The Perturbation Treatment

The Perturiiatioii Treatment.—Based on a statistical solution theory of Longuet-Higgins, a first-order perturbation approach to the free energy of solution has been developed by Luckhurst and Martire and by Tewari et al. This theory appears useful for interpreting the interaction parameters within a series of structurally related solute molecules in the same solvent. It is thus ideally suited for testing data obtained from g.l.c. experiments. The theory involves a comparison of two solutes - a reference one and a perturbed one in a common solvent. The interaction parameter x is shown to be linearly related to Ty( V y where Tf and Ff are the solute critical temperature and the solute molar volume at temperature 0.675 respecively. The term 75/( Ff ) is a measure of the depth of the interaction potential well.  

Plots of X against 75/(Ff) for solute families of n-alkanes, monosubstituted alkanes, disubstituted alkanes, alk-l-enes, and monosubstituted benzenes in solvents n-C24Hso, n-CsoHea, and n-Cs H74 gave straight lines which in these cases gave an average linear correlation coefficient of 0.984. 

The cell theory of Prigogine has been applied by Clark and Schmidt and by Hicks and Young to benzene + polyphenyl systems and n-alkane mixtures respectively. The Flory, Orwoll, and Vrij theory has also been applied by Hicks and Young to n-alkane mixtures. 

Funke et al has used a factor analysis technique to investigate activity coefficients from g.l.c. In this method, the property in question (activity coefficients) is resolved into a linear sum of the minimum number of factors needed to reproduce the experimental results to the required accuracy. Using these factors, it is possible to predict activity coefficients for systems not yet measured. Funke et al. have also suggested that it should be possible to interpret the elements of the matrix into meaningful parameters but this has not yet been achieved. 

Letcher and Marsicano have applied factor analysis to unsaturated hydrocarbon systems and factorized both the logioyia and T results. As expected the fit for Tx was better than for logioyis. 

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Wlien resonances, or near resonances, are present in the 4WM process, the ordering of the field actions in the perturbative treatment (equation (Bl.3.1)), can be highly significant. Though the tliree-colour generators... 

Calculations for nniforin systems showed that the perturbative treatment usually overestimates the fluctuative contribution . Thus more refined, cluster inetods have been developed, such as the ( VM and CFM". They can be extended to nonuniform systems. In particular, the pair cluster (PCA) expression for F c, can be written out analytically" ... 

Although no new numerical information regarding the hydrogen molecule-ion can be obtained by treating the wave equation by perturbation methods, nevertheless it is of value to do this. For perturbation methods can be applied to many systems for which the wave equation can not be accurately solved, and it is desirable to have some idea of the accuracy of the treatment. This can be gained from a comparison of the results of the perturbation method of the hydrogen molecule-ion and of Bureau s accurate numerical solution. The perturbation treatment assists, more-... 

In benzene itself the charge at the point of attack will be not 1.00c, as has been assumed heretofore, but (1.00 + (43/108)5,)c. (Since 5,-is small, the results of the perturbation treatment can be applied directly.) In other molecules the ease of reaction at the ith carbon atom will then be greater than that of benzene if the charge at that point is greater than (1.00 + (43/108)5 )e, and conversely. 

The value o+l <0.4 found for H2 shows that even in the lowest state the molecules are rotating freely, the intermolecular forces producing only small perturbations from uniform rotation. Indeed, the estimated (3vq<135° corresponds to Fo <28 k, which is small compared with the energy difference 164 k of the rotational states j = 0 and j= 1, giving the frequency with which the molecule in either state reverses its orientation. The perturbation treatment shows that with this value of Fo the eigenfunctions and energy levels in all states closely approximate those for the free spatial rotator.9... 

The present approach has been applied to the experiment done by Nelsen et ah, [112], which is a measurement of the intramolecular electron transfer of 2,7-dinitronaphthalene in three kinds of solvents. Since the solvent dynamics effect is supposed to be unimportant in these cases, we can use the present theory within the effective ID model approach. The basic parameters are taken from the above reference except for the effective frequency. The results are shown in Fig. 26, which shows an excellent agreement with the experiment. The electronic couphng is quite strong and the perturbative treatment cannot work. The effective frequencies used are 1200, 950, and 800 cm for CH3CN, dimethylformamide (DMF), and PrCN [113]. 

As a normalized trial function 0 for the determination of the ground-state energy by the variation method, we select the unperturbed eigenfunction r2) of the perturbation treatment, except that we replace the atomic number Zby a parameter Z ... 

The numerical value of S is listed in Table 9.1. The simple variation function (9.88) gives an upper bound to the energy with a 1.9% error in comparison with the exact value. Thus, the variation theorem leads to a more accurate result than the perturbation treatment. Moreover, a more complex trial function with more parameters should be expected to give an even more accurate estimate. 

