Heitler-London approximation

In the Heitler-London approximation, with allowance made only for biquadratic anharmonic coupling between collectivized high-frequency and low-frequency modes of a lattice of adsorbed molecules (admolecular lattice), the total Hamiltonian (4.3.1) can be written as a sum of harmonic and anharmonic contributions ... 

At smaller distances the exchange interaction comes into play. In the Heitler-London approximation there will be two types of exchange integrals without and with excitation transfer. These are... 

The Dlatomics-ln-Molecules Approach. The simple version of the DIM method that we employ is based on the Heitler-London approximation (28). In spirit, it is similar to the London-Eyring approach except that we use accurate diatomic potential curves (29a) rather than an approximate form for the diatomic triplet curve (e.g. the Sato parameter for LEPS surfaces) (29b). 

From the experimental values of dipole moments it is possible, in a number of cases, to make a semi-quantitative evaluation of the weights of the various valence bond structures contributing to a bond (see Chapter 18). These calculations must be regarded as only approximate since the bond is described in terms of the Heider-London theory with the superposition of ionic states. The results cannot, therefore, be more precise than is permitted by the Heitler-London approximation. Nevertheless, the calculations are of significance since they permit an assessment to be made of the more important structures contributing to the bond and thus assist in predicting and explaining the reactivity of bonds. 

The concept of Frenkel excitons (molecular excitons) [54] provides a good starting point for the description of the electronically excited states of the LH complexes. The respective model Hamiltonian reads in Heitler-London approximation... 

By writing the exehange Hamiltonian in Heitler-London approximation as... 

Taking into account the exchange interactions of the fragments 1 and 2 in the Heitler-London approximation (the index k equal to 1 and 2 corresponds to exchange with outer and 3a inner molecular electronic shells respectively). 

Let us note that the above assumption was used by Frenkel and Davydov (9)-(11) by constructing the wavefunctions in the Heitler-London approximation in its simple version. In their theory only the part of the intermolecular interaction was taken into account, which causes the excitation transfer from one molecule to another, and not those which gives the mixing of molecular electronic and vibronic configurations. 

In our book we present methods of computation of Frenkel exciton states in molecular crystals, which are not based on the molecular two-level model and Heitler-London approximation (Ch. 3). The methods allow us, in particular, to obtain the Frenkel exciton spectra for arbitrary strength of the intermolecular interaction, assuming that the interaction does not violate the charge neutrality. However, in this section we use the simplest form of the Heitler-London method to construct the wavefunctions and to obtain some qualitative results on the properties of the spectra which occur by the aggregation of molecules into a crystal. 

To demonstrate this effect we first consider an exciton in a crystal with one molecule in the unit cell having a nondegenerate molecular term /. In this case the exciton energy, in the Heitler-London approximation, is given by the relation (2.12). 

As we have shown above, the exciton energy depends not only on the characteristics of a single molecular term /, as it would follow from an elementary approach based on the Heitler-London approximation, but, in general, depends on all excited states of the molecule. This property is reflected by the fact that, as we have shown, the energies of Coulomb excitons can be expressed in terms of the crystal dielectric tensor, which includes contributions of all resonances. 

Excitonic states in the two-level model. Transition to the Heitler—London approximation... 

Let us first consider the expression (3.19) without the operator H3. Such an approximation corresponds to the Heitler-London approximation. In this approximation the excitation energy operator has the form... 

Exciton states beyond the Heitler London approximation... 

In the previous section we have obtained the crystal excitation energy in the Heitler-London approximation, i.e. neglecting the term H3 in the crystal energy operator (3.19). We now wish to determine the crystal energy without using this simplification. 

The results (3.67) coincide with those obtained by the Heitler-London approximation (3.41). Thus the inequality (3.66) indicates the limits of applicability of the Heitler-London approximation. In molecular crystals for excitations /, which correspond to the lowest electronic excitations of free molecules, the quantity L(k) is mostly less than 103 cm-1, when the quantity Ae + V is of order 3 104 cm-1. Thus the inequality (3.66) is satisfied and the Heitler-London approximation can be used for those states. In some crystals (for example, in anthracene) the second molecular excitation corresponds to the value L (k) 104cm. In such cases corrections to the Heitler-London approximation can be of importance (see also below Section 3.10). 

With the aim of obtaining general expressions for corrections to the Heitler-London approximation in crystals with several molecules per unit cell, we put eqns (3.62) into the form... 

Equations (3.69), with regard to (3.60), yield for the Heitler-London approximation... 

Equations (3.70) and (3.71) coincide, as one might expect, with eqns (3.52) and (3.49) giving the energy and the wavefunctions in the Heitler-London approximation. For computing the excitonic energies within this approximation we notice that the quantities... 

The equality (3.74) allows us to determine improved values of the excitonic energies, when the values E L(k) resulting from the Heitler-London approximation are known. 

In previous sections we discussed crystal excitonic states starting from the assumption that by creation of those states only two stationary molecular states are included the ground state (0), and the excited state (/). This assumption is valid when the excited state (/) is sufficiently far from the remaining excited states. If we deal with a group of near states, then all those states can participate in the creation of excitonic states ( mixing of molecular terms). Theoretical discussion of the above case was given by Craig (12) (13) in the Heitler-London approximation, and by the author (2) by means of second quantization. Below we present the results obtained by second quantization, which permit us to go beyond the HLA. 

If we neglect the operator H3 in (3.88) and thus go to the Heitler London approximation, then... 

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