Wavefunctions description

Any function used as the local part of the wavefunction description must be periodic in the cell dimensions. In a plane-wave basis this is ensured by choosing a Hnear combination of plane-waves with particular reciprocal space vectors ... 

The wavefunction description Eq. (10.3) corresponds closely to the Mulliken charge transfer picture of PT [33] shown in Fig. 10.2, in which a non-bonding (ng) electron of the base is transferred into an antibonding orbital of the acid. A... 

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 ... 

Such a compact MCSCF wavefunction is designed to provide a good description of the set of strongly occupied spin-orbitals and of the CI amplitudes for CSFs in which only these spin-orbitals appear. It, of course, provides no information about the spin-orbitals that are not used to form the CSFs on which the MCSCF calculation is based. As a result, the MCSCF energy is invariant to a unitary transformation among these virtual orbitals. 

A basis set is the mathematical description of the orbitals within a system (which in turn combine to approximate the total electronic wavefunction) used to perform the theoretical calculation. Larger basis sets more accurately approximate the orbitals by imposing fewer restrictions on the locations of the electrons in space. In the true quantum mechanical picture, electrons have a finite probability of existing anywhere in space this limit corresponds to the infinite basis set expansion in the chart we looked at previously. 

You are probably used to this idea from descriptive chemistry, where we build up the configurations for many-electron atoms in terms of atomic wavefunctions, and where we would write an electronic configuration for Ne as... 

Electrons are indistinguishable, they simply cannot be labelled. This means that an acceptable electronic wavefunction has to treat all electrons on an equal footing. Thus, although 1 have so far implied that electron 1 is to be associated with nucleus Ha, and electron 2 with nucleus Hb, 1 must also cater for the alternative description where electron 1 is associated with nucleus Hb and electron 2 with nucleus Ha. 1 therefore have to modify Table 4.1 to Table 4.2. 

In the limit of infinite atom separations, or if we switch off the Coulomb repui. sion between two electrons, all four wavefunctions have the same energy. But they correspond to different eigenvalues of the electron spin operator the first combination describes the singlet electronic ground state, and the other three combinations give an approximate description of the components of the first triplet excited state. 

Even at the equilibrium geometry, it is necessary to treat both of the l+ states in order to get a reasonable description of the electronic wavefunction for the ground state. 

The proposed scenario is mainly based on the molecular approach, which considers conjugated polymer films as an ensemble of short (molecular) segments. The main point in the model is that the nature of the electronic state is molecular, i.e. described by localized wavefunctions and discrete energy levels. In spite of the success of this model, in which disorder plays a fundamental role, the description of the basic intrachain properties remains unsatisfactory. The nature of the lowest excited state in m-LPPP is still elusive. Extrinsic dissociation mechanisms (such as charge transfer at accepting impurities) are not clearly distinguished from intrinsic ones, and the question of intrachain versus interchain charge separation is not yet answered. 

In general, the first excited state (i.e. the final state for a fundamental transition) is described by a wavefunction pt which has the same symmetry as the normal coordinate (Appendix). The normal coordinate is a mathematical description of the normal mode of vibration. 

Consider now spin-allowed transitions. The parity and angular momentum selection rules forbid pure d d transitions. Once again the rule is absolute. It is our description of the wavefunctions that is at fault. Suppose we enquire about a d-d transition in a tetrahedral complex. It might be supposed that the parity rule is inoperative here, since the tetrahedron has no centre of inversion to which the d orbitals and the light operator can be symmetry classified. But, this is not at all true for two reasons, one being empirical (which is more of an observation than a reason) and one theoretical. The empirical reason is that if the parity rule were irrelevant, the intensities of d-d bands in tetrahedral molecules could be fully allowed and as strong as those we observe in dyes, for example. In fact, the d-d bands in tetrahedral species are perhaps two or three orders of magnitude weaker than many fully allowed transitions. 

First of all, the wavefunction has to contain the necessary ingredients to properly describe the phenomenon under investigation for example, when dealing with electronic spectra, it thus has to contain every CSFs needed to account at least qualitatively for the description of the excited states. The zeroth-order wavefunction has then to include a number of monoexcitations from the groimd state occupied orbitals to some virtual orbitals. In that sense, the choice of a Single Cl type of wavefunction as proposed by Foresman et al. [45,46] in their treatment of electronic spectra represents the minimum zeroth-order space that can be considered. 

Density functions can be obtained up to any order from the manipulation of the Slater determinant functions alone as defined in section 5.1 or from any of the linear combinations defined in section 5.2. Density functions of any order can be constructed by means of Lowdin or McWeeny descriptions [17], being the diagonal elements of the so called m-th order density matrix, as was named by Lowdin the whole set of possible density functions. For a system of n electrons the n-th order density function is constructed from the square modulus of any n-electron wavefunction attached to the n-electron system somehow. 

By omitting time-dependent terms, as in the preceding paragraph, the liP ) function may be read as the sum of the unperturbed wavefunction ) and a term which is the product of this function by a linear combination of the electronic coordinates, i.e. the Kirkwood s j) function. Thus, the (r) dipolar factor ensures gauge-invariance. But the role of the dipolar factor g f) in this mixed method is essential on the following point its contribution in the a computation occurs in a complementary (and sometimes preponderant) way to that calculated only from the n) excited states, the number of which is unavoidably limited by the computation limits. But before discussing their number, we have to comment the description of these states. 

Therefore, it appears that the overall agreement obtained for a variety of spectroseopie eonstants is comparable for the two methods while the present method allows us to use a more compact wavefunction. It should also be noted that a good Cl description of a triple bonded system involving a third period atom is much harder to achieve. It can be concluded that the shape of the theoretical potential energy curve reflects its experimental counterpart with acceptable accuracy in the interatomic region of interest. 

---