The reference (zeroth-order) function in the CASPT2 method is a predetermined CASSCF wave function. The coefficients in the CAS function are thus fixed and are not affected by the perturbation operator. This choice of the reference function often works well when the other solutions to the CAS Hamiltonian are well separated in energy, but there may be a problem when two or more electronic states of the same symmetry are close in energy. Such situations are common for excited states. One can then expect the dynamic correlation to also affect the reference function. This problem can be handled by extending the perturbation treatment to include electronic states that are close in energy. This extension, called the Multi-State CASPT2 method, has been implemented by Finley and coworkers.24 We will briefly summarize the main aspects of the Multi-State CASPT2 method. 

An example is the perturbation treatment of the He atom in which we neglect the interelectronic repulsion e2/rl2. The correct zeroth-order wave function for the ground state is... 

At low energies the pseudopotential is too large for the perturbation treatment to be applicable. We can then rewrite the Fourier transform of the pseudopotential in terms of the scattering length a, so that... 

Using MP2(full)/6-31+G geometries (second-order Moller-Plesset perturbation theory with core electrons included in the perturbation treatment). 

These points need emphasizing, for the use of free valence as a measure of reactivity was first introduced on intuitive grounds, based on the Thiele theory of partial valence. On this basis there seems no obvious reason why free valence should serve as a measure of reactivity only for compounds of one type. The perturbational treatment given above shows that the validity of the correlation is due merely to a fortuitous coincidence which holds only for alternant hydrocarbons. 

The localized bond model fails for odd conjugated systems (ions and radicals such as allyl or benzyl) the mesomeric stabilization of these can be estimated by the perturbation treatment of Section VIII. However, the localized bond model can still4 be used for other types of odd systems, i.e. ones where the... 

As already mentioned, the MOs in Eqs. (10a) and (10b) are assumed to be exact, i.e., represented at the complete basis set limit. In practice, the Kohn-Sham equations are converted into the respective self-consistent-field matrix equations in the basis, and the perturbation treatment is carried out from there. Any dependence of the basis set (GIAOs) on the perturbation (Bext in the case of shielding tensors) is in this way naturally covered. 

Consider the problem of wave packet control in a weak laser field. Here wave packet control refers to the creation of a wave packet at a given target position on a specific electronic potential energy surface at a selected time tf. For this purpose, a variational treatment is introduced. In the weak field limit, the wave packet can be calculated by first-order perturbation theory without the need to solve explicitly the time-dependent Schrodinger equation. In strong fields, where the perturbative treatment breaks down, the time-dependent Schrodinger equation must be explicitly taken into account, as will be discussed in later sections. 

As in all perturbational approaches, the Hamiltonian is divided into an unperturbed part and a perturbation V. The operator is a spin-free, one-component Hamiltonian and the spin-orbit coupling operator takes the role of the perturbation. There is no natural perturbation parameter X in this particular case. Instead, J4 so is assumed to represent a first-order perturbation The perturbational treatment of fine structure is an inherent two-step approach. It starts with the computation of correlated wave functions and energies for pure spin states—mostly at the Cl level. In a second step, spin-orbit perturbed energies and wavefunctions are determined. 

Since f o and F are not, in general, eigenfunctions of the respective hamiltonians, an additional term appears in the perturbational treatment, known as the zeroth-order exchange energy or complementary exchange term,61 A r). In regions of small orbital overlap, A may be written as62... 

In practice, the result of the perturbation treatment may be expressed as a series of formulae for the spectroscopic constants, i.e. the coefficients in the transformed or effective hamiltonian, in terms of the parameters appearing in the original hamiltonian, i.e. the wavenumbers tor, the anharmonic force constants , the moments of inertia Ia, their derivatives eft , and the zeta constants These formulae are analogous to equations (23)—(27) for a diatomic molecule. They are too numerous and too complicated to quote all of them here, but the various spectroscopic constants are listed in Table 3, with their approximate relative orders of magnitude, an indication of which parameters occur in the formula for each spectroscopic constant, and a reference to an appropriate source for the perturbation theory formula for that constant. 

It is instructive to consider two of the formulae for the spectroscopic constants in more detail, and for this we choose — and xrs for an asymmetric top, these being respectively the coefficients of (vT + )J, the vibrational dependence of the rotational constant, and (vr + i)(fs + i), the vibrational anharmonic constant quadratic in the vibrational quantum numbers. As for diatomic molecules these two types of spectroscopic constant provide the most important source of information on cubic and quartic anharmonicity, respectively. The formulae obtained from the perturbation treatment for these two coefficients in the effective hamiltonian are as follows ... 

Sadlej AJ, Snijders JG, van Lenthe E, Baerends EJ (1995) Relativistic regular two-component Hamiltonians, four component regular relativistic Hamiltonians and the perturbational treatment of Dirac s equation. J. Chem. Phys. 102 1758-1766... 

In the vicinity of the theta temperature, the perturbation treatment can also be applied to two-dimensional chains, and the following equations can be derived for the expansion factors1 ... 

The condition for the validity of this approach to the perturbation treatment is... 

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