Electronic structure wave-function calculations

Luce T A and Bennemann K H 1998 Nonlinear optical response of noble metals determined from first-principles electronic structures and wave functions calculation of transition matrix elements P/rys. Rev. B 58 15 821-6... 

The adiabatic approximation means the neglect of the nuclear motion in the Schrodinger equation. The electronic structure is thus calculated for a set of fixed nuclear coordinates. This approach can in principle be exact if one uses the set of wave functions for fixed nuclear coordinates as a basis set for the full Schrodinger equation, and solves the nuclear motion on this basis. The adiabatic approximation stops at the step before. (The Born-Oppenheimer approximation assumes a specific classical behavior of the nuclei and hence it is more approximate than the adiabatic approximation.)... 

The left-hand side of Eq. (177) has a structure similar to the electronic gradient vector in variational wave function calculations. Unlike variational calculations, Eq. (177) cannot be used to determine the response parameters t(n) in a CC calculation. However, for the calculation of the nth geometrical derivative W n Eq. (177) eliminates r(n), which would otherwise appear in the calculation. In fact, we show below that the calculation of (3N-6)n response amplitudes t(n) is replaced by the solution of one set of linear equations of similar but simpler structure. By inserting Eq. (176) in Eq. (177) and rearranging terms, we see that X fulfills the equations... 

Abstract. We have calculated the scalar and tensor dipole polarizabilities (/3) and hyperpolarizabilities (7) of excited ls2p Po, ls2p P2- states of helium. Our theory includes fine structure of triplet sublevels. Semiempirical and accurate electron-correlated wave functions have been used to determine the static values of j3 and 7. Numerical calculations are carried out using sums of oscillator strengths and, alternatively, with the Green function for the excited valence electron. Specifically, we present results for the integral over the continuum, for second- and fourth-order matrix elements. The corresponding estimations indicate that these corrections are of the order of 23% for the scalar part of polarizability and only of the order of 3% for the tensor part... 

Resonance hybrids Various possible chemical structures of molecules, each with identical atomic connectivity, but differing in the disposition of electrons. The wave function of the molecule is approximately represented by mixing the wave functions of the contributing structures. The energy calculated for such a mixture is lower, presumably because the representation is more nearly correct than it would be if formally represented by a single structure. 

Unsatisfactory features of these particular calculations are due to the limitations of the HF-RPA procedure to produce quantitatively correct spectroscopic patterns, and to the very small basis set that we adopted in order to carry out evaluations of several thousands of HF ground-electronic-state wave functions and their associated RPA spectra. Such a quantitative inadequacy of the computational model, however, does not alter the physical meaning of the conclusion that can be drawn the appearance of a single broad band in the optical spectrum of Na9 is simply a thermal broadening effect, which, in general, is particularly effective in the presence of several low-lying isomers or in the case of cluster structures characterized by low-frequency vibrations, accompanied by large nuclear displacements due to the low curvature of the BO surface. 

Electron structure calculations often become difficult when transition metals are involved. If the system has incomplete d shells many electron configurations contribute even to the ground state leading to non-dynamical electron correlation. Wave function-based methods with multideterminant references are required for high accuracy. Density functional theory is often successful, but no current functional is reliable for transition metals compounds in general. QMC as a wave function-based method has to use multideterminant... 

It is well known that bulk iron is bcc, and that GGA is needed to find the bcc structure as the ground state in ab initio calculations 39. o. i.42,43 have calculated Fe-bcc bulk with both SZSP and DZSP basis sets. First of all we performed a grid study in order to find the cutoff radius for the pseudoorbitals that minimize the energy of the system. For both SZSP and DZSP basis sets we obtain a cutoff radius of 4.90 a.u. (this basis sets will be the ones to be used later for more complicated Fe systems). The electron density is obtained from the wave functions calculated at 100 k points. No significative changes are obtained including up to 5000 k points. 

The most common form of AIMD simulation employs DFT (see section First Principles Electronic Structure Methods ) to calculate atomic forces, in conjimction with periodic boundary conditions and a plane wave basis set. Using a plane wave basis has two major advantages over atom-centered basis functions (1) there is no basis set superposition error (Boys and Bernardi 1970 Marx and Hutter 2000) and (2) the Pulay correction (Pulay 1969,1987) to the HeUmann-Feynman force, due to basis set incompleteness, vanishes (Marx and Hutter 2000, 2009). 

As shown in Figure 27, an in-phase combination of type-V structures leads to another A] symmetry structures (type-VI), which is expected to be stabilized by allyl cation-type resonance. However, calculation shows that the two shuctures are isoenergetic. The electronic wave function preserves its phase when tr ansported through a complete loop around the degeneracy shown in Figure 25, so that no conical intersection (or an even number of conical intersections) should be enclosed in it. This is obviously in contrast with the Jahn-Teller theorem, that predicts splitting into A and states. 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Semiempirical calculations are set up with the same general structure as a HF calculation in that they have a Hamiltonian and a wave function. Within this framework, certain pieces of information are approximated or completely omitted. Usually, the core electrons are not included in the calculation and only a minimal basis set is used. Also, some of the two-electron integrals are omitted. In order to correct for the errors introduced by omitting part of the calculation, the method is parameterized. Parameters to estimate the omitted values are obtained by fitting the results to experimental data or ah initio calculations. Often, these parameters replace some of the integrals that are excluded. 

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