by Hohenberg and Kohn is valid for any external potential. Kato s theorem is valid not only for the ground state but also for the excited states. Consequently, if the density n, of the f-th excited state is known, the Hamiltonian H is also known in principle and its eigenvalue problem ... 

Figure 33. The stability of yeast glycolysis A Monte Carlo approach. A Shown in the distribution of the largest positive real part within the spectrum of eigenvalues, depicted from above (contour plot). Darker colors correspond to an increased density of eigenvalues. Instances with > 0 are unstable. B The probability that a random instance of the Jacobian corresponds to an unstable metabolic state as a function of the feedback strength 0, . The loss of stability occurs either via in a saddle node (SN) or via a Hopf (HO) bifurcation.

A commonly used quantity to present the information obtained from a first-principles calculation based on the density-functional method is the local density of states (LDOS) at every energy value below the Fermi level at zero absolute temperature. Because every state has an energy eigenvalue, the information with both spatial and energetic distributions is important for many experiments involving energy information. The LDOS p(r, ) at a point r and at an energy level E is defined as... 

Density functional theory (DFT) calculations have been carried out to elucidate the structure and energetics of the various isomeric (Si-N) rings as well as the corresponding anions and dianions. The local minima on the potential energy surfaces were verified by computation of the eigenvalues of the respective Hessian matrices. From these, harmonic vibrational frequencies and the zero-point vibration corrected energetics were calculated. The calculations were carried out for isolated molecules in the gas phase. The theoretical results are expected to be reliable for molecules in non-polar or weakly aprotic polar solvents. 

This transition-state-like point is called a bond critical point. All points at which the first derivatives are zero (caveat above) are critical points, so the nuclei are also critical points. Analogously to the energy/geometry Hessian of a potential energy surface, an electron density function critical point (a relative maximum or minimum or saddle point) can be characterized in terms of its second derivatives by diagonalizing the p/q Hessian([Pg.356]

In Kohn-Sham density functional theory, the ionization potential is the negative of the eigenvalue of the highest occupied Kohn-Sham orbital. 86-88 The IP = —sH0M0 relation holds, however, only for the exact exchange-correlation potential. Numerical confirmations for this relation exist for model systems such as the... 

Fig. 18. Ground state electronic structures for cresyl and o-(methylthio)cresyl phenoxyl radicals. Isosurface representations of molecular orbitals solved by ab initio density functional theory methods for cresyl (ere) and o-(methylthio)cresyl (mtc) phenoxyl radicals. Eigenvalues are listed and for each the SOMO is identihed with an asterisk ( ).

The density functional theory for ensembles is based on the generalized Rayleigh-Ritz variational principle [7]. The eigenvalue problem of the Hamiltonian H is given by... 

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