Diatomic fragment

The most commonly used semiempirical for describing PES s is the diatomics-in-molecules (DIM) method. This method uses a Hamiltonian with parameters for describing atomic and diatomic fragments within a molecule. The functional form, which is covered in detail by Tully, allows it to be parameterized from either ah initio calculations or spectroscopic results. The parameters must be fitted carefully in order for the method to give a reasonable description of the entire PES. Most cases where DIM yielded completely unreasonable results can be attributed to a poor fitting of parameters. Other semiempirical methods for describing the PES, which are discussed in the reviews below, are LEPS, hyperbolic map functions, the method of Agmon and Levine, and the mole-cules-in-molecules (MIM) method. 

The summation over final states in Eq. (4.1) also includes an integration over all the directions of dissociation, k. The boundary conditions of the wavefunction / ,.(r, R k, ) correspond to a pure outgoing wave where the diatomic fragment has vibrational-rotational quantum numbers vj and is associated with the wavefunction Xv ( ) m(r)-... 

Lippincott and Schroeder [1955, 1957] introduced a semiempirical two-dimensional PES and fitted their parameters from experimental data. Further studies in this direction were carried out by Savel ev and Sokolov [1975] and Sokolov and Savel ev [1977], Lautie and Novak [1980], Saitoh et al. [1981], and Emsley [1984]. These studies have shown that an adequate two-dimensional PES can be constructed from Morse functions of diatomic fragments XH and HY and repulsive functions representing the XY interaction. The values of rXH ar,d wXH and isotope effects as a function of R are in agreement with the experimental ones for OH O, OH-N, and NH-- N fragments. The dependencies rXH(/ ) and a>XH(R) collected by Novak [1974] are shown in Figure 6.1. The method of Lippincott and Schroeder [1957] is one of the versions of the general semiempirical method of bond energy-bond order (BEBO) developed by Johnston and Parr [1963] to construct a two-dimensional PES. 

E. Brandas, M. Rittby, N. Elander, Multichannel Complex Scaled Titchmarsh Weyl Theory a Model for Diatomic Fragmentation, Lect. Notes Phys. 325 (1989) 345. 

In the following we will call the a u,n,j) partial photodissociation cross sections.t They are the cross sections for absorbing a photon with frequency u and producing the diatomic fragment in a particular vibrational-rotational state (n,j). Partial dissociation cross sections for several photolysis frequencies constitute the main body of experimental data and the comparison with theoretical results is based mainly on them. Summation over all product channels (n,j) yields the total photodissociation cross section or absorption cross section ... 

R — oo. As we will show below, the partial cross sections for absorbing the photon and producing the diatomic fragment in vibrational channel n are proportional to the square modulus of the overlap of these continuum wavefunctions with the nuclear wavefunction in the electronic ground state (indicated by the shaded areas). Since the bound wavefunction of the parent molecule is rather confined, only a very small portion of the continuum wavefunctions is sampled in the overlap integral. 

In order to keep the formulation as simple as possible we confine the discussion to systems with only two degrees of freedom. The extension to more complex problems is — formally at least — straightforward. We will treat triatomic molecules ABC dissociating into products A+BC. First, we again consider in Section 3.1 the linear model, outlined in Sections 2.4 and 2.5, in which the diatomic fragment vibrates while its rotational degree of freedom is frozen. Subsequently, we treat in Section 3.2 the... 

Hamilton s equations form a set of coupled first-order differential equar tions which under normal conditions can be numerically integrated without any problems. The forces —dVi/dR and —dVi/dr and the torque —dVj/d7, which reflect the coordinate dependence of the interaction potential, control the coupling between the translational (R,P), the vibrational (r,p), and the rotational (7,j) degrees of freedom. Due to this coupling energy can flow between the various modes. The translational mode becomes decoupled from the internal motion of the diatomic fragment (i.e., dP/dt = 0 and dR/dt =constant) when the interaction potential diminishes in the limit R — 00. As a consequence, the translational energy... 

In the preceding two sections we considered resonances induced by temporary excitation of the mode that finally becomes the vibrational mode of the diatomic fragment. In this section we feature two other types of resonant excitation of internal vibrational motion. 

In the pure impulsive model, the total excess energy Eexcess partitions into translational energy of both products, as well as vibrational and rotational energy of the diatomic fragment. In a modified version, Busch and Wilson (1972a) assumed that the B-C bond is infinitely stiff such that vibrational energy transfer is prohibited. Employing conservation of... 

