Term wavefunctions

Term wavefunctions describe the behaviour of several electrons in a free ion coupled together by the electrostatic Coulomb interactions. The angular parts of term wavefunctions are determined by the theory of angular momentum as are the angular parts of one-electron wavefunctions. In particular, the angular distributions of the electron densities of many-electron wavefunctions are intimately related to those for orbitals with the same orbital angular momentum quantum number that is. 

The electron distributions in term wavefunctions and orbitals may be the same or complementary, as shown below... 

The basis for this formula is just the same as described above but, in this case, spin-orbit coupling admixes the higher-lying 2 2(g) term wavefunctions into the ground E(g). The coefficient 2 in Eq. (5.17) rather than the 4 in Eq. (5.16) arises from the different natures of the wavefunctions being mixed together. 

In 1959, C. L. Pekeris, by optimizing a 1,078-term wavefunction for helium, obtained an energy essentially identical to the experimental value. 

Figure 8. Profiles of ground state energy of He obtained by SAM using 10-term wavefunction for (a)fixed and (b)updated annealing runs...

Since Eq. (224) shows that an arbitrary two-electron singlet wavefunction may be expanded in terms of only n CSFs, a possible solution to this problem may lie in the particular choice of these n expansion terms. If there are alternate n-term expansions for the general wavefunction, these might then be used to advantage in the simultaneous descriptions of several states since different CSF expansion spaces would produce different approximations to the other states. It is instructive to consider a possible -term expansion in terms of i i and its single excitations. This -term wavefunction expansion consists of a symmetric C matrix of the form... 

In early literature [208], the term wavefunction collapse was frequently used to describe this property. We avoid it here it causes confusion with wavefunction collapse in the quantum theory of measurement. For this reason, we prefer the term orbital collapse, following [196]. 

Every molecule is made up from the nuclei and electrons of two or more atoms via bonds that result from the overlap of atomic orbitals. Hence, the shapes and properties of atomic orbitals are of paramount importance in dictating the bonding in and properties of molecules. The Bohr model of atoms had electrons moving in specific orbits (hence the term orbitals) around the nucleus. We now view the shapes and properties of atomic orbitals as they are obtained from basic quantum mechanics via solution of the Schrodinger equation. The solutions to the Schrodinger equation are termed wavefunctions, and in their most common implementation these wavefunctions correspond to atomic or molecular orbitals. 

For separable initial states the single excitation terms can be set to zero at all times at this level of approximation. Eqs. (32),(33),(34) together with the CSP equations and with the ansatz (31) for the total wavefunction are the working equations for the approach. This form, without further extension, is valid only for short time-domains (typically, a few picoseconds at most). For large times, higher correlations, i.e. interactions between different singly and doubly excited states must be included. 

For /2(Ar)i7, an extensive CI-CSP simulation was carried out, and the results were compared with those of the simple CSP approximation. Both calculations are for the ultrafast dynamics following excitation of the I2 into the B state. We found that the CI-CSP calculation, including doubly excited configurations , is close to converged for times up to t 500 fs, when 1500 configurations are included. Fig. 2 shows co(t)p, the coefficient of the CSP term and the doubly excited terms in the full CI-CSP wavefunction,... 

It turns out that the CSP approximation dominates the full wavefunction, and is therefore almost exact till t 80 fs. This timescale is already very useful The first Rs 20 fs are sufficient to determine the photoadsorption lineshape and, as turns out, the first 80 fs are sufficient to determine the Resonance Raman spectrum of the system. Simple CSP is almost exact for these properties. As Fig. 3 shows, for later times the accuracy of the CSP decays quickly for t 500 fs in this system, the contribution of the CSP approximation to the full Cl wavefunction is almost negligible. In addition, this wavefunction is dominated not by a few specific terms of the Cl expansion, but by a whole host of configurations. The decay of the CSP approximation was found to be due to hard collisions between the iodine atoms and the surrounding wall of argons. Already the first hard collision brings a major deterioration of the CSP approximation, but also the role of the second collision can be clearly identified. As was mentioned, for t < 80 fs, the CSP... 

VVc can now see why the normalisation factor of the Slater determinantal wavefunction is I v/N . If each determinant contains N terms then the product of two Slater determinants, ldeU rminant][determinant], contains (N ) terms. However, if the spin orbitals form an oi lhonormal set then oidy products of identical terms from the determinant will be nonzero when integrated over all space. We Ccm illustrate this with the three-electron example, k ljiiiidering just the first two terms in the expansion we obtain the following ... 

In order to calculate higher-order wavefunctions we need to establish the form of the perturbation, f. This is the difference between the real Hamiltonian and the zeroth-order Hamiltonian, Remember that the Slater determinant description, based on an orbital picture of the molecule, is only an approximation. The true Hamiltonian is equal to the sum of the nuclear attraction terms and electron repulsion terms ... 

The additional terms in the molecular orbital wavefunction correspond to states with the two electrons in the same orbital, which endows ionic character to the bond The... 

VVe now need to consider how the form of the wavefunction varies with k. The first situation we consider corresponds to fc = 0, where the exponential terms are all equal to 1 and the overall wavefunction becomes a simple additive linear combination of the atomic orbitals ... 

Equation (3.85) T is a translation vector that maps each position into an equivalent ition in a neighbouring cell, r is a general positional vector and k is the wavevector ich characterises the wavefunction. k has components k, and ky in two dimensions and quivalent to the parameter k in the one-dimensional system. For the two-dimensional lare lattice the Schrodinger equation can be expressed in terms of separate wavefunctions ng the X- and y-directions. This results in various combinations of the atomic Is orbitals, ne of which are shown in Figure 3.13. These combinations have different energies. The /est-energy solution corresponds to (k =0, ky = 0) and is a straightforward linear... 

Quantum Mechanics Describes Matter in Terms of Wavefunctions and Energy Levels. Physical Measurements are Described in Terms of Operators Acting on Wavefunctions... 

The time evolved wavefunction, according to this first rule, can be expressed in terms of the vibrational functions fPfi 4ind energies Efi of the N2 ion as... 

The fact that the set of hi is, in principle, complete in r-space allows the full (electronic and nuclear) wavefunction h to have its r-dependence expanded in terms of the hp... 

This wavefunction needs to be expanded in terms of the eigenfunctions of the angular... 

The wavefunction is now written in terms of the eigenfunetions of the angular momentum... 

To express, in terms of Slater determinants, the wavefunctions corresponding to each of the states in each of the levels, one proceeds as follows ... 

---