Orbitals, wave-function calculations

HyperChem can plot orbital wave functions resulting from semi-empirical and ab initio quantum mechanical calculations. It is interesting to view both the nodal properties and the relative sizes of the wave functions. Orbital wave functions can provide chemical insights. 

If a covalent bond is broken, as in the simple case of dissociation of the hydrogen molecule into atoms, then theRHFwave function without the Configuration Interaction option (see Extending the Wave Function Calculation on page 37) is inappropriate. This is because the doubly occupied RHFmolecular orbital includes spurious terms that place both electrons on the same hydrogen atom, even when they are separated by an infinite distance. 

The zero-order ground state spin-orbit wave functions for these systems, obtained as previously described (41), have therefore been used to calculate the magnetic susceptibilities via the Van Vleck equation... 

It is not possible to use normal AO basis sets in relativistic calculations The relativistic contraction of the inner shells makes it necessary to design new basis sets to account for this effect. Specially designed basis sets have therefore been constructed using the DKH Flamiltonian. These basis sets are of the atomic natural orbital (ANO) type and are constructed such that semi-core electrons can also be correlated. They have been given the name ANO-RCC (relativistic with core correlation) and cover all atoms of the Periodic Table.36-38 They have been used in most applications presented in this review. ANO-RCC are all-electron basis sets. Deep core orbitals are described by a minimal basis set and are kept frozen in the wave function calculations. The extra cost compared with using effective core potentials (ECPs) is therefore limited. ECPs, however, have been used in some studies, and more details will be given in connection with the specific application. The ANO-RCC basis sets can be downloaded from the home page of the MOLCAS quantum chemistry software (http //www.teokem.lu.se/molcas). 

An ab initio calculation uses the correct molecular electronic Hamiltonian (1.275) and does not introduce experimental data (other than the values of the fundamental physical constants) into the calculation. A semiempirical calculation uses a Hamiltonian simpler than the correct one, and takes some of the integrals as parameters whose values are determined using experimental data. The Hartree-Fock SCF MO method seeks the orbital wave function 0 that minimizes the variational integral <(4> //el initio method. Semiempirical methods were developed because of the difficulties involved in ab initio calculation of medium-sized and large molecules. We can... 

Because the /-orbitals are fairly well shielded from the valence effects it might appear that a relationship between the Dq and Fr parameters could be calculated. However, it seems that the 4/-orbital wave functions are not known sufficiently well for this purpose, and Dq and Fr are to be treated as independent variables. This makes the interpretation of /-electron system properties, even in high symmetry, a much less straightforward business than for -systems. 

It has not proved mathematically feasible to calculate the electron-electron repulsion that causes this change in orbital-energies for many-electron molecules. It is even difficult to rationalize the qualitative changes in sequence on the basis of the shapes of the 11orbitals. Greater success has been achieved by an approximate method which begins with orbitals characteristic of the isolated atoms present in the molecule, and assumes that molecular orbital wave functions can be obtained by taking linear combinations of atomic orbital wave functions (abbreviated L.C.A.O.). For... 

Contribution /x of ionic terms calculated with best screening constant of — curve 1 a molecular orbital wave function curve 2 a Heitler-London wave function curve 3 atomic orbitals. 

Kohn-Sham orbitals are expanded in a Gaussian basis set, so the problem remains to choose the type of basis set and the number of basis functions. Fortunately, the basis functions used are the same as in wave function calculations, offering maximum compatibility between DFT and ab initio MO calculations. 

Wave functions calculated for the lowest-energy levels of linear Agw are shown in Fig. 9. The coefficient of the 5s orbital for each atom is plotted versus atom number along the chain. It is apparent that the electrons in the lowest-energy level are spread almost uniformly throughout the cluster. As energy increases, the uniformity decreases since nodes appear in the wave function. The uniform spreading of electron density would be predicted by the free electron model. 

We consider a multireference Cl (MRCI) wave function calculated from a set of MCSCF orbitals. The Cl reference state is denoted by CI> at X0 and by CI(g)> at the displaced geometry X0 + g. In addition to the reorthonormalization part, the MRCI orbital connection contains the MCSCF orbital... 