The vector of the electromagnetic field defines a well specified direction in the laboratory frame relative to which all other vectors relevant in photodissociation can be measured. This includes the transition dipole moment, fi, the recoil velocity of the fragments, v, and the angular momentum vector of the products, j. Vector correlations in photodissociation contain a wealth of information about the symmetry of the excited electronic state as well as the dynamics of the fragmentation. Section 11.4 gives a short introduction. Finally, we elucidate in Section 11.5 the correlation between the rotational excitation of the products if the parent molecule breaks up into two diatomic fragments. 

Up to now we have exclusively considered the scalar properties of the photodissociation products, namely the vibrational and rotational state distributions of diatomic fragments, i.e., the energy that goes into the various degrees of freedom. Although the complete analysis of final state distributions reveals a lot of information about the bond breaking and the forces in the exit channel, it does not completely specify the dissociation process. Photodissociation is by its very nature an anisotropic process — the polarization of the electric field Eo defines a unique direction relative to which all vectors describing both the parent molecule and the products can be measured. These are ... 

The last question that we will address concerns the final vibrational and rotational state distributions of the diatomic fragments. Although the excited resonance states can live up to nanoseconds or even microseconds, the final distributions do not follow simple statistical laws which were briefly spoken about in Section 10.3.2. On the contrary, they manifest either prominent propensity rules or dynamical features similar to those discussed in the context of direct dissociation. 

The set of coupled equations must be solved subject to boundary conditions similar to (2.59) with unit outgoing flux in only one particular electronic channel, which we designate by e, and one particular vibrational state n of the diatomic fragment, e.g., e = l,n = 5 for A + BC(n = 5) and e = 2, n = 6 for A + BC(n = 6). In the actual calculation one would subsequently expand the nuclear wavefunctions (.R, r E, e, n) in a set of vibrational basis functions (n = 0,..., nmax) as described in Section 3.1 which leads to a total of 2(nmax + 1) coupled equations. It is not difficult to surmise how complicated the coupled equations will become if the rotational degree of freedom is also included. 

DIM Diatomics in molecules. A semiempricial method used to construct potential energy surfaces of polyatomic molecules from the energy of the diatomic fragments. 

Scheme 1.12 summarizes the elementary entropy/information increments of the diatomic bond indices generated by the MO channels of Eq. (42). They give rise to the corresponding diatomic descriptors, which are obtained from Eq. (32). For example, by selecting i = 1 of the diatomic fragment consisting additionally the = 2,3,4 carbon, one finds the following IT bond indices ... 

Orbital and condensed atom probabilities of diatomic fragments in molecules... 

The molecular probability scattering in the specified diatomic fragment (A, B), involving AO contributed by these two bonded atoms, Xab = (xA, xB), to the overall basis set x = (Xxn is completely characterized by the corresponding P(XabIXab) block [22, 26] of the molecular conditional probability matrix of Eq. (4), which determines the molecular communication system in OCT [46-48] of the chemical bond ... 

In other words, P(A,B z) measures the probability that the electron occupying Xi will be detected in the diatomic fragment AB of the molecule. The inequality in the preceding equation reflects the fact that the atomic basis functions participate in chemical bonds with all constituent atoms, with the equality sign corresponding only to a diatomic molecule, when /AB = /. ... 

These vectors of AO probabilities in diatomic fragment AB subsequently define the condensed probabilities PX(AB) of both bonded atoms in subsystem AB ... 

We finally observe that the effective orbital probabilities of Eqs. (52-54) and the associated condensed probabilities of bonded atoms (Eq. 55) do not reflect the actual AO participation in all chemical bonds in AB, giving rise to comparable values for the bonding and nonbonding (lone-pair) AO in the valence and inner shells. The relative importance of basis functions of one atom in forming the chemical bonds with the other atom of the specified diatomic fragment is reflected by the (nonnormalized) joint bond probabilities of the two atoms, defined by the diatomic components of the simultaneous probabilities of Eqs. (52 and 53) ... 

In Eq. (61) the conditional entropy SAb(Y /x) quantifies (in bits) the delocalization X—> Y per electron so that the total covalency in the diatomic fragment A-B reads as follows ... 

They generate the total information ionicity of all chemical bonds in the diatomic fragment ... 

Figure 5.5 Difference in cohesive energy calculated for a diatomic fragment assuming valence state functions in symmetric and anti symmetric linear combination.

---