Wavefunctions, of orbitals

Note that a particular orbit, therefore, is determined by fixing the values of the parameters a, fi and C j = i. Thus, employing these orbitals, we can construct a single Slater determinant for the configuration [ls 2s], that is the generating wavefunction of orbit... 

For Iran sition metals th c splittin g of th c d orbitals in a ligand field is most readily done using HHT. In all other sem i-ctn pirical meth -ods, the orbital energies depend on the electron occupation. HyperCh em s m oiccii lar orbital calcii latiori s give orbital cri ergy spacings that differ from simple crystal field theory prediction s. The total molecular wavcfunction is an antisymmetrized product of the occupied molecular orbitals. The virtual set of orbitals arc the residue of SCT calculations, in that they are deemed least suitable to describe the molecular wavefunction, ... 

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... 

If we want to determine the specific type of orbital transformation for this transition, we will need to examine the molecular orbitals for the largest components of the transition, indicated by the largest wavefunction coefficients. In this case, this is the relevant entry ... 

The wavefunctions of electrons in atoms are called atomic orbitals. The name was chosen to suggest something less definite than an orbit of an electron... 

FIGURE 1.34 The radial wavefunctions of the first three s-orbitals of a hydrogen atom. Note that the number of radial nodes increases (as n 1), as does the average distance of the electron from the nucleus (compare with Fig. 1.32). Because the probability density is given by ip3, all s-orbitals correspond to a nonzero probability density at the nucleus. 

The VB and MO theories are both procedures for constructing approximations to the wavefunctions of electrons, but they construct these approximations in different ways. The language of valence-bond theory, in which the focus is on bonds between pairs of atoms, pervades the whole of organic chemistry, where chemists speak of o- and TT-bonds between particular pairs of atoms, hybridization, and resonance. However, molecular orbital theory, in which the focus is on electrons that spread throughout the nuclear framework and bind the entire collection of atoms together, has been developed far more extensively than valence-bond... 

The left superscript indicates that the arrangements are all spin triplets. The letter T refers to the three-fold degeneracy just discussed and it is in upper case because the symbol pertains to a many-electron (here two) wavefunction (we use lower-case letters for one-electron wavefunctions or orbitals, remember). The subscript g means the wavefunctions are even under inversion through the centre of symmetry possessed by the octahedron (since each d orbital is of g symmetry, so also is any product of them), and the right subscript 1 describes other symmetry properties we need not discuss here. More will be said about such term symbols in the next two sections. 

In an octahedral crystal field, for example, these electron densities acquire different energies in exactly the same way as do those of the J-orbital densities. We find, therefore, that a free-ion D term splits into T2, and Eg terms in an octahedral environment. The symbols T2, and Eg have the same meanings as t2g and eg, discussed in Section 3.2, except that we use upper-case letters to indicate that, like their parent free-ion D term, they are generally many-electron wavefunctions. Of course we must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term splits into -I- T 2gin octahedral symmetry. Notice that the crystal-field splitting has no effect upon the spin-degeneracy. This is because the crystal field is defined completely by its ordinary (x, y, z) spatial functionality the crystal field has no spin properties. 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

For the one-center valence contribution, there are essentially three factors that control its value (a) the radial wavefunction of the 3d orbitals, (b) the covalent dilution of the 3d orbitals with ligand orbitals, and (c) the occupation pattern of the 3d shell. An additional factor may be low-symmetry induced 3d/4p mixing. We will focus on the first three factors here. 

Figure 15. Snapshots of the two frontier excited-state natural orbitals (computed using the HF-OA-CAS(4/4) S wavefunction) of the excited-state trajectory of cyclobutene shown in Fig. 13. Left panels Before the onset of disrotatory motion, the excited-state wavefunction can be described using a single determinant with one electron in a tt-like orbital (4>a) and one in a 7t -like orbital (4>b). Middle panels During the disrotatory motion the simplest description of the electronic wavefunction requires two determinants. In one determinant both electrons are in the (j)a orbital, and in the other they are both in the (j)b orbital. Both orbitals (<()a and 4>b) show significant cj—it mixing, which is a consequence of the significant disrotatory motion. Right panels When the disrotatory motion is completed, the excited-state wavefunction is described by a single determinant in which both electrons are in the <()b orbital. Note how the shape of the orbitals changes as the initial bonds are broken and the two new tc bonds are formed.

Although HF orbitals are, by definition, the best possible for a singleconfiguration wavefunction, it is actually possible to find a better set of orbitals, called natural orbitals,31 to describe the correlated p(r). The natural orbitals are maximum-occupancy orbitals, determined from itself and guaranteed to give fastest possible convergence to p(r), i.e., consistently higher occupancies n, than HF orbitals inEq. (1.15). For a HF wavefunction the natural orbitals and HF orbitals are equivalent, but for more accurate wavefunctions the natural orbitals allow us to... 

Natural population analysis of transition-metal-fluoride wavefunctions gives orbital populations that are generally consistent with the expected configurations, but with large departures from an idealized ionic picture. Table 2.4 compares the... 

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