Molecular-orbital approaches to edge structures differ for semiconducting and isolating molecular complexes. The latter and transition-metal complexes allow one to minimize solid-state effects and to obtain molecular energy levels at various degrees of approximation (semiempirical, Xa, ab initio). The various MO frameworks, namely, multiple-scattered wave-function calculations (76, 79, 127, 155) and the many-body Hartree-Fock approach (13), describe states very close to threshold (bound levels) and continuum shape resonances. 

In calculating a theoretical photoelectron spectmm, the atomic ionization cross section a. is usually taken so far from the theoretical values calculated for a neutral free atom in the ground state. However, the MO calculation by DV-Xa method is carried out self consistently and provides Q. by Mulliken population analysis using the SCF MO wave function calculated. In the present calculations, the atomic orbital Xj used for the basis function flexibly expands or contracts according to reorganization of the charge density on the atom in molecule in the self-consistent field. Furthermore, excited state atomic orbitals are sometimes added to extend the basis set. In such a case, the estimation of peak intensity of the photoionization using the data of Oj previously published is not adequate. Thus a calculation of the photoionization cross section is required for the atomic orbital used in the SCF calculation in order... 

We wish to compare the valence band density of states (DOS) of f.c.c. and h.c.p. metals with and without stacking faults. We therefore adopt a mixture of the f.c.c. and h.c.p. structures as a representative of the stacking fault structure of either of these structures. To calculate the DOS we summed up the squares of the coefficients of molecular orbital wave functions and convoluted the summed squares with the Gaussian of full width 0.5 eV at half maximum. For these DOS calculations we chose the metals Mg, Ti, Co, Cu and Zn. The model clusters employed here for both the f.c.c. and the h.c.p. structures were made of 13 atoms i.e., a central atom and 12 equidistant neighbor atoms. These structures are shown in Fig. 1. We reproduced the typical electronic structures in bulk materials by extracting the molecular orbitals localized only on the central atom from all the molecular orbitals which contributed - those localized on ligand atoms as well as on the central atom. To perform calculations we take the symmetry of the cluster as C3, and the number... 

Calculations for best fit, using this orbitally degenerate model, employed the parameters D, J ID, and y, the last being close to 0.5 and originating in the orbital wave function, e.g., (j> ( 2) = 2 (1... 

Nitrogen-14 nuclear quadrupole coupling constants in oxazole have been calculated by using the complete neglect of differential overlap method (CNDO/2) including all the valence electrons,232 and from ab initio molecular-orbital wave functions using Gaussian basis sets.234... 

One technical point should be stressed here often encountered in similar work. For the wave function calculation for Li the authors employ a Is core by calculating the closed-shell Li ion yielding an Iso orbital energy of -2.79303 a.u. and the corresponding wave function V (lso)- The valence 2s function was then obtained by solution of the corresponding radial differential equation facilitating the potential of the Iso functions. Obviously there is no orbital- or spin-polarization by the valence electron included, and all these effects come into play only after addition of further terms in the perturbation expansion. This procedure is commonly termed as the method and was... 

A priori it is not clear if effective core potentieds, which have for example been adjusted to reproduce atomic energy differences in wave function based calculations or to reproduce the shape of the valence orbitals outside the core, can successfully be used in density functional calculations. For so-called small core potentieds, where the atomic core has been chosen such that core and valence densities have little overlap, test calculations have shown that results from allelectron and pseudopotential calculations were virtually the same [74]. A related investigation on gold compounds comes to the same conclusion [75]. It is however not recommended to perform density functional investigations with large-core pseudopotentials that have been adjusted in wave function calculations. One example for a leirge-core situation is a transition metal where the vedence d orbiteds are (of course) treated explicitly, while the s emd p orbitals of the same principal quantum number are considered core orbitals. From an energetic view, such a separation seems well justified. However, problems arise since the densites of the s,p, and d orbitals of the same principal quantum number have considerable overlap. 

In order to make DFT a practical and accurate approach, Kohn and Sham reintroduced the use of orbitals in computational DFT. The reason for this is twofold most importantly, the kinetic energy term can be calculated exactly from the orbital approach (but will clearly only give the right number if the orbitals themselves are exact), and the electron density can be simply obtained by summing the square of the orbital wave functions ... 

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